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John Smith Home Page Text to be displayed /h1 end header /center hr size=3 width=75% Draw a horizontal line (hr) width 3, and only over 75% of the width br break (i.e. new line) font size=4 Anything which follows will have a font size of 4 I am an text strong begin boldface undergraduate /strong end boldface student in the a href=http://civil.colorado.edu/index.html establish a link to .../index.html Civil Engineering Link is established whenever user clicks on this text. /a close the anchor Dept. p new paragraph font size=2 From here on, use font size 2 I am currently taking blink start blinking cven-4837 this text will be blinking /blink end blinking which covers br break img src=/bechtel/users5/ceae/Pictures/ball.green.gif Insert a bullet Intro to Unix Text br img src=/bechtel/users5/ceae/Pictures/red.green.gif Internet, HTML Programing

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MATLAB

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GPA Grades

Text header which will span 4 columns Victor E. Saouma Computing Literacy for Undergraduate Engineering Students Draft 7.5 Anatomy of an HTML le 7{5 Undergraduate Text tr Begin new row td Begin new cell Freshman cell entry td Sophomore td Junior td Senior etc.. tr td align=right 3.0 td align=center 3.2 < > < > < > < >< < > > ment < < > >				3.1	3.7 note optional align-

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More Help /h2 /center For more information on html programming, please consult ul begin unnumbered list li List entry a href=http://union.ncsa.uiuc.edu/HyperNews/get/www/html.html HyperText Markup Language (HTML) a text li a href=http://union.ncsa.uiuc.edu/HyperNews/get/www/html/learning.html Learning HTML a li a href=http://union.ncsa.uiuc.edu/HyperNews/get/www/html/guides.html Guides to Writing HTML Documents a /ul end unnumbered list /BODY Closing body tag. /HTML Closes the html tag.

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Victor E. Saouma

Computing Literacy for Undergraduate Engineering Students

World Wide Web

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Victor E. Saouma

Computing Literacy for Undergraduate Engineering Students

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Chapter 8

XMGR 8.1 Introduction xmgr is an extremely powerful graphing tool developed for the Sun workstations. xmgr is capable of producing almost any type of 2-D graph and is capable of performing many advanced mathematical functions, such as; digital ltering, spline curve tting, linear regression, fourier series, and more. It is a very exible program in that it allows the user to de ne how the data is to be read. Because it is exible and very advanced, it is not the easiest program to use. However, it's use is recommended because it can eliminate a considerable amount of programming eort, Fig. ??. A few of xmgr's features are:

     

Polynomial regression, splines, running averages, DFT/FFT, cross/auto-correlation. Plots up to 11 graphs with 15 datasets per graph. User-de ned scaling, tick marks, labels, symbols, line styles, colors. Batch mode for unattended plotting. Read and write parameters used during a session. Support through device independent graphic drivers for HP75xx plotters, PostScript, Sun raster les, and the HP LaserJet II. In general, xmgr requires two input les: 1. A numerical data le containing the data. 2. a parameter le containing information regarding the display of the plot (such as titles, tic mark spacing, etc..). If the plot is being generated for the rst time, xmgr will create the parameter le for you.

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8.2 Starting up XVGR

To start up xmgr, from the shelltool type > xmgr &

8.3 Exiting XVGR

To exit xmgr: 1. Click on File (with left button). This will pop-up a window. 2. With the left mouse button click on Exit.

8.4 Data les

 When there is more than one data set in a le each data set must be terminated by &  Comment lines may be included anywhere as long as they are preceded with a #  Files containing numeric data can have three basic formats: Two column format: This can be used for single or multi-data sets. The rst column contains X axis data and the second contains y axis data. For example:

# Example of a single set of two column format 282.553000 -0.167460 282.595000 -0.126310 282.637000 -0.130120 282.678000 -0.119800 282.720000 -0.037540 282.762000 -0.002380 # Not sure about this last point. experiment went wrong 282.803000 -0.022020 &

Another example may have two separate data sets with dierent X's: # Experimental Data 2. 22. 3.5 25. 4.2 36. 5.8 40 6.2 42. & # Numerical Prediction

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8.4 Data les 1. 2. 3. 4. 5. 6. 7. 8. &

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15. 21. 23. 30. 38. 41. 44. 45.

Multi column data: Quite often several data sets have the same corresponding X axis data. The multi column format is well suited for this data. The rst column contains X values and the remaining columns (up to 15) contain corresponding Y values.

# data 1.0 2.0 3.0 4.0 5.0 6.0 7.0 &

set 1 22.5 20.5 18.4 17.1 15.6 13.4 11.0

20.2 18.3 16.7 14.9 13.0 11.1 8.7

-25.5 -23.3 -20.9 -18.9 -17.1 -15.0 -13.2

IHL format: A 3 column data le with the rst two lines giving one line of alpha and the second an integer value with the number of points to follow. What ever that means!

Sample data les can be found in the directory: /usr/local/lib/xmgr.

8.4.1 Parameter File

The parameter le is best de ned interactively through xmgr and then saved. Do not attempt to edit it. Note that the same parameter le can be used for any data set. If you want your data set to be displayed using a prede ned parameter le, you must rst read in the parameter le before you load in your data. Sample parameter les can be found in /usr/local/lib/xmgr.

8.4.2 Reading an Input File

Once xmgr has been \ red-up", 1. Click on File (with right button). This will pop-up a window. 2. With the left mouse button click on Read Sets. This will pop up a menu with a list of all the les that have been found in the current directory. Victor E. Saouma

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3. With the left mouse button, select the le you wish to be loaded. 4. Underneath the list of les appears File Type. If your data is not in an XY format select File Type with the right mouse button, this will pop up a list of format options, choose the appropriate one with the left mouse button. 5. Once you are done select the Accept button at the bottom of the window, with the left mouse button.

8.5 Generating a Plot

Once you have loaded your data into xmgr, if you have not used a parameter le, you will notice that it automatically plots your data in a prede ned window. In all likelyhood you will need to rede ne the size of this window, as well as create a heading for your plot, and labels for the axises.

8.5.1 Changing the Size if the Plot Window

To rede ne the values of Xmax, Xmin, Ymax, and Ymin: 1. Click on View with the right mouse button. This will pop up a menu. 2. Click on Define world... with the left mouse button. This will pop-up another menu. 3. This last menu will allow you to modify the max and min values of X and Y axises, and also let you modify the placing of tick marks. Another way to do this is to use the Autoscale function located uder View, or click on the AS button at the top of the xmgr window. Both of these options will automatically resize the window to t your data.

8.5.2 De ning Labels for your Plot

To de ne X and Y-axis labels for your plot: 1. Click on View with the right mouse button. This will pop up a menu. 2. Click on Ticks/tick labels... with the left mouse button. This will pop-up another menu. 3. Click on Edit: with the right mouse button, this will allow you to choose which axis you will be modifying. 4. The next line, called Axis label is where you type in your label. 5. when you are done modifying this portion of your plot click on the Accept button with the left mouse button. Victor E. Saouma

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8.5 Generating a Plot

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8.5.3 De ning a Title for your Plot

De ning a heading for your plot is done similarly to de ning labels: 1. Click on View with the right mouse button. This will pop up a menu. 2. Click on Title/subtitle... with the left mouse button. This will pop-up another menu. 3. This last menu will prompt you to type in a title. The subtitle is optional. 4. when you are done modifying this portion of your plot click on the Accept button with the left mouse button.

8.5.4 Plotting Multiple Datasets

To plot another set of data on the axis displayed in the plot window, simply repeat the steps for loading up a le. 1. Click on File (with right button). This will pop-up a window. 2. With the left mouse button click on Read Sets. This will pop up a menu with a list of all the les that have been found in the current directory. 3. With the left mouse button, select the le you wish to be loaded. 4. Underneath the list of les appears File Type. If your data is not in an XY format select File Type with the right mouse button, this will pop up a list of format options, choose the appropriate one with the left mouse button. 5. Once you are done select the Accept button at the bottom of the window, with the left mouse button.

8.5.5 Changing the Plot Symbols

You may also change how your data sets are displayed. For instance if you have multiple datasets on one plot you may wish to have one dataset ploted using a dotted line, and another plotted using a dashed line. Another example would be if you have done some sort of t to your data you may wish to see the actual data points plotted on the tted curve. All this is done from the Symbols option of the View menu. After you have clicked on Symbols you will get a window. If you have multiple data sets, you will need to de ne which one you wish to modify. This can be done by clicking on Select set with the right mouse button, then select the number of the dataset you wish to modify. Xvgr numbers your sets starting with zero for the rst set loaded, one for the second set loaded, etc. If you wish to plot out your data as points Victor E. Saouma

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XMGR

1. 2. 3. 4.

Click on Symbols with the left mouse button. Choose which symbol you wish to represent your data points, with the left mouse button. Click on Style with the right mouse button. Choose None with the left mouse button. This will shut o the line-drawing operation, so your data will be plotted as the symbols you have speci ed. 5. Lastly, click on Accept with the left mouse button, and this will redisplay your data in the new format. If you wish to change the style of the line that represents your data: 1. Chose the data set you wish to modify, as explained above. 2. Click on Style with the right mouse button. 3. Choose one of the options listed, with the left mouse button. 4. Click on Accept with the left mouse button, and this will redisplay your data in the new format. Once you have nished modifying your display, click on Done with the left mouse button.

8.5.6 De ning a Legend for your Plot

If you have multiple data sets plotted on one graph you may wish to de ne a legend for the graph. This can be done as follows: 1. You need to follow the steps in the preceding section for rede ning the plot symbols. 2. Next click on View with the right mouse button, then click on Legends with the left mouse button. 3. On the top line of the window is Legend, click the ON button with the left mouse button. 4. Click Legend location type with the right mouse button, and choose World coordinates. 5. To place the legend on your plot, click Place with the left mouse button, then place the pointer on your plot where you wish the legend to appear and click the left mouse button again. 6. Click on Load comments with the left mouse button to set the legend comments. The default is to name each dataset with the lename that the set was loaded from, if you wish to change this, this can be done from the Symbols menu, by editing the line Legend: on Symbols menu. Sometimes the legend will not end up exactly where you wanted it to be. You can move the legend around by editing the Legend X: and Legend Y: lines on the Legend menu. The world coordinates of the current pointer position are listed at the top of the xmgr window, this will help you with ne tuning the legend position. Victor E. Saouma

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8.6 Plotting Multiple Frames in one Window

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8.6 Plotting Multiple Frames in one Window To view multiple plots in a non-overlapping format you would: 1. Click on View with the right mouse button. 2. Click on Graphs with the right mouse button. 3. Click on Arrange Graphs... with the left mouse button. 4. Select the number of rows, and columns you wish to be displayed. The layout will start from the lower left and go up each column, for example given 3 columns and 2 rows the graphs would be laid out as follows: 1 3 5 0 2 4

Note: the numbering starts at 0. 5. Select the packing method. Use this item when there are several graphs with the same X or Y axis scaling so graphs on the outside of the packing arrangement provide the tick and axis labeling for all graphs in that row, or column or both. 6. The next selections will determine the individual graph size and how much space to leave between each graph. These coordinates are in viewport coordinates. They can be thought of as percentages of the graph window. For example if you were to have one row of two graphs, you would want to de ne the width of each graph as slighlty less than 50% of the graph window, or Graph width as .45, and something similar for graph height. 7. You can adjust the Start at X and Start at Y to center your graphs in the window. This will position the rst graph. 8. When you have nished making your selections click the Accept button with the left mouse button. 9. When you are satis ed with your plot click on Done with the left mouse button. Because of the interactive nature of xmgr, arranging your plots in the graph window may be an iterative process. You may continuously adjust the starting point of the graphs, or the height and width of the plots, etc., until you get the desired results.

8.7 Getting a Hardcopy of your Plot To print a plot that is currently showing in the plot window: 1. Click on File with the right mouse button. 2. Choose Print with the left mouse button. This will automatically send a copy of your plot to the printer. Victor E. Saouma

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XMGR

8.8 Saving Your Plot to a File If you wish to save your plot in a le: 1. Click, with the left mouse button on Printer setup. This will pop up a window. 2. Click with the right mouse button on Print to. This will display two options Printer or File. 3. Click on File with the left mouse button, this will then prompt you for a le name for your plot. 4. Once you have entered the le name, click on the Print button at the bottom of this window with the left mouse button. This will \print" your plot to a le instead of the printer.

8.9 Other Buttons in The XVGR Window

Across the top of the xmgr window you will notice other buttons. These are: Re-Draw: Generally after you have modi ed a plot xmgr will automatically re-draw once you have clicked the Accept button. If this doesn't happen, click Re-draw with the left mouse button. You will also need to Re-draw if you have resized your window. Z: Zoom. E: Expand world. S: Shrink world. AS: Autoscale. AT: Auto-ticks. PU: Push current world. PO: Make top of stack the current world. L: Scroll left. R: Scroll right. U: Scroll up. D: Scroll Down. PZ: Push current world and enable zoom. CY: Cycle through the world stack.

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8.10 Editing your Data in XVGR

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8.10 Editing your Data in XVGR will allow you to \massage" your data: 1. Edit to add, delete, or move points 2. Transformations to perform certain operations such as Integration, Regression, Fourier, Digital Filtering, etc... When you are done, select Files again to store your parameter le (unless you loaded one which has not been changed) and then give it a name under Write parameters. Exit by selecting

Compose

Exit

8.11 References, and Demo Unfortunately there is no reference to xmgr other than the man pages. For a brief demo of the program, type: % /usr/local/lib/xmgr/Demo

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XMGR

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Victor E. Saouma

Computing Literacy for Undergraduate Engineering Students

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Part II

MODULE II (1 cr)MATLAB & MATHEMATICA

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Chapter 9

WEEK I; MATLAB: Basic Introduction 9.1 Background

9.1.1 What is MATLAB? MATLAB is an interactive, matrix-based system for scienti c and engineering calculations. Contrarily to programming languages, such as Fortran, C, or Basic you can solve complex numerical problems without actually writing a program. 2 MATLAB is an abbreviation for MATrix LABoratory. 3 MATLAB is the most widely used software package for interactive numeric computation, graphics, data analysis1. 1

9.1.2 Availability MATLAB is available in the CAD lab (PC/Windows version), as well as in the Bechtel Laboratory. 5 A relatively inexpensive Student Edition is available from Prentice Hall and can be purchased from the Bualo Chip.

4

9.1.3 Accessing MATLAB In the CAD Lab, Matlab is accesible within Windows. 7 In the Bechtel Lab, MATLAB is accesible through by clicking the left mouse button and then selecting MATLAB. Note that the program is licensed to run only on the server (bechtel). 6

1 Other

similar products include IDL, and MATHEMATICA for symbolic computations.

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WEEK I; MATLAB: Basic Introduction

8 If you want to run the program dierently, you need to xhost + on your console rlogin bechtel To connect you to the server setenv DISPLAY yourmachine:0.0 To have the graphics 9

displayed on your workstation

To exit MATLAB, simply type exit or quit.

9.1.4 Help for help you can either type help sqrt (i.e. help command followed by the name of the function, or you may type lookfor square (similar to the man -k in Unix).

10

9.1.5 Basic Features When you run MATLAB, there will be one or more windows on your monitor. The command window is the primary one where you interact with MATLAB, and the MATLAB prompt will look like

11

>>

with a blinking cursor to the right. 12 Basic commands are shown in Table 9.1.

help demo who what clear computer bC exit quit

General help facility run demonstrations list variables in memory list M- les on disk clear workspace type of computer local abort exit MATLAB same as exit

Table 9.1: Basic Commands 13

MATLAB has also a few special characters, Table 9.2

9.1.6 Simple Math 14

To begin with, let us go through a simple example

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9.1 Background

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Special Characters = assignment statement [ used to form vectors and matrices ] see [ ( arithmetic expression precedence ) see ( . decimal point ... continue statement to next line , separate subscripts and function arguments ; end rows, suppress printing % comments : subscripting, vector generation ! execute operating system command Table 9.2: Special Characters >> 29+6 ans= 35 >> 21*2+5^2 ans= 67 >> force=10 force = 10 >> distance=5 distance= 5 >> moment=force*distance moment= 50 >>

Try now the following commands who, whos, clear, who. 15 The basic arithmetic operations are +

addition ; subtraction  multiplication b power n or = left or right division Victor E. Saouma

7.2+9.6 11-92 2.9*4.5 2.6b 0.45 27=5=5n27

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WEEK I; MATLAB: Basic Introduction

9.1.7 Common Mathematical Functions 16

Common mathematical functions are shown in Table 9.3

9.1.8 Statements, Expressions and Variables MATLAB is an interprative language; the expressions you type are interpreted and evaluated. MATLAB statements are usually of the form variable = expression, or simply expression 18 Expressions are usually composed from operators, functions, and variable names. Evaluation of the expression produces (most often) a matrix, which is then displayed on the screen and assigned to the variable for future use. If the variable name and = sign are omitted, a variable ans (for answer) is automatically created to which the result is assigned. 19 A statement is normally terminated with the carriage return. However, a statement can be continued to the next line with three or more periods followed by a carriage return. On the other hand, several statements can be placed on a single line if separated by commas or semicolons. 17

a=2;b=4;c=-240;deltasq=b^2-4*a*c;delta=sqrt(deltasq) delta = 44 x1=(-b+delta)/(2*a) x1 = 10 x2=(-b-delta)... /(2*a) x2 = -12

If the last character of a statement is a semicolon, the printing is suppressed, but the assignment is carried out. This is essential in suppressing unwanted printing of intermediate results. 21 MATLAB is case-sensitive in the names of commands, functions, and variables. For example, Apples is dierent from APPLES. 20

9.1.9 Display Formats 22 23

All computations in MATLAB are performed in double precision. The format of the displayed output is shown in Table 9.4 Once invoked, the chosen format

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9.1 Background

9{5 Elementary Math Functions abs absolute value or complex magnitude angle phase angle sqrt square root real real part imag imaginary part conj complex conjugate round round to nearest integer x round toward zero

oor round toward ;1 ceil round toward 1 sign signum function rem remainder exp exponential base e log natural logarithm log10 log base 10 Trigonometric Functions sin sine cos cosine tan tangent asin arcsine acos arccosine atan arctangent atan2 four quadrant arctangent sinh hyperbolic sine cosh hyperbolic cosine tanh hyperbolic tangent asinh hyperbolic arcsine acosh hyperbolic arccosine atanh hyperbolic arctangent Special Functions bessel bessel function gamma gamma function rat rational approximation erf error function inverf inverse error function ellipk complete elliptic integral of rst kind ellipj Jacobian elliptic integral Table 9.3: Common Mathematical Functions

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WEEK I; MATLAB: Basic Introduction

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format short format long format short e format long e format bank format + format rat format hex

default display 16 digits 5 digits plus exponent 16 digits plus exponent 2 decimal digits positive, negative or zero rational approximation hexadecimal

Table 9.4: Display Options remains in eect until changed. 24 The command format compact will suppress most blank lines allowing more information to be placed on the screen or page. It is independent of the other format commands. sqrt(pi) ans = 1.7725 format long ans ans = 1.77245385090552 format long e ans ans = 1.772453850905516e+000

9.1.10 Printing Text and Matrices 25

The disp command can be used to display both data and text

disp(pi);disp('University of Colorado') 3.1416 University of Colorado

Formatted output can be done through the fprintf command which provides you with better control on the format. It has two arguments: 1. Text and format speci cations which must be enclosed in a single quote. Within the text, the following speci er can be used:

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9.1 Background

9{7 %e %f %g n

n

Exponential notation Fixed point or decimal notation Whichever is shorter New line

2. Matrix to be printed v=153.98; fprintf('Velocity is %f miles an hour',v) Velocity is 153.980000 miles an hourfprintf('Velocity is %f miles an hour\n',v) Velocity is 153.980000 miles an hour fprintf('Velocity is %e miles an hour\n',v) Velocity is 1.539800e+002 miles an hour fprintf('Velocity is %g miles an hour\n',v) Velocity is 153.98 miles an hour fprintf('Velocity is %8.3f miles an hour\n',v) Velocity is

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153.980 miles an hour

Text and string operations, Table 9.5

abs eval num2str int2str setstr sprintf isstr strcomp hex2num

Text and Strings convert string to ASCII values evaluate text macro convert number to string convert integer to string set ag indicating matrix is a string convert number to string detect string variables compare string variables convert hex string to number

Table 9.5: Text and String Operations

9.1.11 Variables 28

Variables are case sensitive, can contain up to 19 characters, and must start with a letter.

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WEEK I; MATLAB: Basic Introduction

MATLAB has several special variables, Table 9.6.

ans eps pi i, j inf NaN clock date

ops nargin nargout

Special Variables answer when expression not assigned

oating point precision

p

;1 1

Not-a-Number wall clock date

oating point operation count number of function input arguments number of function output arguments Table 9.6: Special Variables

9.1.12 MATLAB Workspace 30 When one logs out or exits MATLAB all variables are lost. However, invoking the command save before exiting causes all variables to be written to a non-human-readable disk le named matlab.mat. When one later reenters MATLAB, the command load will restore the workspace

to its former state. 31 To recall previous commands, you can use the cursors keys. 32 MATLAB can execute a sequence of statements stored on disk les. Such les are called \M- les" because they must have the le type of .m as the last part of their lename. 33 There are two types of M- les: Script les are ASCII les which consist of a sequence of normal MATLAB statements. If the le has the lename, say, hw1.m, then the MATLAB command hw1 will cause the statements in the le to be executed. Script les can be edited by a standard editor. Script les can also be used to enter data into a large matrix; in such a le, entry errors can be easily edited out. An M- le can reference other M- les, including referencing itself recursively. Function les Function les provide extensibility to MATLAB. You can create new functions speci c to your problem which will then have the same status as other MATLAB functions.

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9.1 Background

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9.1.13 Saving and Retrieving Data 34 If you want to save the variables in the fn command where fn is the le name. 35 36

workspace before you quit, you must use the save

To retrieve your data next time, load fn. Disk les commands are shown in Table 9.7

chdir delete diary dir load save type what fprintf pack

Disk Files change current directory delete le diary of the session directory of les on disk load variables from le save variables to le list function or le show M- les on disk write to a le compact memory via save

Table 9.7: Disk Files Commands

9.1.14 Arrays 37

Arrays can be entered in several dierent ways:  Entered by an explicit list of elements,  Generated by built-in statements and functions,  Created in M- les,  Loaded from external data les

38

The following session is self explanatory, study it in detail.

x=[2. 6. 12.] x = 2 6

12

y=[3. 5. 1.] y =

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WEEK I; MATLAB: Basic Introduction 3

5

1

x.*y ans = 6

30

12

10 30 60

2 6 12

12

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y' ans = 3 5 1 x*y' ans = 48 x'*y ans = 6 18 36 2*x ans = 4 x(2) ans = 6 x(1:2) ans = 2

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x(3:-1:1) ans = 12

6

2

x=(0:0.1:1) x = Columns 1 through 7 0 0.1000 Columns 8 through 11 0.7000 0.8000

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0.2000

0.3000

0.9000

1.0000

0.4000

0.5000

0.6000

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9.1 Background

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linspace(0,pi,11) ans = Columns 1 through 7 0 0.3142 Columns 8 through 11 2.1991 2.5133 x=[2. 6. 12.] x = 2 6 x.^2 ans = 4 x=[1,2,3] x = 1

39

0.6283

0.9425

2.8274

3.1416

1.2566

1.5708

1.8850

12

36

144

2

3

Array operations are shown in Table 9.8 Element by Element Array Operations Operation Algebraic Form MATLAB Addition ai + bi a+b Substraction ai ; bi a;b Multiplication ai  bi a:  b Right division ai =bi a:=b Left division bi =ai a:nb b i Power ai a:^b Table 9.8: Array Operations

x=[6 2 4 8 -2 12];y=[2 6,3, 1 2,10] y = 2 6 3 1 2 10 size(y) ans = 1 6 x+y ans =

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WEEK I; MATLAB: Basic Introduction 8

8

x.*y ans = 12 12 x./y ans = 3.0000 y.\x ans = 3.0000

7

9

0

22

12

8

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120

0.3333

1.3333

8.0000

-1.0000

1.2000

0.3333

1.3333

8.0000

-1.0000

1.2000

9.1.15 Graphics One of the major strength of MATLAB are its graphics capabilities. They are numerous, and only few will be explored at this early stage. 41 MATLAB can produce both planar plots and 3-D mesh surface plots. To preview some of these capabilities enter the command plotdemo. 42 The plot command creates linear x-y plots; if x and y are vectors of the same length, the command plot(x,y) opens a graphics window and draws an x-y plot of the elements of x versus the elements of y.

40

x=(0:0.1:10); y=sin(x).*exp(-x); plot(x,y) grid xlabel('time [s]') ylabel('Temperature [Deg. F]') print -deps2 plot1.eps

will generate the plot of Fig. 9.1 Note that as new commands are entered, the plot is immediately updated. 43 When in the graphics screen, pressing any key will return you to the command screen while the command shg (show graph) will then return you to the current graphics screen. If you are in the Bechtel Lab, then you can have multiple graphics windows. 44 The command grid will place grid lines on the current graph. 45 Graphs can be given titles, axes labeled, and text placed within the graph commands which take a string as an argument, Table 9.9. 46 To make multiple plots on a single graph x=0:0.1:2*pi;y1=sin(x);y2=cos(x);plot(x,y1,'-',x,y2,'-.');grid

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9.1 Background

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title xlabel ylabel gtext text

graph title x-axis label y-axis label interactively-positioned text position text at speci ed coordinates Graphs plot linear X-Y plot loglog loglog X-Y plot semilogx semi-log X-Y plot semilogy semi-log X-Y plot polar polar plot bar bar charts stairs stairstep graph errorbar add error bars Graph Annotation title plot title xlabel x-axis label ylabel y-axis label grid draw grid lines text arbitrarily position text gtext mouse-positioned text ginput graphics input Graph Window Control axis manual axis scaling hold hold plot on screen shg show graph window clg clear graph window subplot split graph window Table 9.9: 2D Graphics Commands

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WEEK I; MATLAB: Basic Introduction

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Figure 9.2: Multiple Plots in a Single Graph Line colors can also be separately speci ed 48 The command subplot can be used to partition the screen so that up to four plots can be viewed simultaneously. 47

9.1.16 Graphics hardcopy 49 A hardcopy of the graphics screen can be most easily obtained with the MATLAB print which will print or save the graph (see example above). The syntax is

command

PRINT [ -ddevice] [ -options ]

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9.2 Assignment

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where the most common devices are shown in Table 9.10 dps PostScript for black and white printers dpsc PostScript for color printers dps2 Level 2 PostScript for black and white printers dpsc2 Level 2 PostScript for color printers deps Encapsulated PostScript (EPSF) depsc Encapsulated Color PostScript (EPSF) deps2 Encapsulated Level 2 PostScript (EPSF) depsc2 Encapsulated Level 2 Color PostScript (EPSF) dhpgl HPGL compatible with Hewlett-Packard 7475A plotter dljetplus HP LaserJet+ dljet3 HP LaserJet III dcdeskjet HP DeskJet 500C with 1 bit/pixel color dbj10e Canon BubbleJet BJ10e dgif8 8-bit color GIF le format Table 9.10: Printer Devices

9.2 Assignment 9.2.1 Practice

Start by repeating all the examples in this handout.

9.2.2 Work

The force F which moves a body along a straight line path S is the scalar product of F:S = W . In general we may either break down the path into a series of linear segments, or if the path is R curvilinear we must perform a line integral W = S F:dS . Given a force F with components (2,4,6), i.e. F = 2i + 4j + 6k, and the following points Point O A B C D E

Coordinates X Y Z 0 0 0 5 0 0 0 7 0 0 0 11 -200 -500 +1000 6 8 12

Determine the work for each of the following paths Victor E. Saouma

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9{16

Discuss your results.

WEEK I; MATLAB: Basic Introduction Case 1 2 3 4

Path O-A-B-C-E O-C-E O-D-E O-E

9.2.3 Stresses

If an in nitesimal element is subjected to the cartesian stresses shown below,

it can be shown that the stresses along any other orientation are given by: x1 = x +2 y + x ;2 y cos 2 + xy sin 2 x1y1 = ; x ;2 y sin 2 + xy cos 2 and the principal stresses are given by:

1;2 = x +2 y 

s



(9.1-a) (9.1-b)

x ; y 2 +  2

(9.2) xy 2 For x = 12; 300 psi, y = ;4; 200 psi and xy = ;4; 700 psi, 1. Determine the stresses when the element is rotated by +30 degrees (counter-clockwise), 2. Determine the principal stresses 3. Plot the variation of all three stresses in terms of  Victor E. Saouma

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9.2 Assignment

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9.2.4 Measurement Errors, [1]

Let us consider an instrument with a scale graduated from 0-1000 (such as 0 to 1000 volts). If the instrument is guaranteed as belonging to the 3% class, the maximum error is 3% of the full-scale de ection, in this case 30. Hence, if the meter reads 500, the true value can be 30 100 = 6% Hence, anywhere in the range 470-530. The corresponding relative error equals 500 it is clear that the full scale de ection of the measuring instrument should not be much higher than the range of expected values. One frequently used rule of thumb is to havethe expected value between 1/2 and 2/3 of the full scale de ection. For this instrument 1. Calculate and display the per cent error for measured values in increments of 100. 2. Plot the percent error, against the measured values, in the interval 0-1000. Use intervals of 10.

9.2.5 Statistical Analysis; Part I 9.2.5.1 Elements of Statistics

Elementary statistics formulaes will be reviewed, as they are needed to properly understand structural reliability. 51 When a set of N values xi is clustered around a particular one, then it may be useful to characterize the set by a few numbers that are related to its moments (the sums of integer powers of the values): Mean: estimates the value around which the data clusters. 50

 = N1

N X i=1

xi

(9.3)

it is the arithmetic average of all the data points. Expected Value: If data are not available, an expected value is assigned based on experience and judgment, and E (x) = x. Both the mean and the expected values are termed rst moment, and they correspond to the centroid of a probability density distribution. ( R1 = xf (x)dx Continuous systems E (x) = x = P;1 (9.4) N xf (x) Discrete systems i=1

Median: of a sorted series (xi;1 < xi < xi+1 ) is de ned as: xmed = Victor E. Saouma

(

x N2+1 N odd 1 (x N + x N ) N even 2 2 2 +1

(9.5)

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WEEK I; MATLAB: Basic Introduction

Variance: is an indication of the \width" of the cluster: 2 = N 1; 1 Ni=1 (xi ; )2 (9.6) or 2 = N1 Ni=1 (xi ; )2

(9.7)

Note that if N is less than 10, it is more appropriate to use the second equation, otherwise use the rst one. Standard Deviation: is de ned as the square root of the Variance

p

 = 2

(9.8)

Coecient of Variation: is the standard deviation normalized with respect to the mean: V = 

(9.9)

When insucient data are available to accurately compute the moments, the coecient of variation is often estimated on the basis of experience. Covariance: Pairs of random variables may be correlated or independent. If correlated, then the likelihood of y depends on the likelihood of x. Thus, covariance xy measures the combined eect of how two variables vary together.

xy = N1

N X i=1

(xi ; x)(yi ; y )

(9.10)

Correlation Coecient: xy is a nondimentional measure of the degree of correlation xy = xy

x y

(9.11)

A correlation coecient of 1:0 or ;1:0 indicates a perfect linear correlation. A positive value indicates that the variables either increase or decrease together, a negative one indicates that one value increases while the other decreases. A zero value indicates that there is no linear correlation between the variables.

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9.2 Assignment

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Skewness: characterizes the degree of asymmetry of a distribution around its mean. It is de ned in a non-dimensional value. A positive one signi es a distribution with an asymmetric tail extending out toward more positive x  3 (9.12) Skew = 1 N xi ; 

N i=1



Kurtosis: is a nondimensional quantity which measures the \ atness" or \peakedness" of a

distribution. It is normalized with respect to the curvature of a normal distribution. Hence a negative value would result from a distribution resembling a loaf of bread, while a positive one would be induced by a sharp peak:  4 Kurt = 1 N xi ;  ; 3 (9.13)

N i=1



the ;3 term makes the value zero for a normal distribution. The expected value (or mean), standard deviation and coecient of variation are interdependent: knowing any two, we can determine the third. 53 Distribution of variables can be mathematically represented. 54 A Uniform distribution implies that any value between xmin and xmax is equaly likely to occur. 55 The general normal (or Gauss) distribution is given by, Fig. 9.3: 52

(x) = p 1 e; 21 [ 2 56

x; ]2 

(9.14)

A normal distribution N (; 2 ) can be normalized by de ning

y = x ; 

(9.15)

y2 (y) = p1 e; 2 2

(9.16)

and y would have a distribution N (0; 1):

The normal distribution has been found to be an excellent approximation to a large class of distributions, and has some very desirable mathematical properties: 1. f (x) is symmetric with respect to the mean . 57

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−3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.40

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0.00 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 µ 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 x

CDF

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9{20

Figure 9.3: Normalized Gauss Distribution, and Cumulative Distribution Function 2. f (x) is a \bell curve" with in ection points at x =   . 3. f (x) is a valid probability distribution function as: Z

1 f (x) = 1 ;1

4. The probability that xmin < x < xmax is given by:

P (xmin < x < xmax ) =

Z

(9.17) xmax

xmin

f (x)dx

5. Cumulative distribution functions (cdf) of the normal distribution de ned as: Z s x; 2 1 e; 12 [  ] dx (s) = p 2 ;1 and is expressed in terms of the error function (erf). 6. The cdf of normalized normal distribution function is given by: Z s 2 (s) = p1 e; x2 dx 2 ;1 and is usually tabulated in books. Victor E. Saouma

(9.18) (9.19)

(9.20)

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9.2 Assignment

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9.2.5.2 Assignment

1. Generate the data for a normal distribution with a mean of 100 and a standard deviation of 20 and plot the probability distribution function from the mean minus the standard deviation, to the mean plus the standard deviation. 2. Retrieve (through anonymous ftp to bechtel) the following two sets ftp/pub/Structures/cven4837/set1.da and ftp/pub/Structures/cven4837/set2.dat. Those sets contains results of concrete compressive strength fc0 for two dierent ready mix companies. (a) For each set determine the Mean, Standard Deviation, Coecient of variation, Skewness, and Kurtosis. (b) Plot the probability distribution function f (x) (based on a Normal Distribution), plotted from  ; 4: to  + 4: (c) Plot a histogram of the normalized strength (from the input le) with 20 bins. (d) Superimpose the two plots. Note This problem will be revisited later to: a) use MATLAB statistical functions, and b) Apply numerical integration for the evaluation of the probability of the stregth being below a certain value.

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Chapter 10

WEEK II; MATLAB: Matrix Algebra 10.1 Syntax

10.1.1 Matrix Operations 10.1.1.1 Matrix De nition 1

Matrices can be entered in several dierent ways:  Entered by an explicit list of elements, or built from blocks. For example, A=[1 4 6; 2 1 8; -2 7 -9]; B = [A, ones(3,2); zeros(2,3), eye(2)]} will build a 5-by-5 matrix.

 Generated by built-in statements and functions, Table 10.1. For example,

ones(m,n)

produces an m-by-n matrix of ones; if A is a matrix, then ones(A) produces a matrix of ones of the same size as A. If x is a vector, diag(x) is the diagonal matrix with x down the diagonal; if A is a square matrix, then diag(A) is a vector consisting of the diagonal of A.  Created in M- les,  Loaded from external data les

2

The following session is self explanatory,

x=[1,2;6,7] x = 1 2

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eye zeros ones diag triu tril rand hilb magic toeplitz

identity matrix matrix of zeros matrix of ones see below upper triangular part of a matrix lower triangular part of a matrix randomly generated matrix Hilbert matrix magic square see help toeplitz

Table 10.1: Built In Matrix De nition Functions 6 x=[1 2 3 4 5 6 9 10 11] x = 1 4 9

7

2 5 10

3 6 11

\colon operation" is a much more eective way to deal with consecutive numbers than through loops (very slow). For instance

3

x=[3 -1 5] x = 3 -1

5

y=[x;x+1;2*x-1] y = 3 -1 4 0 5 -3

5 6 9

rot90(y) ans = 5 -1 3

6 0 4

9 -3 5

fliplr(y) ans = 5 -1 6 0 9 -3

3 4 5

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10.1 Syntax flipud(y) ans = 5 -3 4 0 3 -1

10{3 9 6 5

y(:,3) ans = 5 6 9 y(2,:) ans = 4

0

6

x(1,:) ans = 3

-1

5

z=[y(1,:);y(3,:)] z = 3 -1 5 5 -3 9 v=[0.2 1.4 2.6] v = 0.2000 1.4000

2.6000

diag(v) ans = 0.2000 0 0

0 0 2.6000

0 1.4000 0

10.1.1.2 Matrix Operations Matrix operations are listed in Table 10.2. 5 The \matrix division" operations deserve special comment. If A is an invertible square matrix and b is a compatible column, vector, then x = Anb is the solution of A  x = b

4

>> A=[2 3;1 4];b=[8;9];AInv=inv(A); >> x=AInv*b x = 1 2 >> x=A\b

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WEEK II; MATLAB: Matrix Algebra

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+ addition ; subtraction  multiplication b power 0 transpose n left division / right division Table 10.2: Matrix Operations x = 1 2

10.1.1.3 Matrix functions 6

Matrix built in functions are listed in Table 10.3.

eig chol svd inv lu qr logm expm sqrtm poly det size norm cond rank

eigenvalues and eigenvectors cholesky factorization singular value decomposition inverse LU factorization QR factorization matrix logarithm matrix exponential matrix square root characteristic polynomial determinant size 1-norm, 2-norm, F-norm, 1-norm condition number in the 2-norm rank Table 10.3: Matrix Functions

EDU>> x=[4 2 3; 1 6 5;-2 5 10] x = 4 2 3 1 6 5

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10.1 Syntax -2

10{5

5

10

EDU>> y=inv(x) y = 0.2318 -0.0331 -0.1325 0.3046 0.1126 -0.1589 EDU>> x*y ans = 1.0000 0.0000 0.0000

-0.0530 -0.1126 0.1457

0 1.0000 0.0000

0.0000 0.0000 1.0000

2.0000 6.0000 -0.9167

3.0000 11.5000 -6.2917

EDU>> lux=lu(x) lux = 4.0000 2.0000 -0.2500 6.0000 0.5000 -0.9167

3.0000 11.5000 -6.2917

EDU>> triu(lux) ans = 4.0000 2.0000 0 6.0000 0 0

3.0000 11.5000 -6.2917

EDU>> tril(lux) ans = 4.0000 0 -0.2500 6.0000 0.5000 -0.9167

0 0 -6.2917

EDU>> tril(lux,-1) ans = 0 0 -0.2500 0 0.5000 -0.9167

0 0 0

EDU>> lu(x) ans = 4.0000 -0.2500 0.5000 EDU>> det(x) ans = 151

EDU>> diag(lux) ans = 4.0000 6.0000

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10{6

WEEK II; MATLAB: Matrix Algebra

-6.2917 EDU>> eig(x) ans = 4.4123 12.9438 2.6440 EDU>> x=[0.5 0.25;0.25 0.5] x = 0.5000 0.2500 0.2500 0.5000 EDU>> [V,D]=eig(x) V = 0.7071 0.7071 -0.7071 0.7071 D = 0.2500 0 0 0.7500 EDU>> x*V-V*D ans = 0 0 0 0

10.1.2 Graphics Revisited 10.1.2.1 Polar Plots

theta=0:pi/60:2*pi;r=2*(1-cos(theta));polar(theta,r);axis square print -deps2 polar.eps

resulting in Fig 10.1

10.1.2.2 3-D mesh plots. 7 Three dimensional mesh surface plots are drawn with the function mesh. mesh(z) creates a three-dimensional perspective plot of the elements of the

The command matrix z . The mesh surface is de ned by the z-coordinates of points above a rectangular grid in the x-y plane.

% Shape Functions for Quadrilateral Quadratic Elements X=-1:1/20:1; Y=X; YT=Y'; XT=X'; N8=0.5*(1-YT.*YT)*(1-X); meshc(X,Y,N8)

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10{7

Draft

90 4 120

60 3 2

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30

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240

300 270

Figure 10.1: Polar Plot of a Cardioid print -deps2 shap8-8.eps c=contour(X,Y,N8); clabel(c) print -deps2 shap8-8-c.eps

and the plots are shown in Fig. 10.2 1 0.1 0.8

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Figure 10.2: Sample of Surface and Contour Plots

10.1.2.3 Animation 8 Animation can be achieved in MATLAB moviein and getframe commands.

(student's edition may not support it) through the

X=-1:1/5:1;

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10{8

WEEK II; MATLAB: Matrix Algebra

Y=X; YT=Y'; XT=X'; N8=0.5*(1-YT.*YT)*(1-X); k=0; M=moviein(11); % set up for 11 frames for fac=-5:1:5 % loop k=k+1; z=fac*N8; surf(X,Y,z); %axis off; axis([-1 1 -1 1 -5 5]) % freeze the axis to user specified values M(:,k)=getframe; % grab the frame into M end movie(M,10) % play back 10 times

will generate the animation of Fig. 10.3 5

0

−5 1 1

0.5 0.5

0 0

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−1

Figure 10.3: Picture Used for Animation Note that alternatively you may generate the image within a loop, and insert a pause after each one.

9

10.1.3 Data Analysis

10.1.3.1 Statistical analysis 10

Statistical analysis prede ned functions are shown in Table 10.4.

EDU>> x=rand(1,10) % generate a uniform distribution x = Columns 1 through 7

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10{9

Draft

max min mean median std sort sum prod cumsum cumprod di hist corrcoef cov cplxpair

maximum value minimum value mean value median value standard deviation sorting sum of elements product of elements cumulative sum of elements cumulative product of elements approximate derivatives histograms correlation coecients covariance matrix reorder into complex pairs

Table 10.4: Statistical Analysis Functions 0.2190 0.0470 Columns 8 through 10 0.8310 0.0346

0.6789

0.6793

0.9347

0.3835

0.5194

0.0535

EDU>> rand('seed',0) %set seed EDU>> rand('uniform') % specify a uniform random distribution EDU>> rand(1,10) ans = Columns 1 through 7 0.2190 0.0470 0.6789 0.6793 0.9347 0.3835 Columns 8 through 10 0.8310 0.0346 0.0535

0.5194

EDU>> rand('seed',0) % reset the seed EDU>> x=randn(1,100); %Generates random numbers with a normal distribution EDU>> hist(x,20) % Histogram with 20 bins EDU>> print -deps2 hist1.eps EDU>> mean(x) ans = 0.0507 EDU>> max(x) ans = 2.9432 EDU>> min(x) ans =

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10{10

WEEK II; MATLAB: Matrix Algebra

-2.0186 EDU>> median(x) ans = 0.0239 EDU>> sum(x) ans = 5.0730 EDU>> std(x) ans = 0.9979 EDU>> sort(x) ans = Columns 1 through 7 -2.0186 -1.8769 -1.7651 ............. Columns 99 through 100 2.3722 2.9432

-1.6984

-1.6853

-1.6237

-1.4921

where the le hist1.eps is shown in Fig. 10.4 14

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2

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Figure 10.4: Histogram

10.1.3.2 Regression Analysis

The following example will generate data which approximately t y = 2x +10, and then through a linear regression determine the parameters. EDU>> x=[0:1:10]; EDU>> mnoise=0.5*rand(1,11) mnoise = Columns 1 through 7

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10.1 Syntax 0.0173 0.0267 Columns 8 through 11 0.2087 0.3434

10{11 0.2649

0.3356

0.2945

0.4652

0.0038

0.1917

0.0334

22.3609

24.0212

EDU>> bnoise=2*rand(1,11); EDU>> y=(2+mnoise).*x+10+bnoise % noisy y=2x+10 y = Columns 1 through 7 11.6923 13.0806 14.7136 18.3146 18.8474 Columns 8 through 11 26.9856 29.2720 30.7453 36.1243 EDU>> coef=polyfit(x,y,1)% get the coefficients coef = 2.3546 10.6048 EDU>> m=coef(1); b=coef(2); yfit=m*x+b;plot(x,yfit,x,y,'o'),title('Linear Regression'),grid EDU>> print -deps2 regres.eps EDU>>

where the le regres.eps is shown in Fig. 10.5. Linear Regression 40

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Figure 10.5: Regression Analysis

10.1.3.3 Signal Processing 11

MATLAB has a library of very powerful signal processing functions, Table 10.5.

10.1.4 Control Flow 12

MATLAB programming is very simple and supports the DO, WHILE, and IF constructs.

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WEEK II; MATLAB: Matrix Algebra

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abs angle conv corrcoef cov deconv t t2 it it2 tshift

complex magnitude phase angle convolution correlation coecients covariance deconvolution radix-2 fast Fourier transform two-dimensional FFT inverse fast Fourier transform inverse 2-D FFT FFT rearrangement

Table 10.5: Signal Processing Functions MATLAB provides ow control statements (such as DO, WHILE and IF) which operate like those in most computer languages. 14 Relations and logical operators in MATLAB are given in Table 10.6.

13

< > <= >=

less than greater than less than or equal greater than or equal == equal = not equal. & and j or  not. Table 10.6: Relations and Logical Operators 15

Note that \=" is used in an assignment statement while \==" is used in a relation.

EDU>> x=4*rand(1,10); EDU>> for k=1:10 if x(k)>=3 disp('A') elseif x(k)<3 & x(k)>=2 disp('B') else disp(' You failed') end end

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10.1 Syntax

10{13

You failed B A You failed You failed A B A B

another example EDU>> max=10;x(1)=1;x(2)=1;k=3; EDU>> while k<max x(k)=x(k-1)+x(k-2); k=k+1; end EDU>> x x = 1 1 2 3 5

16

8

13

21

34

MATLAB has some prede ned relational and logical functions, Table 10.7.

any all nd isnan nite isempty isstr strcmp

logical conditions logical conditions nd array indices of logical values detect NaNs detect in nities detect empty matrices detect string variables compare string variables

Table 10.7: Relational and Logical Functions 17

Control ow is described through Table 10.8

10.1.5 Function Files Function les take one or more external arguments (enclosed in parenthesis) and return one or more output.

18

function hyp=pyt(a,b) % All commented lines following the function definition

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WEEK II; MATLAB: Matrix Algebra

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if elseif else end for while break return pause

conditionally execute statements used with if used with if terminate if, for, while repeat statements a number of times do while break out of for and while loops return from functions pause until key pressed Table 10.8: Control Flow

% will be displayed if the user types help pyt if nargin < 2 disp('invalid number of arguments'; end hyp=sqrt(a.^2+b.^2);

This le should be placed in a disk le with lename pyt.m (corresponding to the function name). The rst line declares the function name, input arguments, and output arguments; without this line the le would be a script le. Then a MATLAB statement c=hyp(4,5), for example, will cause the numbers 4 and 5 to be passed to the variables a and b in the function le with the output result being passed out to the variable c. Since variables in a function le are local, their names are independent of those in the current MATLAB environment. 20 Note that use of nargin (\number of input arguments") permits one to set a default value of an omitted input variable. 21 A function may also have multiple output arguments. For example:

19

function [mean, stdev] = stat(x) % STAT Mean and standard deviation [m n] = size(x); if m == 1 m = n; end mean = sum(x)/m; stdev = sqrt(sum(x.^2)/m - mean.^2);

Once this is placed in a disk le stat.m, a MATLAB command [xm, xd] = stat(x), for example, will assign the mean and standard deviation of the entries in the vector x to xm and xd, respectively. Single assignments can also be made with a function having multiple output arguments. For example, xm = stat(x) (no brackets needed around xm) will assign the mean of x to xm. Victor E. Saouma

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10.2 Assignment

10{15

10.2 Assignment 10.2.1 Practice

Start by repeating all the examples in this handout.

10.2.2 Polar Plot

Plot the following curve (Folium of Descartes) 3a sin  cos  r = sin 3  + cos3  for ;=6    =2, use a = 1.

(10.1)

10.2.3 Animation

The transverse vibration of an undamped simply supported beam straight beam is given by q

v = C2 sin nx L

(10.2)

EI4 Write a function which will accept as argument the mode at thee frequency !n = n2 2 mL shape (n which is an integer), the number of frames N and will generate an animation of the beam vibration. Let C2 vary from ;L=5 to L=5, and take L = 10.

10.2.4 Strain Rosette 10.2.4.1 Theory

Experimentally, strains are measured by a strain gage. The most common type of strain gage is the bonded resistance strain gage shown in Fig. 10.6. These gages use a grid of ne wire or a

Figure 10.6: Bonded Strain Gage metal foil grid encapsulated in a thin resin backing. The gage is glued to the carefully prepared test specimen by a thin layer of epoxy. The epoxy acts as the carrier matrix to transfer the Victor E. Saouma

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10{16

WEEK II; MATLAB: Matrix Algebra

strain in the specimen to the strain gage. As the gage changes in length, the tiny wires either contract or elongate depending upon a tensile or compressive state of stress in the specimen. The cross sectional area will increase for compression and decrease in tension. Because the wire has an electrical resistance that is proportional to the inverse of the cross sectional area, R  A1 , a measure of the change in resistance can be converted to arrive at the strain in the material. Bonded resistance strain gages are produced in a variety of sizes, patterns, and resistance. One type of gage that allows for the complete state of strain at a point in a plane to be determined is a strain gage rosette. It contains three gages aligned radially from a common point at dierent angles from each other, Fig. 10.7. The strain transformation equations to

Figure 10.7: Strain Rosette convert from the three strains at any angle to the strain at a point in a plane are

"a = "xx cos2 a + "yy sin2 a + xy sin a cos a "b = "xx cos2 b + "yy sin2 b + xy sin b cos b "c = "xx cos2 c + "yy sin2 c + xy sin c cos c

(10.3-a) (10.3-b) (10.3-c)

The angles are usually given by

b a b c c = b 0o 60o 120o c

(10.4)

When the measured strains a , b , and c , are measured at their corresponding angles from the reference axis and substituted into the above equations the state of strain at a point may be solved, namely, xx, yy , and xy . The stresses are in turn related to the strains through the elastic constants E and  , Young's modulus and Poisson's ratio (lt0:5)respectively.

xx = (1;2E)(1+ ) [(1 ;  )"xx +  ("yy + "zz )] yy = (1;2E)(1+ ) [(1 ;  )"yy +  ("zz + "xx )] E xy xy = 2(1+ ) Victor E. Saouma

(10.5)

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10.2 Assignment

10{17

Those would de ne the stress tensor 2 6 4

xx xy xz xy yy yz xz yz zz

3

(10.6)

7 5

It can be shown that the solution for principal stresses, i.e. stresses acting on principal planes where there is no shear stress, yields

l(xx ; ) + mxy + nxz = 0 lxy + m(yy ; ) + nyz = 0 lxz + myz + n(zz ; ) = 0

(10.7-a) (10.7-b) (10.7-c)

where l, m and n are direction cosines. Since those are linear and homogeneous equations in l, m and n a solution will exist only if

or



xx ;  xy xz xy yy ;  yz = 0 xz yz zz ; 

(10.8)

3 ; I1 2 ; I2  ; I 3 = 0

(10.9)

where I1 , I2 and I3 are three invariants equal to

I1 = xx + yy + zz I2 = xy2 + xz2 + yz2 ; xxyy ; xxzz ; yy zz  xx xy xz I3 = xy yy yz xz yz zz

(10.10-a) (10.10-b) (10.10-c)

The three roots (1 ; 2 ; 3 ) are the three principal stresses. The direction cosines of the three principal axes are obtained from Eq. 10.7-a-?? by setting  in turn equal to 1; 2 ; 3 and recalling that l2 + m2 + n2 = 1.

10.2.4.2 Assignment Write a function which accepts as input arguments: 1. "a "b and "c from a 60 degrees strain rosette. 2. Young's modulus and Poisson's ratio E , and  . Determines 1. Cartesian strains (Eq. 10.3-a-10.3-c) Victor E. Saouma

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10{18

WEEK II; MATLAB: Matrix Algebra

2. Stresses (Eq. 10.5) 3. Stress invariants (Eq. 10.10-a-10.10-c) 4. Principal stresses (through the eigenvalues in Eq. 10.8) 5. Direction cosines (Eq. ??; Tricky!). Test your program for the values shown in Table 10.9. Note that  equal one microstrain

E  "a a "b b "c c

Mild Steel Aluminum High Strength Concrete 30,000 ksi 70 GPa 31 GPa 0.30 0.33 0.20 600  2,000  -400  0 deg 0 deg 15 deg 500  1,500  -800  45 deg 120 deg 75 deg -200  -1,300  -1,200 90 deg 60 deg 135 deg Table 10.9: Strain Rosette Problems

(10;6 in/in)

10.2.5 Structural Design 10.2.5.1 Theory

The design of a steel truss entail the following steps: 1. Structural analysis in which the equations of equilibrium are applied at each node and written in terems of the unknown internal element forces and known externally applied load. This will result in a system of linear equations of the form [B] fxg = fPg

(10.11)

where fPg is the vector of externally applied load, and fxg the vector of unknown element forces (tension if positive, compression if negative). Note that the unknown vector contains both element forces ass well as reactions. 2. Analysis is performed for dierent loads, and then multiple load cases are coonsidered. Each load case is typically a linear combination of the basic load investigated. 3. Results (internal forces) are tabulated for all load cases, maximum forces determined, and each element is separately designed. Victor E. Saouma

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10{19

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10.2 Assignment

Figure 10.8: Steel Truss

Victor E. Saouma

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10{20

WEEK II; MATLAB: Matrix Algebra

10.2.5.2 Assignment

The truss shown in Fig. 16.1 is to be designed. The statics [B ] matrix can be assembled1 The resulting statics matrix which can be copied from ftp/pub/Structures/cven4837/truss.m is given by b=[... 1., 0. , 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 1. , 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 1. , 0., 0.; ... 0., 0. , .82, 1. , 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., -1.,-.57, .08, 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... -1., 0.,-.82, 0., 0. , 1. , 0., 0., 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., .57, 0., 1. , 0. , 0., 0., 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., -1., 0. , 0.,-.77, 1. , 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0.,-.08, -1., 0., .64, .08, 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., -1., .77, 0., 0. , 1. , 0., 0., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0.,-.64, 0., 1. , 0. , 0., 0., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0.,-1. , 0. , 0.,.73 , 1., ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0.,-.08, -1., 0.,-.68, .08, ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., -1.,-.73, 0., ... 0., 1. , 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0. ,.68 , 0., ... 1., 0. , 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -1., ... 0., 0., .71, 1., 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,-.08, ... -1., 0.,-.71,.08 , 0., 0., 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ... 0., -1.,-.71, 0., 0. , 1. , 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0. , .71, 0., 1. , 0. , 0., 0., 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0., 0., -1., 0. , 0.,.68 , 1.0, 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0., 0.,-.08, -1., 0.,-.74, .08, 0., 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0., 0., 0., 0., -1.,-.68, 0., 0. , 0., 1. , 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0. , .74, 0., 1. , 0., 0., 0.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0., 0.,-1. , 0. , 0., 0., 1.; ... 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ... 0., 0., 0., 0., 0., 0., 0.,-.08, -1., 0., 0., 0.]; r=-[0., 0., 0., -0.9, 0., 0., 0., -1.74, 0., 0., 0., -1.68,... 0., 0., 0., -1.68, 0., 0., 0., -1.68, 0., 0., 0., -0.84];

1 Note

that in assembling the matrix, each column corresponds to an unknown internal element force, or an external reaction. Each row corresponds to an equation of equilibrium.

Victor E. Saouma

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10.2 Assignment s=-[0., 0., 0., l=-[0., 0., 0.,

0., 0., -2.57, 0., -4.8 , 0., 0., 0., 0., 0., 0., 0.,

10{21 0., 0., 0., 0.,

0., 0., -4.97, 0., -4.8 , 0., 0., 0., 0., 0., 0., 0.,

0., 0., 0., -4.8, ... 0., 0., -2.4]; 0., -6.4, 0., ... 0.,-6.15, 0., 0.];

The three load vectors (corresponding to roof, snow, and live loads) are: r=-[0., 0., 0., -0.9, 0., 0., 0., -1.74, 0., 0., 0., -1.68, 0., 0., 0., -1.68, 0., 0., 0., -1.68, 0., 0., 0., -0.84] s=-[0., 0., 0., -2.57, 0., 0., 0., -4.97, 0., 0., 0., -4.8, 0., 0., 0., -4.8 , 0., 0., 0., -4.8 , 0., 0., 0., -2.4] l=-[0., 0., 0., 0., 0., 0., 0., 0., 0., -6.4, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,-6.15, 0., 0.]

Write a MATLAB program which will 1. Read the statics matrix and the load vectors 2. Invert the statics matrix [B] 3. Solve for the roof, snow, and live load forces 4. Obtain the forces for each of the 3 following three load combinations Load Case Roof Live Snow 1 1.4 0 0 2 1.2 1.6 0.5 3 1.2 0.5 1.6 5. Determine the maximum load for each member 6. Select the cross sectional area of each member using allowable = 20 kips/in2 in compression and nal area of each member using allowable = 24 kips/in2 in tension. 7. Tabulate the results as shown in Table 16.1 plus an additional column showing the crosssectional area in in2. Note that Note that the actual results are given by Table 16.1.

10.2.6 Dynamic Response of a Linear Oscillator 10.2.6.1 Theory

Newton's Second Law states that

dmv = f dt

(10.12)

where m is the mass, v the velocity, t time, and f the force. for a constant mass (unlike a rocket) we have f = ma (10.13) Victor E. Saouma

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10{22

WEEK II; MATLAB: Matrix Algebra Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

L0 ; L1 U0 ; L0 U0 ; L1 U0 ; U1 U1 ; L1 L1 ; L2 U1 ; L2 U1 ; U2 U2 ; L2 L2 ; L3 U2 ; L3 U2 ; U3 U3 ; L3 L3 ; L4 U3 ; L4 U3 ; U4 U4 ; L4 L4 ; L5 U4 ; L5 U4 ; U5 U5 ; L5

Load Load Cases Design Load Roof Snow Live 1 2 3 Compr. Tens. 0. 0. 0. 0. 0. 0. 0 0 -8.5 -24. -13. -12. -35. -55. -55. 0 12. 34. 20. 17. 52. 79. 0 79. -9.8 -28. -16. -14. -43. -65. -65. 0 -6.8 -20. -11. -9.6 -30. -45. -45. 0 9.8 28. 16. 14. 43. 65. 0. 65. 7.3 21. 16. 10. 38. 50. 0. 50. -15. -44. -29. -22. -72. -100. -100. 0 -4.6 -13. -3.9 -6.5 -14. -29. -29. 0 15. 44. 29. 22. 72. 100. 0 100. 4. 11. 5.2 5.6 15. 26. 0 26. -18. -52. -32. -26. -83. -120. -120. 0 -2.7 -7.8 -3.6 -3.8 -10. -18. -18. 0 18. 52. 32. 26. 83. 120. 0 120. 1.4 3.9 4.6 1.9 9.8 10. 0 10. -19. -55. -36. -27. -90. -130. -130. 0 -0.97 -2.8 -3.3 -1.4 -6.9 -7.3 -7.3 0 19. 55. 36. 27. 90. 130. 0 130. -0.89 -2.5 4.2 -1.2 5.1 -3.1 -3.1 5.1 -19. -53. -38. -26. -93. -130. -130. 0 0.66 1.9 3.1 0.92 6. 5.3 0 6. Table 10.10: Result of Truss Design

Figure 10.9: Single Degree of Freedom Oscillator Victor E. Saouma

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10.2 Assignment

10{23

If we consider the linear oscillator shown in Fig. 10.9 setting up the sum of the horizontal forces we have mu + cu_ + ku = P sin !t (10.14) where u and u_ are the acceleration and velocity respectively, k the stiness of the spring, P sin !t the driving force. The solution to this problem is of the form

u = et

(10.15)

(m2 + c + k)et = P sin !t

(10.16)

and substituting into Eq. 10.14 we obtain Since the exponential is never equal to zero we have

p2 ; c  c ; 4mk =

or If we de ne

2m

!1=2  1=2  1=2 2 c k k c  = ; 2(km)1=2 m  m 4km ; 1

(10.18)

s

k m Undamped natural frequency c Fraction of critical damping  = pc = 2m! 2 km n

!n = then Eq. 10.14 becomes

A solution of this equation is where

(10.17)

(10.19-a) (10.19-b)

P sin !t u + 2!n u_ + !2 u = m

(10.20)

u = C1 sin !t ; C2 cos !t

(10.21)

1 ; (!=!n )2 C1 = Pk [1 ; (!=!n )2 ]2 + [2!=!n ]2 2!=!n C2 = Pk [1 ; (!=!n )2 ]2 + [2!=!n ]2  = tan;1 CC2 1

(10.22-a) (10.22-b) (10.22-c)

 is the phase lag between the applied force, and the system response. Victor E. Saouma

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10{24

WEEK II; MATLAB: Matrix Algebra

We can also express the displacement, velocities and accelerations as

where

u = Pk Rd sin(!t ; ) u_ = pP Rv cos(!t ; ) km P u = ; m Ra sin(!t ; )

1 [1 ; (!=!n )2 ]2 + [2!=!n ]2

Rd =

q

Rv =

q

!=!n

[1 ; (!=!n )2 ]2 + [2!=!n ]2 (!=!n)2 Ra = q [1 ; (!=!n )2 ]2 + [2!=!n ]2

(10.23-a) (10.23-b) (10.23-c) (10.24-a) (10.24-b) (10.24-c) (10.24-d)

and correspond to the displacement, velocity and acceleration factors respectively.

10.2.6.2 Assignment 1. Generate surface plots showing (a) Rd in terms of 0  !=!n  3 and 0    1:2 (b) Rv in terms of 0  !=!n  3 and 0    5 (c) Ra in terms of 0  !=!n  3 and 0    5 (d)  in terms of 0  !=!n  3 and 0    5 000 = 5:176 lb/in/sec2), is supported by a 60" cantilever with a 2. A 2,000 lbm (or (322:;2)(12) q p stiness k = 1; 065lb=in; The natural frequency is !n = mk = 1; 0655:176 = 14:34 rad/sec. An external force with an amplitude of 250 lb which oscillates at 3 cycles per second is applied. The excitation frequency is ! = 2f = 2(3:1416)(3) = 18:85 rad/sec. The system is damped to 2 percent of critical damping. Generate 4 plots (on the same display) showing the excitation force, displacement, velocity and accelerations for 0  t  5 sec.

10.2.7 Nonlinear Equation 10.2.7.1 Theory

Considering a beam-column subjected to axial and shear forces as well as a moment, Fig. 10.10, dv between the axis taking the moment about i for the beam segment and assuming the angle dx Victor E. Saouma

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10.2 Assignment

10{25 w(x)

P

P x dx y,u

w

M

V+

i

P

V

δV δx dx

δv δx

P

dx

M+ δM δx dx

w

P

θi

P

i j

θj

P

dx

Figure 10.10: Simply Supported Beam Column; Dierential Segment; Eect of Axial Force P of the beam and the horizontal axis is small, leads to   (dx)2 + V + dV  dx ; P  dv  dx = 0 M ; M + dM dx + w (10.25) dx 2 dx dx neglecting the terms in dx2 which are small, and then dierentiating each term with respect to x, we obtain d2 M ; dV ; P d2 v = 0 (10.26) dx2 dx dx2 However, considering equilibrium in the y direction gives dV = ;w (10.27) dx

From beam theory, neglecting axial and shear deformations, we have

d2 v M = ;EI dx 2

(10.28)

Substituting Eq. 10.27 and 10.28 into 10.26, and assuming a beam of uniform cross section, we obtain 4 2 (10.29) EI d v4 ; P d v2 = w

dx

Victor E. Saouma

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10{26

WEEK II; MATLAB: Matrix Algebra

P , the general solution of this fourth order dierential equation to any set of Introdcing k2 = EI boundary conditions is v = C1 sin kx + C2 cos kx + C3 x + C4 (10.30) If we consider a column with one end xed (at x = 0), and one end hinged (at x = L). The boundary conditions are

v = 0; v;xx = 0 atx = 0 v = 0; v;x = 0 atx = L These boundary conditions will yield C2 = C4 = 0, and sin kL ; kL cos kL = 0

(10.31) (10.32)

But since cos kL can not possibly be equal to zero, the preceding equation can be reduced to tan kL = kL

(10.33)

which is a transcendental algebraic equation and can only be solved numerically.

10.2.7.2 Assignment Write a MATLAB function to solve for the rst 5 roots of Eq. 10.33 by zooming into the intersection of the left hand side with the right hand side.

10.2.8 Newton Raphson Method 10.2.8.1 Theory

Given an equation f (x) = 0 with f (x) expandable in a Taylor's series about an initial guess value x0 , we can write f (x) = f (x0 ) + f 0 (x)(x ; x0) +


= 0 (10.34) neglecting high order terms. From this equation we can obtain x = x0 ; ff0((xx0 )) (10.35) 0 or (10.36) xi+1 = xi ; ff0((xxi )) i

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10.2 Assignment

10{27

10.2.8.2 Assignment Write a MATLAB program which will 1. Ask the user (through the input function) (a) Equation f (x) to be solved (b) First derivative of the equation f 0 (x) (c) Range of x for plotting (xmin and xmax). (d) Initial guess for the solution (e) Tolerance 2. Plot the equation for the speci ed range 3. Solve the equation within the speci ed tolerance 4. Display the root of the equation and the number of iterations. Test your program by solving Eq. 10.33 and compare the number of ops ( oating point operations) with the preceding solution.

Victor E. Saouma

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WEEK II; MATLAB: Matrix Algebra

Draft

10{28

Victor E. Saouma

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Chapter 11

WEEK III; MATLAB: \Advanced" Topics 11.1 Background

11.1.1 Numerical Integration and Dierentiation R

The integration of ab F (x)dx is essentially based on passing a polynomial P (x) through given Rb values of F (x) and then use a P (x)dx as an approximation.

1

Z

b a

F (x)dx 

Z

a

b

(11.1)

P (x)dx

2 Using P (x) = F (x) at n points, and recalling the properties of Lagrangian interpolation functions, we obtain

P (x) = l1 (x)F (x1 ) + l2 (x)F (x2 ) +


+ ln(x)F (xn ) n X = li (x)F (xi ) i=1

(11.2-a) (11.2-b)

11.1.1.1 Newton-Cotes Method In Newton-Cotes integration, it is assumed that the sampling points are equally spaced, Fig. 11.6, thus we de ne

3

Z

a

b

P (x)dx =

Z

n bX a i

li (x)dxF (xi ) =

n Z X i

b a

li(x)dxF (xi )

(11.3)

WEEK III; MATLAB: \Advanced" Topics

Draft

11{2

P(ξ) F(ξ)

b -1

0

ξ

1

a ξ

Figure 11.1: Newton-Cotes Numerical integration or

Approximation ab P (x)dx = R ni=1 Wi(n) F (xi ) (11.4) Wi(n) = ab li (x)dx = (b ; a)Ci(n) Weights where Ci(n) are the \weights" of the Newton-Cotes quadrature for numerical integration with n equally spaced sampling points. 4 newton-Cotes constants, and corresponding reminder are shown in Table 11.1, (Bathe 1982). R

P

n C0(n) C1(n) C2(n) C3(n) C4(n) 1 1 2 3 4 5

21 61 87 90

24 63 328 90

1 63 128 90

1 328 90

7 90

Error 10;1 (b ; a)3 F II (x) 10;3 (b ; a)5 F IV (x) 10;3 (b ; a)5 F IV (x) 10;6 (b ; a)7 F V I (x)

Table 11.1: Weights for Newton-Cotes Quadrature Formulas It can be shown that this method permits exact integration of polynomial of order n ; 1, and that if n is odd, then we can exactly integrate polynomials of order n. Hence we use in general odd values of n, 6 For n = 2 over [;1; 1], we select equally spaced points at x1 = ;1 and x2 = 1 to evaluate 5

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11.1 Background

11{3

R1

;1 P (x)dx

P (x) l1 (x) l2 (x) W1(2) (2) W 2 R1 ;1 F (x)dx

= = = = =



P2

i=1 li (x)F (xi ) x;x2 = 1 (1 ; x) xx1;;xx2 2 1 = 1 (1 + x) 2 Rx2 ;x1 R

1 l1 (x)dx = 1 1 (1 ; x)dx = 1 2 R;1 1 1 (1 + x)dx = 1 l ( x ) dx = 2 1 ;1 R; P 1 P (x)dx = 2 2 W (2) F (xi ) = F (;1) + F (1) i=1 i ;1 1 R; 1

which is the trapezoidal rule 7 For n = 3 over [;1; 1], we select equally spaced points at x1 = ;1 x2 = 0, and x3 = 1, to R evaluate ;1 1 P (x)dx P (x) = P3i=1 li (x)F (xi ) l1 (x) = (x(x1 ;;xx22)()(xx1;;xx3 3) ) = 21 x(x ; 1) l2 (x) = (x(x2 ;;xx11)()(xx;2 ;xx3 )3 ) = ;(1 + x)(x ; 1) l3 (x) = (x(x3 ;;xx11)()(xx;3 ;xx2 )2 ) = 21 x(1 + x) R R W1(3) = R;11 l1 (x)dx = R12 ;1 1 x(x ; 1)dx = 13 W2(3) = R;1 1 l2 (x)dx = ;1R1 ;(1 + x)(x ; 1)dx = 34 (3) = 1 l (x)dx = 1 1 x(1 + x)dx = 1 W 3 13 2 ;1 (3) 3 R1 R; P 1 3 W F (xi ) = 1 [F (;1) + 4F (0) + F (1)] F ( x ) dx  P ( x ) dx = i =1 i ;1 ;1 3 which is Simpson's rule

11.1.1.2 MATLAB Examples MATLAB's functions for numerical (de nite) integration and dierentiation are shown in Table 11.2. Numerical Integration trapz trapezoidal numerical integration quad numerical function integration; Simpson's rule quad8 Newton-Cotes 8 panel rule di approximate derivatives

8

Table 11.2: Numerical Integration and Dierentiation 9

For example, let us recall the expression for the center of gravity Z

ydA y= A Victor E. Saouma

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11{4 and moment of inertia

WEEK III; MATLAB: \Advanced" Topics Z

Z

y2 dA

Iyy = x2 dA A Iyy = Iyycg + Ad2x

Ixx = A cg + Ad2 Ixx = Ixx y

(11.6-a) (11.6-b)

and moment of inertia for a given cross section for a rectangular cross (width b, height h) section we would have Z

ydA A Z h b ydy 0 = bh = h

y =

(11.7-a) (11.7-b)

(11.7-c) 2 Determining the moment of inertia Ixx with respect to the neutral axis (which passes through the centroid)

Ixx =

Z

A

y2 dA

Z

= 2b

h=2 2 y dy

0

(11.8-a) (11.8-b)

3 = bh (11.8-c) 12 We now seek to determine the centroid of the triangular cross section shown in Fig. 11.2 where A(;2; 0), B (4; 0) and C (0; 6) First we need to de ne the following functions (note that each one of them must be in a separate le). function y=line1(x) xx=[-2 0];yy=[0,6];coef=polyfit(xx,yy,1); y=coef(1).*x+coef(2); function y=line1x(x) xx=[-2 0];yy=[0 6];coef=polyfit(xx,yy,1); y=coef(1).*x.*x+coef(2).*x; function x=line1y(y) xx=[-2 0];yy=[0,6];coef=polyfit(yy,xx,1); x=-(coef(1).*y.*y+coef(2).*y); function y=line2(x)

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11.1 Background

11{5

6

y

C

C.G. u

A

x

B

Figure 11.2: Center of Gravity and Moments of Inertia of A Triangular Cross-Section xx=[0 4];yy=[6 0];coef=polyfit(xx,yy,1); y=coef(1).*x+coef(2); function y=line2x(x) xx=[0 4];yy=[6 0];coef=polyfit(xx,yy,1); y=coef(1).*x.*x+coef(2).*x; function x=line2y(y) xx=[0 4];yy=[6 0];coef=polyfit(yy,xx,1); x=-(coef(1).*y.*y+coef(2).*y);

and now the following function will determine the areas and coordinates of the centroid a1=quad('line1',-2,0) cgx1=quad('line1x',-2,0)/a1 cgy1=quad('line1y',0,6)/a1 a2=quad('line2',0,4) cgx2=quad('line2x',0,4)/a2 cgy2=-(quad('line2y',0,6)/a2) a=a1+a2 cgx=(quad('line1x',-2,0)+quad('line2x',0,4))/a

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cgy=(quad('line1y',0,6)-quad('line2y',0,6))/a

11.1.2 Nonlinear Equations and Optimization Nonlinear Equations and Optimization

fmin minimum of a function of one variable fmins minimum of a multivariable function fsolve solution of a system of nonlinear equations fzero

(zeros of a multivariable function) zero of a function of one variable

Table 11.3: Nonlinear Equations and Optimization

11.1.3 Ordinary Dierential Equations

The solution of ordinary dierential equations (ODE) can always be reduced to the study of sets of rst order dierential equations. For example

d2y + q(x) dy = r(x) dx2 dx

(11.9)

can be rewritten as two sets of rst order equations

dy dx = z(x) dz = r(x) ; q(x)z (x) dx

(11.10) (11.11)

where z is a new variable. Hence, all problems in ordinary dierential equations are reduced to the study of a set of n coupled rst order dierential equations for the function yi, i = 1; 2;


; n with the general form dyi(x) = f (x; y ;


; y ) i = 1;


; n (11.12) 1 n dx

where the functions fi are known derivatives of the y0 s and the x0 s are the independent variables. This is known as the Cauchy form and can be rewritten as

y10 (x) = f1(x; y) yn0 (x) = fn(x; y)

(11.13) (11.14)

where n corresponds to the order of the ODE. In order to solve the dierential equation, we also need to have its initial (or boundary) values. The simplest method to solve for yn+1 is the Euler method through

yn+1 = yn + hf (xn ; yn ) + O(h2 )

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11.1 Background

11{7

where xn+1  xn + h, and in which a linear approximation is made by using the rst two terms of the Taylor's series expansion. We observe that the formula is unsymmetrical (it advances the solution by h but uses the derivative only at the beginning of that interval). One can improve on Euler's method by Midpoint method where the derivative is taken at midpoint rather than at the beginning of the step, Fig. 11.3.

g1 = f (xn ; yn) yn+ 21 = yn + h2 g1  g = f x + h; y 2

n

yn+1 = yn + hg2

:

2 n+ 21

(11.18) (11.19)

g2

slope = g1

yn



(11.16) (11.17)

yn+1

 ; ; ;slope = g

2

 tn

h=2

-

h=2

tn+1

Figure 11.3: Runge's Midpoint Method

Trapezoid method where we consider the average values of the derivatives, Fig.11.4 g1 = f (xn; yn) g2 = f (xn + h; yn + hg1 ) yn+1 = yn + h2 (g1 + g2 ) + O(h3 )

(11.20) (11.21) (11.22)

hence, through the symmetrization the method is now second order1 . Actually, we have just derived the equations for the second-order Runge-Kutta method. For example if 1 A method

is conventionally called nth order if its error is O(hn+1 ).

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11{8

WEEK III; MATLAB: \Advanced" Topics slope = g1

yn

XXXz g2 yn+1

g 1  : slope = g1 +g2 2

 tn

h

tn+1

Figure 11.4: Runge's Trapezoid Method

dy = x2 + y(x)2 , then xn = 2, yn = 1, h = 0:1 and the dierential equation is dx g1 = x2n + yn2 = 5 g2 = (tn + h)2 + (yn + hg1 )2 = 2:12 + 1:52 = 6:66 yn+1 = yn + h g1 +2 g2 = 1 + 0:1(5:83) = 1:583

(11.23) (11.24) (11.25)

Fourth-Order Runge-Kutta method is given by, Fig. 11.5. g1 = f (xn; yn) g2 = f (xn + h2 ; yn + h2 g1 ) g = f (x + h ; y + h g )

3 n 2 n 2 2 g4 = f (xn +h; yn + g3 )  yn+1 = yn + h g61 + g32 + g33 + g64 + O(h5 )

and it will require four evaluations of the right hand side per step h Fourth-Order Runge-Kutta method is given by g1 = f (xn; yn) g2 = f (xn + h2 ; yn + h2 g1 )

g3 = f (xn + h2 ; yn + h2 g2 ) g4 = f (xn +h; yn + hg3 )  yn+1 = yn + h g61 + g32 + g33 + g64 + O(h5 )

Victor E. Saouma

(11.26) (11.27) (11.28) (11.29) (11.30)

(11.31) (11.32) (11.33) (11.34) (11.35)

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11.1 Background

11{9

g : 2

slope = g1

yn

; ; ;

XXX z



-

tn

g g g ynslope +1 = 61 + 32 + 64

g3

h=2

PPPq g 4 h=2

tn+1

Figure 11.5: 4th Order Runge-Kutta Method

and it will require four evaluations of the right hand side per step h. Note, in many applications x corresponds to the time t, and thus Equation 11.12 can be rewritten as: dxi (t) = f (t; x ;


; x ) i = 1;


; n (11.36)

dt

1

n

dy = 1 + y(x)2 with y(0) = 0, As an illustrative example, let us consider the following ODE dx we seek y(=4), and stepsize h = =4 ' 0:78540 y0 = 0 g1 = 1 h y0 + 2 g1 = 0:39270 g2 = 1:1542 (11.37) y0 + h2 g2 = 0:45326 g3 = 1:2054 h y0 + 2 g3 = 0:94676 g2 = 1:8963 y1 = 0:99687 10

11

In this other example, let us consider the following second order ordinary dierential equation

d2 y = y dy (0) = 0 y (0) = 1 2 t dt e + 1 dt

(11.38)

we use
t = 0:1and solve for the rst step only. But rst we transform it into two coupled equations

z = dy dt dz = y dt et + 1 z (0) = 1 dy = z y(0) = 1 dt

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(11.39-a) (11.39-b) (11.39-c)

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WEEK III; MATLAB: \Advanced" Topics

For the rst step y0 = 1 and z0 = 0 and we seek y1 and z1 . We treat the dependent variables and their derivatives as vectors and each component of each vector must be calculated before proceding. Hence, g1 (y; z; t) = ety+1 ; g2 (y; z; t) = z; (11.40) are components of (g1 and g2 ) the derivative vector ~g . Now we can start by evaluating ~g at (y0 ; z0 ; t0 ). g11 (y0 ; z0 ; t0 ) = e0y+1 = 0:5 (11.41) g 2 (y ; z ; t ) = 0 1 0 0 0

Solve for the locations of the intermediary point z1=2 = z0 +
2t g11 (y0 ; z0 ; t0 ) = 0 + 02:1 (0:5) = 0:025 (11.42) y1=2 = y0 +
2t g12 (y0; z0 ; t0 ) = 1 + 02:1 (0) = 1 Derivatives g21 (y1=2 ; z1=2 ; t1=2 ) = e0:051 +1 = 0:487503 (11.43) g22 (y1=2 ; z1=2 ; t1=2 ) = z1=2 = 0:025 compute again the dependent variables z1=2 = z0 +
2t g21 (y1=2 ; z1=2 ; t1=2 ) = 0 + 02:1 (0:487503) = 0:024375 (11.44) y1=2 = y0 +
2t g22 (y1=2 ; z1=2 ; t1=2 ) = 1 + 02:1 ((0:025) = 1:001250 Derivatives g31 (y1=2 ; z1=2 ; t1=2 ) = 1e:0001250 :05 +1 = 0:488112 (11.45) 2 g3 (y1=2 ; z1=2 ; t1=2 ) = z1=2 = 0:024375 Dependent variables z1 = z0 +
tg31(y1=2 ; z1=2 ; t1=2 ) = 0 + (0:1)(0:488112) = 0:048811 (11.46) y1 = y0 +
tg32 (y1=2 ; z1=2 ; t1=2 ) = 1 + (0:1)(0:024375) = 1:002438 Derivatives g41 (y1 ; z1 ; t1 ) = 1e:002438 0:1 +1 = 0:476179 (11.47) g42 (y1 ; z1 ; t1 ) = z1 = 0:048811 First step is now completed, solve for z1 and y1   z1 = z0 +
t 16 g11 (y0; z0 ; t0 ) + 31 g21 (y1=2 ; z1=2 ; t1=2 ) + 13 g31 (y1=2 ; z1=2 ; t1=2 ) + 16 g41 (y1;(11.48-a) z1 ; t1 )   = 0 + 0:1 61 (0:5) + 13 (0:487503) + 31 (0:488112) + 16 (0:476179) (11.48-b) = 0:048790 (11.48-c)   1 1 1 1 2 2 2 2 y1 = y0 +
t 6 g1 (y0 ; z0 ; t0 ) + 3 g2 (y1=2 ; z1=2 ; t1=2 ) + 3 g3 (y1=2 ; z1=2 ; t1=2 ) + 6 g4 (y1 ;(11.48-d) z1 ; t1 )   (11.48-e) = 0 + 0:1 61 (0) + 13 (0:0:025) + 13 (0:0:024375) + 61 (0:048811) (11.48-f) = 1:002459

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11.1 Background

11{11

There are two MATLAB functions for the solution of a system of ordinary dierential equations, Table 11.4. 12

Dierential Equations ode23 2nd/3rd order Runge-Kutta method ode45 4th/5th order Runge-Kutta-Fehlberg method Table 11.4: Dierential Equations As An illustrative example, let us reconsider the mass-spring-damper system previously analyzed in Sect. 10.2.6.1, the governing dierential equation was given by Eq. 10.14 mx + cx_ + kx = P sin !t (11.49) we now seek to rearrange this equation in Cauchy form suitable for the Runge-Kutta MATLAB solution. 9 x = ; mc x_ ; mk x + P sin !t > = y10 = ; mc y1 ; mk y2 + P sin !t (11.50) y1 = x_ 0 > ; y2 = y1 y = x

13

2

the initial boundary conditions are x = 0 and x_ = 0 or y1 (0) = 0 (11.51) y2 (0) = 0 (11.52) To analyze this problem with MATLAB we rst need to de ne an M- le which we call msd.m function yp=msd(t,y) % This function defines the differential equations, written in Cauchy % form, governing the response of a damped mass-spring system subjected % to a harmonic excitation. % Assign constants p= 2250.; % lbf m= 2000.; % lbm c=250; % k= 1065; % lb/in omega= 120.; % radiands % Initialize the yd matrix [rows, cols]=size(y); yp=zeros(rows,cols); % define the derivatives yp(1)=-c/m*y(1)-k/m*y(2)+p/m*sin(omega*t); yp(2)=y(1);

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Now, to see how the system oscillates, we de ne the initial boundary values t0=0.;y0=[0.;0.]; tf=100;

and then the simulation is run through t0=0.;y0=[0.;0.]; tf=30; [t,y]=ode23('msd',t0,tf,y0); plot(t,y(:,2)),grid % Note that x corresponds to y(2) xlabel('Time, t [sec]') ylabel('Displacement, x [mm]') pause plot(y(:,1),y(:,2)) xlabel('Velocity') ylabel('Displacement')

Which will generate Figs. ?? 0.015

0.015

0.01

0.01

0.005

0.005

Displacement

Displacemnt, x [in]

Draft

11{12

0

−0.005

−0.01 0

0

−0.005

5

10

15 Time, t [sec]

20

25

30

−0.01 −0.02

−0.015

−0.01

−0.005

0 Velocity

0.005

0.01

0.015

0.02

Figure 11.6: Runge-Kutta Solution for the Mass-Spring-Damper System

11.2 Assignment 11.2.1 Practice

Start by repeating all the examples in this handout.

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11.2 Assignment

11{13

11.2.2 Probability

For the data set 1 of the rst homework (ftp/pub/Matlab/set1.dat), determine the probability that the concrete strength is between 3,950 psi and 4,050 psi using the following two approaches: 1. From the computed mean and standard deviation, use Eq. 9.18

P (xmin < x < xmax ) = where

Z

xmax

xmin

(x) = p 1 e; 12 [ 2

f (x)dx

x; ]2 

(11.53) (11.54)

2. From the raw data. 3. Compare the two probabilities.

11.2.3 Moment of Inertias

Determine the Moment of inertias Ixx and Iyy for the triangular section of Fig. 11.2.

11.2.4 Reinforced Concrete Ultimate Stress Distribution

In reinforced concrete (r/c) beams, the strain distribution along the depth is assumed to be linearly varying, positive and negative at the lower and upper extreme bers respectively. At failure, the maximum compressive strain is 0.003. The stress-strain curve of (normal strength) concrete is nonlinear, hence the exact stress distribution is also nonlinear. In order to determine the ultimate load carrying capacity of a r/c beam, all we need is the resultant and location of the resultant force. Through extensive experimental studies, it was determined that a rectangular stress block, Fig. 11.7, would have the same resultant force F

F = 0| :85 f 0 ab {z }c |{z} 

A

(11.55)

acting at a distance a=2 from the top where

a = 1 c 1 = 0:85

(11.56-a) (11.56-b)

In order to assess the accuracy of this equation, the following stress-strain curve was experimentally obtained fc0 2 "max  =  2 " (11.57) " 1 + "max Using this equation, Victor E. Saouma

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11{14

Figure 11.7: Equivalent and Exact Stress Distribution in Reinforced Concrete Beams 1. Determine the resultant force and its location 2. Compare with the actual simpli ed approximation (where the resultant force is at a=2 fro the top and is equal to 0:85fc0 ab). Use: fc0 = 3; 000 lbs/in2, "max = 0:003, c = 4:in.

11.2.5 Mixture Problem, [2]

A 120 gallon tank contains 90 pounds of salt dissolved in 90 gallons of water. Brine containing 2 pounds of salt per gallon is owing into the tank at a rate of 4 gallons per minute. The mixture ows out of the tank at the rate of 3 gallons per minute. The dierential equation that speci es the amount of slat x(t) in pounds in the tank at time t is (11.58) x0 = 8 ; 90 3; t :x The tank is full after 30 minutes. 1. Determine and plot the amount of salt in the tank from time=0 until the tank is full. 2. Determine the amount of time required for the tank to contain 150 pounds of salt. 3. The analytical solution to the dierential equation is 4 (11.59) x(t) = 2(90 + t) ; (9090+ t)3 Victor E. Saouma

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11.2 Assignment

11{15

compare your numerical results with the exact ones.

11.2.6 Dog-Tracking, [1]

A number of interesting curves are trajectories of a point D moving at constant speed while always pointing toward another point M that moves along a known path. Such a method of tracking is known as dog-tracking because it resembles the path followed by a dog chasing his master. Similar curves are those followed by a vessel chasing another one, and by some ground-to-air or air-to-air missiles. Determine and plot the trajectories of a dog chasing his master for each of the following two cases: 1. The master moves on a straight line and (a) Initial dog position (x0 ; y0 )=(0,0) (b) Dog speed vd =10m/s (c) Master position at time t, xm = vm t, and ym =100m (d) Master speed vm =5m/s (e) Initial and nal times t0 = 0 and tf =10 s. 2. The dog is at the center of a circular pond of radius r while his master walks on the bank at constant velocity vm . The dog swims at a constant speed vd always toward his master and (a) Initial dog position (x0 ; y0 )=(0,0) (b) Dog speed vd =2.5m/s (c) Master speed vm =2m/s (d) Radius r = 15m ~ Hint: ~vd = vd jDM D~ M j , this will result into two separate equations (one for the x component and one for the y component).

11.2.7 Ballistic Model, [1]

A simple ballistic model assumes that only two forces act on a projectile, namely gravity and drag. According to this model, the equation of motion are (11.60-a) mvx0 = ;W vvx (11.60-b) mvy0 = ;W vvy ; mg where vx and vy are components of the velocity and v its magnitude. The drag W is given by (11.61) W = c  v2  d2 w2

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11{16

WEEK III; MATLAB: \Advanced" Topics

where cw is the drag coecient and is nearly constant as long as the ight is at subsonic speed,  is the air density, and d the diameter of the projectile. Plot the trajectory, using the following parameters: g=9.81 ms;2 , m=10 kg, cw =0.2, =1.225kgm;3 , d=0.005m. The initial conditions are x0=0m, y0 =0m, v0 =250 m/s and the gun elevation is =6, Use ode23 with a tolerance of 5:10;4 .

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Chapter 12

Week IV: MATHEMATICA 12.1 Background 12.1.1 Introduction

12.1.1.1 What is Mathematica ? Mathematica is a program for symbolic mathematics. 2 With Mathematica, you can perform algebraic operations of various complexities. Hence, rather than numerically solving an Engineering problem, you can solve it algebraically. 3 Because of its power, Mathematica will enable you to easily address problem of such complexities that it would have been impractical to solve otherwise. 4 Finally, Mathematica has a very powerful set of graphics capabilities. 1

12.1.1.2 Availability Mathematica is available in the Bechtel Laboratory. 6 A relatively inexpensive Student Edition can be purchased from the Bualo Chip, and the University used to have a site license for it.

5

12.1.1.3 Front End and Kernel Mathematica consists of two parts: the Kernel which is the computation engine, and the Front End which is the user interface. The Kernel is identical on all computers, however the Kernel may vary. 8 There are two Front Ends supported in the Bechtel Lab:

7

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Week IV: MATHEMATICA

Command-Line Interface: where the current input line is displayed on the last line of the

screen (it can be edited by using the arrow, insert and delete keys). This Front End does not support graphics, and is invoked by the math command. It is a convenient (and only available one) to use if you connect to the server via a remote terminal which does not support graphics (such as a home pc). Files storing Mathematica commands can be created by any ASCII editor and are usually given the le extension .m Notebook Front End: which can contain a mixture of text, graphics, and Mathematica definitions, Fig. 12.1. In this environment there is no need to use an external editor since the Notebook Front End plays also that role. Notebook les usually have a .ma or .mb for ASCII (to facilitate transfer to other computers) or binary les. 9

In both cases Mathematica is used interactively

12.1.1.4 Accessing Mathematica In the Bechtel Lab, Mathematica is accessible through by clicking the left mouse button and then selecting Mathematica. Note that the program is licensed to run only on the server (bechtel). 11 If you want to run the program dierently, you need to xhost + on your console rlogin bechtel To connect you to the server setenv DISPLAY yourmachine:0.0 To have the graphics displayed on your workstation Type math for the Command Line Front End. To exit you simply type Exit. 12 Alternatively, you may simply type mathematica for the Notebook Front End.

10

12.1.1.5 Help For help, simply type ?. For a speci c function, you can type ?Sqrt or ?P*. 14 Note that after each command, you should hit the following two keys simultaneously <Shift> 15 The Notebook Front End also provides extensive help, Fig. 12.2.

13

12.1.1.6 Notebook Front End 12.1.1.6.1 Pointers changes:

Victor E. Saouma

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As the mouse moves around the Notebook, the shape of the pointer

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12.1 Background

Figure 12.1: Notebook Front End for Mathematica

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12{4

Figure 12.2: Notebook Front End Help for Mathematica

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12.1 Background

12{5

Vertical I-bar appears when pointing to text (or input) that can be edited. If you click on

the mouse, the pointer will then blink. Horizontal I-bar appears when pointing to the space between two cells, or before the rst and after the last one. If you click the mouse, a horizontal line is drawn. Anything typed after will be part of a new cell. Cell-bracket Pointer appears when pointing to a cell bracket. If you click the mouse to select a group a cell (or group of cells), then you can copy the cell, paste something else into it, or re-execute the cell. If you double click, it will open or close the cell. Formatted-Cell Pointer appears when pointing to cells that are formatted (such as Mathematica output or graphics which can not be changed). Graphics Pointer appears when pointing inside the bounding box of a selected graphics. You can drag the graphic by depressing the mouse button and then moving the mouse. Sizing Pointers appears when pointing to handles of graphic's bounding box. Click and drag the mouse to rescale the graphic in the direction shown by the pointer. Watch Pointer Mathematica is busy can not be interrupted!.

12.1.1.6.2 Cell Brackets

17 A cell is the basic unit of organization in a Notebook. It can contain text, Mathematica input or output, or other cells. The style of the bracket in the right margin gives an indication of the cell attributes, and its size indicates its extent. Unformatted Cell contains ordinary text that can be edited such as Mathematica input or text. Inactive Cell can not be executed (such as Mathematica output, and they are usually formatted. Initialization Cell can be automatically evaluated when the Notebook is opened Locked Cell means that its content can not be altered unless it is rst unlocked. Closed Group implies that only the rst cell in the group is visible, and all others are not.

12.1.1.7 Brackets, Parentheses, and Braces Brackets are used to specify arguments of functions. Parenthesis are used for grouping. Without them, multiplication and division have a higher precedence than addition and subtraction. Braces are used to specify lists, vectors, and matrices.

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12{6

Week IV: MATHEMATICA

Double Brackets are used for indexing. Comments are enclosed by (* Comment *).

12.1.2 Examples

12.1.2.1 Basic Arithmetic Operation You can perform arithmetic operations with Mathematica as you would perform them with a pocket calculator: 1. Simple

18

In[1]:= 6 + 5 Out[1]= 11

2. or slightly more complex operations: In[2]:=6 ((3+4)^2-(5/6*7)+8 (9+10)) Out[2]=1171

The arguments of all Mathematica functions are enclosed in square brackets, and the names of built-in functions begin with capital letters. 3. and now try to do this on your calculator: In[1]:= 3^99 Out[1]= 171792506910670443678820376588540424234035840667

which is the exact value of 399 .

12.1.2.2 Approximate Numerical Values 19

You can use the Mathematica function N to get approximate numerical results.

= N[3^99] 47 Out[3]= 1.71793 10 In[4]:= 3^99//N 47 Out[4]= 1.71793 10

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12.1 Background

12{7

12.1.2.3 Complex Numbers 20

Mathematica can also handle complex numbers: In[2]:= (2+5 I) (2-3 I) Out[2]= 19 + 4 I

12.1.2.4 Derivatives 21

Here is an example of a derivative.

In[1]:= D[x^2 Sin[a x],x] 2 Out[1]= a x

Cos[a x] + 2 x Sin[a x]

12.1.2.5 Integration In[1]:= Integrate[(x+y)^5,x] 6 4 2 3 3 2 4 x 5 5 x y 10 x y 5 x y 5 Out[1]= -- + x y + ------- + -------- + ------- + x y 6 2 3 2 In[2]:=

Integrate[Sin[x],{x,a,b}]

Out[2]= Cos[a] - Cos[b] In[3]:= Integrate[Sin[x]+Sin[y],{x,0,Pi/2},{y,0,x}] Pi Out[3]= -2 In[4]:= NIntegrate[Sin[x]^2-Sin[2 x],{x, 0,Pi}] Out[4]= 1.5708

12.1.2.6 Algebraic Formulae 22

Mathematica can expand or reduce mathematical expressions

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In[5]:= Expand[(x+y)^5] 5 4 3 2 2 3 4 5 Out[5]= x + 5 x y + 10 x y, + 10 x y + 5 x y + y

12.1.2.7 Solving equations 23

Mathematica can determine the root of equations: In[15]:= Solve[x^2+3x-a==0,x] -3 + Sqrt[9 + 4 a] -3 - Sqrt[9 + 4 a] Out[15]= {{x -> ------------------}, {x -> ------------------}} 2 2

12.1.2.8 Matrices In[28]:= m=Table[(i),{i,9},{j,1}] Out[28]= {{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}} In[26]:= m=Table[(i+j),{i,0,8},{j,1}] Out[26]= {{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}} In[29]:= m=Table[(i),{i,9},{j,2}] Out[29]= {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6}, {7, 7}, {8, 8}, > {9, 9}}

Matrices are presented in Mathematica as a list of lists. Mathematica can add, multiply, determine the inverse, determinant, eigenvalues and eigenvectors of matrices:

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In[37]:= m=Table[(j+i),{i,0,1},{j,2}] Out[37]= {{1, 2}, {2, 3}} In[41]:=Inverse[m] Out[40]= {{-3, 2}, {2, -1}}

or

In[40]:= Inverse[%37]

In[41]:= Det[%] Out[41]= -1 In[43]:= %37.%40 Out[43]= {{1, 0}, {0, 1}}

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In[44]:= a={{12,7,9},{4,3,1},{11,6,3}} Out[44]= {{12, 7, 9}, {4, 3, 1}, {11, 6, 3}} In[45]:= Inverse[a] 3 33 5 1 63 6 9 5 2 Out[45]= {{-(--), -(--), --}, {--, --, -(--)}, {--, -(--), -(--)}} 52 52 13 52 52 13 52 52 13 In[46]:= %2.%1 Out[46]= {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}} In[47]:= Eigenvalues[a]//N Out[47]= {20.4216, -3.21391, 0.79228} In[48]:= Det[a] Out[48]= -52

12.1.2.9 Graphics and Three-Dimensional Plots In[13]:= Plot[Sin[x],{x,0,2 Pi}] Out[13]= -Graphics-

In[10]:= Plot3D[Exp[x 2 y],{x,-2,2},{y,-2,2}] Out[10]= -SurfaceGraphics-

12.1.2.10 Interfacing with Mathematica 12.1.2.10.1 Input

25 Rather than typing all the input commands, you can store them in an ASCII le, the extension .m is recommended. To execute the le, simply type:

In[1]:= << >

where is the le name. If you had an .m extension, you do not need to specify it.

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1

0.5

1

2

3

4

5

6

-0.5

-1

Figure 12.3: Two-dimensional Plot generated from Mathematica

1 0.5

2

0 1

-0.5 -1 -2

0 -1 0

-1 1

2

-2

Figure 12.4: Three-dimensional Plot generated from Mathematica Victor E. Saouma

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12.1.2.10.2 Output

12{11 26

Mathematica can be interfaced

In[10]:= Integrate[x/(1+xsinx),x] 2 x Out[10]= ------------2 (1 + xsinx)

with: Mathematica Form

In[14]:= InputForm[%]

Out[14]//InputForm= x^2/(2*(1 + xsinx))

Fortran

In[12]:= FortranForm[%]

Out[12]//FortranForm= x**2/(2*(1 + xsinx))

C

In[11]:= CForm[%] Out[11]//CForm= Power(x,2)/(2*(1 + xsinx))

TEX

In[13]:= TeXForm[%] Out[13]//TeXForm= {{{x^2}}\over {2 \left( 1 + {\it xsinx} \right) }}

12.1.2.11 Packages Most function in Mathematica are written in C. However, some functions are written in Mathematica itself. Such functions are de ned in les called packages which will allow you to: 1. De ne a function or set of functions that are often used

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2. Hide the implementation from the user 3. Save the functions and reload them when needed.

12.1.2.12 Graphics hardcopy The function PSPrint will generate a postscript le which can be later sent to the laser printer.

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Type: PSPrint[%n]

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If you want to save your graphics use the Mathematica command:

Display["file name",%n]

Where n is the number of your graphics output. Then in shelltool window use the ps x command to create a PostScript le.

12.1.2.13 Input le You can store all your operations into an ASCII le through a text editor, and then \load" it into Mathematica

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See the {\it Mathematica} manuals for further info on input files.

12.1.3 Some Mathematica Commands 12.1.3.1 Basic Operations

Following are the basic operations supported by Mathematica Note that multiplication can be speci ed by either using an asterisk or by leaving a blank space between arguments. x^y power -x minus x/y devide x y z or x*y*z multiply x+y+z add 31

12.1.3.2 Mathematical Functions Following is a list of the most commonly used functions, a complete list is presented at the end of the chapter.

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Sqrt[x] Exp[x] Log[x] Log[b,x] Sin[x], Cos[x] Tan[x] ArcSin[x] ArcCos[x],ArcTan[x] n! Abs[x] Round[x] Mod[n, m] Random[ ] Max[x,y,..], Min[x,y,..] Grad[f] Div[f] Curl[f]

square root exponential natural logarithm logarithm to base b trigonometric functions (with arguments in radians) inverse trigonometric functions factorial (product of integers 1,2,...n) absolute value closest integer to x n modulo m (remainder on division of n by m) pseudorandom number between 0 and 1 maximum,minimum of x,y,... Gradient in the given system Divergence Curl

Note that all Mathematica built-in functions start with an upper case letter, and the arguments are enclosed in square brackets.

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12.1.3.3 Some Mathematica Constants 34

Mathematica also has some physical constants hardwired into the system. Pi 3,14159 E 2.71828 Degree 3.14159/180:degrees to radians conversion factor p;1 I In nity 1

12.1.3.4 Complex Numbers x + Iy Re[z] Im[z] Conjugate[z] Abs[z] Arg[z]

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The complex number x + i y Real part of z Imaginary part of z Complex conjugate of z Absolute value of z The argument phi in the exponential form of a complex number

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12.1.3.5 Recall of Previous Expressions

% Last result %% Next to last result % n Result on output line Out[n]

12.1.3.6 Assignment of Variables

x = value Assign value to x x = y = value Assign value to x and y x = . or Clear[x] Remove assignment of x

12.1.3.7 Brackets

(term) f[x] fa, b, cg v[[i]]

Parentheses for grouping Square brackets for functions Braces for lists Double brackets for indexing

12.1.3.8 Help

?name Display information on name ??name Display more information on name ?Aaaa* Display information on all commands whose names begin with C*

12.1.3.9 Interrupting Mathematica Continue Show Inspect Abort Exit

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Continue calculation Show what MATHEMATICA is doing Inspect current state Abort this particular calculation Exit MATHEMATICA

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12.1.3.10 Transformation of Algebraic Expressions Expand[expr]

Expand an expression into sum of terms Factor[expr] Write expr as a minimal product of factors Simplify[expr] Simplify expression Coecient[expr, form] Coecient of form in expr Numerator[expr] Numerator Denominator[expr] Denominator expr // Short Show one line outline of output Short[expr, n] Show n-line outline of result Eliminate[flhs1 == rhs1,lhs2 == rhs2, ...g, fx, ...g] Eliminate x, ... from the set of simultaneous equations Reduce[flhs1 == rhs1, lhs2 == rhs2, ...g, fx, y, ...g] Give a set of simpli ed equations, including all possible solutions

12.1.3.11 Dierentiation D[f, x] D[f, x1, x2, ...] D[f, fx, ng] Dt[f] Dt[f, x] Limit[f, x->xo]

Victor E. Saouma

Partial derivative of f with respect to x Multiple derivative of f with respect to x1, x2, ... The nth derivative of f with respect to x The total derivative of f The total derivative of f with respect to x The limit of f as x goes to xo

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12.1.3.12 Integration Integrate[f, x]

Inde nite integral of f with respect to x Integrate[f, fx, xmin, xmaxg] De nite integral of f wrt to x from xmin to xmax Integrate[f, fx, xmin, xmaxg, fy, ymin, ymaxg] Multiple integral of f with respect to y and x

12.1.3.13 Summation & Products Sum[f, fi, imin, imax, dig] Sum[f, fi, imin, imaxg, fj, jmin, jmaxg] Product[f, fi, imin, imaxg] fimaxg fi, imaxg fi, imin, imaxg fi, imin, imax, dig

12.1.3.14 Equations 12.1.3.14.1 Preliminaries

x=y x == y x != y x>y xy x
Summation with i incremented by di Nested summation of f with respect to j then i The product of f from imin to imax Iterate imax times without incrementation of a variable Increment i from 1 to imax by steps of 1 Increment i form imin to imax by steps of 1 Increment i from imin to imax by steps of di

Assigns the value of y to x Tests for x = y Unequal Greater than Greater than or equal to Less than Less than or equal to All equal All unequal not and or Execute then if p is true, else otherwise

12.1.3.14.2 Solution

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Solve[lhs == rhs, x] x /. expr expr1 /. expr2 Solve[flhs1 == rhs1, lhs2 == rhs2, ...g, fx, y, ...g]

Solve equation in terms of x Use expr to get values of x Use expr2 to get values for expr1 Solve simultaneous set of equations for x,y,...

12.1.3.14.3 Dierential Equations DSolve[eqns, y[x], x] Solve dierential equation for y[x], taking x as the independent variable DSolve[eqns, y, x] Give sol'n for y in pure functional form

12.1.3.14.4 Numerical Values N[expr] NIntegrate[f. fx, xmin, xmaxg] NSolve[lhs == rhs, x] NDSolve[eqns, y, fx, xmin, xmaxg]

12.1.3.15 Functions

Victor E. Saouma

Numerical value of expr Numerical approximation of integral Numerical approximation of the solution to the equation Solve numerically for y with independent variable x

f[x ] De ne the function f ?f Show de nition of f Clear[f] Clear de nition of f

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12.1.3.16 Vectors & Matrices fa, b, cg Vector (a,b,c) " # a b ffa,bg,fc,dgg Matrix c d Array[a, n] Range[n] Length[list] IdentityMatrix[n] MatrixForm[list] a. b cm Inverse[m] MatrixPower[m, n] Det[m] Transpose[m] Eigenvalues[m] Eigenvectors[m] Eigenvalues[N[m]] Eigenvectors[N[m]]

Build a length-n vector of the form fa[1],... Create a list f1, 2, 3, ..., ng Give number of elements in list Generate n x n identity matrix Display list in matrix form Matrix product Multiply a matrix by a scalar c Matrix inverse nth power of a matrix Determinant Transpose Eigenvalues Eigenvectors and Numerical eigenvalues Numerical eigenvectors

12.1.3.17 Graphics 12.1.3.17.1 Preliminaries Plot[f, fx, xmin, xmaxg] Plot f in terms of x from xmin to xmax Plot[Evaluate[f], fx, xmin, xmaxg First evaluate f then plot it 12.1.3.17.2 Plot Options

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12.1 Background AspectRatio Axes AxesLabel AxesOrigin DefaultFont DisplayFunction Frame FrameLabel FrameTicks GridLines PlotLabel PlotRange Ticks Automatic None All True False PlotPoints MaxBend PlotDivision Show[plot] Show[plot, option->value] Show[plot1, plot2, ...]

12{19 Height-to-width ratio for plot Whether to include axes (True or False) Labels to be put on axes fxlabel, ylabelg or None Point at which axes cross The default font for text in the plot How to display graphics Whether to draw frame around plot Labels to be put around the frame What tick marks to draw if frame is included What grid lines to include An expression to be printed as a label for the plot Plot range of coordintes to include in the plot What tick marks to draw if there is an axes Use internal algorithms Do not include this Include everything Do this Do not do this Minimum number of points at which to sample the function Maximum kink angle between successive segments of a curve Maximum factor by which to subdivide in sampling the function Redraw a plot Redraw plot with changed options Combine several plots

12.1.3.17.3 3 Dimensional Plots Plot3D[f, fx, xmin, xmaxg, fy, ymin, ymaxg] Make 3D plot of f as a function of x and y 12.1.3.17.4 Parametric Plots

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First you must load the ParametricPlot3D package:

In[1]:=<
ParametricPlot[ffx, fyg, ft, tmin, tmaxg] Make a parametric plot ParametricPlot[ffx, fyg, fgx, gyg, ft, tmin, tmaxg] Parametric plot of several curves simultaneously

12.1.3.17.5 File Manipulation <> name expr >>> name !!name Victor E.Save["name", Saouma f, g, ...]

Load an ASCII le containing Mathematica commands Output expr to an ASCII le Append expr to an ASCII le Display contents of an ASCII le Save de nitions for variables or Undergraduate functions in aEngineering le Computing Literacy for Students

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12.1.3.17.6 Generating C, Fortran & TEXFiles

CForm[expr] Write expr as C code FortranForm[expr] Write expr as Fortran code TexForm[expr] Write expr in TEXform

12.1.4 Programming in Mathematica Programming in Mathematica is the process of writing functions, or packages, that can subsequently be executed by simply typing in the name of the package. Writing programs with Mathematica allows you to develop specialized functions in your eld of interest that can then be run from the Mathematica environment.

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12.1.4.1 Building a Package The goal of writing a package is to make the new function behave as much like a Mathematica function as possible. This includes being able to type ?FunctionName for documentation on how to use the new package FunctionName. Also the output from the package should not be dependant on any previous calculations you have done during your Mathematica session. 38 Let's look at a few examples:

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(*This function returns the first n powers of x*) PowerSum[x_,n_] := Block[{i}, Sum[x^i,{i,1,n}] ]

This would then be saved in a le with an extension .m, such as Example.m 1. PowerSum[ ] is the name of the function were are creating. 2. All the variables local to PowerSum[ ] are declared in a Block[ ] statement, this isolates them from any values they might hold globally. 3. For this example, you must make sure that you don't pass in variables that are used locally inside the function. For example:

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In[1]:= <<Example.m In[2]:= PowerSum[x,5] Out[2]= x + x^2 + x^3 + x^4 + x^5

but if we were to call PowerSum with the following: Victor E. Saouma

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12.1 Background

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In[3]:= PowerSum[i,5] Out[3]= 3413

notice that we get a number for our answer and not an expression. This is because the variable i, which we passed in, was captured by the variable (also called i) in the range of the summation. The following example shows how to avoid such a problem. PowerSum::usage = "PowerSum[x,n] returns the sum of first n powers of x." Begin["Private`"] PowerSum[x_,n_] := Block[{i}, Sum[x^i,{i,1,n}] ] End[]

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Note we have done two things: 1. The usage statement de nes a help message to be printed out if the user types ?PowerSum. 2. The local variable i is now created in the context Private` which is not searched when you type in a variable name later on.

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For example, with the revised Example.m you could type in the following:

In[1]:= <<Example.m In[2]:= PowerSum[i,5] Out[2]= i + i^2 + i^3 + i^4 + i^5

we now get the expression and not a numerical value.

12.1.4.2 A Complete Example Here is an example of a package that will plot a random walk of length n. It will start at the origin and then randomly choose a direction to follow for a line of length 1.

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BeginPackage["RandomWalk`"] RandomWalk::usage = "RandomWalk[n] plots a random walk of length n." Begin["`Private`"] RandomWalk[n_Integer] := Block[{loc = {0.0,0.0}, dir, points = Table[0,{n+1}], range = N[{0, 2Pi}]}, points[[1]]=loc; Do[ dir = Random[Real,range]; loc += {Cos[dir],Sin[dir]}; points[[i]] = loc,

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Week IV: MATHEMATICA {i,2,n+1}]; Show[ Graphics[{Point[{0,0}], Line[points]}], Frame->True, AspectRatio->Automatic]

] End[] EndPackage[]

The package is started by BeginPackage["RandomWalk`"] and ended by EndPackage[]. Once a package is complete these statements need to be inserted, but during development of the package it is suggested that they be left out. If they are included in the development stage they may interfere with the debugging process. Next is the usage statement followed by a short description of the package. This is the message that will be displayed when the user requests help for this command. The Begin["`Private`"] and End[] statements are to keep values of variables used outside the package from interfering with the variables used locally by the package. The RandomWalk[n Integer] := statement assigns the name RandomWalk and the parameter n (de ned as an integer) to the function. All the variable declarations and operations are done within the Block[........] statement. The rst part of Block[....] declares some variables: 1. loc = f0.0,0.0g de nes loc as a collection of two values, these will be used as x and y coordinates later on. 2. points is declared as a Table, see Mathematica reference for more details if you are not familiar with this command. Basically, points will be a list going from 0 to n+1. 3. range is exactly that, the N is to give an approximate value of Pi. The rst point of points is then assigned the origin. The semicolon at the end of the line is to separate statements. The next construct is a Do[...], this enable the user to iterate. The format is Do[statement , interator]. This Do loop has more than one statement, so they are separated by semicolons. The iterator is fi,2,n+1g. The following occur within the Do loop: 1. dir is chosen randomly from the range indicated in range and returned as a real. 2. loc is then assigned the cosine and sine of dir 3. points is assigned the value of loc. The next statements; Show[...], and Graphics[..], are straight forward Mathematica commands. For more information on these commands see the Mathematica reference text. Victor E. Saouma

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12.2 Assignment

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12.1.5 List of All Mathematica Functions Table 12.1 contains all Mathematica functions, with the most commonly used ones in boldfaced.

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12.2 Assignment 12.2.1 Practice

Start by repeating all the examples in this handout.

12.2.2 Problems

Write a Mathematica Notebook to execute the following: 1. Solve x2 + 2x + 1 = 0 R Rp 2. Integrate 01 0 x yex2 dydx 3. Solve the following dierential equation y0 (x) = y(x) 4. Plot the function x sin x for x in the range [0; 6], and (a) Change the AspectRatio so that the width of the plot is twice its height. (b) Give the plot the label ``x Sin[x]''. (c) Label the x axis \x", and the y axis \y". 5. Plotsin( sin x + y) for ;3 < x < 3 and ;3 < y < 3 using 33 grid points, no display of the axes. 6. Generate a contour plot of the previous equation. p 7. Write a function which takes a pair x; y and returns x2 + y2 .

12.3 References 1. Wolfram, S. \Mathematica: A System for Doing Mathematics by Computer second edition, Addison-Wesley, 1991. 2. Maeder, R., Programming in Mathematica, Addison Wesley, 1991 3. Blachman, N., Mathematica a Practical Approach, Prentice Hall, 1991. 4. Gray, T., and Glynn J., The Beginner's Guide to Mathematica, Addison-Wesley, 1992. Victor E. Saouma

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Above CoecientList EllipticK Headers yLightSources OneIdentity Re yCollect EllipticLog HermiteH yLighting OpenAppend Read Accumulate ColonForm Encode yHiddenSurf* Limit OpenRead ReadList Accuracy ColumnForm End Hide Line OpenTemporary ReadProtected AccuracyGoal Complement EndAdd Hold LineBreak OpenWrite Real yAddTo yComplex EndOfFile HoldAll LinearProgra* Operate Recall AiryAi ComplexIn ni* EndPackage HoldFirst LinearSolve Optional Rectangle AlgebraicRules Compose EndProcess HoldForm List Options yReduce AlgebraicRule* ComposeSeries EngineeringFo* HoldRest ListContourPl* Or Release Alias CompoundExp* Environment HorizontalForm ListDensityPl* Order Remove All Condition Epilog Hypergeometri* ListPlot OrderedQ Removed yAmbientLig* Conjugate Equal Hypergeometri* ListPlot3D Orderless RenderAll And Constant EquatedTo Hypergeometri* Listable Out R Apart Constants Erf Hypergeometri* Literal Outer RepeatedNull Append ConstrainedMax EulerE Hypergeometri* Locked OutputForm RepeatedString AppendTo ConstrainedMin EulerGamma Hypergeometri* yLog Over ow Replace Apply Construct EulerPhi Hypergeometri* LogIntegral OwnValue ReplaceAll ArcCos Context EvenQ I LogicalExpand yPSPrint ReplaceRepeat* ArcCosh ContextToFile* yExit Identity MachineID PageHeight ResetMedium ArcCot Continuation yExp IdentityMatrix MachineName PageWidth Residue ArcCoth Continue ExpIntegralE If MainSolve yParametri* Rest ArcCsc yContourGr* ExpIntegralEi Im MakeRules Part Resultant ArcCsch ContourLevels yExpand Implies Map Partition Resultant2 ArcSec yContourPlot ExpandAll In MapAll PartitionsP Return ArcSech ContourSpacing ExpandDenom* Increment MapAt PartitionsQ Reverse ArcSin Cos ExpandNumer* Indent MatchBox Pattern Right ArcSinh CosIntegral Exponent Indeterminate MatchQ PatternTest yRoots ArcTan Cosh ExponentStep Inequality MatrixExp Permutations RotateLeft ArcTanh Cot Expression yIn nity yMatrixForm yPi RotateRight Arg Coth ExtendedGCD In x MatrixPower Plain Round ArithmeticGeo* Count ExtraFactors Information MatrixQ yPlot RowReduce Array Csc ExtraTerms Inner Max yPlot3D Rule AspectRatio Csch FaceForm Input MaxBend Plot3Matrix RuleDelayed AtomQ Cubics Factor InputForm MaxIterations PlotColor RuleForm Attributes Cyclotomic FactorComplete InputString MaxMemoryUsed PlotDivision RuleTable Automatic yD FactorInteger Insert MaxRecursion PlotJoined Run Auxiliary DSolve FactorList Integer Medium PlotLabel RunThrough Axes DampingFactor FactorSquareF* IntegerQ MemberQ PlotPoints SameQ AxesEdge Dashing FactorSquareF* yIntegrate MemoryConstra* yPlotRange Save AxesLabel yDebug FactorTerms InterpolatingPo MemoryInUse PlotStyle Scaled AxesStyle yDecompose FactorTermsLi* Interrupt Mesh yPlus Scan Background Decrement Factorial Intersection MeshRange Pochhammer yScienti cFo* BaseForm Default Factorial2 yInverse MeshStyle Point Sec Begin DefaultColor Factors InverseFourier Message PointSize Sech BeginPackage DefaultValue Fail InverseFuncti* MessageName Points Second Below De nition False InverseFuncti* Messages PolyGamma SeedRandom BernoulliB Degree FindMinimum InverseJacobi* Method PolyLog Select BesselI Delimiters yFindRoot InverseJacobi* Min Polygon Sequence BesselJ Denominator First InverseJacobi* MinRecursion PolynomialG* SequenceForm BesselK DensityGraphi* Fit InverseJacobi* Minors PolynomialQ SequenceLimit BesselY DensityPlot FixedPoint InverseJacobi* Minus PolynomialQu* Series Beta Depth Flat InverseJacobi* Mod PolynomialQu* SeriesCoec* BetaRegulariz* yDerivative Flatten InverseJacobi* Mode PolynomialRe* SeriesData Binomial Det Floor InverseJacobi* yModular Position Set Blank DiagonalMatrix yFont InverseJacobi* Modulus Positive SetAccuracy BlankForm DigitBlock FontForm InverseJacobi* MoebiusMu yPostScript SetAttributes BlankNullSequ* Digits For InverseSeries Multinomial Post x SetDelayed BlankSequence Dimensions Format JacobiAmplitu* Multiplicity Power SetOptions Block DirectedIn n* FormatType JacobiCD N PowerMod SetPrecision Bottom Disk yFortranForm JacobiCN NBernoulliB PreDecrement Shading BoxRatios Dispatch Fourier JacobiCS yNIntegrate PreIncrement Share BoxStyle Display Framed JacobiDC NProduct PrecedenceFo* Sh Boxed DisplayFuncti* FreeQ JacobiDN yNRoots Precision Show Break Distribute FromASCII JacobiDS NSum Pre x Sign Byte yDivide FullDe nition JacobiNC NameQ Prepend Signature ByteCount DivideBy FullForm JacobiND Names PrependTo ySimplify C DivisorSigma Function JacobiNS Needs Prime ySin CForm Divisors GCD JacobiP Negative PrimeQ SinIntegral CallProcess Do Gamma JacobiSC Nest Print SingularValues Cancel yDot GammaRegul* JacobiSD NestList PrintForm SingularityDe* Cases DoubleExpone* GaussianQu* JacobiSN NonAssociati* PrintValue Sinh Catalan DoublyIn nite yGegenbaue* JacobiSymbol NonCommutati* ProbablePrimeQ Skeleton Catch DownValue General Jacobian NonConstants Product Slot Ceiling Drop Generic Join NonNegative Prolog SlotSequence CellArray DropFrom Get LCM None PromptForm ySolve Center Dt GoldenRatio Label Normal Protect SolveAlways Character Dump Goto LaguerreL Not Protected Sort Characters yE Gradient LaserPrint Null PseudoInverse Space ChebyshevT EdgeForm Graphics Last NullSpace Put SpaceForm ChebyshevU Edit Graphics3D LatticeReduce NumValue PutAppend SphericalHarm* Check EditDef GraphicsFont LeafCount Number Quartics Splice Chop EditIn GraphicsLeading Left NumberForm Quit ySqrt Circle Eigensystem GrayLevel LegendreP NumberPoint Quotient StartProcess Clear yEigenvalues Greater LegendreQ NumberQ RGBColor StirlingS1 ClearAll Eigenvectors GreaterEqual LegendreType NumberSepar* Random StirlingS2 ClearAttribut* ElapsedTime GroebnerBasis Length Numerator Range String Victor E. Saouma Computing Literacy for Undergraduate Engineering ClipFill yEliminate Hash LerchPhi O Rational StringBreak Close EllipticE HashTable Less OddQ Rationalize StringForm CodeAddress EllipticExp Head LessEqual O Raw StringJoin Coecient EllipticF HeadCompose Level On RawMedium StringLength

yAb

Table 12.1: Mathematica Functions

StringMatchQ StringSkeleton SubValue Subscript Subscripted Subtract

ySubtractFr* ySum

Superscript SurfaceGraphics Switch Symbol Tab Table TableForm TagSet TagSetDelayed TagUnset Take Tan T yTeXForm TensorRank Terms Text TextFont yTextForm TextLeading TextRendering Thickness Thread Through Throw Ticks TimeConstrained Times TimesBy Timing ToASCII ToExpression ToRules ToString yTogether Tolerance TooBig Top TotalHeight TotalWidth Transpose TreeForm yTrigExpand True TrueQ UnAlias Under ow Unequal Union Unique Unprotect UnsameQ Unset UpSet UpSetDelayed UpValue Update Using ValueForm ValueList ValueQ Variables Vector VerticalForm ViewPoint WeierstrassP WeierstrassP* Which While WorkingPrecis* Write WriteString WynnDegree Xor ZeroTest Zeta

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Chapter 13

SAMPLES of MATLAB PROGRAMS This chapter contains the description and listing of a number of simple (and not so simple) MATLAB codes. The source codes can be freely copied from /pub/Structures/cven4837

13.1 arches

13.1.1 Description

SUBDIRECTORY: arches SYNOPSIS: axial, shear, and normal force plots for an arch (Structures) DESCRIPTION: arches.m plots the Bending Moment, Shear Force, and Axial Force for a three pin semi circular arch under uniform load.

CONTENTS;

main program Written by: Brian Rose arches.m

Determine the reactions of the three-hinged arch shown in Fig. 13.1 Solution: Four unknowns, three equations of equilibrium, one equation of condition ) statically determinate. (+ ;
) MzC = 0; (RAy )(140) + (80)(3:75) ; (30)(80) ; (20)(40) + RAx (26:25) = 0 ) 140RAy + 26:25RAx = 2:900 (+ - ) Fx = 0; 80 ; RAx ; RCx = 0

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13{2

Figure 13.1: (+ 6) Fy = 0; (+ ;
) MzB = 0;

RAy + RCy ; 30 ; 20 = 0 (Rax )(60) ; (80)(30) ; (30)(20) + (RAy )(80) = 0 ) 80RAy + 60RAx = 3; 000

(13.1-a)

Solving those four equations simultaneously we have: 2 6 6 6 4

140 26:25 0 1 1 0 80 60

0 0 1 0

0 1 0 0

38 > > 7> < 7 7 5> > > :

RAy RAx RCy RCx

9 > > > = > > > ;

8 > > > <

=> > > :

2; 900 80 50 3; 000

9 > > > = > > > ;

8 > > > <

)> > > :

RAy RAx RCy RCx

9 > > > = > > > ;

8 > > > <

=> > > :

15:1 k 29:8 k 34:9 k 50:2 k

9 > > > = > > > ;

(13.2)

We can check our results by considering the summation with respect to b from the right: (+ 
;) MzB = 0; ;(20)(20) ; (50:2)(33:75) + (34:9)(60) = 0

p

(13.3)

Determine the reactions of the three hinged statically determined semi-circular arch under its own dead weight w (per unit arc length s, where ds = rd). 13.2

Figure 13.2: Victor E. Saouma

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13.1 arches

13{3

The reactions can be determined by integrating the load over the entire structure (+ ;
) MA = 0;

;(VC )(2R) +

Z

=

(1 +{zcos }) = 0 wRd} R |

{z =0 | dP Z  =

moment arm ) VC = wR 2 =0 (1 + cos )d = wR = wR 2 [ ; sin ] j=0 = 2 [( ; sin ) ; (0 ; sin 0)] = 2 wR Next we determine the horizontal reaction Z =  2  (+ ) MB = 0; ;(HC )(R) + (Vc )(R) ; wRd R cos  =0 | {z } | {z } ;


 wR ; wR

) HC

Z

= 2

dP moment arm

(13.4-a)

=0

2 =0 cos d = 2 =  wR ; wR(  ; 0) = 2 wR ; wR[sin ] j=0 2 ;
2 = 2 ; 1 wR

=2

(13.5-a)

By symmetry the reactions at A are equal to those at C Draw the shear and moment diagram for the three hinged statically determined semi-circular arch under its own dead weight w, Fig. 13.3. Solution:

Reactions: Those were determined earlier, Example ??. For the sake of clarity, we repeat their derivation: 1. Starting with CY

(+ 
;) MA = 0;

;(CY )(2R) +

Z

=

wRd} R (1 +{zcos }) = 0 |

{z =0 | dP Z =

moment arm ) CY = wR 2 =0 (1 + cos )d = wR = wR 2 [ ; sin ] j=0 = 2 [( ; sin ) ; (0 ; sin 0)] (13.6-a) = 2 wR 2. Next we determine the horizontal reaction Z =  2  (+ ) MB = 0; ;(Cx )(R) + (Cy )(R) ; wRd R cos  =0 | {z } | {z } ;


) Cx

= 2 wR ; wR 2

=

= 2

=0

cos d

=0  wR ; wR(sin ] j= 2 =  wR ; wR(  =0
2 2 2 ; =  ; 1 wR

2

Victor E. Saouma

Z

dP moment arm

; 0)

(13.7-a)

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13{4

Figure 13.3:

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13.1 arches

13{5

3. By symmetry the reactions at A are equal to those at C Shear Forces: Considering the free body diagram of the arch, and summing the forces in the radial direction (FR = 0): Z    ;( 2 ; 1)wR cos  + 2 wR sin  ; wRd sin  + V = 0 |

{z

Cx

}

(13.8)

=0

| {z }

Cy

   ) V = wR ( 2 ; 1) cos  + ( ; 2 ) sin  

(13.9)

Axial Forces: Similarly, if we consider the summation of forces in the axial direction (FN = 0):

(

; 1)wR sin  +  wR cos  ;

Z



wRd cos  + N = 0  2 ; 1) sin 

(13.10) (13.11)

Moment: Now we can consider the third equation of equilibrium (M = 0): ( 2 ; 1)wR
R sin  ; 2 wR2 (1 ; cos ) + Z  wRd
R(cos  ; cos ) + M = 0 =0   M = wR2 2 (1 ; sin ) + ( ; 2 ) cos 

(13.12) (13.13)

2

N = wR

2

=0 ( ;  ) cos  ; ( 



2

We seek to determine the vertical de ection of the crown of the three hinged statically determined semi-circular arch, Fig. 13.4 under its own dead weight w, Fig. ??. 1. We rst seek to determine the analytical expression of the moment diagram. From statics, it can be shown that the vertical and horizontal reactions are Rv = 2 wR and Rh = ( 2 ; 1)wR. 2. Next considering the free body diagram of the arch, and summing the forces in the radial direction (FR = 0): Z



wRd sin  + V = 0 ;( 2 ; 1)wR cos  + 2 wR sin  ;  =0   V = wR ( 2 ; 1) cos  + ( ; 2 ) sin 

(13.14-a) (13.14-b)

3. Similarly, if we consider the summation of forces in the axial direction (FT = 0): Z



( 2 ; 1)wR sin  + 2 wR cos  ; wRd cos  + N = 0 =0   N = wR ( ; 2 ) cos  ; ( 2 ; 1) sin  Victor E. Saouma

(13.15-a) (13.15-b)

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13{6

Figure 13.4:

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13.1 arches

13{7

4. Now we can consider the third equation of equilibrium (M = 0): ( 2 ; 1)wR
R sin  ; 2 wR2 (1 ; cos ) + Z



wRd
R(cos  ; cos ) + M = 0   M = wR2 2 (1 ; sin ) + ( ; 2 ) cos 

(13.16-a) (13.16-b)

=0

5. The real curvature  is obtained by dividing the moment by EI  2  M wR   = EI EI 2 (1 ; sin ) + ( ; 2 ) cos 

(13.17)

6. The virtual force P will be a unit vertical point in the direction of the desired de ection, causing a virtual internal moment

M = R2 [1 ; cos  ; sin ]

0    2

(13.18)

7. Hence, application of the virtual work equation yields:  2  wR  1

= 2 (1 ; sin ) + ( ; 2 ) cos 
R2
[1 ; cos  ; sin ] Rd |{z} |{z} EI 2 =0 | {z } dx {z } | Z 

2

P

wR4

h

= 16EI 72 ; 18 ; 12 4 = :0337 wR EI

i



M

(13.19-a)

13.1.2 Listing 13.1.2.1

arches.m

Equations used for calculation of shear force, axial force, and bending moment are as follows:

V = !R[( 2 ; 1) cos  + ( ; 2 ) sin ] N = !R[( ; 2 ) cos  ; ( 2 ; 1) sin ] M = !R2 [ 2 (1 ; sin ) + ( ; 2 ) cos ] arches.m le starts here: Victor E. Saouma

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13{8

Figure 13.5: Simply Supported Arch with Uniform Load %Arches is a script that plots the Bending Moment, Shear Force, %and Axial Force for a three pin semi circular arch under uniform %load %clear the command screen clc %prompt the user for some information R=input('Enter the radius of the arch: '); w=input('Enter the load on the of the arch: '); %create a vector of theta from 0 to 90 degrees theta=(0:pi/100:pi/2); %calculate V, N, M for a half the arch V=w*R*((pi/2-1)*cos(theta)+(theta-pi/2).*sin(theta)); N=w*R*((theta-pi/2).*cos(theta)-(pi/2-1)*sin(theta)); M=w*R^2*(pi/2*(1-sin(theta))+(theta-pi/2).*cos(theta)); %since the structure is symmetric, get V,N,M values %for theta from 90 to 180 degrees by flipping V,N,M %notice the negative sign for Shear %also note that if we just flip V,N,M we'll %repeat V(90 degrees),N(90 degrees), M(90 degrees) %so we'll pull those values out V=[V fliplr(-V(1:length(V)-1))]; N=[N fliplr(N(1:length(N)-1))]; M=[M fliplr(M(1:length(M)-1))]; %redefine theta so that it goes from 0 to 180 degrees theta=(0:pi/100:pi);

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13.2 beam1

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%Plot the shear diagram %The axislimit=... commands are there so that I can %tell Matlab to do a polar plot with a radial axis starting from %-V to +V, instead of 0 to +V. %The tickmarks=... command makes it so that the tick labels are nice round numbers %Also note that we are not using the command polar, but instead we're using %polarhg. Polarhg is a more flexible version of polar, but it doesn't come %standard with Matlab subplot(221) axislimits=max(abs(V))*1.2; axislimits=ceil(axislimits/10.^floor(log10(axislimits)))*10.^floor(log10(axislimits))*[-1 1]; tickmarks=(axislimits(1):2*10^floor(log10(axislimits(2))):axislimits(2)); polarhg([theta;theta],[V;0*V],'rlim',axislimits,'rtick',tickmarks); title('Shear Force') %plot the Axial force diagram subplot(222) axislimits=max(abs(N)); axislimits=ceil(axislimits/10.^floor(log10(axislimits)))*10.^floor(log10(axislimits))*[-1,1]; tickmarks=(axislimits(1):2*10^floor(log10(axislimits(2))):axislimits(2)); polarhg([theta;theta],[N;0*N],'rlim',axislimits,'rtick',tickmarks); title('Axial Force') %plot the Moment diagram subplot(223) axislimits=max(abs(M)); axislimits=ceil(axislimits/10.^floor(log10(axislimits)))*10.^floor(log10(axislimits))*[-1,1]; tickmarks=(axislimits(1):2*10^floor(log10(axislimits(2))):axislimits(2)); polarhg([theta;theta],[-M;0*M],'rlim',axislimits,'rtick',tickmarks); title('Bending Moment')

13.2 beam1

13.2.1 Description

SUBDIRECTORY : beam1 SYNOPSIS: shear and bending moment diagram for a simple beam (Structures) DESCRIPTION: BMdemo.m gets the bending moment and de ection for a simply supported beam subjected to either a point load or a distributed load.

CONTENTS:

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13{10 Shear Force 200 90 60 120 0

150

30

−200

180

0

210

330 240

270

300

Axial Force 400 90 60 120 200 0 150 −200 −400 180 210

30 0 330

240

270

300

Bending Moment 3000 90 120 60 1000 150

−1000 −3000

180 210

30 0 330

240

270

300

Figure 13.6: Sample output for arches.m using r = 100 and != 2 BMdemo.m M2deflection.m P2M.m P2V.m w2M.m w2V.m

Written by: Brian Rose

the main program converts bending moments to de ections by integration converts point loads to moment using a formula converts point loads to shear using a formula converts uniform loads to moment using a formula converts uniform loads to shear using a formula

13.2.2 Listing 13.2.2.1

BMdemo.m

Equations used for calculation of shear, bending moment, and de ection diagrams:

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13{11

Draft

13.2 beam1

Figure 13.7: Simply Supported Beams with Various Loadings

 Pointload

R1 = Pb L

R2 = Pa L

M = R1 x

for (x < a) for (x  a)

= R2 (L - x)

V = R1

for (x < a)

= ;R2 ;

for (x  a)

v00 = MEI(i) = v(i ; 1) ; 2dxv(2i) + (i + 1)

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13{12

SAMPLES of MATLAB PROGRAMS  Continuous Load

R1 = 2!bL (2a + b)

R2 = 2!bL (2c + b) M = R1 x

for (x < a)

= R1 x ; !2 (x ; a)

for (a  x  a + b) for (x  a + b)

= R2 (L - x)

V = R1

for (x < a)

= (x ; a)(;bR2 ; R1 ) + R1

= ;R2

for (a  x  a + b) for (x  a + b)

v00 = MEI(i) = v(i ; 1) ; 2dxv(2i) + (i + 1) BMdemo.m le starts here: %BMdemo %this script gets the bending moment and %deflection for a simply supported beam subjected to either %a point load or a distributed load %prompt the user for the load type choice=input('Analyze for point load (1) or a uniform load (2) ? '); %clear the screen clc %do the point load case if choice==1 %prompt the user for some parameters P=input('Enter P: '); a=input('Enter a: '); L=input('Enter L: '); E=input('Enter E: '); I=input('Enter I: '); number_points=input('Enter the number of discretization points: '); %calculate the mesh size

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13.2 beam1

13{13

dx=L/(number_points-1); %get the BM M=P2M(P,a,L,dx); %get the shear V=P2V(P,a,L,dx); %get the deflection v=M2deflection(M,E,I,dx); %do the distributed load case elseif choice==2 %prompt the user for some parameters w=input('Enter w: '); a=input('Enter a: '); b=input('Enter b: '); L=input('Enter L: '); E=input('Enter E: '); I=input('Enter I: '); number_points=input('Enter the number of discretization points: '); %calculate the mesh size dx=L/(number_points-1); %get the deflection M=w2M(w,a,b,L,dx); %get the Shear V=w2V(w,a,b,L,dx); %get the deflection v=M2deflection(M,E,I,dx); end %plot the BM and deflection %make a vector for location along beam x=(0:dx:L); %plot the BM subplot(311) plot(x,M) ylabel('Bending Moment') grid %plot the Shear subplot (312) plot(x,V) ylabel('Shear') grid

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%plot the deflection subplot(313) plot(x,v) ylabel('Displacement') grid

Shear

Bending Moment

4

6

x 10

4 2 0 0 4 x 10 1

2

4

6

8

10

12

2

4

6

8

10

12

2

4

6

8

10

12

0

−1 0 0

Displacement

Draft

13{14

−0.5 0

Figure 13.8: Sample output of BMdemo.m with a pointload using P = 20 kN, a = 6 m, L = 12 m, E = 200 GPa, I = 6e6 mm4 , and 12 discretization points.

13.2.2.2

M2deflection.m

function v=M2deflection(M,E,I,dx) %M2DEFLECTION(M,E,I,dx) returns the deflection of a simply %supported beam, given the descritized moment, M; %E, Young's modulus; I, moment of inertia; and dx, %the mesh size. % %This algorithm integrates the bending moment twice %using the central difference method. %The second derivative is defined as %

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13.2 beam1

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% v(i-1) - 2v(i) + (i+1) %v''= ---------------------% dx^2 % % %Using the central difference method we can w%The central difference method will take on the form Av=b.rite % % v(i-1) - 2v(i) + (i+1) %M(i)/EI = ---------------------% dx^2 % %If you write the above equation for every mesh point %you get the following matrix equation. % % -------% | 1 -2 1 | | v(-1) | | M(0) | % | 1 -2 1 | | v(0) | | M(1) | % | 1 -2 1 | | v(1) | | M(2) | % | . | | v(2) | | . | %1/dx^2 | . | | . | = 1/EI | . | % | . | | . | | . | % | 1 -2 1 | | . | | M(n) | % | 1 | | v(n) | | 0 | % | 1 | | v(n+1)| | 0 | % -------% % |________________________________| |_______| |______| % A v b % %v(-1) and v(n+1) are known as the phantom points,and %they do not represent any real displacements. The last two %lines embody the boundary conditions: v(0)=0 and v(n)=0. By %solving for v in the matrix equation Av=b, you will obtain %the displacements, including the phantom points. %get the number of discretization points m=length(M); %make a diagonals for the matrix firstdiagonal=ones(m+2,1)/dx^2; seconddiagonal=-2*ones(m+1,1)/dx^2; thirddiagonal=ones(m,1)/dx^2; %make a tridiagonal matrix of dimensions (m+2,m+2) A=diag(firstdiagonal,0)+diag(seconddiagonal,1)+diag(thirddiagonal,2); %replace the last row with a bunch of zeros A(m+1:m+2,:)=zeros(2,m+2); %now put ones in the appropriate places A(m+1,2)=1; A(m+2,m+1)=1;

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13{16

SAMPLES of MATLAB PROGRAMS

%make the vector b b=M/(E*I); b(m+1)=0; b(m+2)=0; %make sure that b is a column vector if size(b,1)==1 b=b'; end %solve the system of equations v=(A\b); %discard the phantom points v=v(2:m+1);

13.2.2.3

P2M.m

function M=P2M(P,a,L,dx) %FUNCTION M=P2M(P,A,L,dx) returns the bending moment diagram for a %point load on a simply supported beam % % |P % | % \|/ % __________________________ % /\ /\ % |----A----| % |------------L------------| % %

|-------B-------| |---> x

%get b b=L-a; %get the reactions R1=P*b/L; R2=P*a/L; %make a vector x x=(0:dx:L); %get the moment M=R1*(x=a).*(L-x); %the above statement is a shorthand way of saying %for i=1:length(x) % if x(i) < a

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13.2 beam1

13{17

% M(i)=R1*x(i); % elseif x(i) >= a % M(i)=R2*(L-x(i)); % end %end %Matlab is very fast at evaluating matrix operations %like the shorthand command for M. Matlab is not so fast %at doing if statements and for loops.

13.2.2.4

P2V.m

function V=P2V(P,a,L,dx) %FUNCTION V=P2V(P,A,L,dx) returns the Shear diagram for a %point load on a simply supported beam % % |P % | % \|/ % __________________________ % /\ /\ % |----A----| % |------------L------------| % %

|-------B-------| |---> x

%get b b=L-a; %get the reactions R1=P*b/L; R2=P*a/L; %make a vector x x=(0:dx:L); %get the shear V=(x=a)*R2; %the above statement is a shorthand way of saying %for i=1:length(x) % if x(i) < a % V(i)=R1; % elseif x(i) >= a % V(i)=-R2; % end %end %Matlab is very fast at evaluating matrix operations

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13{18

SAMPLES of MATLAB PROGRAMS

%like the shorthand command for M. Matlab is not so fast %at doing if statements and for loops.

13.2.2.5

w2M.m

function M=w2M(w,a,b,L,dx) %FUNCTION M=w2M(W,a,b,L,dx) returns the bending moment diagram for a %distributed load on a simply supported beam % % % _______ % ||||||| w % __________________________ % /\ /\ % |--A--|--B--| % |------------L------------| % %

|---> x |--------C----|

%get c c=L-a-b; %get the reactions R1=w*b/2/L*(2*c+b); R2=w*b/2/L*(2*a+b); %make a vector x x=(0:dx:L); %get the bending moment M=(x=a)&(x<(a+b))).*(R1*x-w/2*(x-a).^2) + (x>=(a+b)).*(R2*(L-x)); %the above statement is a shorthand way of saying %for i=1:length(x) % if x(i) < a % M(i)=(x(i)*R1); % elseif (x(i) >= a) &| (x(i) < (a+b)) % M(i)=(R1*x(i)-w/2*(x(i)-a).^2); % elseif (x(i) >= (a+b)) % M(i)=(R2*(L-x(i))); % end %end %Matlab is very fast at evaluating matrix operations %like the shorthand command for M. Matlab is not so fast %at doing if statements and for loops.

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13.2 beam1 13.2.2.6

13{19

w2V.m

function V=w2V(w,a,b,L,dx) %FUNCTION V=w2V(W,a,b,L,dx) returns the shear diagram for a %distributed load on a simply supported beam % % % _______ % ||||||| w % __________________________ % /\ /\ % |--A--|--B--| % |------------L------------| % %

|---> x |--------C----|

%get c c=L-a-b; %get the reactions R1=w*b/2/L*(2*c+b); R2=w*b/2/L*(2*a+b); %make a vector x x=(0:dx:L); %get the shear V=(x=a)&(x<(a+b))).*((x-a)*(-R2-R1)/b+R1) - (x>=(a+b))*R2; %the above statement is a shorthand way of saying %for i=1:length(x) % if x(i) < a % V(i)=R1 % elseif (x(i) >= a) & (x(i) < (a+b)) % V(i)=(x-a)*(-R2-R1)/b+R1 % elseif (x(i) >= (a+b)) % V(i)=-R2 % end %end %Matlab is very fast at evaluating matrix operations %like the shorthand command for M. Matlab is not so fast %at doing if statements and for loops.

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13{20

SAMPLES of MATLAB PROGRAMS

13.3 beam2

13.3.1 Description

SUBDIRECTORY: beam2 SYNOPSIS: principle stress plots in a simple beam under midspan point load (Structures) DESCRIPTION: beam under load.m plots the principle stresses in a prismatic rectangular beam under uniform loads, with simple supports, assuming plane stress conditions.

CONTENTS:

beam under load.m

main program

Written by: Brian Rose

13.3.2 Listing 13.3.2.1

beam under load.m

Figure 13.9: Beam Under Continuous Load Major formulas used:

M = !2 x(L ; x) V = !9 L2 ; x) Q = b( d ; y)( y + d ) 2

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2

4

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13.3 beam2

13{21

Draft

xx = ; My I V Q  = Ib

s

2 1;2 = 2xx  4xx +  2

beam under load.m le starts here: %script beam_under_load will plot the principle stresses in a prismatic %rectangular beam under uniform loads, with simple supports. We will %assume plane stress conditions. %beam is depth d, width b, length L and has uniform load of w %define dimensions b=4; d=12; L=20*12; %define the uniform load w=2; %get some cross sectional properties A=b*d; I=b*d^3/12; %set up x and y coordinate system numberx=100; numbery=40; x=0:L/numberx:L; y=-d/2:d/numbery:d/2; %get a mesh of x and y's [x,y]=meshgrid(x,y); %calculate moment and shear M=w/2*x.*(L-x); V=w*L/2-w*x; %calculate the first moment of intertia Q=b*(d/2-y).*(y/2+d/4); %calculate bending stress and shear stress for each point in the beam sb=-M.*y/I; sv=V.*Q/I/b; %calculate the principle and maximum shear stresses tmax=sqrt(sb.^2/4+sv.^2); p1=sb/2+tmax; p2=sb/2-tmax;

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13{22

SAMPLES of MATLAB PROGRAMS

%set the colorscale for the plots %for more details type "help color" and "help colormap" colormap(jet) %plot results %the shading interp takes away ugly gridlines subplot(311) pcolor(x,y,p1) shading interp title('Principle Stress 1') colorbar subplot(312) pcolor(x,y,p2) shading interp title('Principle Stress 2') colorbar subplot(313) pcolor(x,y,tmax) shading interp title('Maximum Shear Stress') colorbar %for those of you who appreciate psychedelic stuff try this %spinmap(20)

13.4 boussinesq 13.4.1 Description

SUBDIRECTORY: boussinesq SYNOPSIS: stress plot for semi-in nite domain under a point load (Geotech) DESCRIPTION: boussinesq.m plots the stress distribution of a point load on a semi-in nite half space

CONTENTS:

main program Written by: Brian Rose boussinesq.m

13.4.2 Listing 13.4.2.1

boussinesq.m

Main formula: Victor E. Saouma

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13.4 boussinesq

13{23 Principle Stress 1 150

5

100 0 50 −5 0

50

100 150 Principle Stress 2

0

200

0

5

−50 0 −100 −5 0

50

100 150 Maximum Shear Stress

−150

200

5 60 40

0

20 −5 0

50

100

150

Figure 13.10: Sample output of with ! = 2; b = 4; d = 12; and L = 20.

0

200

principle stresses

for

beam under load.m

 )3 y = 3P2(sin R2 boussinesq.m le starts here: %script boussinesq plots the stress distribution %of a point load on a semi-infinite half space % % % | P % | % | % \|/ %-----------------------------% % --->x % | % | % |/ y %set up a radial grid %define the r's and theta's for our grid as vectors r=0.01:.002:0.1; theta=0:pi/20:pi; P=100;

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SAMPLES of MATLAB PROGRAMS

Draft

13{24

Figure 13.11: Point Load on Semi-In nite Domain

%Now define a R and THETA for each grid point %this means that R and THETA are both matrices [R,THETA]=meshgrid(r,theta); %get the stress in the y direction stressy=3*P/(2*pi)*(sin(THETA)).^3./R.^2; %take R and THETA and convert them to cartesian coordinates [x,y] = pol2cart(THETA,R); %plot the results (note the -y is so that %it plots upside down pcolor(x,-y,stressy)

%note that since the solution blows up at R=0 and theta=0, %we have left out the solution for that point.

13.5 dodgecity

13.5.1 Description

SUBDIRECTORY: dodgecity SYNOPSIS: statistical analysis of wind power in Dodge City (Energy Management) DESCRIPTION: wind.m makes a histogram of windspeed and does some simple statistics on it. Accompanies Jan Kreider's handout "Wind Power Assessment"

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13.5 dodgecity

13{25

Draft

0 −0.01 −0.02 −0.03 −0.04 −0.05 −0.06 −0.07 −0.08 −0.09 −0.1 −0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Figure 13.12: Stress plot for point Load on Semi-In nite Domain

CONTENTS:

data le data le data le data le data le data le data le data le data le data le data le data le main program Written by: Brian Rose apr aug dec feb jan jul jun mar may nov oct sep wind.m

13.5.2 Listing 13.5.2.1

wind.m

Equation used:

P = 12 v3 wind.m le starts here: Victor E. Saouma

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13{26

SAMPLES of MATLAB PROGRAMS

%Script wind.m %Makes a histogram of windspeed and does some simple statistics` %on it. Accompanies Jan Kreider's handout Wind Power Assessment %load data for all 12 months %use the -ascii extension to tell Matlab that we're reading ascii %files. Matlab will take the data in the file jan and stuff it into %a matrix called jan. Same goes for all the other months load load load load load load load load load load load load

jan feb mar apr may jun jul aug sep oct nov dec

-ascii -ascii -ascii -ascii -ascii -ascii -ascii -ascii -ascii -ascii -ascii -ascii

%Concatentate them into one big matrix called data data=[jan;feb;mar;apr;may;jun;jul;aug;sep;oct;nov;dec]; %now pull out the wind speed from the 7th column of data V=data(:,7); %specify the bins %the bin goes from 0 to the maximum windspeed rounded up to the %next whole number. Try typing 'help ceil' in the matlab window bin=(0:ceil(max(V))); %sort the velocity into bins hours=hist(V,bin); %make a histogram %on your own try doing bar(bin,hours) and bar(midpoint,hours). %the results will be slightly different. midpoint=bin+0.5; bar(midpoint,hours) title('Dodge City') xlabel('Wind Velocity (m/s)') ylabel('Hours') %Compute power. %Note here that the we use .^3 instead of ^3 %because the former is an array operation (means operate %element by element), whereas the latter is a matrix operation. rho=1.124; Power=rho/2*(V.^3);

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%do some statistics on power mean_power=mean(Power) max_power=max(Power) std_power=std(Power) median_power=median(Power)

Dodge City 1400

1200

1000

800

Hours

Draft

13.6 eectiveL

600

400

200

0 0

5

10

15 Wind Velocity (m/s)

20

25

30

Figure 13.13: Sample histogram of windspeed Simple Statistics calculated from wind data: mean_power =

194.0820

max_power =

8.7812e+03

std_power =

311.3429

median_power =

70.2500

13.6 eectiveL

13.6.1 Description

SUBDIRECTORY: eectiveL Victor E. Saouma

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13{28

SAMPLES of MATLAB PROGRAMS

SYNOPSIS: eective length of a sidesway inhibited column of a building (Structures) DESCRIPTION: eectiveL.m returns the eective length factor, K given the G values for the top and bottom of a sidesway inhibited column.

CONTENTS:

effectiveL.m

main function. Note, not a scipt.

Recall that the Euler buckling load was derived for a pinned column. In many cases, a column will have dierent boundary conditions. It can be shown that in all cases, the buckling load would be given by

2 Pcr =  EI2 (KL) where K is called eective length factor, and KL is the eective length. and 2 cr =  E2 KL

(13.20)

(13.21)

r

The ratio KL r is termed the slenderness ratio. The eective length, can only be determined by numerical or approximate methods, and is the distance between two adjacent (real or virtual) in ection points, Fig. 13.14, 13.15 P

P P

P

KL = 0.7L KL = L

L

KL =

1 L L 2

KL < L

L

P P

P

P (a) End rotations unrestrained

(b) End rotations fully restrained

(c) One end restrained, other unrestrained

(d) Partially restrained at each end

Figure 13.14: Column Eective Lengths Victor E. Saouma

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13.6 eectiveL

Draft

13{29

∆ P

P

P

P

KL 2 L

L 0.7L
KL>2L

∆

(a) Braced frame, hinged base P

P

L

0.5L
(b) Unbraced frame, hinged base P

P

L L
(c) Braced frame, fixed base

(d) Unbraced frame, fixed base

Figure 13.15: Frame Eective Lengths The most widely used charts for the eective length determination are those produced by the Structural Stability Research Council. The alignment chart, for an individual column, Fig. 13.16 is shown in Fig. 13.17. It should be noted that this chart assumes that all members are still in the elastic range. The use of the alignment chart involces computing G at each end of the column using the following formula P Ic

Ga = P LIgc Lg

(13.22)

where Ga is the stiness at end a of the column, Ic, Ig are the moment of inertias of the columns and girders respectively. The summation must include only those members which are rigidly connected to that joint and lying in the plane for which buckling is being considered. Hence, once Ga and Gb are determined, those values are connected by a straight line in the appropriate chart, and k is the point of intersection of that line with the midle axis. Alternatively :-)

GAGB   2 +  GA + GB  1 ; =K  + 2 tan =2K = 1 4 K 2 tan =K =K

(13.23)

(Ref. McGuire P. 467). Victor E. Saouma

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SAMPLES of MATLAB PROGRAMS

Draft

13{30

Figure 13.16: Column Eective Length in a Frame

13.6.2 Listing 13.6.2.1

effectiveL.m

function K=effectiveL(Ga,Gb) %K=effectiveL (Gtop, Gbot) returns the effective length %factor, K given the G values for the top and bottom of a %sidesway inhibited column. % %K can be found by solving the following equation numerically % % _ _ _ _ 2 _ _ _ _ %| Ga Gb | | pi | | Ga+Gb | | pi/K | 2 tan(pi/2K) %| ----- | | ---- | + |-------- | | 1 - --------- | + ------------- = 1 %| 4 | | K | | 2 | | tan(pi/K) | pi/K % % %This function uses a root finding scheme, namely the bisection %method to find the root of the following equation. % % _ _ _ _ 2 _ _ _ _ % | Ga Gb | | pi | | Ga+Gb | | pi/K | 2 tan(pi/2K) %R = | ----- | | ---- | + |-------- | | 1 - --------- | + ------------- - 1 % | 4 | | K | | 2 | | tan(pi/K) | pi/K % %

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13{31

Draft

13.6 eectiveL

Sidesway Inhibited 150.

Ga

10. 5. 3. 2. 1. 0.8 0.7 0.6 0.5 0.4 0.3 0.2

K 1.0 0.9 0.8 0.7

0.6

0.1 0.

Sidesway Uninhibited Gb

1 50.

10. 5. 3. 2. 1. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.5

0.

1

Ga

K

100. 50. 30. 20. 10. 9. 8. 7. 6. 5. 4. 3. 2.

1 20. 10. 5. 4. 3.

2.

1.5

1. 0

Gb

1

100. 50. 30. 20. 10. 9. 8. 7. 6. 5. 4. 3. 2. 1.

1.

0

Figure 13.17: Standard Alignment Chart (AISC)

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13{32

SAMPLES of MATLAB PROGRAMS

% note: R is the residual and we want to find K such that R=0; % %For this method, we make two guesses of K, Klower and Kupper, that %bound the correct K. Then, we estimate K as the average of the %two initial guesses. Next we figure out if our if the estimated %K and Kupper or the estimated K and Klower bound the solution. %Then with the two new bounds, we start over again. % %More ambitious people can use the Matlab function fzero

%define upper and lower bounding estimates of K Kupper=.98; Klower=.52; %define the tolerance for rootfinding tolerance=1e-3; %get the residual for the upper and lower bounding estimates Rupper=Ga*Gb/4*(pi/Kupper)^2 + (Ga+Gb)/2*(1-pi/Kupper/tan(pi/Kupper))+2*tan(pi/2/Kupper)/(pi/Kupper)-1; Rlower=Ga*Gb/4*(pi/Klower)^2 + (Ga+Gb)/2*(1-pi/Klower/tan(pi/Klower))+2*tan(pi/2/Klower)/(pi/Klower)-1; %set R to something big so that we can enter the loop R=100*tolerance; %loop until we find the root while abs(R)>tolerance %let our estimate of K be halfway between Kupper and Klower K=(Kupper+Klower)/2; %calculate the residual corresponding to K R=Ga*Gb/4*(pi/K)^2 + (Ga+Gb)/2*(1-pi/K/tan(pi/K))+2*tan(pi/2/K)/(pi/K)-1; %if K and Kupper bound the root let Klower=K if R>0 Klower=K; Rlower=R; %if K and Klower bound the root let Kupper=K else Kupper=K; Rupper=R; end end

13.7 eigenvalues 13.7.1 Description

SUBDIRECTORY: eigenvalues Victor E. Saouma

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13.7 eigenvalues

13{33

SYNOPSIS: Principle stresses and directions of a 3-Dimensional stress element DESCRIPTION: eigenvalues.m prompts the user for stress components of a 3-dimensional stress element and then calculates the principal stresses and directions.

CONTENTS:

main program Written by: Brian Rose eigenvalues.m

13.7.2 Listing 13.7.2.1

eigenvalues.m

Figure 13.18: 3-Dimensional Stress Element %The script "eigenanalysis" prompts the user for stress components %in 3-D. Then is calculates the principle stresses and the %principle directions %clear the screen

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SAMPLES of MATLAB PROGRAMS

clc %prompt user for some numbers %note that I can define the elements of the matrix %without even defining the size of the matrix before hand Stress(1,1)=input('Enter normal stress on x face: '); Stress(2,2)=input('Enter normal stress on y face: '); Stress(3,3)=input('Enter normal stress on z face: '); Stress(1,2)=input('Enter shear stress xy: '); Stress(1,3)=input('Enter shear stress xz: '); Stress(2,3)=input('Enter shear stress yz: '); %Make the stress matrix symmetric %First pull out the upper triangular part of the matrix (without the diagonal). %Second transpose the part we pulled out and add it to the Stress matrix %for more details on the command triu, try typing "help triu" Stress temp=triu(Stress,1) Stress=Stress+temp' %Now do the eigenanalysis [eigenvectors,principle_stress]=eig(Stress); %Display the results as well as the stress matrix %clc disp('The stress tensor is') disp(Stress) disp(' ') disp(' ') for i=1:3 disp('Principle stress number') disp(i) disp(principle_stress(i,i)) disp('Its unit normal is:') disp(eigenvectors(:,1)') disp(' ') disp(' ') end Sample run: >> eigenvalues Enter normal stress on x face: 3 Enter normal stress on y face: 4 Enter normal stress on z face: 1 Enter shear stress xy:

2

Enter shear stress xz:

5

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13.7 eigenvalues

13{35

Enter shear stress yz:

8

Stress = 3 0 0

2 4 0

5 8 1

2 0 0

5 8 0

2 4 8

5 8 1

temp = 0 0 0

Stress = 3 2 5 The stress 3 2 5

tensor is 2 5 4 8 8 1

Principle stress number 1 1.4822 Its unit normal is: -0.8460 0.5314

0.0442

Principle stress number 2 -6.4509 Its unit normal is: -0.8460 0.5314

0.0442

Principle stress number 3

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13{36

SAMPLES of MATLAB PROGRAMS

12.9688 Its unit normal is: -0.8460 0.5314

0.0442

13.8 in ltration 13.8.1 Desctiption

SUBDIRECTORY: in ltration SYNOPSIS: in ltration of water in soil where ponding has occured (Hydrology) DESCRIPTION: in ltration.m calculates the the depth of in ltration of water in a soil medium after ponding has occured. The problem is taken from "Applied Hydrology" by V.Chow, D. Maidment, and L. Mays page 120, example 4.4.2.

CONTENTS:

infiltration.m

main program

Written by: Brian Rose

13.8.2 Listing

13.8.2.1 in ltration.m Equation used:

F = Fp +
 ln[

 ++ FF ] + k(t ; tp ) p in ltration.m le starts here: %script infiltration will calculate the the depth of infiltration %of water in a soil medium after ponding has occured. The problem is %taken from "Applied Hydrology" by V.Chow, D. Maidment, and L. Mays %page 120, example 4.4.2. % %The equations is % % psiDtheta + F % F - Fp - (psiDtheta)*ln ---------------- = K (t-tp). % psiDtheta + Fp % % %We will solve this equation for F by rewriting the above as % % psiDtheta + F

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13.9 montecarlo

13{37

% F = Fp + (psiDtheta)*ln ---------------- + K (t-tp). % psiDtheta + Fp % %First we will guess an F. Then we will plug this F into the right hand %side of the equation. This will give us a new F. We will then %plug this new F into the right hand side and so on. %define the contants Fp=0.85 psiDtheta=5.68 K=0.65 tp=.17 t=1 %make an initial guess of F F=.2; %iterate for i=1:1000 F=Fp+psiDtheta*log((psiDtheta+F)/(psiDtheta+Fp))+K*(t-tp); end %display F F Sample run: >> infiltration Fp =

0.8500

psiDtheta = K =

5.6800

0.6500

tp =

0.1700

t =

1

F =

3.0176

13.9 montecarlo 13.9.1 Description

SUBDIRECTORY: montecarlo SYNOPSIS: montecarlo simulation of a simple beam under midspan point load (Structures and Statistics)

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13{38

SAMPLES of MATLAB PROGRAMS

DESCRIPTION: montecarlo.m does a monte carlo simulation for a simply supported beam

under a point load at the midspan. The load, P and the yield stress, fy are normally distributed.

CONTENTS:

main program Written by: Brian Rose montecarlo.m

13.9.2 Listing 13.9.2.1

montecarlo.m

%script montecarlo does a monte carlo simulation for a simply %supported beam under a point load at the midspan %the load, P and the yield stress, fy are normally distributed %define a sample size samplesize=5000; %define the properties of the beam L=480; S=150; %define the mean and standard deviation for %yield stress and load meanfy=36 meanP=40 stdfy=1 stdP=1 %get a normally distributed set of stress values and %a normally distributed set of load values %note that randn gives me a standard normal %distribution so I have to convert to a normal %distribution fy=meanfy+stdfy^2*randn(samplesize,1); P=meanP+stdP^2*randn(samplesize,1); %calculate the max stress in a simple span beam loaded in the center M=P*L/4; f=M/S; %define X=capacity/demand X=fy./f; %plot f and fy subplot(211) bin=(30:.1:42); plot(bin,hist(f,bin)/samplesize,bin,hist(fy,bin)/samplesize) legend('computed stress','yield stress');

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xlabel ('stress (ksi)') ylabel('PDF') grid %plot X subplot(212) bin=(-0.5:0.1:2); plot(bin,hist(log(X),bin)/samplesize); legend('computed stress','yield stress'); xlabel ('ln of C/D (ksi)') ylabel('PDF of ln(C/D)') legend off grid

%get the safety index %note that log denotes the natural log beta=mean(log(X))/std(log(X)) %get the probability of failure failureP=sum(f>fy)/length(f)

0.06 0.05 computed stress yield stress

PDF

0.04 0.03 0.02 0.01 0 30

32

34

36 stress (ksi)

38

40

42

0.8

PDF of ln(C/D)

Draft

13.9 montecarlo

0.6 0.4 0.2 0 −0.5

0

0.5 1 ln of C/D (ksi)

1.5

2

Figure 13.19: Sample output for a monte carlo simulation for a simply supported beam under a point load at the midspan.

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SAMPLES of MATLAB PROGRAMS

13.10 3dplot

13.10.1 Description

SUBDIRECTORY : 3Dplots SYNOPSIS: takes elevation readings and plots them in 3D (Surveying) DESCRIPTION: pix.m will plot the elevation data for some mountains. The elevations are

contained in the le mtns.mat. Matrices x and y contains x and y coordinates of each elevation reading. Matrix z contains the elevations. The rst part of the code sets up buttons that allows the user to change the perspective of the plots: the rst slider changes the azimuth and the second slider changes the elevation. The second part of the code cylces through each of type of plot.

CONTENTS:

creates the a ctitius elevations readings for a mountain contains the elevation reading of mountains the main program takes the elevation readings and smooths them out Written by: Brian Rose cont.m mtns.mat pix.m smooth.m

13.10.2 Listing 13.10.2.1

cont.m

[x,y]=meshdom(0:.1:2*pi,0:.1:2*pi); a=rand(1,5); b=rand(1,5); a=3.5.^(a+.1); b=3.5.^(b+.1); z=zeros(size(x)); for i=1:size(a,2) z=z+log(abs(sin(a(i)*x)))+log(abs(cos(b(i)*y))); end z=z+rand(size(z)); mesh(x,y,smooth(z,2)) figure(1)

13.10.2.2

pix.m

%script pix.m %will plot the elevation data for some mountains. %The elevations are contained in the file mtns.mat. %Matrices x and y contains x and y coordinates of %each elevation reading. Matrix z contains the elevations. %The first part of the code sets up buttons that allows

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13.10 3dplot

13{41

%the user to change the perspective of the plots: the first %slider changes the azimuth and the second slider changes %the elevation. The second part of the code cylces through %each of type of plot. %clear screens, etc clear figure(1) clg %load the elevation data load mtns %set up the buttons for the 3D plots hAZ=uicontrol('style','slider','position',[.7 .95 .3 .05],... 'units','normalized','min',0,'max',360,... 'callback','[az,el]=view; az=get(gco,''val''); view(az,el);'); hEL=uicontrol('style','slider','position',[.7 .89 .3 .05],... 'units','normalized','min',0,'max',180,... 'callback','[az,el]=view; el=get(gco,''val''); view(az,el);'); hdef=uicontrol('style','pushbutton','position',[.88 .83 .12 .05],... 'units','normalized','String','Default',... 'callback','view(-37.5, 30)'); %do the 3D plots mesh(x,y,z) title('mesh(x,y,z)') xlabel('press any key to continue') pause meshz(x,y,z) title('meshz(x,y,z)') xlabel('press any key to continue') pause contour3(z) title('contour3(z)') xlabel('press any key to continue') pause meshc(x,y,z) title('meshc(x,y,z)') xlabel('press any key to continue') pause surf (x,y,z) title('surf (x,y,z)') pause surfc(x,y,z)

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13{42

SAMPLES of MATLAB PROGRAMS

title('surfc(x,y,z)') xlabel('press any key to continue') pause surfl(x,y,z) title('surfl(x,y,z)') xlabel('press any key to continue') pause waterfall (x,y,z) title('waterfall (x,y,z)') xlabel('press any key to continue') %clear the figure window clg %do the 2D plots contour(x,y,z) title('contour(x,y,z)') xlabel('press any key to continue') pause pcolor (x,y,z) title('pcolor (x,y,z)') colorbar xlabel('press any key to continue') pause

13.10.2.3

smooth.m

function newz=smooth(z,weight) [m,n]=size(z); if ~exist('weight') weight=5; end newz=z; for i=2:m-1 for j=2:n-1 newz(i,j)=(z(i-1,j-1)+ end end

z(i-1,j+1)+

z(i+1,j-1)+

z(i+1,j+1)+ weight*z(i,j))/(4+weight);

13.11 zec

13.11.1 Description

SUBDIRECTORY: zec Victor E. Saouma

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13.11 zec

13{43 mesh(x,y,z)

4

x 10 1.2

1.1

1

0.9

0.8 15000 15000

10000 10000 5000

5000 0

0

press any key to continue

Figure 13.20: Sample plot from pix.m

SYNOPSIS: statistical analysis for heating energy of a building (Energy Management) DESCRIPTION: zec.m does a linear regression to nd a relationship between CONTENTS: function that performs linear regression main program contains heat and temperature data Written by: Brian Rose linregress.m zec.m zecdata

13.11.2 Listing 13.11.2.1

linregress.m

function [p,Rsquared,sigmabeta1]=linregress(x,y) %[p,Rsquared,sigmabeta1]=linregress(x,y) performs a linear regression on x and y. %p is a vector where y=p(1)+p(2)*x. % %[p,Rsquared,sigmabeta1]=linregress(x,y) will return R^2 as well as the standard %deviation in p(2)

%make x and y into row vectors

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13{44

SAMPLES of MATLAB PROGRAMS

if size(x,1)<size(x,2) x=x'; end if size(y,1)<size(y,2) y=y'; end %print an error message is x and y are not the same lengths if size(x)~=size(y) disp('x and y need to be the same size') return end n=length(x) %linear regress. A contains a column of ones then the x values A=[ones(length(x),1) ,x]; %find p p=inv(A'*A)*A'*y; %get r, which is the residual r=y-(p(1)+p(2)*x);

%get SSE, SST, and Rsquared SSE=sum(r.^2); SST=sum((y-mean(y)).^2); sigmasquared=SSE/(length(y)-2); sigmasquaredbeta1=sigmasquared/sum((x-mean(x)).^2); sigmabeta1=sqrt(sigmasquaredbeta1); Rsquared=1-SSE/SST;

13.11.2.2

zec.m

Emperical linear relationship between heating energy and outdoor temperature: HeatEnergy = k1 + k2 Temperature

k1 and k2 may be found using regression features of Matlab. Regression improves as the correlation coecient(R-squared) value is found closer to '1.0'. zec.m le starts here: %script zec does a linear regression to find a relationship between %the heat energy and the outside temperature of a building. This script %uses data from the Zachry Engineering Center at Texas A&M %load the data

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13.12 Stress/Strain Programs

13{45

load -ascii zecdata %extract only the columns we need energy=zecdata(:,11); temperature=zecdata(:,5); %do a linear regression. You can use the built in function polyfit %but that will not give Rsquared or delta. Note that delta is %the standard deviation of the slope. [p,Rsquared,delta]=linregress(temperature,energy); k1=p(1) k2=p(2) Rsquared delta %plot the results %plot the raw data plot(temperature,energy,'ro') hold on %plot the equation obtained from the linear regression x=[min(temperature) max(temperature)]; y=k1+k2*x; plot(x,y,'g-') hold off grid legend('raw data','linear regression') ylabel('Heat energy (MMBtu/hr)') xlabel('Temperature (F)') title('Heating vs. Temperature Plot')

Sample run of it zec.m: >> zec n =

744

k1 =

7.6999

k2 =

-0.0764

Rsquared = delta =

0.6639

0.0020

13.12 Stress/Strain Programs 13.12.1 Description

SUBDIRECTORY : Stress/Strain Programs Victor E. Saouma

Computing Literacy for Undergraduate Engineering Students

13{46

SAMPLES of MATLAB PROGRAMS

Draft

Heating vs. Temperature Plot 6

Heat energy (MMBtu/hr)

5

4

3

2

1

raw data linear regression

0 30

35

40

45

50 55 Temperature (F)

60

65

70

75

Figure 13.21: Sample output for zec.m

SYNOPSIS: stress and strain analysis for 2-D and 3-D elements DESCRIPTION: Two main startup commands are used for dierent analysis: 1. stressanalysis.m is a program which provides a menu driven interface to execute 2-D and 3-D stress/strain calculations. 2. stressdecomp.m creates a menu for 3-Dimensional analysis of stresses including options to Input Stresses, nd Normal and Shear stress components, Principal Invariants, plot the Principal Mohr's Circle with principal stresses and absolute max shear stresses, nd Mean and Deviatoric stresses, nd deviatoric principal invariants, and plot the Deviatoric principal Mohr's circles.

CONTENTS:

Program startup commands. stressanalysis.m - Start stress/strain analysis program. stressdecomp.m - Start stress decomposition program.

Print commands. elemprint.m circprint.m

- Save figure to file and ask to print. - Save Mohr's Circles and data to files.

2-Dimensional Stress/Strain Analysis.

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13.12 Stress/Strain Programs stress2dmenu.m planestress2d.m planestrain2d.m definitions.m principalangle.m maxshearangle.m plotelement.m plotprincipal.m plotmaxshear.m strainplotelement.m subplots2d.m transform.m rotelement2d.m plotcircle.m mohrcircle.m

13{47 -

Plane stress/plane strain menu options. Plane stress menu. Plane strain menu. Definitions of plane stress and plane strain. Principal stresses/strains and directions. Maximum shear stress/strain and directions. Plot of unit stress element. Plot of stress/strain principal values. Plot of stress/strain maximum shear values. Plot of unit strain element. Plot original, principal, & max shear elements. Transformation equations. Rotate element by an angle theta. Plot 2-D Mohr's Circle. 2-D Mohr's Circle with rotated angle.

3-Dimensional Stress/Strain Analysis. stress3dmenu.m - Stress/Strain menu options. stress3d.m - Stress menu. strain3d.m - Strain menu. plotelement3d.m - Plot of unit stress/strain element. plotcircle3d.m - Plot 3-D Mohr's Circle. Generalized 2-D and 3-D & Misc. functions. hlp.m - General help to use menus in stress programs. arrow.m - Draw a line with an arrowhead. parammenu.m - Menu driven way of entering variables. principal.m - Find Principal stresses/strains. straintostress.m - Convert from strain to stress. stresstostrain.m - Convert from stress to strain. 3-D Stress Decomposition. mkmatrix.m normcomp.m shrcomp.m invariant.m mndevcomp.m devinvariant.m -

Build a 3-D stress matrix. Normal stress component. Shear stress component. Principal invariants. Mean and Deviatoric stresses. Deviatoric principal invariants.

Written by: Melissa Holland

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SAMPLES of MATLAB PROGRAMS

13.12.2 Sample Output:

How to access program: At Matlab prompt type: p=path path(p,'/bechtel/users1/graduate/hollanmm/matlab/') stressanalysis or stressdecomp Typing stressanalysis will cause the following menu to appear:

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13.12 Stress/Strain Programs

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Figure 13.22: First menu encountered after starting stressanalysis Next, choosing 2-Dimensional Analysis from this menu brings up the next menu.

Figure 13.23: Menu encountered after clicking on 2-Dimensional Analysis button Finally, choosing Plane Stress button brings up the next menu(see next gure).

Figure 13.24: Menu encountered after clicking on Plane Stress button. These menus allow easy movement through the program. Clicking on other buttons in the previous menus will bring up other menus or ask for dierent information at the matlab prompt. Following are some examples which used stressanalysis program menus and input prompts to solve them. Victor E. Saouma

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SAMPLES of MATLAB PROGRAMS

13.12.2.1 Example 1: Given: A state of plane strain exists at a point in a member, with the nonzero strain components xx = ;0:200, yy = 0:040, xy = ;0:900. Objective:

(a) Determine the principle strains in the (x,y) plane. (b) Determine the maximum shear strain in the (x,y) plane. (c) Show schematically the deformed shape of a rectangular element with the original undeformed element. (d) Plot Mohr's circle of strain.

Results: 2-D ELEMENT DATA Original Values: XX YY XY

= -0.2 = 0.04 = -0.9

Principal Values: (a)

S1 S2

= 0.827965 = -0.987965

Maximum Shear Values:

(b)

Avg Max shear

= -0.08 = 1.81593

2-D MOHR'S CIRCLE DATA Inputs: P1 P2

= (-0.2,-0.9) = (0.04,0.9)

C

= (-0.08,0)

Tmax Tmin

= (-0.08,0.907965) = (-0.08,-0.907965)

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Principal Values: S1 S2

= (0.827965,0) = (-0.987965,0)

Principal Angle = 131.20 degrees

ORIGINAL Orientation

MAXIMUM SHEAR Orientation

PRINCIPAL Orientation

y y

y

x

x

x

Theta = 131.2 degrees

Theta = 86.2 degrees Theta = 0 degrees

Figure 13.25: Example 1(c): Original, Principal, Maximum Shear Stresses, and Deformed shapes. 2*Theta = 262.4 degrees 1

Tmax P2

0.8 0.6 0.4 Gamma xy/2

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13.12 Stress/Strain Programs

0.2 0

2*Theta

S2 C

S1 Ex

−0.2 −0.4 −0.6 −0.8 P1 Tmin

−1 −1

−0.5

0

0.5

1

Figure 13.26: Example 1(d): Mohr's circle of 2-D stress.

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SAMPLES of MATLAB PROGRAMS

13.12.2.2 Example 2: Given: A rectangular strain rosette is cemented to a free surface of a structural member

made of alluminum alloy(7075 T6/  = 0:33 and E = 72.0 GPa). Under load, the strain readings are a = 0:00250, b = 0:00140, and c = ;0:00125 from which it may be found that xx = 0:00250, yy = ;0:00125, and xy = 0:000775. Note: a free surface means zz = zx = zy = 0, therefore, we will treat as a 2-Dimensional stress case. However, eventhough the stress acts as if 2-Dimensional, strain is 3-Dimensional in this case.

Objective:

(a) Determine principal stresses. (b) Determine the orientation of the volume element on which the principal stresses in the plane of the rosette act. (c) Determine maximum shear stress. (d) Determine the orientation of the volume element on which max acts. (e) Plot Mohr's circle of stress.

Results: 2-D ELEMENT DATA Original Values: XX YY XY

= 1.68668e+08 = -3.43396e+07 = 4.19549e+07

Principal Values: (a)

S1 S2

= 1.76997e+08 = -4.26685e+07

Maximum Shear Values:

(c)

Avg Max shear

= 6.71642e+07 = 1.09833e+08

2-D MOHR'S CIRCLE DATA Inputs: P1 P2

Victor E. Saouma

= (1.68668e+08,4.19549e+07) = (-3.43396e+07,-4.19549e+07)

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13.12 Stress/Strain Programs

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C

= (6.71642e+07,0)

Tmax Tmin

= (6.71642e+07,1.09833e+08) = (6.71642e+07,-1.09833e+08)

Principal Values: S1 S2

= (1.76997e+08,0) = (-4.26685e+07,0)

Principal Angle = 11.23 degrees PRINCIPAL Orientation S2

ORIGINAL Orientation Sig y

MAXIMUM SHEAR Orientation Savg

Txy

y x

y

S1 x

Sig x

y x Tmax Theta = −33.77 degrees Savg

Theta = 0 degrees

Theta = 11.23 degrees

Figure 13.27: Example 2(b,d): Original, Principal and Maximum Shear Stresses.

13.12.2.3 Example 3: Given: Let the state of stress at a point be given by xx = ;120 MPa, yy = 140 MPa, zz = 66 MPa, xy = 45 MPa, yz = ;65 MPa, and zx = 25 MPa. Objective: (a) Determine the principal stresses. (b) Determine the maximum shear stress. (c) Plot Mohr's circle.

Results: 3-D MOHR'S CIRCLE DATA Inputs: X Y

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= -120 = 140

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SAMPLES of MATLAB PROGRAMS 2*Theta = 22.46 degrees

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8

x 10

Tmax 1

Tau xy

0.5

0

P1

2*Theta

S2 C

S1 Sigma x

P2

−0.5

−1 Tmin −5

0

5

10

15 7

x 10

Figure 13.28: Example 2(e): Mohr's circle of 2-D stress. Z XY XZ YZ

= = = =

66 45 25 -65

Principal Values: (a)

S1 S2 S3

= (180.207,0) = (40.0573,0) = (-134.264,0)

Maximum Shear Values: Tmax1 Tmax2 Tmax3 (b)

= (-47.1035,87.1608) = (22.9714,157.236) = (110.132,70.0748)

Tmax = 157.236

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13.12 Stress/Strain Programs

Tmax2 150

100

Tmax1 Tmax3

Tau xy

50

S3

0

S2

S1 Sigma x

−50

−100

−150

−150

−100

−50

0

50

100

150

200

Figure 13.29: Example 3(c): Mohr's circle of 3-D stress.

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SAMPLES of MATLAB PROGRAMS

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Part III

MODULE III (1 cr)Fortran 90

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Chapter 14

Day I: INTRODUCTION For a better understanding of fortran, this chapter will begin with a brief overview of the architecture of a computer, role of its software, and nally status of fortran. 2 The next section will be devoted to a brief review of fortran (with which the reader is asumed to have a certain familiarity), highlight those features which may not have been covered in an introductory course, and terminate with recommendations for programming style and documentation. 1

14.1 Introduction

14.1.1 Computer Architecture and Software 14.1.1.1 Computer Hardware

Digital computers are concerned with the processing of data1 abd their basic architecture has seen little if any fundamental changes since the conceptual model rst proposed by Von Neumann in a landmark paper in 1946:

3

Storage: of data and commands:  Data as well as commands are internally stored in binary form (0 or 1).  The most elementary unit of data is the bit.  A group of eight bits form a byte. A byte can have 28 = 256 dierent values, and is

1 Early

usually used to store a character (extended ASCII).  In most present day computers, four bytes form a word.  Early PC's had 16 bit word, most other computers have 32 bits, and supercomputers have 64 bits. on Engineers used extensively analog computers.

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14{2

Day I: INTRODUCTION  Storage is accomplished on any one of the following:

1. cache memory, extremely fast, it acts as a buer between the CPU and the core memory. 2. Core memory, very fast, usually around 8 Mb (106 ) bytes for PC's and at least 32 Mb for workstations. 3. Disk space, about 1 Gb. 4. Other peripherals such as magnetic tapes, CD-ROM, scanner, laser printer.  Most computers have a Virtual Memory, that is when core memory is full, data is temporarily stored on disk by the operating system. Central Processing Unit or CPU:  The heart of the computer is the CPU or central processing unit. It is the device which performs actual data processing.  For instance it may take the contents of two speci ed locations in memory, add them together and return the result to a third location.  Operands are temporarily stored in registers in the CPU.  The CPU has an arithmetic and a logic unit(s) to perform various basic operations (such as add, multiply, substract, divide, and others) on operands stored in the registers.  It was found that in most cases only a limited number of CPU operations were triggered. Hence, most modern CPU architecture are based on RISC architecture or Reduced Instruction Set Computers, where we have CPU capable of performing fewer operations but much faster2 . Instructions are orders on how data is to be manipulated:  The instructions control directly the movement of data words between the main memory and the registers of the CPU.  The instructions are stored themselves just like data in the main memory which is partitioned.  In most computers, a special register points to the location in memory which contains the next instruction to be executed. When the current one has been executed, the next one is fetched from memory, decoded and its orders executed.  This process can be streamlined by vectorizing the operations through a pipeline, this is done on SuperComputers with vectorization.  Through networking, you now can have as many computers on your workstations as you have windows. 2 Hence

a program compiled on a RISC machine is likely to generate a much bigger object le.

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14.1 Introduction

14{3

 Finally, you may have access to multiple CPU on one computer, or better yet multiple

CPU on multiple computers all through you workstation. Input/Output: has drastically changed over the years:  Access to computers: 1. Computing started as one mainframe executing only one program at a time. The program was either punched on a tape, or later on cards. 2. In the early 70's we had one mainframe (such as IBM) or minicomputer (such as VAX), with numerous concurrent users connected through terminals. 3. Later came the workstations in which each user has his/her own computer. 4. Finally, with the new generation of workstations/software, a user can have access to more than one computer or CPU.  A typical modern workstation has: 1. a 1,012 by 1,012 8 bit graphical display (i.e a resolution of 1; 012  1; 012 pixels, each one capable of assuming up to 28 = 256 colors). 2. A keyboard, and a mouse 3. Unix Operating system 4. At least 12 Mb of core memory 5. 200 Mb of disk space 6. Access to a local area network (usually ethernet) 7. X based window management and access to laser printer, plotter, CD-ROM, tape drive, digitizer, scanner, speaker, modem.  Computers are often networked together, a commonly used standard is ethernet. Increasingly the distinction between PC and workstation is being blurred. Currently, most new PC's are computationally more powerful than the immediate past generation of workstations. 5 Major distinction between PC and Workstation is highlighted by table 14.1.

4

14.1.1.2 Computer Software

For the computer to operate, it must also have: Operating System: which is a collection of programs to control the operation of the hardware and application programs.  Early on each computer vendor had its own operating system. Most recently, Unix3 has emerged as the uncontested de facto standard operating system for workstations, 3 Unix

is a trademark of AT&T.

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14{4

Day I: INTRODUCTION Feature PC Workstation Processor Intel Motorola Memory 8-16 Mb 32-64 Mb Operating System Windows 95/NT Unix/Motif RISC N Y Monitor 15"-17" 19"-21" Resolution 1024x1024 1024x1024 Bit Planes 8 8-24 Graphics Accelerators Y Y Networking Novell Ehternet Editors Word Vi, Emacs, Crisp Virtual Memory N Y Primary Application Everything Scienti c Computing Source of Software Microsoft Public-Domain, GNU, Vendors Graphics Open GL Open GL, X Table 14.1: PC Vs Workstation

mainframes, and supercomputers. It is also increasingly used on PC's for scienti c computing (through Linux).  Unix is for the most part written in C, and comes in two major avors: Berkeley Unix, or AT&T System V Unix. Window Management: for workstations  Early on (about 1980) each vendor had his own proprietary windowing management.  Through original funding from IBM, and Digital, MIT has developed X as a uni ed windowing system for workstations through the Athena project.  X was found to be very powerful, portable, but relatively cumbersome to use. Hence around X two major competing products have evolved: Motif: which has been developed by the Open Software Foundation, (OSF) which groups most major vendors with exception to Sun. OpenWindow: developed by AT&T and Sun. Note that each one of the two can also be used on the other.  Numerous software have been modi ed to run under X and its Graphical User Interface (GUI). Low Level Language: such as assembler. High-level languages: such as fortran. Fortran source code is written in an ASCII le, the compiler will then translate it into an object-code, and nally object-codes are all put together by the loader or linker. Victor E. Saouma

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14.2 From Source to Binary

14{5

\Very High-level languages" MATLAB, Mathematica, IDL, MathCad.

14.1.2 Fortran Past, Present, and Future

Despite numerous challenges, fortran remains the uncontested language of choice for scienti c/engineering programming. Eventhough over 34 years old, fortran has continuously been rejuvinated through revisions of its standards:  Fortran (FORmula TRANslator) was originally developed by John Backus from IBM in 1957. As the rst ever high level language, it was simple to learn and use.  In 1966 Fortran was standardized by ANSI (American National Standard Institute) and resulted in Fortran 66 (or sometimes called Fortran IV).  Again in 1977 the Fortran standard was revisited and modi ed to account for either the ellimination of some features, or the addition of others.  Many manufacturers have developed their own extension to the standards. While in many cases it made programing easier, in most cases it was an attempt to \capture" users to keep their code on a particular machine4 .  Recently, Fortran underwent a major revision which resulted in Fortran-90. For economic reasons, it was made upwardy compatible (i.e it recognizes most of Fortran-77 syntax, even though the use of many of them is discouraged). Fortran-90 not only addresses some of the features found in more \modern" languages such as C, but is also designed to facilitate the full exploitation of modern supercomputers/workstations.

14.2 From Source to Binary There are two major steps in creating an executable code: Compiling: which takes as input the source code(s) written (in ASCII) by the developped and converts it into an object code. Usually, the former has extensions such as .f, .F or .for, and the later .obj. Linking: Which takes the object les written by the user, and link them not only among themselves bu also with standard (fortran) and non-standard (such as graphics) libraries. This process results in an executable which on Unix is by default called a.out, or given an extension .exe in Windows. 4 You should never use non-standard Fortran statements. 6

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Day I: INTRODUCTION

14.3 Sample Example 7

To begin with, use your favourite editor to create the following le ello.f5

program Hello print *, 'Hello, world' end program Hello

8

Next save the le, and type

f90 hello.f a.out

14.4 Fortran Compilers 14.4.1 Compilers 9

In the Bechtel Lab, you have access to the fortran compilers listed in Table 14.2. Compiler Debugger Description f77 f90 f77 xlf90

Sun Workstations None

Fortran 77 Nag Fortran 90

xldbx xldbx

Fortran 77 Fortran 90

dbx

IBM Workstations PC

MicroSoft Power Station Fortran Compiler Table 14.2: Fortran Compilers in the Bechtel Lab It should be noted that the Fortran-90 compiler f90 on the Sun is only a \pseudo-compiler" in the sense that the code is rst internally translated into C and then compiled with a C compiler. This was one of the earliest Fortran-90 compilers developed by the Numerical Analysis Group in the UK until a true Fortran-90 is developed. 11 Debugger will be separately discussed in a subsequent lecture along with the Make command (which enables the programmer to easily keep track of modi ed les in the process of developping a large code). 12 The general form of the f90 (or f77) command is: 10

> f90 list-of-options list-of-files-and-libraries

5 This classic example was rst introduced by XX & xx in their introductory book on C, and has since then become the opening example in most introductory computer language tutorials.

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14.5 Flow Charts, Macro-Code

14{7

14.4.2 File Extensions The following le extensions are recognized by most compilers: .f, .for Standard fortran input le. .F Fortran le with ccp directives which will be run through a ccp preprocessor (more about this later). .o Object le. an extension of .o instead of the .f, .F or .for. 13

14.4.3 Workstation Compilers 14 On the Sun or IBM workstation, the simplest way to compile your code is to type followed by the le names (with extension), and this will result in le a.out which the executable (hence compilation and linking are performed automatically). For example: > f77 filename.f

As in most programs, the fortran compiler can be invoked with a number of options. Those are too numerous to list, and most of them are only used in very special cases. Table 14.3 lists the most commonly used ones, additional information may be obtained through the man command. 15

14.4.4 Compiling with Microsoft Power Station

14.5 Flow Charts, Macro-Code

It is highly advisable that prior to programing, a owchart describing the program be prepared. 17 Fig. ?? illustrates the symbols commonly associated with various operations.

16

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14{8

-C -g -o -u -U -c -d -D -F

-k -O -O2 -O3 -p -pg -v -w

Highly Recommended Performs run-time checking of array bounds and character substring expressions. Produces debug information. Names the executable le instead of a.out. Require explicit declarations for each variable by making the default type for all variables unde ned. Distinguish between upper and lower case letters. Others After assembling, do not proceed with linking. Leave each assembled object le with the extension .o. These may be linked at a later time. Leaves temporary les produced by cpp, instead of deleting them. Compiles lines of Fortran code that have a D in column 1. Apply the appropriate preprocessors to each FORTRAN source le but do not proceed with compilation. Leave each preprocessed source le with the extension .f. States that Fortran code is in free-form input format. Optimizes code generated by the compiler. Optimizes code (this is the same as -O). Performs the -O level optimizations and perform additional optimizations that are memory or compile time intensive. Generates simple pro ling support code. Generates pro ling support code. Provides more extensive pro ling than -p. Displays verbose information on the compiler's progress. Suppresses all warning messages. Table 14.3: Fortran Compiler Options

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Chapter 15

WEEK 1: Basic Operations 15.1 Background 15.1.1 Preliminaries

15.1.1.1 Program Layout 1

The layout of a program should be as follows

program hello_world .... end program hello_world

15.1.1.2 Characters 2

Acceptable characters in fortran are shown in Table 15.1 a-z, A-Z, 0-9,

& ! ;

characters Continuation Comment; Line following ! is ignored by compiler Separator Table 15.1: Characters

Statements can start in column 1 and extend up to 2640 characters including blanks, or 40 lines.

3

15.1.1.3 Variable Types 4

The following variable types are recognized by fortran

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15{2

WEEK 1: Basic Operations

Integers they usually have a range from ;2n;1 to 2n;1 + 1, where n is the word size (for 16

bits, the range is from -32768 to +32767). Real is a decimal oating point characterized by a mantissa and an exponent such as 1.56E-03, that is 1:56  10;3 , or 123.4567 On 32 bit computers, real have 7 signi cant digits. Double Precision A real number which is stored in two words. 1.56D-03, that is 1:56  10;3 , or 123.45678901234 On 32 bit computers, double precision reals have 14 signi cant digits. Complex (12.,5) represents (12+5i). Logical (.TRUE. or .FALSE.) String 'this is a long string' 5 Note that integers and reals are stored in one word which consists of 32 bits, a double precision real uses two words.

15.1.1.4 Labels 6

A label is a numeric pointer to a statement

delta=b**2-4.*a*c if(delta>0)go to 100 ! Note this form of branching should be avoided .... 100 continue ! 100 is the label

Usage of labels (and correspondingly of assigned and conditional branches) is strongly discouraged unless unavoidable.

7

15.1.2 Data Declaration Type Statement: integer,

real, double precision, complex, logical, character. Rea(0:100,-1,10). type should be

member that the range need not start from 1, i.e real prefered over dimension. ! Declare variables real :: alpha, one, x, y, z, length integer :: i, kounter, kode double precision :: w character :: file_name, output_file logical :: test

Implicit Statement: There are three possibilities: 1. By default in fortran all variables are real except those beginning with letter in the range I to N unless otherwise declared. Victor E. Saouma

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2. Stipulate implicitly all the variables beginnig with a certain letter as being of a certain type implicit real*8 (a-h,o-z) ! declare all reals to be double precision

3. Explicitly declare all the variables through implicit none real :: x, y, ...

This is by far the most suitable form of declaration as it will facilitate program debugging (where 1, l, i, L may be easily confused and create debugging nightmares). Initialization: Often time a variable may have to be initialized to a certain value. This can be done in either two ways: 1. Using the parameter statement integer,parameter:: year=1996 real,parameter::pi=3.141592654,one=1.0e+00 complex,parameter :: i=(0,1)

and during execution, these constants can not be rede ned (as a matter of fact, substitution id done during compilation). 2. Through the type statement declaration integer:: year=1996

Contrarily to the previous example, variable year can be over-written during execution. 8

Logical or \Boolean" variables can be declared

logical :: on, off, correct, wrong, small, large, true, false

those values (stored in only one bit) can only take the values .true. or .false. on=.true.; false=.false.; small=.true.;large=.false.

9

Logical or \Boolean" variables can be declared

logical :: on, off, correct, wrong, small, large, true, false

those values (stored in only one bit) can only take the values .true. or .false. on=.true.; false=.false.; small=.true.;large=.false.

Victor E. Saouma

Computing Literacy for Undergraduate Engineering Students

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15{4

WEEK 1: Basic Operations

15.1.3 Basic Operations

15.1.3.1 Assignment Statement 10

The basic assignment statement is of the form

variable=expression

which means: 1) evaluate mean \equals". x=x+y pi=3.14159 feet=12.*inches

expression

rst, and 2) assign it to

variable.

Thus = does not

| evaluate x+y, and then assign result to x | set the value of pi

15.1.3.2 Arithmetic Statements 11

Basic arithmetic operations are shown in Table 15.2 + Addition ; Subtraction  Multiplication  Exponentiation = Division Table 15.2: Basic Arithmetic Operations

Order of operations: 1) exponentiation; 2) multiplication, division; 3) addition, substraction. For example A+B**2/C is equivalent to A+((B**2)/C) Truncation occurs for integer division: 12

8/3 = 2

15.1.3.3 Logical Operators Logical variables can be operated by operators in which .not. negates or changes the logical value of the single variable it operates on. 14 Other operators are binary and are placed between two logical variables and used to compare their states. 15 Order of precedence is .not., .and., .or., .eqv. and .neqv. Parenthesis can be used. 16 A complete \truth table" for logical operators is shown in 15.3. 13

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Computing Literacy for Undergraduate Engineering Students

15.1 Background

Draft

on .true. .true. .false. .false.

15{5 off .true. .false. .true. .false.

on.and.off .true. .false. .false. .false.

on.or.off .true. .true. .true. .false.

on.eqv.off .true. .false. .false. .true.

on.neqv.off .false. .true. .true. .false.

Table 15.3: \Truth Table" for Logical Operators

15.1.4 Intrinsic Functions Common arithmetic intrinsic functions are shown in Table 18 Table 15.5 lists intrinsic functions that inquire about the numerical environment. 19 Table ?? lists intrinsic functions that manipulate the numerical characteristics of variables in the real model. 20 An important feature of all of these intrinsic functions is that they are generic and may be used to obtain information about any kind of integer or real supported by the Fortran 90 implementation. 17

15.1.4.1 Character Expression 21

The only allowed character expression is the concatenation operator //

c='Student Name' d='John Smith' e='c//': '//d

will give e='Student

Name:

John Smith'

15.1.4.2 Relational Operators 22

23

The following relational operators can operate on numeric expressions and variables < Less than <= Less than or equal to == Equal to = = Not equal to > Greater than >= Greater than or equal to A logical variable can be set equal to a logical expression

logical:: converged ; real::epsilon=1.e-03 ...

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Computing Literacy for Undergraduate Engineering Students

Draft

15{6

WEEK 1: Basic Operations Function

sqrt(x) abs(x) sin(x) cos(x) tan(x) sinh(x) cosh(x) tanh(x) asin(x) acos(x) atan(x) exp(x) log(x) log10(x) int(x) real(i) dble(x) cmplx(x) cmplx(x,y) aimag(x) anint(x) nint(x) ceiling(x) floor(x) sign(x,y) mod(i,j) max(x,y,...) min(x,y,...) max(x,y,...) minval(ax) maxval(ax) product(ax) sum(ax) len(strx) len trim(strx) trim(strx) index(strx,stry)

Description Square root of x Absolute value of x Sine of angle x (in radians) Cosine of angle x (in radians) Tangent of angle x (in radians) Hyperbolic sine of x Hyperbolic cosine of x Hyperbolic tangent of x Arcsine of x Arcscosine of x Arctangent of x e raised to the power x Natural log of x Common log of x Converts a real into an integer Converts an integer into an integer Converts x to double precision x + 0i x + yi Imaginary part of x Round x to nearest whole number Round x to nearest integer Least integer  x Greatest integer  x Transfer sign of y to jxj Remainder or modulo functio Maximum of x; y; ::: Minimum of x; y; ::: Maximum of x; y; ::: Minimum value in array ax Maximum value in array ax Product of values in array ax Sum of values in array ax Length of character string strx Length of string strx without trailing blanks strx without trailing blanks Position of stry in string strx

Table 15.4: Common Arithmetic Intrinsic Functions Victor E. Saouma

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15.1 Background

Function

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digits(x) epsilon(x) huge(x) minexponent(x) maxexponent(x) precision(x) radix(x) range(x) tiny(x)

Description q for an integer argument, p for a real argument b1;p for a real argument Largest in the integer or real model Minimum value of e in the real model Maximum value of e in the real model Decimal precision (real or complex) The base b of the integer or real model Decimal exponent range (real, complex, or integer) Smallest positive value in the real model

Table 15.5: Intrinsic Numerical Functions for Numeric Inquiry

Function

exponent(x) fraction(x) nearest(x) rrspacing(x) set exponent(x) spacing(x)

Description Value of e in the real model Fractional part in the real model Nearest processor number in a given direction Reciprocal of relative spacing near argument Set the value of e to a speci ed value Model absolute spacing near the argument

Table 15.6: Numeric Manipulation Functions

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converged=abs(error)<epsilon ! if rhs is true, converged=.true. otherwise it is .false. if(converged)xf=x ! set xf equal to x only if converged=.true.

15.1.5 Control Statements 15.1.5.1 Branches

Branches came in two types: Unconditional go to should be used as scarcely as possible, unless you want your code to look like a spaghetti plate. Assigned go to: (not to be confused with the unconditional go to), is seldom used but can be quite powerful. It is actually written in two parts: assign and then go to 24

go 10 go go 20 go

to (10,20,30) kode ! go to 10 if kode is negative, 20 if zero, and 30 if positive x=x*x to 40 to 40 x=2.e+00*x to 40 ........ 40 continue

15.1.5.2

if

Statement

If statements come in two forms Logical if in which a single statement is executed if the operand is true 25

if(grade>3.)letter_grade='A' ! Note that letter_grade is a character if(converged)xf=x ! Note that converged is a logical

Block if is to be used when multiple statements may have to be executed, or if a series of if checks must be performed. An if statement invoking a block can be given a name. It is highly recommended to properly indent a block if (usually by two extra spaces). if(i=2)then k=k+3 elseif(i.eq.4)then ! Note the old form of .eq. inherited from fortran 77 ! and which should be replaced by = k=2*k else k=0 endif ... grade: if(average>3)then

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let_grad='a' elseif(average<3.and.average>2)then let_grad='b' endif grade

15.1.5.3

case

To branch into dierent parts according a very simple criterion we use the case statement. It can choose a suitable case for a scalar argument of type integer, logical or character.

26

select case (ivar) case (:-1) write *, 'negative number' case (0) write *, ' zero' case (1:9) write *, ' digit ', ivar case (10:99) write *, ' number ', ivar case default write *, ' too great number' end select decades: select case (nyear) case(:1980) decade='past' case(1980:1989) decade='1980-1989' case(1990:1996) decade='1990-1996' case default decade='future' end select decades

!

all negative numbers

!

zero case

!

one-digit number

!

two-digit number

!

all remaining cases

Note that overlapping arguments are not permitted. This means that one single argument may not satisfy more than one of the cases of case. The default case does not have to be included. If no valid case is found the execution will continue with the rst statement after the end select. 28 It is recommended to use the case instead of an assigned or computed go to statement, or an arithmetic if statement.

27

15.1.5.4

do

loops

There are essentially two type of do loops Endless Loop (do while) is a loop in which a sequence of statements are executed as long as a certain condition isis met. It has the form do while logical expression statement 1 29

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... statement n end do

Iterative loop do involves initialization of a counter, automatic increment of the counter by an increment at the end of the loop, and repetition of the statements within the loop as long as the counter does not exceed the user speci ed upper value. do index,limit,increment statement 1 ... statement n end do

where index, limit and increment are the running index (which can not be overwritten within the loop), the upper limit, and the increment. If the increment is not speci ed, the default is one. index, limit and increment can all be either integers or reals, positive and/or negative. 30 31

A loop may be exited through exit A loop may also be given a name

program mean implicit none real::f_c,sum,mean integer::ndata sum=0.0 count: do read*, f_c if(f_c<=0.)exit sum=sum+f_c ndata=ndata+1 end do count mean=sum/real(ndata)

32

Finally, do loops can be nested.

15.1.6 Input/Output

15.1.6.1 Devices Default Devices/Read/Write Units must be assigned to input and output devices. On most systems, they are 5 for the keyboard, and 6 for the screen. 33 Hence, by default, the basic format of read write formats becomes read*, a,b,c print*, a,b,c

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Selected Input/Output Devices In the most general case, input data will be read from,

and output writen into user speci ed les. Thoose les must rst be de ned through the open statement which takes the following form open (unit=unit number,file=' le name', status=status) where unit number is an integer number assigned to the device, ' le name' is a character string giving the le name, and status is a character string which can be 'old', 'write', or 'read'.

15.1.6.2 Formated I/O 33 The format for the read statement is as follows read(unit number,format label)variable list where unit number is the device number through the open statement, and format label is the format label as de ned below.

speci ed

34 The format for the write statement is as follows write(unit number,format label)expression list where unit number is the device number ed through the open statement, and format label is the format label as de ned below. 35

speci-

Note that we can use the default devices for both read and write

open(unit=1,file='input.dat') open(unit=2,file='output.dat') write(*,*)' About to start reading from unit 1' ! May be placed for debugging purposes read(1,800)a,b,c write(2,900)a,b,c

Format De nition A format can be de ned in anyone of the following ways: Statement Label: referring to a format statement 900

write(6,900)A format(f10.2)

Character Variable provides a exible way to dynamically de ne the format: character*(*)form1, form2 parameter (form1='(f10.3)', form2='(f10.4)') ... if(...)then write(6,form1)a else write(6,form2)a endif

Edit Desciptors: Beside the familiar Iw, Fw.d, and Ew.d forms, there are a couple of others with which you should familiarize yourself: Iw.m speci es the width of the eld (w) and the number of digits to be printed (m); i.e I5.3 for 99 will be printed as bb099

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mPEw.d is a scale factor which should be used with the E format exclusively (eventhough

you can use it with the F format). It will print m signi cant digits to the left of the decimal point. Hence x = 123:5  103 will be printed as .1235E+06 with e10.4, and 1.235E+05 with 1pe10.3 Gw.d is a general format to be used when magnitudes are not known in advance, and where an F conversion is preferred where possible to an E format. Tabulation: is an input and output eld which can take anyone of three forms: Tn, TRn and TLn. Respectively they mean that the next part of the I/O should begin at position n (global), or n to the right or left of the current position (relative). Its use should be encouraged for tabulation of results.

15.1.7 Portability

Following are some additional guidelines which should be followed to ensure portability of your program: 1. Don't ever use variable names which exceed 6 characters. Be careful as some compilers may accept them, but would internally truncate any character following the sixth (hence temperin and temperout would be treated as one single variable (temper) 2. Even if the compiler allows it, do not insert more than one statement per line. Avoid the non-standard semi-colon (;) as a separator 3. Do not insert in-line comments following your fortran statement and the exclamation mark (!) 4. For logical constants, use the full descriptor .true. and .false. rather than .t. and .f.

5. Do not use mixed-mode relational expressions such as x=a/I 6. Do not use non-integer subscript in arrays such as in x(a/b)=..., instead round it explicitely as in x(nint(a/b))=... 7. Do not use the do while or do until or most importantly the do end do constructs. They are not (yet) part of the standard. 8. Do not use the read or write statements without specifying a unit. 9. Do not use underscore or currency symbol $ in variable names.

15.1.8 Style

1. Nested loops should be arranged such that the inner most loop should reference the rst subscript, and the outer most loop, the last subscript.

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do

k=1,50 do l=1,20 a(l,k)=b(l,k)+c(l,k) end do end do

2. All common blocks and sub-system control variables de ned via parameter statements are de ned in include les. The naming convention for include les is based on their contents; include les containing common blocks have the extension .cmn and include les containing parameters have the extension .par. The dierent extensions allow the developer to determine the contents of an include le without using a text editor. 3. Fortran is not a case sensitive language. However, lower case code is allowed. All source code should be in lower case, this is much easier in a Unix environment which is case sensitive, with the exception of the following exceptions:  Global or common variables begin with an upper case letter; all other characters are lower case.  Variables de ned via parameter statements are all upper case. This allows the developer to quickly determine where a particular variable comes from. 4. All variables should be declared; and the FORTRAN standard for variable types should be ignored. To insure that all variables are declared the -u option should be used when the source code is compiled on the Sun. Explicit declaration of all variables eliminates typographical errors and the omission of function subroutine type declarations, both of which are fairly tricky bugs to locate. 5. Variable names should be selected in a logical manner by choosing descriptive names. All variables should have no more than 6 characters. Also a common "syntax rule" should be adopted where a single basic variable name is employed with dierent pre xes depending on its usage in the program: (a) Pre x i,j, or l should be used to indicate a DO loop variable (b) Pre x k will indicate a counter (c) Pre x m will indicate a maximum value (d) Pre x n will indicate a given number For example ipoin, npoin, mpoin will indicate a particular point, number of points, and maximum number of points respectively. 6. All do loops should end on separate continue statements and the code between a do statement and the corresponding continue statement should be indented. This is more a matter of readability than anything else, but often comes in very handy when examining a particularly long do loop. Victor E. Saouma

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7. Ideally, a subroutine should have only one return statement. This is particularly true with long subroutines where multiple return statements make it dicult to follow the logic of the subroutine. It can be especially dicult to debug a subroutine with multiple return statements and for these reasons the use of one return statement per subroutine is strongly advocated. If there are multiple conditions that require returning to the calling subroutine the eect of multiple return statement can be duplicated with one labeled return statement and goto statements using the label corresponding to the return statement. The statement label will appear in a compiled listing with cross-references easily allowing a programmer to pinpoint the return conditions. 8. Ideally, the numbers used as statement labels in a given subroutine should appear in increasing order and the numbers should be evenly spaced. The statement labels found in most subroutines are multiples of ten. When there is a statement label associated with the return statement this statement label is generally 999. Statement labels associated with format statements are generally numbers between 700 and 990, inclusive. Statement labels between 700 and 790 are associated with debug prints; statement labels between 800 and 890 are associated with echoing input data; and statement labels between 900 and 990 are associated with error, warning, and informational messages written to the formatted output le. Associating a particular number or range of numbers with a certain type of format statement makes it easy to locate and/or recognize these statements when examining a compiled listing with cross-references. 9. Insert blanks in long expressions to facilitate their reading 10. Whenever there is a division, perform a check to determine whether the divisor is zero. 11. The use of extensions to the FORTRAN 77 standard should be avoided at all costs for the sake of portability. Extensions to the standard are generally identi ed in some way in the FORTRAN manual for a given compiler. If in doubt, consult the manual to be sure.

15.1.9 Debugging

1. Learn how to use your local debugger. 2. If possible, compile your program on another computer/compiler. You will be surprised to nd out how \forgiving" your computer may be. 3. The most common causes of Fortran errors are: (a) In nite loops (b) Division by zero (c) Erroneous array subscripts causing a location outside the allowable range. i. If it addresses a memory location associated with instruction, then a sudden crash may result.

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15.2 Assignment

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ii. If it addresses another variable location, then that particular one will be overwritten. As the program will not crash, this error is more dicult to locate unless you specify array subchecking as a compiler option. (d) Use of uninitialized variable. You can not always assume them to be zero.

15.2 Assignment

15.2.1 Beam on Elastic FOundation De ection

The defelection of a beam, length L, and exural rigidity EI on elastic foundation with stiness k by a central point load P is given by (L ; x)) ; sin( L)icosh( x) cos( (L ; x))] h v = 2P [sinh( L) cos( x) cosh( (15.1) k sinh2 ( L) ; sin2 ( L) q

where = 4 EIk Find the de ection for x = 0, 2, and 5 m for P = 100 kN, k = 4  103 kN/m2 and EI = 106 kN/m2

15.2.2 Quadratic Equation

Write a program to read in three numbers a; b and c, assumed to be the coeceints of the quadratic equation ax2 + bx + c = 0 (15.2) and calculate and print the roots. Account for the three possibilities 1. Two dierent real roots 2. One (double) root 3. A pair of complex conjugate roots

15.2.3 Statistical Analysis

Retrieve (through anonymous ftp to bechtel) the following set ftp/pub/Matlab/set1.dat. It contains results of concrete compressive strength fc0 for two dierent ready mix companies. Determine the Mean, Standard Deviation, Coecient of variation, Skewness, and Kurtosis. (Refer to MATLAB lecture notes).

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Chapter 16

WEEK 2: ARRAYS 16.1 Background 16.1.1 De nition

An array is de ned to have a shape given by its number of dimensions, called rank1 and extent for each one of these.

1

16.1.2 Declaration Whereas the declaration of scalar quantities is optional (unless we use implicit arrays must be declared. For example

2

none0,

all

real, dimension(20):: force real, dimension(20,20):: k

Alternatively we can use the following format real :: force(20) real :: k(20,20)

It is conventially assumed that the rst subscript of a two-dimensional array represents the row, and the second the column. 4 In fortran, arrays are stored column-wise, i.e. internally all arrays (no matter what their rank is) must be stored sequentially as a vector, hence stiffness will be stored as

3

k(1,1),k(2,1),...,k(20,1),k(1,2),k(2,2),...k(20,2),k(1,3),...k(20,20)

5

Array subscript can be negative, zero, and/or positive:

{\tt real:: vector(-10:22)}

1 Note

that the rank of an array is the number of dimensions and has nothing to do with the mathematical rank of a matrix.

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is an example of one dimensional array where the subscript may range from -10 to +22. {\tt real dimension(-10:25, 20:40):: grid}

is a two dimensional array.

16.1.3 Array Initialization Arrays can be initialized in a compact Implicitely for an entire array 6

real, dimension(5)::load=0.0 ! vector load is initialized to zero

Explicitely For each array value

real dimension (5):: load=(/10., -20., 15., 25, -32./)

Do loop Through a loop

real, dimension(3)::load=(/(2*i,i=1,3) /)

Note that speci cation of constants must be enclosed in \(/" and \/)" delimiters.

16.1.4 Array Constructors 7

Assignments statements for array elements can be performed as for scalar

k_struc(25,3)=k_elem(1,3,5)

8

Alternatively, we can \construct" the elements of a one dimensional array using a shorthand:

load=(/10., -20., 15., 25, -32./)

is equivalent to

load(1)=10.; load(2)=-20.; load(3)=15.; load(4)=25; load(5)=-32.

16.1.5 Array Sections Sections of arrays can be separately accessed. Array sections select part of an array consisting of regularly spaced array elements taken parallel to the dimensions of the source array. 9

real, real, real, real, real, real,

dimension(6,6) dimension(3,3) dimension(6) dimension(6) dimension(6) dimension(36)

k_node_i k_left_column k_top_row k_linearized k_diagonal

:: :: :: :: :: ::

k k_node_i k_left_column k_top_row diagonal k_linearized

= = = =

k(1:3,1:3) k(:,6) k(1,:) pack(k,.true.) ! pack is an intrinsic function which ! collapses k into a vector = k(1:36:6)

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16.1 Background

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16.1.6 Dynamic Memory Allocation 10 Dynamic allocation is useful when reading a data le of a size that is not known until runtime. The array would need to be de ned as allocatable. This is done by simply adding allocatable to the declaration line, with the dimensions speci ed as \:". Next, the array must be allocated once the dimensions are known, but before anything is read into the array. A second argument can be used with the allocate routine, in which the error status is set according to the success of the call. This should always be checked, to be sure there was enough memory for the allocation. real, dimension(:,3), allocatable :: nodal_coordinates integer, dimension(:,2), allocatable :: element_connectivity integer :: error read*, number_nodes, number_elements ! Get the matrix dimensions allocate(nodal_coordinates(number_nodes, 3), stat = error) allocate(element_connectivity(number_elements, 2), stat = error) if (error .ne. 0) then print*, "Allocation failed" stop endif read*, nodal_coordinates read*, element_connectivity ... ! Use arrays ... deallocate(nodal_coordinates, element_connectivity) ! Release memory when finished

16.1.7 11

do

Loops & Vector Operations

The do loops is often unnecessary for most array operations 1. Rows, columns, matrices, etc. may be assigned to each other as long as the dimensions are the same. All of the following assignments are valid: ! Initialize A and B to dummy arrays of the same dimension integer, dimension(5) :: a = (/ 1, 2, 3, 4, 5 /) integer, dimension(8) :: b = (/ 1, 2, 3, 4, 5, 6, 7, 8 /) integer, dimension(8) :: c ... c = b ! set vector c equal to vector b a = b(4:8) ! Assign to a the 4th through 8th elements of b b(3:7) = c(2:6) + a

2. For example, if we have an array (any dimensions) and want to initialize it to any scalar, say 0:0, rather than \walking" through each element with nested do loops, we can simply say \matrix = scalar", and every element of \matrix" will then be assigned the value of \scalar". 3. Some do loops can be replaced with a call to the dot product intrinsic function. Victor E. Saouma

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real, dimension(4) :: a, b c = dot_product(a,b)

4. Arrays or matrices can be accessed in non-ascending order, and portions can be extracted using appropriate subscripts. To repeat the dot product of the previous example with array b reversed, the following could be done: c = dot_product(a,b(n:1:-1))

16.1.8

where

Construct

This is essentially an extension of the if, then, else constructo for scalars, it is a powerful way to check/modify each and every cell in an array.

12

real, dimension(5,20) ::x,y real, dimension(-2:2,20) ::z : z = 4.0*y*sqrt(x) where ( x >= 0.0 ) z = 4.0*y*sqrt(x) elsewhere z = 0.0 end where

16.1.9 Intrinsic Functions Fortran 90 has over 100 intrinsic functions, the most important ones operating also on arrays are listed below. c 1993, Note that this section is adapted from Fortran 90 for the Fortran 77 programmer by Bo Einarsson2 and Yurij I Shokin3. Permission (was) granted to copy and/or print this le as long as the copyright notice and this permission is included on all copies. This le is also available as one book in Swedish and one in Russian. A complete textbook in Swedish on Fortran 90 has been prepared, and its translation into English is in progress.

13

16.1.9.1 Vector and matrix multiplication dot product(a, b)

length)

Computes the scalar product of two vectors (which should have the same

Computes the matrix product of two matrices which must have consistent ranks (such as (m,l) and (l,n)

matmul(a,b)

2 National Supercomputer Centre, University of Linkoping, S-581 83 L, Sweden. 3 Institute of Computational Technologies, Prospekt Lavrentyeva 6, Russian Academy

Division, SU-630090 Novosibirsk 90, Russia.

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16.1 Background

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16.1.9.2 Array Functions Returns a logical value that indicates whether all relations in mask are true, along only the desired dimension if the second argument is given. any(mask, dim) Returns a logical value that indicates whether any relation in mask is true, along only the desired dimension if the second argument is given. count(mask, dim) Returns a numerical value that is the number of relations in mask who are true, along only the desired dimension if the second argument is given. maxval(array, dim, mask) Returns the largest value in the array array, of those that obey the relation in the third argument mask if that one is given, along only the desired dimension if the second argument dim is given. minval(array, dim, mask) Returns the smallest value in the array array, of those that obey the relation in the third argument mask if that one is given, along only the desired dimension if the second argument dim is given. product(array, dim, mask) Returns the product of all the elements in the array array, of those that obey the relation in the third argument mask if that one is given, along only the desired dimension if the second argument dim is given. sum (array, dim, mask) Returns the sum of all the elements in the array array, of those that obey the relation in the third argument mask if that one is given, along only the desired dimension if the second argument dim is given.

all(mask, dim)

16.1.9.3 Array Inquiry Logical function which indicates if the array is allocated. lbound(array, dim) Function which returns the lower dimension limit for the array. if dim (dimension) is not given as an argument, we get an integer vector, if dim is included, we get the integer value with exactly that lower dimension limit, for which we asked. shape(source) Function which returns the shape of an array source as an integer vector. size(array, dim) Function which returns the number of elements in an array array, if dim is not given, and the number of elements in the relevant dimension if dim is included. ubound(array, dim) Function similar to lbound which returns the upper dimensional limits.

allocated(array)

16.1.9.4 Array Construction merge(tsource, fsource, mask) Function which joins two arrays. It gives the elements in tsource if the condition in mask is true and fsource if the condition in mask is false.

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WEEK 2: ARRAYS The two elds tsource and fsource have to be of the same type and the same shape. The result is also of this type and this shape. Also mask must have the same shape. Following is a rather complete example of the use of merge which also uses reshape from the next section in order to build suitable test matrices. Note that the two subroutines write array and write l array are test routines to write matrices which in the rst case are of a real type, in the second case of a logical type. implicit none interface ! This shall be explained nex week subroutine write_array (a) real :: a(:,:) end subroutine write_array subroutine write_l_array (a) logical :: a(:,:) end subroutine write_l_array end interface real, dimension(2,3) :: tsource, fsource, result logical, dimension(2,3) :: mask tsource = reshape( (/ 11, 21, 12, 22, 13, 23 /), & (/ 2, 3 /) ) fsource = reshape( (/ -11, -21, -12, -22, -13, -23 /), & (/ 2,3 /) ) mask = reshape( (/ .true., .false., .false., .true.,& .false., .false. /), (/ 2,3 /) ) result = merge(tsource, fsource, mask) call write_array(tsource) call write_array(fsource) call write_array(mask) call write_array(result) end subroutine write_array (a) real :: a(:,:) do i = lbound(a,1), ubound(a,1) write(*,*) (a(i, j), j = lbound(a,2), ubound(a,2) ) end do return end subroutine write_array

subroutine write_l_array (a) logical :: a(:,:) do i = lbound(a,1), ubound(a,1) write(*,"(8l12)") (a(i, j), j= lbound(a,2), ubound(a,2)) end do return end subroutine write_l_array

The following output is obtained Victor E. Saouma

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16.1 Background 11.0000000 21.0000000

16{7 12.0000000 22.0000000

13.0000000 23.0000000

-11.0000000 -12.0000000 -21.0000000 -22.0000000

-13.0000000 -23.0000000

t f 11.0000000 -21.0000000

f t

f f

-12.0000000 22.0000000

-13.0000000 -23.0000000

pack(array, mask, vector) Packs an array to a vector with the control of mask. The shape of the logical array mask has to agree with the one for array or mask must be a scalar. If vector is included, it has to be an array of rank 1 (i.e. a vector) with at least as many elements as those that are true in mask and have the same type as array. If mask is a scalar with the value true then vector instead must have the same number of elements as array.

The result is a vector with as many elements as those in array that obey the conditions if vector is not included (i.e. all elements if MASK is a scalar with value true). In the other case the number of elements of the result will be as many as in vector. The values will be the approved ones, i.e. the values which ful l the condition, and will be in the ordinary Fortran order. If vector is included and the number of its elements exceeds the number of approved values, the lacking values required for the result are taken from the corresponding locations in vector. The following example is based on the modi cation of the one for merge, but only the results are given array 11.0000000 21.0000000

12.0000000 22.0000000

13.0000000 23.0000000

vector -11.0000000 -21.0000000 -12.0000000 -22.0000000 -13.0000000 -23.0000000

mask t f

f t

f f

pack(array, mask) 11.0000000 22.0000000

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WEEK 2: ARRAYS pack(array, mask, vector) 11.0000000 22.0000000 -12.0000000 -22.0000000 -13.0000000 -23.0000000

Returns an array of the same type as the argument source with the rank increased by one. The parameters dim and ncopies are integer. If ncopies is negative the value zero is used instead. If source is a scalar, then spread becomes a vector with ncopies elements that all have the same value as source. The parameter dim indicates which index is to be extended. it has to be within the range 1 and 1+(rank of source), if source is a scalar then dim has to be one. The parameter ncopies is the number of elements in the new dimensions. unpack(vector, mask, array) Scatters a vector to an array under control of mask. the shape of the logical array mask has to agree with the one for array. The array vector has to have the rank 1 (i.e. it is a vector) with at least as many elements as those that are true in mask, and also has to have the same type as array. If array is given as a scalar then it is considered to be an array with the same shape as mask and the same scalar elements everywhere. The result will be an array with the same shape as mask and the same type as vector. The values will be those from vector that are accepted (i.e. those ful lling the condition in mask), taken in the ordinary fortran order, while in the remaining positions in array the old values are kept.

spread(source, dim, ncopies)

16.1.9.5 Array reshape

Constructs an array with a speci ed shape shape starting from the elements in a given array source. If PAD is not included then the size of source has to be at least product (shape). If pad is included it has to have the same type as source. If order is included, it has to be an integer array with the same shape as shape and the values must be a permutation of (1,2,3,...,N), where N is the number of elements in shape, it has to be less than, or equal to 7. The result has of course a shape shape and the elements are those in source, possibly complemented with pad. The dierent dimensions have been permuted at the assignment of the elements if order was included, but without in uencing the shape of the result.

reshape(source, shape, pad, order)

! program to test the optional arguments to reshape interface subroutine write_matrix(a) real, dimension(:,:) :: a end subroutine write_matrix

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16.1 Background

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end interface real, dimension (1:9) :: b = (/ 11, 12, 13, 14, 15, 16, 17, 18, 19 /) real, dimension (1:3, 1:3) :: c, d, e real, dimension (1:4, 1:4) :: f, g, h integer, dimension (1:2) :: order1 = (/ 1, 2 /) integer, dimension (1:2) :: order2 = (/ 2, 1 /) real, dimension (1:16) :: pad1 = (/ -1, -2, -3, -4, -5, -6, -7, -8, & & -9, -10, -11, -12, -13, -14, -15, -16 /) c = reshape( b, (/ 3, 3 /) ) call write_matrix(c) d = reshape( b, (/ 3, 3 /), order = order1) call write_matrix(d) e = reshape( b, (/ 3, 3 /), order = order2) call write_matrix(e) f = reshape( b, (/ 4, 4 /), pad = pad1) call write_matrix(f) g = reshape( b, (/ 4, 4 /), pad = pad1, order = order1) call write_matrix(g) h = reshape( b, (/ 4, 4 /), pad = pad1, order = order2) call write_matrix(h) end subroutine write_matrix(a) real, dimension(:,:) :: a write(*,*) do i = lbound(a,1), ubound(a,1) write(*,*) (a(i,j), j = lbound(a,2), ubound(a,2)) end do end subroutine write_matrix

The output from the above program is as follows. 11.0000000 12.0000000 13.0000000

14.0000000 15.0000000 16.0000000

17.0000000 18.0000000 19.0000000

11.0000000 12.0000000 13.0000000

14.0000000 15.0000000 16.0000000

17.0000000 18.0000000 19.0000000

11.0000000 14.0000000 17.0000000

12.0000000 15.0000000 18.0000000

13.0000000 16.0000000 19.0000000

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15.0000000 16.0000000 17.0000000 18.0000000

19.0000000 -1.0000000 -2.0000000 -3.0000000

-4.0000000 -5.0000000 -6.0000000 -7.0000000

11.0000000 12.0000000 13.0000000 14.0000000

15.0000000 16.0000000 17.0000000 18.0000000

19.0000000 -1.0000000 -2.0000000 -3.0000000

-4.0000000 -5.0000000 -6.0000000 -7.0000000

11.0000000 15.0000000 19.0000000 -4.0000000

12.0000000 16.0000000 -1.0000000 -5.0000000

13.0000000 17.0000000 -2.0000000 -6.0000000

14.0000000 18.0000000 -3.0000000 -7.0000000

16.1.9.6 ARRAY manipulation The shift functions return the shape of an array unchanged, but move the elements. cshift(array, shift, dim) Performs circular shift by shift positions to the left if shift is positive and to the right if it is negative. If array is a vector the shift is being done in a natural way, if it is an array of a higher rank then the shift is in all sections along the dimension dim. If dim is missing it is considered to be 1, in other cases it has to be a scalar integer number between 1 and n (where n equals the rank of array). The argument shift is a scalar integer or an integer array of rank n-1 and the same shape as the array, except along the dimension dim (which is removed because of the lower rank). Dierent sections can therefore be shifted in various directions and with various numbers of positions. eoshift(array, shift, boundary, dim) Performs shift to the left if shift is positive and to the right if it is negative. Instead of the elements shifted out new elements are taken from boundary. If array is a vector the shift is being done in a natural way, if it is an array of a higher rank, the shift on all sections is along the dimension dim. If dim is missing, it is considered to be 1, in other cases it has to have a scalar integer value between 1 and n (where n equals the rank of array). The argument shift is a scalar integer if array has rank 1, in the other case it can be a scalar integer or an integer array of rank n-1 and with the same shape as the array array except along the dimension dim (which is removed because of the lower rank). The corresponding applies to boundary which has to have the same type as the array. If the parameter boundary is missing we have the choice of values zero, .false. or blank being used, depending on the data type. Dierent sections can thus be shifted in various directions and with various numbers of positions. A simple example of the above two functions for the vector case follows, both the program and the output.

14

real, dimension(1:6)

Victor E. Saouma

:: a = (/ 11.0, 12.0, 13.0, 14.0, &

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16{11 15.0, 16.0 /)

real, dimension(1:6) :: x, y write(*,10) a x = cshift ( a, shift = 2) write(*,10) x y = cshift (a, shift = -2) write(*,10) y x = eoshift ( a, shift = 2) write(*,10) x y = eoshift ( a, shift = -2) write(*,10) y 10 format(1x,6f6.1) end 11.0 13.0 15.0 13.0 0.0

12.0 14.0 16.0 14.0 0.0

13.0 15.0 11.0 15.0 11.0

14.0 16.0 12.0 16.0 12.0

15.0 11.0 13.0 0.0 13.0

16.0 12.0 14.0 0.0 14.0

Another example b = (/ 11.0, 12.0, 13.0, 14.0, 15.0, 16.0 /) 11.0 14.0 17.0

12.0 15.0 18.0

13.0 16.0 19.0

z = reshape( b, (/3,3/) )

17.0 11.0 14.0

18.0 12.0 15.0

19.0 13.0 16.0

x = cshift (z, shift = 2)

13.0 16.0 19.0

11.0 14.0 17.0

12.0 15.0 18.0

x = cshift ( z, shift = 2, dim = 2)

14.0 17.0 11.0

15.0 18.0 12.0

16.0 19.0 13.0

x = cshift (z, shift = -2)

17.0 0.0 0.0

18.0 0.0 0.0

19.0 0.0 0.0

x = eoshift ( z, shift = 2)

13.0 16.0 19.0

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 11.0

0.0 0.0 12.0

0.0 0.0 13.0

Victor E. Saouma

x = eoshift ( z, shift = 2, dim = 2)

x = eoshift ( z, shift = -2)

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Transposes a matrix, which is an array of rank 2. It replaces the rows and columns in the matrix.

transpose (matrix)

16.1.9.7 Array location Returns the position of the greatest element in the array array, if mask is included only for those which ful l the conditions in mask. The result is an integer vector! minloc(array, mask) Returns the position of the smallest element in the array array, if mask is included only for those which ful l the conditions in mask. The result is an integer vector! Note that the two subroutines write array and write l array are test routines to write matrices which in the rst case are of a real type, in the second case of a logical type.

maxloc(array, mask)

16.2 Assignment

16.2.1 Structural Design 16.2.1.1 Theory

The design of a steel truss entail the following steps: 1. Structural analysis in which the equations of equilibrium are applied at each node and written in terems of the unknown internal element forces and known externally applied load. This will result in a system of linear equations of the form [B] fxg = fPg

(16.1)

where fPg is the vector of externally applied load, and fxg the vector of unknown element forces (tension if positive, compression if negative). Note that the unknown vector contains both element forces ass well as reactions. 2. Analysis is performed for dierent loads, and then multiple load cases are coonsidered. Each load case is typically a linear combination of the basic load investigated. 3. Results (internal forces) are tabulated for all load cases, maximum forces determined, and each element is separately designed.

16.2.1.2 Assignment

The truss shown in Fig. 16.1 is to be designed. The statics [B ] matrix can be assembled4 The resulting statics matrix can be copied from 4 Note

that in assembling the matrix, each column corresponds to an unknown internal element force, or an external reaction. Each row corresponds to an equation of equilibrium.

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16.2 Assignment

Figure 16.1: Steel Truss

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ftp/pub/Structures/cven4837/b-truss.dat and its inverse from ftp/pub/Structures/cven4837/b-truss-inv.dat

The three load vectors (corresponding to roof, snow, and live loads) are:

r=-[0., 0., 0., -0.9, 0., 0., 0., -1.74, 0., 0., 0., -1.68, 0., 0., 0., -1.68, 0., 0., 0., -1.68, 0., 0., 0., -0.84] s=-[0., 0., 0., -2.57, 0., 0., 0., -4.97, 0., 0., 0., -4.8, 0., 0., 0., -4.8 , 0., 0., 0., -4.8 , 0., 0., 0., -2.4] l=-[0., 0., 0., 0., 0., 0., 0., 0., 0., -6.4, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,-6.15, 0., 0.]

and are stored in

ftp/pub/Structures/cven4837/truss-load.dat

Write a Fortran-90 program which will 1. Read the [B] matrix and it inverse. 2. Check that the product of those two matrices is the identity matrix. 3. Solve for the roof, snow, and live load forces 4. Obtain the forces for each of the 3 following three load combinations Load Case Roof Live Snow 1 1.4 0 0 2 1.2 1.6 0.5 3 1.2 0.5 1.6 5. Determine the maximum load for each member 6. Select the cross sectional area of each member using allowable = 20 kips/in2 in compression and nal area of each member using allowable = 24 kips/in2 in tension. 7. Tabulate the results as shown in Table 16.1 plus an additional column showing the crosssectional area in in2. Note that Note that the actual results are given by Table 16.1.

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16.2 Assignment

Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

L0 ; L1 U0 ; L0 U0 ; L1 U0 ; U1 U1 ; L1 L1 ; L2 U1 ; L2 U1 ; U2 U2 ; L2 L2 ; L3 U2 ; L3 U2 ; U3 U3 ; L3 L3 ; L4 U3 ; L4 U3 ; U4 U4 ; L4 L4 ; L5 U4 ; L5 U4 ; U5 U5 ; L5

Load Load Cases Design Load Roof Snow Live 1 2 3 Compr. Tens. 0. 0. 0. 0. 0. 0. 0 0 -8.5 -24. -13. -12. -35. -55. -55. 0 12. 34. 20. 17. 52. 79. 0 79. -9.8 -28. -16. -14. -43. -65. -65. 0 -6.8 -20. -11. -9.6 -30. -45. -45. 0 9.8 28. 16. 14. 43. 65. 0. 65. 7.3 21. 16. 10. 38. 50. 0. 50. -15. -44. -29. -22. -72. -100. -100. 0 -4.6 -13. -3.9 -6.5 -14. -29. -29. 0 15. 44. 29. 22. 72. 100. 0 100. 4. 11. 5.2 5.6 15. 26. 0 26. -18. -52. -32. -26. -83. -120. -120. 0 -2.7 -7.8 -3.6 -3.8 -10. -18. -18. 0 18. 52. 32. 26. 83. 120. 0 120. 1.4 3.9 4.6 1.9 9.8 10. 0 10. -19. -55. -36. -27. -90. -130. -130. 0 -0.97 -2.8 -3.3 -1.4 -6.9 -7.3 -7.3 0 19. 55. 36. 27. 90. 130. 0 130. -0.89 -2.5 4.2 -1.2 5.1 -3.1 -3.1 5.1 -19. -53. -38. -26. -93. -130. -130. 0 0.66 1.9 3.1 0.92 6. 5.3 0 6. Table 16.1: Result of Truss Design

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Chapter 17

WEEK III; PROGRAM UNITS; POINTERS 17.1 Background

17.1.1 Program Units A fortran 90 code is a modular hierarchical object de ned by fundamental blocks. 2 Each program unit may have one or more external procedures. Those procedures (function and/or subroutine) are nearly independent pieces segments of the code. They can be written separately, make it easier to debugg the code, can be reused in other programs, and can be reused in the same code. 3 Program and procedure can share data through the module unit.

1

17.1.1.1 Main Program Always assign a header, i.e. program stress. 5 End the program similarly: i.e. end program stress. 6 Always return control to the main program upon termination (good programming practice), and hence there should be no more than one stop statement in your code. 7 For abnormal completion, you may append a character string after the stop 4

stop 'Incomplete data, abnormal completion'

8 There should be few executable statements in the main program. Calls to subroutines should be used instead.

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17.1.1.2 Functions Often times an application program needs to have a relatively simple operation to be performed. If that operation is to be repeated, and returns a single variable as a result of an operation on one or more variables then we should use the function procedure which has the general form

9

function-type

function

function name(argument list)

the function must also end with a return and endfunction-name statements. 10 The function arguments referenced in the main program are the actual arguments. They should match in type, number and order the dummy ones in the function statement. 11 Functions can be de ned within a subroutine if necessary. 12 If one of the function arguments is an array, its dimensions must be speci ed in both the main program and the function. 13 A function may call other functions. 14 Because a function returns a single value, the function name should appear in the type statement in the main program. 15 The intent of the use of each variable should be de ned according to whether it is only to be used as an input variable (in), output variable (out), or both (inout): 16 Following is a main program program statistic ! declare variables implicit none real :: average real allocatable :: fc(:) integer :: error, k, n ! open the data file and read number of data points open(unit=1,file='strength.dat') read(1,*) n ! allocate space for array x allocate(fc(n), stat = error) if (error .ne. 0) then print*, "Allocation failed" stop endif ! read the data points read(1,*)(fc(i),i=1,n) ! print the average (which is to be determined from a separate function) print*, average(fc,n) ! end program statistic

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17.1 Background

17{3

which references the following function real function average(x,n) ! This function determines the average value of an array x ! variable declaration implicit none integer :: i, n real :: x(n), sum=0.0 intent(in) :: x,n ! Note the intent declaration as in do i=1,n ! Note that we could have used the intrinsic function sum sum=sum+x(i) ! instead of this loop end do average=sum/real(n) return end function average

17.1.1.3 Recursive Subprograms 17 Fortran 90 supports recursion. If a subprogram calls itself, directly or indirectly, the keyword recursive must appear in the subprogram statement.

Recursive functions must also declare a result variable to avoid ambiguity with array-valued functions that are directly recursive. The result variable is used to hold the function result for each function invocation; the function name is used to invoke the function itself. Consequently, the recursive function's name should never appear on the left side of an assignment statement. An example of a recursive factorial function is shown below.

18

recursive function factorial( n ) result( f ) integer f integer, intent( in ) :: n if ( n <= 0 ) then f = 1 else f = n * factorial( n-1 ) end if end function factorial

17.1.1.4 Subroutines Subroutines are procedures used to perform operations which are too complex to be performed by functions (if several values need to be returned, or values are simply to be read and passed back). 20 Just as we have intrinsic functions, fortran 90 has intrinsic subroutines which are included in the compiler. You should consult your compiler manual to determine which intrinsic subroutines are pre-de ned. 21 The general format to reference a subroutine is through the following executable statement

19

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22

subroutine-name (argument list)

In turn, the rst line in a subroutine should look like subroutine

subroutine-name(argument list)

Contrarily to functions, a subroutine use the argument list to identify both input and output values to a subroutine. 24 A subroutine may return zero, one or many values. 25 Declaration of all parameters should include the intent attribute. 26 The subroutine should include a return and end subroutine 27 It is advisable to have input arguments preceding output ones. 28 Following is a main program 23

program statistic ! declare variables implicit none real :: average, fc_min, fc_max real allocatable :: fc(:) integer :: error, k, n ! open the data file and read number of data points open(unit=1,file='strength.dat') read(1,*) n ! allocate space for array fc allocate(fc(n), stat = error) if (error .ne. 0) then print*, "Allocation failed" stop endif ! read the data points read(1,*)(fc(i),i=1,n) ! call the subroutine which will compute the average, minimum and maximum values call stat(fc, n & average, fc_min, fc_max) ! print results print*, "Average compressive strength ',average print*, "Minimum compressive strength ',fc_min print*, "Maximum compressive strength ',fc_max end program statistic

which references the following subroutine subroutine stat(x, n & average, x_min, x_max) ! This subroutine determines the average minimum and maximum value of an array x ! variable declaration

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17.1 Background

17{5

implicit none integer :: i, n real :: x(n), x_min=x(1), x_max=x(1), average, sum=0.0 intent(in) :: x,n intent(out) :: average, xmin, xmax ! do i=2,n sum=sum+x(i) if ( x(i) < x_min ) x_min = x(i) if ( x(i) > x_max ) x_max = x(i) end do average=sum/real(n) return end subroutine stat

17.1.1.5 yModules A module is a collection of declarations and subprograms. Each modules has two components: 1. The speci cation declares variable types, de nes derived types, and marks what is visible or hidden from calling programs and subprogram interfaces 2. The subprogram part actually provides the code for subroutines and functions involved in a module. More about modules later.

29

17.1.1.6 y Interface An interface speci cation should be de ned at the beginning of the main program so the compiler will know exactly how best to treat each argument of a subroutine or function. To use a subroutine de ned as above, the following would need to be placed at the beginning of the main program: More about modules later.

30

17.1.2 Derived Data Type There are two types of data types in fortran Intrinsic such as integer, real, double-precision, logical, complex and character. An array de ned through an intrinsic data type can be of only one type, i.e. we can not mix two types in the same array. Derived or user de ned data types (similar to records or structures in other languages) are available in Fortran 90. 31

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In a derived data type we can group dierent data types together. 33 To assign values to the component of a derived data type, we use the structure identi er followed by a percent sign (%) followed by the component identi er. 34 The following example should be self-explanatory. 32

! define element data type type element_type character*10 :: type real :: length, area, young integer :: first_node, second_node end type element-type ! type (element_type) :: element(100) do i=1,100 read(5,*)element(i)%type, element(i)%first_node, element(i)%second_node, & element(i)%length, element(i)%area, element(i)%young end do

17.1.3 Pointers

17.1.3.1 Pointer Variable 35 A pointer target

variable, called pointer is a mere alias to another variable which is called the

real, pointer :: pntr real, target :: x=10., y=20., z

So far we have not yet speci ed what they are.

17.1.3.2 Pointer Assignment 36

To assign a value to a pointer we use the operator specifed by =>

pntr=>x z=pntr

will result in pntr=10 and z=10 37 A pointer can be nulli ed nullify(pntr}

A pointer can point to other pointers, and hence to their targets, or to a static object that has the target attribute: 39 There is also an internal function associated in order to investigate if a pointer is associated (and if it is associated with a certain target) and a statement nullify in order to terminate the association.

38

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17.1 Background

17{7

if ( associated (a) ) write(*,*) ' a is associated ' if ( associated (a, b) ) write(*,*) ' a is associated with b ' nullify (a)

40

Yet another example

real, target real, pointer a = 10. p => a q => p a = 2.

:: a :: p, q

Here the value of q equals 2. since both p and q point towards the same variable a and that one has just changed its value from 10. to 2. We now make a simple variation. real, target :: a, b real, pointer :: p, q a = 10. b = 2. p => a q => b

Now both the values of a and p are equal to 10. and the values of both b and tt q are 2. If we now give the statement q=p, all four variables will get the value 10., which means that an ordinary assignment of pointer variables has the same eect as the conventional assignment b=a. If we instead give a pointer association q=>p, then the three variables a, p and q all have the value 10. while b contains the value 2. In the second case q only points to the same variable as p while in the rst case q becomes the same as p, and the value addressed by q becomes equal to the value addressed by p. 41 Important application of pointers are lists and trees, and especially dynamic arrays. 42 Note that a pointer has both type and rank, and that these must agree with the corresponding target. This increases the security at the use of pointers, it is therefore not possible by mistake to let a pointer change values of variables of other (dierent) data types. The fact that you have to specify that a variable can be a target also increases both security and eciency of the compilation. real, real, a => c =>

target pointer b(4,:) b(:,4)

:: b(10,10) :: a(:), c(:) ! vector a becomes the fourth row ! and vector c becomes the fourth ! column of the matrix b

or real, target :: matrix(100,100) real, pointer :: window(:,:) integer :: n1, n2, m1, m2 window => matrix(n1:m1, n2:m2)

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If you later wish to change a dimension of the partial matrix window you only need to make a new pointer association. Please note that the indices in window are not from n1 to m1 and from n2 to m2 but from 1 to m1-n1+1 and from 1 to m2-n2+1.

17.1.3.3 Dynamic Memory Allocation for Pointers 43

Pointers can also be assigned using allocate

(pointer)

which will assign a target memory location to the pointer. integer, target :: x integer, pointer :: ptr_1, ptr_2 ... ptr_1=>x allocate(ptr_2)

17.1.3.4 Pointers to Arrays There does not exist arrays of pointers directly in Fortran 90, but you can construct such facilities by creating a new data type.

44

! declare the derived data type as pointer type pointer_element real pointer :: ptr end type pointer_element ! ! declare variables implicit none integer :: i real, target, dimension(5) :: load=(/10., -20., 15., 25, -32./) type(pointer_element) :: y(5) ! do i=1,5 y(i)%ptr=>load(i) end do

Yet another example which stores a lower (or left) triangular matrix with rows with varying length.

45

! define a new data type row type row real, pointer :: r(:) end type ! specify the lower triangular matrices l as vector ! of rows with varying length integer :: n type(row) :: l(n) ! allocate the matrix

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17.1 Background do i = 1, n allocate (l(i)%r(1:i)) end do

17{9 ! various length of rows

Finally, pointers allow the creation of a new type of data structure linked data structures. This is similar to an array because it is a group of nodes that contain information and have an order (i.e. there is a rst, second order and so on). However, a linked list diers from the array in that we do not use a subscript to reference a node in the linked list, instead we use a pointer variable. 47 The linked list is built by associating pointers to the nodes on the linked list as the need arises, and we do not have to declare the maximum size of the linked list. Each node in the linked list has a pointer contained within it that points to the next node in the list.

46

17.1.3.5 Pointers as a Tool for Linked Lists 48 Pointers may be used with scalar or array quantities of any type and are used to construct dynamic structures such as linked lists and trees. 49 The following program illustrates how a dynamic data structure can be declared and manipulated. program LinkedList type node real data type( node ), pointer :: next end type node type( node ), pointer :: list, current nullify( list )

! Initialize list to point to no target.

! Place two elements in the list. allocate( list ) ! call random_number( list%data ) ! allocate( list%next ) ! call random_number( list%next%data ) ! nullify( list%next%next ) ! ! Output the list. current => list do while ( associated( current ) ) print *, current%data current => current%next end do end program LinkedList

Victor E. Saouma

Reserve space for first node. Initialize data portion. Reserve space for second node. Initialize data portion. Initialize next to point to no target.

! Assign target of list to target of current.

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17.2 Assignment 17.3 Mendeleev The Mendeleev table contains 106 known elements. Each element is characterized by name character of length 12 symbol character of length 2 atomic number which is an integer mass a double precision number number isotopes an integer indicating the total number of isotopes for the current element weight which is a list of number isotopes double precision number. A sample (input) data le would read as: Hydrogen 1.007825 Helium 3.016030 .....

H 1 1.00794 3 2.014000 3.016050 He 2 4.00260 5 4.002600 5.012220 6.018886

8.033920

Write a program isoweight which will 1. Open a data le mendelev.dat containing all the pertinent data 2. create a derived data type mendeleev in which weight is an allocatable array and pointer 3. Loop over all the elements, and for each one (a) Read name, symbol, atomic number, mass, and number of isotopes (b) Allocate sucient memory for each isotopic weight (based on the number of isotopes read). (c) Read the individual weight into the allocated memory 4. Print the results in the following format 0 1 2 3 4 5 6 123456789012345678901234567890123456789012345678901234567890 Name Symbol Number Mass Isotope Weights -------------------------------------------------------------Hydrogen H 1 1.00794 1 1.007825 2 2.014000 3 3.016050 Helium

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17.4 Linked List

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4.002600 5.012220 6.018886 8.033920

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17.4 Linked List 1. Write a subroutine named new order that is referenced with the following statement call new order(order,item,first,last)

the subroutine should add a node at the end of the list containing the integer order number and the integer item number 2. Write a subroutine named print orders that is referenced as call print orders(first)

this shubroutine should print all the information in the orders linked list. If the list is empty, print an appropriate message. 3. Write a subroutine order filled that is referenced as call order filled(first,order)

whish should delete the node in the linked list that corresponds to the order number. If the correspondingentry is not in the linked list, print an error message. Write a program which uses all three above subroutines to accept information from the keyboard, either to add data to the end of the linked list, or to remove data from the linked list. Assume that the terminal interaction begins with this message: Enter N to add new order to list D to delete order from list Q to quit program

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WEEK III; PROGRAM UNITS; POINTERS

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Computing Literacy for Undergraduate Engineering Students

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Chapter 18

WEEK 4: MAKE, DEBUGGER 18.1 Debugging options Sometimes there are mysterious things going on within our programs. The code looks like it should execute properly, but for some reason the code bombs, or produces questionable results. These \reasons" are called \bugs". The FORTRAN compiler has a few options to help in tracking these bugs down, or debugging you code. These options are extremely helpful for debugging code under development, but they should not be used for production programs. The extra code and expanded symbol table that these options generate will increase run time. Also available is a more extensive debugging tool called Debugger, see section 18.4. The following options can be given to create information need for debugging and pro ling tools. Note that these options must be given both in the compile and link steps if you compile and link separately. -C Generates additional code for run-time array bounds checking. When the program is executing , it will print a warning message to the screen if a reference to an array element occurs that falls outside of the arrays declared boundaries. -g Generates an expanded symbol table for the debugger. -p Compiles and links the program for pro ling with prof. A program compiled with the -p option will generate a le called mon.out when executed. This le contains program execution statistics. The pro ler prof will read this le and produce a table describing your program's execution. To pro le a program you need to compile each source le with the -p option, and also specify this option in the link step. An example: f77 -p -c file1.c f77 -p -c file2.c f77 -p file1.o file2.o

Now the program can be executed, and the le mon.out will be generated. The pro ler then be run to make the information in mon.out readable. Giving one of the commands:

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WEEK 4: MAKE, DEBUGGER prof prof mon.out prof mon.out > prof.out

will run the pro ler. The last command will put the output in a le called prof.out, this le can then be viewed in an editor. -pg Compiles and links the program for pro ling with gprof. This option is the same as the -p option, except that the le generated is called gmon.out and it should be used with the program gprof.

18.2 FORTRAN Preprocessor There is a preprocessor that understands directives like #include to include les and #de ne to de ne symbolic constants. The f77 compiler will automatically run the preprocessor on all les with an .F extension (and not on les with an .f or .for extension). The #include directive is used to include another le in the text processed by the compiler. (This will not aect the original source le). If the line: #include "definitions.h"

is present in a source le, the preprocessor will replace the line with the contents of the le de nitions.h before continuing. The #de ne directive is used to de ne symbolic constants. If the line: #define PI 3.1415926

is present in a source le, all uses of the symbol PI after this line will be replaced with 3.1415926.

18.2.1 System Calls

As with most compilers, it is possible to make system commands from your Fortran program. The use of this feature can make a program very powerful when used to access other tools available on the system. Since the system calls are not standard among compilers and computer systems, they should be used cautiously if the program is to be at all portable. Usage: CALL SYSTEM(string) where string is an operating system command or program call. System calls are especially good for making programs more \user friendly". For example, many programs prompt for input le names. It would be convenient if a program provided a list of les in the current directory that have the correct extension. Following is a short Fortran routine that lists all les having the extension \.dat". C

PROGRAM PROMPT CHARACTER*40 FILENAME USE A SYSTEM CALL TO PROVIDE A LIST OF POSSIBLE INPUT FILES

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18.3 Make C

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CALL SYSTEM('ls *.dat') PROMPT USER FOR INPUT DATA FILE NAME WRITE(*,*)' ENTER THE INPUT DATA FILE NAME > ' READ(*,1) FILENAME FORMAT(A40)

18.3 Make When developing a program, the executable le must be continuously recreated as the source code is modi ed and recompiled. This can become a hassle, especially if your program is lengthy. If your modi cations have only been in a few of the subroutines making up your program, it is a considerable waste of time to recompile the entire program. This is where the make command is useful. A program can be created from several source les, each le containing one or more subroutines. Make will only recompile those les that have been modi ed since the last compile and will relink all the object les and create a new executable. To use make you need a description le that contains the information make needs to create your program. This information includes the names of the source les, the compiler you wish to use, the compiler options, and so on. This le is called the make le. While this section will not go into the million and one uses for make it will show you how to build a make le. Basically, a make le consists of a number of de nitions and a number of targets. The de nitions are used to specify le names, which compiler and linker to use, ags to use when compiling, etc. The targets is what tells make what to do when invoked. A target speci es a dependency and a list of command. The dependencies are used to tell make when to remake something, and the commands are used to tell make what to do to remake the target. In the following example, the target $(PROGRAM) is dependent on the les in $(OBJECTS), and the command to make the program from the object les is speci ed on the next line (note that the command lines must start with a tab character). Make is partially intelligent, it understands how to make an object le from a source le, for example a .o le from a .f le. Thus when it sees that $(PROGRAM) is dependent on a number of object (.o) les, it will nd the source (.f) les in the directory and create the .o les automatically. The following is an example of a simple make le. The lines starting with # are comments, the numbers in parentheses refers to the more detailed explanations following the example. # (1) Name of the program. PROGRAM = sample # (2) List of source files making up the program. SOURCES.f = \ file1.f \

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WEEK 4: MAKE, DEBUGGER file2.f \ file3.f \ file4.f

# (3) Linker to use. LINK = $(LINK.f) # (4) Flags to the compiler and linker. FFLAGS += -g # (5)Source and object files. SOURCES = $(SOURCES.f) OBJECTS = $(SOURCES.f:%.f=%.o) # (6) Default target. all: $(PROGRAM) # (7) Program target. $(PROGRAM): $(OBJECTS) $(LINK) -o $(PROGRAM) $(OBJECTS) # (8) Clean target. clean: $(RM) $(OBJECTS) core # (9) Dependencies. file1.o: file1.inc file2.o: file1.inc file2.inc file3.o: file1.inc

The make le and the source les must reside in the same directory, and the object les and the executable le will be put in that directory. 1. Here the name of the program is speci ed. In our example, the program is called sample. 2. This section de nes the sources to be used in the program. In this example the source les are in FORTRAN, if they were in C, the extensions would all be .c. If there is more than one le, follow the lename with a backslash, hit return, then tab and type in the next lename, etc. 3. This section speci es the source language. Specify LINK.f for FORTRAN and LINK.c for C. Victor E. Saouma

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18.3 Make

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4. Specify the compiler ags, FFLAGS for FORTRAN, CFLAGS for C. Here the -g invokes the debugger, see section 18.4 for more information about this option. For more information on other compiler options see the respective section on the compiler you are interested in using. 5. We will now de ne SOURCES to contain all the source les listed in SOURCES.f. The $ means to use all that was de ned for SOURCES.f and de ne SOURCES. Next we will de ne OBJECTS, to contain all the object les. The % is considered a wildcard by make, so it will take all the .f les in SOURCES.f and create the corresponding .o les. 6. The default target is usually called all. If make is invoked without any arguments, the rst target is processed. Here the all target will just invoke the $(PROGRAM) target to make the program. 7. The $(PROGRAM) target is the target to create the executable le by linking all the object les and creating the executable with the name you have de ned in PROGRAM. 8. The clean target is used to clean up the directory of the program. Giving the command make clean will remove all the object les and the core le, if any. 9. The last section of the make le is to de ne any dependencies. For example if you have used an include le in your subroutines, the object le is dependent on both the source le and the include le. So, if either the source le or the include le has changed, you will want your object le updated. Once your make le is complete, the command to create your program is simply: > make

This looks for a le called make le in whatever directory you are in. If you want to call your description le something other than make le you would use the -f option as follows > make -f filename

To make another target than the default one, specify the target on the command line to make: > make target

Usual targets are all (usually default), the program name, both of which makes the executable le, and clean, with cleans up in the directory. If you need more information with creating your make le the on-line man pages may help. Also the book Managing Programs with make by Andrew Oram and Steve Talbott, O'Reilly and Associates, Inc. may prove very helpful. Victor E. Saouma

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18.4 Debugger Written by Johannes Hermanrud

18.5 Introduction to Debugging A debugger is a program that makes it easy to debug programs. Using a debugger, one can:  run the program in a normal fashion,  step through the program line by line,  run the program, but stop the execution at speci ed places (by setting breakpoints),  continue execution after a stop,  see the contents of variables, either at speci ed times or continuously, and  see the call stack, i.e. see which functions has been called to reach the current point of execution. This information is valuable in deciding which parts of the program are actually executed, and what the values of the program's variables are at dierent points during the execution. When a program has a serious error, such as a segmentation fault, the operation system will create a le called core in the current working directory. This le can be used with a debugger to nd out where in the program the error occurred. See the command where for details.

18.5.1 The debugger

Debugger is a tool that can be used with C, FORTIN, Pascal, Modula-2 and assembler (See section 18.5.2 for how to run the debugger). Debugger opens a window on the screen containing a menu bar and ve subwindows.

Menu Bar There are ve menu options each with their corresponding submenus. Each menu and submenu are brie y described below:

 the Program menu allows the users to debug a new program, go to a speci c line or

function in the current program, edit the source code for the current program using the vi editor, compile the source with or without command line arguments and list all of the program modules.  the Breakpoint menu makes it possible for the programmer to place a breakpoint anywhere in the program so that the execution will stop at a certain point. Any breakpoints which have been set may also be removed through this menu.  the Execution menu contains many of the same commands that are present in the Command Subwindow which allow you to step through the program or to run the program until completion. Victor E. Saouma

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18.5 Introduction to Debugging

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 the Stack menu displays which function the execution is in and at which line. This menu will also allow the programmer to go up or down one call level.  the Data menu contains options to allow one to display the value of variables and to evaluate variables.  the Props menu is a way in which the default properties of the debugger may be changed.

>From top to bottom the subwindows are (refer to gure 18.1). Status Subwindow. This area displays information about the state of the debug session. The following information is given:  the name of the le currently displayed in the source subwindow,  the name the function the debugger last stopped in,  the number of the line last stopped at,  the numbers of the rst and last lines displayed in the source subwindow. This information is updated when a command has been executed and when the window is resized. Source Subwindow. This text subwindow displays the source les of the program currently being debugged. The program is initially loaded by using the debug command in the command subwindow. The source le containing the main program code will appear in this window. The text can be scrolled using the scrollbar. The le to be displayed can also be selected using the le command. Also, the func command can be used to display the le containing a speci ed function. The line at which the execution of the program is stopped is marked by an arrow. Breakpoints are marked by a stop sign. Buttons Subwindow. This area contains buttons that can be selected to perform a number of commands. Many of the commands require associated arguments such as line numbers, function or subroutine names, etc. These arguments can be de ned by marking the lines or function names with the mouse in the source window. Marking is performed with the left and middle mouse buttons (same as with the text editor) and is done prior to selecting the function button. Command Subwindow. Debugger reads commands from this area and the program output that is directed to the standard output device (screen prompts, etc.) is displayed here. The text can be scrolled using the scollbar. To type commands in this subwindow, rst activate the text cursor by placing the mouse cursor in the window and pressing the left button on the mouse. Display Subwindow. This area shows the values of variables speci ed with the display command. The text can be scrolled using the scollbar. The display command needs to be used only once for a single variable, as the display of the variable will be updated Victor E. Saouma

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WEEK 4: MAKE, DEBUGGER throughout the execution of the program. To get the display window to pop-up either type display variable name or select "Display Window..." from the data menu.

Figure 18.1: Debugging Tool

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18.5 Introduction to Debugging

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18.5.2 Running the debugger

Debugger can be started using the Bechtel Lab menu from the programming tools submenu. When the debugger option is selected the debugger window will appear on the screen. The easiest way to load your program is to choose the le from the program loader menu which automatically appears when the debugger is called up. Keep in mind that the program name that you choose must be the name of an executable. At this point the source code of your program should appear in the window. The next step is to set stopping points within the program. Important Note: In order for debugger to work with your program, your program must have been compiled using the -g option. For example: % f77 -g -o bobs_program bobs_program.f

18.5.3 Debugger Commands

This section describes some of the most used debugger commands. These commands must be typed in in the command subwindow. Some commands are also available on a button, these are marked (button) in the following

debug le Debug the le le instead of the le currently being debugged. run [args] (button) Start execution of the current program using the arguments args. If no

arguments are given, the arguments of any previous runs are used. The execution will stop at any breakpoints set with the stop command. rerun [args] Start execution of the current program using the arguments args. If no arguments are given, the program is run without arguments. The execution will stop at any breakpoints set with the stop command. next [n] (button) Execute the next n source lines, counting a function call as a single statement. If n is not given, one line is executed. step (button) Execute the next n source lines, stepping into functions. If n is not given, one line is executed. cont (button) Continue execution from where it stopped. The execution will stop at any breakpoints set with the stop command. stop at n (button) Stop execution at line n if that line is reached during execution. The easiest way to set stopping points is to mark the line in the source sub-window using the mouse. Then click on the STOP button using the left mouse button. A \stop sign" will then appear in the source window. stop in function (button) Stop execution at the rst line in function function when that function is entered.

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where List the functions that have been called to reach the current point of execution. When

debugging a program that has a core le, this command can be used when starting the debugging session to see where the core dump occurred. display variable Display the value of the variable variable in the display subwindow. This command only needs to be performed once for a single variable. When the variable changes value, the displayed value will re ect this. The Display* button will display a pointer. clear [n] Clear breakpoints on line n, if n is not given, clear the breakpoint last stopped at. change directory - cd path When the debugger is started from the menu system, the current directory is your home directory. Generally it is best to run the debugger in the directory in which your program resides, particularly if your program must read any data les. To change from your home directory to a subdirectory called homework/CVEN5555 type : (debugger) cd homework/CVEN5555

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Bibliography [1] A. Biran and M. Breiner. MATLAB for Engineers. Addisoon-Wesley, 1995. [2] D.M. Etter. Engineering Problem Solving with MATLAB. Prentice-Hall, 1993.