Structural Engineer's Pocket Book

Structural Engineer’s Pocket Book

This Page Intentionally Left Blank

Structural Engineer’s Pocket Book Fiona Cobb

AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO

Elsevier Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 200 Wheeler Rd, Burlington, MA 01803 First published 2004 Copyright ª 2004, Fiona Cobb. All rights reserved The right of Fiona Cobb to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1T 4LP. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publisher Permissions may be sought directly from Elsevier’s Science and Technology Rights Department in Oxford, UK: phone: (þ44) (0) 1865 843830; fax: (þ44) (0) 1865 853333; e-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’ British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 7506 5638 7

For information on all Elsevier Butterworth-Heinemann publications visit our website at http://books.elsevier.com Typeset by Integra Software Services Pvt. Ltd, Pondicherry, India www.integra-india.com Printed and bound in Great Britain

Contents Preface

ix

Acknowledgements

xi

1

2

3

General Information Metric system Typical metric units for UK structural engineering Imperial units Conversion factors Measurement of angles Construction documentation and procurement Drawing conventions Common arrangement of work sections Summary of ACE conditions of engagement

3 4 5 6 8 10 11

Statutory Authorities and Permissions Planning Building regulations and standards Listed buildings Conservation areas and Tree preservation orders Archaeology and ancient monuments Party Wall etc. Act CDM

13 14 17 18 19 21 24

Design Data Design data checklist Structural form, stability and robustness Structural movement joints Fire resistance periods for structural elements Typical building tolerances Historical use of building materials Typical weights of building materials Minimum imposed floor loads Typical unit floor and roof loadings Wind loading Barrier and handrail loadings

25 26 29 30 31 32 34 38 41 43 44

1 2

vi

4

5

6

Contents

Selection of materials Selection of floor construction Transportation Temporary works toolkit

46 47 48 52

Basic and Shortcut Tools for Structural Analysis Load factors and limit states Geometric section properties Parallel axis theorem and Composite sections Material properties Coefficients of linear thermal expansion Coefficients of friction Sign conventions Beam bending theory Deflection limits Beam bending and deflection formulae Clapeyron’s equations of three moments Continuous beam bending formulae Struts Rigid frames under lateral loads Plates Torsion Taut wires, cables and chains Vibration

55 56 60 61 64 65 66 67 68 69 76 78 79 81 84 88 89 91

Geotechnics Geotechnics Selection of foundations and retaining walls Site investigation Soil classification Typical soil properties Preliminary sizing Trees and shallow foundations Contamined land

92 93 94 95 96 100 109 113

Timber and Plywood Timber Timber section sizes Laminated timber products Durability and fire resistance Preliminary sizing of timber elements

117 119 120 122 125

Contents

7

8

9

vii

Timber design to BS 5268 Timber joints

127 135

Masonry Masonry Geometry and arrangement Durability and fire resistance Preliminary sizing of masonry elements Masonry design to BS 5628 Masonry design to CP111 Lintel design to BS 5977 Masonry accessories

141 143 147 148 152 166 168 170

Reinforced Concrete Reinforced concrete Concrete mixes Durability and fire resistance Preliminary sizing of concrete elements Reinforcement Concrete design to BS 8110 Reinforcement bar bending to BS 8666 Reinforcement estimates

175 177 179 180 182 185 205 207

Structural Steel Structural steel Mild steel section sizes and tolerances Slenderness Durability and fire resistance Preliminary sizing of steel elements Steel design to BS 5950 Steel design to BS 449 Stainless steel to BS 5950

208 210 239 242 246 249 261 269

10 Composite Steel and Concrete Composite steel and concrete Preliminary sizing of composite elements Composite design to BS 5950

275 277 281

11 Structural Glass Structural glass Typical glass section sizes and thicknesses Durability and fire resistance Typical glass sizes for common applications Structural glass design Connections

284 287 288 289 291 293

viii

Contents

12

Building Elements, Materials, Fixings and Fastenings Waterproofing Basement waterproofing Screeds Precast concrete hollowcore slabs Bi-metallic corrosion Structural adhesives Fixings and fastenings Cold weather working Effect of fire on construction materials Aluminium

295 296 299 300 301 302 304 307 308 310

Useful Mathematics

314

13

Useful Addresses

320

Further Reading

331

Sources

336

Index

339

Preface As a student or graduate engineer it is difficult to source basic design data. Having been unable to find a compact book containing this information, I decided to compile my own after seeing a pocket book for architects. I realised that a Structural Engineer’s Pocket Book might be useful for other engineers and construction industry professionals. My aim has been to gather useful facts and figures for use in preliminary design in the office, on site or in the IStructE Part 3 exam, based on UK conventions. The book is not intended as a textbook; there are no worked examples and the information is not prescriptive. Design methods from British Standards have been included and summarized, but obviously these are not the only way of proving structural adequacy. Preliminary sizing and shortcuts are intended to give the engineer a ’feel’ for the structure before beginning design calculations. All of the data should be used in context, using engineering judgement and current good practice. Where no reference is given, the information has been compiled from several different sources. Despite my best efforts, there may be some errors and omissions. I would be interested to receive any comments, corrections or suggestions on the content of the book by email at [email protected]. Obviously, it has been difficult to decide what information can be included and still keep the book a compact size. Therefore any proposals for additional material should be accompanied by a proposal for an omission of roughly the same size – the reader should then appreciate the many dilemmas that I have had during the preparation of the book! If there is an opportunity for a second edition, I will attempt to accommodate any suggestions which are sent to me and I hope that you find the Structural Engineer’s Pocket Book useful. Fiona Cobb

This Page Intentionally Left Blank

Acknowledgements Thanks to the following people and organizations: Price & Myers for giving me varied and interesting work, without which this book would not have been possible! Paul Batty, David Derby, Sarah Fawcus, Step Haiselden, Simon Jewell, Chris Morrisey, Mark Peldmanis, Sam Price, Helen Remordina, Harry Stocks and Paul Toplis for their comments and help reviewing chapters. Colin Ferguson, Derek Fordyce, Phil Gee, Alex Hollingsworth, Paul Johnson, Deri Jones, Robert Myers, Dave Rayment and Andy Toohey for their help, ideas, support, advice and/or inspiration at various points in the preparation of the book. Renata Corbani, Rebecca Rue and Sarah Hunt at Elsevier. The technical and marketing representatives of the organizations mentioned in the book. Last but not least, thanks to Jim Cobb, Elaine Cobb, Iain Chapman for his support and the loan of his computer and Jean Cobb for her help with typing and proof reading.

This Page Intentionally Left Blank

        

                    

                                                                                  "# $%        ' *   *       +  $  

 



Z          

  

       <=%  #=%>    ?  Q   [  [  = \ ˆ %]> ^



  

      _        '   [     

 

   

 

# === === === # === === #=== ==# ===# ======# =========#

#  #= #  #=< #  #=> #  #= % #  #= > #  #= < #  #=

       _                                         # ` ˆ # {$%   # Z` ˆ # {$%

%

      Q `[ }[

         Z    |     } [   ~ $ $   }     *     |    ~           Z     |

   €        ‚      ‚          Z             ‚    

[ [$> [{$> [{ [{ [{$ [{$% [{$% [{$% {$% [{$%   % >  %  %$  ƒ  ƒ >  > %  %   

‡     

>

     }                                                                                 >%% $%        ~          ^               „                                     '  #  #    #    #   #      #   #   #   #   #  

#]<=   %%=   >  #%   #$#% 

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

<ƒ=   ƒ"ƒ=     #ƒƒ  

#$#ƒƒ  

 # 

#     #   #    #    #   # 

#    #   #  †    #     #    #    

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

%%ƒ=   ##%   #ƒ   #<   #$#<  

"   ƒ    %   #$%    #$%=   ˆ %]    ˆ #$%]     ˆ #$#]%"   

  #    #  # 

ˆ <="=  ˆ <==  ˆ < 

ƒ

      Q `[ }[

 !    ‡          }        ^         [              ?       

   ' 

# [ # 



#  # 



ˆ %%=ˆ  ˆ = "ƒ%  

ˆ =ƒˆ>< [ ˆ #=#< 



#  # #

# 

#  # 



# % # % # %

ˆ ==> >] 

ˆ >%"#  ˆ #= ƒ 

ˆ %ˆƒ  ˆ =>=ƒ"  ˆ = #ƒƒ 

  

# > # > # >

#  > # > # >

"

ˆ =====<#  > ˆ >ˆ>% > ˆ #>=" >

# [$> ˆ ==<%ƒ% $> # 

$> ˆ =]ˆ%ƒ  $>

# $> #  $>

#{ # [{

ˆ =%%ƒ"  ˆ =#==ƒ  

#  #  

# 

ˆ ===#ˆ>  % ˆ #=]<ƒ % ˆ ## < %

$  

# {$% # {$% # {$% # [{$%

 

ˆ ˆ ˆ ˆ

# [{$ # [{$

#  % # % # %

#ƒˆ $ % ==<ƒ]  $ % ==%=" $% === >  $%

ˆ #< > = > ˆ ==%"> > ˆ =]<ƒ< >

ˆ #<=% [$> ˆ #>% 

$> ˆ ƒƒƒ" { ˆ  <ƒ [{

# $ % #  $ % # $% #  $%

ˆ ˆ ˆ ˆ

ˆ <"ˆ> $ ˆ ==>=ˆ  $

# $ #  $

ˆ ==#ƒ< [{$ ˆ >%< [{$

ˆ =]>]<  

#  

 

# {

     

# {$%

ˆ #ƒˆ $ %

# $ %

# [{$%

ˆ #ƒˆ =>% $ %

# $ %

$ 

 

# > # >

$ 

    

# ƒ # ƒ

ˆ <#=#  #= ˆ <#=#  #=

 \

ˆ <ƒˆ% % ˆ == % % ˆ ="><# %

ˆ %ƒ=>  #= ˆ %ƒ=>  #=

< > < %

 >  >

#  > #  >

 ƒ  ƒ

#  ƒ #  ƒ

ˆ ‰#" ‡ >%Š €

 €

===<" {$% #ˆƒƒ {$% ƒ]"" {$% #=]> [{$%

ˆ #>ˆ< { ˆ <"  #= {$% ˆ <"  #= [{$%

>

<

ˆ #< > = > ˆ #<> >

ˆ ƒ#< %== ƒ ˆ ƒ#<% ƒ ˆ ‰ 

>%$#"Š \

ˆ

‡     

                        ^

%   

    _      ' #    ˆ =   #   ˆ <=0   #0   ˆ <=00  

   

\                   

                                                p  

ˆ #"=  

# 

ˆ

#"= #"= ˆ ˆ ˆ] #]0 ƒƒ00  >:#ƒ#<

%&!             

?  y   

=

?  y   

=

 y

=

 y

#

 y

=

 < >=

 ƒ ƒˆ

# % p > % # p >

# p % # p % #

 > <= p > % # % p >

 % = # = 1

<

      Q `[ }[

                                       _                           [                 

           [           †   ‹                                                 [       [                         Œ  Q                                

   Œ`   Q                               

                                    _          „                               [           [                       [                                                      ^                                

                      ‹                       

               [               

‡     

]

                                          [               [                            [         

                             [                        ~        

     [                     _ „          „    Œ  [Q

"  '           

                    „               [   ‹            |   }            [               Œ  Q   [   

    ?        _           

            [  _      

        „                   ?                                                     

( `              [                                                  †         

   `         [        [                                     _    ‹                                   {                                         

                             [                [          

"

      Q `[ }[

")  !  |                   [     ‹                                                 \ ‹                    } {  ]ˆ#  [        } ## %   } >=" ?               „ $                 _    Z     

                      >=    _   ƒˆ   „            ‹           '                          „ |                          ?       [          '   *   Ž               ' #'%== #'ˆ==     #'%ˆ==    „      +- Ž                ‹    „                             #'ˆ=   #'%=               "  Ž                               ' #'%= #'#= #'ˆ #'%  #'#                                      €  „         Œ  Q                                 [                        [               [    [                      

        

‡     

  cut section/slab edge/element to be highlighted elevations/infill details demolished structure under/hidden gridline/centre line outline of boundaries/adjacent parts limit of partially viewed element/cut-backline not at intersection breakline straight + tube

.

Existing brickwork

New brickwork

New blockwork

Stonework

Concrete

Sawn softwood

Hardwood

Insulation

Subsoil

Hardcore

Mortar/ screed/ plaster

Plywood

Glass

Steel

Damp proof course or membrane

$/     

Stairs

Ramp

Landscape slope

Arrow indicates ‘up’

Slope/pitch

#=

      Q `[ }[



    )     \ ?

    ~ [    }   ~ [ \?~                                 „     [                                 _                 [    '                                _    \?~    †               '

 (  *    

?# ?>

  „     Q

   

?% ?ƒ

    \   Q    

 %: * ; * !

\#

|

\%

\>

?  Ž   

\ƒ

?  Ž   ‚  $       $



\ˆ

‚  $     $

|#

  $ † $    `    

 

|%

_  $ 

|ƒ

|   

"  ) 

|> |ˆ %    *    

# > ˆ

     ‚    `       

% ƒ <

€  [         \   

#  

€# €>

} [$[   Z

  

€%

   

 $ *      ;

‡#

‡%

  $   

‡>

  $    Z$ [ 

‚#

|  

‚%

 

< "    

                        

    €#=    } [$[   ‘       _          $ = \`\ #

"

‡     

##

$

  %        ?  \        ?\                     ^  ?\ \       ?   }# >   %==%                     [                     ?      {      ?   }#              [  ?Ž*     ‚     }  ?  ‚}?

#;    $ 

? 

                    * \                                

$ '

    

\    [                                       \               [    

           „ `       [                       \    ` „ }      [                                   \    

                         

(>    $ 

    

                  ?          _       

                    ?                      \                                      `     [             * \                       \ `  `                  

$ "

|  

|           

      `

  ?   * \    `                           \ `  `                  

#%

      Q `[ }[

$

  %       ?   (>    ?   $ %

€   

|    ‹         „                                        ?                ~ [         

$ #

`  

  

|                             |      }   ‚   ?   `                   \     

    ~ [ _               [   [                             `    |  Q ‚[ ?      ‘ ’  \|Z ‚   ?   * \                

$ 

    

$ .

  

?  * \                     $          „ ?                |                                            

    $ @

Z†

$ 

\      

$ 

?   

$ = ?\ #

"

?  \    * \                       

                                                   \                         \                         ?                                   ?  * \     _                   ~ [     \    ?          [                    ?   * \           \    \[   [    _                              

     „            

                     

        `    [     \   

      

       [ ?  * \                       Z[              * \         

           (  `

           Q                      ^                |                   ^   `

                 

         *  |   \         

%             

    [       ~      \   `

  ? # =     Q    

        ~ `  `

  ` ‡   { ``‡ Z   ` ‡   { Z`‡ ‚  ` ‡   { ‚`‡                      |  `  Z  |`Z

$    €  Z          

    [  



           \   `

  ?   # ]    `

  * }     \   ?    ? # ]          _      {  `

  ` ‡   {``‡                        `

  ?  { `?{                  

                     

    `

  {   # #                   ?  [            

    { ‹     ?    ^    {     _          `

  `   ``                  

#ƒ

      Q `[ }[

'       }              ‚           

                         }                            ^                       ?  `           _         

_   [            Π Q 

    Π Q                                                       }                                    [

%          ~             ##"      

  *  }   ?            }   ? # "ƒ           }   ‚   %===  ?   |       |`Z                    ?      Œ   Q        [          [   Œ   Q ƒ"      [   \  ‹      }   ‚   ?                                  

 !    +  

   ?# *  ?% ‡   Z   ?>   ?ƒ |    \ ' €     `     ‚   Z  " _    % ‚    `    #  

 ‘  . |     ~ | @ ‘ `    ?     ‚   ‡    \    €    `  ?   €  | ` ‡†  Ž Z    ` 

<   H Z    ~ [ 

   ?     `  

#ˆ

$   }          _          ###         ‚ }            

             }     ? # ˆ     }        ‚   # =            # = Ž  _               _   ?           [                                   

           [       ^  [           

      }                                

  }                             [         Œ‹ Q   

   '  " % #  . @   ( J < $

‡     |   €   Z        €  `    Z      €  ‘ `            *    ‡  €  `       ‚   Z  ‚         \    €    `    }   |        €      Z   ‘†  €  |   ~   |    €      [   ‚   `   }

 

#<

      Q `[ }[

              {        }   ‚ ‹   {       # ]      `

    }   ‚ ‹   {     ?     # =“  }   ‚   { %===      [ Ž                     {     _   }       {            |    €     ` 

                 

                               ~ ?           Π Q     ?       Π  Q  

    [      Œ   Q     „     [   }                  \        [              

 ;    '  " % #  . @   ( < 

      ‡   Z    ~ [  `       ‚   Z     €   \    €    `     

  ‚   ‡     ~  }    

‘ `    ?    *`‡  |      ?       ‘ ~    ?   €  | ` ‡† 

   ?     `  

#]

 ;    ^     Œ         Q                              _       }                                _          ‹       *                        [  

                                      ‘ 

%       ‘  ‘       \?|~  ~  [              Œ         Q ?        #]==        #]==   #"ƒ=                ?           >=                 '  ”           _ ˆ== ===     

      #> ===  ~ ‡     ”          ‘    „             ? †         

$   ‘                _   ?      #"ƒ=                  ƒ= ===            ' ? }   \ ‡  ?       

              _                ‡  \                    _             

           ‘    ‘   {     _    

            [      [  {     }   | }        >=             

    "ˆ==               ' ? }•   }       }#   }%           ^

 

#"

      Q `[ }[

 !   *                       Œ         Q                                 "ˆ==            ~ <==      >=  {              „                                                 \                      [                    Q   

 !   *               [    `     `                 ? ` [  

                          

    Q    ?   ^                                                 

            [   ?   ‹           [     

                   [           ?  ‹    ?                     

        ˆ   < [             ^          

     ~          

                      `                [       [  

]ˆ      #ˆ          #==    

           €     –%= ===       [  

       

   ?     `  

#

      ?          ~      ?   Z     ?  ?  ? # ]    ‘  Z     ?  „ {   # ˆ   {     ?    ^                    #= ===                                                    ?                                                               Œ  Q        ˆ     [  \  *    — [ ‘                                                 ‹                          '   ‘  ‘    \?|~  ~   ‘  {        ^                          [                                  ``‡ #<       ~ {``‡ #"      `` <  {                   ' # ?             [  

     % ?                  

     > ?            ƒ            _        ˆ                    

     <                              

%=

      Q `[ }[

~  Π      Q            

            ‘  ‘    \?|~ 

~   ‘  {     ?                          

                    ' #                                                             

         % |[          > €          [        $          [   ƒ {            Q       _                         ˆ Z Ž           _                            _   

                [          

     _   < `‹_   [                     ‡               

                               

   ‹                     

{                 _       ‹ [               ~                Œ   Q €        

            _ ‹                                                   [  [                

                                              _         ‹    

     _        Q                    `                              ˜

   ?     `  

%#

( 

K   `  ~  ? # <       # ]          ~ 

%==%           {           *              |       `  ~ ?           

    [   _              “     ‹                            $ _   >              ~ [   

       ?     ?                     `  ~ ?          [  ?          `    ' Œ`  ~Q   Œ`  €  ~Q `  ~              „                   `  €  ~                   “         [    ? `                       {       # %   <   ?             

           _      `      ?          {  ?„     “           

     ‹                         ‹                     ?„     

 ?                                                         ?                    {            [ |   {            ?    [      {            ?„      ? {                            Q         Q      “       {          ?        “         [                       [  {           _   „        Œ         Q    {      ?„                [       {          [    [  

              {  #ƒ          ?„            [        `  ~ ?                 {   

%%

      Q `[ }[

  {       ?„          #ƒ               ?„                                      ?„               

       „                    `  ~ ?       

                  _     [               {                       {                                       ˜                      _  [     [ ~          `  ~ ?          

    $           [    

  “          „     Q       [  “ 

                [                  [        

           „           {         [     #ƒ      \  \     ?           

     ?         ? ?„      [      Œ Q             ?„         [          ?                     [ 

         [        #ƒ Q {            ?„     Q     

        [                             ?     [               „           Q  [                         

   ?     `  

%>

$  N ? ) ;    ;   {                                  „ —                     „        „       { 

     N         

$  O ? )   :  )

         _  `  ~                |`\          “    

     $

    `  ~  $                       Z   [    _         [ 

                     ? ?        `  ~         Q       {   O             [

$  Q ? :!   ;  ;  {        N           _                    >    „     Q        [      „    Q      <    „     Q        [        „     ƒˆ         Q              „   {                        ?„      ?„                   Œ   Q Ž              ?  [     ?„               [           $ = |‚ #

]

%ƒ

      Q `[ }[

"  \   | ’ Z   \|Z ‚   # ƒ                          ‹     ?   \  `       ‘    _         ‚              

     `         ‹              

      „      `     

                 `              †         [               „                 

     [  `      ‹                          „‹                                   „                   _  [       [                                     Q      [                                                 [                          „         `                    `                             $    „ ?         

                                      [                            ?           [      

3 Design Data Design data checklist The following design data checklist is a useful reminder of all of the limiting criteria which should be considered when selecting an appropriate structural form:

. . . . . . . . . . . . .

Description/building use Client brief and requirements Site constraints Loadings Structural form: load transfer, stability and robustness Materials Movement joints Durability Fire resistance Performance criteria: deflection, vibration, etc. Temporary works and construction issues Soil conditions, foundations and ground slab Miscellaneous issues

26

Structural Engineer’s Pocket Book

Structural form, stability and robustness Structural form It is worth trying to remember the different structural forms when developing a scheme design. A particular structural form might fit the vision for the form of the building. Force or moment diagrams might suggest a building shape. The following diagrams of structural form are intended as useful reminders:

TRUSSES

Couple

Tied rafter

Howe (>10 m steel/ timber)

Double howe (8–15 m steel/ timber)

Bowshing

Thrust

Northlight (>5 m steel)

Bowshing (20–40 m steel)

King post

Northlight (5–15 m steel)

Queen post

Fink (>10 m steel/ timber)

Double fink (5–14 m timber) (8–13 m steel)

Scissor (6–10 m steel/ timber)

Double scissor (10–13 m steel/ timber)

Fan (8–15 m steel)

French truss (12–20 m steel)

Umbrella (~13 m steel)

Saw tooth (~5 m steel)

Pratt

Warren

Modified warren

Howe

Fink

Modified fink

GIRDERS

Double lattice

Vierendeel

Design Data

PORTAL FRAMES

All fixed

2 pin

2 pin mansard

3 pin

ARCHES

Thrust

Tied

3 pin

SUSPENSION

Cable stay

Suspension

Closed suspension

WALLS Solid

Piers

Chevron

Diaphragm

TIMBER

Ply/ply stressed skin

Ply web

Ply/timber stressed skin

Flitched

RETAINING WALLS

Embedded

Cantilever

Gravity or reinforced earth

27

28

Structural Engineer’s Pocket Book

Stability Stability of a structure must be achieved in two orthogonal directions. Circular structures should also be checked for rotational failure. The positions of movement and/or acoustic joints should be considered and each part of the structure should be designed to be independently stable and robust. Lateral loads can be transferred across the structure and/or down to the foundations by using any of the following methods:

. Cross bracing which carries the lateral forces as axial load in diagonal members. . Diaphragm action of floors or walls which carry the forces by panel/plate/shear action. . Frame action with ‘fixed’ connections between members and ‘pinned’ connections at the supports.

. Vertical cantilever columns with ‘fixed’ connections at the foundations. . Buttressing with diaphragm, chevron or fin walls. Stability members must be located on the plan so that their shear centre is aligned with the resultant of the overturning forces. If an eccentricity cannot be avoided, the stability members should be designed to resist the resulting torsion across the plan.

Robustness and disproportionate collapse All structural elements should be effectively tied together in each of the two orthogonal directions, both horizontally and vertically. This is generally achieved by specifying connections in steel buildings as being of certain minimum size, by ensuring that reinforced concrete junctions contain a minimum area of steel bars and by using steel straps to connect walls and floors in masonry structures. It is important to consider robustness requirements early in the design process. The Building Regulations require buildings of five or more storeys (excluding the roof) to be designed for disproportionate collapse. This is intended to ensure that accidental damage to elements of the building structure cannot cause the collapse of a disproportionately large area of a building. The disproportionate collapse requirement for public buildings with a roof span of more than 9 m appears to have been removed from the regulations. Typically the Building Regulations require that any collapse caused by the failure of a single structural element should be limited to an area of 70 m2 or 15% of any storey area (whichever is the lesser). Alternatively the designer can strengthen the structure to withstand the ‘failure’ of certain structural supports in order to prevent disproportionate collapse. In some circumstances the structure cannot be arranged to avoid the occurrence of ‘key elements’, which support disproportionately large areas of the building. These ‘key elements’ must be designed as protected members (to the code of practice for the relevant structural material) to provide extra robustness and damage resistance.

Design Data

29

Structural movement joints Joints should be provided to control temperature, moisture, acoustic and ground movements. Movement joints can be difficult to waterproof and detail and therefore should be kept to a minimum. The positions of movement joints should be considered for their effect on the overall stability of the structure.

Primary movement joints Primary movement joints are required to prevent cracking where buildings (or parts of buildings) are large, where a building spans different ground conditions, changes height considerably or where the shape suggests a point of natural weakness. Without detailed calculation, joints should be detailed to permit 15–25 mm movement. Advice on joint spacing for different building types can be variable and conflicting. The following figures are some approximate guidelines based on the building type:

Concrete

25 m (e.g. for roofs with large thermal differentials)– 50 m c /c.

Steel industrial buildings

100 m typical–150 m maximum c /c.

Steel commercial buildings

50 m typical–100 m maximum c /c.

Masonry

40 m–50 m c /c.

Secondary movement joints Secondary movement joints are used to divide structural elements into smaller elements to deal with the local effects of temperature and moisture content. Typical joint spacings are: Clay bricks

Up to 12 m c/c on plan (6 m from corners) and 9 m vertically or every three storeys if the building is greater than 12 m or four storeys tall.

Concrete blocks

3 m–7 m c/c.

Hardstanding

70 m c/c.

Steel roof sheeting

20 m c/c down the slope, no limit along the slope.

30

Structural Engineer’s Pocket Book

Fire resistance periods for structural elements Fire resistance of structure is required to maintain structural integrity to allow time for the building to be evacuated. Generally, roofs do not require protection. Architects typically specify fire protection in consultation with the engineer. Minimum period of fire resistance minutes

Building types

Basement storey including floor over

Ground or upper storey

Depth of a lowest basement

Height of top floor above ground, in a building or separated part of a building

>10 m <10 m >5 m

<18 m <30 m <120 m

301

602

902

301

301

603

n/a

60

301

60

90

1205

90 60

60 60

301 301

60 301

90 60

X 1205

not sprinklered sprinklered

90 60

60 60

60 301

60 60

90 60

X 1205

Assembly & recreation

not sprinklered sprinklered

90 60

60 60

60 301

60 60

90 60

X 1205

Industrial

not sprinklered 120 sprinklered 90

90 60

60 301

90 60

120 90

X 1205

Storage and other nonresidential

not sprinklered 120 sprinklered 90

90 60

60 301

90 60

120 90

X 1205

n/a 60

151 301

151 60

Residential flats and maisonettes

90

60

Residential houses

n/a

Institutional residential4

90

Office

not sprinklered sprinklered

Shops & commercial

Car park for open sided light vehicles all others

n/a 90

151 90

1202 n/a

60 1205

NOTES: X Not permitted 1. Increased to 60 minutes for compartment walls with other fire compartments or 30 minutes for elements protecting a means of escape. 2. Reduced to 30 minutes for a floor in a maisonette not contributing to the support of the building. 3. To be 30 minutes in the case of three storey houses and 60 minutes for compartment walls separating buildings. 4. NHS hospitals should have a minimum of 60 minutes. 5. Reduced to 90 minutes for non-structural elements. 6. Should comply with Building Regulations: B3 section 12.

Source: Building Regulations Approved Document B (1991).

Design Data

Typical building tolerances SPACE BETWEEN WALLS Brickwork ± 20 mm Blockwork ± 21 Timber ± 32

Steel Timber

WALL VERTICALITY

COLUMN VERTICALITY

10 mm 10 17 11

Maximum

Steel 6 mm Timber 10 In situ concrete 12 Precast concrete 10

Maximum

VERTICAL POSITION OF BEAMS

Steel Timber In situ concrete Precast concrete

± 12 mm ± 12

In situ concrete ± 18 Precast concrete ± 13

In situ concrete ± 24 Precast concrete ± 18

Brickwork Blockwork In situ concrete Precast concrete

SPACE BETWEEN COLUMNS

± 20 mm ± 20 ± 22 ± 23

PLAN POSITION

VERTICAL POSITION OF FLOORS

In situ concrete ± 15 mm Precast concrete ± 15

FLATNESS OF FLOORS 3 m straight edge max

Brickwork Steel Timber In situ concrete Precast concrete Source: BS 5606: 1990.

± 10 mm ± 10 ± 10 ± 12 ± 10

In situ concrete Floor screed

5 mm 5

31

32

Structural Engineer’s Pocket Book

Historical use of building materials

1714

1800

1837

1901

Post Wars

Inter Wars

Edwardian

Victorian

Georgian including William IV

Masonry and timber

1919

1945

MASONRY Bonding timbers Non hydraulic lime mortar 84

Mathematical tiles 50s

Hydraulic lime mortar

30s 90s

96

60s

Clinker concrete blocks 00s

Cavity walls

50

10

51

Pressed bricks

70s

Flettons

20

Concrete bricks

50s

Dense concrete blocks 20s

Sand line bricks

20s

Stretcher bond

40s

Mild steel cavity wall ties

45s

60s

Galvanised steel cavity wall ties

80s 65

Stainless steel cavity well ties

53 60s

Aerated concrete blocks

TIMBER Trussed timber girders

33

King + queen post trusses

50

92 50s

50

Wrought iron flitched beams Belfast trusses

10s

70 60

40s 50s

Trussed rafters

60s

Ply stressed skin pannels Mild steel flitched beams

Source: Richardson, C. (2000).

40s

80s

Design Data

33

1714

1800

1837

1901

1919

Post Wars

Inter Wars

Edwardian

Victorian

William IV

including

Georgian

Concrete and steel

1945

CONCRETE Limecrete/Roman cement

96

Jack arch floors

96

80s 62 24

Portland cement

51

30 30s

70s

Filler joists

80

Clinker concrete 54

RC framed buildings

30 97 20s

RC shells + arches

25

Hollow pot slabs 00s

Flat slabs

80

31 32

Lightweight concrete

50 50

Precast concrete floors Composite metal deck slabs

52

64

Woodwool permanent shutters

69

90s

Waffle/coffered stabs

60s

Composite steel + concrete floors with shear keys

70s

CAST IRON (CI) + WROUGHT IRON (WI) CI columns CI beams WI rods + flats WI roof trusses WI built up beams WI rolled sections

20s

92

30s

96

65 80

10s 37 40 50s

90s

‘Cast steel’ columns

10s

MILD STEEL 80

Plates + rods

90s

Riveted sections Hot rolled sections Roof trusses Steel framed buildings

60

83 90s 96

55

Welds

38

Castellated beams

50

High strength friction grip bolts (HSFG)

60

Hollow sections

13

STAINLESS STEEL Bolts, straps, lintels, shelf angles, etc.

Source: Richardson, C. (2000).

70s

34

Structural Engineer’s Pocket Book

Typical weights of building materials Material

Aggregate Aluminium Aluminium bronze Asphalt Ballast Balsa wood Bituminous felt roofing Bitumen Blockboard Blockwork Books Brass Brickwork

Bronze Cast stone Cement Concrete

Coal Chalk Chipboard Chippings Clay Copper

Description

Cast alloy Longstrip roofing

Thickness/ quantity of unit

Unit load kN/m2

Bulk density kN/m3 16 27

0.8 mm

0.022 76

Roofing – 2 layers Paving

25 mm

0.58 21

see Gravel 1 3 layers and vapour barrier

0.11 11–13

Sheet Lightweight – dense On shelves Bulk Cast Blue Engineering Fletton London stock Sand lime Cast

18 mm

0.11 10–20 7 8–11 85 24 22 18 19 21 83 23 15 10 18 24 9 22 7

Aerated Lightweight aggregate Normal reinforced Loose lump

Flat roof finish Undisturbed Cast Longstrip roofing

1 layer

0.05 19 87

0.6 mm

0.05

Design Data

Cork Double decker bus Elephants Felt Glass Glass wool Gold Gravel Hardboard Hardcore Hardwood

Hollow clay pot slabs

Granulated see Vehicles Adult group Roofing underlay Insulating Crushed/refuse Clear float Quilt

1

50 mm

3.2 0.015 0.05

16 25 100 mm

0.01 194 16 21

Loose Undisturbed

Greenheart Oak Iroko, teak Mahogany Including ribs and mortar but excluding topping

6–8 19 10 8 7 6 12

300 mm thick overall

100 mm thick overall Iron Ivory Lead

Lime

Linoleum Macadam Magnesium MDF Mercury Mortar Mud Partitions

15

Cast Wrought Cast Sheet 1.8 mm Sheet 3.2 mm Hydrate (bags) Lump/quick (powder) Mortar (putty) Sheet 3.2 mm Paving Alloys Sheet

Plastered brick Medium dense plastered block Plaster board on timber stud

6

72 77 19 114 0.21 0.36 6 10 18 0.05 21 18 8 136 17–18 17–20

102 þ 2  13 mm 100 þ 2  13 mm

2.6 2.0

21 16

100 þ 2  13 mm

0.35

3

35

36

Structural Engineer’s Pocket Book

Typical weights of building materials – continued Material

Description

Patent glazing Pavement lights Perspex Plaster

Single glazed Double glazed Cast iron or concrete framed Corrugated sheets Lightweight Wallboard and skim coat Lath and plaster Traditional lime plaster Sheet Expanded sheet

Plywood Polystyrene Potatoes Precast concrete planks

Quarry tiles

Roofing tiles

Sand Screed Shingle Slate Snow

Thickness/ quantity of unit

Unit load kN/m2

Bulk density kN/m3 25

100 mm

0.26–0.3 0.52 1.5

13 mm 13 mm

0.05 0.11 0.12

12 9

19 mm

0.25 20 7 2 7

Beam and block plus 50 mm topping Hollowcore plank Hollowcore plank Solid plank and 50 mm topping

150–225 mm

1.8–3.3

150 mm 200 mm 75–300 mm

2.4 2.7 3.7–7.4

Including mortar bedding Clay – plain Clay pantile Concrete Slate Dry, loose Wet, compact Sand/cement Coarse, graded, dry Slab Fresh Wet, compacted

12.5 mm

0.32

0.77 0.42 0.51 0.30

minimum 0.6 minimum 0.6

Softwood

Soils

Battens for slating and tiling 25 mm tongued and grooved boards on 100  50 timber joists at 400 c/c 25 mm tongued and grooved boards on 250  50 timber joists at 400 c/c Loose sand and gravels Dense sand and gravels Soft /firm clays and silts Stiff clays and silts

19 19 24 28 16 19 22 19 28 1 3 6

0.03 0.23

0.33

16 22 18 21

Design Data

Stainless steel roofing Steel Stone Granite Limestone

Marble Sandstone

Slate Terracotta Terrazzo Thatch Timber

Longstrip

0.4 mm

0.05

78

Mild

78

Cornish (Cornwall) Rublislaw (Grampian) Bath (Wiltshire) Mansfield (Nottinghamshire) Portland (Dorset) Italian Bramley Fell (West Yorkshire) Forest of Dean (Gloucestershire) Darley Dale or Kerridge (Derbyshire) Welsh

26 25 21 22 22 27 22 24 23–25 28

Paving Including battens see Hardwood or Softwood

20 mm 305 mm

Vehicles

London bus New Mini Cooper Rolls Royce Volvo estate

73.6 kN 11.4 kN 28.0 kN 17.8 kN

Water

Fresh Salt

0.43 0.45

Cast Longstrip roofing

18 22

10 10–12 6

Woodwool slabs Zinc

37

72 0.8 mm

0.06

38

Structural Engineer’s Pocket Book

Minimum imposed floor loads The following table from BS 6399: Part 1 gives the normally accepted minimum floor loadings. Clients can consider sensible reductions in these loads if it will not compromise future flexibility. A survey by Arup found that office loadings very rarely even exceed the values quoted for domestic properties. The gross live load on columns and/or foundations from sections A to D in the table, can be reduced in relation to the number of floors or floor area carried to BS 6399: Part 1. Live load reductions are not permitted for loads from storage and/or plant, or where exact live loadings have been calculated. Type of activity/occupancy for part of the building or structure

Examples of specific use

UDL kN/m2

Point load kN

A Domestic and residential activities (also see category C )

All usages within self-contained dwelling units. Communal areas (including kitchens) in blocks of flats with limited use (see Note 1) (for communal areas in other blocks of flats, see C3 and below)

1.5

1.4

Bedrooms and dormitories except those in hotels and motels

1.5

1.8

Bedrooms in hotels and motels Hospital wards Toilet areas

2.0

1.8

Billiard rooms

2.0

2.7

Communal kitchens except in flats covered by Note 1

3.0

4.5

Balconies

Single dwelling units and communal areas in blocks of flats with limited use (see Note 1)

1.5

1.4

Guest houses, residential clubs and communal areas in blocks of flats except as covered by Note 1

Same as rooms to which they give access but with a minimum of 3.0

1.5/m run concentrated at the outer edge

Hotels and motels

Same as rooms to which they give access but with a minimum of 4.0

1.5/m run concentrated at the outer edge

Operating theatres, X-ray rooms, utility rooms

2.0

4.5

Work rooms (light industrial) without storage

2.5

1.8

Offices for general use

2.5

2.7

Banking halls

3.0

2.7

Kitchens, laundries, laboratories

3.0

4.5

Rooms with mainframe computers or similar equipment

3.5

4.5

Machinery halls, circulation spaces therein

4.0

4.5

Projection rooms

5.0

Determine loads for specific use

Factories, workshops and similar buildings (general industrial)

5.0

4.5

Foundries

20.0

Determine loads for specific use

Catwalks

–

1.0 at 1 m c/c

Balconies

Same adjacent rooms but with a minimum of 4.0

1.5 kN/m run concentrated at the outer edge

B Offices and work areas not covered elsewhere

Fly galleries (load to be distributed uniformly over width)

4.5 kN/m run

–

Ladders

–

1.5 rung load

Design Data C Areas where people may congregate

Public, institutional and communal dining rooms and lounges, cafes and restaurants (see Note 2)

C1 Areas with tables

Reading rooms with no book storage

2.5

4.5

Classrooms

3.0

2.7

Assembly areas with fixed seating (see Note 3)

4.0

3.6

Places of worship

3.0

2.7

Corridors, hallways, aisles, etc. (foot traffic only)

3.0

4.5

Stairs and landings (foot traffic only)

3.0

4.0

Corridors, hallways, aisles, etc. (foot traffic only)

4.0

4.5

Corridors, hallways, aisles, etc., subject to wheeled vehicles, trolleys, etc.

5.0

4.5

C2 Areas with fixed seats

C3 Areas without obstacles for moving people

Corridors, hallways, aisles, stairs, landings, etc. in institutional type buildings (not subject to crowds or wheeled vehicles), hostels, guest houses, residential clubs, and communal areas in blocks of flats not covered by Note 1. (For communal areas in blocks of flats covered by Note 1, see A) Corridors, hallways, aisles, stairs, landings, etc. in all other buildings including hotels and motels and institutional buildings

Stairs and landings (foot traffic only)

2.0

2.7

4.0

4.0

Industrial walkways (light duty) Industrial walkways (general duty) Industrial walkways (heavy duty)

3.0 5.0 7.5

4.5 4.5 4.5

Museum floors and art galleries for exhibition purposes

4.0 (see Note 4)

4.5

Balconies (except as specified in A)

Same as adjacent rooms but with a minimum of 4.0

1.5/m run concentrated at the outer edge

Fly galleries

4.5 kN/m run distributed uniformly over width

–

C4 Areas with possible physical activities (see clause 9)

Dance halls and studios, gymnasia, stages

5.0

3.6

Drill halls and drill rooms

5.0

9.0

C5 Areas susceptible to overcrowding (see clause 9)

Assembly areas without fixed seating, concert halls, bars, places of worship and grandstands

5.0

3.6

Stages in public assembly areas

7.5

4.5

D Shopping areas

Shop floors for the sale and display of merchandise

4.0

3.6

39

40

Structural Engineer’s Pocket Book

Minimum imposed floor loads – continued Type of activity/ occupancy for part of the building or structure

Examples of specific use

UDL kN/m2

Point load kN

E Warehousing and storage areas. Areas subject to accumulation of goods. Areas for equipment and plant

General areas for static equipment not specified elsewhere (institutional and public buildings)

2.0

1.8

Reading rooms with book storage, e.g. libraries

4.0

4.5

General storage other than those specified

2.4 per metre of storage height

7.0

File rooms, filing and storage space (offices)

5.0

4.5

F

G

Stack rooms (books)

2.4 per metre of storage height (6.5 kN/m2 min)

7.0

Paper storage for printing plants and stationery stores

4.0 per metre of storage height

9.0

Dense mobile stacking (books) on mobile trolleys, in public and institutional buildings

4.8 per metre of storage height (9.6 kN/m2 min)

7.0

Dense mobile stacking (books) on mobile trucks, in warehouses Cold storage

4.8 per metre of storage height (15 kN/m2 min) 5.0 per metre of storage height (15 kN/m2 min)

Plant rooms, boiler rooms, fan rooms, etc., including weight of machinery

7.5

4.5

7.0 9.0

Ladders

–

1.5 rung load

Parking for cars, light vans, etc. not exceeding 2500 kg gross mass, including garages, driveways and ramps

2.5

9.0

Vehicles exceeding 2500 kg. Driveways, ramps, repair workshops, footpaths with vehicle access, and car parking

To be determined for specific use

NOTES: 1. Communal areas in blocks of flats with limited use refers to blocks of flats not more than three storeys in height and with not more than four selfcontained dwelling units per floor accessible from one staircase. 2. Where these same areas may be subjected to loads due to physical activities or overcrowding, e.g. a hotel dining room used as a dance floor, imposed loads should be based on occupancy C4 or C% as appropriate. Reference should also be made to Clause 9. 3. Fixed seating is seating where its removal and use of the space for other purposes is improbable. 4. Museums, galleries and exhibition spaces often need more capacity than this, sometimes up to 10 kN/m2.

Source: BS 6399: Part 1: 1996.

Design Data

41

Typical unit floor and roof loadings Permanent partitions shown on the floor plans should be considered as dead load. Flexible partitions which may be movable should be allowed for in imposed loads, with a minimum of 1 kN/m2. 1.5/2.5 kN/m2 (1.0) 0.15 0.2 0.15 Domestic/ 2.0/4.0 kN/m2 office totals

Timber floor

Live loading: domestic/office (Office partitions) Timber boards/plywood Timber joists Ceiling and services

Timber flat roof

Snow and access Asphalt waterproofing Timber joists and insulation Ceiling and services

0.75 kN/m2 0.45 0.2 0.15 Total 1.55 kN/m2

Timber pitched roof

Snow Slates, timber battens and felt Timber rafters and insulation Ceiling and services

0.6 kN/m2 0.55 0.2 0.15 Total 1.5 kN/m2

Internal RC slab

Live loading: office/ classroom/corridors, etc. Partitions 50 screed/75 screed/raised floor Solid reinforced concrete slab Ceiling and services

t

External RC slab

t

Metal deck roofing

Live loading: snow and access/office/bar Slabs/paving Asphalt waterproofing and insulation 50 screed Solid reinforced concrete slab Ceiling and services

Live loading: snow/wind uplift Outer covering, insulation and metal deck liner Purlins – 150 deep at 1.5 m c/c Services Primary steelwork: light beams/trusses

2.5/3.0/4.0 kN/m2 1.0 (minimum) 1.2/1.8/0.4 24t 0.15 Total – kN/m2 0.75/2.5/5.0 kN/m2 0.95 0.45 1.2 24t 0.15 Total – kN/m2 0.6/ 1.0 kN/m2 0.4 0.3 0.1 0.5–0.8/0.7–2.4 Total – kN/m2

42

Structural Engineer’s Pocket Book

Typical ‘all up’ loads For very rough assessments of the loads on foundations, ‘all up’ loads can be useful. The best way is to ‘weigh’ the particular building, but very general values for small-scale buildings might be: Steel clad steel frame

5–10 kN /m2

Masonry clad timber frame

10–15 kN/m2

Masonry walls and precast concrete floor slabs

15–20 kN/m2

Masonry clad steel frame

15–20 kN/m2

Masonry clad concrete frame

20–25 kN/m2

Design Data

43

Wind loading BS 6399: Part 2 gives methods for determining the peak gust wind loads on buildings and their components. Structures susceptible to dynamic excitation fall outside the scope of the guidelines. While BS 6399 in theory allows for a very site-specific study of the many design parameters, it does mean that grossly conservative values can be calculated if the ‘path of least resistance’ is taken through the code. Unless the engineer is prepared to work hard and has a preferred ‘end result’ to aim for, the values from BS 6399 tend to be larger than those obtained from the now withdrawn wind code CP3: Chapter V: Part 2. As wind loading relates to the size and shape of the building, the size and spacing of surrounding structures, altitude and proximity to the sea or open stretches of country, it is difficult to summarize the design methods. The following dynamic pressure values have been calculated (on a whole building basis) for an imaginary building 20 m  20 m in plan and 10 m tall (with equal exposure conditions and no dominant openings) in different UK locations. The following values should not be taken as prescriptive, but as an idea of an ‘end result’ to aim for. Taller structures will tend to have slightly higher values and where buildings are close together, funnelling should be considered. Small buildings located near the bases of significantly taller buildings are unlikely to be sheltered as the wind speeds around the bases of tall buildings tends to increase.

Typical values of dynamic pressure, q in kN/m2 Building location

Maximum q for prevailing south westerly wind kN/m2

Minimum q for north easterly wind kN/m2

Arithmetic mean q kN/m2

Scottish mountain-top Dover cliff-top Rural Scotland Coastal Scottish town City of London high rise Rural northern England Suburban South-East England Urban Northern Ireland Rural Northern Ireland Rural upland Wales Coastal Welsh town Conservative quick scheme value for most UK buildings

3.40 1.69 1.14 1.07 1.03 1.02 0.53 0.88 0.83 1.37 0.94 –

1.81 0.90 0.61 0.57 0.55 0.54 0.28 0.56 0.54 0.72 0.40 –

2.60 1.30 0.87 0.82 0.80 0.78 0.45 0.72 0.74 1.05 0.67 1.20

NOTE: These are typical values which do not account for specific exposure or topographical conditions.

44

Structural Engineer’s Pocket Book

Barrier and handrail loadings Minimum horizontal imposed loads for barriers, parapets, and balustrades, etc. Type of occupancy for part of the building or structure

Examples of specific use

Line load kN/m

UDL on infill kN/m2

Point load on infill kN

A Domestic and residential activities

(a) All areas within or serving exclusively one dwelling including stairs, landings, etc. but excluding external balconies and edges of roofs (see C3 ix)

0.36

0.5

0.25

(b) Other residential (but also see C)

0.74

1.0

0.5

(c) Light access stairs and gangways not more than 600 mm wide

0.22

n/a

n/a

(d) Light pedestrian traffic routes in industrial and storage buildings except designated escape routes

0.36

0.5

0.25

(e) Areas not susceptible to overcrowding in office and institutional buildings. Also industrial and storage buildings except as given above

0.74

1.0

0.5

C Areas where people may congregate: C1/C2 areas with tables or fixed seating

(f) Areas having fixed seating within 530 mm of the barrier, balustrade or parapet

1.5

1.5

1.5

(g) Restaurants and bars

1.5

1.5

1.5

C3 Areas without obstacles for moving people and not susceptible to overcrowding

(h) Stairs, landings, corridors, ramps

0.74

1.0

0.5

(i) External balconies and edges of roofs. Footways and pavements within building curtilage adjacent to basement/sunken areas

0.74

1.0

0.5

C5 Areas susceptible to overcrowding

(j) Footways or pavements less than 3 m wide adjacent to sunken areas

1.5

1.5

1.5

(k) Theatres, cinemas, discotheques, bars, auditoria, shopping malls, assembly areas, studios. Footways or pavements greater than 3 m wide adjacent to sunken areas

3.0

1.5

1.5

B and E Offices and work areas not included elsewhere including storage areas

(l) Designated stadia*

See requirements of the appropriate certifying authority

D Retail areas

(m) All retail areas including public areas of banks/building societies or betting shops. For areas where overcrowding may occur, see C5

1.5

1.5

1.5

F/G Vehicular

(n) Pedestrian areas in car parks including stairs, landings, ramps, edges or internal floors, footways, edges of roofs

1.5

1.5

1.5

(o) Horizontal loads imposed by vehicles

See clause 11. (Generally F 5 150 kN)

* Designated stadia are those requiring a safety certificate under the Safety of Sports Ground Act 1975

Source: BS 6399: Part 1: 1996.

Design Data

45

Minimum barrier heights Use

Position

Single family dwelling

(a) Barriers in front of a window (b) Stairs, landings, ramps, edges of internal floors (c) External balconies, edges of roofs

All other uses

(d) Barrier in front of a window (e) Stairs (f) Balconies and stands, etc. having fixed seating within 530 mm of the barrier (g) Other positions

*Site lines should be considered as set out in clause 6.8 of BS 6180.

Source: BS 6180: 1999.

Height mm 800 900 1100 800 900 800* 1100

46

Structural Engineer’s Pocket Book

Selection of materials Material

Advantage

Disadvantage

Aluminium

Good strength to dead weight ratio for long spans Good corrosion resistance Often from recycled sources

Cannot be used where stiffness is critical Stiffness is a third of that of steel About two to three times the price of steel

Concrete

Design is tolerant to small, late alterations Integral fire protection Integral corrosion protection Provides thermal mass if left exposed Client pays as the site work progresses: ‘pay as you pour’

Dead load limits scope Greater foundation costs Greater drawing office and detailing costs Only precasting can accelerate site work Difficult to post-strengthen elements Fair faced finish needs very skilled contractors and carefully designed joints

Masonry

Steelwork

Provides thermal mass The structure is also the cladding Can be decorative by using a varied selection of bricks Economical for low rise buildings Inherent sound, fire and thermal properties Easy repair and maintenance Light construction reduces foundation costs Intolerant to late design changes Fast site programme Members can be strengthened easily Ideal for long spans and transfer structures

Timber

Traditional/low-tech option Sustainable material Cheap and quick with simple connections Skilled labour not an absolute requirement Easily handled

Skilled site labour required Long construction period Less economical for high rise Large openings can be difficult Regular movements joints Uniform appearance can be difficult to achieve Design needs to be fixed early Needs applied insulation, fire protection and corrosion protection Skilled workforce required Early financial commitment required from client to order construction materials Long lead-ins Vibrations can govern design Limited to 4–5 storeys maximum construction height Requires fire protection Not good for sound insulation Must be protected against insects and moisture Connections can carry relatively small loads

Selection of floor construction 800

5

700

6

Depth (m)

600

11

500

2

400 1

300

9 8

7 3

200 100 0

4 10 2

4

6

8

10

12

14

16

18

20

Span (m) Timber joists at 400 c/c Stressed skin ply panel One way reinforced concrete slab Precast prestressed concrete plank Precast double tee beams Coffered concrete slab

7. 8. 9. 10. 11.

Beam + block floor Reinforced concrete flat slab Post tensioned flat slab Concrete metal deck slab Composite steel beams 47

1. 2. 3. 4. 5. 6.

48

Structural Engineer’s Pocket Book

Transportation Although the transport of components is not usually the final responsibility of the design engineer, it is important to consider the limitations of the available modes of transport early in the design process using Department for Transport (DfT) information. Specific cargo handlers should be consulted for comment on sea and air transport, but a typical shipping container is 2.4 m wide, 2.4–2.9 m high and can be 6 m, 9 m, 12 m or 13.7 m in length. Transportation of items which are likely to exceed 20 m by 4 m should be very carefully investigated. Private estates may have additional and more onerous limitations on deliveries and transportation. Typical road and rail limitations are listed below as the most common form of UK transport, but the relevant authorities should be contacted to confirm the requirements for specific projects.

Rail transportation Railtrack can carry freight in shipping containers or on flat bed wagons. The maximum load on a four axle flat wagon is 66 tonnes. The maximum height of a load is 3.9 m above the rails and wagons are generally between 1.4 and 1.8 m high. All special requirements should be discussed with Railtrack Freight or Network Rail.

Road transport The four main elements of legislation which cover the statutory controls on length, width, marking, lighting and police notification for large loads are the Motor Vehicles (Construction & Use) Regulations 1986; the Motor Vehicles (Authorization of Special Types) General Order 1979, the Road Vehicles Lighting Regulations 1989 and the Road Traffic Act 1972. A summary of the requirements is set out below.

Height of load There is no statutory limit governing the overall height of a load; however, where possible it should not exceed 4.95 m from the road surface to maximize use of the motorway and trunk road network (where the average truck flat bed is about 1.7 m). Local highway authorities should be contacted for guidance on proposed routes avoiding head height restrictions on minor roads for heights exceeding 3.0 m–3.6 m.

Weight of vehicle or load Gross weight of vehicle, W kg

Notification requirements

44 000 < W  80 000 or has any axle weight greater than permitted by the Construction & Use Regulations

2 days’ clear notice with indemnity to the Highway and Bridge Authorities

80 000 < W  150 000

2 days’ clear notice to the police and 5 days’ clear notice with Indemnity to the Highway and Bridge Authorities

W > 150 000

DfT Special Order BE16 (allow 10 weeks for application processing) plus 5 days’ clear notice to the police and 5 days’ clear notice with indemnity to the Highway and Bridge Authorities

Design Data

49

Width of load Total loaded width*, B m

Notification requirements

B  2.9

No requirement to notify police

2.9 < B  5.0

2 days’ clear notice to police

5.0 < B  6.1

DfT permission VR1 (allow 10 days for application processing) and 2 days’ clear notice to police

B > 6.1

DfT Special Order BE16 (allow 8 weeks for application processing) and 5 days’ clear notice to police and 5 days’ clear notice with indemnity to Highway and Bridge Authorities

* A load may project over one or both sides by up to 0.305 m, but the overall width is still limited as above.

Loads with a width of over 2.9 m or with loads projecting more than 0.305 m on either side of the vehicle must be marked to comply with the requirements of the Road Vehicles Lighting Regulations 1989.

Length of load Total loaded length, L m

Notification requirements

L < 18.75

No requirement to notify police

18.75  L <27.4

Rigid or articulated vehicles*. 2 days’ clear notice to police

(rigid vehicle) L > 27.4

DfT Special Order BE16 (allow 8 weeks for application processing) and 5 days’ clear notice to police and 5 days’ clear notice with indemnity to Highway and Bridge Authorities

(all other trailers) L > 25.9

All other trailer combinations carrying the load. 2 days’ clear notice to police

* The length of the front of an articulated motor vehicle is excluded if the load does not project over the front of the motor vehicle.

Projection of overhanging loads Overhang position

Overhang length, L m

Rear

L < 1.0

No special requirement

1.0 < L < 2.0

Load must be made clearly visible

2.0 < L < 3.05

Standard end marker boards are required

L > 3.05

Standard end marker boards are required plus police notification and an attendant is required

Front

Notification requirements

L < 1.83

No special requirement

2.0 < L < 3.05

Standard end marker boards are required plus the driver is required to be accompanied by an attendant

L > 3.05

Standard end marker boards are required plus police notification and the driver is required to be accompanied by an attendant

50

Typical vehicle sizes and weights Vehicle type

Weight, W kg

3.5 tonne van

3500

7.5 tonne van

Length, L m

Width, B m

Height, H m

Turning circle m

5.5

2.1

2.6

13.0

7500

6.7

2.5

3.2

14.5

Single decker bus

16 260

11.6

2.5

3.0

20.0

Refuse truck

16 260

8.0

2.4

3.4

17.0

2 axle tipper

16 260

6.4

2.5

2.6

15.0

Van (up to 16.3 tonnes)

16 260

8.1

2.5

3.6

17.5

Skiploader

16 260

6.5

2.5

3.7

14.0

Fire engine

16 260

7.0

2.4

3.4

15.0

Bendy bus

17 500

18.0

2.6

3.1

23.0

51

52

Structural Engineer’s Pocket Book

Temporary works toolkit Steel trench prop load capacities Better known as ‘Acrow’ props, these adjustable props should conform to BS 4704 or BS EN 1065. Verticality of the loads greatly affects the prop capacity and fork heads can be used to eliminate eccentricities. Props exhibiting any of the following defects should not be used:

. . . . .

A tube with a bend, crease or noticeable lack of straightness. A tube with more than superficial corrosion. A bent head or base plate. An incorrect or damaged pin. A pin not properly attached to the prop by the correct chain or wire.

Steel trench ‘acrow’ prop sizes and reference numbers to BS 4074 Prop size/reference*

0 1 2 3 4

Height range Minimum m

Maximum m

1.07 1.75 1.98 2.59 3.20

1.82 3.12 3.35 3.96 4.87

*The props are normally identified by their length.

Steel trench prop load capacities A prop will carry its maximum safe load when it is plumb and concentrically loaded as shown in the charts in BS 4074. A reduced safe working load should be used for concentric loading with an eccentricity, e  1.5 out of plumb as follows:

Capacity of props with e  1.5 (KN) Height m

£2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75

Prop size 0, 1, 2 and 3

17

16

13

11

10

–

–

–

–

Prop size 4

–

–

17

14

11

10

9

8

7

53

Design Data

Soldiers Slim soldiers, also known as slimshors, can be used horizontally and vertically and have more load capacity than steel trench props. Lengths of 0.36 m, 0.54 m, 0.72 m, 0.9 m, 1.8 m, 2.7 m or 3.6 m are available. Longer units can be made by joining smaller sections together. A connection between units with four M12 bolts will have a working moment capacity of about 12 kNm, which can be increased to 20 kNm if stiffeners are used.

Slimshor section properties Area cm2

Ixx cm4

Iyy cm4

Zxx cm3

Zyy cm3

rx cm

ry cm

Mmax kNm

19.64

1916

658

161

61

9.69

5.70

38

x

Slimshor compression capacity x-

m 5m =2 m , e 8m xis = 3 xa ,e xis xa

Allowable load (kN)

x-

m 5m =2 m , e 8m xis = 3 ya ,e xis ya

y-

y-

150 140 120 100 80 60 40 20 0 2

4

6

8

10

Allowable bending moment (kNm)

Effective length (m) e = eccentricity of load Factor of safety = 2.0

50 40 30 Use hi-load waler plate

20 10 0 20

40

60

80

100

120

140

160

Allowable axial load (kN) Factor of safety = 1.8

Mmax kNm 7.5

y

54

Structural Engineer’s Pocket Book

Slimshor moment capacity Source: RMD Kwikform (2002).

Ladder beams Used to span horizontally in scaffolding or platforms, ladder beams are made in 48.3f 3.2 CHS, 305 mm deep, with rungs at 305 mm centres. All junctions are saddle welded. Ladder beams can be fully integrated with scaffold fittings. Bracing of both the top and bottom chords is required to prevent buckling. Standard lengths are 3.353 m (110 ), 4.877 m (160 ) and 6.400 m (210 ). Manufacturers should be contacted for loading information. However, if the tension chord is tied at 1.5 m centres and the compression chord is braced at 1.8 m centres the moment capacity for working loads is about 8.5 kNm. If the compression chord bracing is reduced to 1.5 m centres, the moment capacity will be increased to about 12.5 kNm. The maximum allowable shear is about 12 kN.

Unit beams Unit beams are normally about 615 mm deep, are about 2Z.5 times stronger than ladder beams and are arranged in a similar way to a warren girder. Loads should only be applied at the node points. May be used to span between scaffolding towers or as a framework for temporary buildings. As with ladder beams, bracing of both the top and bottom chords is required to prevent buckling, but diagonal plan bracing should be provided to the compression flange. Units can be joined together with M24 bolts to make longer length beams. Standard lengths are 1.8 m (60 ), 2.7 m (90 ) and 3.6 m (120 ) Manufacturers should be contacted for loading information. However, if the tension chord is tied at 3.6 m centres and the compression chord is braced at 2.4 m centres the moment capacity for working loads is about 13.5 kNm. If the compression bracing is reduced to 1.2 m centres, the moment capacity will be increased to about 27.5 kNm. The maximum allowable shear is about 14 kN.

                            '                                         

    [       _  Ÿ                          Œ Q'       +$- Ž             [     ‹                                 $!;     +$$- Ž |               ?            %=  #==              [                      ' 

);    ;    Ž ~          ‹                         {                        ‹      

                            [                 

                       Ž ~   [                                              

             [            Ž                                                   [                          Q                             [‹                     [                   [           [  

      

O









  

  

 + :K €

%

 %

 %

ƒ #%

ƒ #%

ˆƒ ><



 %

 %

> #%

> #%

  > ƒ #<  # ƒ > #%   > 

% ƒ

 %

 %

ƒ <ƒ

ƒ <ƒ

ƒ >%

b cx cy b

d cx cy b

d

Cx

Cy

ˆ<

$ 

 %

 >

 %

> ><

 %

 %

ƒ

> ƒ"

>  > …#ˆ% ‡ %=% †

d cx cy b

t

b

%

…

%†%

… %†ƒ #%

ƒ

… %†ƒ #%

…

†> 

cx cy b

    $     ˆ                _    ˆ $

ˆ]

$ 



O   %

t d

:







:  



 %

 %

…ƒ

%# †

 %

 %

>

 %

%% # ‡ … %…%# ‡ …

%†%

…

ˆ"

      ?  

ƒ

   … <ƒ

%†ƒ †

…ƒ

 + :K-  € … <ƒ

%†ƒ †

†> 

… ƒ

cx cy

%# ‡ % …

t1

…

% †… #%

%# †>

%# >

…

% †… #%

%# †>

%# >

… %# †%> #%

%># 

%>  >

t2

d cx cy b t1

t2 d cx cy b

 …

… % †

%# †

%# †%% %# †% †

>

… %# †%>  #% % ‡%# %   % ‡% … %# †  %%

> … ‡ %† >

# ‡ …

t1

# †%

 # 

 # ‡ # … % %

# †% %

 %

#> ‡ % …  #% ‡ # 

cx d

t2

‡ % …

# †>

# > %

#  %   # † 

# %

… #%

# †>%

#> 

%>  >

%

cy b

t2 d cx t1

cy

% ‡ …

% †#

% % … % †#% %

% # … # †%% %

% >

% †# > … %# †> #% %    # % % †#  %

…

  ‡ % % ‡ …

# > ‡ … # †%> #%  % ‡# %    % % ‡… # †%  %

># 

%>  >

b

    $     ˆ                _    ˆ $

ˆ

<=

      Q `[ }[

(

 :  

 ˆ

 ˆ



… 

 †% ‡



\

~ '     _                  _               

              ‹ _ \         

     ?                             ?              ‹                — Q    Œ   Q      [                   a ˆ

# %

€      a  #ˆ €    a  >ˆ

          _        [             ' # ˆ a %            

                 

}           ? 

<#

   ‘  '           '      

    ?  '                 ‹  '                 ?              %= \

(       

$ )   * 

?      ?     ?      † ?}# ?}% \ }  {    Ž  }  † `   † Z     >=ƒ*     _ %%=ˆ ‡    

}[    

~    

%] %]# ]]

        *

O

$

      *

O

(    

(     

 *

O

   ‚

   *

O

%      ƒ

<

%ˆˆ

=>ƒ

 %ˆ

 ˆ"

>=Ž<=

]=

%<<

=>%

#>=Ž%ˆ=

ƒ<

=>= ˆ==Žˆ = <ƒ=Ž]== %%=Ž>"ˆ ƒ<=Žƒ"= ƒ#=Žˆ =

#"Žƒ= #>Ž%=

#%=

#]=Ž%== %ˆ=Ž><= <=Ž>%ˆ % =Ž>== #]=Ž#ƒ=

" "ƒˆ "ƒ

< #=% #==

>" >]> >

=>ˆ =>ˆ =>ƒ

"%Ž"< ""

<Ž#%= ##<

><Žƒƒ ƒ>

=>ƒ =>>

"%Ž< = Ž

%==Ž">= ƒ#=

ˆŽ<= #ˆ

]"ˆ ]"Ž"=

%=ˆ %=ˆ

"%% ]<

=> =>

%]ˆŽ>ˆˆ %#=

ƒ>=Ž<%= ˆ%=Ž]%=

%=Ž%% ƒˆ

]"Ž"=

%=ˆ

]<

=>

ƒ<=

<ƒ=Ž"ƒ=

%=

]% ]>ˆ

#>= #]=

ƒ" <"

Ž =%<

Ž #"=

#ˆ=$<== %<=$]"=

Ž #=Ž#ƒ

]ƒŽ]"

# =

]ˆ

=>

%#=

>ƒ=

>ˆ

”                

>=Ž#ˆ

<%

      Q `[ }[

(      /       

$ )   * 

        *

O

\    ]ˆ  \   \   [ \  [ €   ‡  ‡ 

%=

ƒ#ˆ

%ƒ ˆŽ%=

#]Ž>#

%%ˆŽ%" #ˆ %ƒ" %ˆŽ%ˆ<

ˆŽ>= #=” ]ƒ ]=Ž]ƒ

=%% =%Ž=%]

#=Ž = #ˆ= #=== #===

%< %=Ž% %<Ž%

ƒ=Ž]= %=Ž]= ˆ=Ž#==

=%Ž=> =%Ž=> =%Ž=>

]=Ž%"= %=Ž%== ˆ=Ž#"=

‡   *  Z 

(    

  

 *

O

     

   *

O #]ˆ=

=#Ž=%

#=Ž]= >Ž%=

#== >=Ž = ƒˆ 

 #%=Ž#ˆ=   

(      ;” 

$ )   * 

        *

O

     

   *

O

    ! 

   *

O

    

  *

O

? } }     |   Z  [    `    

<ˆ ]ƒ ]# ˆ< ƒ"Žˆ< ˆƒ <ƒŽ]% ˆ> ƒˆ ƒ> <%

#= #= #ˆ ## ##Ž#> " ##Ž#% "Ž#= ] ]Ž Ž#ƒ

<= <=Ž##= "ˆŽ = ƒ=Žˆƒ ƒˆŽ]> <= ˆ<Ž"] ƒ#" ƒ=Žƒ> ><Ž<% <%Ž#=<

ƒ" %]Žˆƒ <]Ž]ƒ #]Ž>% ƒ Ž]ƒ ƒˆ %]Žˆ= %#Žƒ% %= #"Ž> %<Žƒ<

#= "Ž#ƒ #>Ž#" "Ž## ]ƒŽ"" < #%Ž#" ˆ%Ž ] ƒ" ƒ>Ž" ""Ž#ˆ

”                   

 

}           ? 

<>

(          

$ )   * 

        *

O

     

   *

O

%      ? ƒ

`  ‘| `\ `\ † `   ` _ ?  `€ `   {

‚  _ 

{   \    ^  ƒ `   ‡ `\ 

Ž#ƒ #>Ž#ƒ #>Ž#ƒ #=Ž#> #% ##]Ž#% %#Ž%% #% ##ˆ # #<Ž%=

=ˆˆŽ# %ƒŽ>= ==# >Ž>> >> %]Ž>% =>Ž=< %%Žƒ %Ž>ˆ ===%Ž=# %= =]Ž%= %ƒ= #%ˆ #ƒ

%=Ž>] ƒ=Ž<= #ˆ= >ˆŽ<" "=Ž = ˆ=Ž"= %=Ž>ˆ ˆ=Ž<= <=Ž##= ]Ž%= <"Ž%== >ˆŽ%ƒ >ˆ > ==

%=Ž#== %==

#ƒ

> < %Ž" #== #==Ž#>= ˆ= #==Ž"== ƒ #ƒ %" #ƒŽ%=

<ƒ

      Q `[ }[

      :  ?       _     ˆ a…_  †    ?            ^  ' ˆ \  >ˆ \ _       [ #ˆ \  <= \               

 ?    ?      † }  }  † \ €  \  

~    

Z     Ž       Ž 

 * ~ Ž      

~ Ž        

¡    Ž      Ž     Ž     Ž    \   Ž        \   Ž     ` \  [ \   [ `   ‡‚`  $   ‚ `\ {

?

 N„

Q

*

%ƒ #] #"Ž# %= #] "Ž #=Ž## #% #% #" #= % > >= %< "Ž#= >Žƒ ƒŽ< ]Ž#% #=Ž#ƒ ]Ž" #"Ž%# ˆŽ" <Ž#% <=Ž]= #"Ž%ˆ ƒ%Ž]% "=Ž#== >=Ž"=

}           ? 

            ˆ m                

 

     

Z   Z    ~   ‚      {   `€   Z   Z

  

 Z

        

=#ˆŽ=<= =%=Ž=<= =%ˆŽ=ˆ= =]=Ž= = =>=Ž=ˆ= ==ˆŽ=%= ==% =<=Ž=]= =ˆ= =%ˆŽ#==

<ˆ

<<

      Q `[ }[

$  !  P W

X Z

T Y

~                                         ?        [       '

M

S

M N

N S

M

S N

N S

M

?                                             ?               ?   _    

  [        ΠQ          

       [        ?              ‹[   

}           ? 

<]

' ;   Z '  '

ˆ

ˆ



%  %

 

%    ‘[Q    — Q       ˆ s$e — Q                     s      e         ‘[Q              ˆ t$g   t        g       ` Q  n ˆ e  $e_         _                      

 '  ˆ $…%…# ‡ ††       ? n      =  #ˆ      =>   =ˆ  

%  ;     s  ˆ ˆ    ~                  s                     _                     ˆ $                     s ˆ $ 

. †    ;  tˆ

  

~  t    †                                              “                                            ‘ †                              <=¢   [            _

<"

      Q `[ }[

"       !       

 $%ˆ=

*   

 $><=

|   „

 $><=  #ƒ 

|     

 $ˆ==

\   

 $#"=

                             

 $<==

`           

 $%==

`                  

 $#==

   †               

$>==

       ‹   

$>==

             

$ˆ==

             

    

                

      

‘ †                          †     

 $ˆ==

\             †

 $#]ˆ

\             †

 $%ˆ=

' ;                [{       [{                  [{$

    

< 

P

? ˆ } ˆ

 ˆ

%    ƒ

f? ˆ f} ˆ

:     % #<

d ˆ

> ƒ"

RB

RA

? ˆ

P b

a c RA

 %

: 



RB

} ˆ

   

 ˆ

 

f? ˆ

 … ‡ † <

f} ˆ

 … ‡ † <

~  > ; … ‡ †  >… ‡ † %]  … ‡ †     ˆ >

d ˆ

<

]=

' ;         ?       

P

P

C

RA

: 



%  

? ˆ } ˆ 

 ˆ 

f? ˆ f} ˆ

 … %

f? ˆ f} ˆ

% %ƒ

†

d ˆ

 …>% %ƒ

d ˆ

ˆ> >"ƒ

a

b

a RA

:    

< 

C RB

RB

? ˆ } ˆ

 %

 ˆ

 "

ƒ% †

b

W

a

c RB

RA

 ˆ … ‡ =:ˆ†   ˆ … ‡ †   ?

? ˆ 

f? ˆ

 % … <

f} ˆ

…# † <

% ‡ †

~     ˆ  ‡ 

x

*  ˆ

=:ˆ ‡  

} ˆ …#

†

 ˆ …%

W

? ˆ } ˆ RA

 %

 ˆ

 <

f? ˆ f} ˆ

ˆ% <

%

…#

††

 ƒ %ƒ 

 ƒ… ‡ †> ‡ <% % ‡    % > … ‡ ƒ †   ˆ ƒ  % %  % d ˆ

d ˆ

> <=

RB

]#

]%

' ;         ?   < 

    

? ˆ

 >

} ˆ

% >

? ˆ

 % … %

: 



%  

 ˆ

…% % † >%

} ˆ

% %

Ž

:    

d ˆ

ˆ> >"ƒ

W

RA

RB x

b

% †

Ž

ˆ

w/unit length

RA

C x

d   

a

w = kN/m

RB

} ˆ

 … ‡ †% %

%

%ƒ

%

… ‡ † … † "%    % _    ˆ # % %  \ ˆ



d   ˆ

ƒ

%% #% ‡ #> ‡ %% % ‡ %% #%

 …>> ‡ ƒ% %ƒ

> †

P

? ˆ

% … ‡ %† >

? ˆ

% %

} ˆ

% … ‡ %† >

} ˆ

% %

\ ˆ

%% % >

b

a C RA

RB

f? ˆ f} ˆ =

d ˆ

f? ˆ f} ˆ =

d ˆ

%> % >… ‡ %†%

 

? ˆ } ˆ RA

C

 %

? ˆ } ˆ

RB

\ ˆ

b

a RA

B

C

? ˆ

> >"ƒ

 %ƒ

P

? ˆ 

 #%



f? ˆ = % f} ˆ f\ ˆ %

d\ ˆ

  %  ‡ % >

]>

]ƒ

' ;         ?       

<  ? ˆ 

RA

B

b c

RA

RA

RA

c

RB

? ˆ

 %

…>% % † %> % …% ‡ † } ˆ %>

… ‡ † %% % …% ‡ † \ ˆ %>

ˆ " > } ˆ "

 "  | ˆ  =:<%*   ? #%"

? ˆ

P a

:  

? ˆ

? ˆ

? ˆ

%  

:    

f? ˆ = % f} ˆ  <

d ˆ

f? ˆ = f} ˆ

%  ƒ

f? ˆ = f} ˆ

% ƒ"

> "

p >  ˆ  % d\_ ˆ #=% p > … ‡ !†>  >  % d?\ ˆ > …>% % †%  p %    <  % d\} ˆ < % ‡  d_ ˆ

>  =:ˆ"*   ? #"ˆ

P

RA

a

b

D

RB

c

> %     ‡ } ˆ  % ? ˆ

? ˆ } ˆ

 % 

f? ˆ = f\ ˆ

 … ‡ † ƒ

d\ ˆ d| ˆ

  %  ‡ > ƒ %   =:<< %]

]ˆ

]<

      Q `[ }[

    &     \  Q                     ‹          

X2 X1 A2

A1

Free moment diagram

L2

L1

M1 = 0

M1 = 0

Fixing moment diagram

M2

Resultant bending moment diagram

M2

 &        # # # % > % # # % % % …% > † ‡ ‡ %% ‡ ˆ< ‡ ‡ < ‡ #  # % % # # % % # %

~ '          ΠQ                                                            

}           ? 

]]

 = !           # # % % # # ‡ %% …# ‡ % † ‡ > % ˆ < ‡ # %

  # ˆ % ˆ > ˆ =   # ˆ % ˆ >

#   >    ' [   _    †        [ 

                           %

# = N ‡  ‡ „   # # % % ‡ %% …# ‡ % † ˆ > # %        % ˆ

…># ‡ >% † "…# ‡ % †

                               "        " %                 " % [      

]"

      Q `[ }[

   ; ;     Z                             

    [{           ‚ £    ~ Z  £    ~  * w

0.125

1.250

w

0.100

0.100

1.100

0.400

1.100

0.025

0.080

w

w

0.107

0.071

0.107

w 0.077

0.929

1.143

0.393

w 0.036

0.036

0.077

w 0.080

0.393

w

0.400

0.375

0.375

0.070

0.070

1.143

w

w

w 0.188

1.375

w

w

w

0.150

w 2

w 2

0.116

0.170

0.339

0.894

0.116

1.214

0.339

w 2

0.161

0.107

0.161 0.170

w

w

w

w

0.350

0.175

0.100

1.150

0.175

1.150

0.350

0.150

1.214

0.313

0.313

0.156

0.156

w 2

0.167 0.330

0.111

w 2

w 2

w 2

0.056

w 2

w 2

w 2

0.143

0.096

0.143

1.190

0.122

0.056

0.119

0.357

0.367

1.133 w 2

0.119

0.357

1.133

0.033

w 2

w 2

0.133

0.122

w 2

w 2

w 2

0.133

0.367

w 2

0.190

w 2

0.905

0.330

1.330

0.111

}           ? 

]

$     [          _           [                     '   Q          `

Ž‚            [    } ƒƒ 

%! 

L

0.85L

0.7L

L

2L

%       [  '

     [   '

%

 ˆ  %   % %  %  s ˆ   ˆ %  ˆ   

         [ _                 

"=

      Q `[ }[

( ?< ;  `

Ž‚ 

 [  '     …s ‡ s … ‡ #††  ‡  … ‡ #† %  `‚ ˆ s s  % %

 %     ˆ =:> #==

s                s         

(  )  

;  

W kN/m

Z_         a ˆ    a* _ ˆ    %

 #



 

Z_       s _ ˆ

  ‡ 

Z_  

 % ‡  "

d_ ˆ

}           ? 

"#

<      ‚                ?                                                 

 „   _      _                          

            [                                                       †                     †       

<   )   ;

                      

    €   _               

                                          †     ?                                 

I=∞

Fh 3

F

h

F 3

F 3

F 3

F 3

B.M.D

"%

      Q `[ }[

<   ) ;          +?                    _           _         _                         [             _     ?                            †      _    

      †                      _    

Assumed point of contraflexure

b b/2

Fh 4

F

Fh 4

Fh 4

Fh 2

Fh 4

Fh 4

Fh 4

h

F

F

F

2(n – 1)

(n – 1)

2(n – 1)

B.M.D

n = number of columns

 >    ) ;          €   ‹         _                                          _                  

                     

      _     ?                                               _                                                                  _                                   _    

      ?   FULL FRAME EXAMPLE b/2 F1

F1

h F2

F1 4

F1 4 F1 2

F2 h F1 + F2 2

F3

F3

h 2 h

b PART FRAME EXAMPLE

ΣF 4

F1 + F2 4 F1 + F2 2

F1 2 F1 4 F1 + F2 2 F 1 + F2 2

F1 4

F1 + F2 4

F1 + F2 + F3

F1 + F2 + F3

F1 + F2 + F3

2 ΣF 2

2 ΣF 4

2

ΣF 4

ΣF 2

ΣF 4

F1 2

F1 4

F1 + F2 4

F1 + F2 2

B.M.D

b F1 4

F1h 8

F2 F1 + F2 4

F1h 4 Mb

Mb

Mb (F1 + F2)h 8

where Mb =

F1h 8

(F1 + F2)h 4

(F1 + F2)h 8

">

(2F1 + F2)h 8 B.M.D

Mb

"ƒ

      Q `[ }[

(  ¤  Q             |              —             

     [                                   [         

       [  \ [      _        €                            ?                                                        \ [                                 —           [{$                              ?            [                                      ?          Œ     Q           ?         

                                      '

Simple support

i=1 i=3

i=4

i=2

b

a

Fixed support

Column support

}           ? 

"ˆ

           _         ‡  

£ =        £ #  _            _                   #Q   #Q £ #   #          

m′ = m3

m′ = m4 M

Lr

€_           #               †           Œ      Q   % %  ˆ        <  :  ˆ " …# ‡ # † ‡ …# ‡ % †

€                ' %  ˆ   …# ‡ # † ‡ …# ‡ % †

%  ˆ   …# ‡ > † ‡ …# ‡ ƒ †

         '  ˆ    " #‡ ‡  

€ _             ˆ  ˆ       ˆ % =%ƒ [{=: €            ˆ =% [{$

"<

      Q `[ }[

$              _               —                 [                }             _ ‹                  [  

%  ˆ p p # ‡ % ‡ # ‡ ƒ

i4 a

m

i1

i3

i2

   #ˆ    " #‡ ‡  

b

m

a

i3

i1 i2 > 0

%  ˆ p p # ˆ # ‡ # ‡ # ‡ >   

b

      > ‡ #% ‡ % % # ‡        

      ˆ p # ˆ >   # ‡ # ‡ > ‡ % # ‡ %  % 

m

a

%  ˆ p p # ‡ # ‡ # ‡ >

i1 i2 > 0

b

€   _      #   %

 ˆ =:<

c m

i1

a

… ‡ † # ‡ … ‡ † % ‡‡‡

i2

b

d

d c m′

m

c

m′

a

c

#ˆ

#=

=:#ˆ #‡

>  #= ˆ    " %‡ ‡   %

 % … ‡ % † < }        

#0 ˆ

  ˆ =:>ˆ; # ˆ #0 ˆ

% #<

}           ? 

m

m′ c h

#ˆ

%  =:> >

  ˆ =:>>; # ˆ #0 ˆ

% #ˆ % m

 % ; #0 ˆ  <



=:>>

  % > %

% ˆˆ



"]

""

      Q `[ }[

           '

 t f ˆ ˆ $  

~         $          t              f                               g               f    ‹             Z                    $                         †                                                   $ ˆ  %             $ ˆ %% ˆ  ‡  

           $ ˆ pƒ =>%         $ ˆ ˆƒ =><   ƒ ƒ         $ ˆ p…  

 †=>%  ¤ ˆ %p >        t ˆ =        [              

                        

                                       [                >            '   $ ˆ #>  > &      $   =>    t ˆ =$: $  [            [                      €  ‹                                     

"

}           ? 

 )/ ;                ‹ ~    _      †                   _                    _                              
&   & ˆ =  '  ( ) _ 

   \  ?    \      _   *      Z          ?        

      ‘   

‘ †  

Z_      |    

 

 ;  )  †    Tmax

w

H y

H h

V

V

x L ˆ

ƒ… > † %

'ˆ

 %

 "    # ‡ &% >

)ˆ

% "

p _ ˆ ) # ‡ #<&%  >% ƒ & ‡


ˆ


 

  ) # ‡ #<& > 

=

      Q `[ }[

 

 ;  )     Tmax H h

H

V

θ

V

y x

L ˆ

ƒ… % † %

)ˆ

'ˆ

)(  ‡  %

_ ˆ )

     y # ‡

 "&% ƒ >  y

% "


 

  % # ‡ ( ‡ ƒ&

  #<& ) # ‡ >  ƒy 

D

}           ? 

#

;  ~                        ‘                                                                   „      [    ‹  [                         |                          Ž             ‹      ~                         ‹                            ƒˆ ‘†                 _         ?                   *  ‘†                +           d              ,     …, ˆ =† #                                          [       _    ;

%    & / 

‡             

 + d  # , *ˆ %p #

‡                    

                                  _          \             \            €_          

*ˆ

# %p

#" * ˆ p d  # ƒ" *ˆ %p #>  #  *ˆ ƒ #ƒ   *ˆ #ƒ  # > > %p #  =:ˆ<  *ˆ ƒ %p #  >:ˆ]  *ˆ %p #ƒ *ˆ

€       $   %ˆ       [                                "  $ = }  ? # ]"

   ‡                                                                           

                                      '

. } ˆ >=      . } "==ƒ  €  |   } "==%  ‚   ~ |  .   ]  ‡  |                  '

. ^                   

. ?             .                                                 Œ€  ‚ Q            

.                 †                       

 

. ?                 g £ %  >                       

 

.                    

.

.

.

   %ˆ  |           \                           €                 [    [                                      €             ƒˆ=        ‡                             ‡        [              [                      |              

        „                                _  

             

‡ 

>

$         )

  [   

                

                

   

                 

#      †        €            Ž    ‹      ‹ $    `        

    [   >                Œ   Q                  ]ˆ   %===     ˆ   #==                     ‹                   † :     

 †      # /  

      

#



   !          

$ /  

      

. ) ;  * 

  

$  

   !     

      

/  :;  

`     

`     

€ 

    

`   



}      

.!  

`     

}      



}      



}  

  

}      

< )

    †         †    

 †     "   !

$   !

    

  ” ! ; 

.

‡      

   

.

‡      

   

.

|                

   

.

‚    

   

.

^             \      |     

.

      

.

.

     

.

^             \     

.

|  

\      |  

.

  )  

. . .   )          

! 

. .

.

      |  

” ~ [               [        

.

  

.

|  

.

\      |  

.

ƒ

      Q `[ }[

$ !                                       †             [    } ˆ> =' `  %        [     #ˆ              ?                                                      ?  

           [    [  >         [      [      [

$

     !  &     †     $   

$

 

     )  %:! 

$

)    

"      

.

`          _  

.

           

.

              

.

      

          _  

.

         

.

|      †        

.

                 _ 



.

           

.

* ‹                

.

                        [

.

\ 

      

.

                    

.

|      †        

.

Z                     

‡ 

ˆ

$           †            }       Sieve size mm

0.002 0.006 0.020 0.06 0.2

Description

Silt

0.6

Sand

2.0

6.0

20

60

Gravel

200

Cobbles

Boulders

$    ;   † ?                _        } ˆ >=                                                     '    ‡‚?*     ‡‚?*      ?{|   ?{|  ‡‚?*    ?{|  ‡‚?* \ ?{|  ‡‚?*   *  \*?—    

  ˆ¢  

  ‡‚?*

%=¢Žˆ=¢  

‡‚?* $?{|

     

%=¢Žˆ=¢   

    ?{|  ?{|  ‡‚?*

  ˆ¢   

  ˆ¢  #ˆ¢Ž>ˆ¢ 

  ?{|  ‡‚?*    ?{|  ‡‚?* ‡   *  \*?—

ˆ¢Ž#ˆ¢  >ˆ¢Ž<ˆ¢  

ˆ¢Ž%=¢  

ˆ¢Ž#ˆ¢    ˆ¢  #ˆ¢Ž>ˆ¢  >ˆ¢Ž<ˆ¢   

  ˆ=¢      

$    ;   ‘  

?         

‘   

?       _    

   

?                     

~ 

\      [     $          † €        [    Œ  Q    

€  

} [           

  

   _     

€   

‚ †              

?   

   _     †    

<

      Q `[ }[

                                                                        

              €                                  

       ‘                           [       ~                                        ‹                     

              \           ‹                   

            Q   Q

"   /   „

Failure envelope

Shear stress τ φ′

Over consolidated clays c′ > 0

c′ = 0 generally

Effective stress circle

Direct stress σ

‡ 

]

 :                 " 

   * Z    |     

$(”  ; )

%!       

'   ) 

"  ) 

 * 

 * 

=Žƒ ƒŽ#= #=Ž>= >=Žˆ= >ˆ=

%<Ž%" %"Ž>= >=Ž>< ><Žƒ% ƒ%Žƒ<

 #< #<Ž#" #"Ž# # Ž%# %#

 #ƒ #ƒŽ#< #<Ž#] #]Ž# #

; 



” ?  _                |       ' \  ƒ== [{$%

€                     '  ˆ #ˆ ‡ =ˆ #ˆ

 :             !                        |                       _                     Q                              €                Q  †     

          >= [{$% $   

  > ; 

(  : ( ƒ

'   ) ;   * 

%!       0

\     

‘ Z  *

>>ˆ %ˆŽ>ˆ #=Ž%ˆ

#<Ž%% #<Ž%= #<Ž%=

#"Ž%ƒ %%Ž%< %<Ž>ƒ

"

      Q `[ }[

   /  ˆ „ Shear stress τ

Effective stress circle

Failure envelope φu = 0

Cu

Cu > 0 Direct stress σ

 :             !    " 

     * O

'   ) ;   * 

         

>#ˆ= #==Ž#ˆ=

# Ž%%

€     €       

]ˆŽ#== ˆ=Ž]ˆ ƒ=Žˆ=

#]Ž%=

           

%=Žƒ= %=

#<Ž#

      \  ƒ:ˆ      _     >=           `    ` 

  !      ' < +'<" 

'<

  €    $   \   *    \   

=ˆŽ# #ˆŽ% >Žˆ #=Ž#ˆ %=Ž%ˆ ƒ=Žˆ=

‡ 

                                                                                                _          

          _       _               ‹       $   

" 

      

" 

      

   

*    

>ˆŽƒ=

*    

ƒˆ

 

*    

ƒ=Žƒˆ

*    

%=Ž%ˆ

(

*    

#ˆ

*    

ƒˆ

  *$ 

€          *   

#]Ž#

`  

#ˆŽ#

#ˆ %=Ž%ˆ

    

# ƒ=Žˆ=

$

\     

>ˆŽƒ= >ˆŽƒˆ

*      

>=Ž>ˆ %ˆ

!

     

>ˆŽƒˆ ƒ=Žƒˆ

ƒ= # Ž%%

Z      

>=Žƒˆ

*      $    Z     

| 

>ˆ

~

ƒˆ

'   

%ˆŽ>=

#==

      Q `[ }[

(   †   

);  ;     " 

$ ;    * 

#   * 

ˆ===

€     [

   [     *                   ‘  [ [

  %===   #===   <==

       

>==Ž<==

 

#ˆ=Ž>==

€  

]ˆŽ#ˆ=

    

 ]ˆ

      



\          % Z             % *          % \  %

<==

Z     % *  % €      $  

#==Ž>==  #== %=Žƒ=

      

%ˆŽˆ=

       $



`     $



  >===

%==Ž<==  %== >==‡

}    [             ‚        _              ‘  [ Ž 

                  \                       _        † ‚        _ 

‘  [ Ž              ‘  [ Ž       ‘  [ Ž              \               ‹            }         [                    

   \               

{' #                } "==ƒ %               !     #            !        

$ = } "==ƒ' # "<

#=#

‡ 

J        

)     &    

);  ;   ' .

€           g ˆ %=  >= -0       ‹      g        !                                   g             ' - ˆ `  ' -

 ‡ -0  ‡ =:ˆg!g g 

#:> ‡ -0  ‡ =:ƒg!g ˆ g 

? _                g       

  

      = ˆ #= #ˆ %= %ˆ >= >ˆ ƒ=

'   ” 

&



ˆ= <ˆ "ˆ ##= #ˆˆ %#= >== ƒˆ= ]ˆ=

#= #ˆ %ˆ ƒ= <ˆ #=ˆ #"ˆ >ƒ= <ˆ=

== == == #ƒ >ˆ "= #]= ƒ== "=

”        }   ‘  # <#

$   &    

);  ;   ' . €          †Q           [   $      - ˆ %

           g ˆ %ˆ

$    ! - ˆ #=

`       g ˆ >

- ˆ ]

        g ˆ >

-  ˆ -

%$         

#=%

      Q `[ }[

J                   \                                              _                             _      

     _       >==  #"== [{      >==  <==  ˆ==  %=== [{      %]ˆ  ƒ==        %]ˆ  %===    >==  #ˆ== [{         \€?   >==  <==    ˆ=  ˆ== [{      ]ˆ  %"=        %=                       

Pile capacity (kN)

Working pile loads for CFA piles in granular soil (N = 15) 2000 1800 1600 1400 1200 1000 800 600 400 200 0

900 φ 750 φ 600 φ 450 φ 300 φ 150 φ 5

10

15 20 Pile length (m)

25

30

Pile capacity (kN)

Working pile loads for CFA piles in granular soil (N = 25) 2000 1800 1600 1400 1200 1000 800 600 400 200 0

900 φ 750 φ 600 φ 450 φ 300 φ 150 φ 5

10

15 20 Pile length (m)

25

30

#=>

‡ 

Working pile loads for CFA piles in cohesive soil (Cu = 50)

Pile capacity (kN)

900 800 900 φ

700 600 500

750 φ 600 φ

400 300 200 100 0

450 φ 300 φ 150 φ 5

10

15 20 Pile length (m)

25

30

Pile capacity

Working pile loads for CFA piles in cohesive soil (Cu = 100) 1800 1600 1400

900 φ

1200

750 φ

1000

600 φ

800 600 400 200 0

450 φ 300 φ 150 φ 5

10

15 20 Pile length (m)

25

30

#=ƒ

      Q `[ }[

$  ;       + ˆ

   a  ‡ g  g 

~                                                        _    a ˆ =>  =<                  a ˆ =>         a ˆ =ƒˆŽ=ˆ        *    ˆ                   

              [  %ˆ      >=   

     ;                    %ˆ                ?  [       '

ˆ#





#

 ( ‰#…"

#† ‡ "…# =#"

#†Š

~  (             #   "      

           

!    {  [                          \                                                

     €                                           [   ˆ )g=             )   [                [    g                             

                              

                                    [    

‡ 

#=ˆ

(       ?                     

                                 [                                                   g    ,             ˆ

  -0 ‡  -0  ,  d g

~  -             [  } †                          -0            ,    †          ,             d                0                      g ˆ %ˆ  >   !   &”

`   ` 



ˆ

O„

Oˆ „ ˆ €„

#< % < #]ˆ

## %ƒ ˆ> #ƒ"

] %= ƒˆ #>=

” } †  # <#                

   

    d   ,                [  ^  # "ƒ  ' (  *   

    *    *0

  $  ‚  

 $  \     $  `    $   $ 

=ˆŽ= =]Ž= #= ="Ž#= ="Ž=



      

    †    *     *

|        |       }      ¤ 

#==Ž%== =]ˆŽ#%ˆ =]=Ž#== =ˆ=Ž=]=

?                  %=     [               Œ[Q      ‡    [          ##= === [{$%       #=  %=       [    [     #= [{$%       #=  %ˆ [{$%          %ˆ  ]= [{$%          ]=  ##= [{$%          $ = ^  €‘ # "ƒ ‚        ?\

#=<

      Q `[ }[

(   `                     [     

    ]ˆ        „ #ˆ=                          „   †       €   `          [                                            ' (    +

-

>==

>ˆ=

ƒ==

ƒˆ=

ˆ==

ˆˆ=

<==

]ˆ=

(    +

-

]==

"==

==

#===

##==

#%==

#ƒ==

#<==

‡ 

#=]

< )

 ‚ [ Q                         \ Q        _             ‚ [ Q                                                 [    Œ Q   Œ Q                [               Œ Q ?       †      =%ˆŽ#                      %Žƒ          #=Ž#ˆ        ?                  †                                   

   ‚ [ Q                                 ‚ [ Q                 [  

         †            ' s ˆ ,s €    , ˆ ,        , ˆ ,         , ˆ , ,  #

 f

, ˆ

   ,     ‹     \‚ ˆ          €

…# …#

 f†  f†



 

, ˆ 



# …# ‡  f† ˆ , …#  f†    



 ,     =>ˆ      =<     =ˆ  =<         #=  %"          *       ,                           ‚ [ Q              [                         Œ [Q                   [                [  [            †    p           [   ˆ % 0 =…g , †           ˆ % =g       

#="

      Q `[ }[

(   †   )

 !  )

 Ž          <=Ž"=     

   (   ;  )

 Ž     #<                   _           ?        } "==% €       Z  [             }   ’ ` Z   [ ?     [           % ;  )

 Ž Z      _                           ?  _ 

                            [            %=    _        _        _     

‡ 

#=

  

)                                            ‹      {  ‘  }   \  {‘}\                     [          †                     

   %==%                     [  {‘}\         _      _                                                    =%                    =ˆ               #%ˆ          ~     _      [             )  

(  :/ ( = &   ± (    * Z  ‘

#=Ž%=¢ #=Žƒ=¢ >ƒ=¢

=]ˆ = #=

$ = {‘}\ %==%                    [  {‘}\

##=

      Q `[ }[

               }                   ~     

            

'  !  $

 

 $ ”

'    

 $ ”

 $  .” ”



.

 #"Ž%ƒ    #" [ #<Ž%ƒ

` ~

%ˆŽ%" #<Ž%ƒ

\ 

#"Ž%=

 

?  ?

#"

* 

#%

\

%=

?

#"

*

%%

\



#ˆ

%> #=

Z Z 

 `        

~ 

"Ž#" ##

|   `   

%=

? }  

%= #"

` ` 

#% #=

}[ " \

 Ž#] ‘ #= ‘  #ƒ   ‘  #] ‘  %=    )

} }  ‘

%= #ƒ #%

%< %% %=

~   >= — #%

#"

~ #%

Z  Z 



” €           {‘}\       

$ = {‘}\ %==%                    [  {‘}\

‡ 

###

$         !    (                 )                      [                      {‘}\                      ˆ=      ‹    

$         ;    D/H 0.2

0

0.4

0.6

0.8

1.0

1.2

Minimum depth 1.0 m 1.0

2.0

w Lo

Mo de rat M e od er at e

1.5

Hi

3.0

gh

h

2.5 Hig

Depth of tree influence (m) (minimum depth of shallow foundations (m))

0.5

3.5 Broad leaf water demand Coniferous water demand

$ = {‘}\ %==%                    [  {‘}\

##%

      Q `[ }[

$         ;    D/H 0.2

0

0.4

0.6

0.8

1.0

1.2

Minimum depth 0.9 m 1.0

de

rat e

w Lo

e

at er

gh

Mo

M

Hi

od

Hig h

Depth of tree influence (m) (minimum depth of shallow foundations (m))

0.5

Broad leaf water demand Coniferous water demand

$        ) ;    0.2

0

0.4

0.6

D/H 0.8

1.0

1.2

0.5

er at e

w Lo

2.0

h

1.5

e rat

de

Mo

M

od

1.0

Hig

Depth of tree influence (m) (minimum depth of shallow foundations (m))

Minimum depth 0.75 m

gh

Hi

2.5

3.0

3.5

Broad leaf water demand Coniferous water demand

$ = {‘}\ %==%                    [  {‘}\

‡ 

##>

    \                                               

                             `  ? # =     `  ?      

                 ?    ‹                 Œ Q  

                                          ?                                                                  ŒQ     Œ Q                            ‚                     ‹           ŒQ  # = ?        [     [                  \   *  _  ?  \*?        ‡              $ ‹     ~            _   ‡          $      ‚    

     ˆˆ                   ?   ~                ‡        {                ^        \‚\*  ¡           \‚\* ^   |      }      Œ   Q             [             

                [                              ‚                            

##ƒ

      Q `[ }[



                                                                  

   

( ;       

Œ_    Q       Z  “      [“             ŒQ    [ †  

? †      † “     $$   

\          [  

‡  [“ “   “   

       [

Z‹    

  

     

\   “   “   [

?

       

%     %    

    

_ $       

\   _  _             † 

€   _  

?                    

€     

€     “   [        

\   

\                  

` ‹    

`       Ž              

\

    

?   [“   [        

¡_     

\              

 †_ 

`  

\      

?     †   †  

       [ 

? [     

`_ 

\  †  [  

‚      

        

`   

?  _    ~

‚    

~         [     

` †    

‡    Ž  

     

‚    [          

~  †_   _      _   _   

‡ 

##ˆ

$ !         [     

       [       [   

     

      [                         _     ‹             _           ‡         |          €   ‚  ?  |€‚?      \*?                                          _       ‹             ‚                                             _   †                     #==                               }       %ˆ   #= === %     ===%                 >=            #== %           ##=             

         _                  [        _       

 ˜

##<

      Q `[ }[

<   &                                                 ‹

                       [                                      [    ‘                              

     '

%:!  _                                    _         _          _       Œ ‹Q                

   [        $      ‚          _  „    

'  \    _     [                    [                                                „            [

   

                              † $   

   

  

     „                                      

(    \                      _          

     \                    ‹          ‘                                                                                                      _                    

     ˆ   } ˆ%<"   \#ƒ  \ƒ=   \         |>=  |]=    |                    {$%                 *  |       }                                \#<                \%ƒ   \%ƒ          [                          \#<

 ;   ~‹                    ‹                  [             '      Ž `        **               Z       %  ƒ>  [         ( ;  Ž \         }   ‹ |      

                 ‹   *     #;;  Ž              Z|€ €                 ‹     

##"

      Q `[ }[

$

     "

#%  #=] [{$>          ƒ   < [{$>

    ?            ‹                       ^           #ƒ¢                                  Q    ?      ^           #]Ž%>¢                      ^    

                         [    Z          %=¢     [ $   [                ‹                 >Žƒ¢                 [           } ˆ%<"' `  %      \ # %   >                #%¢  %=¢   ¥%=¢  

   `

##

 ;   † $   ;   †             _       ˆˆ     _         ‹   ' †”















"



'



N„O

O

N„



N„



N„Q

€

N„Q

€





#== #== #== #== #==

>" ˆ= <> ]ˆ #==

>" ˆ= <> ]ˆ #==

<>> ">> #=ˆ= #%ˆ= #<<]

%ƒ# ƒ#] <<% >" #<<]

>#] ƒ#] ˆ%ˆ <%ˆ ">>

=ƒ< #=ƒ %=" >ˆ% ">>

%" %" %" %" %"

##= #ƒƒ #"% %#] %"

#ˆ= #ˆ= #ˆ= #ˆ= #ˆ= #ˆ=

>" ˆ= <> ]ˆ #== #ˆ=

ˆ] ]ˆ ƒ ##% #ˆ= %%ˆ

#ƒ%ˆ #"]ˆ %><> %"#> >]ˆ= ˆ<%ˆ

><# <%ˆ % #ƒ=< %ˆ== ˆ<%ˆ

#=< #ƒ=< #]]% %#= %"#> ƒ%#

=< #ˆ< >#> ˆ%] #%ˆ= ƒ%#

ƒ>> ƒ>> ƒ>> ƒ>> ƒ>> ƒ>>

##= #ƒƒ #"% %#] %" ƒ>>

#]ˆ #]ˆ #]ˆ #]ˆ

>" ˆ= <> ]ˆ

<< "] ##= #>#

# ƒ= %ˆˆ% >%#< >"%"

ƒ%# ]% ##ˆ" #<ƒ#

#< ] %%>> %"#ƒ >>ˆ=

="= #"% ><ˆ <#ˆ

ˆ=ˆ ˆ=ˆ ˆ=ˆ ˆ=ˆ

##= #ƒƒ #"% %#]

%== %== %== %== %== %== %==

>" ˆ= <> ]ˆ #== #ˆ= %==

]< #== #%< #ˆ= %== >== ƒ==

%ˆ>> >>>> ƒ%== ˆ=== <<<] #==== #>>>>

ƒ"# ">> #>%> #"]ˆ >>>> ]ˆ== #>>>>

%ˆ>> >>>> ƒ%== ˆ=== <<<] #==== #>>>>

= # %=" ƒ#] ]=> #<<] ˆ<%ˆ #>>>>

ˆ]] ˆ]] ˆ]] ˆ]] ˆ]] ˆ]] ˆ]]

##= #ƒƒ #"% %#] %" ƒ>> ˆ]]

%%ˆ %%ˆ %%ˆ %%ˆ

>" ˆ= <> ]ˆ

"ˆ ##% #ƒ# #<"

>%=< ƒ%# ˆ>#< <>%"

ˆƒ% >" #ƒ"" %#=

><=] ƒ]ƒ< ˆ "= ]##

#=> %>ƒ ƒ< ] #

<ˆ= <ˆ= <ˆ= <ˆ=

##= #ƒƒ #"% %#]

%ˆ= %ˆ= %ˆ= %ˆ=

ˆ= ]ˆ #== %ˆ=

#%ˆ #"] %ˆ= <%ˆ

ˆ%=" ]"#> #=ƒ#] %<=ƒ%

#=ƒ% %>ƒƒ ƒ#<] %<=ƒ%

<ˆ#= ]<< #>=%# >%ˆˆ%

%<= "] %="> >%ˆˆ%

]%% ]%% ]%% ]%%

#ƒƒ %#] %" ]%%

>== >== >== >== >==

ˆ= ]ˆ #== #ˆ= >==

#ˆ= %%ˆ >== ƒˆ= ==

]ˆ== ##%ˆ= #ˆ=== %%ˆ== ƒˆ===

#%ˆ= %"#> ˆ=== ##%ˆ= ƒˆ===

##%ˆ= #<"]ˆ %%ˆ== >>]ˆ= <]ˆ==

>#> #=ˆˆ %ˆ== "ƒ>" <]ˆ==

"<< "<< "<< "<< "<<

#ƒƒ %#] %" ƒ>> "<<

”     _    

$ = } ˆ%<"' `  %' #

#

    ;     } { >><        †       \ #     Œ Q   \ %     Œ Q           \ #   #   ‡>        #==    %   ‡ƒ    ‹      #==  € \ %           #==   #   #ˆ        #==       } ˆ%<"                 

     ΠQ                          

  

#%=

      Q `[ }[

   ;   ( )  `            „       



         `   ^       ?  \  ‚  €     €     ‚    €   

   } ˆ%<"                                [               [             ^ †           †  "0  ƒ0 %ƒƒ=  #%%=         ‹    ^  '                    Z     |   #"                #%       \               #                  €             }                 #% #"   %ƒ  [                   

     ;      ƒ>  [              *               „               =   %ƒ=       %<ˆ    % =      }             %=ˆ=    >#  ‹   \           =Ž%==      =Žƒ%=                *        =   [     

   `

#%#

  !  ; +**            >                     #"     %<                  ^    }                

                                            €

     ^ ‹ **     ^ ‹+ **            ' "*) +

-

   +

OH



‰

€ˆ

ˆN

ˆH

Q

Q‰

Hˆ

O„„ OOˆ OQ„ „„ Q„ €„„ €ˆ„ ˆ„„ Q„„

. .

. . .

. . . .

. . . . .

. . . . . .

. . . . . . .

. . . . . . . .

. . . . . . . . .

. . . . . . . . .

  

$+

$+

$+

$+

$+

$+

$+

$+



$ = €

  %==%

#%%

      Q `[ }[

";     ";  |                             [          [  ‹

      |                             [             **                    

     _           _        

         ‚                     [       [    [        

          

#  €        _            %=¢       %= \   >= \ ^          ƒ= \   ‹   [                        ~       

                                              ' Š"   Ž ./  #"&                                 ‹                €      ‹[         ?      [                    [     ‹[             Š   ? 0" 0.0 ./" ^                      ^            ƒ=Žˆ=¢        [           [    [                     [                            

   `

#%>

    [     ^                      [                

                          [                ‹            ^  ' Š

  ;  Ž              [                 

         >Žˆ   “       _ %     Z                  %=         Š  ;  )!  Ž "/# /0"

" /0.# 0"* " ~     [            "/# /0"

                     

        >Žˆ        #     Š( )   ;  Ž /& /""/&     [       [            _  [              [  [         ‹    ƒ  ‹          #ˆ    Š" ) ;  Ž 1&0 /# /*02 0&/#       ‹  [          [                         [  

         Z    ¤  |                _ "            >    Š   ;  Ž )0/.& 3//&                                                   ?       

   

 „                >ˆ         [          #=    $ = }‚ | % ‚   

 >=]   >ƒˆ ‚       }  

#%ƒ

      Q `[ }[

#                  ?             }     %>=Ž%ˆ= \        [             _                       _    ‘                              _                         

                    } ˆ%<"' `  ƒ       

            Œ     Q Z    } ˆ%<"    

    %=   >=     ƒ=   <=    _             [  %ˆ   >=     ˆ=   <=     [  [ „

            #ˆ   >=     >=   <=   *  _         #ˆ   =  

   `

#%ˆ

(   †   ;      *      )  " 

   +

-

|  ˆ=   „  ƒ==   $    ˆ=   „  ƒ==   $ ‚  ˆ=   „  ƒ==   $ }$              ‚     `   [  

$%ƒ • %ˆ  ˆ= $#ˆ $%ƒ $#=  #ˆ Z  #==     $ˆ  " $#=  #ˆ $>=  ƒ=

\

     _                             †        †  ‘            [    ˆ=       

  

    

( )        [                   „                             `                  „                              

         <==    |*  =]ˆ [{$%   ƒ==    |*  #ˆ [{$%    >==    |*     #ˆ [{$%            [                 [       Ž#%  [   |*  =]ˆ [{$%    #%Ž#"  [   |*  #ˆ [{$%   [         "Ž                               ƒƒ%

#%<

      Q `[ }[

"    ‹                

       „ † 

  \#<    ƒ==    

6.0

50 0× 63 30 0 × 75 25 5 × 63 × 22 25 2

Total working load (kN/m2)

7.0

5.0

28 0× 2 50 20 25 × 17 0 × 50 17 5 × 50 5× 6 50 3 15 0× 50 12 5× 50

4.0 3.0 2.0

100

× 50

1.0 0 1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Span (m)

$ ;       ;     !  "”  * O

NK„ NKˆ OK„ OKˆ K„

$   + -    

" ;   + -    

Œ

ˆ„

Q

Hˆ

Œ

ˆ„

Q

Hˆ

%% # #] #< #ˆ

>= %< %> %% %#

>" >> >= %" %<

ƒ] ƒ= >< >ƒ >%

>= %< %ƒ %% %#

> >ƒ ># % %]

ˆ% ƒˆ ƒ= >] >ˆ

<> ˆƒ ƒ ƒ< ƒ>

”        $%ƒ=   [          

   `

#%]

 ;   '$ ˆOQŒ } ˆ%<"' `  %' %==%                  ?                                      

    '$ ˆOQŒ= ( O $ ; 

  

$;     * K

$

 





?

k

`     

\ 

t

  

# }  

 

‡ 

?

`       



Z     ‚    





` 

a

?     



?  



Z  

 

$ = } ˆ%<"' `  %' %==%

#%"

$   ;    %    ;  !   N  O $  

NQ O€ "€„ "ˆ„

' 

  

;k *

O

  

  

k *

O

 >   

  

k *

O

       # 

? *

O

 

? *

O

ˆ> ]ˆ #%ˆ #<=

>% ƒˆ ]ˆ <

<" ] #%< #ˆ%

%% %ƒ > ƒˆ

#] # >= >ˆ

$ 

    tk *

O

         *

O

 

  *

O

=<] =]# %== %%=

""== #= "== #= "== #ˆ ===

ˆ"== ]%== ]ˆ== #% <==

!   * 

>]= ƒ%= ]== ]"=

{' # ~                               †      ~   

                                     %            ' \ #  #%¢ \ %  %=¢   \ > ¥%=¢ > € \ >                %     #>  } ˆ%<"                           ƒ                       '                     ˆ tk $> Z             ˆ $%=     ˆ $#<

$ = } ˆ%<"' `  %' %==%

   `

#%

. † 

                       \%ƒ         \%ƒ                  ƒ>  [  †      ; 

  

' 

   Nˆ

  

   NQ

   

   NH

        NŒ

$ 

    N‰

      O„

ƒ ˆ ] #= #ˆ %=

#%< #>ƒ #> #ƒ> #ƒ" #ˆ%

#%< #>ƒ #> #ƒ> #ƒ" #ˆ%

#=ƒ

#ˆˆ

%>ƒ

#=]

$ = } ˆ%<"' `  %' %==%

#>=

      Q `[ }[

$   )     '$ ˆOQŒ   

!   } ˆ<%"                        _    } ˆ%<"           ^      %==%  



 ;   



 





$      N  )  )   O

  

 :  )  * O

  €

        ) =  ˆ > "] "] #%< #%ˆ ƒ’ˆ ## ## %>< #"= ƒ’ˆ #]ˆ #]ˆ ˆ#=

ˆƒ #ƒ=ƒ ƒƒ<<

==ˆƒ ==]> =#="

 "      )   ) =  ˆ > = = #%ˆ ƒ’ˆ #%= #%= #"ˆ ˆ < ’ ] #"= #"=

#>ˆ %ƒ= ˆƒ=

<=" #ƒƒ= ƒ"<=

==ƒƒ ==ˆ" =="ˆ

# ;>  ) =  = ] "" #%= ]’ ##ˆ #"= ## ’ #> #]# %ƒ #> #ˆ ’ #] %%

"" ##ˆ #]# %%

#% %%= ƒ"] "]ƒ

ˆ<" #%]= ƒ#]= #==#=

==<% =="# =##< =#ˆ%

#    ) =  = > ˆ ’ ] "< #%= ƒ ˆ ] ’ ##ˆ #"= < ]  ## ’ #> #]# %ƒ= "  ## #> ’ #] %"#

"< ##ˆ #]# %"#

#%> %%= ƒ"] #>%=

ˆ>= #%ˆ= ƒ#]= #"ƒ"=

==ˆ> ==<< == " =#ˆ"

$       !   N  O $  

' 

   s;k *

O

  

   sk *

O

   

   sk *

O

      

$ 

  

%

%

# 

      # 

 

s? s? tk tk   *

O *

O *

O *

O *

O  $  J

#]ˆ #>%

#%# "

#ƒ" ###

% #ˆ

{'     **      ˆ#= [$>

$ = } ˆ%<"' `  %' %==% €

  %==%

#ƒ #ƒ

%= %>

#ˆ =<

## ˆ== "><=

   `

#>#

$  ? :    ;   "    

:  *

{          

                          >=                   

    [  „                   

    [  „     [        <               

% > ƒ ˆ < ]

     "       "   



*     ‡     # Ž {         _  

                      

#==

Z     ‡    ‡    

#%ˆ

       ‡  ‡   ‡  >

#]ˆ

     ‡  ‡  %  ‡  ‡   ‡  %

#ˆ=

{' # € \>       } <>  > ˆ #ˆ  |*   

          %  ‹        #ˆ                   ¥ˆ= >    ‹        ˆ                    ˆ=

$ = } ˆ%<"' `  %' %==%

#>%

      Q `[ }[

"      :  ; H  )      

 ; N€

1.20 1.15

K7 or K14

1.10 1.05 1.00 0.95

K14

0.90

K7

0.85 0.80

0

200

400

600

800

1000

1200

Width or depth of member (mm)  >     Π~      

                 „          _   <#=  " ˆ ##             "        „                

   

   `

#>>

%!       ; %   

 *

‚             

‚               

 

‚              

‚                             

‚                      

=] ="ˆ #= #ˆ %=

‡            #"=    

        %ˆ=                         

   ;    KNO                     

                     #%           #%            sk    ‹  

1.0 0.9 0.8 0.7 K 12

0.6 0.5

C1

6,

0.4 0.3

C2

4o

D4

0

0.2

rD

50

0.1 0

50

150

100 L/ry

$ = } ˆ%<"' `  %' %==%

200

250

#>ƒ

      Q `[ }[

"       K‰

‡             ===>      

                                      #ƒ       ‹                   „     ‡     ‹         ;     ;

      # % > ƒ

$ = } ˆ%<"' `  %' %==%

‰ $ ) 

.) 

#== ##ƒ #%# #%ƒ

#== #=< #=" ##=

   `

#>ˆ

 ; ‹               „  ¤      _          †               „    ¤      _       ‹                      

  ‹         „                                  ‹     ="      _   

       ;   ; ‹               _ |         =" ‘                  ˆ

5d 10d

20d

20d

( ;        ‹   !   N  O  ˆ   ˆ=  "        " ˆ        „  ˆ= ˆ =     #=              €    

               =]     €  ‹      ##ˆ 

#><

      Q `[ }[

'           ;   ; ‹    



'    $ )  +  >

-

 € ˆ Q

.)  +>

-

$  

$  

$  



NQ

O€

 

 



"€„

"ˆ„

>< ƒ" <= ]%

=>=ˆ =ƒ > =]## = <#

=>%ˆ =ˆ> =]ˆˆ #=%#

%ƒ >% ƒ= ƒ"

=ƒ<ƒ =]] ##ˆƒ #ˆ ƒ

=ˆ#ƒ ="<> #%]" #]<ˆ

'           ;   )  ‹    )  



  









'  ”  $ )  +  >

-

.)  +>

-

NQ

O€

"€„  "ˆ„

Q

> ƒ

ˆ= ]ˆ

=%ˆ< =><=

=%<] =><=

=% ˆ =><=

NO

> ƒ

ˆ= ]ˆ

=%"< =ƒ>=

=% < =ƒƒ<

=>ˆ% =ˆƒˆ

NŒ

> ƒ

ˆ= ]ˆ

=>ƒƒ =ƒ"ˆ

=>ˆˆ =ˆ=#

=ƒ#] =<=#

ON

> ƒ

ˆ= ]ˆ

=>ˆ =ˆ%=

=>]ƒ =ˆ>]

=ƒˆ< =<ƒ#

” ?        „    €

      ‹   } ˆ%<"' `  %' %==%'  <>

$ = } ˆ%<"' `  %' %==%' ? _ ‡

   `

#>]

$) ‹          „            } #%#=   ‹                      [ f             [            f$%           ~       [    

                '  $   [    [        %f    [            ^               }    

  ) 

5d 3d

10d

10d

10d

( ;      ) ‹    !   N  O  ˆ   ˆƒ  " " ˆ         „  ˆƒ ˆ =     #=                   €                     =]    

#>"

      Q `[ }[

'       )   ;   ; ‹  $) $) 

$  



{ { { { { {

>ƒˆ ƒ#] ƒ"" ˆˆ <>= ]=#

< " #= #% #ƒ #<

$  



#% #< %% >ˆ >" ƒƒ

'     .) 

$ )  NQ

O€

"€„  "ˆ„

=%]# =ƒ=] =ˆ%< =]>" =ˆ"ˆ #==ˆ

=>=> =ƒƒ= =ˆ<" =] < = %] #= =

=> < =ˆ"ƒ =]ƒ] #=ˆ> #%%] #ƒƒ%

'       )   ;   )  ‹    )  



$)

'   ” 

$) 

$  



 

)





$ ) 

.) 

NQ

O€

"€„

"ˆ„

NO

{ { { { {

< " #= #% #ƒ

>ƒˆ ƒ#] ƒ"" ˆˆ <>=

>" >" >" >" ƒƒ

=%#ˆ =>=> =>ˆ< =ƒ]" =ˆˆ#

=%ƒ% =>%< =>"ƒ =ˆ%% =<=>

=%"ƒ =ƒ%< =ˆ## =]#% =">=

=>%= =ƒ = =ˆ % =">ƒ = %<

NŒ

{ { { { {

< " #= #% #ƒ

>ƒˆ ƒ#] ƒ"" ˆˆ <>=

ˆ] ˆ] ˆ] ˆ] ˆ]

=%"] =>"] =ƒ% =ˆƒ< =]<ƒ

=% => =ƒˆˆ =ˆ"< ="%

=>ˆ# =ƒ"< =ˆ<] =]ˆˆ ##=

=>"# =ˆƒ% =<>" ="<> #%""

” _        „    €

      ‹   } ˆ%<"' `  %' %==%'  <"

$ = } ˆ%<"' `  %' %==%' ? _ ‡

   `

#>

'  ‹         „     [       } { %=" "‹#         } ƒ>%= }          

%           ~                   [   =%ˆ                  ?                 

7d

4d 4d

4d

Minimum 4d 1.5d 4d 4d 4d 1.5d

Loaded edge parallel to grain

Loaded edge perpendicular to grain

4d 4d 4d 4d 1.5d 5d 4d 4d

5d 4d

  ;   ( ;      ;  ‹    !   N  O  ˆ   ˆ]  " " ˆ        „  ˆ] ˆ # …>…" #††=#==    #=                  ˆ] ˆ =]     #=                  ˆ] ˆ #=                   „          [  =>      %ˆ                   #%ˆ €      

           

               _    

      

#ƒ=

      Q `[ }[

'         €KQ ;    )  ;  ;     ; 

 

 ;  +

-

'          €KQ ;      )  ;”  ;    + "    (

    Œ

NO

NQ

(     O„

Œ

NO

NQ

NQ

ƒ] ]% ]

#%% #ƒ< #ƒ<

#"= %<" >#>

%>= >ˆ% ƒ<>

%]> ƒ# ˆ<ƒ

##> #> #>

#ˆ< %> %]

# # % > > ƒ

O„ %# >>< ƒˆ%

O€

ƒ] ]% ]

#>> #ˆˆ #ˆˆ

%=ƒ % > >ƒ%

%ˆ > ] ˆ=ˆ

>= ƒ]> <>]

#%> #ƒ] #ƒ]

#]< %<ƒ >=]

%#< >>= ƒƒ>

%ƒ] >] ˆ##

"€„

ƒ] ]% ]

#"> # # # #

>=" ƒ=% ƒ%#

> % ˆ " < >

ƒ<] ]#< >%

#"> # # # #

>=" ƒ=% ƒ%#

> % ˆ " < >

ƒ<] ]#< >%

"ˆ„

ƒ] ]% ]

%#% %#% %#%

>]" ƒ<< ƒ<<

ƒ"# < % "=

ˆ]> "]" #="%

%#% %#% %#%

>]" ƒ<< ƒ<<

ƒ"# < % "=

ˆ]> "]" #="%

” _       

       } ˆ%<"' `  %' %==%'  ]< ]] ]   "=

$ = } ˆ%<"' `  %' %==%' ? _ ‡

! Z

      ^   ‚                             Z

                   „    

           

          

         Z

          _     ~   _      

 Œ     „ Q           Œ  [Q ?  [                 „         

              

        [            [   

$

       ;         ^            [           } [         |     %%ˆ   %" [{$> \  [    _               [                    \ ?    ¥]= {$%“    ƒˆ¢   \ }'    ¥ˆ= {$%“    ]=¢       ; \        [        [               [    [ |     #]   %# [{$>   ;  \    [            ˆ   %= [{$>   [    “     [         “   [         }[    [  ==#Ž== ¢   [    [       ==>¢           [               $             [       [  [ [      } ˆ> =                } "% "                  _                                   

   _                 [          ‹                   ††                           *      _                         [ 

#ƒ%

      Q `[ }[

        + ! '$-

 

 ;  

    !  *

O

€    [ } > %#

     ?      } €   [      \  \ #  #ˆ \ # \ % \ > \ #ˆ €  [  [    

 ƒˆ¢ ]=¢ #=Ž>=¢

¥]= ¥ˆ= #=Žˆ=

\    [ } #"]

%=Ž%ƒ¢

#ˆŽ%ˆ¢ %=Žƒ=¢

\ %  ]

\    [ } <=]>' ` #

] #ƒ %= #=ˆ #ˆŽ%ˆ >Ž%=

#ƒŽƒ"ˆ

]Ž%=

\   [ } <=]>' ` #

|   |   *

] #=Ž>= >ˆ ] #= %" >ˆ ƒ ]

‚     } <ƒˆ]

|  

       [

{    } ˆ> =   } "% "

                             

 

#ˆŽ#==

Z



#ƒ>

     '  ;  †     ^  [  %#ˆ  #=%  <ˆ      ‹    †  %%ˆ  ##%  ]ˆ      ^ [  †  ƒƒ=  %#ˆ   [    ]ˆ  %#ˆ      ‹    †  ƒˆ=  %%ˆ        [        [  

Frog

Perpend

Stretcher

65 215

215

102 440 Varies

 ‘    _   ‘         [     %= [            „    [  Z        [  #ƒ=  [     _ %= [  ‘          [                                     [   

   [    

#ƒƒ

      Q `[ }[

>      :          

:  

 :      ; !  

*  '   

%'ˆ

             _      

`†† ”'  '   

#'%Ž>'%

         

        

” `††                         

                            \          ‹      

    _       \       |    _   ‹                         

    [    

          „   _                 

  _          [      ‹       %#ˆ                           

   :        

"     

    +    ; !  -

 !  *

O

 =

 =

   =

 = )  †

;

$

#'='> #'¦'> #'§'ƒ  ƒ§ #'#'ˆ  < #'%'" 

Ž Ž #'%§  >§ #'ƒ  ˆ #'ˆ§  <§

Ž Ž #'>  ƒ #'ˆ  < #']  "

Ž #<= <ˆ >< #ˆ

Ž ##= ƒˆ %ˆ #=

{' # Z_            ~                      ‹                 % ?           _     [   [     > \ ''                  '    †       

$ = } ˆ<%"' `  #' #

%

Z



#ƒˆ

$  ;   €                     [     

„   

English bond

Flemish bond

English garden wall bond

Flemish garden wall bond

Stretcher bond

#ƒ<

      Q `[ }[

 !  ‹     )  >;  Z   „                       †  [                     [  _              „            [             _     [         [               „       ' 

\  [

 :   †  ‹  %       ! 

  ‹  



: 

 

=  # >'#

#%   _ 

#<

#ˆŽ#"    „       ƒˆ=   $

%%

#"Ž%=    „       %%ˆ   $

%ˆ

\    [

]ˆŽ     [

#=

>'#

\    [

<     [

#=

%'#

\   [

<Ž]     [

#=

%'#

#ˆŽ#"    „       ƒˆ=   $

%%

#"Ž%=    „       %%ˆ   $

%ˆ

<       

#=

{     

>'#

{' # \   „             _  %   †  „          „         

  „                          #%                        [

               [      [    _    _   [  „            _  [             _ [         {  #   „          

       #=       

            

Z



#ƒ]

";     ";  |   

                          ~  [               [ } [               _        †   [ } ˆ%<"' `  >             

          _        †'

";         ;     

 

 

  

   #

'   %   ;     >

      $_     |`\

?  [$#ˆ {$%  [



€* €{ Z* Z{ *  {

_   |`\ $    $ 

] {$%   [ $%= {$%  [



€*  €{ Z*  Z{        

} [      

   } ]ƒ>

      [ ?



€* 

     

] {$%   [ $>= {$%  [

  

€*  €{

`  _

      [$ %= {$%  

  

€*  €{

    

 [ $>= {$%  [



€*  €{

Z     

 

  

      [ $ %= {$%  

  

€    

      [ ?

  

€*  €{ Z*  Z{    

#ˆ=       

{' #                  _          [    % €      [       Z                   > {      [             *          ƒ ‚        _                           

#  ? 

                                `        [         

     [        [                       _      #==   [             %          ƒ     ‹    *              [                         

    

#ƒ"

      Q `[ }[

(   †         *     

"  #*  !

%      ) 

(      

      Ž  [ 

)$<Ž"ˆ#

)$%=  $%=

)$%%  $%ˆ

      Ž  [ 

)$ƒˆŽ<ƒ#

)$#=  $%=

)$#%  $%ˆ

Ž

)$%=  $>=

)$%%  $>ˆ

Ž

)$%=  $>ˆ

)$%%  $ƒ=

)$#ƒ

Ž

 

_    %   %

_         „      _      

)$#=

‚   



       [   

)$#=Ž#ˆ

 

     [   

)$%ˆŽƒ

Ž

Ž

)$" )$## Ž Ž

)$#"Ž%% )$ˆˆ $%=Ž>= $#=Ž#<

Ž Ž $>=Ž<= Ž

    \   Z

  $   ‚    [  

{' # |      _     %                   †      )        >      [          _      

Z



#ƒ

   )

       )

   Nˆ„



Permissible working stress (N/mm2)

1.8 1 100 Block (10 N/mm2)

1.6

2 102 Brick (20 N/mm2 in 1:1:6 mortar)

1.4

3 100 Block (7 N/mm2)

1.2

4 102 Old brick in lime mortar 5 102 Brick/75 cavity/ 100 Block (10 N/mm2)

1.0 0.8 5 6 7 8

1 2 3

0.6 0.4

6 140 Block (10 N/mm2) 7 102 Brick/75 cavity/ 100 Block (7 N/mm2)

4

8 140 Block (7 N/mm2)

0.2 0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Effective height (m)

        )

   Nˆ„

 1.2 9 330 Old stock brick in lime mortar 10 215 Brick (20 N/mm2 in 1:1:6 mortar) 11 215 Block (10 N/mm2)

Permissible stress (N/mm2)

1.0

0.8

12 215 Block (7 N/mm2) 13 215 Old stock brick in lime mortar

0.6 10

9

0.4 11 12 13

0.2

0 1.0

2.0

3.0 4.0 Effective height (m)

5.0

6.0

#ˆ=

      Q `[ }[

(   †  : ! )

                    ‹           >ˆ                        [ ~  „              >ˆ   

   †                 ƒˆ  *                 ‹   

               ‡      

         [          |     

            ' #                %              ' #=¢   _             _      } ˆ<%"  Œ*  Q       %ˆ¢        >           _                        „    

#ˆ#

Z



Height /Effective thickness (H/te)

  > ;     

90 80

End restraints required

60

Outside these areas the walls are unstable Top + end restraint required

40

Top restraint required

30 20 0

Top or end restraint required 20 40 60 80 100 110 Length/Effective thickness (L /te)

120

      !       

} \[ ‡   *  *   Z         

 *

O

  *

O

$ *

O

' *

O

"ˆ ## << ˆ>] #=] <ƒƒ ˆ>] %#ˆ "ˆ"

"< Ž >% %] #= >% ## =ˆ ##

ƒ> Ž ˆƒ ƒ> >" ˆƒ >% ## >%

Ž Ž #=] <ƒ ˆƒ Ž ˆƒ %# ˆƒ

                                  #= ~                             }                                 [                                               

       }            

     }   ‚    }‚ ?                                                                  [                  $ = } ˆ<%"' `  >' %==#“ ‘ ¤? # #=

#ˆ%

      Q `[ }[

    '$ ˆQOŒ (       

   ; 

|    |     |          |     ?  

"

  



%  )

#ƒ  = #ƒ  = #ƒ  =

#< Ž Ž

Ž #ƒ” #%”

#% #% Ž

#% = ˆ  #=ˆ

#% =>ˆ  #=ˆ

#%” =>ˆ

#% Ž

” }            †      #ˆ¢          ==#ˆ[            ==#ˆ[           

                   [  g ˆ >ˆ       _           

   g ˆ >=                    ƒ   ƒ  } ˆ<%"                             

    '$ ˆQOŒ= ( N $ ; 

*

 

$;    

$

 

,



k

\ 

, }   2

?

 ` 

?

`     „  `       „ 



  ' m

          _             ˆ *[_k =*[_? 

a

`       m     †                             ‹      „     [      „    

$ = } ˆ<%"' `  #' #

%

Z



#ˆ>

  $    !          ;   + *

O   

   

 !    *

O ˆ

N„ $ 

Nˆ

O„ #  

OHKˆ

ˆ

ˆ„   '

H„   

N„„   

%ˆ %ˆ %ˆ %ˆ

ƒƒ ƒ% ƒ# >ˆ

<= ˆ> ˆ= ƒƒ

]ƒ <ƒ ˆ" ˆ%

% ] ]# <%

##ƒ ƒ "ˆ ]>

#ˆ= #%% #=< =

# % #ˆ# #># #="

%ƒ= #"% #ˆˆ #%]

$    !        ;    + *

O        

 !    *

O HK„

N„

Nˆ

O„

ˆ

N„„

     

 

) ;   %" >ˆ ˆ=  %" >ˆ ˆ=

<ƒ <ƒ

"ƒ "%

#=< #==

#%" ##<

#"" #]=

N€„

     

 

) ;   %> % ƒ%  %> % ƒ%

ˆ> ˆ>

]= <

" "ƒ

#=] ]

#ˆ] #ƒ%

ONˆ

    ;  )

+   ) N„„

  ;     #ƒ #] %ˆ >%  #ƒ #] %ˆ >%

ƒ% ƒ#

ˆ> ˆ=

<ƒ ˆ"

ƒ "ˆ

OKŒ

Kˆ

ˆK„

{   ' # €                =% % *[       g ˆ =] ‡ #ˆ  % ~    [   #=%  [ *[      g ˆ ##ˆ       [  

        } ˆ<%"  %>#ƒ > {    

   [          [        

   

               ]ˆ¢      ‚   

                     ˆ=¢          [  \    

$ = } ˆ<%"' `  #' #

%

#ˆƒ

      Q `[ }[

%! 

wp t1

t

t2 te = t

te = t 1 te = t 2 te = 2/3(t1 + t2)

tp

t sp te = tk

greater of

$      )

  ;  <    +  -   )  =

<     )

/  =

Q N„ O„ *        _     

$ = } ˆ<%"' `  #' #

%

N

O



#= #= #=

#ƒ #% #=

%= #ƒ #=

Z



#ˆˆ

%!    *          

     †        %ˆ                            

       †                             

                         } ˆ%<"' `  #'   %"%   ? _ \      

                        

       

 _ %] €     =  [                   _ %=

H

H

H

H DPC He = H

He = 2.5H

He = 0.75H

SIMPLE RESTRAINT

He = 2.0H

ENHANCED RESTRAINT

. †     !  In houses <3 storeys: timber floor or roof bearing >90 mm or joist hangers and straps at 1.2 m centres

t

All other buildings: in situ or precast concrete floor or roof (irrespective of direction of span) bearing a minimum of 90 mm or t/2

Any type of floor at same level

t /2 or 90 PROVISION OF ENHANCED RESISTANCE

    !  t

t

Masonry tied with metal ties at 300 mm c/c minimum

Masonry stitched or bonded into place >t

>t

≥10t

≥10t

PLAN – SIMPLE RESISTANCE

PLAN – ENHANCED RESISTANCE

$ = } ˆ<%"' `  #' # } ˆ%<"' `  >' %==#

%“

#ˆ<

      Q `[ }[

Slenderness factor for vertical load (β)

$      b 1.2 e = 0.05t

1.0

e = 0.1t 0.8 e = 0.2t 0.6 e = 0.3t 0.4 0.2 0 5

10

15

20

25

30

Effective slenderness (heff /teff) e = eccentricity t = thickness

    )

     ~'

,

\  '



,

ˆ

…b*[ † g



ˆ

…b*[ †g g

          [        b     

    g         g            [    *[              

 ?                *[            

Z



#ˆ]

$ 

A *  * ˆ g

A

B C

 \                  *  ˆ =] {$%   [         =ˆ {$%   [        *  ˆ =>ˆ {$%  ] {$% [ ' \         †         '        †       †   ' *  ˆ …=:>ˆ‡=:< †{=%   _   #]ˆ {$%          *  ˆ …=:#ˆ‡=:< † {=%   _   #ƒ {$%        

#ˆ"

      Q `[ }[

   †    †  _                                _   

Thurso

Dingwall

4

Inverness Fort William

Aberdeen

Pitlochry Dundee

Oban

3

Glasgow Edinburgh

4

Ayr Dumfries

Londonderry Enniskillen Belfast

2

Carlisle

3

Douglas

Lancaster York Leeds Hull Grimsby

0

100

200

300

Manchester

Conway

Aberystwyth

Norwich Yarmouth Kings Lynn Lowestoft Cambridge Ipswich

London Reading Southampton Brighton Dover Taunton Bristol

Barnstaple

Plymouth Penzance

Leicester

Brecon Cheltenham Swansea Oxford Cardift

Ilfracombe

Skegness

Derby

Shrewsbury Cardigan

Lincoln

2

Portsmouth

1

#ˆ

Z



:        : ! )

 ” + O † 

. (   ; !  

'



 C

A

" D

%

# I

G

E



. J

 K

L

N

ˆK€ N„KŒ

##= =

#]ˆ #>=

%<ˆ #]ˆ

%=ˆ #ˆˆ

>%= %ƒ=

>%= >%=

"ˆ ]=

#ƒ= # ˆ #== #ˆˆ

O

ˆK€ N„KŒ

ˆ "=

#ƒ= ##ˆ

%#= #>ˆ

#]ˆ #>=

%]= # =

>%= %"=

]ˆ <=

#=ˆ #]= = #>=



ˆK€ N„KŒ

"ˆ ]=

#%ˆ #==

#ˆˆ ##ˆ

#ƒˆ ##=

%%= #ƒˆ

>=ˆ %ƒˆ

<ˆ ˆ=

ˆ #ƒˆ ]ˆ ##ˆ

€

ˆK€ N„KŒ

"= <ˆ

##= =

#>= #=ˆ

#%ˆ ˆ

#"= #%ˆ

%]= %#ˆ

<= ƒ=

"ˆ #%ˆ <ˆ #==

”                   #==  [ 

 [          #ƒ=   _               %=¢

{   

   

        _ ˆ=    [  ‚   

         %=¢  

               _ <=    [  $ = } ˆ%<"' `  >' %==#

#<=

      Q `[ }[

  :         _      

  [       

    [                 

                        ~       _        

    |`\   [                               [     

  

\  [        '   ]¢  ]¢   #%¢   #%¢

(     (

   ; ‹  : 

(    ; ‹  : 

   *

O

   *

O



=] =ˆ

=ˆ =ƒ

=ƒ

=>

\        [ ] {$%    [' #==    #ƒ=    %#ˆ    ?  [     [ ' #=ˆ {$% #ƒ= {$%

'     / ‚

   *

O



%= #ˆ

#ˆ ##

=>ˆ =>>

=>> =><

##

=

=><

=>>

*

O

*

O

   *

O

=>

=

=>>

=%ˆ =%] =>%

=< =<< =]

=ƒ# =ƒ= =ƒ=

=%ˆ =%ˆ

=]ˆ =

=>> =>=

”       ˆ f[_k =f[_?        *[_k           ' f[_k   ˆ f[_k =g ‡ = ‡[

$ = } ˆ<%"' `  #' #

%

Z



#<#

       )

   _      

      „          _            „      _                                 } ˆ<%"' `  #  a     _                                         _            ' k ˆ mag [ % [                †        ? ˆ k $            [    _          

    :     )

  † 

? ˆ



 *[_?

g

    :     )

 !

k ˆ



 *[ _ k g ‡ =: [

 _          „                        ‹                      ‹                 

    :     )

 !

k ˆ

g  %  %g

                    

      [         ~  g               

#<%

      Q `[ }[

'     

 )

 

µα

Simple support α

h

α Fixed support

µα L

(     

Ž    

     / 

„K„

„Kˆ„

„KHˆ

NK„„

NKOˆ

NKˆ„

NKHˆ



A

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

==># ==>% ==>ƒ ==>ˆ ==>" ==ƒ= ==ƒ> ==ƒˆ ==ƒ"

==ƒˆ ==ƒ] ==ƒ ==ˆ# ==ˆ> ==ˆ< ==<# ==<ƒ ==<]

==ˆ ==<# ==<ƒ ==<< ==< ==]> ==]] =="= =="%

==]# ==]> ==]ˆ ==]] =="= =="> =="] ==" == #

==] =="# =="> =="ˆ =="" == = == > == ˆ == ]

=="ˆ =="] ==" == # == > == ˆ == " =#== =#=#

== = == % == > == ˆ == ] == =#=# =#=> =#=ƒ

B

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

==%ƒ ==%ˆ ==%] ==%" ==>= ==># ==>ƒ ==>ˆ ==>]

==>ˆ ==>< ==>] ==> ==ƒ% ==ƒƒ ==ƒ] ==ƒ ==ˆ#

==ƒ< ==ƒ] ==ƒ ==ˆ# ==ˆ> ==ˆˆ ==ˆ] ==ˆ ==<#

==ˆ> ==ˆˆ ==ˆ< ==ˆ" ==ˆ ==<# ==<> ==<ˆ ==<<

==ˆ ==<= ==<# ==<% ==<ƒ ==<< ==<] ==<" ==]=

==<% ==<> ==<ˆ ==<< ==<] ==< ==]= ==]# ==]%

==<ˆ ==<< ==<] ==<" ==< ==]# ==]% ==]> ==]ƒ

C

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

==%= ==%# ==%% ==%> ==%ƒ ==%ˆ ==%] ==% ==>=

==%" ==% ==># ==>% ==>ƒ ==>ˆ ==>" ==> ==ƒ=

==>] ==>" ==> ==ƒ= ==ƒ# ==ƒ> ==ƒƒ ==ƒˆ ==ƒ<

==ƒ% ==ƒ> ==ƒ> ==ƒƒ ==ƒ< ==ƒ] ==ƒ" ==ƒ ==ˆ=

==ƒˆ ==ƒ< ==ƒ] ==ƒ" ==ƒ ==ˆ= ==ˆ# ==ˆ% ==ˆ%

==ƒ" ==ƒ" ==ƒ ==ˆ= ==ˆ# ==ˆ% ==ˆ> ==ˆ> ==ˆƒ

==ˆ= ==ˆ= ==ˆ# ==ˆ# ==ˆ% ==ˆ> ==ˆƒ ==ˆƒ ==ˆˆ

D

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

==#> ==#ƒ ==#ˆ ==#< ==#] ==#" ==%= ==%% ==%>

==%# ==%% ==%> ==%ˆ ==%< ==%" ==># ==>% ==>ƒ

==% ==># ==>% ==>> ==>ˆ ==>] ==> ==ƒ= ==ƒ#

==>ˆ ==>< ==>" ==> ==ƒ= ==ƒ% ==ƒ> ==ƒƒ ==ƒ<

==ƒ= ==ƒ= ==ƒ# ==ƒ> ==ƒƒ ==ƒˆ ==ƒ] ==ƒ" ==ƒ

==ƒ> ==ƒ> ==ƒƒ ==ƒˆ ==ƒ< ==ƒ" ==ƒ ==ˆ= ==ˆ#

==ƒˆ ==ƒ< ==ƒ] ==ƒ] ==ƒ" ==ˆ= ==ˆ# ==ˆ# ==ˆ%

'



"



Z

 %

#<>

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

===" === ==#= ==## ==#% ==#ƒ ==#] ==#" ==%=

==#" ==# ==%# ==%> ==%ˆ ==%" ==>% ==>ˆ ==>"

==>= ==>% ==>ˆ ==>] ==ƒ= ==ƒƒ ==ƒ ==ˆ% ==ˆˆ

==ƒ% ==ƒƒ ==ƒ< ==ƒ ==ˆ> ==ˆ] ==ˆ% ==<ƒ ==<"

==ˆ# ==ˆƒ ==ˆ< ==ˆ ==<% ==<< ==]# ==]ƒ ==]]

==ˆ ==<% ==<ƒ ==<] ==]= ==]ƒ ==]" =="# ==">

==<< ==<" ==]# ==]> ==]< =="= =="ƒ =="< =="

F

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

===" ===" === ==#= ==## ==#> ==#ˆ ==#< ==#"

==#< ==#] ==#" ==%= ==%% ==%ƒ ==%] ==% ==>#

==%< ==%] ==% ==># ==>> ==>< ==> ==ƒ# ==ƒƒ

==>ƒ ==>< ==>] ==> ==ƒ% ==ƒƒ ==ƒ" ==ˆ= ==ˆ%

==ƒ# ==ƒ% ==ƒƒ ==ƒ< ==ƒ" ==ˆ# ==ˆƒ ==ˆˆ ==ˆ]

==ƒ< ==ƒ" ==ƒ ==ˆ# ==ˆ> ==ˆ< ==ˆ" ==<= ==<%

==ˆ# ==ˆ% ==ˆƒ ==ˆˆ ==ˆ] ==ˆ ==<% ==<> ==<ˆ

G

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

===] ===" ===" === ==#= ==## ==#> ==#ƒ ==#<

==#ƒ ==#ˆ ==#< ==#] ==# ==%# ==%> ==%ˆ ==%<

==%% ==%> ==%ƒ ==%< ==%" ==>= ==>% ==>> ==>ˆ

==%" ==% ==># ==>% ==>ƒ ==>< ==>" ==> ==ƒ#

==>> ==>ƒ ==>ˆ ==>] ==>" ==ƒ= ==ƒ% ==ƒ> ==ƒƒ

==>] ==>" ==> ==ƒ= ==ƒ% ==ƒ> ==ƒˆ ==ƒ< ==ƒ]

==ƒ= ==ƒ# ==ƒ% ==ƒ> ==ƒƒ ==ƒ< ==ƒ] ==ƒ" ==ƒ

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

===ˆ ===< ===< ===] ===" === ==#= ==## ==#>

==## ==#% ==#> ==#ƒ ==#ˆ ==#] ==# ==%# ==%%

==#" ==# ==%= ==%% ==%ƒ ==%ˆ ==%" ==% ==>#

==%ƒ ==%ˆ ==%] ==%" ==>= ==>% ==>ƒ ==>< ==>]

==% ==>= ==>% ==>> ==>ˆ ==>< ==> ==ƒ= ==ƒ#

==>> ==>ƒ ==>ˆ ==>] ==>" ==ƒ= ==ƒ% ==ƒ> ==ƒƒ

==>< ==>] ==>" ==ƒ= ==ƒ# ==ƒ> ==ƒˆ ==ƒ< ==ƒ]

E

#



.

H

#<ƒ

      Q `[ }[

'     

 )

  ?   (     

Ž    

     / 

„K„

„Kˆ„

„KHˆ

NK„„

NKOˆ

NKˆ„

NKHˆ



NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

===ƒ ===ƒ ===ˆ ===ˆ ===< ===] ===" === ==#=

=== ==#= ==#= ==## ==#> ==#ƒ ==#< ==#] ==#

==#ˆ ==#< ==#] ==# ==%= ==%% ==%ƒ ==%< ==%"

==%# ==%% ==%> ==%ˆ ==%< ==%" ==># ==>% ==>ƒ

==%< ==%] ==%" ==>= ==># ==>> ==>ˆ ==>] ==>"

==>= ==># ==>% ==>> ==>ˆ ==>] ==> ==ƒ= ==ƒ%

==>> ==>ƒ ==>ˆ ==>] ==>" ==ƒ= ==ƒ% ==ƒ> ==ƒƒ

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

=== ==#= ==#% ==#> ==#ˆ ==#" ==%# ==%ƒ ==%]

==%> ==%< ==%" ==>% ==>< ==ƒ% ==ˆ= ==ˆˆ ==<%

==ƒ< ==ˆ= ==ˆƒ ==<= ==<] ==]] == = == " =#="

==]# ==]< =="> == # =#== =#>> =#># =#ƒƒ =#<=

== < =#=> =### =#%# =#>ˆ =#ˆ> =#]] =# ƒ =%#ƒ

=#%% =#># =#ƒ% =#ˆ< =#]> =# ˆ =%%ˆ =%ƒƒ =%<

=#ˆ# =#<% =#]ˆ =# # =%## =%>] =%]% =% < =>%ˆ

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

=== ==#= ==## ==#% ==#ƒ ==#< ==# ==%# ==%ƒ

==%# ==%> ==%ˆ ==%" ==># ==>ˆ ==ƒ# ==ƒˆ ==ˆ=

==>" ==ƒ# ==ƒˆ ==ƒ ==ˆƒ ==<# ==< ==]ˆ =="%

==ˆ< ==<= ==<ˆ ==]= ==]] =="ˆ == ] =#=ƒ =##%

==]ƒ ==] =="ƒ == # == =#= =#%# =#% =#>

== # == ] =#=> =##= =## =#>= =#ƒƒ =#ˆ% =#<%

=#=" =##> =#%= =#%" =#>" =#ƒ =#<ƒ =#]> =#">

NK„„ „K‰„ „KŒ„ „KH„ „KQ„ „Kˆ„ „K€„ „Kˆ „K„

===< ===] ===" === ==#= ==#% ==#ƒ ==#< ==#"

==#ˆ ==#] ==#" ==%# ==%> ==%] ==>% ==>ˆ ==>

==% ==>% ==>ƒ ==>" ==ƒ% ==ƒ" ==ˆˆ ==<= ==<<

==ƒƒ ==ƒ] ==ˆ# ==ˆ< ==<# ==<" ==]" =="ƒ == %

==ˆ ==<> ==<] ==]> =="= ==" =#== =#=" =##<

==]> ==]" =="ƒ == = == " =#=" =#%# =#% =#>"

=="" == > == =#=< =##ˆ =#%< =#> =#ƒ" =#ˆ"

I

@

J 

K 

L



{' # *      m   $    % ~                $                            € _      ?    $   =>                     $    #]ˆ       † 

$ = } ˆ<%"' `  #' #

%

Z



#<ˆ

                                                                                  

  

" ;  *

O

|     

#:%ˆ*[ g

‘                             >ƒ } ˆ<%"

? =ƒ”    

b*[ g

               ƒˆ     =ƒ               

$ = } ˆ<%"' `  #' #

%

#<<

      Q `[ }[

    (NNN \`###   Œ  [ Q              ?                      ‹              } ˆ<%"           

  

'  !          ; + *

O"  

     ; !  

. '  !   #   *

O OHKˆ €Kˆ ˆOK„ Q‰K„ ‰QKˆ OKŒ HK„ N„Kˆ O„Kˆ   '    $  #  

 =  = +'$ ˆQOŒ   |  [ Ž #'='>  \   Ž#'#'<  \   Ž #'%'  { ‹       †† $    Ž ='#'> ‘    Ž ='#'% { ‹      Ž ='#'>

] #ƒ #ƒ #ƒ

=%" =%" =%" =%#

=]= =]= =ˆˆ =ƒ

#=ˆ = ˆ ="ˆ =]=

#<ˆ #>= ##ˆ = ˆ

%=ˆ #<= #ƒˆ ##ˆ

%ˆ= #"ˆ #<ˆ #ƒ=

>ˆ= %ˆ= %=ˆ #]=

ƒˆˆ >#= %ˆ= %=ˆ

ˆ"ˆ >"= >#= %ƒ=

#ƒ %"%

=%# =ƒ =]= =%# =ƒ% =ˆˆ

= ˆ =]=

##ˆ #ƒ= #]= %=ˆ =]ˆ ="ˆ #=ˆ ##ˆ

%ƒ= #ƒ=

{' # €                =% %               g ˆ =] ‡ #ˆ  % *                 

$ = \`###' # ]=

Z



#<]

           $  

< " #= #% #ƒ #< #" %= %% %ƒ %< %]

$     /  :



%    *Q

 *€

 *

 *   *O

#=== = ˆ= =" = ="ƒ= =]"= =]>= =<]= =<%= =ˆ<= =ˆ#= =ƒˆ= =ƒ>=

#=== = >= ="ˆ= =]"= =]== =<>= =ˆˆ= =ƒ"= =ƒ== =>>= =%ˆ= =%%=

#=== = %= =">= =]ˆ= =<<= =ˆ"= =ƒ = =ƒ#= =>%= =%ƒ= Ž Ž

#=== = #= ="#= =]%= =<%= =ˆ>= =ƒ>= =>ƒ= =%ƒ= Ž Ž Ž

#=== =""ˆ =]]# =<ˆ] =ˆƒ% =ƒ%" =>#ƒ =%== Ž Ž Ž Ž

   ?  \`###               #ˆ                       

         [                              [        €          _        =ƒ% {$%  [  [         =] {$%         Œ   Q            $ = \`###' # ]=

#<"

      Q `[ }[

   '$ ˆ‰HH } ˆ ]]              

          ƒˆ             ><                     

                              

                   

                 =<   =%  

                    =<  

             =<   

         ~

 [   _             _         

Interaction zone

Load triangle

60°

45°

L 1.1L

*    ' #     

        

     Ž   

   †       % ?                     ƒˆ   

     > ‘              

   †       

    

Z



#<

~                                †                 †              †                       ‘                 †                      ‹             ¨Ž—   [     ?        †                   ƒˆ      ¨Ž—             ?                _' Opening

Partition

Floor

Lintel

45°

Masonry in load triangle

Floor load in load triangle

Floor load in interaction zone W/2

W/2

W Paint load from partition

W/2

X

$ = } ˆ ]]' `  #' # "#

Y

Y

Additional load from structure above opening in interaction zone

#]=

      Q `[ }[

    @   ¤        [            

      „   ¤        } <#]"      ‹ †              {  <==   

                          

   \   „               

   „                 [      „    [    [                     _            >]ˆ  >=       †   [                             „            „      _           „              Q               

$    ; Z

                                                       >=  ˆ             #= [{        †          _    „   _         

    _     %  ?    

           ˆ [{$          _  \ ‹            

            $     }   ‚     } "#=>' `  #         _        

(                 

                             

           ƒˆ                                       

Z



#]#

(  >   ;           [              

                 [  ‹         =   >>     =#ˆ        

          ƒ"    :    

;   *

 )



 



#   

NKˆ

OKN

OKH

#==

#== #ƒ= %#ˆ

=ˆ =ˆ =ˆ

#ƒ] >#< ˆ=ƒ

ˆ< #]= >#>

%" "" #"ˆ

#< ˆ> ##%

#= >ˆ ]ƒ

=] %ƒ ˆ%

=ƒ #] >"

#ƒ=

#== #ƒ= %#ˆ

=ˆ =ˆ =ˆ

>== ƒ>< <"=

##ˆ %>% ƒ%%

ˆ #%= %<=

>ˆ ]% #ˆ"

%> ƒ] #=ˆ

#ˆ >> ]>

## %ƒ ˆƒ

#ˆ=

#== #ƒ= %#ˆ

#= #= #=

>=" ƒ<# ]#"

##" %>" ƒƒˆ

<= #%ƒ %<]

>< ]ƒ #<%

%> ƒ #=]

#< >ƒ ]ˆ

## %ƒ ˆˆ

# =

#== #ƒ= %#ˆ

#= #= #=

ƒ=% ˆ"% ><

#ˆƒ % ƒ< ˆ"#

] #ˆ% >ƒ=

ƒ] % %=<

>= <= #>]

%# ƒ# <

#ƒ >= ]=

%#ˆ

#== #ƒ= %#ˆ

#= #= #=

ƒ%% <<ƒ #=ˆˆ

#<% >ˆ> <ˆˆ

"> #"> ƒ=>

ƒ ˆ ##=< %ƒƒ

># ]% #<%

%# ˆ= ##ƒ

#ˆ >< ">

   ;)  

„K‰

K

K‰

€Kˆ

$ =    %==% {        „        \                 

#]%

      Q `[ }[

(      `    †                                           `                 ˆ=   #<ˆ                     

                                              ‡ *  

%:K !





N*$ N„„

#=%

K . ! ;    :: *



 €







=Ž#=ˆ #==

=<Ž#ˆ #<Ž#" # Ž%# %%Ž%ƒ %ˆŽ>= >#Ž>< >]Žƒ"

%ˆ %ˆ %ˆ %" %" >= >=

"< < #=# ##" #% #ˆ> #"%

#>ˆ< %ƒ#% >=] ƒ>=% <>]< #=%%ƒ ##<ƒ=

%=]= >=# >ˆƒ# ƒˆ] ˆ #] "%> "]>

#= #< #ˆ #] %ƒ %ƒ %ƒ

>> ƒ ˆ] ]ƒ < #>> #ƒ%

#=%

=Ž#=ˆ #== #%ƒ #%]# #ˆ%# #]]# #]]# %=]# %=]# %=]#

=<Ž#ˆ #<Ž#" # Ž%# %%Ž%ƒ %ˆŽ%] %"Ž>= >#Ž>< >]Žƒ%

%" %" %" %" >= >= >= >=

#=] #>] #ƒ" #ˆ #<" #"% #"% #"%

%< " >#=% ƒ ƒƒ ]>%= ]]"< ##<ƒ= ##<ƒ= ##<ƒ=

>>]< >ˆ"# ƒ"<# <%"] <]#> "]> "]> "]>

%ƒ >= >ˆ >ˆ ƒ= ƒ= >< >=

ˆƒ ˆ" ]" #=% #= #ƒ% #ƒ% #ƒ%

#=%

=Ž#=ˆ #== #ˆ%# #]]# %=]# %=]#

=<Ž#ˆ #<Ž%# %%Ž%ƒ %ˆŽ%]

>= >= >= >=

#ˆ< #<" #"% #"%

ˆ%ˆ" ]]"< ##<ƒ= ##<ƒ=

ˆ# = <]#> "]> "]>

ƒˆ ƒˆ ƒˆ ƒ=

"ƒ #= #ƒ% #ƒ%

#=%

=Ž#=ˆ #ƒ=

"] ##% #%ƒ #>< #<# #<# #<ƒ# #<ƒ#

=<Ž#% #>Ž#ˆ #<Ž#" # Ž%# %%Ž%ƒ %ˆŽ%] %"Ž>< >]Žƒ%

%ˆ %ˆ %ˆ %" %" >= >= >=

"< < #=# ##" #% #> #<" #"%

ˆ # #< %ˆ#ƒ >ˆ = ˆˆ=% ˆ" # <<% #=%=ˆ

#<]# %ˆ]= >=]= ƒ=% ˆ>%ˆ ˆ]=# ˆ  ] ] <#

#% #% #ƒ #] #] # %ƒ %ƒ

%] ƒ# ƒ <ˆ "< % ] #%

#=%

=Ž#=ˆ #ƒ= ### #>< #> # #<ƒ# #<ƒ# # ƒ#

=<Ž#% #>Ž#ˆ #<Ž#" # Ž%# %%Ž%ƒ %ˆŽ%]

%" >= >= %" >= >=

#=] #%] #ˆ< #ˆ #<" #"%

%#ƒ> >"ƒƒ ƒ>%< <%ˆƒ <<% #=%=ˆ

%"]ƒ ƒ>#> ƒˆ>% ˆ<> ˆ  ] ] <#

#" %ƒ %" >% >ƒ >ƒ

ƒ< ]= ]> # ] #%

h

h

95 88 100

N*’." N„„



);    )  

 



#%ƒ #>< #ƒ #]ƒ %=ƒ %=]#

95 88 100

N*." N„„

‘::  

h

95 88 100

N*$  N„„

h

95 88 125

N*."  N„„

h

95 88 125

Z



#]>

ˆ*N„„O #=%

=Ž#=ˆ

#%ˆ _

#ˆ> %%" %%" %%" %%"

=<Ž#" # Ž%ƒ %ˆŽ> ƒ=Žƒ" =<Žƒ"

%ˆ %ˆ >= >= >=

#<% # % %%ƒ %%ƒ %>#

"## # ˆ% %>ƒ" %>ƒ" %ƒƒ"

" ]" #> ˆˆ #<]ˆ# #<]ˆ# #">"ˆ

>< ƒ" ˆ= >" ƒˆ

#ƒˆ %%< %]% %]% % "

#=%

=Ž#=ˆ

#%ˆ _

%#> %#> %#> %#> %#> %#> %#> %#>

#%Ž>= >#Žƒ" ˆ# ˆƒ ˆ] <= <> <<

Ž Ž Ž Ž Ž Ž Ž Ž

ƒ> ƒ> ƒ> ƒ> ƒ> ƒ> ƒ> ƒ>

ƒ="" ƒ<>= ƒ<>= ƒ<>= ƒ<>= ƒ<>= ƒ<>= ƒ<>=

>#>%= >%"] >%"] >%"] >%"] >%"] >%"] >%"]

ˆ "> ]ˆ <ˆ ˆ" ˆ> ƒ] ƒ>

ˆ<> ˆ>ƒ ˆ>ƒ ˆ>ƒ ˆ>ƒ ˆ>ƒ ˆ>ƒ ˆ>ƒ

%==Ž%#ˆ

ˆˆ ˆˆ #==

=<Ž#ˆ #<Ž#" # Ž%]

%ˆ >= >=

ˆ< <" "ˆ

# %ƒ #=%

ƒ> ˆƒ" #ƒ ƒ

< < #=

=] =" %ƒ

#=%

<= ##= %#=

=<Ž#% #>Ž#" # Ž%]

>= >= %"

>ˆ ƒ] <ˆ

#] <" > <

h

95 88 100

Q*N„„

207

6 250

45

‰

h

200

N„

h

90

><] % >=>%

ƒ " #=

=ˆ #ˆ ƒ

#]ƒ

      Q `[ }[

(      ?    

%:K ! K . ! ;    :: ‘::





*



  €  







NN

#=% 50

#ˆ= %%ˆ %%ˆ

=<Ž#" # Ž%ƒ %ˆŽ>=

%ˆ %ˆ >=

ˆ< ]# "ˆ



);    )   

 

% % >ƒ=" #< ]ˆƒ ˆ  < %= %= ]>#ˆ %%

ˆˆ ] ##"

h

100

{' #                % *ˆ   *##              

      > *<       %=>  #>>}>=      

  ƒ  *#$        *#$ ##=    ##=Ž#%ˆ  *#$ #>=  #>=Ž#ƒˆ    *#$ #ˆ=  #ˆ=Ž#<ˆ  ˆ ‡        

          #==     < *       ?      _   _   *$>%ˆ   „     [  

$ = ‡  %==>

" #  $   ‚                                      [         `   \       #"%ƒ  ¤ ?   * ‘  [      €    Z    *   _         |         ?    #"]=                   # ==Ž# #= \                                    _     \          '   *              #ƒ== \          ‡                  _            _                \    #=  %=      [    ]ˆ¢   _   \                ‹    ~                 $                   _      [     [               <   ‚                           * ˆ ƒ<= {$%     [   * ˆ %ˆ= {$% ‘      ‹                                        [                    *            #  %¢                 [                       : _ 

~ [              

#  )  ‡                   [            [               €  [                       [      †   

#]<

      Q `[ }[

$

    

"

#]  %ƒ [{$>            

 !       

|           ˆ ]  <= {$%

`    "  #ˆ¢    ‚       _ 

           _                 ‹  %"     ' %ƒ  >% [{$% * ‹        >=  ˆ=¢    ‹          : 

"  #%  #=

<

\

$ ?                      [       [                   {    ‹    [    ==>¢       =¢          

                                                  †            ƒ=¢   "=¢                >=        >=           sf$   s                           f            #=   >%  ^      %" 

#]]

‚   \  

  : \   _                    

         _           _ „      ‹ ‹

               ? `   Z_         _                              _ ? |  Z_                          _        ‹                 [  _        ‡   |  Z_    _  \  `    Z_    ‹                `   \  „         ^           [         _  } ˆ>%"       _                   } {  ==#   ‹                  ‡   |  _    _  ‡{ €{|  ‚\         } ˆ>%"         } { %=<   } "ˆ==   [ 

   

"   :  '$ ˆOŒ= ( N "

:

%

„

%

N

%



 !  *

O

]ˆ

#=

%=

    

     * 

:  )*  

    



^      [ 

#%=

$

  #=

}          |    [        

#]ˆ

$

]ˆ

#]ˆ #]ˆ

$

$

#=Žˆ= ]ˆ

Z             

#]ˆ

$

]ˆ

#]ˆ

$

#%ˆ

# " O/ / €  €

>ˆ

€   

    ' % > ƒ  ƒ?

 

=<

]ˆ

< „

>=

‚     

%]ˆ

=<ˆ

ˆ=Ž#==

< ˆ

>ˆ

>==

=<=

ˆ=Ž#==

< €„

ƒ=

>%ˆ

=ˆˆ

ˆ=Ž#==

< ˆ„

ˆ=

ƒ==

=ƒˆ

ˆ=Ž#==

$ = } ˆ>%"' `  #' #

]'  #> 

#]"

      Q `[ }[

  ; :     \                                   

       _           _                 _                  [                                          ˆ=   ]ˆ   '   ; : !   ;       =  = O„



( ;;    /  *

O

#'#ˆ'> #'%'ƒ #'>'<

ƒ= >= %=

  ;       : \         [  [     _                    _                _     _   } ˆ>%"            ' N    

:  &          

%:     ;  

#= >  #=  %= >  %=  ˆ= >  ˆ= 

ƒ= > "= > %== >

Z        }    [    [ 

?    ŒQ    [         _                 } #""#         <= >     _                     ‘              ?                ~          ]   %"                     

            %= \  % \                             \                 %= {$%      '

. ?                      ?  . .

              > {$%      ‹        ~                        #ˆ¢                                                                 > {$%             

                _      [‹     $ = } ˆ>%"' `  #' #

]

‚   \  

#]

";     ";  :      '$ ŒNN„  

\                

  _               †           ‹         „      $!   _                    †           !     _       ‹     _  

         †         !       _      ‹   \      †    #          _                $  _                     



               ?                   

   !   ;  :     %:    

  !  

  



Z Z          Z   

%ˆ Ž Ž Ž Ž

%= >ˆ Ž Ž Ž

%= >= ƒ= ˆ= Ž

%= %ˆ >= ƒ= Ž

%= %= %ˆ >= ˆ=

:   )*   

=<ˆ

=<=

=ˆˆ

=ˆ=

=ƒˆ

      * 

%]ˆ

>==

>%ˆ

>ˆ=

ƒ==

 )    

\>=

\>ˆ

\ƒ=

\ƒˆ

\ˆ=

$ = } "##=' `  #' #

]

#"=

      Q `[ }[

       !        ;

<& 

„Kˆ

NK„

NKˆ

OK„

K„

€K„

   

:   

Z      \  ”

#ˆ= %=

%== %=

%ˆ= %=

>== %ˆ

ƒ== %ˆ

ƒˆ= %ˆ



 +„K€  Nƒ  -

Z    [  \  ”

#== %=

#%= %=

#ƒ= %=

#<= %ˆ

%== %ˆ

%ƒ= %ˆ

' 

Z     \      ” \      ”

%== %= %=

%== %= %=

%== %= %=

%== >= ƒ=

%ƒ= ƒ= <=

%"= ˆ= ]=

$ ; )    

Z    [  \      ” \      ”

]ˆ %= %=

ˆ %= %=

##= %ˆ %=

#%ˆ >ˆ %ˆ

#ˆ= ƒˆ >ˆ

#]= ˆˆ ƒˆ

<;;  ; +      -

Z     [  Z     \      ” \      ”

]ˆ #%ˆ %= %=

ˆ #%ˆ %= %=

##= #%ˆ >ˆ %=

#%ˆ #%ˆ ƒˆ >ˆ

#ˆ= #ˆ= ˆˆ ƒˆ

#]= #]ˆ <ˆ ˆˆ

#   

” \                [    ¥>ˆ         

   [   

$ = } "##=' `  #' #

]

(   †         *     

Ž!

   $   

  

 !

   

     

   €  \         <==  $ \      ==Ž#ˆ==  $ `     

ˆŽ< <Ž## ƒŽ" <Ž#ƒ

$%%Ž>= $%ƒŽ>ˆ $%] $%>

$%"Ž>< $>ƒŽƒ= $>< 456

$]Ž#= Ž $]Ž#= $

"Ž#ƒ

$#ˆŽ%=

$#"Ž%ƒ

$]

Ž#=

$>ˆŽƒ=

$>"Žƒˆ

$#=Ž#%

‚     ¥%ˆ=  €  

>Ž#=

$#%

$#ˆ

$<

ˆŽ#ˆ

*$#=

$#%

$<

\  

%ˆŽ"

)$#=Ž%=

)$#=Ž%=

)$#=

~

%Žƒ

)$>=Ž>ˆ

)$ƒˆ

)$#ˆŽ#"

‚   

%Ž"

Ž

Ž

)$#=Ž#ƒ

%  

{' #%ˆ             [      

‚   \  

#"#

(   † '  ?    $                                   

    \[      Ž         $ „            % {$%        $   ;   

         =¢   [      

   (    ;              †                 =< {$%           ?             [  _      =ƒˆ             [     ˆ    $#]ˆ ?                „                                  #%ˆ      #ˆ=     %==                         € [              '  ˆ $#ˆ $#"  $%#      ‚\>ˆ        #¢ %¢  >¢     

#"%

      Q `[ }[

<          * ˆ %ˆ= {$%      * ˆ ƒ<= {$%      

     ; ;   +* Q

Œ

N„

NO

NQ

O„

Oˆ

O

=%%%

=> ˆ

=<#<

="""

#ˆ]

%ƒ<<

>"ˆƒ

<>#>

€„

"<ƒ

<    +

O-      ;  ;  ;

N O  € ˆ Q H Œ ‰

'  

Q

Œ

N„

NO

NQ

O„

Oˆ

O

€„

%" ˆ] "ˆ ##> #ƒ# #]= # " %%< %ˆƒ

ˆ= #=# #ˆ# %=# %ˆ# >=% >ˆ% ƒ=% ƒˆ%

] #ˆ] %>< >#ƒ > > ƒ]# ˆˆ= <%" ]=]

##> %%< >> ƒˆ% ˆ<ˆ <] ] % =ˆ #=#"

%=# ƒ=% <=> "=ƒ #==ˆ #%=< #ƒ=] #<=" #"#=

>#ƒ <%" ƒ% #%ˆ] #ˆ]# #""ˆ %# %ˆ#> %"%]

ƒ # "% #ƒ]> # <> %ƒˆƒ % ƒˆ >ƒ>< > %] ƒƒ#"

"=ƒ #<=" %ƒ#> >%#] ƒ=%# ƒ"%ˆ ˆ<>= <ƒ>ƒ ]%>"

#%ˆ] %ˆ#> >]]= ˆ=%] <%"> ]ˆƒ= "] < #= =ˆ> ## >#=

<    +

O* -    ;  $



'  

Q

ˆ„ Hˆ N„„ NOˆ Nˆ„ NHˆ O„„ OOˆ Oˆ„

ˆ<ˆ >]] %"> %%< #"" #<% #ƒ# #%< ##>

Œ #==ˆ <]= ˆ=> ƒ=% >>ˆ %"] %ˆ# %%> %=#

N„

NO

NQ

O„

Oˆ

O

€„

#ˆ]# #=ƒ] ]"ˆ <%" ˆ%ƒ ƒƒ > > >ƒ >#ƒ

%%<% #ˆ=" ##># =ˆ ]ˆƒ <ƒ< ˆ<ˆ ˆ=> ƒˆ%

ƒ=%# %<"# %=## #<=" #>ƒ= ##ƒ #==ˆ " ƒ "=ƒ

<%"> ƒ#" >#ƒ% %ˆ#> %= ƒ #] ˆ #ˆ]# #> < #%ˆ]

"#] <ˆƒˆ ƒ = > %] >%]% %"=ˆ %ƒˆƒ %#"% # <>

Ž #= ]%> "=ƒ% <ƒ>ƒ ˆ><% ƒˆ < ƒ=%# >ˆ]ƒ >%#]

Ž Ž #% ˆ<< #= =ˆ> ">]" ]#"# <%"> ˆˆ"ˆ ˆ=%]

‚   \  

#">

<     '$ €€Œ #; 

  ) " 



(



  ) 

O*

" 



(





O*

 * O

$&  Ž ‘    ?> > #= %== ?%ˆ% " %== ?# > ] %== ?#ƒ% < %== ? " ˆ %==

> > %ˆ% # > #ƒ% "

#= " ] < ˆ

%== %== %== %== %==

> > %ˆ% # > #ƒ% "

<#< > ˆ >=% %%% #ˆƒ

$  Ž ‘    }#># #% #== }]"ˆ #= #== }ˆ=> " #== }>"ˆ ] #== }%"> < #== }# < ˆ #==

##># ]"ˆ ˆ=> >"ˆ %"> # <

" " " ] ] ]

%== %== %== %== %== %==

%ˆ% %ˆ% %ˆ% # > # > # >

#= "#ƒ ˆ > ƒˆ> >]> >=ˆ

   Ž ‘    \]"ˆ #= #== \<>< #== \ˆ=> " #== \>"ˆ ] #== \%"> < #==

]"ˆ <>< ˆ=> >"ˆ %">

< < ˆ ˆ ˆ

ƒ== ƒ== ƒ== ƒ== ƒ==

]=" ]=" ƒ ƒ ƒ

<]% ˆˆˆ ƒ>ƒ >ƒ# %<#

ˆ %ˆ

%== #==

" ƒ

#ˆƒ =]]

  Ž Z  | " ˆ |ƒ %ˆ $   †

%== #==

" ƒ

  )

  )

$ 

*  ƒ" 

~ %ƒ 

##ˆ% %

$ = } ƒƒ"<' # "ˆ

#"ƒ

      Q `[ }[

$      $  / !

O

$  *  ; / ! *!

O*

K 





 / !



  

Q Œ N„

O



NO N„„

NOˆ

Nˆ„

NHˆ

O„„

OOˆ

Oˆ„

OHˆ

„„

%%<

=ˆ<= #=== #ˆ"= %%<=

=ƒƒ" ="== #%<ƒ #"="

=>]> =<<] #=ˆ> #ˆ=]

=>%= =ˆ]# = => #% #

=%"= =ˆ== =] = ##>=

=%ƒ =ƒƒƒ =]=% #==ƒ

=%%ƒ =ƒ== =<>% = =ƒ

=%=ƒ =><ƒ =ˆ]ˆ ="%%

=#"] =>>> =ˆ%] =]ˆ>

>>

="ƒ= #ˆ== %>]= >> =

=<]% #%== #" < %]#%

=ˆ<= #=== #ˆ"= %%<=

=ƒ"= ="ˆ] #>ˆƒ # >]

=ƒ%= =]ˆ= ##"ˆ #< ˆ

=>]> =<<] #=ˆ> #ˆ=]

=>>< =<== = ƒ" #>ˆ<

=>=ˆ =ˆƒˆ ="<% #%>>

=%"= =ˆ== =] = ##>=

ƒˆ%

##%= %=== >#<= ƒˆ%=

=" < #<== %ˆ%" ><#<

=]ƒ] #>>> %#=] >=#>

=<ƒ= ##ƒ> #"=< %ˆ">

=ˆ<= #=== #ˆ"= %%<=

=ƒ " ="" #ƒ=ƒ %==

=ƒƒ" ="== #%<ƒ #"="

=ƒ=] =]%] ##ƒ #<ƒƒ

=>]> =<<] #=ˆ> #ˆ=]

<]"

#<"= >=== ƒ]ƒ= <]"=

#>ƒƒ %ƒ== >] % ˆƒ%ƒ

##%= %=== >#<= ƒˆ%=

= <= #]#ƒ %]= >"]ƒ

="ƒ= #ˆ== %>]= >> =

=]ƒ] #>>> %#=] >=#>

=<]% #%== #" < %]#%

=<## #= # #]%ƒ %ƒ<ˆ

=ˆ<= #=== #ˆ"= %%<=

ˆ< #== #ˆ"

"ƒ #ˆ= %>]

€

##% %== >#<

Q

#<" >== ƒ]ƒ

‚   \  

#"ˆ

    '$ ŒNN„ (               ; 

    !

"

|              |               |                 

%



!

'

!

'

 ) 

#ƒ

#=

#<

==

#ƒ

Ž

#ƒ

#=

Ž

Ž

#ƒ

#ƒ

#%

#%

#%

#%

#%

#%

%!                                         [        

"  ;  "     ;  )    

  $

   



    

=

 ##

 %

$ = } "##=' `  #' #

Ž

]

        #<

      

    

 #ƒ

 #%:ˆ ˆ

Ž

#"<

      Q `[ }[

      ;  

 ˆ =#ˆ< %        #=¢      

#    !  +-   : +-  ˆ

  

%$ $

„K„€

„K„ˆ„

„K„H„

„K„‰„

„KNN„

„KN„

„KN€ˆ

„KNˆQ

= ˆ=

= ƒ#

= #ˆ

=""]

="ˆ]

="%ˆ

=] "

=]]]

=#>

=#ˆ

=#

=%ˆ

=>%

=>

=ƒˆ

=ˆ=

~  % ˆ        ˆ    _ 

           ;                       ?    ˆ $‰=: ˆ…% †* Š                     0 -/  ˆ …

=:#ˆ<† / % $‰=: ˆ* …

0 †Š             

-/  ˆ =:#ˆ</ % $‰=: ˆ* %Š ‡ 0&        #=¢

%&!  ;     :     ;  #  ; 

$   

  

 !

 ;

 ‡  $ˆ

 ‡  $]



 ;

 ‡  $#=

 ‡  $#>



~   ˆ      ˆ                 

\  ,    €  ,   =              _  $  =        _                

     =  ¥      _                } "##='   >ƒƒˆ $ = } "##=' `  #' #

]

‚   \  

#"]

$   ;         ˆ '4n 

$            2     †            

1.3 d = 125 (mm) d = 150 d = 175 d = 200 d = 225 d = 250 d = 300 d = 400

Shear capacity,Vc (N/mm2)

1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3

0

0.5

1.0

1.5

2.0 2.5 100As % bd

3.0

3.5

4.0

#          ;         + *

O-

#        ;  !

       ;  ! +

O-

2   =ˆ2     

Z    [           

             

     

* ˆ %ˆ= {$%

=ˆ2   2   =ƒ ‡ 2

Z    [            

   =ƒ {$%

…=:ƒ p  ‡ 2 † < 2 < %=:"    ˆ {=

* [               =]ˆ        {          #ˆ=         [ ”

n >

=:%n  #==

n >

=:ƒn  =: ˆ*n

n >

n n …n n † =: ˆ*n

”} ‹            [         ˆ=¢   

       ‹         } "##='   >ƒˆ<

$ = } "##=' `  #' #

]

#""

      Q `[ }[

"     ;            ~             ˆ   =%ƒ       

                 

"          )        ; %  * ;   

     

     

=

$

 %:ˆ

     

      

    

 #>

 ##:ˆ

 #ˆ:ˆ

 #ˆ:ˆ

Ž

> ˆ

Ž

 %

  

$    

 ##:ˆ

     %ˆ < #>

Ž

     

~                     

"     ) )        ; ~   =  #:ˆ                          _ ˆ b    ˆ b        '    

  

\     Z

Ž      

\     Z

Ž     

\     Z

) ‹    

       !

        !

   ”

\     Z

 ˆ #:= 

 ˆ #:% 

 ˆ #:ˆ 

# >%

# %>

# #

# >#

# ƒ#

# %

# %ˆ

# ƒ#

# %<

# %=

# #]

# %]

# >ƒ

# %<

# %>

# >ˆ

# %<

# #]

# #>

# %]

# >>

# %%

# #"

# >ˆ

# %#

# #ˆ

# #%

# %%

# %]

# %=

# #]

# %

”                                                   ˆ ]ˆ¢                  =ˆ  

#             ;      2                  †              >#<  } "##=             $ = } "##=' `  #' #

]

‚   \  

#"

"     ; €                                             

           ?   †      

    } "##=          ‹                „        

$   ;        ; ?        } "##=         

          

                   =#ˆ    ˆ ƒ  =p    ˆ     ?    ‹                                     '

(             ;  %  * ;    $    

  

                 

                       



=

% ##

% %=

% #=

%% #>

% ##

%% #ˆ

  



=

% ##

% %=

% #=

% %=

% ##

% %=

   |*  [{$%               [{$   

Z          _         _ ˆ =#ˆ %            

     €  >#>  } "##=          „   [               _

";   ;      ;                        *                        ]ˆ¢                     €       ˆˆ¢                                                 } "##=       =  %:=  

       =%                 <<                   =ƒ         

# =

      Q `[ }[

(        ;                         '                                                    €             ' ' ˆ ##ˆ'        ' ˆ #%ˆ'         ' ˆ #%ˆ'                ' ˆ #ƒ'               

(       ;             ['  ˆ ' =7    7 p                     ="   ˆ {$% `                ['  ˆ ' =7    7                   [  ˆ #   #ˆ                  =]ˆ            [              2 } "##='   >]]<             

                                  [              $  „         

‚   \  

# #

$     } "##=     $         $%ˆ=                 $ˆ==  %=       #= 

' *     ;  $    

<    )  ‡ NK„

#    )  „K

 ! $      

] %= %<

ˆ< #<= %="

€     = > =:>                   



);  *  ?  $ ˆ #  %  >  ƒ  }  $  ~ ' #

              #=  # ˆ #==    #   #=

%

       

>

       

ƒ

             <=¢   

          ƒ ˆ ##ˆ %*              * ˆ >     $ = } "##=' `  #' # ]

# %

            2.0

Modification factor for tension reinforcement F2

1.9 1.8 1.7

M = 0.5 bd 2

1.6 1.5

0.75

1.4

1.0

1.3 1.2

1.5

1.1

2.0

1.0 3.0

0.9

4.0 5.0 6.0

0.8 100

120

140

160

180

200

Service stress, fs =

2fyAs req 3As prov

220

(N/mm2)

240

260

280

            

Modification factor for compression reinforcement, F3

1.6

1.5

1.4

1.3

1.2

1.1

1.0 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

100As′ bd

# >

# ƒ

      Q `[ }[

                              ¥ ƒ            ‡                 <=          

                                             

                           ˆ b                _ =   = ?       Œ[Q         #ˆ        #=       \   _           Œ  Q

%!    +-      %     ;    

%          

   N

   O

   

' ; ' ; ' ;    N QZ Q 

                

  ”

=]ˆ

#%=

="=

#>=

= =

#<=

   O QZ Q 

          

  

  ”

="=

#>=

="ˆ

#ˆ=

= ˆ

#"=

    Q`

Q 



= =

#<=

= ˆ

#"=

#==

$

   € Q€ Q  

$

%%=

$

$

$

$

”\               

$ = } "##=' `  #' #

]

#       

KU

M FU = Me

KL + KU + 0.5KB

KU

W kN/m

KB KL M FL = Me

KU

Total factored load WT kN/m

Unfactored dead load WD kN/m KB1

KL KL + KU + 0.5KB

KU M FU = Mes KL + KU + 0.5K1 + 0.5K2

KB2 KL

KL M FL = Mes KL + KU + 0.5K1 + 0.5K2

Stiffness, k = I L Me = Fixed end beam moment Mes =Total out of balance fixed end moment

# ˆ

# <

      Q `[ }[

      \                 _        [              _  

      _          _              $%=                     

              _                 _   #=¢         ~             [    

  _  

    '  ˆ =ƒ*  ‡ =]ˆ *                         

                          >">  } "##=

': ;     ~        _           _         _                   _ ~                 Ž _            Ž _' 

 

           Ž _     ˆ  ‡



     

         Ž _     ˆ  ‡

b   

b  

  b '

  

===

=#=

=%=

=>=

=ƒ=

=ˆ=

=<=



#==

=""

=]]

=<ˆ

=ˆ>

=ƒ%

=>=

$ = } "##=' `  #' #

]

<        

1.8 e

1.6

b

h = 20

d = 0.95 h Asc 2

1.4 1. pf y =u Fc

Bars excluded

Asc 2

4

1.2

Bars included in calculating Asc

h d

p=

Asc bh

2

1. 0

1. 0.

N bhFcu

1.0 8

0.8

6

0.

0.6 4

0. 2

0.

0.4 0 0.

0.2 Design as beam 0

0.2

0.3 m bh 2Fcu

0.4

0.5

0.6

0.7

# ]

$ =   %==%

0.1

# "

1.8 e

1.6

b

h = 20

d = 0.85 h Asc 2

1.4

h d 4

1.

1.2

Bars excluded Asc 2

=

cu

pf y

F

Bars included in calculating Asc

p=

Asc bh

2

1. 0

1.

1.0 N bhFcu 0.8

8

0. 6

0. 4

0.

0.6 2

0.

0.4 0

0.

0.2 Design as a beam 0

0.1

0.2

0.3

0.4 m bh 2Fcu

$ =   %==%

0.5

0.6

0.7

1.8

b

h = 0 e 2

1.6 1.4

Bars included in calculating Asc Bars excluded

h d Asc 2

1.2 pf y

p=

Asc bh

=

u

Fc

4

1.

1.0

2

1. 1.

0.8

0

N bhFcu

d = 0.75 h

Asc 2

8

0. 6

0. 4

0.

0.6 0

0.

2

0.

0.4 0.2

Design as a beam 0

0.1

0.3

0.4 m bh 2Fcu

0.5

0.6

#

$ =   %==%

0.2

%==

      Q `[ }[

e= h 20

1.4 1.2

hs = 0.9 h

h hs

pf

1.0

F

y =1 cu

p=

1. 1. 2 0 0. 8

0.8 0.6

6 0.

0.

4

0.

0.4

.4

N bhFcu

0.

2

0

0.2 Design as a beam 0

0.1

0.2

0.3 M bh2Fcu

$ =   %==%

0.4

4Asc πh 2

‚   \  

1.4 e h = 20

1.2

hs

pf y

1.0

u

4 1.

1.

0

2

1. 8 0.

0.

p=

=

Fc

N 0.8 bhFcu 0.6

6

4

0.

0.

0.4

hs = 0.8 h

h

2

0 0.

0.2

Design as a beam 0

0.1

0.2

0.3 M bh2Fcu

$ =   %==%

0.4

4Asc πh 2

%=#

%=%

      Q `[ }[

e= h 20

1.4 1.2

u

1. 4

1. 2 0 8

=

Fc 1.

p=

0. 0.

6 0.

0.4

hs

pf y

1.0 0.8 N bhFcu 0.6

hs = 0.7 h

h

4

0.

0.

2

0

0.2 Design as a beam 0

0.1

0.2

0.3 M bhFcu

$ =   %==%

0.4

4Asc πh 2

‚   \  

%=>

$               '$ ŒNN„ ‡             

          Z                     [    _    ‹                                          

       €           $     €                    €          

  ˆ =:#>¢    ˆ =:%¢    ˆ =:ƒ¢ 

:       €  €       €  †      ?         †     

 _  _  _  _

ˆ ƒ¢  ˆ <¢  ˆ "¢  ˆ #=¢ 

%=ƒ

      Q `[ }[

$      :   ;) ;     _            >#%##% } "##=     [   =>                 

#>ˆ        #<=                   _ >  ]ˆ=  ‚        [  [         =%ˆ¢                             

  ;   }                                                    }        

                   |   %            €     f                ƒ=f >"f   >ˆf  >= {$% >ˆ {$%  ƒ= {$%       

         #ƒ             %f                  %f      „      ]ˆ  
‚   \  

%=ˆ

<   ; ;  '$ ŒQQQ } "<<<                               

    /    ; 

)      ; +

 <

 

 ;  / "

     / #

    ;   / 

    / 

     / #

    ;   / 

    / 

< " #= #% #< %= %ˆ >% ƒ=

#% #< %= %ƒ >% ƒ= ˆ= <ƒ "=

%ƒ >% ƒ= ƒ" <ƒ "= #== #%" #<=

##= ##ˆ #%= #%ˆ #>= #<= %== %<= >%=

#% #< %= %ƒ >% ]= "] ##% #ƒ=

%ƒ >% ƒ= ƒ" <ƒ #ƒ= #]ˆ %%ƒ %"=

##= ##ˆ #%= #%ˆ #>= # = %ƒ= >=ˆ >"=

~  ‡  ‚   ‡  %ˆ=     ‡     ‡  ƒ<=?  ƒ<=}    %        } ƒƒƒ 

$ = } "<<<' %===

%=<

      Q `[ }[

' ;     '$ ŒQQQ

R

A A

(B ) L = A + (B ) – r/ – d 2

L=A (C)

(B ) L = A + (B ) – R/ – d 2

A B

A

B

L = A + (C ) B

B

C

(C )

D (E)

A B

A

D B

C L = A + B + C + D + (E ) – 2r – 4d C

E B D

C L = A + 2B + C + (E )

A

B

D

C L = A + B + C + D + (E ) – 2r – 4d A

D

B

L = 2 (A + B + C ) – 5r – 5d /2

C = no. of turns

$ = } "<<<' %===

A

E

(E )

B

A L = Cπ(A-d)

(D)

C L = A + B + C + (D ) – 3r/ – 3d 2

A L = A + B + (C )

(C)

L = 2A + 3B + 17d A B D

A B

D

L =A + B + (E )

(C )

B L = A + B + (C ) – r – 2D

(C )

A L = A + 0.57B + (C ) – 1.57 d A

A

L=A

R

‚   \  

%=]

<     Œ*[                                                   Q                   

                  

                  | {' ‡                   [        |' ‚          [       

                    \          „    [  +                               † ?          ^ 

                                  [       [            ?    [                 ‹            ' 

"=Ž##= [$>

‚\   

]=Ž = [$>

   

#ˆ= [$>

` $ 

##ˆ [$>

\  

#ˆ=Žƒˆ= [$>

‡   

%>= [$>

}

%%= [$>

‚   

##= [$>

 

#>ˆ [$>

~

<ˆ [$>

Œ? Q         ' ‘     

#%ˆ [$>

\ 

ˆ [$>

   

"ˆ [$>

$ = `  ’ Z  %==#

9 Structural Steel The method of heating iron ore in a charcoal fire determines the amount of carbon in the iron alloy. The following three iron ore products contain differing amounts of carbon: cast iron, wrought iron and steel. Cast iron involves the heat treatment of iron castings and was developed as part of the industrial revolution between 1800 and 1900. It has a high carbon content and is therefore quite brittle which means that it has a much greater strength in compression than in tension. Typical allowable working stresses were 23 N/mm2 tension, 123 N/mm2 compression and 30 N/mm2 shear. Wrought iron has relatively uniform properties and, between the 1840s and 1900, wrought iron took over from cast iron for structural use, until it was in turn superseded by mild steel. Typical allowable working stresses were 81 N/mm2 tension, 61 N/mm2 compression and 77 N/mm2 shear. ’Steel’ can cover many different alloys of iron, carbon and other alloying elements to alter the properties of the alloys. The steel can be formed into structural sections by casting, hot rolling or cold rolling. Mild steel which is now mostly used for structural work was first introduced in the mid-nineteenth century.

Types of steel products Cast steel Castings are generally used for complex or non-standard structural components. The casting shape and moulding process must be carefully controlled to limit residual stresses. Sand casting is a very common method, but the lost wax method is generally used where a very fine surface finish is required.

Cold rolled Cold rolling is commonly used for lightweight sections, such as purlins and wind posts, etc. Work hardening and residual stresses caused by the cold working cause an increase in the yield strength but this is at the expense of ductility and toughness. Cold rolled steel cannot be designed using the same method as hot rolled steel and design methods are given in BS 5950: Part 5.

Hot rolled steel Most steel in the UK is produced by continuous casting where ingots or slabs are pre-heated to about 1300 C and the working temperatures fall as processing continues through the intermediate stages. The total amount of rolling work and the finishing temperatures are controlled to keep the steel grain size fine – which gives a good combination of strength and toughness. Although hollow sections (RHS, CHS and SHS) are often cold bent into shape, they tend to be hot finished and are considered ‘hot rolled’ for design purposes. This pocket book deals only with hot rolled steel.

Structural Steel

Summary of hot rolled steel material properties Density

78.5 kN/m3

Tensile strength

275–460 N/mm2 yield stress and 430–550 N /mm2 ultimate strength

Poisson’s ratio

0.3

Modulus of elasticity, E

205 kN/mm2

Modulus of rigidity, G

80 kN/mm2

Linear coefficient of thermal expansion

12  10

6 

/ C

209

210

Structural Engineer’s Pocket Book

Mild steel section sizes and tolerances Fabrication tolerances BS 4 covers the dimensions of many of the hot rolled sections produced by Corus. Selected rolling tolerances for different sections are covered by the following standards: UB and UC sections: BS EN 10034 Section height (mm)

h  180

180 < h  400

400 < h  700

700 < h

Tolerance (mm)

þ3/ 2

þ4/ 2

þ5/ 3

5

Flange width (mm)

b  110

110 < b  210

210 < b  325

325 < b

Tolerance (mm)

þ4/ 1

þ4/ 2

4

þ6/ 5

Out of squareness for flange width (mm)

b  110

110 < b

Tolerance (mm)

1.5

2% of b up to max 6.5 mm

Straightness for section height (mm)

80 < h  180

180 < h  360

360 < h

Tolerance on section length (mm)

0.003L

0.0015L

0.001L

RSA sections: BS EN 10056–2 Leg length (mm)

h  50

50 < h  100

100 < h  150

150 < h  200

200  h

Tolerance (mm)

1

2

3

4

þ6/ 4

Straightness for section height

h  150

h  200

200 < h

Tolerance along section length (mm)

0.004L

0.002L

0.001L

PFC sections: BS EN 10279 Section height (mm)

h  65

65 < h  200

200 < h  400

400 < h

Tolerance (mm)

1.5

2

3

4

Out of squareness for flange width

b  100

100 < b

Tolerance (mm)

1.5

2.5% of b

Straightness for section height

h  150

150 < h  300

300 < h

Tolerance along section length (mm)

0.005L

0.003L

0.002L

Hot finished RHS, SHS and CHS sections: BS EN 10210 Straightness:

0.2%L

Depth, breadth of diameter:

1% (min 0.5 mm and max 10 mm)

Squareness of side for SHS and RHS: Twist for SHS and RHS:

90  1

2 mm þ 0.5 mm per m maximum

Structural Steel

211

Examples of minimum bend radii for selected steel sections The minimum radius to which any section can be curved depends on its metallurgical properties, particularly its ductility, cross sectional geometry and end use (the latter determines the standard required for the appearance of the work). It is therefore not realistic to provide a definitive list of the radii to which every section can be curved due to the wide number of end uses, but a selection of examples is possible. Normal bending tolerances are about 8 mm on the radius. In cold rolling the steel is deformed in the yield stress range and therefore becomes work hardened and displays different mechanical properties (notably a loss of ductility). However, if the section is designed to be working in the elastic range there is generally no significant difference to its performance.

Section

Typical bend radius for S275 steel m

610  305 UB 238 533  210 UB 122 305  165 UB 40 250 150  12.5 RHS 305  305 UC 118 300  100 PFC 46 150  150  12.5 SHS 254  203 RSJ 82 191  229 TEE 49 152  152 UC 37 125  65 PFC 15 152  127 RSJ 37

40.0 30.0 15.0 9.0 5.5 4.6 3.0 2.4 1.5 1.5 1.0 0.8

Source: Angle Ring Company Limited (2002).

212

Structural Engineer’s Pocket Book

Hot rolled section tables Universal beams – dimensions and properties

UB designation

Mass Depth Width Thickness Root Depth Ratios for Second moment h/t per of of radius between local buckling of area better fillets known metre section section Flange Web Axis x–x Axis y–y in Web Flange BS449 as h b s t r d b/2t d/s Ix Iy D/T kg/m

mm

mm

mm

mm

mm

mm

y

486.6 436.9 392.7 349.4 314.3 272.3 248.7 222

1036.1 1025.9 1016 1008.1 1000 990.1 980.2 970.3

308.5 305.4 303 302 300 300 300 300

30 26.9 24.4 21.1 19.1 16.5 16.5 16

54.1 49 43.9 40 35.9 31 26 21.1

30 30 30 30 30 30 30 30

867.9 867.9 868.2 868.1 868.2 868.1 868.2 868.1

2.85 3.12 3.45 3.77 4.18 4.84 5.77 7.11

28.9 32.3 35.6 41.1 45.5 52.6 52.6 54.3

914  419  388 914  419  343

388 343.3

921 911.8

420.5 418.5

21.4 19.4

36.6 32

24.1 24.1

799.6 799.6

5.74 6.54

914  305  289 914  305  253 914  305  224 914  305  201

289.1 253.4 224.2 200.9

926.6 918.4 910.4 903

307.7 305.5 304.1 303.3

19.5 17.3 15.9 15.1

32 27.9 23.9 20.2

19.1 19.1 19.1 19.1

824.4 824.4 824.4 824.4

838  292  226 838  292  194 838  292  176

226.5 193.8 175.9

850.9 840.7 834.9

293.8 292.4 291.7

16.1 14.7 14

26.8 21.7 18.8

17.8 17.8 17.8

762  267  197 762  267  173 762  267  147 762  267  134

196.8 173 146.9 133.9

769.8 762.2 754 750

268 266.7 265.2 264.4

15.6 14.3 12.8 12

25.4 21.6 17.5 15.5

686  254  170 686  254  152 686  254  140 686  254  125

170.2 152.4 140.1 125.2

692.9 687.5 683.5 677.9

255.8 254.5 253.7 253

14.5 13.2 12.4 11.7

610  305  238 610  305  179 610  305  149

238.1 179 149.2

635.8 620.2 612.4

311.4 307.1 304.8

610  229  140 610  229  125 610  229  113 610  229  101

139.9 125.1 113 101.2

617.2 612.2 607.6 602.6

533  210  122 533  210  109 533  210  101 533  210  92 533  210  82

122 109 101 92.14 82.2

457  191  98 457  191  89 457  191  82 457  191  74 457  191  67

98.3 89.3 82 74.3 67.1

y y y y y y y

1016  305  487 1016  305  437 1016  305  393 1016  305  349 1016  305  314 1016  305  272 1016  305  249 1016  305  222

cm4

cm4

cm

1021400 909900 807700 723100 644200 554000 481300 408000

26720 23450 20500 18460 16230 14000 11750 9546

19 21 23 25 28 32 38 46

37.4 41.2

719600 45440 625800 39160

25 28

4.81 5.47 6.36 7.51

42.3 47.7 51.8 54.6

504200 15600 436300 13300 376400 11240 325300 9423

29 33 38 45

761.7 761.7 761.7

5.48 6.74 7.76

47.3 51.8 54.4

339700 11360 279200 9066 246000 7799

32 39 44

16.5 16.5 16.5 16.5

686 686 686 686

5.28 6.17 7.58 8.53

44 48 53.6 57.2

240000 205300 168500 150700

8175 6850 5455 4788

30 35 43 48

23.7 21 19 16.2

15.2 15.2 15.2 15.2

615.1 615.1 615.1 615.1

5.4 6.06 6.68 7.81

42.4 46.6 49.6 52.6

170300 150400 136300 118000

6630 5784 5183 4383

29 33 36 42

18.4 14.1 11.8

31.4 23.6 19.7

16.5 16.5 16.5

540 540 540

4.96 6.51 7.74

29.3 38.3 45.8

209500 15840 153000 11410 125900 9308

20 26 31

230.2 229 228.2 227.6

13.1 11.9 11.1 10.5

22.1 19.6 17.3 14.8

12.7 12.7 12.7 12.7

547.6 547.6 547.6 547.6

5.21 5.84 6.6 7.69

41.8 46 49.3 52.2

111800 98610 87320 75780

4505 3932 3434 2915

28 31 35 41

544.5 539.5 536.7 533.1 528.3

211.9 210.8 210 209.3 208.8

12.7 11.6 10.8 10.1 9.6

21.3 18.8 17.4 15.6 13.2

12.7 12.7 12.7 12.7 12.7

476.5 476.5 476.5 476.5 476.5

4.97 5.61 6.03 6.71 7.91

37.5 41.1 44.1 47.2 49.6

76040 66820 61520 55230 47540

3388 2943 2692 2389 2007

26 29 31 34 40

467.2 463.4 460 457 453.4

192.8 191.9 191.3 190.4 189.9

11.4 10.5 9.9 9 8.5

19.6 17.7 16 14.5 12.7

10.2 10.2 10.2 10.2 10.2

407.6 407.6 407.6 407.6 407.6

4.92 5.42 5.98 6.57 7.48

35.8 38.8 41.2 45.3 48

45730 41020 37050 33320 29380

2347 2089 1871 1671 1452

24 26 29 32 36

Structural Steel

b

s h d r

Radius of gyration

Elastic modulus

Plastic modulus

t

Buckling Torsional Warping Torsional Area of parameter index constant constant section

Axis x–x Axis y–y Axis x–x Axis y–y Axis x–x Axis y–y rx ry Zx Zy Sx Sy u

H

J

A

dm6

cm4

cm2

21.1 23.1 25.5 27.9 30.7 35 39.9 45.7

64.4 55.9 48.4 43.3 37.7 32.2 26.8 21.5

4299 3185 2330 1718 1264 835 582 390

620 557 500 445 400 347 317 283

0.885 0.883

26.7 30.1

88.9 75.8

1734 1193

494 437

1601 1371 1163 982

0.867 0.866 0.861 0.854

31.9 36.2 41.3 46.8

31.2 26.4 22.1 18.4

926 626 422 291

368 323 286 256

9155 7640 6808

1212 974 842

0.87 0.862 0.856

35 41.6 46.5

19.3 15.2 13

514 306 221

289 247 224

610 514 411 362

7167 6198 5156 4644

959 807 647 570

0.869 0.864 0.858 0.854

33.2 38.1 45.2 49.8

11.3 9.39 7.4 6.46

404 267 159 119

251 220 187 171

4916 4374 3987 3481

518 455 409 346

5631 5000 4558 3994

811 710 638 542

0.872 0.871 0.868 0.862

31.8 35.5 38.7 43.9

7.42 6.42 5.72 4.8

308 220 169 116

217 194 178 159

7.23 7.07 7

6589 4935 4111

1017 743 611

7486 5547 4594

1574 1144 937

0.886 0.886 0.886

21.3 27.7 32.7

14.5 10.2 8.17

785 340 200

303 228 190

25 24.9 24.6 24.2

5.03 4.97 4.88 4.75

3622 3221 2874 2515

391 343 301 256

4142 3676 3281 2881

611 535 469 400

0.875 0.873 0.87 0.864

30.6 34.1 38 43.1

3.99 3.45 2.99 2.52

216 154 111 77

178 159 144 129

22.1 21.9 21.9 21.7 21.3

4.67 4.6 4.57 4.51 4.38

2793 2477 2292 2072 1800

320 279 256 228 192

3196 2828 2612 2360 2059

500 436 399 356 300

0.877 0.875 0.874 0.872 0.864

27.6 30.9 33.2 36.5 41.6

2.32 1.99 1.81 1.6 1.33

178 126 101 75.7 51.5

155 139 129 117 105

19.1 19 18.8 18.8 18.5

4.33 4.29 4.23 4.2 4.12

1957 1770 1611 1458 1296

243 218 196 176 153

2232 2014 1831 1653 1471

379 338 304 272 237

0.881 0.88 0.877 0.877 0.872

25.7 28.3 30.9 33.9 37.9

1.18 1.04 0.922 0.818 0.705

121 90.7 69.2 51.8 37.1

125 114 104 94.6 85.5

cm

cm

cm3

cm3

cm3

cm3

40.6 40.4 40.2 40.3 40.1 40 39 38

6.57 6.49 6.4 6.44 6.37 6.35 6.09 5.81

19720 17740 15900 14350 12880 11190 9821 8409

1732 1535 1353 1223 1082 934 784 636

23200 20760 18540 16590 14850 12830 11350 9807

2800 2469 2168 1941 1713 1470 1245 1020

0.867 0.868 0.868 0.872 0.872 0.873 0.861 0.85

38.2 37.8

9.59 9.46

15630 13730

2161 1871

17670 15480

3341 2890

37 36.8 36.3 35.7

6.51 6.42 6.27 6.07

10880 9501 8269 7204

1014 871 739 621

12570 10940 9535 8351

34.3 33.6 33.1

6.27 6.06 5.9

7985 6641 5893

773 620 535

30.9 30.5 30 29.7

5.71 5.58 5.4 5.3

6234 5387 4470 4018

28 27.8 27.6 27.2

5.53 5.46 5.39 5.24

26.3 25.9 25.7

x

Source: Corus Construction (2002). Copyright Corus UK Ltd – reproduced with the kind permission of Corus UK Ltd.

213

214

Structural Engineer’s Pocket Book

Universal beams – dimensions and properties

UB designation

Mass Depth Width Thickness Root Depth Ratios for per of of radius between local buckling fillets metre section section Web Flange Flange Web

h

b

s

t

r

d

kg/m

mm

mm

mm

mm

mm

mm

457  152  60 457  152  52

59.8 52.3

454.6 449.8

152.9 152.4

8.1 7.6

13.3 10.9

10.2 10.2

406  178  74 406  178  67 406  178  60 406  78  54

74.2 67.1 60.1 54.1

412.8 409.4 406.4 402.6

179.5 178.8 177.9 177.7

9.5 8.8 7.9 7.7

16 14.3 12.8 10.9

406  140  46 406  140  39

46 39

403.2 398

142.2 141.8

6.8 6.4

356  171  67 356  171  57 356  171  51 356  171  45

67.1 57 51 45

363.4 358 355 351.4

173.2 172.2 171.5 171.1

356  127  39 356  127  33

39.1 33.1

353.4 349

305  165  54 305  165  46 305  165  40

54 46.1 40.3

305  127  48 305  127  42 305  127  37

Second moment of area

h/t better known Axis x–x Axis y–y in BS449 as D/T Ix Iy

b/2t

d/s

407.6 407.6

5.75 6.99

50.3 53.6

25500 21370

795 645

34 41

10.2 10.2 10.2 10.2

360.4 360.4 360.4 360.4

5.61 6.25 6.95 8.15

37.9 41 45.6 46.8

27310 24330 21600 18720

1545 1365 1203 1021

26 29 32 37

11.2 8.6

10.2 10.2

360.4 360.4

6.35 8.24

53 56.3

15690 12510

538 410

36 46

9.1 8.1 7.4 7

15.7 13 11.5 9.7

10.2 10.2 10.2 10.2

311.6 311.6 311.6 311.6

5.52 6.62 7.46 8.82

34.2 38.5 42.1 44.5

19460 16040 14140 12070

1362 1108 968 811

23 28 31 36

126 125.4

6.6 6

10.7 8.5

10.2 10.2

311.6 311.6

5.89 7.38

47.2 51.9

10170 8249

358 280

33 41

310.4 306.6 303.4

166.9 165.7 165

7.9 6.7 6

13.7 11.8 10.2

8.9 8.9 8.9

265.2 265.2 265.2

6.09 7.02 8.09

33.6 39.6 44.2

11700 9899 8503

1063 896 764

23 26 30

48.1 41.9 37

311 307.2 304.4

125.3 124.3 123.4

9 8 7.1

14 12.1 10.7

8.9 8.9 8.9

265.2 265.2 265.2

4.47 5.14 5.77

29.5 33.1 37.4

9575 8196 7171

461 389 336

22 25 28

305  102  33 305  102  28 305  102  25

32.8 28.2 24.8

312.7 308.7 305.1

102.4 101.8 101.6

6.6 6 5.8

10.8 8.8 7

7.6 7.6 7.6

275.9 275.9 275.9

4.74 5.78 7.26

41.8 46 47.6

6501 5366 4455

194 155 123

29 35 44

254  146  43 254  146  37 254  146  31

43 37 31.1

259.6 256 251.4

147.3 146.4 146.1

7.2 6.3 6

12.7 10.9 8.6

7.6 7.6 7.6

219 219 219

5.8 6.72 8.49

30.4 34.8 36.5

6544 5537 4413

677 571 448

20 23 29

254  102  28 254  102  25 254  102  22

28.3 25.2 22

260.4 257.2 254

102.2 101.9 101.6

6.3 6 5.7

10 8.4 6.8

7.6 7.6 7.6

225.2 225.2 225.2

5.11 6.07 7.47

35.7 37.5 39.5

4005 3415 2841

179 149 119

26 31 37

203  133  30 203  133  25

30 25.1

206.8 203.2

133.9 133.2

6.4 5.7

9.6 7.8

7.6 7.6

172.4 172.4

6.97 8.54

26.9 30.2

2896 2340

385 308

22 26

cm4

cm4

203  102  23

23.1

203.2

101.8

5.4

9.3

7.6

169.4

5.47

31.4

2105

164

22

178  102  19

19

177.8

101.2

4.8

7.9

7.6

146.8

6.41

30.6

1356

137

23

152  89  16

16

152.4

88.7

4.5

7.7

7.6

121.8

5.76

27.1

834

89.8

20

127  76  13

13

127

76

4

7.6

7.6

96.6

5

24.1

473

55.7

17

yAdditional sizes to BS4 available in UK.

215

Structural Steel

b

s h d r

Radius of gyration

Elastic modulus

Plastic modulus

Axis x–x Axis y–y

Axis x–x

Axis y–y

Axis x–x

Axis y–y

Buckling parameter

Torsional index

Warping constant

u

x

t

Torsional constant

Area of section

H

J

A

dm6

cm4

cm2

rx

ry

Zx

Zy

Sx

Sy

cm

cm

cm3

cm3

cm3

cm3

18.3 17.9

3.23 3.11

1122 950

104 84.6

1287 1096

163 133

0.868 0.859

37.5 43.9

0.387 0.311

33.8 21.4

76.2 66.6

17 16.9 16.8 16.5

4.04 3.99 3.97 3.85

1323 1189 1063 930

172 153 135 115

1501 1346 1199 1055

267 237 209 178

0.882 0.88 0.88 0.871

27.6 30.5 33.8 38.3

0.608 0.533 0.466 0.392

62.8 46.1 33.3 23.1

94.5 85.5 76.5 69

16.4 15.9

3.03 2.87

778 629

75.7 57.8

888 724

118 90.8

0.871 0.858

38.9 47.5

0.207 0.155

19 10.7

58.6 49.7

15.1 14.9 14.8 14.5

3.99 3.91 3.86 3.76

1071 896 796 687

157 129 113 94.8

1211 1010 896 775

243 199 174 147

0.886 0.882 0.881 0.874

24.4 28.8 32.1 36.8

0.412 0.33 0.286 0.237

55.7 33.4 23.8 15.8

85.5 72.6 64.9 57.3

14.3 14

2.68 2.58

576 473

56.8 44.7

659 543

0.871 0.863

35.2 42.2

0.105 0.081

15.1 8.79

49.8 42.1

13 13 12.9

3.93 3.9 3.86

754 646 560

127 108 92.6

846 720 623

196 166 142

0.889 0.891 0.889

23.6 27.1 31

0.234 0.195 0.164

34.8 22.2 14.7

68.8 58.7 51.3

12.5 12.4 12.3

2.74 2.7 2.67

616 534 471

73.6 62.6 54.5

711 614 539

116 98.4 85.4

0.873 0.872 0.872

23.3 26.5 29.7

0.102 0.085 0.072

31.8 21.1 14.8

61.2 53.4 47.2

12.5 12.2 11.9

2.15 2.08 1.97

416 348 292

37.9 30.5 24.2

481 403 342

60 48.5 38.8

0.866 0.859 0.846

31.6 37.4 43.4

0.044 0.035 0.027

12.2 7.4 4.77

41.8 35.9 31.6

10.9 10.8 10.1

3.52 3.48 3.36

504 433 351

92 78 61.3

566 483 393

141 119 94.1

0.891 0.89 0.88

21.2 24.3 29.6

0.103 0.086 0.066

23.9 15.3 8.55

54.8 47.2 39.7

10.5 10.3 10.1

2.22 2.15 2.06

308 266 224

34.9 29.2 23.5

353 306 259

54.8 46 37.3

0.874 0.866 0.856

27.5 31.5 36.4

0.028 0.023 0.018

9.57 6.42 4.15

36.1 32 28

8.71 8.56

3.17 3.1

280 230

57.5 46.2

314 258

88.2 70.9

0.881 0.877

21.5 25.6

0.037 0.029

10.3 5.96

38.2 32

8.46

2.36

207

32.2

234

49.8

0.888

22.5

0.015

7.02

29.4

7.48

2.37

153

27

171

41.6

0.888

22.6

0.01

4.41

24.3

6.41

2.1

109

20.2

123

31.2

0.89

19.6

0.005

3.56

20.3

5.35

1.84

22.6

0.895

16.3

0.002

2.85

16.5

74.6

14.7

84.2

89.1 70.3

Source: Corus Construction (2002). Copyright Corus UK Ltd – reproduced with the kind permission of Corus UK Ltd.

216

Structural Engineer’s Pocket Book

Universal columns – dimensions and properties

UC designation

Mass Depth Width Thickness Root Depth Ratios for per of of radius between local buckling metre section section fillets Web Flange Flange Web

Second moment of area

h/t better known in BS449 as

Axis x–x Ix

Axis y–y Iy

cm4

cm4

6.1 6.89 8.11 9.48 10.9 12.8 15.8

274800 226900 183000 146600 122500 99880 79080

98130 82670 67830 55370 46850 38680 30990

6 7 8 9 9 11 13

6.94 7.83 8.95 10.5

17.6 20.2 23.6 27.9

66260 57120 48590 40250

23690 20530 17550 14610

14 15 17 20

246.7 246.7 246.7 246.7 246.7 246.7 246.7

3.65 4.22 5.01 6.22 7.12 8.22 9.91

9.21 10.7 12.9 15.6 17.9 20.6 24.9

78870 64200 50900 38750 32810 27670 22250

24630 20310 16300 12570 10700 9059 7308

8 9 11 13 15 17 20

12.7 12.7 12.7 12.7 12.7

200.3 200.3 200.3 200.3 200.3

4.18 5.16 6.31 7.41 8.96

10.4 13.1 15.6 19.4 23.3

30000 22530 17510 14270 11410

9870 7531 5928 4857 3908

9 11 13 15 18

20.5 17.3 14.2 12.5 11

10.2 10.2 10.2 10.2 10.2

160.8 160.8 160.8 160.8 160.8

5.1 5.97 7.25 8.17 9.25

12.7 16.1 17.1 20.4 22.3

9449 7618 6125 5259 4568

3127 2537 2065 1778 1548

11 12 15 16 18

8 11.5 6.5 9.4 5.8 6.8

7.6 7.6 7.6

123.6 123.6 123.6

6.71 8.13 11.2

15.5 19 21.3

2210 1748 1250

706 560 400

14 17 22

h

b

s

t

r

d

b/2t

kg/m

mm

mm

mm

mm

mm

mm

356  406  634 356  406  551 356  406  467 356  406  393 356  406  340 356  406  287 356  406  235

633.9 551 467 393 339.9 287.1 235.1

474.6 455.6 436.6 419 406.4 393.6 381

424 418.5 412.2 407 403 399 394.8

47.6 42.1 35.8 30.6 26.6 22.6 18.4

77 67.5 58 49.2 42.9 36.5 30.2

15.2 15.2 15.2 15.2 15.2 15.2 15.2

290.2 290.2 290.2 290.2 290.2 290.2 290.2

2.75 3.1 3.55 4.14 4.7 5.47 6.54

356  368  202 356  368  177 356  368  153 356  368  129

201.9 177 152.9 129

374.6 368.2 362 355.6

374.7 372.6 370.5 368.6

16.5 14.4 12.3 10.4

27 23.8 20.7 17.5

15.2 15.2 15.2 15.2

290.2 290.2 290.2 290.2

305  305  283 305  305  240 305  305  198 305  305  158 305  305  137 305  305  118 305  305  97

282.9 240 198.1 158.1 136.9 117.9 96.9

365.3 352.5 339.9 327.1 320.5 314.5 307.9

322.2 318.4 314.5 311.2 309.2 307.4 305.3

26.8 23 19.1 15.8 13.8 12 9.9

44.1 37.7 31.4 25 21.7 18.7 15.4

15.2 15.2 15.2 15.2 15.2 15.2 15.2

254  254  167 254  254  132 254  254  107 254  254  89 254  254  73

167.1 132 107.1 88.9 73.1

289.1 276.3 266.7 260.3 254.1

265.2 261.3 258.8 256.3 254.6

19.2 15.3 12.8 10.3 8.6

31.7 25.3 20.5 17.3 14.2

203  203  86 203  203  71 203  203  60 203  203  52 203  203  46

86.1 71 60 52 46.1

222.2 215.8 209.6 206.2 203.2

209.1 206.4 205.8 204.3 203.6

12.7 10 9.4 7.9 7.2

152  152  37 152  152  30 152  152  23

37 30 23

161.8 157.6 152.4

154.4 152.9 152.2

d/s

D/T

217

Structural Steel b

s h d r t

Radius of gyration

Elastic modulus

Plastic modulus

Buckling parameter

Torsional index

u

x

Axis x–x rx

Axis y–y ry

Axis x–x Zx

Axis y–y Zy

Axis x–x Sx

Axis y–y Sy

cm

cm

cm3

cm3

cm3

cm3

18.4 18 17.5 17.1 16.8 16.5 16.3

11 10.9 10.7 10.5 10.4 10.3 10.2

11580 9962 8383 6998 6031 5075 4151

4629 3951 3291 2721 2325 1939 1570

14240 12080 10000 8222 6999 5812 4687

7108 6058 5034 4154 3544 2949 2383

0.843 0.841 0.839 0.837 0.836 0.835 0.834

Warping constant

Torsional Area of constant section

H

J

A

dm6

cm4

cm2

5.46 6.05 6.86 7.86 8.85 10.2 12.1

38.8 31.1 24.3 18.9 15.5 12.3 9.54

13720 9240 5809 3545 2343 1441 812

808 702 595 501 433 366 299

16.1 15.9 15.8 15.6

9.6 9.54 9.49 9.43

3538 3103 2684 2264

1264 1102 948 793

3972 3455 2965 2479

1920 1671 1435 1199

0.844 0.844 0.844 0.844

13.4 15 17 19.9

7.16 6.09 5.11 4.18

558 381 251 153

257 226 195 164

14.8 14.5 14.2 13.9 13.7 13.6 13.4

8.27 8.15 8.04 7.9 7.83 7.77 7.69

4318 3643 2995 2369 2048 1760 1445

1529 1276 1037 808 692 589 479

5105 4247 3440 2680 2297 1958 1592

2342 1951 1581 1230 1053 895 726

0.855 0.854 0.854 0.851 0.851 0.85 0.85

7.65 8.74 10.2 12.5 14.2 16.2 19.3

6.35 5.03 3.88 2.87 2.39 1.98 1.56

2034 1271 734 378 249 161 91.2

360 306 252 201 174 150 123

11.9 11.6 11.3 11.2 11.1

6.81 6.69 6.59 6.55 6.48

2075 1631 1313 1096 898

744 576 458 379 307

2424 1869 1484 1224 992

1137 878 697 575 465

0.851 0.85 0.848 0.85 0.849

8.49 10.3 12.4 14.5 17.3

1.63 1.19 0.898 0.717 0.562

626 319 172 102 57.6

213 168 136 113 93.1

9.28 9.18 8.96 8.91 8.82

5.34 5.3 5.2 5.18 5.13

850 706 584 510 450

299 246 201 174 152

977 799 656 567 497

456 374 305 264 231

0.85 0.853 0.846 0.848 0.847

10.2 11.9 14.1 15.8 17.7

0.318 0.25 0.197 0.167 0.143

137 80.2 47.2 31.8 22.2

110 90.4 76.4 66.3 58.7

6.85 6.76 6.54

3.87 3.83 3.7

273 222 164

309 248 182

140 0.848 112 0.849 80.2 0.84

13.3 16 20.7

0.04 0.031 0.021

91.5 73.3 52.6

19.2 10.5 4.63

47.1 38.3 29.2

Source: Corus Construction (2002). Copyright Corus UK Ltd – reproduced with the kind permission of Corus UK Ltd.

218

Structural Engineer’s Pocket Book

Rolled steel joists – dimensions and properties

Inside slope ¼ 8o RSJ designation

254  203  82 203  152  52 152  127  37 127  114  29 127  114  27 102  102  23 102  44  7 89  89  19 76  76  13

Radius Depth Mass Depth Width Thickness Ratios for between local buckling per of of fillets metre section section Web Flange Root Toe Flange Web

h

b

s

kg/m

mm

mm

mm mm

t

mm

82 52.3 37.3 29.3 26.9 23 7.5 19.5 12.8

254 203.2 152.4 127 127 101.6 101.6 88.9 76.2

203.2 152.4 127 114.3 114.3 101.6 44.5 88.9 76.2

10.2 8.9 10.4 10.2 7.4 9.5 4.3 9.5 5.1

19.6 15.5 13.5 9.9 9.9 11.1 6.9 11.1 9.4

19.9 16.5 13.2 11.5 11.4 10.3 6.1 9.9 8.4

r1

r2

d

b/2t

d/s

mm 9.7 166.6 7.6 133.2 6.6 94.3 4.8 79.5 5 79.5 3.2 55.2 3.3 74.6 3.2 44.2 4.6 38.1

5.11 4.62 4.81 4.97 5.01 4.93 3.65 4.49 4.54

16.3 15 9.07 7.79 10.7 5.81 17.3 4.65 7.47

Second moment of area Axis x–x

Axis y–y

Ix

Iy

cm4

cm4

12020 4798 1818 979 946 486 153 307 158

2280 816 378 242 236 154 7.82 101 51.8

h/t better known in BS449 as

D/T

13 12 12 11 11 10 17 9 9

219

Structural Steel

b

98°

r1 h d

t

Radius of gyration Axis x–x

Elastic modulus Axis y–y

Plastic modulus

Axis x–x

Axis y–y

Axis x–x

Axis y–y

3

3

3

cm3

cm

cm

cm

10.7 8.49 6.19 5.12 5.26 4.07 4.01 3.51 3.12

4.67 3.5 2.82 2.54 2.63 2.29 0.907 2.02 1.79

947 472 239 154 149 95.6 30.1 69 41.5

cm

224 107 59.6 42.3 41.3 30.3 3.51 22.8 13.6

cm

1077 541 279 181 172 113 35.4 82.7 48.7

371 176 99.8 70.8 68.2 50.6 6.03 38 22.4

Buckling parameter

Torsional index

Warping constant

Torsional constant

u

x

H

J

A

dm6

cm4

cm2

0.312 0.0711 0.0183 0.00807 0.00788 0.00321 0.000178 0.00158 0.000595

152 64.8 33.9 20.8 16.9 14.2 1.25 11.5 4.59

105 66.6 47.5 37.4 34.2 29.3 9.5 24.9 16.2

0.89 0.891 0.866 0.853 0.868 0.836 0.872 0.83 0.852

11 10.7 9.33 8.76 9.32 7.43 14.9 6.57 7.22

Area of section

220

Structural Engineer’s Pocket Book

Parallel flange channels – dimensions and properties

PFC designation

Mass Depth Width Thickness Per of of metre section section

D kg/m

mm

B

Root Depth Ratios for local radius between buckling

Web Flange t T r

nd

mm

mm

mm

mm

mm

11

Flange b/t

Web d/t

Second moment of area

h/t better known in BS449 as

Axis x–x

Axis y–y

D/T

4

cm4

cm

430  100  64 64.4

430

100

19

15

362

5.26

32.9

21940

722

380  100  54 54.0

380

100

9.5

17.5

15

315

5.71

33.2

15030

643

23 22

300  100  46 45.5 300  90  41 41.4

300 300

100 90

9 9

16.5 15.5

15 12

237 245

6.06 5.81

26.3 27.2

8229 7218

568 404

18 19

260  90  35 260  75  28

34.8 27.6

260 260

90 75

8 7

14 12

12 12

208 212

6.43 6.25

26 30.3

4728 3619

353 185

19 22

230  90  32 230  75  26

32.2 25.7

230 230

90 75

7.5 6.5

14 12.5

12 12

178 181

6.43 6

23.7 27.8

3518 2748

334 181

16 18

200  90  30 200  75  23

29.7 23.4

200 200

90 75

7 6

14 12.5

12 12

148 151

6.43 6

21.1 25.2

2523 1963

314 170

14 16

180  90  26 180  75  20

26.1 20.3

180 180

90 75

6.5 6

12.5 10.5

12 12

131 135

7.2 7.14

20.2 22.5

1817 1370

277 146

14 17

150  90  24 150  75  18

23.9 17.9

150 150

90 75

6.5 5.5

12 10

12 12

102 106

7.5 7.5

15.7 19.3

1162 861

253 131

13 15

125  65  15

14.8

125

65

5.5

9.5

12

82

6.84

14.9

483

80

13

100  50  10

10.2

100

50

5

8.5

9

65

5.88

13

208

32.3

12

221

Structural Steel

b

r1 s d

h

t

Radius of gyration

Elastic modulus

Elastic NA

Plastic NA

Buckling parameter

Torsional index

Warping constant

Torsional constant

Area of section

Axis x–x

Axis y–y

Axis x–x

Axis y–y

cy

Axis x–x

Axis y–y

kg/m

mm

mm

mm

mm

mm

mm

ceq

u

x

H

J

A

cm4

cm4

16.3

2.97

1020

97.9

2.62

1222

176

0.954

0.917

22.5

0.219

63

82.1

14.8

3.06

791

89.2

2.79

933

161

11.9 11.7

3.13 2.77

549 481

81.7 63.1

3.05 2.6

641 568

148 114

0.904

0.932

21.2

0.15

45.7

68.7

1.31 0.879

0.944 0.934

17 18.4

0.081 0.058

36.8 28.8

58 52.7

10.3 10.1

2.82 2.3

364 278

56.3 34.4

2.74 2.1

425 328

102 62

1.14 0.676

0.942 0.932

17.2 20.5

0.038 0.02

20.6 11.7

44.4 35.1

9.27 9.17

2.86 2.35

306 239

55 34.8

2.92 2.3

355 278

98.9 63.2

1.69 1.03

0.95 0.947

15.1 17.3

0.028 0.015

19.3 11.8

41 32.7

8.16 8.11

2.88 2.39

252 196

53.4 33.8

3.12 2.48

291 227

94.5 60.6

2.24 1.53

0.954 0.956

12.9 14.8

0.02 0.011

18.3 11.1

37.9 29.9

7.4 7.27

2.89 2.38

202 152

47.4 28.8

3.17 2.41

232 176

83.5 51.8

2.36 1.34

0.949 0.946

12.8 15.3

0.014 0.008

13.3 7.34

33.2 25.9

6.18 6.15

2.89 2.4

155 115

44.4 26.6

3.3 2.58

179 132

76.9 47.2

2.66 1.81

0.936 0.946

10.8 13.1

0.009 0.005

11.8 6.1

30.4 22.8

5.07

2.06

77.3

4

1.58

41.5

18.8

2.25

89.9

33.2

1.55

0.942

11.1

0.002

4.72

18.8

1.73

48.9

17.5

1.18

0.942

10

0

2.53

13

9.89

Plastic modulus

cm

Source: Corus Construction (2002). Copyright Corus UK Ltd – reproduced with the kind permission of Corus UK Ltd.

222

y

a

v t x

Rolled steel equal angles – dimensions and properties RSA designation

Mass per metre

DBT

u

Root radius

Toe radius

Distance of centre of gravity

Second moment of area

Radius of gyration

r1

r2

Cx & Cy

Axis x–x, y-y

Axis u–u

Axis v–v

Axis x–x, y–y

Axis u–u

Axis v–v

Axis x–x, y–y

kg/m

mm

mm

cm

cm4

cm4

cm4

cm

cm

cm

cm3

200  200  24 200  200  20 200  200  18 200  200  16

71.3 60.1 54.4 48.7

18 18 18 18

4.8 4.8 4.8 4.8

5.85 5.7 5.62 5.54

3356 2877 2627 2369

5322 4569 4174 3765

1391 1185 1080 973

6.08 6.13 6.15 6.18

7.65 7.72 7.76 7.79

3.91 3.93 3.95 3.96

237 201 183 164

150  150  18 150  150  15 150  150  12 150  150  10

40.2 33.9 27.5 23.1

16 16 16 16

4.8 4.8 4.8 4.8

4.38 4.26 4.14 4.06

1060 909 748 635

1680 1442 1187 1008

440 375 308 262

4.55 4.59 4.62 4.64

5.73 5.78 5.82 5.85

2.93 2.95 2.97 2.98

120  120  15 120  120  12 120  120  10 120  120  8

26.7 21.7 18.3 14.8

13 13 13 13

4.8 4.8 4.8 4.8

3.52 3.41 3.32 3.24

448 371 316 259

710 588 502 411

186 153 130 107

3.63 3.66 3.69 3.71

4.57 4.62 4.64 4.67

100  100  15 100  100  12 100  100  10 100  100  8

21.9 17.9 15.1 12.2

12 12 12 12

4.8 4.8 4.8 4.8

3.02 2.91 2.83 2.75

250 208 178 146

395 330 283 232

105 86.4 73.7 60.5

2.99 3.02 3.05 3.07

90  90  12 90  90  10 90  90  8 90  90  7 90  90  6

16 13.5 10.9 9.6 8.3

11 11 11 11 11

4.8 4.8 4.8 4.8 4.8

2.66 2.58 2.5 2.46 2.41

149 128 105 93.2 81

235 202 167 148 128

62 52.9 43.4 38.6 33.6

þ

11.9 9.7 7.4

11 11 11

4.8 4.8 4.8

2.33 2.25 2.16

87.7 72.4 56

139 115 88.7

36.5 30.1 23.3

þ

80  80  10 80  80  8 80  80  6

c y a

Elastic modulus

mm  mm  mm

þ

u r1

c

t v

r2

x

Area of section

D/T

A cm2

8 10 11 13

90.8 76.6 69.4 62

99.8 84.6 68.9 58

8 10 13 15

51.2 43.2 35 29.5

2.34 2.35 2.37 2.38

52.8 43.1 36.4 29.5

8 10 12 15

34 27.6 23.3 18.8

3.76 3.81 3.84 3.86

1.94 1.95 1.96 1.97

35.8 29.3 24.8 20.2

7 8 10 13

28 22.8 19.2 15.6

2.7 2.73 2.75 2.76 2.76

3.4 3.43 3.46 3.47 3.48

1.75 1.76 1.77 1.77 1.78

23.5 19.9 16.2 14.2 12.3

8 9 11 13 15

20.3 17.2 13.9 12.3 10.6

2.4 2.42 2.44

3.03 3.05 3.07

1.55 1.56 1.58

15.5 12.6 9.6

8 10 13

15.2 12.3 9.4

þ

70  70  10 70  70  8 70  70  6

10.3 8.4 6.4

11 11 11

4.8 4.8 4.8

2.08 2 1.92

57.1 47.4 36.8

90.3 75 58.2

24 19.7 15.4

2.08 2.1 2.12

2.62 2.65 2.67

1.35 1.36 1.37

11.6 9.49 7.24

7 9 12

13.2 10.7 8.2

60  60  10 60  60  8 60  60  6 þ 60  60  5

8.8 7.2 5.5 4.6

11 11 11 11

4.8 4.8 4.8 4.8

1.84 1.76 1.67 1.62

34.7 28.9 22.6 19.2

54.7 45.7 35.7 30.2

14.7 12.1 9.45 8.06

1.76 1.78 1.8 1.8

2.21 2.24 2.26 2.26

1.15 1.15 1.16 1.17

8.33 6.82 5.21 4.37

6 8 10 12

11.2 9.12 7 5.91

þ

50  50  8 50  50  6 50  50  5 þ 50  50  4 þ 50  50  3

5.9 4.6 3.9 3.1 2.4

11 11 11 11 11

4.8 4.8 4.8 4.8 4.8

1.51 1.42 1.37 1.32 1.25

16 12.6 10.7 8.72 6.6

25.3 19.9 16.9 13.7 10.3

6.78 5.28 4.51 3.71 2.88

1.46 1.47 1.48 1.48 1.47

1.83 1.85 1.86 1.85 1.83

0.949 0.954 0.958 0.963 0.968

4.59 3.52 2.95 2.37 1.76

6 8 10 13 17

7.52 5.8 4.91 4 3.07

þ

45  45  6 45  45  5 45  45  4 þ 45  45  3

4.1 3.5 2.8 2.2

11 11 11 11

4.8 4.8 4.8 4.8

1.3 1.25 1.2 1.13

8.95 7.63 6.22 4.71

14.1 12 9.79 7.37

3.76 3.21 2.65 2.05

1.31 1.32 1.31 1.3

1.65 1.65 1.65 1.63

0.851 0.853 0.857 0.86

2.8 2.35 1.88 1.4

8 9 11 15

5.2 4.41 3.6 2.77

þ

40  40  6 40  40  5 40  40  4 þ 40  40  3

3.6 3.1 2.5 1.9

11 11 11 11

4.8 4.8 4.8 4.8

1.18 1.13 1.08 1.01

6.1 5.21 4.25 3.22

9.63 8.22 6.7 5.04

2.57 2.19 1.8 1.4

1.15 1.15 1.15 1.14

1.45 1.45 1.45 1.43

0.747 0.748 0.75 0.752

2.16 1.81 1.45 1.08

7 8 10 13

4.6 3.91 3.2 2.47

þ

2.3 1.9 1.5

11 11 11

4.8 4.8 4.8

0.89 0.84 0.78

2.02 1.65 1.25

3.19 2.61 1.96

0.846 0.691 0.53

0.832 0.829 0.816

1.05 1.04 1.02

0.539 0.536 0.532

0.956 0.764 0.561

6 8 10

2.91 2.4 1.87

1.9 1.6 1.2

11 11 11

4.8 4.8 4.8

0.78 0.73 0.67

1.09 0.894 0.672

1.72 1.42 1.06

0.462 0.372 0.281

0.673 0.668 0.654

0.846 0.841 0.823

0.438 0.431 0.423

0.634 0.504 0.367

5 6 8

2.41 2 1.57

þ þ þ þ þ

þ þ

þ þ

þ þ

30  30  5 30  30  4 30  30  3

þ þ þ

25  25  5 25  25  4 25  25  3

þ þ

+British Standard sections not produced by Corus. Source: Corus Construction (2002). Copyright Corus UK Ltd – reproduced with the kind permission of Corus UK Ltd.

223

224

Structural Engineer’s Pocket Book

Rolled steel unequal angles – dimensions and properties

RSA designation

Mass per metre

DBT mm  mm  mm

Root radius

Toe radius

Distance of centre of gravity

Distance of centre of gravity

Angle x–x to u–u axis

Second moment of area

r1

r2

Cx

Cy

Tan a

Axis x-x

Axis y–y

kg/m

mm

mm

cm

cm

cm4

cm4

200  150  18 200  150  15 200  150  12 200  100  15 200  100  12 200  100  10

47.2 39.7 32.1 33.9 27.4 23.1

15 15 15 15 15 15

4.8 4.8 4.8 4.8 4.8 4.8

6.34 6.22 6.1 7.17 7.04 6.95

3.86 3.75 3.63 2.23 2.11 2.03

0.549 0.551 0.553 0.26 0.263 0.265

2390 2037 1667 1772 1454 1233

1155 989 812 303 251 215

150  90  15 150  90  12 150  90  10

26.7 21.6 18.2

12 12 12

4.8 4.8 4.8

5.21 5.09 5

2.24 2.12 2.04

0.354 0.359 0.361

764 630 536

207 172 147

150  75  15 150  75  12 150  75  10

24.9 20.2 17

11 11 11

4.8 4.8 4.8

5.53 5.41 5.32

1.81 1.7 1.62

0.254 0.259 0.262

715 591 503

120 100 86.3

125  75  12 125  75  10 125  75  8

17.8 15 12.2

11 11 11

4.8 4.8 4.8

4.31 4.23 4.14

1.84 1.76 1.68

0.354 0.358 0.36

355 303 249

96 82.5 68.1

100  75  12 100  75  10 100  75  8

15.4 13 10.6

10 10 10

4.8 4.8 4.8

3.27 3.19 3.1

2.03 1.95 1.87

0.54 0.544 0.547

189 162 133

90.3 77.7 64.2

100  65  10 100  65  8 100  65  7

12.3 10 8.8

10 10 10

4.8 4.8 4.8

3.36 3.28 3.23

1.63 1.56 1.51

0.41 0.414 0.415

154 127 113

51.1 42.3 37.7

8.3 7.3 6.3

8 8 8

4.8 4.8 4.8

2.55 2.5 2.46

1.56 1.52 1.48

0.544 0.545 0.546

65.8 58.5 50.9

31.5 28.1 24.5

7.4 5.7

7 7

2.4 2.4

2.53 2.44

1.29 1.21

0.43 0.436

52.4 40.9

18.6 14.6

6.8 5.2 4.4

6 6 6

2.4 2.4 2.4

2.12 2.04 2

1.37 1.3 1.26

0.569 0.575 0.577

34.9 27.4 23.3

17.8 14.1 12

þ

4 3.4

6 6

2.4 2.4

2.2 2.16

0.72 0.68

0.252 0.256

18.3 15.7

þ

1.9

4

2.4

1.36

0.62

0.38

þ

80  60  8 80  60  7 80  60  6

þ þ þ

75  50  8 75  50  6

þ þ

65  50  8 65  50  6 65  50  5

þ þ þ

60  30  6 60  30  5 40  25  4

3.86

3.05 2.64 1.15

225

Structural Steel

r2 u v

t x

x

R1 t

U

cx t v

cy

Second moment of area

Radius of gyration

Elastic modulus

Area of section

Axis u–u

Axis v–v

Axis x–x

Axis y–y

Axis u–u

Axis v–v

Axis x–x

Axis y–y

cm4

cm4

cm

cm

cm

cm

cm3

cm3

2922 2495 2044 1879 1544 1310

623 531 435 197 162 138

6.3 6.34 6.38 6.41 6.45 6.48

4.38 4.42 4.45 2.65 2.68 2.7

6.97 7.02 7.07 6.6 6.65 6.68

3.22 3.24 3.26 2.13 2.15 2.17

175 148 120 138 112 94.5

104 87.8 71.4 39 31.9 26.9

11 13 17 13 17 20

60.1 50.6 40.9 43.1 34.9 29.4

844 698 595

127 104 89.1

4.74 4.78 4.81

2.47 2.5 2.52

4.99 5.03 5.06

1.93 1.95 1.96

78 63.6 53.6

30.6 25 21.2

10 13 15

34 27.6 23.2

756 626 534

79.2 65.2 55.7

4.75 4.79 4.82

1.95 1.98 2

4.89 4.93 4.96

1.58 1.59 1.6

75.5 61.6 52

21.1 17.3 14.7

10 13 10

31.7 25.7 21.7

392 336 275

58.8 50.2 41.2

3.95 3.98 4

2.06 2.08 2.09

4.16 4.18 4.21

1.61 1.62 1.63

43.4 36.7 29.7

17 14.4 11.7

10 13 16

22.7 19.2 15.5

230 197 163

49.5 42.2 34.7

3.1 3.12 3.14

2.14 2.16 2.18

3.42 3.45 3.48

1.59 1.59 1.61

28.1 23.8 19.3

16.5 14 11.4

8 10 13

19.7 16.6 13.5

175 144 128

30.2 24.9 22.1

3.14 3.17 3.18

1.81 1.83 1.84

3.35 3.38 3.39

1.39 1.4 1.41

23.2 18.9 16.6

10.5 8.56 7.56

10 13 14

15.6 12.7 11.2

80.2 71.4 62.2

17.1 15.2 13.2

2.49 2.5 2.51

1.72 1.73 1.74

2.75 2.76 2.77

1.27 1.27 1.28

12.1 10.6 9.19

7.09 6.26 5.41

10 11 13

10.6 9.35 8.08

60.1 47.1

10.9 8.48

2.36 2.38

1.4 1.42

2.52 2.55

1.07 1.08

10.5 8.1

5 3.86

9 13

9.44 7.22

43.1 34 29

9.62 7.49 6.38

2.01 2.04 2.05

1.44 1.46 1.47

2.24 2.27 2.28

1.06 1.07 1.07

7.97 6.14 5.18

4.92 3.8 3.21

8 11 13

8.61 6.59 5.55

19.3 16.6

2.01 1.72

1.9 1.91

0.774 0.783

1.95 1.96

0.629 0.632

4.82 4.08

1.34 1.14

10 12

5.09 4.3

0.692

1.26

0.685

1.33

0.532

1.46

0.612

10

2.45

4.32

þBritish

D/T

A cm2

Standard sections not produced by Corus.

Source: Corus Construction (2002). Copyright Corus UK Ltd – reproduced with the kind permission of Corus UK Ltd.

226

B Y

D X

X t

Hot finished rectangular hollow sections – dimensions and properties RHS designation

Mass per metre

Area of section

Radius of gyration

Y

Ratios for local buckling

Second moment of area

Elastic modulus

Flange

Web

Axis x–x

Axis y–y

Axis x–x

Torsional constants

b/t

d/t

Ix

Iy

rx

cm4

cm4

cm 1.79 1.77 1.76 1.74 1.72 1.67

1.19 1.17 1.16 1.14 1.13 1.08

4.73 5.43 5.68 6.16 6.6 7.49

3.48 3.96 4.13 4.45 4.72 5.26

5.92 6.88 7.25 7.94 8.59 10

4.11 4.76 5 5.46 5.88 6.8

11.7 13.5 14.2 15.4 16.6 19

5.73 6.51 6.8 7.31 7.77 8.67

Axis y–y

ry

Zx

Zy

Sx

Sy

J

cm

cm3

cm3

cm3

cm3

cm4

m

0.154 0.152 0.152 0.151 0.15 0.147

346 293 277 250 228 189

T

mmmm

mm

50  30 50  30 50  30 50  30 50  30 50  30

2.5 3 3.2 3.6 4 5

2.89 3.41 3.61 4.01 4.39 5.28

3.68 4.34 4.6 5.1 5.59 6.73

9 7 6.37 5.33 4.5 3

17 13.7 12.6 10.9 9.5 7

11.8 13.6 14.2 15.4 16.5 18.7

60  40 60  40 60  40 60  40 60  40 60  40 60  40 60  40

2.5 3 3.2 3.6 4 5 6 6.3

3.68 4.35 4.62 5.14 5.64 6.85 7.99 8.31

4.68 5.54 5.88 6.54 7.19 8.73 10.2 10.6

13 10.3 9.5 8.11 7 5 3.67 3.35

21 17 15.7 13.7 12 9 7 6.52

22.8 26.5 27.8 30.4 32.8 38.1 42.3 43.4

12.1 13.9 14.6 15.9 17 19.5 21.4 21.9

2.21 2.18 2.18 2.16 2.14 2.09 2.04 2.02

1.6 1.58 1.57 1.56 1.54 1.5 1.45 1.44

7.61 8.82 9.27 10.1 10.9 12.7 14.1 14.5

6.03 6.95 7.29 7.93 8.52 9.77 10.7 11

9.32 10.9 11.5 12.7 13.8 16.4 18.6 19.2

7.02 8.19 8.64 9.5 10.3 12.2 13.7 14.2

25.1 29.2 30.8 33.8 36.7 43 48.2 49.5

9.73 11.2 11.7 12.8 13.7 15.7 17.3 17.6

0.194 0.192 0.192 0.191 0.19 0.187 0.185 0.184

272 230 217 195 177 146 125 120

80  40 80  40 80  40 80  40 80  40 80  40 80  40 80  40

3 3.2 3.6 4 5 6 6.3 8

5.29 5.62 6.27 6.9 8.42 9.87 10.3 12.5

6.74 7.16 7.98 8.79 10.7 12.6 13.1 16

10.3 9.5 8.11 7 5 3.67 3.35 2

23.7 22 19.2 17 13 10.3 9.7 7

54.2 57.2 62.8 68.2 80.3 90.5 93.3 106

18 18.9 20.6 22.2 25.7 28.5 29.2 32.1

2.84 2.83 2.81 2.79 2.74 2.68 2.67 2.58

1.63 1.63 1.61 1.59 1.55 1.5 1.49 1.42

13.6 14.3 15.7 17.1 20.1 22.6 23.3 26.5

9 9.46 10.3 11.1 12.9 14.2 14.6 16.1

17.1 18 20 21.8 26.1 30 31.1 36.5

10.4 11 12.1 13.2 15.7 17.8 18.4 21.2

43.8 46.2 50.8 55.2 65.1 73.4 75.6 85.8

15.3 16.1 17.5 18.9 21.9 24.2 24.8 27.4

0.232 0.232 0.231 0.23 0.227 0.225 0.224 0.219

189 178 160 145 119 101 97.2 79.9

5.22 5.94 6.2 6.67 7.08 7.89

Axis x–x

m2/m

DB

cm2

Axis y–y

Approx length per tonne

Thickness

A

Axis x–x

Surface area of section

Size

kg/m

Axis y–y

Plastic modulus

C cm3

76.250.8 76.250.8 76.250.8 76.250.8 76.250.8 76.250.8 76.250.8 76.250.8

3 3.2 3.6 4 5 6 6.3 8

5.62 5.97 6.66 7.34 8.97 10.5 11 13.4

7.16 7.61 8.49 9.35 11.4 13.4 14 17.1

13.9 12.9 11.1 9.7 7.16 5.47 5.06 3.35

22.4 20.8 18.2 16.1 12.2 9.7 9.1 6.53

56.7 59.8 65.8 71.5 84.4 95.6 98.6 113

30 31.6 34.6 37.5 43.9 49.2 50.6 57

2.81 2.8 2.78 2.77 2.72 2.67 2.66 2.57

2.05 2.04 2.02 2 1.96 1.91 1.9 1.83

14.9 15.7 17.3 18.8 22.2 25.1 25.9 29.6

11.8 12.4 13.6 14.8 17.3 19.4 19.9 22.4

18.2 19.2 21.3 23.3 28 32.2 33.4 39.4

13.7 14.5 16 17.5 20.9 23.9 24.8 29

62.1 65.7 72.5 79.1 94.2 108 111 129

19.1 20.1 22 23.8 27.8 31.2 32 36.1

0.246 0.246 0.245 0.244 0.241 0.239 0.238 0.233

178 167 150 136 111 95 91.1 74.6

9050 9050 9050 9050 9050 9050 9050 9050

3 3.2 3.6 4 5 6 6.3 8

6.24 6.63 7.4 8.15 9.99 11.8 12.3 15

7.94 8.44 9.42 10.4 12.7 15 15.6 19.2

13.7 12.6 10.9 9.5 7 5.33 4.94 3.25

27 25.1 22 19.5 15 12 11.3 8.25

84.4 89.1 98.3 107 127 145 150 174

33.5 35.3 38.7 41.9 49.2 55.4 57 64.6

3.26 3.25 3.23 3.21 3.16 3.11 3.1 3.01

2.05 2.04 2.03 2.01 1.97 1.92 1.91 1.84

18.8 19.8 21.8 23.8 28.3 32.2 33.3 38.6

13.4 14.1 15.5 16.8 19.7 22.1 22.8 25.8

23.2 24.6 27.2 29.8 36 41.6 43.2 51.4

15.3 16.2 18 19.6 23.5 27 28 32.9

76.5 80.9 89.4 97.5 116 133 138 160

22.4 23.6 25.9 28 32.9 37 38.1 43.2

0.272 0.272 0.271 0.27 0.267 0.265 0.264 0.259

160 151 135 123 100 85.1 81.5 66.5

10050 10050 10050 10050 10050 10050 10050 10050

3 3.2 3.6 4 5 6 6.3 8

6.71 7.13 7.96 8.78 10.8 12.7 13.3 16.3

8.54 9.08 10.1 11.2 13.7 16.2 16.9 20.8

13.7 12.6 10.9 9.5 7 5.33 4.94 3.25

30.3 28.3 24.8 22 17 13.7 12.9 9.5

110 116 128 140 167 190 197 230

36.8 38.8 42.6 46.2 54.3 61.2 63 71.7

3.58 3.57 3.55 3.53 3.48 3.43 3.42 3.33

2.08 2.07 2.05 2.03 1.99 1.95 1.93 1.86

21.9 23.2 25.6 27.9 33.3 38.1 39.4 46

14.7 15.5 17 18.5 21.7 24.5 25.2 28.7

27.3 28.9 32.1 35.2 42.6 49.4 51.3 61.4

16.8 17.7 19.6 21.5 25.8 29.7 30.8 36.3

88.4 93.4 103 113 135 154 160 186

25 26.4 29 31.4 36.9 41.6 42.9 48.9

0.292 0.292 0.291 0.29 0.287 0.285 0.284 0.279

149 140 126 114 92.8 78.8 75.4 61.4

10060 10060 10060 10060 10060 10060 10060 10060

3 3.2 3.6 4 5 6 6.3 8

7.18 7.63 8.53 9.41 11.6 13.6 14.2 17.5

9.14 9.72 10.9 12 14.7 17.4 18.1 22.4

17 15.7 13.7 12 9 7 6.52 4.5

30.3 28.3 24.8 22 17 13.7 12.9 9.5

124 131 145 158 189 217 225 264

55.7 58.8 64.8 70.5 83.6 95 98.1 113

3.68 3.67 3.65 3.63 3.58 3.53 3.52 3.44

2.47 2.46 2.44 2.43 2.38 2.34 2.33 2.25

24.7 26.2 28.9 31.6 37.8 43.4 45 52.8

18.6 19.6 21.6 23.5 27.9 31.7 32.7 37.8

30.2 32 35.6 39.1 47.4 55.1 57.3 68.7

21.2 22.4 24.9 27.3 32.9 38.1 39.5 47.1

121 129 142 156 188 216 224 265

30.7 32.4 35.6 38.7 45.9 52.1 53.8 62.2

0.312 0.312 0.311 0.31 0.307 0.305 0.304 0.299

139 131 117 106 86.5 73.3 70.2 57

12060 12060 12060 12060 12060 12060

3.6 4 5 6 6.3 8

9.66 10.7 13.1 15.5 16.2 20.1

12.3 13.6 16.7 19.8 20.7 25.6

13.7 12 9 7 6.52 4.5

30.3 27 21 17 16 12

227 249 299 345 358 425

76.3 83.1 98.8 113 116 135

4.3 4.28 4.23 4.18 4.16 4.08

2.49 2.47 2.43 2.39 2.37 2.3

37.9 41.5 49.9 57.5 59.7 70.8

25.4 27.7 32.9 37.5 38.8 45

47.2 51.9 63.1 73.6 76.7 92.7

28.9 31.7 38.4 44.5 46.3 55.4

183 201 242 279 290 344

43.3 47.1 56 63.8 65.9 76.6

0.351 0.35 0.347 0.345 0.344 0.339

104 93.7 76.1 64.4 61.6 49.9

227

Source: Corus Construction (2002). Copyright Corus UK Ltd – reproduced with the kind permission of Corus UK Ltd.

228

B Y

D X

X t

Hot finished rectangular hollow sections – dimensions and properties RHS designation

Size

Thickness

DB

T

mm  mm

mm

Mass per metre

Radius of gyration

Y

Area of section

Ratios for local buckling

Second moment of area

Elastic modulus

Plastic modulus

Torsional constants

Flange

Web

A

b/t

d/t

Axis x–x Ix

Axis y–y Iy

Axis x–x rx

Axis y–y ry

Axis x–x Zx

Axis y–y Zy

Axis x–x Sx

Axis y–y Sy

J

cm4

cm4

cm

cm

cm3

cm3

cm3

cm3

cm4

cm2

8 10

10.8 11.9 14.7 17.4 18.2 22.6 27.4

13.7 15.2 18.7 22.2 23.2 28.8 34.9

19.2 17 13 10.3 9.7 7 5

30.3 27 21 17 16 12 9

276 303 365 423 440 525 609

147 161 193 222 230 273 313

4.48 4.46 4.42 4.37 4.36 4.27 4.18

3.27 3.25 3.21 3.17 3.15 3.08 2.99

46 50.4 60.9 70.6 73.3 87.5 102

36.7 40.2 48.2 55.6 57.6 68.1 78.1

55.6 61.2 74.6 87.3 91 111 131

42 46.1 56.1 65.5 68.2 82.6 97.3

301 330 401 468 487 587 688

150  100 150  100 150  100 150  100 150  100 150  100 150  100 150  100

4 5 6 6.3 8 10 12 12.5

15.1 18.6 22.1 23.1 28.9 35.3 41.4 42.8

19.2 23.7 28.2 29.5 36.8 44.9 52.7 54.6

22 17 13.7 12.9 9.5 7 5.33 5

34.5 27 22 20.8 15.8 12 9.5 9

607 739 862 898 1087 1282 1450 1488

324 392 456 474 569 665 745 763

5.63 5.58 5.53 5.52 5.44 5.34 5.25 5.22

4.11 4.07 4.02 4.01 3.94 3.85 3.76 3.74

81 98.5 115 120 145 171 193 198

64.8 78.5 91.2 94.8 114 133 149 153

97.4 119 141 147 180 216 249 256

73.6 90.1 106 110 135 161 185 190

160  80 160  80 160  80 160  80 160  80 160  80 160  80 160  80

4 5 6 6.3 8 10 12 12.5

14.4 17.8 21.2 22.2 27.6 33.7 39.5 40.9

18.4 22.7 27 28.2 35.2 42.9 50.3 52.1

17 13 10.3 9.7 7 5 3.67 3.4

37 29 23.7 22.4 17 13 10.3 9.8

612 744 868 903 1091 1284 1449 1485

207 249 288 299 356 411 455 465

5.77 5.72 5.67 5.66 5.57 5.47 5.37 5.34

3.35 3.31 3.27 3.26 3.18 3.1 3.01 2.99

76.5 93 108 113 136 161 181 186

51.7 62.3 72 74.8 89 103 114 116

94.7 116 136 142 175 209 240 247

58.3 71.1 83.3 86.8 106 125 142 146

3.6 4 5 6 6.3

Approx length per tonne

C

kg/m

12080 120  80 120  80 120  80 120  80 120  80 120  80

Surface area of section

cm3

m2/m

m

59.5 65 77.9 89.6 92.9 110 126

0.391 0.39 0.387 0.385 0.384 0.379 0.374

92.7 83.9 68 57.5 54.9 44.3 36.5

660 807 946 986 1203 1432 1633 1679

105 127 147 153 183 214 240 246

0.49 0.487 0.485 0.484 0.479 0.474 0.469 0.468

66.4 53.7 45.2 43.2 34.7 28.4 24.2 23.3

493 600 701 730 883 1041 1175 1204

88.1 106 122 127 151 175 194 198

0.47 0.467 0.465 0.464 0.459 0.454 0.449 0.448

69.3 56 47.2 45.1 36.2 29.7 25.3 24.5

5 6 6.3 8 10 12 12.5

22.6 26.8 28.1 35.1 43.1 50.8 52.7

28.7 34.2 35.8 44.8 54.9 64.7 67.1

17 13.7 12.9 9.5 7 5.33 5

37 30.3 28.7 22 17 13.7 13

1495 1754 1829 2234 2664 3047 3136

505 589 613 739 869 979 1004

7.21 7.16 7.15 7.06 6.96 6.86 6.84

4.19 4.15 4.14 4.06 3.90 3.89 3.87

149 175 183 223 266 305 314

101 118 123 148 174 196 201

185 218 228 282 341 395 408

114 134 140 172 206 237 245

1204 1414 1475 1804 2156 2469 2541

172 200 208 251 295 333 341

0.587 0.585 0.584 0.579 0.574 0.569 0.568

44.3 37.3 35.6 28.5 23.2 19.7 19

250  150 250  150 250  150 250  150 250  150 250  150 250  150 250  150

5 6 6.3 8 10 12 12.5 16

30.4 36.2 38 47.7 58.8 69.6 72.3 90.3

38.7 46.2 48.4 60.8 74.9 88.7 92.1 115

27 22 20.8 15.8 12 9.5 9 6.38

47 38.7 36.7 28.3 22 17.8 17 12.6

3360 3965 4143 5111 6174 7154 7387 8879

1527 1796 1874 2298 2755 3168 3265 3873

9.31 9.27 9.25 9.17 9.08 8.98 8.96 8.79

6.28 6.24 6.22 6.15 6.06 5.98 5.96 5.8

269 317 331 409 494 572 591 710

204 239 250 306 367 422 435 516

324 385 402 501 611 715 740 906

228 270 283 350 426 497 514 625

3278 3877 4054 5021 6090 7088 7326 8868

337 396 413 506 605 695 717 849

0.787 0.785 0.784 0.779 0.774 0.769 0.768 0.759

32.9 27.6 26.3 21 17 14.4 13.8 11.1

300  200 300  200 300  200 300  200 300  200 300  200 300  200 300  200

5 6 6.3 8 10 12 12.5 16

38.3 45.7 47.9 60.3 74.5 88.5 91.9 115

48.7 58.2 61 76.8 94.9 113 117 147

37 30.3 28.7 22 17 13.7 13 9.5

57 47 44.6 34.5 27 22 21 15.8

6322 7486 7829 9717 11820 13800 14270 17390

3396 4013 4193 5184 6278 7294 7537 9109

11.4 11.3 11.3 11.3 11.2 11.1 11 10.9

8.35 8.31 8.29 8.22 8.13 8.05 8.02 7.87

421 499 522 648 788 920 952 1159

340 401 419 518 628 729 754 911

501 596 624 779 956 1124 1165 1441

380 451 472 589 721 847 877 1080

6824 8100 8476 10560 12910 15140 15680 19250

552 651 681 840 1015 1178 1217 1468

0.987 0.985 0.984 0.979 0.974 0.969 0.968 0.959

26.1 21.9 20.9 16.6 13.4 11.3 10.9 8.67

400  200 400  200 400  200 400  200 400  200 400  200 400  200

6 6.3 8 10 12 12.5 16

55.1 57.8 72.8 90.2 107 112 141

70.2 73.6 92.8 115 137 142 179

30.3 28.7 22 17 13.7 13 9.5

63.7 60.5 47 37 30.3 29 22

15000 15700 19560 23910 28060 29060 35740

5142 5376 6660 8084 9418 9738 11820

14.6 14.6 14.5 14.4 14.3 14.3 14.1

8.56 8.55 8.47 8.39 8.3 8.28 8.13

750 785 978 1196 1403 1453 1787

514 538 666 808 942 974 1182

917 960 1203 1480 1748 1813 2256

568 594 743 911 1072 1111 1374

12050 12610 15730 19260 22620 23440 28870

877 917 1135 1376 1602 1656 2010

1.18 1.18 1.18 1.17 1.17 1.17 1.16

18.2 17.3 13.7 11.1 9.32 8.97 7.12

450  250 450  250 450  250 450  250 450  250

8 10 12 12.5 16

85.4 106 126 131 166

109 135 161 167 211

28.3 22 17.8 17 12.6

53.3 42 34.5 33 25.1

30080 36890 43430 45030 55710

12140 14820 17360 17970 22040

16.6 16.5 16.4 16.4 16.2

10.6 10.5 10.4 10.4 10.2

1337 1640 1930 2001 2476

971 1185 1389 1438 1763

1622 2000 2367 2458 3070

1081 1331 1572 1631 2029

27080 33280 39260 40720 50550

1629 1986 2324 2406 2947

1.38 1.37 1.37 1.37 1.36

11.7 9.44 7.93 7.62 6.04

500  300 500  300 500  300 500  300 500  300

8 10 12 12.5 16

97.9 122 145 151 191

125 155 185 192 243

34.5 27 22 21 15.8

59.5 47 38.7 37 28.3

43730 53760 63450 65810 81780

19950 24440 28740 29780 36770

18.7 18.6 18.5 18.5 18.3

12.6 12.6 12.5 12.5 12.3

1749 2150 2538 2633 3271

1330 1629 1916 1985 2451

2100 2595 3077 3196 4005

1480 1826 2161 2244 2804

42560 52450 62040 64390 80330

2203 2696 3167 3281 4044

1.58 1.57 1.57 1.57 1.56

10.2 8.22 6.9 6.63 5.24

Source: Corus Construction (2002). Copyright Corus UK Ltd – reproduced with the kind permission of Corus UK Ltd.

229

200  100 200  100 200  100 200  100 200  100 200  100 200  100

230

Structural Engineer’s Pocket Book

D Y

D X

X t Y

Hot finished square hollow sections – dimensions and properties Mass Area of Ratios for per section local buckling metre

SHS designation

Size DB mm  mm

Thickness T mm kg/m

A cm2

Flange b/t

Second Radius moment of of area gyration

Web d/t I cm4

Elastic Plastic modulus modulus

Torsional constants

r cm

Z cm3

J cm4

S cm3

C cm3

m2/m

m

0.154 0.152 0.152 0.151 0.15 0.147

346 293 277 250 228 189

40  40 40  40 40  40 40  40 40  40 40  40

2.5 3 3.2 3.6 4 5

2.89 3.41 3.61 4.01 4.39 5.28

3.68 4.34 4.6 5.1 5.59 6.73

13 10.3 9.5 8.11 7 5

13 10.3 9.5 8.11 7 5

8.54 9.78 40.2 11.1 11.8 13.4

1.52 1.5 1.49 1.47 1.45 1.41

4.27 4.89 5.11 5.54 5.91 6.68

5.14 5.97 6.28 6.88 7.44 8.66

13.6 15.7 16.5 18.1 19.5 22.5

50  50 50  50 50  50 50  50 50  50 50  50 50  50 50  50

2.5 3 3.2 3.6 4 5 6 6.3

3.68 4.35 4.62 5.14 5.64 6.85 7.99 8.31

4.68 5.54 5.88 6.54 7.19 8.73 40.2 10.6

17 13.7 12.6 10.9 9.5 7 5.33 4.94

17 13.7 12.6 10.9 9.5 7 5.33 4.94

17.5 20.2 21.2 23.2 25 28.9 32 32.8

1.93 1.91 1.9 1.88 1.86 1.82 1.77 1.76

6.99 8.08 8.49 9.27 9.99 11.6 12.8 13.1

8.29 9.7 10.2 11.3 12.3 14.5 16.5 17

27.5 32.1 33.8 37.2 40.4 47.6 53.6 55.2

10.2 11.8 12.4 13.5 14.5 16.7 18.4 18.8

0.194 0.192 0.192 0.191 0.19 0.187 0.185 0.184

272 230 217 195 177 146 125 120

60  60 60  60 60  60 60  60 60  60 60  60 60  60 60  60

3 3.2 3.6 4 5 6 6.3 8

5.29 5.62 6.27 6.9 8.42 9.87 10.3 12.5

6.74 7.16 7.98 8.79 10.7 12.6 13.1 16

17 15.7 13.7 12 9 7 6.52 4.5

17 15.7 13.7 12 9 7 6.52 4.5

36.2 38.2 41.9 45.4 53.3 59.9 61.6 69.7

2.32 2.31 2.29 2.27 2.23 2.18 2.17 2.09

12.1 12.7 14 15.1 17.8 20 20.5 23.2

14.3 15.2 16.8 18.3 21.9 25.1 26 30.4

56.9 60.2 66.5 72.5 86.4 98.6 102 118

17.7 18.6 20.4 22 25.7 28.8 29.6 33.4

0.232 0.232 0.231 0.23 0.227 0.225 0.224 0.219

189 178 160 145 119 101 97.2 79.9

70  70 70  70 70  70 70  70 70  70 70  70 70  70 70  70

3 3.2 3.6 4 5 6 6.3 8

6.24 6.63 7.4 8.15 9.99 11.8 12.3 15

7.94 8.44 9.42 10.4 12.7 15 15.6 19.2

20.3 18.9 16.4 14.5 11 8.67 8.11 5.75

20.3 18.9 16.4 14.5 11 8.67 8.11 5.75

59 62.3 68.6 74.7 88.5 401 104 120

2.73 2.72 2.7 2.68 2.64 2.59 2.58 2.5

16.9 17.8 19.6 21.3 25.3 28.7 29.7 34.2

19.9 21 23.3 25.5 30.8 35.5 36.9 43.8

92.2 97.6 108 118 142 163 169 200

24.8 26.1 28.7 31.2 36.8 41.6 42.9 49.2

0.272 0.272 0.271 0.27 0.267 0.265 0.264 0.259

160 151 135 123 400 85.1 81.5 66.5

80  80 80  80 80  80 80  80 80  80 80  80 80  80

3.2 3.6 4 5 6 6.3 8

7.63 8.53 9.41 11.6 13.6 14.2 17.5

9.72 10.9 12 14.7 17.4 18.1 22.4

22 19.2 17 13 10.3 9.7 7

22 19.2 17 13 10.3 9.7 7

95 105 114 137 156 162 189

3.13 3.11 3.09 3.05 3 2.99 2.91

23.7 26.2 28.6 34.2 39.1 40.5 47.3

27.9 31 34 41.1 47.8 49.7 59.5

148 164 180 217 252 262 312

34.9 38.5 41.9 49.8 56.8 58.7 68.3

0.312 0.311 0.31 0.307 0.305 0.304 0.299

131 117 406 86.5 73.3 70.2 57

90  90 90  90 90  90 90  90 90  90 90  90

3.6 4 5 6 6.3 8

9.66 10.7 13.1 15.5 16.2 20.1

12.3 13.6 16.7 19.8 20.7 25.6

22 19.5 15 12 11.3 8.25

22 19.5 15 12 11.3 8.25

152 166 200 230 238 281

3.52 3.5 3.45 3.41 3.4 3.32

33.8 37 44.4 51.1 53 62.6

39.7 43.6 53 61.8 64.3 77.6

237 260 316 367 382 459

49.7 54.2 64.8 74.3 77 90.5

0.351 0.35 0.347 0.345 0.344 0.339

104 93.7 76.1 64.4 61.6 49.9

10.8 11.9 14.7 17.4 18.2 22.6 27.4

13.7 15.2 18.7 22.2 23.2 28.8 34.9

24.8 22 17 13.7 12.9 9.5 7

24.8 22 17 13.7 12.9 9.5 7

212 232 279 323 336 400 462

3.92 3.91 3.86 3.82 3.8 3.73 3.64

42.3 46.4 55.9 64.6 67.1 79.9 92.4

49.5 54.4 66.4 77.6 80.9 98.2 116

328 361 439 513 534 646 761

62.3 68.2 81.8 94.3 97.8 116 133

0.391 0.39 0.387 0.385 0.384 0.379 0.374

92.7 83.9 68 57.5 54.9 44.3 36.5

100  100 3.6 100  100 4 100  100 5 100  100 6 100  100 6.3 100  100 8 100  100 10

6.22 7.1 7.42 8.01 8.54 9.6

Surface Approx area of length section per tonne

231

Structural Steel

D Y

D X

X t Y

SHS designation

Size DB mm  mm

Mass Area of Ratios for Second Radius Elastic Plastic Torsional per section local buckling moment of modulus modulus constants metre of area gyration

Thickness T A mm kg/m cm2

Flange b/t

Web d/t

I cm4

Surface Approx area of length section per tonne

r cm

Z cm3

S cm3

J cm4

C cm3 m2/m

m

120  120 4 120  120 5 120  120 6 120  120 6.3 120  120 8 120  120 10 120  120 12 120  120 12.5

14.4 47.8 21.2 22.2 27.6 33.7 39.5 40.9

18.4 22.7 27 28.2 35.2 42.9 50.3 52.1

27 21 17 16 12 9 7 6.6

27 21 17 16 12 9 7 6.6

410 498 579 603 726 852 958 982

4.72 4.68 4.63 4.62 4.55 4.46 4.36 4.34

68.4 83 96.6 100 121 142 160 164

79.7 97.6 115 120 146 175 201 207

635 777 911 950 1160 1382 1578 1623

101 122 141 147 176 206 230 236

0.47 0.467 0.465 0.464 0.459 0.454 0.449 0.448

69.3 56 47.2 45.1 36.2 29.7 25.3 24.5

140  140 5 140  140 6 140  140 6.3 140  140 8 140  140 10 140  140 12 140  140 12.5

21 24.9 26.1 32.6 40 47 48.7

26.7 31.8 33.3 41.6 50.9 59.9 62.1

25 20.3 19.2 14.5 11 8.67 8.2

25 20.3 19.2 14.5 11 8.67 8.2

807 944 984 1195 1416 1609 1653

5.5 5.45 5.44 5.36 5.27 5.18 5.16

115 135 141 171 202 230 236

135 159 166 204 246 284 293

1253 1475 1540 1892 2272 2616 2696

170 198 206 249 294 333 342

0.547 0.545 0.544 0.539 0.534 0.529 0.528

47.7 40.1 38.3 30.7 25 21.3 20.5

150  150 5 150  150 6 150  150 6.3 150  150 8 150  150 10 150  150 12 150  150 12.5

22.6 26.8 28.1 35.1 43.1 50.8 52.7

28.7 34.2 35.8 44.8 54.9 64.7 67.1

27 22 20.8 15.8 12 9.5 9

27 22 20.8 15.8 12 9.5 9

1002 1174 1223 1491 1773 2023 2080

5.9 5.86 5.85 5.77 5.68 5.59 5.57

134 156 163 199 236 270 277

156 184 192 237 286 331 342

1550 1828 1909 2351 2832 3272 3375

197 230 240 291 344 391 402

0.587 0.585 0.584 0.579 0.574 0.569 0.568

44.3 37.3 35.6 28.5 23.2 19.7 19

160  160 160  160 160  160 160  160 160  160 160  160 160  160 160  160

5 6 6.3 8 10 12 12.5 16

24.1 28.7 30.1 37.6 46.3 54.6 56.6 70.2

30.7 36.6 38.3 48 58.9 69.5 72.1 89.4

29 23.7 22.4 17 13 10.3 9.8 7

29 23.7 22.4 17 13 10.3 9.8 7

1225 1437 1499 1831 2186 2502 2576 3028

6.31 6.27 6.26 6.18 6.09 6 5.98 5.82

153 180 187 229 273 313 322 379

178 210 220 272 329 382 395 476

1892 2233 2333 2880 3478 4028 4158 4988

226 264 275 335 398 454 467 546

0.627 0.625 0.624 0.619 0.614 0.609 0.608 0.599

41.5 34.8 33.3 26.6 21.6 18.3 17.7 14.2

180  180 180  180 180  180 180  180 180  180 180  180 180  180 180180

5 6 6.3 8 10 12 12.5 16

27.3 32.5 34 42.7 52.5 62.1 64.4 80.2

34.7 41.4 43.3 54.4 66.9 79.1 82.1 102

33 27 25.6 19.5 15 12 11.4 8.25

33 27 25.6 19.5 15 12 11.4 8.25

1765 2077 2168 2661 3193 3677 3790 4504

7.13 7.09 7.07 7 6.91 6.82 6.8 6.64

196 231 241 296 355 409 421 500

227 269 281 349 424 494 511 621

2718 3215 3361 4162 5048 5873 6070 7343

290 340 355 434 518 595 613 724

0.707 0.705 0.704 0.699 0.694 0.689 0.688 0.679

36.7 30.8 29.4 23.4 19 16.1 15.5 12.5

232

Structural Engineer’s Pocket Book

D Y

D X

X t Y

Hot finished square hollow sections – dimensions and properties – continued SHS designation

Mass Area of Ratios for Second Radius Elastic Plastic Torsional per section local buckling moment of modulus modulus constants metre of area gyration

Size DB mm  mm

Thickness T A mm kg/m cm2

Flange b/t

Web d/t

200  200 200  200 200  200 200  200 200  200 200  200 200  200 200  200

5 6 6.3 8 10 12 12.5 16

30.4 38.7 36.2 46.2 38 48.4 47.7 60.8 58.8 74.9 69.6 88.7 72.3 92.1 90.3 115

37 30.3 28.7 22 17 13.7 13 9.5

37 30.3 28.7 22 17 13.7 13 9.5

250  250 250  250 250  250 250  250 250  250 250  250 250  250 250  250

5 6 6.3 8 10 12 12.5 16

38.3 48.7 45.7 58.2 47.9 61 60.3 76.8 74.5 94.9 88.5 113 91.9 117 115 147

47 38.7 36.7 28.3 22 17.8 17 12.6

300  300 300  300 300  300 300  300 300  300 300  300 300  300

6 6.3 8 10 12 12.5 16

55.1 57.8 72.8 90.2 107 112 141

70.2 73.6 92.8 115 137 142 179

350  350 350  350 350  350 350  350 350  350

8 10 12 12.5 16

85.4 106 126 131 166

400  400 400  400 400  400 400  400 400  400

8 10 12 12.5 16

97.9 122 145 151 191

I cm4

Surface Approx area of length section per tonne

r cm

Z cm3

S cm3

J cm4

C cm3 m2/m

m

2445 2883 3011 3709 4471 5171 5336 6394

7.95 7.9 7.89 7.81 7.72 7.64 7.61 7.46

245 288 301 371 447 517 534 639

283 335 350 436 531 621 643 785

3756 4449 4653 5778 7031 8208 8491 10340

362 426 444 545 655 754 778 927

0.787 0.785 0.784 0.779 0.774 0.769 0.768 0.759

32.9 27.6 26.3 21 17 14.4 13.8 11.1

47 38.7 36.7 28.3 22 17.8 17 12.6

4861 5752 6014 7455 9055 10560 10920 13270

9.99 9.94 9.93 9.86 9.77 9.68 9.66 9.5

389 460 481 596 724 844 873 1061

447 531 556 694 851 1000 1037 1280

7430 8825 9238 11530 14110 16570 17160 21140

577 681 712 880 1065 1237 1279 1546

0.987 0.985 0.984 0.979 0.974 0.969 0.968 0.959

46.1 21.9 20.9 16.6 13.4 11.3 10.9 8.67

47 44.6 34.5 27 22 21 15.8

47 44.6 34.5 27 22 21 15.8

10080 10550 13130 16030 18780 19440 23850

12 12 11.9 11.8 11.7 11.7 11.5

672 703 875 1068 1252 1296 1590

772 809 1013 1246 1470 1525 1895

15410 16140 20190 24810 29250 30330 37620

997 1043 1294 1575 1840 1904 2325

1.18 1.18 1.18 1.17 1.17 1.17 1.16

18.2 17.3 13.7 11.1 9.32 8.97 7.12

109 135 161 167 211

40.8 32 26.2 25 18.9

40.8 32 26.2 25 18.9

21130 25880 30430 31540 38940

13.9 13.9 13.8 13.7 13.6

1207 1479 1739 1802 2225

1392 1715 2030 2107 2630

32380 39890 47150 48930 60990

1789 2185 2563 2654 3264

1.38 1.37 1.37 1.37 1.36

11.7 9.44 7.93 7.62 6.04

125 155 185 192 243

47 37 30.3 29 22

47 37 30.3 29 22

31860 39130 46130 47840 59340

16 15.9 15.8 15.8 15.6

1593 1956 2306 2392 2967

1830 2260 2679 2782 3484

48690 60090 71180 73910 92440

2363 2895 3405 3530 4362

1.58 1.57 1.57 1.57 1.56

10.2 8.22 6.9 6.63 5.24

Source: Corus Construction (2002). Copyright Corus UK Ltd – reproduced with the kind permission of Corus UK Ltd.

233

Structural Steel

D Y

X

X Y

t

Hot finished circular hollow sections – dimensions and properties CHS designation

Mass Area of Ratio for Second Radius of Elastic Plastic Torsional per section local moment gyration modulus modulus constants metre buckling of area

Outside Thickness diameter t D

A kg/m

D/t

cm2

mm

mm

21.3 26.9

3.2 3.2

1.43 1.87

1.82 2.38

6.7 8.4

33.7 33.7 33.7 33.7

3 3.2 3.6 4

2.27 2.41 2.67 2.93

2.89 3.07 3.4 3.73

42.4 42.4 42.4 42.4

3 3.2 3.6 4

2.91 3.09 3.44 3.79

48.3 48.3 48.3 48.3 48.3 48.3

2.5 3 3.2 3.6 4 5

60.3 60.3 60.3 60.3 60.3 60.3

2.5 3 3.2 3.6 4 5

76.1 76.1 76.1 76.1 76.1 76.1 76.1 76.1

I

r

Z

S

J

cm4

cm

cm3

cm3

cm4

Surface Approx area of length Per section tonne

C cm3

m2/m

m

0.768 0.65 1.7 0.846

0.722 1.27

1.06 1.81

1.54 3.41

1.44 0.067 2.53 0.085

700 535

11.2 10.5 9.4 8.4

3.44 3.6 3.91 4.19

1.09 1.08 1.07 1.06

2.04 2.14 2.32 2.49

2.84 2.99 3.28 3.55

6.88 7.21 7.82 8.38

4.08 4.28 4.64 4.97

0.106 0.106 0.106 0.106

440 415 374 341

3.71 3.94 4.39 4.83

14.1 13.3 11.8 10.6

7.25 7.62 8.33 8.99

1.4 1.39 1.38 1.36

3.42 3.59 3.93 4.24

4.67 4.93 5.44 5.92

14.5 15.2 16.7 18

6.84 7.19 7.86 8.48

0.133 0.133 0.133 0.133

343 323 290 264

2.82 3.35 3.56 3.97 4.37 5.34

3.6 4.27 4.53 5.06 5.57 6.8

19.3 16.1 15.1 13.4 12.1 9.7

9.46 11 11.6 12.7 13.8 16.2

1.62 1.61 1.6 1.59 1.57 1.54

3.92 4.55 4.8 5.26 5.7 6.69

5.25 6.17 6.52 7.21 7.87 9.42

18.9 22 23.2 25.4 27.5 32.3

7.83 9.11 9.59 10.5 11.4 13.4

0.152 0.152 0.152 0.152 0.152 0.152

354 298 281 252 229 187

3.56 4.24 4.51 5.03 5.55 6.82

4.54 5.4 5.74 6.41 7.07 8.69

24.1 20.1 18.8 16.8 15.1 12.1

19 22.2 23.5 25.9 28.2 33.5

2.05 2.03 2.02 2.01 2 1.96

6.3 7.37 7.78 8.58 9.34 11.1

8.36 9.86 10.4 11.6 12.7 15.3

38 44.4 46.9 51.7 56.3 67

12.6 14.7 15.6 17.2 18.7 22.2

0.189 0.189 0.189 0.189 0.189 0.189

281 236 222 199 180 147

2.52 3 3.2 3.6 4 5 6 6.3

4.54 5.78 5.41 6.89 5.75 7.33 6.44 8.2 7.11 9.06 8.77 11.2 10.4 13.2 10.8 13.8

30.4 25.4 23.8 21.1 19 15.2 12.7 12.1

39.2 46.1 48.8 54 59.1 70.9 81.8 84.8

2.6 2.59 2.58 2.57 2.55 2.52 2.49 2.48

10.3 12.1 12.8 14.2 15.5 18.6 21.5 22.3

13.5 16 17 18.9 20.8 25.3 29.6 30.8

78.4 92.2 97.6 108 118 142 164 170

20.6 24.2 25.6 28.4 31 37.3 43 44.6

0.239 0.239 0.239 0.239 0.239 0.239 0.239 0.239

220 185 174 155 141 114 96.4 92.2

88.9 88.9 88.9 88.9 88.9 88.9 88.9 88.9

2.5 3 3.2 3.6 4 5 6 6.3

5.33 6.36 6.76 7.57 8.38 10.3 12.3 12.8

6.79 8.1 8.62 9.65 10.7 13.2 15.6 16.3

35.6 29.6 27.8 24.7 22.2 17.8 14.8 14.1

63.4 74.8 79.2 87.9 96.3 116 135 140

3.06 3.04 3.03 3.02 3 2.97 2.94 2.93

14.3 16.8 17.8 19.8 21.7 26.2 30.4 31.5

18.7 22.1 23.5 26.2 28.9 35.2 41.3 43.1

127 150 158 176 193 233 270 280

28.5 33.6 35.6 39.5 43.3 52.4 60.7 63.1

0.279 0.279 0.279 0.279 0.279 0.279 0.279 0.279

188 157 148 132 119 96.7 81.5 77.9

114.3 114.3 114.3 114.3 114.3 114.3 114.3

3 3.2 3.6 4 5 6 6.3

8.23 8.77 9.83 10.9 13.5 16 16.8

10.5 11.2 12.5 13.9 17.2 20.4 21.4

38.1 35.7 31.8 28.6 22.9 19.1 18.1

163 172 192 211 257 300 313

3.94 3.93 3.92 3.9 3.87 3.83 3.82

28.4 30.2 33.6 36.9 45 52.5 54.7

37.2 39.5 44.1 48.7 59.8 70.4 73.6

325 345 384 422 514 600 625

56.9 60.4 67.2 73.9 89.9 105 109

0.359 0.359 0.359 0.359 0.359 0.359 0.359

121 114 102 91.9 74.2 62.4 59.6

234

Structural Engineer’s Pocket Book

D Y

X

X Y

t

Hot finished circular hollow sections – dimensions and properties – continued DHS designation

Mass Area of Ratio for Second Radius of Elastic Plastic Torsional per section local moment gyration modulus modulus constants metre buckling of area

Outside Thickness diameter D t

A

D/t

Surface Approx area of length per section tonne

I

r

Z

S

J

cm4

cm

cm3

cm3

cm4 cm3

m2/m

m

320 357 393 481 564 589 720 862

4.83 4.81 4.8 4.77 4.73 4.72 4.66 4.6

45.8 51.1 56.2 68.8 80.8 84.3 103 123

59.6 66.7 73.7 90.8 107 112 139 169

640 713 786 961 1129 1177 1441 1724

91.6 102 112 138 162 169 206 247

0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439

92.8 82.8 74.7 60.2 50.5 48.2 38.5 31.3

52.6 46.8 42.1 33.7 28.1 26.7 21 16.8 14 13.5

566 632 697 856 1009 1053 1297 1564 1810 1868

5.84 5.82 5.81 5.78 5.74 5.73 5.67 5.61 5.54 5.53

67.2 75.1 82.8 102 120 125 154 186 215 222

87.2 97.7 108 133 158 165 206 251 294 304

1131 1264 1394 1712 2017 2107 2595 3128 3620 3737

134 150 166 203 240 250 308 372 430 444

0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529

76.8 68.4 61.7 49.7 41.6 39.7 31.6 25.6 21.6 20.8

29.6 35.4 37.1 46.7 57.7 68.5 71.2

38.7 32.3 30.7 24.2 19.4 16.1 15.5

1320 1560 1630 2016 2442 2839 2934

6.67 6.64 6.63 6.57 6.5 6.44 4.42

136 161 168 208 252 293 303

178 211 221 276 338 397 411

2640 3119 3260 4031 4883 5678 5869

273 322 337 416 504 586 606

0.609 0.609 0.609 0.609 0.609 0.609 0.609

43 36 34.3 27.3 22.1 18.6 17.9

26.4 31.5 33.1 41.6 51.6 61.3 63.7 80.1

33.6 40.2 42.1 53.1 65.7 78.1 81.1 102

43.8 36.5 34.8 27.4 21.9 18.3 17.5 13.7

1928 2282 2386 2960 3598 4200 4345 5297

7.57 7.54 7.53 7.47 7.4 7.33 7.32 7.2

176 208 218 270 328 383 397 483

229 273 285 357 438 515 534 661

3856 4564 4772 5919 7197 8400 8689 10590

352 417 436 540 657 767 793 967

0.688 0.688 0.688 0.688 0.688 0.688 0.688 0.688

37.9 31.7 30.2 24 19.4 16.3 15.7 12.5

29.5 35.3 37 46.7 57.8 68.8 71.5 90.2

37.6 45 47.1 59.4 73.7 87.7 91.1 115

48.9 40.8 38.8 30.6 24.5 20.4 19.6 15.3

2699 3199 3346 4160 5073 5938 6147 7533

8.47 8.43 8.42 8.37 8.3 8.23 8.21 8.1

221 262 274 340 415 486 503 616

287 341 358 448 550 649 673 837

5397 6397 6692 8321 10150 11880 12290 15070

441 523 547 681 830 972 1006 1232

0.768 0.768 0.768 0.768 0.768 0.768 0.768 0.768

33.9 28.3 27 21.4 17.3 14.5 14 11.1

mm

mm

kg/m cm2

139.7 139.7 139.7 139.7 139.7 139.7 139.7 139.7

3.2 3.6 4 5 6 6.3 8 10

10.8 12.1 13.4 16.6 19.8 20.7 26 32

13.7 15.4 17.1 21.2 25.2 26.4 33.1 40.7

43.7 38.8 34.9 27.9 23.3 22.2 17.5 14

168.3 168.3 168.3 168.3 168.3 168.3 168.3 168.3 168.3 168.3

3.2 3.6 4 5 6 6.3 8 10 12 12.5

13 14.6 16.2 20.1 24 25.2 31.6 39 46.3 48

16.6 18.6 20.6 25.7 30.6 32.1 40.3 49.7 58.9 61.2

193.7 193.7 193.7 193.7 193.7 193.7 193.7

5 6 6.3 8 10 12 12.5

23.3 27.8 29.1 36.6 45.3 53.8 55.9

219.1 219.1 219.1 219.1 219.1 219.1 219.1 219.1

5 6 6.3 8 10 12 12.5 16

244.5 244.5 244.5 244.5 244.5 244.5 244.5 244.5

5 6 6.3 8 10 12 12.5 16

C

235

Structural Steel

D Y

X

X Y

DHS designation

Mass Area of Ratio for Second Radius of Elastic Plastic Torsional per section local moment gyration modulus modulus constants metre buckling of area

Outside Thickness diameter D t

A

D/t

t

Surface Approx area of length per section tonne

I

r

Z

S

J

cm4

cm

cm3

cm3

cm4 cm3

m2/m

m

9.48 9.44 9.43 9.37 9.31 9.24 9.22 9.1

277 329 344 429 524 615 637 784

359 428 448 562 692 818 849 1058

7562 8974 9392 11700 14310 16790 17390 21410

554 657 688 857 1048 1230 1274 1569

0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858

40.3 25.3 24.1 19.1 15.4 12.9 12.5 9.86

11.3 11.2 11.2 11.2 11.1 11 11 10.9

393 468 490 612 751 884 917 1136

509 606 636 799 986 1168 1213 1518

12740 15140 15860 19820 24320 28640 29690 36780

787 935 979 1224 1501 1768 1833 2271

1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02

25.4 21.3 20.3 16 12.9 10.8 10.4 8.23

10550 13200 16220 19140 19850 24660

12.4 12.3 12.2 12.2 12.1 12

593 742 912 1076 1117 1387

769 967 1195 1417 1472 1847

21090 26400 32450 38280 39700 49330

1186 1485 1825 2153 2233 2774

1.12 1.12 1.12 1.12 1.12 1.12

18.4 14.6 11.7 9.83 9.45 7.46

64.5 50.8 40.6 33.9 32.5 25.4

15850 19870 24480 28940 30030 37450

14.1 14.1 14 14 13.9 13.8

780 978 1205 1424 1478 1843

1009 1270 1572 1867 1940 2440

31700 39750 48950 57870 60060 74900

1560 1956 2409 2848 2956 3686

1.28 1.28 1.28 1.28 1.28 1.28

16.1 12.7 10.2 8.57 8.24 6.49

89.2 113 140 168 175 222

72.5 57.1 45.7 38.1 36.6 28.6

22650 28450 35090 41560 43140 53960

15.9 15.9 15.8 15.7 15.7 15.6

991 1245 1536 1819 1888 2361

1280 1613 1998 2377 2470 3113

45310 56890 70180 83110 86290 107900

1983 2490 3071 3637 3776 4723

1.44 1.44 1.44 1.44 1.44 1.44

14.3 11.3 9.07 7.59 7.3 5.75

99.3 126 156 187 195 247

80.6 63.5 50.8 42.3 40.6 31.8

31250 39280 48520 57540 59760 74910

17.7 17.7 17.6 17.5 17.5 17.4

1230 1546 1910 2265 2353 2949

1586 2000 2480 2953 3070 3874

62490 78560 97040 115100 119500 149800

2460 3093 3820 4530 4705 5898

1.6 1.6 1.6 1.6 1.6 1.6

12.8 10.1 8.14 6.81 6.55 5.15

kg/m cm2

mm

mm

273 273 273 273 273 273 273 273

5 6 6.3 8 10 12 12.5 16

33 39.5 41.4 52.3 64.9 77.2 80.3 101

42.1 50.3 52.8 66.6 82.6 98.4 102 129

54.6 45.5 43.3 34.1 27.3 22.8 21.8 17.1

3781 4487 4696 5852 7154 8396 8697 10710

323.9 323.9 323.9 323.9 323.9 323.9 323.9 323.9

5 6 6.3 8 10 12 12.5 16

39.3 47 49.3 62.3 77.4 92.3 96 121

50.1 59.9 62.9 79.4 98.6 118 122 155

64.8 54 51.4 40.5 32.4 27 25.9 20.2

6369 7572 7929 9910 12160 14320 14850 18390

355.6 355.6 355.6 355.6 355.6 355.6

6.3 8 10 12 12.5 16

54.3 68.6 85.2 102 106 134

69.1 87.4 109 130 135 171

56.4 44.5 35.6 29.6 28.4 22.2

406.4 406.4 406.4 406.4 406.4 406.4

6.3 8 10 12 12.5 16

62.2 78.6 97.8 117 121 154

79.2 100 125 149 155 196

457 457 457 457 457 457

6.3 8 10 12 12.5 16

70 88.6 110 132 137 174

508 508 508 508 508 508

6.3 8 10 12 12.5 16

77.9 98.6 123 147 153 194

C

Source: Corus Construction (2002). Copyright Corus UK Ltd – reproduced with the kind permission of Corus UK Ltd.

236

Structural Engineer’s Pocket Book

Mild steel rounds typically available Bar Weight diameter kg/m mm

Bar Weight Bar Weight Bar Weight diameter kg/m diameter kg/m diameter kg/m mm mm mm

6 8 10 12

16 20 25 32

0.22 0.39 0.62 0.89

1.58 2.47 3.85 6.31

40 45 50 60

9.86 12.5 15.4 22.2

65 75 90 100

26.0 34.7 49.9 61.6

Mild steel square bars typically available Bar size mm

Weight kg/m

Bar size mm

Weight kg/m

Bar size mm

Weight kg/m

8 10 12.5 16 20

0.50 0.79 1.22 2.01 3.14

25 30 32 40 45

4.91 7.07 8.04 12.60 15.90

50 60 75 90 100

19.60 28.30 44.20 63.60 78.50

Structural Steel

237

Mild steel flats typically available Bar

Weight Bar

Weight Bar

Weight Bar

Weight Bar

Weight

size mm

kg/m

size mm

kg/m

kg/m

size mm

kg/m

size mm

kg/m

13  3 13  6 16  3 20  3 20  5 20  6

0.307 0.611 0.378 0.466 0.785 0.940

45  6 45  8 45  10 45  12 45  15 45  20

100  15 100  20 100  25 100  30 100  40 100  50

11.80 15.70 19.60 23.60 31.40 39.30

160  10 160  12 160  15 160  20 180  6 180  10

12.60 15.10 18.80 25.20 8.50 14.14

120  10 120  12 120  15 120  20 120  25 130  6 130  8

9.42 11.30 14.10 18.80 23.60 6.10 8.16

20  10 25  3 25  5 25  6 25  8 25  10

25  12 30  3 30  5 30  6 30  8 30  10 30  12

30  20 35  6 35  10 35  12 35  20 40  3

40  5 40  6 40  8 40  10 40  12 40  15 40  20

40  25 40  30 45  3

1.570 0.589 0.981 1.18 1.570 1.960 2.360 0.707 1.180 1.410 1.880 2.360 2.830 4.710 1.650 2.750 3.300 5.500 0.942 1.570 1.880 2.510 3.140 3.770 4.710 6.280 7.850 9.420 1.060

45  25 50  3 50  5 50  6 50  8 50  10

50  12 50  15 50  20 50  25 50  30 50  40 55  10

2.120 2.830 3.530 4.240 5.295 7.070 8.830 1.180 1.960 2.360 3.140 3.93 4.71 5.89 7.85 9.81 11.80 15.70 4.56

60  8 60  10 60  12 60  15 60  20 60  25

3.77 4.71 5.65 7.07 9.42 11.80

65  20 65  25 65  30

10.20 12.80 15.30

60  30 65  5 65  6 65  8 65  10 65  12 65  15

14.14 2.55 3.06 4.05 5.10 6.12 7.65

size mm

65  40 20.40 70  8 4.40 70  10 5.50 70  12 6.59 70  20 11.0 70  25 13.70

75  6 3.54 75  8 4.71 75  10 5.90 75  12 7.07 75  15 8.84 75  20 11.78

75  25 14.72 75  30 17.68 80  6 3.77 80  8 5.02 80  10 6.28 80  12 7.54 80  15 9.42

80  20 80  25 80  30 80  40 80  50 90  6

12.60 15.70 18.80 25.10 31.40 4.24

100  8 100  10 100  12

6.28 7.85 9.42

90  10 7.07 90  12 8.48 90  15 10.60 90  20 14.10 90  25 17.70 100  5 3.93 100  6 4.71

110  6 5.18 110  10 8.64 110  12 10.40 110  20 17.30 110  50 43.20 120  6 5.65

130  10 130  12 130  15 130  20 130  25 140  6

140  10 140  12 140  20 150  6 150  8 150  10 150  12

10.20 12.20 15.30 20.40 25.50 6.60 11.00 13.20 22.00 7.06 9.42 11.80 14.10

150  15 17.70 150  20 23.60 150  25 29.40

180  12 180  15 180  20 180  25 200  6 200  10

200  12 200 x 15 200  20 200  25 200  30 220  10 220  15

220  20 220  25 250  10 250  12 250  15 250  20

250  25 250  40 250  50 280  12.5 300  10 300  12 300  15

300  20 300  25 300  40

17.00 21.20 28.30 35.30 9.90 15.70 18.80 23.60 31.40 39.20 47.20 17.25 25.87 34.50 43.20 19.60 23.60 29.40 39.20 49.10 78.40 98.10 27.48 23.55 28.30 35.30 47.10 58.80 94.20

238

Structural Engineer’s Pocket Book

Hot rolled mild steel plates typically available Thick-

Weight

ness

Thick-

Weight

ness

mm

kg/m2

mm

3

23.55

10

3.2

25.12

12.5

4

31.40

5

39.25

6 8

Thick-

Weight

ness kg/m2

Thick-

Weight

ness

Thick-

Weight

ness

mm

kg/m2

mm

kg/m2

78.50

30

235.50

55

431.75

90

706.50

98.12

32

251.20

60

471.00

100

785.00

15

117.75

35

274.75

65

510.25

110

863.50

20

157.00

40

314.00

70

549.50

120

942.00

47.10

22.5

176.62

45

353.25

75

588.75

130

1050.50

62.80

25

196.25

50

392.50

80

628.00

150

1177.50

mm

kg/m2

Durbar mild steel floor plates typically available Basic size mm

Weight kg/m2

Basic size mm

Weight kg/m2

2500  1250  3 3000  1500  3

26.19

3000  1500  8 3700  1830  8 4000  1750  8 6100  1830  8

65.44

2000  1000  4.5 2500  1250  4.5 3000  1250  4.5 3700  1830  4.5 4000  1750  4.5

37.97

2000  1000  10 2500  1250  10 3000  1500  10 3700  1830  10

81.14

2000  1000  6 2500  1250  6 3000  1500  6 3700  1830  6 4000  1750  6

49.74

2000  1000  12.5 2500  1250  12.5 3000  1500  12.5 3700  1830  12.5 4000  1750  12.5

2000  1000  8 2500  1250  8

65.44

The depth of pattern ranges from 1.9 to 2.4 mm.

100.77

Structural Steel

239

Slenderness Slenderness and elastic buckling The slenderness () of a structural element indicates how much load the element can carry in compression. Short stocky elements have low values of slenderness and are likely to fail by crushing, while elements with high slenderness values will fail by elastic (reversible) buckling. Slender columns will buckle when the axial compression reaches the critical load. Slender beams will buckle when the compressive stress causes the compression flange to buckle and twist sideways. This is called Lateral Torsional Buckling and it can be avoided (and the load capacity of the beam increased) by restraining the compression flange at intervals or over its full length. Full lateral restraint can be assumed if the construction fixed to the compression flange is capable of resisting a force of not less than 2.5% of the maximum force in that flange distributed uniformly along its length.

Slenderness limits Slenderness,  ¼ Le =r where Le is the effective length and r is the radius of gyration – generally about the weaker axis. For robustness, members should be selected so that their slenderness does not exceed the following limits: Members resisting load other than wind Members resisting self-weight and wind only Members normally acting as ties but subject to load reversal due to wind

  180

  250

  350

240

Structural Engineer’s Pocket Book

Effective length for different restraint conditions Effective length of beams – end restraint Conditions of restraint at the ends of the beams

Effective length Normal loading

Destabilizing loading

Compression flange laterally restrained; beam fully restrained against torsion (rotation about the longitudinal axis)

Both flanges fully restrained against rotation on plan

0.70L

0.85L

Compression flange fully restrained against rotation on plan

0.75L

0.90L

Both flanges partially restrained against rotation on plan

0.80L

0.95L

Compression flange partially restrained against rotation on plan

0.85L

1.00L

Both flanges free to rotate on plan

1.00L

1.20L

Partial torsional restraint against rotation about the longitudinal axis provided by connection of bottom flange to supports

1.0L þ 2D

1.2L þ 2D

1.2L þ 2D Partial torsional restraint against rotation about the longitudinal axis provided only by the pressure of the bottom flange bearing onto the supports

1.4L þ 2D

Compression flange laterally unrestrained; both flanges free to rotate on plan

NOTE: The illustrated connections are not the only methods of providing the restraints noted in the table.

Source: BS 5950: Part 1: 2000.

Structural Steel

241

Effective length of cantilevers Conditions of restraint

Effective length

Support

Cantilever tip

Continuous with lateral restraint to top flange

Free

3.0L

7.5L

Top flange laterally restrained

2.7L

7.5L

Torsional restraint

2.4L

4.5L

Lateral and torsional restraint

2.1L

3.6L

Free

2.0L

5.0L

Top flange laterally restrained

1.8L

5.0L

Torsional restraint

1.6L

3.0L

Lateral and torsional restraint

1.4L

2.4L 2.5L

Continuous with partial torsional restraint

Continuous with lateral and torsional restraint

Restrained laterally, torsionally and against rotation on plan

Normal loading

Destabilizing loading

Free

1.0L

Top flange laterally restrained

0.9L

2.5L

Torsional restraint

0.8L

1.5L

Lateral and torsional restraint

0.7L

1.2L

Free

0.8L

1.4L

Top flange laterally restrained

0.7L

1.4L

Torsional restraint

0.6L

0.6L

Lateral and torsional restraint

0.5L

0.5L

Cantilever tip restraint conditions Free

Top flange laterally restrained

Torsional Restraint

Lateral and torsional restraint

Source: BS 5950: Part 1: 2000.

Effective length of braced columns – restraint provided by cross bracing or shear wall Conditions of restraint at the ends of the columns

Effective length

Effectively held in position at both ends

0.70L 0.85L 0.85L 1.00L

Effectively restrained in direction at both ends Partially restrained in direction at both ends Restrained in direction at one end Not restrained in direction at either end

Effective length of unbraced columns – restraint provided by sway of columns Conditions of restraint at the ends of the columns

Effective length

Effectively held in position and restrained in direction at one end

1.20L 1.50L 2.00L

Other end effectively restrained in direction Other end partially restrained in direction Other end not restrained in direction

Source: BS 5950: Part 1: 2000.

242

Structural Engineer’s Pocket Book

Durability and fire resistance Corrosion mechanism and protection 4Fe þ 3O2 þ 2H2O ¼ 2Fe2O3 . H2O

Iron/Steel þ Oxygen þ Water ¼ Rust

For corrosion of steel to take place, oxygen and water must both be present. The corrosion rate is affected by the atmospheric pollution and the length of time the steelwork remains wet. Sulphates (typically from industrial pollution) and chlorides (typically in marine environments – coastal is considered to be a 2 km strip around the coast in the UK) can accelerate the corrosion rate. All corrosion occurs at the anode ( ve where electrons are lost) and the products of corrosion are deposited at the cathode (þve where the electrons are gained). Both anodic and cathodic areas can be present on a steel surface. The following factors should be considered in relation to the durability of a structure: the environment, degree of exposure, shape of the members, structural detailing, protective measures and whether inspection and maintenance are possible. Bi-metallic corrosion should also be considered in damp conditions.

Durability exposure conditions Corrosive environments are classified by BS EN ISO 12944: Part 2 and ISO 9223, and the corrosivity of the environment must be assessed for each project. Corrosivity category and risk

Examples of typical environments in a temperate climate* Exterior

Interior

C1 – Very low

–

Heated buildings with clean atmospheres, e.g. offices, shops, schools, hotels, etc. (theoretically no protection is needed)

C2 – Low

Atmospheres with low levels of pollution. Mostly rural areas

Unheated buildings where condensation may occur, e.g. depots and sports halls

C3 – Medium

Urban and industrial atmospheres with moderate sulphur dioxide pollution. Coastal areas with low salinity

Production rooms with high humidity and some air pollution, e.g. food processing plants, laundries, breweries, dairies, etc.

C4 – High

Industrial areas and coastal areas with moderate salinity

Chemical plants, swimming pools, coastal ship and boatyards

C5I – Very high (industrial)

Industrial areas with high humidity and aggressive atmosphere

Buildings or areas with almost permanent condensation and high pollution

C5M – Very high (marine)

Coastal and offshore areas with high salinity

Buildings or areas with almost permanent condensation and high pollution

* A hot and humid climate increases the corrosion rate and steel will require additional protection than in a temperate climate.

BS EN ISO 12944: Part 3 gives advice on steelwork detailing to avoid crevices where moisture and dirt can be caught and accelerate corrosion. Some acidic timbers should be isolated from steelwork. Get advice for each project: Corus can give advice on all steelwork coatings. The Galvanizers’ Association, Metal Sprayers Association and paint manufacturers also give advice on system specifications.

Structural Steel

243

Methods of corrosion protection A corrosion protection system should consist of good surface preparation and application of a suitable coating with the required durability and minimum cost.

Mild steel surface preparation to BS EN ISO 8501 Hot rolled structural steelwork (in mild steel) leaves the last rolling process at about 1000 C. As it cools, its surface reacts with the air to form a blue-grey coating called mill scale, which is unstable, will allow rusting of the steel and will cause problems with the adhesion of protective coatings. The steel must be degreased to ensure that any contaminants which might affect the coatings are removed. The mill scale can then be removed by abrasive blast cleaning. Typical blast cleaning surface grades are: Sa 1 Sa 2 Sa 21/2 Sa 3

Light blast cleaning Thorough blast cleaning Very thorough blast cleaning Blast cleaning to visually clean steel

Sa 21/2 is used for most structural steel. Sa 3 is often used for surface preparation for metal spray coatings. Metallic and non-metallic particles can be used to blast clean the steel surface. Chilled angular metallic grit (usually grade G24) provides a rougher surface than round metallic shot, so that the coatings have better adhesion to the steel surface. Acid pickling is often used after blast cleaning to Sa 21/2 to remove final traces of mill scale before galvanizing. Coatings must be applied very quickly after the surface preparation to avoid rust reforming and the requirement for reblasting.

Paint coatings for structural steel Paint provides a barrier coating to prevent corrosion and is made up of pigment (for colour and protection), binder (for formation of the coating film) and solvent (to allow application of the paint before it evaporates and the paint hardens). When first applied, the paint forms a wet film thickness which can be measured and the dry film thickness (DFT – which is normally the specified element) can be predicted when the percentage volume of solids in the paint is known. Primers are normally classified on their protective pigment (e.g. zinc phosphate primer). Intermediate (which build the coating thickness) and finish coats are usually classified on their binders (e.g. epoxies, vinyls, urethanes, etc.). Shop primers (with a DFT of 15–25 mm) can be applied before fabrication but these only provide a couple of weeks’ worth of protection. Zinc rich primers generally perform best. Application of paint can be by brush, roller, air spray and airless spray – the latter is the most common in the UK. Application can be done on site or in the shop and where the steel is to be exposed, the method of application should be chosen for practicality and the surface finish. Shop applied coatings tend to need touching up on site if they are damaged in transit.

244

Structural Engineer’s Pocket Book

Metallic coatings for structural steel De-greased, blast cleaned (generally Sa 21/2) and then acid pickled steel is dipped into a flux agent and then into a bath of molten zinc. The zinc reacts with the surface of the steel, forming alloys and as the steel is lifted out a layer of pure zinc is deposited on outer surface of the alloys. The zinc coating is chemically bonded to the steel and is sacrificial. The Galvanizers’ Association can provide details of galvanizing baths around the country, but the average bath size is about 10 m long  1.2 m wide  2 m deep. The largest baths available in 2002 in the UK are 21 m  1.5 m  2.4 m and 7.6 m  2.1 m  3 m. The heat can cause distortions in fabricated, asymmetric or welded elements. Galvanizing is typically 85–140 mm thick and should be carried out to BS EN ISO 1461 and 14713. Paint coatings can be applied on top of the galvanizing for aesthetic or durability reasons and an etch primer is normally required to ensure that the paint properly adheres to the galvanizing.

Hot dip galvanizing

Thermal spray

Degreased and blast cleaned (generally Sa 3) steel is sprayed with molten particles of aluminium or zinc. The coating is particulate and the pores normally need to be sealed with an organic sealant in order to prevent rust staining. Metal sprayed coatings are mechanically bonded to the steel and work partly by anodic protection and partly by barrier protection. There are no limits on the size of elements which can be coated and there are no distortion problems. Thermal spray is typically 150–200 mm thick in aluminium, 100–150 mm thick in zinc and should be carried out to BS EN 22063 and BS EN ISO 14713. Paint coatings can be applied for aesthetic or durability reasons. Bi-metallic corrosion issues should be considered when selecting fixings for aluminium sprayed elements in damp or external environments.

Weathering steel Weathering steels are high strength, low alloy, weldable structural steels which form a protective rust coating in air that reaches a critical level within 2–5 years and prevents further corrosion. Cor-ten is the Corus proprietary brand of weathering steel, which has material properties comparable to S355, but the relevant material standard is BS EN 10 155. To optimize the use of weathering steel, avoid contact with absorbent surfaces (e.g. concrete), prolonged wetting (e.g. north faces of buildings in the UK), burial in soils, contact with dissimilar metals and exposure to aggressive environments. Even if these conditions are met, rust staining can still affect adjacent materials during the first few years. Weathering bolts (ASTM A325, Type 3 or Cor-ten X) must be used for bolted connections. Standard black bolts should not be used as the zinc coating will be quickly consumed and the fastener corroded. Normal welding techniques can be used.

Stainless steel Stainless steel is the most corrosion resistant of all the steels due to the presence of chromium in its alloys. The surface of the steel forms a self-healing invisible oxide layer which prevents ongoing corrosion and so the surface must be kept clean and exposed to provide the oxygen required to maintain the corrosion resistance. Stainless steel is resistant to most things, but special precautions should be taken in chlorinated environments. Alloying elements are added in different percentages to alter the durability properties: SS 304

18% Cr, 10% Ni

Used for general cladding, brick support angles, etc.

SS 409

11% Cr

Sometimes used for lintels

SS 316

17% Cr, 12% Ni, 2.5% Mo

Used in medium marine/aggressive environments

SS Duplex 2205

22% Cr, 5.5% Ni, 3% Mo

Used in extreme marine and industrial environments

Structural Steel

245

Summary of methods of fire protection System

Typical thickness2 for 60 mins protection

Advantages

Disadvantages

Boards Up to 4 hours’ protection. Most popular system in the UK

25–30 mm

Clean ‘boxed in’ appearance; dry application; factory quality boards; needs no steel surface preparation

High cost; complex fitting around details; slow to apply

Vermiculite concrete spray Up to 4 hours’ protection. Second most popular system in the UK

20 mm

Cheap; easy on complex junctions; needs no steel surface preparation; often boards used on columns, with spray on the beams

Poor appearance; messy application needs screening; the wet trade will affect following trades; compatibility with corrosion protection needs to be checked

Intumescent paint Maximum 2 hours’ protection. Charring starts at 200–250 C

1–4 mm1

Good aesthetic; shows off form of steel; easy to cover complex details; can be applied in shop or on site

High cost; not suited to all environments; short periods of resistance; soft, thick, easily damaged coatings; difficult to get a really high quality finish; compatibility with corrosion protection needs to be checked

Flexible blanket Cheap alternative to sprays

20–30 mm

Low cost; dry fixing

Not good aesthetics

Concrete encasement Generally only used when durability is a requirement

25–50 mm

Provides resistance to abrasion, impact, corrosion and weather exposure

Expensive; time consuming; heavy; large thickness required

Concrete filled columns Used for up to 2 hoursprotection or to reduce intumescent paint thickness on hollow sections

–

Takes up less plan area; acts as permanent shutter; good durability

No data for CHS posts; minimum section size which can be protected 140  140SHS; expensive

Water filled columns Columns interconnected to allow convection cooling. Only used if no other option

–

Long periods of fire resistance

Expensive; lots of maintenance required to control water purity and chemical content

Block filled column webs Up to 30 minutes protection

–

Reduced cost; less plan area; good durability

Limited protection times; not advised for steel in partition walls

NOTES: 1. Coating thickness specified on the basis of the sections’ dimensions and the number of sides that will be exposed to fire. 2. Castellated beams need about 20% more fire protection than is calculated for the basic parent material.

246

Structural Engineer’s Pocket Book

Preliminary sizing of steel elements Typical span/depth ratios Element

Typical span (L) m

Beam depth

Primary beams/trusses (heavy point loads) Secondary beams/trusses (distributed loads) Transfer beams/trusses carrying floors Castellated beams Plate girders Vierendeel girders

4–12 4–20 6–30 4–12 10–30 6–18

L/10–15 L/15–25 L/10 L/10–15 L/10–12 L/8–10

Parallel chord roof trusses Pitched roof trusses Light roof beams Conventional lattice roof girders Space frames (allow for l/250 pre-camber)

10–100 8–20 6–60 5–20 10–100

L/12–20 L/5–10 L/18–30 L/12–15 L/15–30

Hot rolled universal column

single storey 2–8 multi-storey 2–4 single storey 2–8 multi-storey 2–4 4–10 9–60

L/20–25 L/7–18 L/20–35 L/7– 28 L/20–25 L/35–40

Hollow section column Lattice column Portal leg and rafter (haunch depth <0.11)

Preliminary sizing Beams There are no shortcuts. Deflection will tend to govern long spans, while shear will govern short spans with heavy loading. Plate girders or trusses are used when the loading is beyond the capacity of rolled sections.

Columns – typical maximum column section size for braced frames 203 UC

Buildings 2 to 3 storeys high and spans up to 7 m.

254 UC

Buildings up to 5 storeys high.

305 UC

Buildings up to 8 storeys high or supports for low rise buildings with long spans.

354 UC

Buildings from 8 to 12 storeys high.

Columns – enhanced loads for preliminary axial design An enhanced axial load for columns subject to out of balance loads can be used for preliminary design: Top storey: Intermediate storey:

Total axial load þ 4Y Total axial load þ 2Y

Y þ 2X X YþX X

Where X X and Y Y are the net axial load differences in each direction.

Trusses with parallel chord Axial force in chord, F ¼ Mapplied =d where d is the distance between the chord centroids.  P Ac d2=4 where Ac is the area of each chord.

Itruss ¼

For equal chords this can be simplified to Itruss ¼ Ac d2=2:

Structural Steel

247

Portal frames The Institution of Structural Engineers’ Grey Book for steel design gives the following preliminary method for sizing plastic portal frames with the following assumptions: . Plastic hinges are formed at the eaves (in the stanchion) and near the apex, therefore . . . .

Class 1 sections as defined in BS 5950 should be used. Moment at the end of the haunch is 0.87Mp. Wind loading does not control the design. Stability of the frame should be checked separately. Load, W ¼ vertical rafter load per metre run.

r

h

L

Horizontal base reaction, H ¼ HFRWL

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

0.20

2.0

1.5

1.0

Span/eaves height (L/h)

Rise/span (r/L)

0.15

0.10

0.05

0 0.06

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.76

HFR Horizontal force factor for stanchion base

Design moment for rafter, Mp rafter ¼ MPRWL2 Also consider the high axial force which will be in the rafter and design for combined axial and bending!

248

Structural Engineer’s Pocket Book

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

10.0 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.7 5.5 5.0

Span/eaves height (L/h) 0.20

Rise/span (r /L)

0.15

0.10

0.05

0 0.015

0.020

0.025

0.030

0.035

0.040

0.045

MPR rafter ratio

Design moment for stanchion, Mp

2 stanchion ¼ MPLWL

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5

10.0

Span/eaves height (L /h) 0.20

Rise/span (r /L)

0.15

0.10

0.05

0 0.03

0.04

0.05

0.06

MPL stanchion ratio

Source: IStructE (2002).

0.07

0.08

Structural Steel

249

Steel design to BS 5950 BS 5950: Part 1 was written to allow designers to reduce conservatism in steel design. The resulting choice and complication of the available design methods has meant that sections are mainly designed using software or the SCI Blue Book. As the code is very detailed, the information about BS 5950 has been significantly summarized – covering only grade S275 steelwork and using the code’s conservative design methods.

Partial safety factors Load combination

Load type Dead

Imposed

Dead and imposed

1.4 or 1.0

1.6

Dead and wind

1.4 or 1.0

–

Dead and wind and imposed

1.2

1.2

Dead and crane loads

1.4

–

Dead and imposed and crane loads

1.2

Crane V ¼ 1.4

Wind

Crane loads

Earth and water pressures

–

–

1.4

1.4

–

–

1.2

–

–

–

V ¼ 1.6 H ¼ 1.6 V and H ¼ 1.4 V ¼ 1.4 H ¼ 1.2 V and H ¼ 1.4

–

Crane H ¼ 1.2 Dead and wind and crane loads

1.2

–

1.2

1.2

–

Forces due to temperature change

–

1.2

–

–

–

Exceptional snow load due to drifting

–

1.05

–

–

–

Source: BS 5950: Part 1: 2000.

250

Structural Engineer’s Pocket Book

Selected mild steel design strengths Steel grade

Steel thickness less than or equal to mm

Design strength, py N/mm2

S275

16 40 63

275 265 255

S355

16 40 63

355 345 335

Generally it is more economic to use S275 where it is required in small quantities (less than 40 tonnes), where deflection instead of strength limits design, or for members such as nominal ties where the extra strength is not required. In other cases it is more economical to consider S355.

Ductility and steel grading In addition to the strength of the material, steel must be specified for a suitable ductility to avoid brittle fracture, which is controlled by the minimum service temperature, the thickness of steel, the steel grade, the type of detail and the stress and strain levels. Ductility is measured by the Charpy V notch test. In the UK the minimum service temperature expected to occur over the design life of the structure should be taken as 5 C for internal steelwork or 15 C for external steelwork. For steelwork in cold stores or cold climates appropriate lower temperatures should be selected. Tables 4, 5, 6 and 7 in BS 5950 give the detailed method for selection of the appropriate steel grade. Steel grading has become more important now that the UK construction industry is using more imported steel. The latest British Standard has revised the notation used to describe the grades of steel. The equivalent grades are set out below:

Current grading references BS 5950: Part 1: 2000 and BS EN 100 25: 1993 Grade

Charpy test temperature C

Steel use

Superseded grading references* BS 5950: Part 1: 1990 and BS 4360: 1990 Max steel thickness mm

Grade

Charpy test temperature C

Steel use

Max steel thickness mm <100 >100 N/mm2 N/mm2

S275

Untested

Internal only

25

43 A

Untested

S275 JR Room temp. 20 C

Internal only

30

43 B

Room temp. 20 C

S275 J0 0 C

Internal External

65 54

43 C

0 C

Internal External

94 78

43 D

S275 J2

20 C

20 C

Internal

50

25

External

30

15

Internal

50

25

External

30

15

Internal External

n/a 80

60 40

Internal External

n/a n/a

n/a 90

* Where the superseded equivalent for grades S355 and S460 are Grades 50 and 55 respectively.

Source: BS 5950: Part 1: 2000.

Structural Steel

251

Section classification and local buckling Sections are classified by BS 5950 depending on how their cross section behaves under compressive load. Structural sections in thinner plate will tend to buckle locally and this reduces the overall compressive strength of the section and means that the section cannot achieve its full plastic moment capacity. Sections with tall webs tend to be slender under axial compression, while cross sections with wide out-stand flanges tend to be slender in bending. Combined bending and compression can change the classification of a cross section to slender, when that cross section might not be slender under either bending or compression when applied independently. For plastic design, the designer must therefore establish the classification of a section (for the given loading conditions) in order to select the appropriate design method from those available in BS 5950. For calculations without capacity tables or computer packages, this can mean many design iterations. BS 5950 has four types of section classification: Class 1: Class 2: Class 3: Class 4:

Plastic Cross sections with plastic hinge rotation capacity. Compact Cross sections with plastic moment capacity. Semi-compact Cross sections in which the stress at the extreme compression fibre can reach the design strength, but the plastic moment capacity cannot be developed. Slender Cross sections in which it is necessary to make explicit allowance for the effects of local buckling.

Tables 11 and 12 in BS 5950 classify different hot rolled and fabricated sections based on the limiting width to thickness ratios for each section class. None of the UB, UC, RSJ or PFC sections are slender in pure bending. Under pure axial compression, none of the UC, RSJ or PFC sections are slender, but some UB and hollow sections can be: UB SHS amd RHS (hot rolled) CHS Where D ¼ overall depth, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ¼ 275=Py .

Slender if d/t > 40e Slender if d/t > 40e Slender if D/t > 80e2 t ¼ plate thickness, d ¼ web depth, py ¼ design strength,

For simplicity only design methods for Class 1 and 2 sections are covered in this book. Source: BS 5950: Part 1: 2000.

252

Structural Engineer’s Pocket Book

Tension members Bolted connections: Pt ¼ (Ae 0.5a2) py Welded connections: Pt ¼ (Ae 0.3a2) py If a2 ¼ Ag a1 where Ag is the gross section area, Ae is the effective area (which is the net area multiplied by 1.2 for S275 steel, 1.1 for S355 or 1.0 for S460) and a1 is the area of the connected part (web or flange, etc.).

Flexural members Shear capacity, Pv Pv ¼ 0.6py Av Where Av is the shear area, which should be taken as: tD AD=ðD þ BÞ t (D T ) 0.6A 0.9A

for rolled I sections (loaded parallel to the web) and rolled T sections for rectangular hollow sections for welded T sections for circular hollow sections solid bars and plates

t ¼ web thickness, A ¼ cross sectional area, D ¼ overall depth, B ¼ overall breadth, T ¼ flange thickness.

If d=t > 70 for a rolled section, or >62 for a welded section, shear buckling must be allowed for (see BS 5950: clause 4.4.5). Source: BS 5950: Part 1: 2000.

Structural Steel

253

Moment capacity MC The basic moment capacity (Mc) depends on the provision of full lateral restraint and the interaction of shear and bending stresses. Mc is limited to 1.2py Z to avoid irreversible deformation under serviceability loads. Full lateral restraint can be assumed if the construction fixed to the compression flange is capable of resisting not less than 2.5% of the maximum compression force in the flange, distributed uniformly along the length of the flange. Moment capacity (Mc) is generally the controlling capacity for class 1 and 2 sections in the following cases: . . . .

Bending about the minor axis. CHS, SHS or small solid circular or square bars. RHS in some cases given in clause 4.3.6.1 of BS 5950. UB, UC, RSJ, PFC, SHS or RHS if  < 34 for S275 steel and  < 30 for S355 steel in Class 1 and 2 sections, where  ¼ LE =r


Low shear (Fv < 0.6Pv)

Mc ¼ pyS

High shear (Fv > 0.6P v) Mc ¼ py (S 

Where  ¼ 2 PFvv Pv.

1

2

rSn)

and Sv ¼ the plastic modulus of the shear area used to calculate

Lateral torsional buckling capacity Mb Lateral torsional buckling (LTB) occurs in tall sections or long beams in bending if not enough restraint is provided to the compression flange. Instability of the compression flange results in buckling of the beam, preventing the section from developing its full plastic capacity, Mc. The reduced bending moment capacity, Mb, depends on the slenderness of the section, LT. For Class 1 and 2 sections, LT ¼ .

A simplified and conservative method of calculating Mb for rolled sections uses D=T and LT to determine an ultimate bending stress pb (from the following graph) where Mb ¼ pbSx for Class 1 and 2 sections. Source: BS 5950: Part 1: 2000.

254

Structural Engineer’s Pocket Book

Ultimate bending strengths for rolled sections, pb

270 260 250 240 230 220 Ultimate bending stress, pb (N/mm2)

210 200 190

D = 5 T

180 170 160 150 140 10

130 120 110 100

15

90 80

20

70

25

60

30 35 40 45 50

50 40 25

50

75

100

125

150

175

Slenderness (Le/ry)

200

225

250

Structural Steel

255

Compression members The compression capacity of Class 1 and 2 sections can be calculated as Pc ¼ Agpc, where Ag is the gross area of the section and pc can be estimated depending on the expected buckling axis and the section type for steel of 40 mm thickness.

Strut curve for value of pc

Type of section

Axis of buckling x–x

y–y

Hot finished structural hollow section

a

a

Rolled I section

a

b

Rolled H section

b

c

Round, square or flat bar

b

b

Rolled angle, channel or T section/paired rolled sections/compound rolled sections

Any axis: c

Ultimate compression stresses for rolled sections, pc

Ultimate compression stresses for rolled sections, pc

280 260 240

Ultimate compressive stress, pc (N/mm2)

220 200 c

180

b

a Strut curve

160 140 120 100 80 60 40 20 0 50

100

150 200 250 Slenderness (Le/ry)

300

256

Structural Engineer’s Pocket Book

Combined bending and compression Although each section should have its classification checked for combined bending and axial compression, the capacities from the previous tables can be checked against the following simplified relationship for section Classes 1 and 2:

My F Mx þ < 1:0 þ P Mcx or Mb Mcy

Section 4.8 in BS 5950 should be referred to in detail for all the relevant checks.

Structural Steel

257

Connections Welded connections

W

W

The resultant   2of combined longitudinal and transverse forces should be checked: FL 2 FT þ < 1:0 : PL PT

Ultimate fillet weld capacities for S275 elements joined at 90 Leg length s mm

Throat thickness a ¼ 0.7s mm

Longitudinal capacity* PL ¼ pw kN/mm

Transverse capacity* L ¼ pwaK kN/mm

4 6 8 12

2.8 4.2 5.6 8.4

0.616 0.924 1.232 1.848

0.770 1.155 1.540 2.310

* Based on values for S275, pw ¼ 220 N/mm2 and K ¼ 1.25.

258

Structural Engineer’s Pocket Book

Bolted connections Limiting bolt spacings 1.25D

2.5D 1.25D 2.5D 1.25D

Rolled, machine cut or flame cut, sawn or planed edge.

Direct shear W

W Single shear

W

W Double shear

Simple moment connection bolt groups e P F1 F2 F3

X4 X3

X2 X1

F4

X3

X2

X1

X4

  P Mcap ¼ no. rowsx1 of bolts Pt x2i V ¼ Pnt

Fn ¼ Pt xnxn 1 Where x1 ¼ max xi and xi ¼ depth from point of rotation to centre of bolt being considered, Pt is the tension capacity of the bolts, n is the number of bolts, V is the shear on each bolt and F is the tension in each bolt. This is a simplified analysis which assumes that the bolt furthest from the point of rotation carries the most load. As the connection elements are likely to be flexible, this is unlikely to be the case; however, more complicated analysis requires a computer or standard tables.

Bolt capacity checks

For bolts in shear or tension see the following tabulated values. For bolts in shear and tension check: ðFv =Pv Þ þ ðFt =Pt Þ  1:4 where F indicates the factored design load and P indicates the ultimate bolt capacity.

Structural Steel

259

Selected ultimate bolt capacities for non-pre-loaded ordinary bolts in S275 steel Diameter of bolt, f mm

Tensile Tension Shear stress capacity capacity area kN Single Double mm2 kN kN

Grade 4.6 6 20.1 8 36.6 10 58 12 84.3 16 157 20 245 24 353 30 561

3.9 7.0 11.1 16.2 30.1 47.0 67.8 107.7

3.2 5.9 9.3 13.5 25.1 39.2 56.5 89.8

Grade 8.8 6 20.1 8 36.6 10 58 12 84.3 16 157 20 245 24 353 30 561

9.0 16.4 26.0 37.8 70.3 109.8 158.1 251.3

7.5 13.7 21.8 31.6 58.9 91.9 132.4 210.4

Bearing capacity for end distance ¼ 2f kN Thickness of steel passed through mm 5

6

6.4 11.7 18.6 27.0 50.2 78.4 113.0 179.5

13.8 18.4 23.0 27.6 36.8 46.0 55.2 69.0

15.1 27.5 43.5 63.2 117.8 183.8 264.8 420.8

13.8 18.4 23.0 27.6 36.8 46.0 55.2 69.0

8

10

12

15

20

16.6 22.1 27.6 33.1 22.1 29.4 36.8 44.2 27.6 36.8 46.0 55.2 33.1 44.2 55.2 66.2 44.2 58.9 73.6 88.3 55.2 73.6 92.0 110.4 66.2 88.3 110.4 132.5 82.8 110.4 138.0 165.6

41.4 55.2 69.0 82.8 110.4 138.0 165.6 207.0

55.2 73.6 92.0 110.4 147.2 184.0 220.8 276.0

6.6 22.1 27.6 33.1 22.1 29.4 36.8 44.2 27.6 36.8 46.0 55.2 33.1 44.2 55.2 66.2 44.2 58.9 73.6 88.3 55.2 73.6 92.0 110.4 66.2 88.3 110.4 132.5 82.8 110.4 138.0 165.6

41.4 55.2 69.0 82.8 110.4 138.0 165.6 207.0

55.2 73.6 92.0 110.4 147.2 184.0 220.8 276.0

NOTES: 2 mm clearance holes for f < 24 or 3 mm clearance holes for f < 24. . Tabulated tension capacities are nominal tension capacity ¼ 0.8A p which accounts for prying forces. t t . Bearing values shown in bold are less than the single shear capacity of the bolt. . Bearing values shown in italic are less than the double shear capacity of the bolt. . Multiply tabulated bearing values by 0.7 if oversized or short slotted holes are used. .

. .

Multiply tabulated bearing values by 0.5 if kidney shaped or long slotted holes are used. Shear capacity should be reduced for large packing, grip lengths or long joints.

260

Structural Engineer’s Pocket Book

Selected ultimate bolt capacities for non-pre-loaded countersunk bolts in S275 steel Diameter of bolt, f mm

Tensile Tension Shear stress capacity capacity area kN Single Double mm2 kN kN

Grade 4.6 6 20.1 8 36.6 10 58 12 84.3 16 157 20 245 24 353

3.9 7.0 11.1 16.2 30.1 47.0 67.8

3.2 5.9 9.3 13.5 25.1 39.2 56.5

Grade 8.8 6 20.1 8 36.6 10 58 12 84.3 16 157 20 245 24 353

9.0 16.4 26.0 37.8 70.3 109.8 158.1

7.5 13.7 21.8 31.6 58.9 91.9 132.4

Bearing capacity for end distance ¼ 2f kN Thickness of steel passed through (mm) 5

6

8

10

12

15

20

6.4 11.7 18.6 27.0 50.2 78.4 113.0

8.6 – – – – – –

11.3 12.9 – – – – –

16.8 20.2 21.9 – – – –

22.4 27.6 31.1 34.5 – – –

27.9 35.0 40.3 45.5 55.2 62.1 –

36.2 46.0 54.1 62.1 77.3 89.7 85.6

50.0 64.4 77.1 89.7 114.1 135.7 140.8

15.1 27.5 43.5 63.2 117.8 183.8 264.8

8.6 – – – – – –

11.3 12.9 – – – – –

16.8 20.2 21.9 – – – –

22.4 27.6 31.1 34.5 – – –

27.9 35.0 40.3 45.5 55.2 62.1 –

36.2 46.0 54.1 62.1 77.3 89.7 85.6

50.0 64.4 77.1 89.7 114.1 135.7 140.8

NOTES: . Values are omitted from the table where the bolt head is too deep to be countersunk into the thickness of the plate. . 2 mm clearance holes for f <24 or 3 mm clearance holes for f <24. . Tabulated tension capacities are nominal tension capacity ¼ 0.8A p which accounts for prying forces. t t . Bearing values shown in bold are less than the single shear capacity of the bolt. . Bearing values shown in italic are less than the double shear capacity of the bolt. . . .

Multiply tabulated bearing values by 0.7 if oversized or short slotted holes are used. Multiply tabulated bearing values by 0.5 if kidney shaped or long slotted holes are used. Shear capacity should be reduced for large packing, grip lengths or long joints.

Structural Steel

261

Steel design to BS 449 BS 449: Part 2 is the ‘old’ steel design code issued in 1969 but it is (with amendments) still current. The code is based on elastic bending and working stresses and is very simple to use. It is therefore invaluable for preliminary design, for simple steel elements and for checking existing structures. It is normal to compare the applied and allowable stresses. BS 449 refers to the old steel grades where Grade 43 is S275, Grade 50 is S355 and Grade 55 is S460.

Notation for BS 449: Part 2 Stress subscripts

Symbols f P l/r D t

Applied stress Permissible stress Slenderness ratio Overall section depth Flange thickness

c or bc t or bt q b e

Compression or bending compression Tension or bending tension Shear Bearing Equivalent

Allowable stresses The allowable stresses may be exceeded by 25% where the member has to resist an increase in stress which is solely due to wind forces – provided that the stresses in the section before considering wind are within the basic allowable limits. Applied stresses are calculated using the gross elastic properties of the section, Z or A, where appropriate.

Allowable stress in axial tension Pt Form

Steel grade

Sections, bars, plates, wide flats and hollow sections

43 (S275)

Source: BS 449: Part 2: 1969.

Thickness of steel mm

Pt N/mm2

t  40

170

40 < t  100

155

262

Structural Engineer’s Pocket Book

Maximum allowable bending stresses Pbc or Pbt Form

Steel grade

Thickness of steel mm

Pbc or Pbt N/mm2

Sections, bars, plates, wide flats and hollow sections Compound beams – hot rolled sections with additional plates Double channel sections acting as an I beam

43 (S275)

t  40

180

40 < t  100

165

Plate girders

43 (S275)

170 155

Slab bases

All steels

t  40 40 < t  100

185

Upstand webs or flanges in compression have a reduced capacity and need to be checked in accordance with clause 20, BS 449. These tabulated values of Pbc can be used only where full lateral restraint is provided, where bending is about the minor axis or for hollow sections in bending. Source: BS 449: Part 2: Table 2: 1969.

Structural Steel

263

Allowable compressive bending stresses The maximum allowable bending stress is reduced as the slenderness increases, to allow for the effects of buckling. The reduced allowable bending stress, Pbc, can be obtained from the following graph from the ratio of depth of section to thickness of flange (D/T ) and the slenderness ð ¼ Le =rÞ:

180 170 160

Allowable compressive bending stress, P bc (N/mm2)

150 140 D =5 T

130 120 110 100

10

90 80 15

70 20

60

25

50

30

40

35 40 45 50

30 25 50

75 100 125 150 175 200 225 250 275 Slenderness (le /ry)

264

Structural Engineer’s Pocket Book

Allowable compressive stresses For uncased compression members, allowable compressive stresses must be reduced by 10% for thick steel sections: if t > 40 mm for Grade 43 (S275), t > 63 mm for Grade 50 (S355) and t > 25 mm for Grade 55 (S460). The allowable axial stress, Pc, reduces as the slenderness of the element increases as shown in the following chart:

180

160

140

Allowable compressive stress, Pc (N/mm2)

120

100

80

60

40

20

0 50

100

150

200

250

300

350

Structural Steel

265

Allowable average shear stress Pv in unstiffened webs Form

Steel grade

Thickness mm

Pv* N/mm2

Sections, bars, plates, wide flats and hollow sections

43 (S275)

d  40 40 < d  100 d 63 63 < d  100 d  25

110 100 140 130 170

50 (S355) 55 (S460)

* See Table 12 in BS 449: Part 2 for allowable average shear stress in stiffened webs.

Section capacity checks Combined bending and axial load Compression:

Tension:

fbc fc fbcx þ y  1:0 þ Pc Pbcx Pbcy

ft fbt  1:0 þ Pt Pbt

and

fbc fbcx þ y  1:0 Pbcx Pbcy

Combined bending and shear  2 p 2 p 2 þ 3fq2 Þ or fe ¼ ðfbc fe ¼ ðfbt þ 3fq2 Þ and fe < Pe and ðfbc =Po Þ2 þ fq0 =P0q  1:25

Where fe is the equivalent stress, fq0 is the average shear stress in the web, Po is defined in BS 449 subclause 20 item 2b iii and Pq0 is defined in clause 23. From BS 449: Table 1, the allowable equivalent stress Pe ¼ 250 N/mm2 for Grade 43 (S275) steel < 40 mm thick.

Combined bending, shear and bearing p 2 p 2 fe ¼ ðfbt þ fb2 þ fbt fb þ 3fq2 Þ or fe ¼ ðfbc þ fb2 þ fbc fb þ 3fq2 Þ and  2  0 0 2   fbc =Po þ fq =Pq þ fcw =Pcw  1:25

Source: BS 449: Part 2: 1969.

fe < Pe and

266

Structural Engineer’s Pocket Book

Connections Selected fillet weld capacities for Grade 43 (S275) steel Leg length s mm

Throat thickness a = 0.7s mm

Weld capacity* kN/mm

4 6 8 12

2.8 4.2 5.6 8.4

0.32 0.48 0.64 0.97

* When a weld is subject to a combination of stresses, the combined effect should be checked using the same checks as used for combined loads on sections to BS 449.

Selected full penetration butt weld capacities for Grade 43 (S275) steel Thickness mm

Shear capacity kN/mm

Tension or compression capacity* kN/mm

6 15 20 30

0.60 1.50 2.00 3.00

0.93 2.33 3.10 4.65

* When a weld is subject to a combination of stresses, the combined effect should be checked using the same checks as used for combined loads on sections to BS 449. Source: BS 449: Part 2: 1969.

Structural Steel

267

Allowable stresses in non-pre-loaded bolts Description

Bolt grade

Axial tension N/mm2

Shear N/mm2

Bearing N/mm2

Close tolerance and turned bolts

4.6 8.8

120 280

100 230

300 350

Bolts in clearance holes

4.6 8.8

120 280

80 187

250 350

Allowable stresses on connected parts of bolted connections (N/mm2) Description

Allowable stresses on connected parts for different steel grades N/mm2 43 (S275)

50 (S355)

55 (S460)

Close tolerance and turned bolts

300

420

480

Bolts in clearance holes

250

350

400

Source: BS 449: Part 2: 1969.

268

Structural Engineer’s Pocket Book

Selected working load bolt capacities for non-pre-loaded ordinary bolts in grade 43 (S275) steel Diameter of bolt, f mm

Tensile Tension Shear stress capacity capacity area kN Single Double mm2 kN kN

Bearing capacity for end distance ¼ 2f kN

5

6

8

10

12

6 8 10 12 16 20 24

20.1 36.6 58 84.3 157 245 353

1.9 3.5 5.6 8.1 15.1 23.5 33.9

1.6 2.9 4.6 6.7 12.6 19.6 28.2

3.2 5.9 9.3 13.5 25.1 39.2 56.5

7.5 10.0 12.5 15.0 20.0 25.0 30.0

9.0 12.0 15.0 18.0 24.0 30.0 36.0

12.0 16.0 20.0 24.0 32.0 40.0 48.0

15.0 20.0 25.0 30.0 40.0 50.0 60.0

18.0 24.0 30.0 36.0 48.0 60.0 72.0

30

561

53.9

44.9

89.8

37.5 45.0 60.0 75.0 90.0 112.5 150.0

6 8 10 12 16 20

20.1 36.6 58 84.3 157 245

4.5 8.2 13.0 18.9 35.2 54.9

3.8 6.8 10.8 15.8 29.4 45.8

7.5 13.7 21.7 31.5 58.7 91.6

7.5 10.0 12.5 15.0 20.0 25.0

24 30

353 561

79.1 125.7

66.0 104.9

132.0 209.8

Thickness of steel passed through 15

20

Grade 4.6 22.5 30.0 30.0 40.0 37.5 50.0 45.0 60.0 60.0 80.0 75.0 100.0 90.0 120.0

Grade 8.8 9.0 12.0 15.0 18.0 24.0 30.0

12.0 16.0 20.0 24.0 32.0 40.0

15.0 20.0 25.0 30.0 40.0 50.0

18.0 24.0 30.0 36.0 48.0 60.0

22.5 30.0 30.0 40.0 37.5 50.0 45.0 60.0 60.0 80.0 75.0 100.0

30.0 36.0 48.0 60.0 72.0 90.0 120.0 37.5 45.0 60.0 75.0 90.0 112.5 150.0

NOTES: . 2 mm clearance holes for f < 24 or 3 mm clearance holes for f < 24. . Bearing values shown in bold are less than the single shear capacity of the bolt. . Bearing values shown in italic are less than the double shear capacity of the bolt. . Multiply tabulated bearing values by 0.7 if oversized or short slotted holes are used. . Multiply tabulated bearing values by 0.5 if kidney shaped or long slotted holes are used. .

Shear capacity should be reduced for large packing, grip lengths or long joints.

Bolted connection capacity check for combined tension and shear f t fs þ  1:4 Pt Ps

Structural Steel

269

Stainless steel to BS 5950 Stainless steels are a family of corrosion and heat resistant steels containing a minimum of 10.5% chromium which results in the formation of a very thin self-healing transparent skin of chromium oxide – which is described as a passive layer. Alloy proportions can be varied to produce different grades of material with differing strength and corrosion properties. The stability of the passive layer depends on the alloy composition. There are five basic groups: austenitic, ferritic, duplex, martensitic and precipitation hardened. Of these, only austenitic and Duplex are really suitable for structural use.

Austenitic Austenitic is the most widely used for structural applications and contains 17–18% chromium, 8–11% nickel and sometimes molybdenum. Austenitic stainless steel has good corrosion resistance, high ductility and can be readily cold formed or welded. Commonly used alloys are 304L (European grade 1.4301) and 316L (European grade 1.4401).

Duplex Duplex stainless steels are so named because they share the strength and corrosion resistance properties of both the austenitic and ferritic grades. They typically contain 21–26% chromium, 4–8% nickel and 0.1–4.5% molybdenum. These steels are readily weldable but are not so easily cold rolled. Duplex stainless steel is normally used where an element is under high stress in a severely corrosive environment. A commonly used alloy is Duplex 2205 (European grade 1.44062).

270

Structural Engineer’s Pocket Book

Material properties The material properties vary between cast, hot rolled and cold rolled elements. Density

78–80 kN/m3

Tensile strength

200–450 N/mm2 0.2% proof stress depending on grade.

Poisson’s ratio

0.3

Modulus of elasticity

E varies with the stress in the section and the direction of the stresses. As the stress increases, the stiffness decreases and therefore deflection calculations must be done on the basis of the secant modulus.

Shear modulus

76.9 kN/mm2

Linear coefficient of thermal expansion

17  10 6/ C for 304L (1.4301) 16.5  10 6/ C for 316L (1.4401) 13  10 6/ C for Duplex 2205 (1.4462)

Ductility

Stainless steel is much tougher than mild steel and so BS 5950 does not apply any limit on the thickness of stainless steel sections as it does for mild steel.

Structural Steel

271

Elastic properties of stainless steel alloys for design The secant modulus, Es ¼ Esi ¼   E m  1þk

ðEs1 þ Es2 Þ , where 2

f1 or 2 Py

where i = 1 or 2, k ¼ 0:002E=Py and m is a constant. Values of the secant modulus are calculated below for different stress ratios ðfi =Py Þ

Values of secant modulus for selected stainless steel alloys for structural design Stress Secant modulus ratio* fi Py

kN/mm2 304L

316L

Duplex 2205

Longitudinal Transverse Longitudinal Transverse Longitudinal Transverse 0.0

200

200

190

195

200

205

0.2

200

200

190

195

200

205

0.3

199

200

190

195

199

204

0.4

197

200

188

195

196

200

0.5

191

198

184

193

189

194

0.6

176

191

174

189

179

183

0.7

152

173

154

174

165

168

* Where i ¼ 1 or 2 for the applied stress in the tension and compression flanges respectively.

Typical stock stainless steel sections There is no UK-based manufacturer of stainless steel and so all stainless steel sections are imported. Two importers who will send out information on the sections they produce are Valbruna and IMS Group. The sections available are limited. IMS has a larger range including hot rolled equal angles (from 20  20  3 up to 100  100  10), unequal angles (20  10  3 up to 200  100  13), I beams (80  46 up to 400  180), H beams (50  50 up to 300  300), channels (20  10 up to 400  110) and tees (20  20  3 up to 120  120  13) in 1.4301 and 1.4571. Valbruna has a smaller selection of plate, bars and angles in 1.4301 and 1.4404. Source: Nickel Development Institute (1994).

272

Structural Engineer’s Pocket Book

Durability and fire resistance Suggested grades of stainless steel for different atmospheric conditions Stainless steel grade

Location Rural

Urban

Industrial

Marine

Low Med High Low Med High Low Med High Low Med High 304L

3

3

3

3

3

(3)

(3)

(3)

X

3

(3)

X

O

O

O

O

3

3

3

3

(3)

3

3

(3)

O

O

O

O

O

O

O

O

3

O

O

3

(1.4301) 316L (1.4401) Duplex 2205 (1.4462)

Where: 3 ¼ optimum specification, (3) ¼ may require additional protection, X ¼ unsuitable, O ¼ overspecified.

Note that this table does not apply to chlorinated environments which are very corrosive to stainless steel. Grade 304L (1.4301) can tarnish and is generally only used where aesthetics are not important; however, marine Grade 316L (1.4401) will maintain a shiny surface finish.

Corrosion mechanisms Durability can be reduced by heat treatment and welding. The surface of the steel forms a self-healing invisible oxide layer which prevents ongoing corrosion and so the surface must be kept clean and exposed to provide the oxygen required to maintain the corrosion resistance. Pitting Mostly results in the staining of architectural components and is not normally a structural problem. However, chloride attack can cause pitting which can cause cracking and eventual failure. Alloys rich in molybdenum should be used to resist chloride attack. Crevice corrosion nuts and washers.

Chloride attack and lack of oxygen in small crevices, e.g. between

Bi-metallic effects The larger the cathode, the greater the rate of attack. Mild steel bolts in a stainless steel assembly would be subject to very aggressive attack. Austenitic grades typically only react with copper to produce an unsightly white powder, with little structural effect. Prevent bi-metallic contact by using paint or tape to exclude water as well as using isolation gaskets, nylon/Teflon bushes and washers.

Fire resistance Stainless steels retain more of their strength and stiffness than mild steels in fire conditions, but typically as stainless steel structure is normally exposed, its fire resistance generally needs to be calculated as part of a fire engineered scheme. Source: Nickel Development Institute (1994).

Structural Steel

273

Preliminary sizing Assume a reduced Young’s modulus depending on how heavily stressed the section will be and assume an approximate value of maximum bending stress for working loads of 130 N/mm2. A section size can then be selected for checking to BS 5950.

Stainless steel design to BS 5950: Part 1 The design is based on ultimate loads calculated on the same partial safety factors as for mild steel.

Ultimate mechanical properties for stainless steel design to BS 5950 Alloy type

Steel

European

Minimum

Ultimate

Minimum

desig-

grade

0.2%

tensile

elongation

nation

(UK grade)

proof

strength

after

stress

N/mm2

fracture

N/mm2 1

Basic austenitic

X5CrNi

304L

18-9

(1.4301)

Molybdenum

X2CrNiMo

316L

austenitic2

17-12-2

(1.4401)

Duplex

X2CrNi

Duplex

MoN

2205

22-5-3

(1.4462)

%

210

520–720

45

220

520–670

40

460

640–840

20

NOTES: 1. Most commonly used for structural purposes. 2. Widely used in more corrosive situations. The alloys listed in the table above are low carbon alloys which provide good corrosion resistance after welding and fabrication. As for mild steel, the element cross section must be classified to BS 5950: Part 1 in order to establish the appropriate design method. Generally this method is as given for mild steels; however, as there are few standard section shapes, the classification and design methods can be laborious. Source: Nickel Development Institute (1994).

274

Structural Engineer’s Pocket Book

Connections Bolted and welded connections can be used. Design data for fillet and butt welds requires detailed information about which particular welding method is to be used. The information about bolted connections is more general.

Bolted connections Requirements for stainless steel fasteners are set out in BS EN ISO 3506 which split fixings into three groups: A = Austenitic, F = Ferritic and C = Martensitic. Grade A fasteners are normally used for structural applications. Grade A2 is equivalent to Grade 304L (1.4301) with a 0.2% proof stress of 210 N/mm2 and Grade A4 is equivalent to Grade 316L (1.4401) with a 0.2% proof stress of 450 N/mm2. There are three further property classes within Grade A: 50, 70 and 80 to BS EN ISO 3506. An approximate ultimate bearing strength for connected parts can be taken as 460 N/mm2 for preliminary sizing.

Ultimate stress values for bolted connection design Grade A property class Shear strength* Bearing strength* Tensile strength* N/mm2 N/mm2 N/mm2 50

140

70 (most common) 80

510

210

310

820

450

380

1000

560

* These values are appropriate with bolt diameters less than M24 and bolts less than 8 diameters long. Sources: Nickel Development Institute (1994).

% $ &     $  \                  # <=                  †         

           ;          '           [                 #%ˆŽ#ˆ=  [  

     [       [           [                 [     ˆ=Ž<=       ##ˆŽ#]ˆ      %ˆ  ><  `      ]ˆŽ#==  [  ˆ=Ž%==      >Ž"  $ )  ‡       †             ‹                         

  [          

                                *                               $   #     ˆ   #%=         [          *                             _  ~    

           Π Q   %                

   %ˆŽ>      [   [                 

              >$ƒ       

%]<

      Q `[ }[

$

                                {      ‚\ >=Žˆ=       ‚\ %ˆŽƒ= "

{      %ƒ [{$>       #] [{$>

    …% ˆ $ = †

{    

a ˆ <       a ˆ #"    

*   

a ˆ #=     a ˆ %ˆ    

$ 

%]ˆ                 ƒ= 

                >ˆˆ                             

";     .                           

. \                 [        . .

. .

    [                     ‚                        [          =#                

                  ?                   [              |[                              [             `       [    _         %              

          

    [

\    \  

%]]

(   †          *   %  

  

    +   ;  ;       ; 

( 

( 

$ 

      

<Ž#=

"Ž#"

$#

$%>

       

<Ž#=

"Ž#"

$%%



%$€   

¥#%

¥#%

$#ˆ

$%ˆ

€_  $  

¥#%

¥#%

$%ˆ  

$>%  

\  \    f ˆ %($>    #ˆf $ (    

$

<Ž#<

$#]

$%=

`      

¥#%

¥#%

$#%

$#<

$ 

‘    $#=  _   %(

%]"

      Q `[ }[

(   †                                [     #<= {$%  %]ˆ  %#= {$%  >ˆˆ                ‹  ?          †           <=Ž]=¢    ‹  \            #"  %%    #              |‹                [   

 :        

  ;  ‚                                ~  (          †      =<(    #ˆ(          =](   %(                      ‘         #ˆ(                              ‘         (            %(                                                    #ˆ  %

(      ; † ;    $OHˆ     )   €  * O !   N  * O     (  

' †       

$  

K    ;   

$  ;



' †      ) (  ;

$  ;

$ )  * 

(  ;

$  ;

$ )  * 

<

" #% #ˆ

% % %

>== >== >==

ƒˆ]  #ˆ% } <] ˆ>>  %#= } % <#=  %% } #=#

ƒ=<  #]" } ˆƒ <#=  %% } ##> ]<%  %<] } #]>

=%< =ƒˆ =<ƒ

%ˆƒ  %ˆƒ \ #>% >=ˆ  >=ˆ \ #ˆ" >=ˆ  >=ˆ \ # "

%ˆƒ  %ˆƒ \ #>% >ˆ<  ƒ=< \ %"] >ˆ<  ƒ=< \ ˆˆ#

=<# #= # ]

"

" #% #ˆ

> > >

%<] %<] %<]

ˆ>>  %#= } % <#=  %% } #%ˆ ]<%  %<] } #ƒ]

>ˆ<  #]# } ˆ] <#=  %% } #=# ]<%  %<] } #]>

=>> =ƒ" =]ˆ

>=ˆ  >=ˆ \ # " >=ˆ  >=ˆ \ %"> >ˆ<  ƒ=< \ %"]

%ˆƒ  %ˆƒ \ #=] >=ˆ  >=ˆ \ %"> >ˆ<  ƒ=< \ ƒ<]

=<ˆ #>= # ƒ

" #% #ˆ

> > >

>== >== >==

<#=  %% } #=# <"<  %ˆƒ } #ƒ= ]<%  %<] } #]>

ƒ=<  #]" } ˆƒ <#=  %% } ##> ]<%  %<] } #]>

=># =ƒ =<

>=ˆ  >=ˆ \ %ƒ= >ˆ<  ƒ=< \ %"] >ˆ<  ƒ=< \ > >

%ˆƒ  %ˆƒ \ #>% >ˆ<  ƒ=< \ %"] >ˆ<  ƒ=< \ ˆˆ#

=]ƒ #%= %#=

#=

" #% #ˆ

ƒ ƒ ƒ

%ˆ= %ˆ= %ˆ=

<"<  %ˆƒ } #ƒ= ">"  % % } #]< #ƒ  >=ˆ } %=#

>ˆ<  #]# } ˆ] <#=  %% } #=# ]<%  %<] } #]>

=ƒ= =ˆˆ =">

>ˆ<  ƒ=< \ %"] >ˆ<  ƒ=< \ > > >ˆ<  ƒ=< \ ƒ<]

%ˆƒ  %ˆƒ \ #=] >=ˆ  >=ˆ \ %"> >ˆ<  ƒ=< \ ƒ<]

=] #ƒ< %#"

#%

" #% #ˆ

ƒ ƒ ƒ

>== >== >==

]<%  %<] } #]> #ƒ  >=ˆ } %%ƒ #ƒ  >=ˆ } %"

ƒ=<  #]" } ˆƒ <#=  %% } ##> ]<%  %<] } #]>

=ƒ= =ˆ< =]]

>ˆ<  ƒ=< \ ƒ<] >ˆ<  ƒ=< \ <>ƒ #ƒ  ƒ# } %"

%ˆƒ  %ˆƒ \ #>% >ˆ<  ƒ=< \ %"] >ˆ<  ƒ=< \ ˆˆ#

= > #ƒ %=#

\[         ƒˆ ‘† \        $ ><=  %ˆ 

%]

%"=

(      ; † ;    $OHˆ  )   €  * O !   N  * O         

$  

K    ;   

$  ;



   ) (  ;

$  ;

$ )  * 

(  ;

$  ;

$ )  * 

<

" #% #ˆ

% % %

>== >== >==

ƒˆ]  #ˆ% } <= ˆ>>  %#= } "% ˆ>>  %#= } %

>ˆ<  #]# } ˆ] ˆ>>  %#= } #= ]<%  %<] } #ƒ]

=%] =ƒ> =ˆˆ

%ˆƒ  %ˆƒ \ " %ˆƒ  %ˆƒ \ #>% >=ˆ  >=ˆ \ #ˆ"

%ˆƒ  %ˆƒ \ " >=ˆ  >=ˆ \ %ƒ= >ˆ<  ƒ=< \ ƒ<]

=ƒ# = # #<<

"

" #% #ˆ

> > >

%<] %<] %<]

ƒˆ]  #ˆ% } "% <#=  %% } #=# <#=  %% } #%ˆ

>ˆ<  #]# } ˆ# ˆ>>  %#= } #=# ]<%  %<] } #>ƒ

=% =ƒ< =ˆ

>=ˆ  >=ˆ \ #ˆ" >=ˆ  >=ˆ \ %ƒ= >=ˆ  >=ˆ \ %">

%ˆƒ  %ˆƒ \ " >=ˆ  >=ˆ \ %ƒ= >ˆ<  ƒ=< \ > >

=ˆ> ##= #<<

" #% #ˆ

> > >

>== >== >==

<#=  %% } #=# <#=  %% } #%ˆ ]<%  %<] } #ƒ]

>ˆ<  #]# } ˆ] ˆ>>  %#= } #= ]<%  %<] } #ƒ]

=>% =ƒ] =ˆ

>=ˆ  >=ˆ \ # " >=ˆ  >=ˆ \ %"> >ˆ<  ƒ=< \ %"]

%ˆƒ  %ˆƒ \ " >=ˆ  >=ˆ \ %ƒ= >ˆ<  ƒ=< \ ƒ<]

=ˆƒ #=ƒ #]ˆ

#=

" #% #ˆ

ƒ ƒ ƒ

%ˆ= %ˆ= %ˆ=

<#=  %% } #%ˆ ]<%  %<] } #ƒ] ">"  % % } #]<

>ˆ<  #]# } ˆ# ˆ>>  %#= } #=# ]<%  %<] } #>ƒ

=>< =ˆ> =<ˆ

>=ˆ  >=ˆ \ %ƒ= >ˆ<  ƒ=< \ %"] >ˆ<  ƒ=< \ >ƒ=

%ˆƒ  %ˆƒ \ " >=ˆ  >=ˆ \ %ƒ= >ˆ<  ƒ=< \ > >

=<< #%= #"=

#%

" #% #ˆ

ƒ ƒ ƒ

>== >== >==

]<%  %<] } #ƒ] ">"  % % } # ƒ #ƒ  >=ˆ } %%ƒ

>ˆ<  #]# } ˆ] ˆ>>  %#= } #= ]<%  %<] } #ƒ]

=>] =ˆ> =<ƒ

>ˆ<  ƒ=< \ > > >ˆ<  ƒ=< \ ˆˆ# >ˆ<  ƒ=< \ <>ƒ

%ˆƒ  %ˆƒ \ " >=ˆ  >=ˆ \ %ƒ= >ˆ<  ƒ=< \ ƒ<]

=] #%< # "

(  

\[         ƒˆ ‘†

\    \  

%"#

     '$ ˆ‰ˆ„ } ˆ ˆ=' `  >                             } ˆ ˆ=' `  #   } "##=' `  #                [                              _ Ž  

                                            ‹                                   †   [                                                   [      

Be Ds

Shear stud Mesh for shear and crack resistance

Dp

d Optional reinforcement for fire resistance

D t

Secondary beam

T Primary beam

%!  ; ;       '

! ˆ           * $ ƒ

      '

! ˆ    ="         * $ ƒ

 '

! ˆ               

%"%

      Q `[ }[

             ;  ?              [                             [           =ˆ ' \         ˆ =ƒˆ * ! (         ˆ .   

( 

 

         [     ˆ   

  .

      [          ˆ .     Z              `          `                 ?       

(      )  ?  :     ;= # #    ˆ  ( ‡ ( %

 …( ( † %



(      )  ?  :    = # ˜ # #)  ˆ

 (  …( ‡ ( † ‡ % %

… ‡  †% ƒ

(      )  ?  :   );= < ™ <)  ˆ  ‡

 …( ‡ ( ‡ ( † %

% ƒ

<        ;   

  ˆ 

…

* †

 % 

% #

\    \  

%">

$         ˆ =< .                  =ˆ  ‘                     _                                            [      €     ¥ =ˆ                         

                      ˆ  } ˆ ˆ=' `  >    #           "= [{  #== [{                                      [  =¢           ?                      [                      †                        



                      
$!;  €                                        [ p                 * £ #"$ d   d                                           [   

ƒ ‘†Žƒˆ ‘†

                ‹             

        †                   

     

      |                    %ˆ     

                                           

  _                                         '

. ‡                    

 

. .

. .

.

.

.

      

        ‡                       [           [                

                             \ [             ~                   ‹                           

   [           

    Z              ‹                        

  ‘             ‡             ‹       ‡                 „               [                   [   ?

          Π Q                       [           

                       “ 

       

         

       * ‹      [             [                                 [            #ˆ       [   ` _            

    [         

                         ‡                      [      [  [                      =                                             [  _                    

 ‡       [                    

   ‡

%"ˆ

        *    ‡      '    ]%¢        #>¢        #=¢            ƒ¢  _       [        "=¢    %=¢        #ˆ== \               #=ˆ= \    

                                          # ˆ  `[                                _    [                                                                <== \                                                   [      €                

  %== \   

      Œ

Q                   ‹                        Œ Q   Œ Q                €                       ?   

    [          ?

                       „ 

 *

      

               ]== \                               „                                      [   

                      [               

                              [    ^     Œ Q        _     †                            

                          „     [   [                      

             _            

                         

[            Œ‘ [ Q    } <%=<                       ƒ=    ˆ=            [  ‘                             [

           „     

(  *       

                    “                               [        #ˆ  %    

              

    ‘            [                 Œ‘ [ Q         [   

                                                           [                      [ 

%"<

      Q `[ }[

.              [   {           ~                      ?                           {                    _    _                _  ‘ [         [  {                             [                  >== \ ‘ [        „            %   "    ‡ 

|{               ‡              {    _     ƒ 

          [            #  ƒ== 



                                            `}       =>"                       #%                                 `            †    *                 ‹  ‹            [            [                                [                              *  

                    †        [   

             

                         ΠQ              

      *                      ]= \                   ‹  

$

     "

%ˆŽ%< [{$>

 ! 

 ˆ #=== {$%

                '     

                  †               [ €               #ˆ== {$%            †            ' ƒˆ {$%  

 "= {$%       #%= {$%     `       

        

]=Ž]ƒ [{ %

(   

=%%Ž=%ˆ

      : 

"  #=

< 

$ \

   ‡

%"]

      †           †                                 †   _  † Z                    †        `[      ‡  €   ‘  ‡ +‡    } ‡ [   [            ? [       †          

 *      _  †  >%#=   <===       >%#=   "===                   ‹   

       

>”

ƒ

ˆ”

<

"”

 * O

]ˆ

#==

#%ˆ

#ˆ=

%==

#=

#%

#ˆ

#

%ˆ

%ˆ=

>==

>]ˆ

ƒ]ˆ

<%ˆ

” ‡        †    

 *

     †              †           %ˆ   

     _            

$ †'   

ƒ

ˆ

<

"

#=

#%

#ˆ

#ˆ==  %%==

%===  ƒ%==

%===  ƒ%==

%===  ƒ%==

%===  ƒ%==

%===  ƒ%==

#]==  ƒ%==

# #ˆ==  ƒ%==

” *  †         ^       

.  *     {        "  [      Z                #=    #%    †             

.      †      †    [        %===   <=== 

    *     †                †             

!   \         ^   `[          ]ˆ=   #%  “        #===   #ˆ            #ˆ==   #   ‘         \                        >==   #=     #==   ƒ   <  

%""

      Q `[ }[

";     ";  ‡                           [   ‡  

               [      _                [  _    ˆ       „         %ˆŽ>=        _     |                                  

#  €            <=          

                                              ‡                                               _       

 

       [ `       [   ` _       [                 #%   %   

   ‡

%"

    †   

          †   `[                                     †             _                                     

 

    ; . †     *

    '



=>< =]ƒ #ˆ= >==

#% #ˆ # %ˆ

” € ##   

    

    

 ; ;  ;)           

  "  * O

(   

Q

'

Œ

'

N„

NO

=ˆ #= #ˆ

=%ˆ =ˆ= #ˆ=

#ƒ= = = Ž

#]ˆ #ƒˆ Ž

%#= #]ˆ #%=

%ƒ= %=ˆ #<=

” {        

  ¥#ˆ           \ ?  } <%=<

         "  * O

(   

#ˆ ˆ= ƒ= ƒ=

#ƒ >< ƒˆ ƒ=

   +  ‡ ;   -'

‡

# ‡ #= %ˆ ‡ #ˆ %ˆ ‡ %ˆ %ˆ ‡ #=

  

|    |   \

    

” }     †  # %       =>  #ˆ                      [       # %          _  †$         

% =

      Q `[ }[

  

          

  



  *   ) 

' NK„„  * O

NKOˆ  * O

NKˆ„  * O

NKHˆ  * O

 %= %=Ž%ˆ %ˆŽ>= >=Ž>ˆ >ˆŽƒ=

# $#%= # $#<= %ˆ$#"= %ˆ$%>= %ˆ$%"=

# $#>= %ˆ$#<= %ˆ$%== Ž Ž

%ˆ$#%= %ˆ$#]= Ž Ž Ž

%ˆ$#>= Ž Ž Ž Ž

” ?                  

  )

     †              %>                        _     `  †        #"     †    % %       _           _         %  %                   “ %  >ˆ      _        >ˆ      $ = `[  %==%

   ‡

% #

$    $

     . `                            

. ‡             . |    [         |         . . . . . . . . .

. . . . . . . . .

       $#ˆ=      †          $><= ‡  [                [        €                                  ‡  

                          

           †      _           

              ^  

                 

  ‹                        [               

                       

                ‹      ‡                    ‡    ‹     

    [                        € _         ‹                                                    

                                                                                             ‹               [    [             

         \       $                                  ‡  †         †             †             ~        †                                                 \[              

% %

      Q `[ }[

                                { #>ƒ]ƒ Œ‡  }  Q                                                                } <%<% }   ‚   `  { ‡   ‡†  €  |  `[  | ‡   “    ‡        ‡ 

}      ?      ? #%"" ‡  

                                                             _                  ?            ~                  

  

);                                      { #>ƒ]ƒ    

 ;  *

O

    

     

  

);  ;  *

O

?



ƒˆ

*       

<ˆ %ˆ

] %=

‘  

]=

*       

>ˆ %ƒ

%= >=

#%=

*       

>= %ƒ

ƒ= ˆ=

  

   ‡

% >

   \

                                   \       

             ‡            

                 #     [  \          =%Ž=>       \{\      

     #=Ž#ˆ 

$                                     ‡[                      _                            =ƒ%Ž#ˆ {$%           [ 

#      €  

                            Z  _  

      †   [                      €                           ˆ $m             m ˆ =%

.  ?

                         ‡   ‡†  €                    >=   #ˆ    [                        ƒ ‘                                [ 

'     } 

                      `   

                                           $                   _        

      

            „

% ƒ

      Q `[ }[

    !              _                 _                                                       *   >Z                $ 

$                      

                     ‹   $        „  † |      „ 

               „            <   <    _       >'#   _       _                    ?      „              $#==                  „            „          „  

              | \                | \   " ˆ             ‹        ‹              | \   >                            [               | \         ' " )     

š 

   *

O

   

#   *

O

    

  

);    *

O

‰ +O -

===#ƒ

 *   

= ˆ

 ‹ $   |

                   „  

=#ƒ=

 ‹ $   |

          „              

=%#=

Œ‰ˆ +N -

====

* ‹ $ 

$

$

=<"

 ‹ $   * ‹ $ 

=#=ˆ ==##

 *   

#ƒ=

 ‹ $   |

                   „  

 ‹ $   |

          „               * ‹ $ 

=#ƒ=

$

 ‹ $   * ‹ $ 

=#ƒ= ===]

$

$ = | \   %==%

#=]

=%#=

 ( )  * ! * +,(  + (   ?                   ‹                                   

"      ?      |`\             _                    \         

                                                |`\       _        |`\  ' . . . .

}             }              }                       {  „                  

|                 ' . €_                |`\

                       _         . ‹         _              „   . ‚ |`\              [  \           

                     [   |`\             |`\  

"     ; |      |`Z                                                                                          

              _  

% <

      Q `[ }[

'  )  }          [                           [                 _    [   ?  

                      [                    ‹                                    [   }                ‹                                              

  

'  )   '$ ŒN„O } "#=%                                   #  } "#=%' # =                  \‚? ‚  #> 

   ;  )                                  '          _           _                                     “                [       \                          [                ‹                                                         [  [          ‘                                           } "==]                               ?   \  `         } "##=       Œ  Q        _             

                            " |                                   |          _         [                                 |                              ?                          

'  )   '$ ŒN„O   ;   '$ ŒN„O

'  

(    )  

#     





N

\  [     _           [

              ¥<ˆ¢

  ' Ž ‚\  } "##=   [    => 

`                 „  Z  [ \            

   } "#=%      [        [          }   ‚  

O

~ [     

           ‚    

{                ˆ >ˆŽˆ=¢

  

‚       _            $     Z  [                

  ' Ž ‚\  }"==]

`                 „  Z  [ \               ?   [   [            

  

{              $                   ‘ [

  ' Ž ‚\  } "==]

`                 „  Z  [ \            

  ?   [    



€

       [    

         

?        

|                       ƒ=Ž<=¢

                  

    ˆ >ˆ¢  [ Ž ˆ=¢     

   Ž       

?          [              [‹    ‘  

  

 [                    

[

  ' Ž ‚\  } "==]    



‘ [ Z  [                

   Ž          

  

 [       |`Z

Z  [ ?                _            

{' #    ˆ  [       ' ˆ                ˆ       % ‚                      _                                    ^     ƒ=Žˆ=¢          "ˆ¢   _    

=

% ]

$ = } "#=%' #

% "

      Q `[ }[

<  )  €         [ ?      [              „                [     ?         _                

   _            ?  

                   `                      

               {                      

                [         

}     Z  €_    €  

%

$                                           $            ‘               [            [                  [   [ ‹                                                  '                            

"   |                                            _         [    [        [      €       [                                         [ 

  *   |     [                           [                              < [                    

†† €                     [     „                   _  $    $    '                    ]                       ?  

                                                          ?                     ƒ [  %ˆ     [      ]ˆ¢

    ?                    [            

$                                          [                  [                 $    

>==

      Q `[ }[

(   

)   ;                                                       _           #>=                               _  $>==   [                                  Z         #==        

     ]ˆ          [     ` [  

  #%==                        ##"=  ## =           „     [         {

  [                      _          [  ‹        

  



)  )    #

 

        '



) 



 >



  

 

)



 * O



NKˆ



ˆ

N„

 * O

 * O

 * O

 * O

Nˆ„

  %

%>>

ˆˆŽ]ˆ

ˆ=Ž]=

ƒˆŽ<=

O„„

  %

% ƒ

]ˆŽ#==

]=Ž"ˆ

<=Ž]ˆ

ƒˆŽ<=

OQ„

%

> ]

#==Ž#%=

"ˆŽ##=

]ˆŽ#==

<=Ž"=

O„

%

> ]

#%=Ž#ƒˆ

##=Ž#>=

#==Ž##ˆ

"=Ž ˆ

€„„

%

ƒ">

#ƒˆŽ#]ˆ

#>=Ž#ˆˆ

##ˆŽ#ƒ=

>ˆŽƒˆ

ˆŽ##ˆ

%

”?    #ˆ [{$                  [ ‹ $ =    %==% {        „        \                 

}     Z  €_    €  

>=#

'> 

     ~           ΠQ      

  

                                

                                  Œ   Q                †     †              [ ?         _              

               

                                       „             [     ~                                                 €   [            [                  

  !  ?  Z   ¡  ?    \           \  

* 

\      † {[           \

>=%

      Q `[ }[

$ !                                         \                   ?                                                                       |                

!    _  

‡         „          “      [                

                                   ‚          

`  

       [  _ ‡                      [           <= \ Z                                   ?                           [ 

? 

                                         _                            „      

` 

`  _         _                _   ‘    _      [              

‚  ‹   ‚€    ‹   ‹   `‚€

           \                     _                     _                _             

     „     

` ‹  `€

     Œ  Q        \                                ‚€   `‚€  

Z ‹ ‹ ?                   Z€          „     =#        ‹    #= \ € \ 

|     [              Z€   €            [

`       

*   ‹              

 

?  

|                                                              ?          $  

    

}     Z  €_    €  

>=>

$           ‹             ‹          „                                        



$  

  !

 

#           % ‡     „                     >                        ƒ ?                        

_                

 _       `                 [      

$    

# |     % Z           „           >                

_              $ ~                                           

‘   

#    $†         % |     > *         

    †    ƒ             ˆ                    

_           ?           †   

$  

€  ' # ?           % ?  

   _           

 ' # |         % ‡   > ?            

€  ' # |      %  [      

> ?     † ƒ `          

_           ? †            

   #   %' > ‡   ƒ ?      $      ;

# ‚        % |                             %= > `                  ƒ          

_       

  ‚€   `‚€            |                  '

(   ;   

# |        % ?           

      > ‚           

_        

                       

 

# |              ?          [       %        

         _                  

~}`Ž~ `    } `  }‚Ž} ‚  Z‚ŽZ  ‚  {Ž  

>=ƒ

      Q `[ }[

#:   ?           _            

     ~                    _  _                     ^

$    )    '$ NO„O





ˆ= ]ˆ #== #%ˆ #ˆ=

"  + )  +)- 

##  >= 

#= >>ˆ

><ˆ

.

. .

.

" ƒ=

] ƒˆ

< ˆ=

. .

.

. .

ˆ ˆ<

.

$  )  )  '$ NON„





%ˆ ˆ= ]ˆ #== #%ˆ

"  +  +- 

<  >ƒ" 

] >ˆ=

" ƒ#]

#= ƒ""

#% ˆˆ

#ƒ <>=

#< < ƒ

. . .

. .

. . .

. . . . .

. . . . .

. . . .

. . . .

$   > )  '$ €NH€ ‹                             

AB

B

Metals + plastics

D

U Metals

BF Plastics

>=ˆ

}     Z  €_    €  

$  $Ž  ;  ;   '$ €N‰„  '$ Q‰O Bolt head

Shank

Thread

Nut

Nut

Dome head nut

Pitch – the distance between points of threads

  



Z< Z" Z#= Z#% Z#< Z%= Z%ƒ Z>=

  



#== #%ˆ #ˆ= #]ˆ %== %ˆ= >== >ˆ=

: 

   +; 

: 

  



: 

)    

   

   

#= #> #] # %ƒ >= >< ƒ<

##ˆ #ˆ= # < %# %]] >ƒˆ ƒ#< ˆ>#

ƒ>]ˆ ˆ"]ˆ ]ƒˆ= "ƒˆ= #=ƒˆ= #> == #ˆ == %==ˆ=

ˆ>]ˆ <"]ˆ "ƒˆ= #=ƒˆ= #>ˆˆ= #<ˆˆ= # <ˆ= %ƒ"ˆ=

 

 ;) 



#ˆ %= %ˆ >= ƒ= ˆ= <= ]ˆ

   

O

%=# ><< ˆ"= "ƒ> #ˆ]= %ƒˆ= >ˆ>= ˆ<#=

  †” +#  %  )  '$ €O„

  



Ž  



 



<< = ##= #ƒ= #"= %%= %<= >>=

#%ˆ #]= %#= %ƒ= >== >]= ƒƒ= ˆ<=

#< #< %= %ˆ >= >= ƒ= ƒ=

” *      €  €   €  ‡       } ƒ>%=

'



'  † Q

Œ

>= ˆ= ]= #== #%= #ƒ= #ˆ= #"=

. .

. . .

N„

NO

NQ

O„

O€

. . . .

. . . .

. . . . . .

. . . . .

. . . .

. .

”         

Z< Z" Z#=   Z#%                

$       2(D

15°

15°

Podger

Spanner

) +1

3.2 D

>=<

      Q `[ }[

$    )  '$ €NŒ ?   Z>  Z%=             [                      [ Z      

               '

Pozidrive

Slotted

Cheese

Hexagonal

Cross slot

Round

Phillips

Pig nose

Counter sunk

Square

Allen key

Fillister

$    )  '$ NON„

                       

  ”



"  QKOˆ

HK‰

‰KˆO

NOKˆ

%ˆ >]ˆ ˆ= ]ˆ "]ˆ #== ##% #%ˆ #ˆ= %==

. . . . . . . .

. . . . . .

. . .

.

.

.

. .

. .

. . .

$  )   ;   '$ €€‰

Fillet

Seam

Vee butt

Double vee butt

Spot

Spot flush one side Spot flush two sides

}     Z  €_    €  

>=]

  ) )  \               

     “                      [               _                      

  €                     ‹    # \ ?         _    † _            [  ‘                             [  . \                % \       

      _    *           

        .   [ 

                     ˆ \        #= \         [   ˆ \       . \                                          €  [                   \       ˆ {$%           . Z_         $                                       _        <= \           [   

_                   

'   €    [  [ [      _         [ 

  Z                  % \ ~ [ _      % \    [        [                     _  #      ˆ  <                  ?     

                    

     

     [       . } [            % \          

   _    } [      

         <                             . `[  [  [      [             [          $   [ ?      ‹       [ [            ‘                         ]                    [ [     [             .   [        _       _     [    _             [  _          † 

>="

      Q `[ }[

%                                   [                                                                                             ? #ˆ= \         %ƒ= \      ƒ==Žˆ== \ `\        

 †                ? <==Ž"== \               ? ==Ž#=== \                                                                ‚        

       [        

<    \    [  [            _   

                        [     $      €               #==Ž>== \      "ˆ¢          >==Žˆ== \       ƒ=¢             ˆ== \          ?               >=Žˆ=      _                                                 ?   >== \          [      ƒˆ=Žˆ== \             }               ]=¢   ‹   

}     Z  €_    €  

>=

(                      Z                ‹    _          

        €        >ˆ=Žƒ== \                

 ;      #%=Ž#ˆ= \ [   %==Ž%ˆ= \               ƒ== \ \

                                   \

              *       

              †       

')  } [            #=== \         [                                 \                  _                [     ‘ [            [               [ 

$ )          %= \       ˆ=¢  ˆˆ= \    #=== \   #=¢          }                      

                             [      <== \ 

            [ \  [                \

     [               ?  _     

    ƒˆ= \ 

       <== \     "=¢         "== \ 

     <=¢     

>#=

      Q `[ }[

  

?     _                      ? ‹

                 †  [                   [        ?                              [   [                     '                     [                  „                €                             [               _   ‘                                                                                                ~          ‹                       [                                   \                   

$

     "

%]# [{$>

(   

=>%

      / 

]= [{$%

     /

%> [{$%

      : 

%ƒ  #=

< 

$ \

          

 ‘   

ƒ <

{ ‹    €  ‘

‘   Ž    ‘   Ž    €  ?

     

$

       

  '$ ŒNNŒ                  _          



. ; 

> ; 

 

    ”

  



<=<>

ƒ <

  _            

       

<="%

ƒ <

ˆ=">

"; 

 :K      )  +ƒ-

   $ *

O

$ * $ *

O

$! *

O

#Ž#ˆ= #Ž#ˆ=

}

= ˆ=

<ˆ #<=

"ˆ #]ˆ

ƒ= ˆ

    _  

#Ž#ˆ= #Ž%= %=Ž#ˆ=

}

= ˆ= ˆ=

##ˆ %ˆˆ %]=

#ƒˆ %]ˆ % =

]= #ˆˆ #<=

 € ‘%%

    ‚          [

=%Ž"= >Ž%ˆ =%Ž<

?

= = ƒˆ

#=ˆ #>= %>ˆ

#ˆ= #]= %]=

<ˆ ]ˆ #ƒ=

*Z ˆ

€

Z             

Ž

?

Ž

            

       ? _  '

*Z <

€

‡  _ 

Ž

}

Ž

 

    \  

ƒ=Ž#%=

]=Ž#ƒ=

%ˆŽ]ˆ

Ž

”}  ?    _                  

]

>##

$ = } "##"' `  #' #

>#%

      Q `[ }[

";  \

            } "##"' `  %         ?  } \

          ?                  `       }            [     >                                               

         '     “     ‹ ‹     _‹  “ [              “                 }

                

#    ?              ?          Œ Q           _   [        %==  %ˆ= \ ƒ== \              <== \ #%== \                  [                              

   `                          

$  †  :       '$ NNQN $   

<  † +

 

: 

         \

      

>=  >=  %ˆ ˆ=  >"  >   ƒ <=  >=  ˆ   < <=  >=  ƒ   < ˆ=  >"  >

#%=  #%=  #= #ƒ=  #=ˆ  "ˆ   ## %ƒ=  #==     #> #<=  "=  ]   ## #%=  =  #=

‚   [   <ˆŽ#ˆˆ          >        #ˆ 

}     Z  €_    €  

>#>

$   '$ ŒNNŒ= ( N (         } "##"                                                |                                    } "##"    ŒQ  

(        

N

|  ~     

#%=  ="= #>> #%= #==

$         ;  O

   ;  |                                                       

        

#= #=

        

="

        

=<

        

=ƒ

(             ‹       



 ;

@ 

‚     ~ } $ 

#% #% #%

#% #>  #< >=



       '$ ŒNNŒ ?  } ˆ ˆ=                ‹           ?               ˆ . =g                                 [         

     $ = } "##"' `  #' #

]

.  !                 …  !† ˆ   !    ! …  !† ˆ   !    !  …  !† ˆ

   ! #    !

$        ‡  ! ˆ %  #% … ‡ !†  #% … ‡ !† 

 ! ˆ %  #% … ‡ !†  #% …

!†

 ‡  ! ˆ %  #% … ‡ !†  #% … 

 ! ˆ

%  #% …

‡

!†  #% …

!† !†

 … ‡ !†   !  … !†  ! ˆ   !

 ‡  ! ˆ 

(      %   ! ˆ  …

!† ‡  … ‡ !†

%   ! ˆ …

!†

%   ! ˆ …

…

!†

!† ‡ … ‡ !†

       )      % ˆ %    % ˆ %

 %

%

 % ˆ %   % ˆ #  % ˆ

#

#

%  % %   %

 % ‡ % ˆ # % ˆ  % ‡ #

  Z

<         D A

b

c B

C a `        '  '

% ‡ % ˆ %    ˆ ˆ   !  

 ˆ

\  '

%  &…& †…& †…& †;     ˆ … ‡  ‡ †=%

% ˆ % ‡ %

% 

% ˆ % ‡ % ‡ %  (  ˆ

% ‡ % % %

>#ˆ

>#<

      Q `[ }[

$  

5 4

3

√2 1 45°

1

2

60° 1

30°

f

√3

b e

a

c d a = c = e d f b

  Z

>#]

 ;    J % ‡  ‡  ˆ =

ˆ



p % ƒ %

% ‡ % ‡  % ˆ … ‡ †% % 

>

% ˆ … ‡ †… >

 ˆ …

†… ‡  ‡  % †

( ) 

†

 ˆ ‡

%

 ˆ  



… † ˆ 

            #= …#= †  #= … #= †  #=#=   ˆ %:]#"%" #=  ˆ %:>=%ˆ #= 

  ˆ

#= 

%&   !   %

%

%

 %  % ‡ ˆ# % %

%

 ‡ ˆ

b

a

a

>#"

      Q `[ }[

. ; 

(; 

% %

 % ˆ 

% ˆ# %

a

   ˆ



% ‡

 % # ƒ % L Arc of circle

d

R Centre of circle

  Z

<          2 / …/2† ˆ / ‡ 2      /  # / ˆ % 2  2 2 

/

2 



  2 / …/2† ˆ /2 ‡ / ‡ 2     

…/2† ˆ /



…2†



/ 



…2†

$       "  ˆ "" 

"‡# "‡#



"  ˆ

 #   ˆ  



#  ˆ   

   ˆ  



  ˆ

   ˆ    



  ˆ

   ˆ  …# ‡  † 



   ˆ … 

   ˆ   



   ˆ

   ˆ 

 



   ˆ  

   ˆ %  



   ˆ

   ˆ 

% 



   ˆ  … †

#  ˆ p # %



%   ˆ  



%   ˆ

# p  ˆ 

# %

 



#

  

#

 



#

  

#

#

#  ˆ p # % ˆ

# # ‡ %



ˆ

# # ‡ %



 

" 6ˆ #

 6ˆ =

  

 > =;  6ˆ = #†

 

 … †

#  ˆ 

# ‡ %

 

#

#



 jj<#

>#

.   !  †     #   }  ‘  \  ‚ €  ~ }   }#ˆ #{   [

' =#%# ƒˆ< ##=> _' ="]= #>] "ƒƒ>

    $   ?

Q   ‘ % \     * 

\ƒ ˆ‘}       [

' =%= ]%>< > >ƒ _' =%= ]>% ><]]

;      ? ‘  ? ‚ ‘  ˆ# `?    [

' =#] ƒ ><"]#] _' =#] ƒ ><" ]"

 <  '  +<']> ‘   *  ~#{ < ' =%= ]ˆ"= ˆ"<#    [ _' =%= ]ƒ>< ˆ%< ;  < !       +<?  ‘  %>] }   ‚ }  ‹  ‹ ' =#%"> ˆ>##%<       |# >} _' =#%"> ˆ<"%%"    [        !   % ~ ‘  `     *  {# "`  [ 

' =%= ]>ˆ "=== _' =%= ]>ˆ ="<>

       % ?  ‘  #% \_   *  ~# +*   [

' =%= ]%%% <ˆˆ] _' =%= ]%%% =]ˆ=

    (  $!  +($#< ‚         ‘# %}   [

' =#># %%# ˆ _' =#># %%# ==<#

' "!       +'"~ ‘  ~ [ ~   } [  *ƒ %|¨  [  [

' =#>ƒƒ ""ˆ<ˆ# _' =#>ƒƒ " <#%

  ? 

>%#

' !  $     +'$>> € ~    ‘    ‡% ' =#ƒ>" >ˆ"ˆ#ƒ "}~ _' =#ƒ>" ]ƒ%ˆ<ˆ  [ '  ; ‚}? << `   ` *  ~#{ ƒ?|  ‹   ' '    ›  +''` }_ # ˆ } [  *  ‡   ~  ‘  ~|% ]{‡   [ '      +'\   ‘    ?   \   } [ ‚‡ƒˆ <—   [

' =%= ]ˆ"= ˆˆ>> _' =%= ]<># #"=% ' =# %> <<ˆ>== _' =# %> <<ˆ>=#

' =#>ƒƒ ]<%<]< _' =#>ƒƒ ]<#%#ƒ

'    $ )      +'$ƒ ~ \  *  ~#? % ' =%= ]"> "ˆ<<      _' =%= ] ]< #<>ƒ ' ; <   ‚ *  {~# %|}  [

' =%= ]ƒ#% ]<]< _' =%= ]ƒ#% ] ˆƒ

' >#    #  +' ###= ‡   \     }   ' }#ˆ >? _' ' (   #  +'(#<= \    * *# #€} '     _'

=#%# ƒˆ< >>%% =#%# ƒˆ< #> ƒ =##< %ˆ> <#<# =##< %ˆ# ƒˆ<"

' <;;      +'<< } ` ‚     *  \%? >¤ ' =%= ]ƒˆ] ˆ=ƒ=   [ _' =%= ] ]% ==" ' $   +'$]= \  Q ‚ *  ~< ‚     ' $  $    *   ‘  > Z  ?    #= %+¤   [ ' $   +'$>" \[ ‘ ‚ *  ~ƒ ƒ?* ‹ ' $  ^  ‘  ]] \ ‚ ~  

~> +‘  ‹ 

' =%= "]ƒ# #%># _' =%= "]ƒ# ƒˆˆˆ ' =##ƒ % = =""" _' =##ƒ % = =" ]

' =%= " _' =%= "

< ==# < ]==#

' =# =% ]#]]" _' =# =% ]#]]"

>%%

  ? 

' )  '  ~ ‡   ~  ~|# >+?   

' =# %> %%<ƒ%% _' =# %> %%<="#

'   (!  "  (     +'("# ‡  ‘    ‡ |  |# #` ' =#>>% %%ˆ#==  [ _' =#>>% %%ˆ#=# '   %<     *  ~\# ]}¡      [

' =%= ]< % ƒ=== _' =%= ]ˆ"= <ƒ#

'  < !  $! } [  *  ‡   ~  ~|% ]¤‚   [

' =# %> <<ƒ<<ƒ _' =# %> <<ƒ= "

'  < %;   +'<%} [  *  ‡   ~  ~|% ]¤‚   [

' =# %> <<ƒ=== _' =# %> <<ƒ]"]

'  $! <         +'$< } [  *  ~ } [  } [ ‚‡#% ]?‘   [ " ?   .     \  }   \ ` [ \  \€#= >{+   [

' =#>ƒƒ ƒ%<ˆ## _' =#>ƒƒ ƒ"]ˆ]ˆ

' =% %=ˆ= =%== _' =% %="% <>]ˆ

 +       <  $ ` [ ‘  %# ` [ ‚   [ ' =#]>% ƒˆ==== ^  {#> #¨‚ _' =#]>% ƒˆˆ #]  [    #  ƒ] }   ‚ ~ }  ‚ ~ }  }]= <`—   

' =#%# <=# <> = _' =#%# <=# <> #

 "!    ?   }   }ƒ" ]+}

' =#ˆ%] <<ƒ#ƒ _' =#ˆ%] ˆ"ˆ=]=



       '  %!   +'%  }   ## — [ ‚ *  # ' =%= ] <= %ƒ== ]{¨ _' =%= ] <= %ƒƒƒ   [   <    +<? ‘  %>ˆ ? ‚ ?  ‘  ‡#% ƒ||       [

' =#%ˆ% >%#>=% _' =#%ˆ% >>> =#

  $  \   ‘    ?   \   } [ ‚‡ƒˆ <—     [

' =#>ƒƒ ƒ<<==] _' =#>ƒƒ ƒ<<=="

  ? 

   #:    *   ‘  Z  ?    #= %+¤  

>%>

' =##ƒ %<< >="ƒ _' =##ƒ %<] = #=

    <         +<<  Q ‡ *  ~#` >? ' =%= ]%%% "" #    [ _' =%= ]%%% #]="   "!      +"       %%ƒ *  ‚  ?  ‘  ?*# #?+   [ Ž<$     ` }_ #    {  *    |{#< #}`              '  # # \   ‚ \  ^  \  ‡ ‡*ˆ> "|€ ‹‹  [ "   >ƒ }   *  ~\% ]|*     [

' =#]%] ]>#%== _' =#]%] ]>#%#<

' =#]%ƒ ƒ=ˆ=<= _' =#]%ƒ ƒ=ƒ%%ƒ ' =#%ƒ% ˆ]"%]" _' =#%ƒ% ˆ]"%">

' =%= ]ƒ%= ˆ%== _' =%= ]ƒ%= ˆ>==

%  . %>   ‚ *  ~#¨ #?} ' =%= ] ]> >===  ‹   [ _' =%= ] ]> >==# %!    . $! +%.$- ?    \   ‘  #=Ž#" ?   } ' =%" =ˆƒ >=>ƒ }% "‡}    [ %!    ‚ ‘  ~  |   ?† ~ ?   }  }#% ƒ|     ‹   [ %   †    +%` }_ ƒ#< {* #"== ?[ ?[  {      %  $  $ !  '  +% > :%ƒ# ‚  Q?  *‹##ˆ= * _   ‹ _ 

' =#ƒˆƒ <%ƒƒ== _' =#ƒˆƒ "]"<#ˆ

' ‡># ]%ˆ##ƒ#<# _' ‡># ]%ˆ##>]"> ' ‡>ˆ% %< #= >= ˆ= _' ‡>ˆ% %< #= >= ˆ#

#        %&    ' =%= "<<ˆ ˆ]%] ? ‘  } [ ‚  

_' =%= "<<ˆ <ƒƒ] ‘ 

 \‚] ]¤‡‡    [

>%ƒ

  ? 

#    '  €Z} ‘    #ƒ$#ˆ ‡  ¤   *  ~\#{ >|`   [

' =%= ]%ƒ% ]ˆ"> _' =%= ]ƒ=ƒ =% <

#   (  $  €  \  "> \  \ ‚ }[  ^  }‚> #{‚   [

' =%= "<<> = ƒ] _' =%= "<<> = ƒ

# (      +#(} \  % `  ‡   *  # "{|  [ #   % %<Ž%"      *  {# ]¤+  [  !†    ~  Q \  ˆ<   ‚   \ ~ Z  }]% #—   [     < €†     *  ~#` <|¨       [     † #  ƒƒ }   ‘   *  # #¨}   [      ;    \ ‘  [  *  ‘ ~ ‘`#ƒ ƒ{|   [ .   $ %:! +.$%‚ \  %   [ }  *  # ‘   [ .  $   *   ‘    `    ‘ #‘  ‹   [

' =%= ] =% ˆ>== _' =%= ] =% ˆ>=# ' =%= ]ƒ = #ˆˆˆ _' =%= ]ƒ = =""# ' =#%# >ˆˆ "">" _' =#%# >ˆˆ "]%]

' =%= ]>"] #]%= _' =%= ]>"] #]%# ' =%= ]ƒ=> ]#]] _' =%= ]>ˆ] ]ƒˆ" ' =#ƒ ƒ ˆ<ˆ#"= _' =#ƒ ƒ ˆ<ˆƒ"]

' =%= ]]#] <=== _' =%= ]]#] <]#] ' =#># <<" "]=] _' =#># <<" "<<

.  < *  Q 

€ *  ~\%? >`‘     [   '    +'%Ž#=ƒ     

 ^#] #}   [$ 

' =#>]% ]ƒˆˆ]] _' =#>]% ]ƒ"%"%

   ! % +%#Ž] ‡  ‡    *  ~#` >??   [

' =%= ]%%% ]]%% _' =%= ]%%% ]ˆ==

' =%= ] #] """" _' =%= ] ˆˆ =##=

  ? 

>%ˆ

   $ % +$%##  }     *  ~#¨ "}‘    [

' =%= ]%>ˆ ƒˆ>ˆ _' =%= ]%=# #ˆ]

      ! ƒ= {  ‚ *  \# =‘}     [

' =%= ]>>% >"%= _' =%= ]">> #><

   Ž *  ‚ } [  } [ ‚‡#% %¡   [

' =#>ƒƒ ƒ%=%ƒ% _' =#>ƒƒ "ˆƒ ƒ>

  '  $   + '$Z  ‘  \   \ {    {# >‚    [

' =# # %>% ˆ ƒ _' =# # %>% ˆ]#ƒ

  #   "      + #"#? { ‚  \    Z_ ' =#]"ƒ ƒˆ<] _' =#]"ƒ ƒ<###" ~#" >|‘  ‹      <  (     \   ‘  ˆ<Ž<ƒ *     ' =%= ]<=" ˆ= *  \%? ƒ¤¨ _' =%= ]<=" ˆ="#   . >'    + .'}  [ ‘  \ ?   ?  } [ ‘`< ˆ?`   [  "!    + " ‘ ?   }   {ƒ" ]+}    Ž $! ‚ ‚    ƒ|‘      [ (  ( ) #    €  ‘  #=    }   }#" <*   [

' =#ƒ ƒ ƒ>ƒƒ]] _' =#ƒ ƒ ]%"ˆ%#

' =#ˆ%] ˆ"ƒ ]]] _' =#ˆ%] ˆ"ˆ ˆ<% ' =%> "=] %=== _' =%> "=] %ƒˆ% ' =#%# %>] ##%> _' =#%# %>] ##%ƒ

( <    +(<" ~   ‚    Z_ ~## ' =%= "<#ƒ ƒ"== "*| _' =%= " ƒ> ƒ=]ˆ    [ (   <;; !  $!/ ' (  #  +'(#< } ` ‚     *  \%? >¤ ' =%= ]ƒˆ] ˆ===  [ _' =%= ]ƒˆ] ˆ=ƒˆ (    ;  ; ?    ‚ ‚ ‚   } }>ƒ ˆ+€    [

' =%" ƒ=<> %="% _' =%" ƒ=<> %=">

>%<

  ? 

J (     #ˆ< } [  ` ‚ *  ~#~ ‚   <   * )  < ‚ [ ‘       *  {~# %¨    [$   [  < :   ' \   ‘    ?   \   } [  ‚‡ƒˆ <—    [

' =%= ]]>= "# ƒ _' =%= ]]>= ƒ>ˆˆ

' =%= ]ˆˆ] "=== _' =%= ]ˆˆ] ===

' =#>ƒƒ ]%ˆ]>% _' =#>ƒƒ ]]ƒ ]<

<         $   +<$#ˆ ‚         ‘# %} ' =#># %% ]ˆƒˆ    [ _' =#># %%" %#"" <    '  +<'<< `   ` *  ~#} #?|   

' =%= ]ˆ"= ˆˆ>> _' =%= ]%ˆˆ #ˆƒ#

<      $!  +<$#% ‡  ‡    *  ~#` >?| ' =%= ]%%% ]===    [ _' =%= ]>>ƒ >"== <  $       }  }   ^     ?   \ ` [ \  \€# >{}    <  $      +<$% Z  \  } }] #{¡     [

' =% %="] ƒ]ˆ> _' =% %="] ƒ %<

' =%" =>% >]<= _' =%" =%> ]>#>

$     (     '  >]     *  # <|— ' =%= ]>]] #<ƒƒ   [ _' =%= ]%ƒ] ˆ% < $  $ !  $! ‚ %=ƒ  

  \   %#] `   # ƒ|`   [ $  Ž +!  .$Ž` }_ %  {  {‚> #‡{   $     +$ ` [ } [  ‚ ? } [ *ˆ ]+{ ‹  [

' =##ƒ %%ƒ %%ƒ= _' =##ƒ %]> =ƒƒƒ

' ="]= <== ˆˆ%% _' ="]= <== ˆˆ>> ' =#>ƒƒ <%>>ƒˆ _' =#>ƒƒ <%% ƒƒ

  ? 

$  #   ' +$#'\   ‘  ˆ<Ž<ƒ *     *  \%? ƒ¤¨  ‹    [

>%]

' =%= ]<=" ˆ="= _' =%= ]<=" ˆ="#

  $   $ %    #" ‘  ~ ~  ' =#]" "ƒ% "%% ~ [  \>ˆ { _' =#]" "ƒ% %%  ;  #  %< _    *  ~#— ƒ*  [

' =%= ]"> #" # _' =%= ] >= == ƒ

<"    [  *  ‘    ‘ ~ ‘`#ƒ ƒ{|    [

' =#ƒ ƒ ˆ<>= # _' =#ƒ ƒ ˆ<ˆƒ"]

  $     #ˆ   \   ?   { 

{{> >‚?  [ [

' =#<=ƒ ƒ=ˆ<<< _' =#<=ƒ ƒ=ˆ<<<

  $  # `   ‡   }  ` [ *  ~ƒ #    ‹  [       # +  ?

Q ‡ *   ~#‘ }    [  @    #] ¤  *    ‘    \   `%" ˆ¨  „   [   (  #  %" Z [ ` ‡   *    {‡># <*‚

' =%= " _' =%= "

ƒ #=# ˆ ƒ" ˆ

' =%= ]>ƒƒ #"ƒƒ _' =%= ]>ƒƒ #"<< ' =#ƒ"] ">ƒ=>ƒ _' =#ƒ"] ">%%>%

' =#ƒ]< ˆ<>]=] _' =#ƒ]< ˆ] >#ƒ

>%"

  ? 

    !   >Z ‘  %" ‡  ¤[   Z  Z#ˆ ƒ`? $ [   <    } ‚   ~ Z  |—ƒ ‘     [    )*     ^  \  ƒ#Žˆ# ^  ‚ *  

 ^%% ]¡   ' '    ?_   b % ]=ˆ=#ˆ }   ‡   ‹  

' =#<# %>< "ˆ== _' =#<# %>] ##=ˆ

' =#%# ˆˆ] ]%ƒ# _' =#%# ˆ%% ƒˆˆˆ ' ="== <#ƒ#<

' ‡ƒ ]%ˆ% ˆ=>= _' ‡ƒ ]%ˆ% ˆ=>%">

'< '  (  \  ‚ ?         #< >}`  ‹     [

' =#]"ˆ %%%%"" _' =#]"ˆ %ƒ==%

'    %:    ‚ }   _   ¨#< ]{

' =#% ˆ ƒˆƒˆƒˆ _' =#% ˆ ƒˆƒ< =

 *  +-  % ‚    ~      \  ~  _ ‚‘#= %`}  

' =#% > ƒƒ]"]" _' =#% > ƒƒ]""=

 \  ^ * `        \  \€"> %~¤  

' =% %=>> ] == _' =% %="< >#]"

Ž<$   ~ ‚ \  {{#] ˆ?    $     

' =#ˆ>< ƒ=%#%# _' =#ˆ>< ƒ=ƒ###

 `      \   Z  ="ƒ== ‡   }   

  

' •>ƒ > "ƒ= ƒƒ]=

" )    Z  }   ` [ \ |   ? \    \ˆ ‚‡   

' =#<]< ˆ%"=== _' =#<]< ˆ%"==#

%  ¡  $`  [  ‚b #" ƒƒ==   ?   [

' ‡ƒ> ]%ˆ%" ƒ= _' •ƒ> ]%ˆ%" ƒ%ƒ

  ? 

%     < "   #= \    ?   ` [ ‚ *  {~#= ]‚ # #  +.   -  # } ‚ ‘  ~ ‚  _ ‚Z> =¤‘    [ #  ˆ   \    ‚      

 ^ˆ {{ 

  . '    }   Z^ƒ> *¡   ‹ [  .   ‡       ` [ Z  Z  Z%ƒ #~     †     ?   ‚ \   ‡  {`ƒƒ #¨—  [

>%

' =%= " <# <=<< _' =%= " <# #ƒ## ' =#]=" >]ƒˆ>ƒ _' =#]=" >ƒ=ˆ##

' =%= " =# >ƒ== _' =%= "ƒ%% ><

' ="]= ˆ%ˆ"%ˆ" _' =#%>ƒ ]<%=ƒ= ' =#<# <ˆ> >=>= _' =#<# <ˆ> >=>#

' =#<>> ƒ"<ƒ<# _' =#<>> ƒ"<ƒ ˆ

$   ?  ‚  }   ~ Z  }" #}} ‹  

' =#%# >%< ˆ=== _' =#%# >%< ˆ==ˆ

@       >  ` [ €  ‚ ‘ ‘ ‘`% ]+   [

' =#ƒƒ% "ƒ #== _' =#ƒƒ% %> %"]

   ~ ~ ‡  \ ‘    ?*] #¤}  [

' =#]=] >ˆ""== _' =#]=] >ˆ" ==

(   %< Z[ ‚ *  #ƒ }   (     `  ‚  ‘  ~?#= > [  

' =#]ƒƒ < %=== _' =#]ƒƒ <#>=ƒƒ

( *"! '

    ‘   ‚ } ‘ \  \}> "‘{  

' =# ˆƒ ]"=<"] _' =# ˆƒ ]"% #%

J      +J\   ~ Z

  }   ` [ Z  { {‡# ]¤¡

' =#<%> ƒ#<>== _' =#<%> ƒ#<>=>

' =%= ]ˆ># ƒ<== _' =%= ]ˆ># ƒ<#=

>>=

  ? 

<  $ "  Z €  ‚ ~  ?  ?  |   |< #‘{   <" ) 

} [  *  ?  ~ ~ "}~    [ [ $   $  ; }    ‘  ~ }  \    \> %{|  ‹ ‹ $($ ;  $) \      [ ~ Z  }<< %?  $  $   #: ‘_ ‚ \ _      ‘ ~ ‘`#% >  $   `†† #> >ˆ=#=    `           ~    ?  |   |< ƒ`‘    [  ;     ‚ ~ }  ~ Z  }]= }    [

' =#>>ˆ >== _' =#>>ˆ >=="""

' =# %% ]ƒ>]ƒ> _' =# %% ]ƒ>ƒ== ' =%ƒ ]< ˆƒ]ƒ== _' =%ƒ ]< ˆƒ]]

' =#%# ˆˆˆ ""ˆˆ _' =#%# ˆˆˆ ""<<

' =#ƒ ƒ ƒ<%>%% _' =#ƒ ƒ ƒ<ƒ]ƒ]

' ‡> =ƒ _' ‡> =ƒ

=ˆ=#== =ˆ= <ƒ

' =#>>ˆ ><=<=# _' =#>>ˆ ><==#ƒ

' =#%# ˆˆ> ˆ>"ƒ _' =#%# ˆ== ˆ= ˆ

K @K   ~ [ ^ ‚ } }*% %?*   [

' =#%=ƒ ˆ%#]]# _' =#%=ƒ >"%##ˆ

‘ %!    ` }_ #<ˆ  ~ [ \>ˆ "¡| †      [

' =# %< <%ƒ << _' =# %< <%ƒ %<

+  #( $   N      ?\ # " " 0"  0"& 0* 2  +#" !6 %    ?  \        }[ *  # "   2  "+ "O& *" !00, ƒ   }   ‹‘ 

 \`\ # "    ) "+& *0# 0##0" "+#" 0* 0,  0"& %    \`\ || { #

#'   # !& & 0* (& +" "  0"& 0" //& }

‘   #

 0" )/"O& ,00, ?   ` 

O $     (   |‚ #

]    Q [ ."0 00, ‘Z

‘ %==# "+ "+ ) \ * " 0"&/ 0"Q ( +/ 0"& 6]]^Q ..02 0 0*   ‘ ‘ %==# ) \ * " 0"&/ 0" ‘     ^   '    [ $   [ $   [ $   [    [ $    [ `\ # <    . "Q 0##" 0"     6]]_ `   ’  \ 

 " " } <ƒ"' # ]= / 0*  +& 0* !/  "+  & } } ˆ<=<' #

= /  0 / " !/  "+

} <#"=' #

ˆ 0 0*   *0 0 2 ! & " " 0/ !/  "+&Q }

} <> 0 "+ *0 !/  "+& `  #' # < 0 0*   *0 ( " #.0& 0& `  %' # ] 0 0*   *0  " 0& `  >' # "" 0 0*   *0 #.0& 00* 0& }

>>%

€  ‚ 

\‚? ‚  ### # "< // "02 0" 0*   0" !/  "+& \‚? ‘   #

] 0" )/"O& //& 000, ?   ` 

                '   [ *  { # <]  " .& 0* // (& +" }  *  ? ‚ #

]  & *0  & " !/ & ` " "0/ 0" ? 

Z   ~ # <ƒ  #"& 0* // `  ‚   \ %===  |  ‡  O& $0/" O**„„ ˆ<Žˆ  „**„„ ><Ž>  Q*€*„„ >=Ž>#

€

' $      $  

}  ? # ]" {  €         |  Q  // "+ j " K ‰*   ˆQ %ƒˆŽ%ˆ> \   ¤ ‚ ’ €

  ‚ ? # \  ¤ #

 " "+ " "+ ( !00, Z ` 

> "  "+ "O& ( !00,Q }   ‹‘ 



‡  ¤ Z ’  [  ` # ‘ ‘  #

= " & 0*  & >     \

ƒ // "& &  #. ? 

¤   ^ ~ # ]% {   " 0#/ *0 & \    \   ?  Z   ‘ ‡ #

< // " && "& &Q }   ‹‘ 



Z ~ ‘ ’ }  ¤ ‘ # "]  "*0 0" (& +" >    Z‹     \ #   *

ˆ #.O& "+ " "+ {00, #==   Z‹‡   

~ ‚ ‘ # <# &  " &  (& +" 0* & " &  ’ ‘  

ˆ

 

} †    ‡ # <# * }   \   |   ` €   0Q 0*  | "" 0" 0"*" 0" 0  " &Q  &  K O ##Ž#% } ˆ >=' # "# 0 0*   *0  "2& + 0"& } } "==ƒ' # "< 0 0*   *0 0/" 0"& } } "==%' #

ƒ 0 0*   *0   " "+ //& }

\  ‚ € # || { #

> 0  " & ˆ   \ ‘

]'   " 0"  (& +" }

     ?  # %    ‘Z

] " # / " 0"  ( &.0& 0* 0"# " 0 &

€  ‚ 

>>>

     ?  %==% 0"# ""& " 0 [ 0 0" 0* 000+  ( " ", '/& *0 )/#"& \*‚ ‚   ‘Z ‘   ¤ }   # <# ? ‡   €    }   \ (" & 0"  "& / !/ " K NN ? ‘   ¤ }   # <" ? ‚  _  €    }   \ (" & 0"  "& / !/ " K OŒ ? \  `   €       # ]" (" & 0"  "& / !/ " K O \‚\* # "] / " 0"  20.#" 0* 0"# " " %    ‡   { ˆ $"> | ^ ‚  # "=      Z  `  0Q 0*  0"*" 0"  # 0" 0* 0"# " "   \     'O #Ž#ƒ {‘}\  0" )0/&j!/  "+ 0/"  "&  † ^ ’ `[ ‚ } # ~

< 0  " & " "+ " "+   >   

   Z ¤ %==# 0/" 0" (& +" " 0"&/ 0" ]   `  

Q  ;  ( )  } ˆ%<"' `  %' %==% // 7& 0*  # } || { #

ˆ'   < (& +" 0*  # //& }

†   \ ’ }  ¤ ? %==%  # (& +"&O "/ >    }[

H   } ˆ ]]' `  #' # "#  "&Q 0 *0 &&&&#" 0* 0 } } ˆ<%" 0 0* .  *0 #&0" `  #' # % // /& 0* /" "*0 #&0" `  %' %==#  & \ 0#.0""&} & +" \ 0,#"& . } \`###' # ]= 0 0*   *0  (& +" 0* &0" " !/  "+ //& } \   ~ ‡  ‡ ’ }[ ¤ ^ # "] // &0" (& +"&O "/} %    }` `   }[ || { #

<'   ] (& +" 0* &0" //& }

‘  ¤ #

ˆ  0" ,0" \      ` 

‘ ¤ ? # #=  00+ 0* !/  "+ 0"&   ?  ‚   |  `   %=== "/ *0  (& +" 0*  " &0" " !/  "+ //& #

] \$ 

Œ <    } ˆ>%"' `  #  ƒ' # } ƒƒ">' # "ˆ } ƒƒ">' #

] 0" . *  0" " & "+ } "    *0   "*0#" 0* 0" }

>>ƒ

€  ‚ 

} "##=' `  #' # 0"&/ 0" }

] // 7& 0* 0"Q 0 0*   *0 (& +" "

} "<<<' %=== . *  0" *0 &/ "+}  #"& 0" "+} " "+ " / "+ 0* &  "*0#" *0 0" } || { #

%'   % (& +" 0* 0" //& }

|| { #

%Žƒ'   >  -/   " "+ " 0" " "+ //& }

"/ *0  (& +" 0*  "*0 0" !/  "+ //& %==% \$  Z ~ ‘ ’ }  ¤ ‘ # "]  "*0 0" (& +" >    Z  {  ? Z # ]] 0. & 0* 0" ` 

‰

$

} { ‚ ’ }   } ? %==# // (& +" 0*  "&&   \ ‹      `% # } ƒƒ ' `  %' # <     7& 0* //  " !/  "+ } } ˆ ˆ= // 7& 0* 0, " !/  "+& `  #' %=== 0 0*   *0 (& +" ` 0 "   0"& `  ˆ' # " 0 0*   *0 (& +" ` 0 0#  " /+  0"& } || { #

>'   ƒ (& +" 0*  //& }

Z‡   ¤ ’ ?   \ # % // 0, (& +" 0  #   0 %    }   ‹‘ 

 "/ *0  (& +" 0* 0, !/  "+ //& %==% \$  {[ |      # #%=##

ƒ (& +" "/ *0 //  "&&  {|

  ‡ ~ ’ ^  ` ‚ #  

ƒ   (& +"O& "/ ˆ   }[

\ %==# 0, (& +" /  0 ! |]|~[  6[ ~~~ '0/# 6  0" 0.j  & " # .  & <    \      `%=%

N„    $    } ˆ ˆ= // 7& 0* 0, " !/  "+& `  >' # = 0 0*   *0 (& +" ` 0#.0&  0"&/ 0" `  ƒ' # ƒ 0 0*   *0 (& +" ` 0#.0&  &   0*    "+ } || { #

ƒ'   ˆ (& +" 0* 0#.0&   " 0" //& }

{ ` ~ ’ * *  # "< (& +" & *0 0#.0&   " 0" !#& \   \ # = 0##" 0" ! |]|~[  5[  0" 5Q6 0#.0&  !#&  \ ‹      `]"

€  ‚ 

>>ˆ

NN   && " "  "+ "  !/ " %=== `[    { #>ƒ]ƒ ‡  }   Ž |  ‡ `  `  # !& & *0 (& +" `  % (& +" *0 7" *0# ( & / 0& } // 7& 0* && " !/  "+& #

  

NO '  %   }‚ %==% # "&/ 0" 20  "+  &,&Q 00   /  /..0 "+ !/ j

"+ +/ 0" -/ #"& >    }‚ %<% \‚\ * } "==]' # "] 0 0*   *0 (& +" 0* 0" //& *0  " "+ -/0/&  -/ & } } "#=%' #

= 0 0" 0* &//& + "&  *0#  +0/" }

} "##"' `  #' #

# // 7& 0* /# " /#Q 0 0*   *0 (& +" }

\‚? # ˆ j& & "+ &#" 0"&/ 0" `  +/ Q *+/ "+ " "  & "+ &#"& + "&  " #."&& \‚? ‚  #>  \‚? # " &} 00 "+ "  " && `  0"} 0"&/ 0" \  """ \‚? ‚  #"ƒ || { #

'    (& +" 0* /# " /# //& }

‡   }‚ `  ' ! ( +&& 00 !/  "+ / & ’ 00 .  / & /  0  // 7& 0* & 2& #

  

‚  ¤ ‚ ’ €

 ‚ * %==% /# " /# //& ~

 ?\ # " " 0"  0"& 0* 2  +#" !6 %    ?  \        ‡            ?  ‚  \  * %==% .  " 

*0 & & & 0"& } †   ‡ # <# * }   \   |   ` €   0Q 0*  | "" 0" 0"*" 0" 0  " &Q  &  K O ##Ž#% }  ? # ]" {  €         |    // "+ " K ‰*   ˆQ %ƒˆŽ%ˆ>  # ! ( +&& ]]} 5~€} 5^| _        ‚       }   ‚    } ƒƒ ' `  %' # <     7& 0* //  " !/  "+ }  % ##   #  } ƒƒ">' # "ˆ } ƒƒ">' # } ˆ%<"' `  %' #

"  *  *0   "*0#" 0* 0" }  #

# // 7& 0*  # } ? _ |

} ˆ%<"' `  %' %==% // 7& 0*  # }  " #] #  %= %# %ƒ   _     %" ># >>   >ƒ } ˆ>%"' `  #' # ]     =#$#%$%==> /  0 &. * "+ 0" }  #>    #ˆ } ˆ<=<' #

= /  0 / " /  "+ €  ƒ

} ˆ<%"' `  #' # % // /& 0* /" "*0 #&0" }  # ˆ    _     %   > €  > } ˆ<%"' `  >' %==#  & " 0#.0""&} & +" " 0,#"& . }    €  < } ˆ ˆ= `  #' %=== // 7& 0* 0, " !/  "+&Q 0 0*   *0 (& +" ` 0 "   0"& }  %  #ƒ   %% } ˆ ]]' `  #' # "#  "&Q 0& *0 &&&&#" 0* 0 } €  #   ƒ } <#"=' #  #

ˆ 0 0*   *0 0 2 ! & " " 0/ !/  "+& }

} <> 0 "+ *0 !/  "+& `  #' # 0& }  #   ƒ

< 0 0*   *0 ( " #.0&

} "==ƒ' # "< 0 0*   *0 0/" 0"& }  #  } "#=%' # = 0 0" 0* &//& + "&  *0#  +0/" }  #   \‚? ‚  #>    } "##=' `  #' # ] // 7& 0* 0"Q 0 0*   *0 (& +" " 0"&/ 0" }  %# >> >ƒ >  >#  >%=   €  >%

 

>>]

} "##"' `  #'# # // 7& 0* /# " /#Q 0 0*   *0 (& +" }  ># >%   >>    _     %# %% ƒ#   ƒ% } "<<<' %=== . *  0" *0 &/ "+}  #"& 0" "+} " "+ " / "+ 0* &  "*0#" *0 0" }  > !/  "+ +/ 0"&}  ! #

# ‘Z  ?% ? _ ?

\  \   %==% //  0"& 0 ! ^ \  \`###' # ]= 0 0*   *0  (& +" 0* &0" " !/  "+ //& }  >   ƒ \`\ #

" "+#" 0* 0,  0"& %    \`\   

|‚ #

]    Q [ ."0 00, ‘Z

| \   %==% "+ 2/& *0 & +" 0* &// &  0" 30 "& €

  %==% ' & 0" & %& " + &&&& ‘   ¤ }   # <# ? ‡   €    }   \ (" & 0"  "& / !/ " K NN ? ‘   ¤ }   # <" ? ‚  _  €    }   \ (" & 0"  "& / !/ " K OŒ ? \  `   €       # ]" (" & 0"  "& / !/ " K O ‘ ¤ ? # #=  00+ 0* !/  "+ 0"&   ?  ‚   |  `   %===  ¨¨ ‡ * %==% 0 "+  *0 &  "& ^  € ‘ # "ƒ *      ‚  0 "+& 0* #.0& /# "& & " (& +" 0*   0/" 0"&   Z  ¤ ‚ \  "=Ž "  #   % "/ *0  (& +" 0*  "*0 0" !/  "+ //& %==% \$  ‚   \   \  | \  ? _ "/ *0  (& +" 0* 0, !/  "+ //& %==% \$   ##> €  #% #> #ƒ   #ˆ {‘}\ %==%  0" )0/& !/  "+ 0/"  "& ? _ ƒ%}   ƒ%\ {[ |      # ƒ (& +" "/ *0 //  "&&  {| #%=##  ># >#% >ˆ ><   ?# `[  %=== && " "  "+ "  !/ "  > ƒ ˆ < ]   " ‚   \ %===  |  ‡  O& $0/" O**„„ ˆ<Žˆ  „**„„ ><Ž>  Q*€*„„ >=Ž># ‚Z| ^[  %==% 0 "+  " & *0 /.  # 0 &    %==% 0 "+  *0 .& 0"  "& " &. 0 000 .",&

  Œ  ? Q            †       

This Page Intentionally Left Blank

, ?        # > #  0"  0"& 0* "++#" +#" !6‚ ##Ž#% >># >>< ?  „  %" % ?    ˆ% ?     #=] ?  >%=Ž>= ?  %"ƒ %"" % ƒ >=%Ž> ?„      %#Ž> ?_  #]ˆ ?  <# #]ˆ #]< #]" >=] ?     ƒ" ?    >#]Ž#" Œ? Q  ƒ% %=< ?   & `    ?   #] #" # Ž%= ?       >#=Ž#>   >=>   $   ƒ<   %ƒƒ 

    >#%     >=" >#%    >ƒ <# <ƒ ?   Z     ?  ?  ? # ]  # Ž%= ?    ?  ‚  \  * %## >%" >>< ?  ˆ %#= %%%Žˆ %%ƒ %ˆˆ %]# >#% ?     <# "ƒ ?

  %"ƒ %"ˆ %"< %"] %"  % # % % ?   \  `  \|Z ‚   %ƒ >>#      %" %"# %"ƒ % % ?     ?  #" ?  # Ž%= ?  %] #<] #<" #]% ?  < ## #] %% >= % ˆ % < % ?     #"# #"        #"% #"> #"< #"] %=>     ƒ <] ]< %#%Ž>ˆ %">   % ˆ<Ž  <] ##  %#%Ž>ˆ %ˆ% %ˆˆ  % > ƒ Œ?       Q %= ?  \        ?\ ##Ž#% >%= >># ?     %ƒ% %]% ?      <ƒ #]= #]  %<  %]% %]> %]ƒ ?_   %" <]  

  %<] 

 #<]

      #"# # <Ž%=%

 %>  %ƒ< %ƒ] %ˆ# %ˆ< %<# %<ˆ   ] Ž"= ?_    „  " }  ƒˆ }   ƒƒ ƒˆ }

  ƒƒ ƒˆ }     %=ƒŽˆ %=<   %><Ž] %ˆ% %ˆ> %ˆˆ

    #]ˆ #"% #"] # # # ]Ž  %=>Ž< %]<    %]# }  >= % <Ž" }    \  Q    ]<Ž]      ]"     < Ž]ˆ ]"

      %==Ž% %=>   <] }   #]# }        %]ˆ %]]Ž#=     ]<Ž"     %>   #"<      ƒ< %"=  % #   ˆƒ       #%= #%#   #]# %ƒˆ %ƒ< %ˆ> %"#

    #"< # # %=>

      #"= #"# #"ˆŽ] %==Ž> %=<

     "#Ž>  $  # # %ƒˆ %]]    %]#   ˆƒ    %#= %#%Ž#ˆ %ˆ# %ˆ> %]]     ># }     #== #=# #=%Žˆ  

  %ˆ"Ž#= %<" %]ƒ }   ‘     #=#    % ƒ #==Žˆ }    > }      % #== #<ˆ #<] %<ˆ %<] % # % > } [ ƒ }    % # % %   %## %ƒ] %ˆ# %ˆ> %ˆ< %<%Ž> %<ˆ

 %##

      %=ˆŽ<

>ƒ=

 _

  %=<    %]> }      % << <] ]<Ž] %ˆ>    # <Ž%=%     #"" #"  >==   "=    #<%Žƒ }          >#>  % % 

 #ˆ#  %ˆ>Žƒ   ##] #%" #%  #>= } †  # <# #=ˆ >>% >>< }_    # <Ž%=% }     < #= }‹ 

  %]% >=# }     ##< }[    <% <ƒ #ƒ# #ƒ% #ƒ> #ƒ] #ˆ> #<=     #ƒ]   ƒ<  #ˆ> >== >=    „  %  #ƒ<     #ˆ] }[ [ ># #ƒ  #ˆ> } 

  >=ƒ >=ˆ     >#%  %"" %"  % = % >  %ˆ% %ˆ"Ž<= %<]Ž" %]% %]ƒ   #> Žƒ= }  ? # ]" # >>% >>< }   %% #ƒˆ >#> }   % > #=% #=ƒ #=ˆ }  ƒ }   ` _ %"ƒ %"" }     %#Ž> }   %" ˆƒ # ƒ %ƒ# %ƒ< } [  >=] } [ >ƒ <% #ƒ#Ž> #ƒ< #ƒ] #ˆ] #<<Ž] } [ [     #ƒˆ     #ƒ] >= _     #<=Ž#    >#     #ƒ  #ˆ> }  ?     ƒ"Ž }   ‘   #=# }  ?    _   >## >%" }     > ƒ ˆ }     &    }    #ˆ# >%# }    ##> } [      >#= >#>        %"#  % # 

 #ƒ#    %#%Ž>ˆ %>  %ˆ# %ˆ% %ˆ> %ˆˆ %<>   ] Ž"=   #>>

}   ? # "ƒ #ƒ }      #ƒŽ#< >>]   #ƒ ?   |   #ƒ >=     #ƒ #<      %" Œ   Q    #ƒ #<  % %    #ƒ #<   #<   #]=   [ #< 

  #ˆ }   ‚   %=== #ƒ }   ‚   ?   #% #ƒ }   ‚   { %=== #< }   ‚   {       # ]  #< }   ‚    }‚ #%> #ˆ# >%% >>ˆ >>< }     ? # ˆ    #ˆ }        ‚   # =    #ˆ }   

  #ˆ }    #]   % ˆŽ>=> >>ˆ     >= >=" _       >=ƒŽ<      %ƒ      #ˆˆ  #]    %#Ž>     % ƒ> #>> #ƒ" #ˆ% #]= } [   ƒ >ƒŽ] } [   < ] " }    > }   ’ ` Z #=" }    %" \ " Ž = \?|~ ~ #] # Ž%= \    [ #ƒ# #ƒ% #ƒ< #<= \  }   ‚ \}‚ " \  >== \       %"     %ƒ#     ]] 

 #ƒ"

      #"= #"<

    >    # \  > \  [ >= ƒƒ \      &      \  >% <# <ƒ %=" %]= >=% >#= \   #ƒ< #ƒ" #ˆ= #ˆ  #]%Žƒ % ˆ % < % " \?~ \ ?

    ~ [    }   ~ [ #=

 _ \|Z \   | ’ Z   ‚   # ƒ #% %ƒ >># \   [ #ƒ] \  >ƒ #]ˆ #]] #]" #]  % \     #ƒ# #ƒƒ #ƒ< #<< \       %%% %%ƒ \  <% >#% \    #ƒ #ˆ #< *  |   \  #> ‹   #ˆ \  " \      <# #ˆ< #<=Ž# #]" %"> % % \    %ˆ= \        ##ƒ %]% \     "" %=" %>>Žƒ %ˆ# %ˆ% %ˆ> >#] \      %" \‚? ‚  % ]Ž" >>% >>ˆ \  #ƒ# #ƒ< \ 

  % ƒ \  Q    ]<Ž] \ >ƒ ƒŽ#=# #=ƒ #=] #= \  [ <% <ƒ #ƒ# #ƒ% #ƒƒ #<=    „  %  #ƒ< \   >=< \      ˆ < \  %ƒ>Žˆ >#% \    ƒ> #<<Ž] >>> >>] & &0 ?   \  ` “    \        #=ˆ #=] \      <ˆ \      ƒ< #"= \   ] " #=# #=> #=] ###Ž#% \   %=" %## %]= \   [  >=] \          %]]       #ˆ> #<<     %ƒ#      # ˆ       #%=    #ˆ<       >"    # =

      #"= #"# #" Ž = # ƒŽ%=% %=> %=<

     "#Ž>  %ƒˆ %ƒ<    >#    %#= %#<Ž#] %ƒˆ %ˆ# %ˆ> %]]   >#      "ƒŽˆ \   %ˆ< %<ˆ %<" >#% \     %  >= %=< %]" \ ?

    ~ [    }   ~ [ \?~ #= \   #ƒ #ˆ #< \     >=% } ˆ ˆ= %"#Ž>

>ƒ#

  %]<     %]< %"#      ƒ< %]ˆ %"=     †  %]]Ž    <% %]<   <=  $  %]]       <= %]ˆŽ"> >>ƒ \   [    #>> \   ˆ% %ˆˆŽ< %<< %"# \       ˆ> \    #>> %ˆˆ %<ƒ \          %="  %"ƒ 

 #ƒ# #ƒ% #ƒƒ #ˆ%Ž> #ˆ<

      #]< #]] # # # > %=> >="  %ˆ#   <% \     <= "=     %="  % # % ƒ 

 #ˆ# #<= #<< #<]

      #]= #]< # # # > %=>  %>  %ˆ# %<>Žˆ %]#   #%" #%  #>= \     #<ˆ #<] %"# % > \     >=>   $   ƒ<        ˆ  %]ˆŽ"> >>ƒ     #]"     #]  >=]    >% _   #]]Ž"  #=%Ž> #=ˆ    >ƒ <% <ƒ     % < & &0 `    “ `     “ ‚      \   [ <% #ƒ# #ƒ% #ƒ> #ƒ] #ˆ> #<=    „  %  #ƒ< \      %  ># ƒ% %=< \       ?\ ##Ž#% >># >>< \

     >=  %"ƒ %"" %"  % = % # % >Žƒ

   %"   ˆ>  %ˆ% %ˆ]Ž#= %<<Ž" %]ƒ   #%ˆ & &0 } 

 “ ¤  \     #" \ 

     < ] #=]  ƒ < ] #=ƒ #= \   ]

>ƒ%

 _

\  

   ]  #% %ƒ \   | ’ Z   \|Z ‚   # ƒ #% %ƒ >># \    ##Ž#% %= \     ##>Ž#< \   *  _  ?  \*? ##> ##ˆ \     ]" \       \€?  #=%Ž> \     #"" \    <Ž] ##Ž#%           %=     %ƒ         #" \  _  "# "% \     ƒ ˆ \   #]% # = % > \

   #]  %ƒ%Žƒ %<  %]% %]> %"" >=# >#% \  ‡   %#= %#>Ž>ˆ %ƒ% %ƒƒ >%> >%" >>] \ Q   #=] \`###' # ]= #<<Ž] >>> >>] \`\ # " #= >># >>] \ [     >=" >=  %"ƒ 

 #ƒ# #ƒ<   "ƒ "<Ž]      %  #ƒ< %]< %"# %"> >==

    #=] \    %ƒ \  #]< # # %"< % ƒ >=] \   [   ] Ž"= \     %" %ƒ# \    "" ##  %## %ˆ# %ˆ% >#> \   #ˆˆ \      <% #ˆ# #]" %ƒ= \   %= |    % ˆ |  #ˆ% #<= #<# #"ˆ %ƒ  %"> >#ƒ |[  ƒ# ƒ< #%< %]ˆ %]< %"% %"> % Œ|  Q     #ƒ #< |    > ƒ #=% |

 #"# # #Ž> %ƒ< %]"  " Ž =    # <    >="    < Ž]<  %"" % # % > % ƒ  <" #>ƒ 

 #ƒ<    # #Ž>

      # %]  %"> 

   "=  "ƒ

    #=]

   "#Ž>   %

 #"  %]ˆ % "    %]=   #>ƒ  % |      "ƒ |  [ #ˆ |      >#=  %"< 

 #ƒ#

      #]< %]<  %=  %]=   ##" #%"  % ƒ |          €   ‚  ?  |€‚? ##ˆ >># |        | ƒ] >>% |    >==  $  #%ˆ #"= # # %ƒˆ %]]    %< %#%Ž%= %ˆ# %ˆ% %]] |   #>% #"< |

   # <Ž%=%        %"#Ž>  %"ƒ % #Ž>   #<"Ž     #]        << ##] >#> 

 #ˆ%Ž<   %ƒ Ž<"  [   #=#Ž]

      #"ˆŽ%=>    %< Ž]ƒ   #%]Ž>ƒ |     ] |  %ˆŽˆ> >>#Ž% |    #"ˆ %=#Ž%    # <Ž%=%  "ˆ  #"" #"    %ƒ" |   <Ž] ##Ž#% %ƒ |    #<ˆ #"% %ˆ= %ˆ# %"# |   << %ˆ" % ƒ |     _ #]] #]" |    " |   # Ž%= |    %" %  >== |    ># |     %" % # |    % <  ]" "= #<"Ž  #]# %ƒˆ |    { #< |    <Ž] |    ƒ< | \   * % ƒ >%" >>] |   % < % " % |    <Ž] |   < ##Ž#% ##     "Ž         %ƒ |    % > #=% #=ˆ |   #%%

 _ |    < |  %=" %## %ˆ= %<  %]= | _    <= %<  %]= %]# %]% %]> |  #%%Ž> #ƒ] #] Ž"= %ƒ%Žƒ %]% %]< %"" >#% >#> |    %>"      % #=ˆ #=] #ˆ% #"ˆ %ƒ    %" ˆ% #ˆ< #<] # <     #]  # = %"# %"  % >    #"ˆ # <    #ˆˆ              < ]  #=# #=ˆ     ]  "ˆ #>> #ˆˆ # ƒ %> Žƒ#          #=# #=ˆ #=]     < "   [  #ˆƒ     <=Ž% << ""       <]  %"ƒ % # % %

      #"  %<# %]#  % ƒ      <= <% %]>      #="     ]] # ˆ      % #      ~           #  %=       #ƒ     #] 

     #>   ‘  ‘ #] # Ž%= >%>      ?  ##> >%> >>% >>>        ‘    ‘ #] # Ž%= >%>      `  ? # = ##>  $     %#= %%%Žˆ %%ƒ %ˆˆ %]# >#%   Q   ] Ž"=   % ##] % % >># >>% >>> >>ƒ >>ˆ      &    _                 #  %=       #==        ##<    %#Ž>     ƒ _   <ƒ #ƒƒ #ƒ< #]< %=  %]= %"ˆ >#= _       #ˆ= #ˆ _   ƒ# %"  % # _   >## >#% €  #ƒ# #ƒ% €    <] % #=# #=ƒ #=ˆ     >#>        %"#

>ƒ>

 % % % ƒ 

 #ˆ# #ˆ% #<#

      #"ˆ %=<  %ƒ  %]> €   [   "ƒ "<Ž]    <# <% %]>  %"ƒ % # % ƒ 

 #<= €   >=ƒŽ< € ## €   `   #=< >%ƒ €

    <ƒ %<  %]ƒ €   ##] €  <% <> <ƒ <] #]ˆ %"ƒ >#% €   %= € %#% %#ƒ %#< %#" %%= %ˆ] %<< >=< €        ˆ      ]    % #=       < & &0 \“  €

  #%# >%  >>] €  % = % # €        >="Ž  >#%        %]< %"#    #]# #"= >==     ##ƒ  %"" 

 #ƒ]    %ƒˆ ><]    %]%      >=   #%ƒ €_  >=ƒŽ< & &0 } 

 “ { „ “   „  €_ "ˆ €    %#= %#%Ž%# %%<Ž>% %ˆ#Ž> %<%Ž> %"# %"%

      #"< # # %"%

   %>  %ƒ= %ƒ#    %]# €   %>] %ˆˆ €  ƒ# €  #" Ž = €     ># €  #<< #<] €_    ƒ# €_    #>% %ˆ% €_     #ƒ# #ƒƒ #ˆ% #<=Ž# % ˆ €  <% <ƒ %"ˆ %"] €    %    ƒ< %]ˆ %"=     ƒ# ƒ% ƒ< %]ˆ >=#      ƒ]    % <

>ƒƒ

 _

€  Ž    |     %>"   ># „  #%<    %"      #ˆˆ      >"Žƒ=   %    #]=    ƒ#     >#    #     % < % " €  > ƒ #== #=# %=< & &0 €   €    %<Ž]  # % > ƒ € [  ˆ% €  [ #]ˆ #"# %]ˆ >=] €   Œ Q  ƒ%     % ƒ #==Žˆ } "==ƒ %  > ƒ #=%  % > #=  ###Ž#%       %   %        %"       >"    %>     †  #==Ž<

 > ƒ %=<   >  % > ƒ #=# #= Ž#%  %> & &0 €  €    %" €    %ƒ<   # ˆ # <   %] %ƒ<Ž]

 "#Ž>    "% #"  # < €    ]] €       "Ž €     #<# €        # %]  %"> €   <ˆ # #=ƒ #=ˆ % > €    

  % ƒ €   > #=ƒ €    % #ƒ] #]  >== >=] €  [ #%% >=% ‡  #ˆ= ‡  †  #<  #]= #]% %ƒƒ >=# ‡  † Q ?  %ƒ% >%ƒ ‡[ % # >=# ‡   

   ‡? " ‡      ˆˆŽ ‡    %= ‡     %"" ‡  %Ž##< >>%Ž> ‡   %< %ƒˆŽ< %<%

‡ %"ƒŽ ƒ >>ˆ   %"ƒ %"" % ƒ >=>   %"]   %""   <% <ƒ #]ˆ %""     %"" >=" †  % =    % =    %"ˆŽ<    <% <ƒ %"<  †   [  %"] %" Ž#= % #    % %    % ƒ   %"" ‡   ‡†  €   % % % > >%ƒ ‡       ##] #%= #%  #>ƒ ‡  „  >=> >#> ‡      #]   %ˆ= %<#Ž"    %<  %]% %]> %]ƒ   #%"Ž  #>= #>% #>> #ƒ= ‡        #] ‡    ˆ < #== #=% #=ˆ ‡   > ˆ  #== #=# ‡   # > # %%% %%ƒ ‡       > #=" ‡     ƒ] ‡      %  > ‡    % < ‘     ƒƒ ‘        #<< ‘    % ‘  ##] ##" #%% #%> #>ƒ #>ˆŽ< #>" ‘    %]] ‘    #% %ƒ >>= ‘    _   ‘ %ƒ #ƒ> ‘    %< ‘ [  %"< %"] % # ‘    >#= >## ‘     † ˆ%    "# "% #"# # ƒ    % =     %#=       „  %  %     ˆ=Ž#  #ˆˆ #<# ‘      ˆ ‘ ?     ƒ"Ž ‘  Z     ?  „ {   # ˆ # ‘    #] # Ž%= ‘         >%Ž> #<< #<] #]ˆ %=" ‘    << "< #" ‘  %]] %]" %"#  % >

 _  

  %ˆ" %<] %<"  „  #>ˆ #>] #> ‘       %=" %#= %>>Žˆ %ˆ# %ˆ% %ˆ>

    %=" %#= %%<Ž  %ˆ# %ˆ% %ˆ>    %=" %#= %>=Ž% %ˆ# %ˆ> ‘   % ‘  #] ‘     <# <] ˆ ‘[Q  <] ‘ †    <" ‘ †         <] %"> ‘    †  %ƒ> ‘      %=" %#= %%<Ž>ˆ %ƒˆ %ˆ# %ˆ% %ˆ> %ˆˆ ‘     %="Ž]ƒ %ˆˆ %]= %]# ‘ ¤ ? # #= #ˆ# >>> >>]  ‡ * #]% #]ƒ >%  >>] \‚\*  ##> >>>    % #    > ƒ ˆ   >"Žƒ% ƒƒ #ˆ% #"ˆ %ƒ  %"ƒ >#ƒ      #= ># #=% #=ˆ %]ˆ        %  >= %=<          ##>Ž#<   [ #%>    #% #ƒ #< %% % < "& / 0" 0* // "+ "&O  !00,   %ƒ<Ž]        >= %=<     >#            < ]  #=# #=ˆ     ƒ# %"        &             #Ž% ƒ ˆ       %ƒˆ %"" >## 

       ##<  % #    „  "     <] "ƒ      %== %ƒ<Ž] % % >>ƒ >>ˆ >>] ¤  #=ˆ ¤  Q      "ƒŽ] >>% ¤    %" %     >#>   >=> >#>

 #>ˆŽ< >=ƒ 

 "#   #>]Ž" >=ƒŽˆ   %"" % ƒ   %       %  #ƒ<   #>ˆŽƒ=

>ƒˆ

¤    #< Ž]= ¤

  %#]Ž#" %ˆ#   ƒ< #%ˆ #%< #>% ^ ‚  # "= ##> >>> ^   %" ^    ##" #%% ^  € ‘ # "ƒ #=ˆ >>] *  ˆƒ *   %"< %"] %"" %"  % # *      ##] #%=Ž# #%  #>ƒ *       ** ##] #%= #>= *   ##> ##ˆ *       <]      #=] 

 #ƒƒ #ƒ" #ˆˆ #<# #<%Žƒ  "ƒ

   "#Ž>  %"   "= *     #ˆˆ %> Žƒ# %ˆ> *      [  *} %>  %ˆ> *      %# *  |   \  #> *     ##Ž#% *      † ˆ%    # ƒ _  >=ƒŽ<    %%<Ž>ˆ %ˆ#    >==   ˆƒ   #<" #]#Žƒ    #ˆ#  #=%Ž> #=ˆ  "ˆ   ˆ>   <  ]< #%<   ]    ""   ˆ>  # > ƒ     ƒ Žˆ=    #  #ˆˆ & &0     *    #"< *     # %"> *    #ƒ# #ƒƒ #<< #<] *  ˆˆ ##] >## *     %ˆ *    ƒ *       _   <> #]< %=  %]= %"< >#= *  #<"Ž  #]#Žƒ *    #]

>ƒ<

 _

*      ˆ#  „ #%<  % #    >== „    #< Ž]=   ˆ%  #ƒ  #ˆ= #<] *  " Ž =    #"# %ƒ<    %ˆ< %<ˆ %<" >#>     #<ˆ #<] %"# % >  #ˆ% #<= #<# #"ˆ %ƒ" %"> >#>     < Ž]ˆ    % <  ]" "= #<"Ž  #]# %ƒˆ    ˆˆ ##] # ˆ >#>   >"Žƒ#    ƒ#  %"" % % % ƒ  >"Žƒ% ƒƒ #ˆ% #"ˆ %ƒ  %"ƒ >#> „  #>ˆŽƒ=   #<"Ž  #]#Žƒ     #>% #>> #>ƒ    #>#    ƒ# %] Ž =   % <  ]" "ˆ #%ˆ #<"Ž  %ƒˆ %"> %"

      #"ˆ %]<

 ƒ#  #" Ž =  #==  %ƒ  #ˆ%   % #%] # ˆ %]" % %  %  ƒ]Ž" & &0 ?_ “ *  “ ~    *              # Ž%=       #ƒ #ˆ #<      #"        % <     #] 

     #> #"         #" *   " *  }   ? #ƒ *      %ˆ] %]# %"> Z    >=< Z     < ] %ƒ Z    %ƒ % < Z       ] ## Z     #ƒ> Z      %]ƒ Z

 #= #ƒ#Ž]ƒ >>%   #< Ž]ƒ   $   ƒ< } ˆ<%" #ƒƒ #ˆ# #ˆ%Ž<ˆ \`###' # ]= #<<Ž] >>>

 [   #ƒ#   #ƒ]     #ƒ] >= _     #ƒ# #ƒƒ #ˆ% #<=Ž#     <ˆ   #ƒ>Ž<    >%    #ˆ= #<"Ž  #]#   #ƒ]     †  #ƒ"Žˆ%

   #ƒ" #ˆ     #ƒ]     #ˆ# #ˆ]   [ #ƒ<  $[   #ƒ"     #ƒ% Z

    %" %  ƒ% #ƒ< Z        #"> 

 #ƒƒ     %#%Ž>ˆ  #Žƒ    # Z  ?   \  `  %" %"# %"ƒ % %    >="Ž    >%Ž>    <#Žˆ   ƒ<  >ƒŽ] Z    >#ƒŽ# Z_    <"Ž]< Z     ˆ      ƒ ˆ    > ƒ ˆ    #Ž% ƒ ˆ Z   # Z   % ˆ % < % Z [      ƒ< #%< %]ˆ % "

   ƒ# Z    ?  %ƒ% Z   %ƒƒ >#% Z     ##ƒ    >=" _  >=ˆ    >ƒ <# <ƒ <ˆ Z   #Ž% ƒ ˆ Z    %><Ž] %ˆ% %ˆˆ        #]ˆ %=< 

    %ƒ> >=%    %>] %ˆˆ    >% %=]  %>" %ˆ# %ˆ%    <# <ƒ

    %>< %ˆˆ  †      %#=Ž>"   #]= Z   ` ‡   { Z`‡ #>

 _ Z  #  %= Z   %]< Z         >=  %"<   <#Ž>

      #]< >="  %=  %#>Ž>ˆ %]=      ˆˆ ˆ  <]   %  #%" #>=  % ƒ    # Z     %=  >#= Z    >=%  [  >=]      % ƒ    „  %  #ƒƒ   %  % ƒ <   ##" #%% #%"     %  #ƒ] #]] % ˆŽ Z          %"#Ž>     #"<   ˆƒ   ˆ>  %ˆ>   ˆƒ Z      << <]Ž]ˆ ]" \  Q    ]<Ž]    # ƒŽˆ       #"#   %<Ž]   #]%Žƒ

      #"ˆŽ< #" Ž =

     "#Ž>  ƒ      "ƒŽ] #<# Z      % <] ]" "" # Z       #ƒ# #ƒƒ #ƒ< #<< >=      % ˆ     >="     >=<     #ƒ# #ƒƒ #<< #<] 

    #ˆ> #ˆ]   

 #ˆ#     #ƒ] Z  ?  †    ‡     # ]  ƒ] Z  \   ’  ‚   # "< ƒ] Z   „  %  #ƒ< Z   % = Z ‹     &   Z    ]] { „  #>ˆŽ< >=ƒ {  ‘  }   \  {‘}\ #=  ##= ### ##% >%ˆ >>> >>]

>ƒ]

{  `

  ` ‡   {``‡ #> # {      # %]  %"> {    > {  [    #=ƒ #=ˆ {   ##> {   _ #"< %"# %"% {[ |      {| %]#Žƒ >%ˆ >>ƒ >>] {[       %"ˆŽ< % # { ‹      >= >" ƒƒ %]" {               #  %=       #< |    €     ` 

 #<     #] 

     #> {     }   | #] {  ?„      %#Ž> {   ]    |  `  Z  |`Z #> #ƒ >>#  >= >" ƒƒ %]"    #ˆ= #<"Ž  #]# %]"       %" #ˆ% #<= #<# #<%Žƒ #""     „  "      <# #ƒ#      < ] #=]      \‚ #=]      >  ƒ= ƒƒ       ƒ       <] `   > #=# %=< `  #]# `    %ƒ% %ƒ> %ƒˆ >== >=] >## `         #<%Žƒ _     #ˆ= #ˆ _     #<#         ##] #%# 

 #ƒ< #ƒ" #ˆ= #ˆ% #<# #<%Žƒ     #ˆ% #<%Žƒ

      #""   [  %] ƒ< #%ˆ     #ˆ" #ˆ `  _   ˆ `    

 `€\ %#= %%=Ž# %ˆ# %ˆ> `  ƒƒ ƒˆ `      << #ˆ% #"ˆ %ƒ  %]> %"# >#> `  †    ˆ `   ##] `   ƒ# #ˆ# %] Ž"# `     ] `     %# `     %# `  ~ ?  %% `  ~  ? # < %#Ž> >># >>]

>ƒ"

 _

`     #=] #=" `[   <] ` ˆ  #== `    [ #ƒ] `       ƒ# `    <<  % % % ƒ 

 #ƒ  #<< #<] #]#  %<#Žˆ %<]   ##] #%] #>ˆ #>] #> `   #ƒˆ `

Ž‚    ] Ž"= `     >  ƒ= `  > #ˆ# #ˆ> #ˆƒ #ˆˆ #<< #<" `   #=ˆ  #=< %=<    #=%Ž<   % ƒ      #=ƒ _    #=%Žˆ

  [    #=ƒ #=ˆ   >   #=% #=ƒ    #=< `[  ^ * %"ˆ %"] % = % % >%  >>ˆ >>] `

 „  "# `

   "= `  ƒ# `   ># `    "#Ž> `

  ?  { `?{ #> `

    }   ‚   { 

   ?     # = #< `

  * }     \  

?    ? # ] #> `

  {   # # #> `

  ` ‡   { ``‡ #> # `

  `   `` #> # `

     ## #> #" # Ž%= `

      ` %ƒ `   %ƒ<Ž] %ƒ `    ˆˆ %#>Ž>ˆ %ˆ> `    %ˆ# %ˆ> %"#Ž% `    << ƒ `  _ ` ] " #= ` <> #]ˆ >=> >=ˆ `    %ƒˆ %ƒ< %<% ` "ƒŽ] #>  #]ƒ %>" %ˆ# %ˆ% %]# >## `  ˆ> ` ##]Žƒ= >>>   >=% } ˆ%<" #>=Ž>   %] „  #>< #>"   #%= #>=   [   %] ƒ< #%ˆ `    >=ƒ `   % <  ]" "ˆ #%ˆ #<"Ž  %ƒˆ %"> % = ` Q  <#Ž% <] %=  %]= %"< >#= `   ""

`   ##>Ž#< %ƒ% %]% `  <> <ƒ `      `} %"< `      < `    %] %ƒ<Ž] `   \  #]ˆ `      ƒ< #"= `††    #ƒ# #ƒƒ #<< ` ‹    ##Ž#% `     #=   ƒ# ƒ< %]ˆ %]< %  >==    >==  #=% #=ˆ    ># `      _ #]] #]" `    ƒ `      ƒ< #=% #]# >= `  ’ Z  %==# %=< `      „  % `       < ] %ƒ `    <Ž] ## `    #]%Žƒ #"# ` „   ƒ" ` „     " `    <# %]> %]ƒ % # `       #=" `     %" `     # = ` _ %"ƒ %"" +      < +     >ƒŽ] ‚      <] %"] ‚      % %#>Ž>ˆ %ƒ= ‚    > ƒ %=< ‚  #>% %ƒ< %ƒ] ‚      ƒ] ‚  >=] ‚ ‚ [ Q   #=] ‚          < Ž]ˆ ]"      #ˆˆ %> Žƒ# %ˆ>

     "# "% ‚  [            %= ‚      >= ‚      #"< # # %=>  %=" %#= %%<Ž  %ˆ# %ˆ% %ˆ> ‚   ] #<< ‚  ` ‡   { ‚`‡ #> ‚      #]ˆŽ%=< >>>Žƒ  #"= #"# #"ˆŽ] %==Ž> %=< } "##= #]  #"= #"ˆŽ%=>    #"= #"# #" Ž = # ƒŽ%=% %=> %=<     #]ˆ #]] #]" #]   #] Ž"=     #"= >=" >=   [ #]ˆ

 _    %> „   %" _   #]]Ž"  #=%     †  #"=Ž# #""    #]<

    #"= %=<   [ #]<  ƒ# ƒ% ƒ< #"= #"# #""Ž = %=> %=< %]< %"#Ž> %  $  #"= %]]      "ƒ ‚      % #"% #"> #"ƒ #"]   #]ˆ #"% #"] # # # ]Ž  %=>Ž< %]<  %=<  [ #]ˆ #"ƒ #"]  #]ˆ #"> %]< %"# %">    %=>   % #"% #"ƒ #"] ‚  [       % < %      ##> ##< ‚     #] ‚     #] >= >" ƒƒ ‚    %=" %"ˆ >= ‚   >= >" ƒƒ ‚    } "==% % #="   %]   % #=]     †  #="

      #"= %=<   >     #ƒ] % < ‚}? ‚     }  ?  ## >%< ‚   \ %=== >%Ž> >>% >>] ‚   "#Ž> ‚ %=" ‚[   #% %ƒ ##ˆ ##< ‚Z| ^[  ˆ> >>= >>] ‚   ? # ]% ƒ" ‚     ƒ]Žˆ= ‚  *  ‚   # "  ƒ] ‚   %" #]= %> ‚[  #== ‚   $     %#= %%%Žˆ %%ƒ %ˆˆ %]# „ %#]Ž#" ‚ %  >= ƒ# #ˆˆ #]= % ˆ ‚   %#%Ž%ˆ ‚     }  ?  ‚}? ## >%< ‚   %ƒ%Žƒ  & €  “ ‘      %"ƒ %"     << "ˆ "< #"   ##ˆ #]"

>ƒ

  > ƒ ˆ #==    <   ˆƒ   < %=ƒ  !/ !00, %==% %ƒ  >>ƒ             #  %=       #ˆ     #] 

     #>   ># %   „  #>]Ž" >=ƒŽˆ      ƒ"     %     ##"  _  #]  %ƒ% %]% >#%      %]= %]#        ƒ <] ]< %#%Ž>ˆ %">       „  %       >#%   % ˆˆŽ"       <]    %##   %ˆ=  <=    % ˆ<Ž  ##  %#=Ž>"  %=" %#= %%<Ž>ˆ %ƒˆ %ˆ# %ˆ% %ˆ> %ˆˆ   %#=Ž>"   % ƒ ˆˆ <]  #>=    ˆˆŽ      <<   ˆ>    %]#   ## Ž%# ‹   #ˆ ‹    >=ˆ      ƒ     * <<   % ƒ < #=  #ƒ< _  ˆ     % > ƒ #=# #= Ž#%    ˆˆ ˆ] ˆ        <] „  #>< #>" #ƒ= %ˆ"   ˆ>

       #"ˆ

     "%    %<ˆ %<]   ˆƒ   #"] %ˆ% %ˆ"Ž<= %<ˆ %<< %<" %"% %"> % =     %"    << #"" # = %ˆ>   [     #"ƒ       <=Ž# <] "" #%" %]=

>ˆ=

 _

   #"] %"# %">     <] ""      

  %]ƒ 

 #ˆ# #ˆ]  ƒ < ] " #=ƒ   <% & &0           

     <] ""     %="        %"%Ž>  % ƒ 

 #ˆ#

      #"] #"" # =  < "  %ˆ> %<ˆ   #%ˆ #%" #%  #>=    %]ˆ %]" %"# %">   %ƒ#  >= >" ƒƒ   [ ] ###Ž#% ##" #ƒ< #]< % "     #]ˆ #"# >=]   #Ž% ƒ ˆ   † ˆ      <ˆ   %"" % ƒ  > ˆ ]Ž#== #=    #=ƒ     ##Ž#% } ˆ >= % ƒ      ##ˆ     > ƒ   " †      >#%        %]]Ž"= _       #ˆ= #ˆ    #==Ž<  %"] %" Ž = % # 

 #ƒ"Žˆ%

      #"=Ž# #""

    #=]Ž"    %]> %]ƒ    %#=Ž>" %ƒˆŽ]    #%ˆŽ< [ "Ž [    #=ƒ #=ˆ   [   "ƒ "<Ž]      ƒ< %]ˆ >==   >==   ƒ# ƒ%  %<%      "ƒŽ] & &0 \      

 #ˆ# #ˆˆ #ˆ< #<]

      # ƒ # <  %> Žƒ# %ˆ# %ˆˆ %<>Žƒ   #># #>>   ˆ>    % # 

  #]]    %ƒ  >=<  #"= %">  ##] ##" #%% #%> #%ˆ #>= #>ƒ #>ˆŽ< #>"  ‡     ##> ##ˆ          #==   ˆ   ] " #=# #=ƒ #=] ###Ž#%    ˆ    ƒ < ] #=ƒ #=     ##>Ž#<    <Ž]     >   %      % ƒ <   ƒ   _ ] " #=    <Ž

      >   % ƒ < #=     ƒ < ] " #=ƒ   [ ] ###Ž#%     ƒ   #=ˆ    ##<    <Ž"   ˆ> 

    >=ˆ             <  ]< \  Q    ]<Ž]  %"  % #    >==   #<"Ž  #]#Žƒ 

 #ƒ" #ˆ= #<# #<"Ž   ]]

      #"=Ž# #"ˆ #"" #" Ž = # # %]ˆ %]< %]"Ž"=  $  #%ˆ #ƒ" #"= # # %ƒˆ %]]  %ƒˆ %ƒ<   #%ˆ #%< #>ƒ  %    # Œ         Q #] #"   <#Ž>   < #= ##Ž#% ##  %=ƒ     #=#   [     >=      %>< %ˆ> %ˆˆ       ‘ %=" %#= %>=Ž% %ƒ  %ˆ#     %=  %" %  << # ƒ %ˆ>      >=> } ˆ ˆ= %< Ž]ƒ 

  %]ƒ

 _ 

   %ƒƒ %<  %]% %]>   %]% %""     %]#     %]%   %<  %]% %]> %]ƒ    >> „    #<     †  %]> %]ƒ    <# <ƒ

      #]   %]#   #]=   „  #ƒ<    # # %=< %"    %ƒ< %ƒ]    `    `  < "    ? #%"" % % } ƒ %= } #"] #ƒ% } >=" " } ƒƒ  ]  %<#Ž" >=ˆ >>> >>ˆ } ## % " } #%=% >=ƒ } #%#= #>] >=ƒ >=< } #""# #]" } >< % >=ˆ } > %# #ƒ% } ƒ#]ƒ >=ˆ } ƒ#"> >=< } ƒ# = >=ˆ } ƒ>%= #>  >=ˆ } ƒ><= %ƒ } ƒ]=ƒ ˆ# } ˆ%<" ##] ##" ##  #%ƒ #%]Ž>ƒ #>< #>" #ƒ= #ƒ] #ˆˆ #ˆ  >>> >>< } ˆ>%" #]] #]" >>> >>< } ˆ> = #ƒ# #ƒ% } ˆƒƒ  %#% %#ƒ %#< %#" %%= } ˆ<%" #ƒƒ #ˆ# #ˆ%Ž<ˆ #<< >>> >>< } ˆ >= % ˆ >>% } ˆ ˆ= %=" %ƒ= %ƒ# %ƒ Ž<= %< Ž]ƒ %"#Ž> >#> >>ƒ >>< } ˆ ]] #<"Ž  >>> >>< } <=]> #ƒ% } <#]" #< } <#"=' #  ƒˆ >># >>< } <%=< %"ˆ } <%<% % % } <>  >"Žƒ= ƒ> ƒƒ #>= >># >>< } <ƒˆ] #ƒ% } "==% % #=" >>% } "==ƒ % #== >>% >>< } "==] % < % " >>ˆ } "#=% % ]Ž" >>ˆ } "#=> #]= } "##= #]  #"= #"ˆŽ%=> % < >>ƒ >>< } "##" >#=Ž#% >>ˆ >>] } "% " #ƒ# #ƒ% } "ˆ== #]] } "<<< %=ƒŽˆ >>ƒ >>] } { %=] #]] } { >>< ## } { #=<ˆ ˆ#

>ˆ#

} { #==%ˆ %ˆ= } { #==>ƒ %#= } { #==ˆ<Ž% %#= } { #=#ˆˆ %ƒƒ } { #=%#= %#= } { #=%]  %#= } { %=" "Ž# #> } { %%=<> %ƒƒ } {  >ˆ=< %]ƒ } {  ]ˆ#  " } {  "ˆ=# %ƒ> } {  ==# #]] } {  #% ƒƒ %ƒ% } {  #ƒ]#> %ƒƒ       #ƒŽ#< \ ?

    ~ [    }   ~ [ #= |{ %"<  { #>ƒ]ƒ % % >>ˆ   ˆ ##] >>ƒ   ] % >>>  %%> %ƒ>   ##ˆ #]"   † ˆ    %"ƒ % #   #==        #ƒ >>#  %="Ž]ƒ >>ƒ   >=>   $   ƒ< } ƒƒ  %<#Ž" } ˆ ˆ= %=" %ƒ= %ƒ# %ƒ"Ž<=        ˆ  %]ˆŽ"> >>ƒ        #]ˆ %=ƒ %=< 

  %" ˆ> %ˆ% %ˆ]Ž<= %<<Ž" %]ƒ 

   #]  %ƒ#Ž> %<  %]% %]>   %=" %## %ˆ= %<  %]=   %ƒ%Žƒ %]%     %ƒˆ %]% >=     #   ƒ% %]<   %ˆ= %<#Ž" %<  %]% %]> %]ƒ „    #< Ž]=   #]%Žƒ  #=%Ž> #=ˆ  #>  %>" %ˆ# %ˆ% %]#     †  %ƒˆŽ]

   %    %ˆ# %ˆ<   %"       %ƒ>Žƒ   <=      ˆ%   %< & &0 ‡  † “ Z “        %  >#       >=    <= %]"   <]

>ˆ%

 _

  Ž   

      # # # ˆ

   "#Ž>    %]=    ""    #  #ˆƒ   

 >ƒ <% <ƒ #ƒ# #ƒ% #ƒ< #ˆ# #ˆ>      >= >" ƒ=        #ˆ=      %"     >=     #<"       >" ƒ=    „  #ƒ<

     "%Ž>     #ˆˆ    %ƒˆ %ƒ<    %#=   %" #]=       <=   <# 

 #ƒ# #ƒ% #ƒƒŽˆ #ƒ] #ˆ# #ˆ%Ž> #ˆ]   <% ##]  <#Ž> << #ˆ# & &0 }     “ \     “ \     “ €_    “    “            >#=       <]    %ˆ< %<ˆ %<"    ˆ  %]"    <# "=   <#   <] "" <    [  ] Ž"=  % ƒ  <# "= & &0 `     $    ""   [    %] ƒ< #%ˆ      > #=#      ˆˆŽ # >>%      <ˆ      %ˆ %<Ž"     %"ƒŽ ƒ       „  %  #ƒ<     %="Ž]ƒ   ] Ž"= %ˆˆ   #>% %]ˆ %]" %"# %">      <Ž] #=    "% #"  # <        #ƒ]        < Ž]"   %"% %">     #"ˆ # # %]]  % # % >

  #<"Ž  #]#Žƒ   #<# #<%Žƒ  "ƒŽˆ  #"" #"  >==    #  #ƒ" #ˆˆ #<%Žƒ       %ƒ>Žƒ % " >=#Ž%    >    %#Ž>     %]   #ƒ> #]# %  >%  >><     "      < "Ž  ##Ž#% %ƒ ##      # = #ˆ      >=#     >=  >## >#%  [ [ >=<     >=<    >=]Ž"   [ #%%  %"% %">    „  %  #ƒ< % "  %ƒ  %ˆ=  ƒ    %"ƒ %"ˆ %"< %"] % % % >     [ [ ˆ#Ž>       <Ž] #% %ƒ    >=      <#Ž> #]= #]< %=  %]= %]> %]ƒ %"<      

  %ˆ"Ž<= %<" >=ˆ     %="  %"ƒ % # % ƒ 

 #ƒ# #ˆ# #<= #<#

      #"< # #Ž% %=>      <<  %<# %<ˆ %]#   #%" #%  #>=    %ˆ"Ž<= %<< %<" %"%     ˆ> " Ž = %ƒˆ    [ #=] #="     #>% %ˆ%  †Q    #=#   \    %ˆ=        #]] #]"  %"ƒ % # % >     #% ƒ    `    < "   #ˆ#  _  ]       #%ƒ   _   <ƒ #ƒƒ #ƒ< #]< %=  %]= %"< >#=   [ %"ƒ %"ˆ      %ƒƒ [  >ƒŽ]     >##  %"] %" Ž = % > 

 #ƒ" #ƒ  #ˆ# #ˆƒ #ˆˆ #ˆ<

 _     %#%Ž>ˆ %>  %ƒˆ %ˆ= %ˆ# %<#Žˆ %<"

      #"=Ž# %]<  $[   #ƒ"    %]=    #%# #%ˆ #>=     ""   #= >ƒ ##]Žƒ= >>>   >=% >=>   $   ƒ< } ˆ%<" ##] ##" ##  #%ƒ #%]Ž>ƒ #>< #>" #ƒ=    ˆ   #%%Ž>     #%ƒ >=     ƒ#  %]    ƒ#   ƒ%   [ #%%     #%"Ž  #>= #>% #>> #ƒ=    >%   [ #%> „  ˆ> #>ˆŽƒ= >=ƒ >=ˆ >=< „ ƒ< #%ˆ #%<      ##" #%% #%"  #=ˆ     ƒ#     †  #%ˆŽ<    <% <ƒ <ˆ ##"   >=> >=ƒ >=ˆ    ##"  † ## Ž%#     <%   [ ##"  $  #%ˆ     ##] #%" #>= #><    ># ##         #%<   %< #>%   #%"    #%%     ##] #%%   #    %#" %%% %%ƒ    ># #=< ##  %#=Ž## %<ˆ % # [ ˆ%Žƒ    << "" %"< % #    %" "" #%" %#>Ž>ˆ %>  %ƒ= %ƒ# %ˆ>    <     %"ƒ %"ˆ %"< %"] %"  % = % # % % % >    %=" %]=    \   `

  ? # = #>    \   `

  ?   # ] #> _      ##>Ž#<       &           ƒ"Žˆ# % #        %ˆ] %]#   `     ` #"

>ˆ>

 

   % ƒ #= Ž#%    #= Ž#%    >#ƒŽ#ˆ  _  ]     ##ˆ ##<       ˆ >#>   %< #>% %ƒˆ %ƒ< %]]   "" >##        ƒ Žˆ=  %#= %>        ##] %ˆ>Žƒ   "ƒ #"<    * << #"ˆ   <] #%ˆ %]> %]ƒ %"# %"     <=Ž% << #ˆ# #"% %]> %]ƒ      <Ž"     % #%] # ˆ %]" % %         ]]    ˆƒ    >ƒŽ]      #ƒ%        ƒ ˆ    > ƒ ˆ    #Ž% ƒ ˆ  _ #      } %#= %#%Ž#ˆ %ˆ# %ˆ> %]]        \ %#= %#<Ž#] %ƒˆ %ˆ# %ˆ> %]] ?      #]   #<]    ƒ"Žˆ#    ##] #%=Ž#         %ƒˆ         %"     <" "#   „  #ƒ<    #ƒ" #ƒ  #ˆ> #ˆ< % "   ># ˆ%    # %]" %]  %"#    % > ƒ ~ &      ~ ~   #>% ~       #<ˆ        #<%Žƒ   #ƒ< #ƒ" #ˆ= #ˆ  #]%Žƒ % ˆ % < % "       #ˆ> #ˆ< #<=     % ˆ   %] _     #<=Ž# % ˆ     #<#  % =    #ˆˆ #ˆ<    #ƒƒ    „  #ƒ<

>ˆƒ

 _

~ Ž       %#Ž>

      #"= %=<   #ˆƒ    #]=    >#   >#     % ˆ % < % " % & &0 ‚    ~   ƒ= ~   %#> %#ˆ %#] %#  %%# ~  #>  >=ˆ ~    

 #ƒ# #ƒ% #ƒ]    _ #]ˆ #]< #]] #]  >=]      %""    #ˆ% #"ˆ %ƒ  % <      #= Ž#% ~    %  #ƒ] #]] % ˆŽ ~   ˆ #%% #ƒ] #]  %ƒƒ %]% ~        %]" %"# %"% %">    [   #%ˆ    %#% %#ƒ %#< %#" %%= %%<Ž>% %ˆ# %ˆ% %<% %<ˆ ~     >#=      >ƒŽ]  [ < ] "        %] Ž"=   <  %"ƒ % = % #    >==   #<" #]%Žƒ     %><Ž"  #>=

      #"%

 % >     ƒ]Žˆ# ~      >##        %]ˆ 

  %ˆ% %ˆ] %<< >=<    %<  %]% %]> %]ƒ ~  #%% ~ `  #> ~   #]#Žƒ    %#= %#% %#ƒ %#< %#" %%= %ˆ#     ˆ=Ž# ~    ƒ>     >#%  \`> ƒ>  %"ƒ % = 

 #ƒ" #ˆ% #ˆ" #ˆ  #]= >=]

      #"ˆ  %>  %ƒ  %<#   #>>  % ~  "  #"> >=ƒ ~ &  ~ [     %=" %## >#= ~ [   #= ~ [   > ~      >% <# <> %=" —     "ƒŽ] #<# —    %=" >= —   <# "= %## — Q      <= <] %]> % ƒ ¡       * ##> >>= ¡    %ƒƒ >="