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— sini/rsin^cos0)e 2 (17.15) d3 = cos \\r sin 0ex + sin \jr sin 0e2 + cos 0e3, where 0 is restricted to the interval [0, n\ The strains and shears which take the triads d1? d2, d3 into the triad g1? g2, g3 are infinitesimal. An approximate method for describing them is as follows. Let us denote the displacements of a point on the central axis by u, v, w, in the directions e1? e 2 , and e 3 , respectively. The derivative w{z) represents the elongation in the direction of the axis, but W may be neglected compared to unity; the derivatives u(z) and v(z) are now approximate measures of the angle formed by d3 and e 3 , obtained by means of a rotation about e2 and one about ej (Figure 17.1). The Euler angle cp may be taken as an approximate measure of the angle formed by d! with e 2 , or the one formed by d2 and e t . The introduction of the new quantities u, v, and (p permits us to replace Euler's equations (17.15) by linearized versions: which read 26 sin# + j 2 s i n ^ s i n 0 + zcos0. From (17.15) we obtain the curvatures by means of the formulae already employed: K = n/2. If in (21.35) we put ft = 0, we recover the solution for a cantilevered beam loaded at its tip by a load always directed vertically;34 alternatively, the case f$ — 1 corresponds to a load always acting perpendicular to the tangent at the tip. The case a^ — a-\- (n/2) requires special comment. It corresponds to a beam loaded by a follower force, which is horizontal and compressive in the undeformed rectilinear configuration, but acts in the direction of the tangent to the deflection curve at the tip of the bar, as soon as this moves from the straight configuration. For ap = a-\- (TV/2), the parameter k must necessarily be zero, and the expression for cp(l), as given by (21.34b), loses its meaning. Since the solution 0 = 0 is an equilibrium state, other bent configurations arise because the fundamental state is not stable. The natural procedure for characterizing the stability of the elastic system is that of studying the equilibrium of configurations that are slightly bent with respect to the rectilinear state. However, as first pointed out by Nikolai (1928, 1930) and later by Beck (1952), the linearized criterion of stability is incapable of determining the critical value of the axial load, so generating a paradox in the classical theory of elastic stability. The explanation of this paradox rests on the fact that, in the presence of nonconservative forces, the static criterion of stability may prove inadequate, and instead we must use a dynamic criterion, which takes account of the circumstance that the passage from the fundamental configuration to a nearby bent configuration is not slow but sudden and, as a consequence, the inertial terms cannot be neglected. The approximate methods for evaluating the deflection of a Kirchhoff beam subject to a distributed load are commonly based on the device of representing the slope 6 by means of trigonometric functions. However, the degree of approximation is much more accurate if we apply an expansion of the solution in terms of Tchebychev polynomials (Schmidt and Da Peppo 1971). The equation of an initially horizontal beam of length L, of constant flexural rigidity B, loaded by a vertical force P at S = 0 and by a uniform load w per unit length is Already solved by Bisshopp and Drucker (1945). = u" cos(^? + T/T) + v" sin((^ + T/^) — (1 — cos 0) cos #>, / / (24.16) or 2
we may write (14.28) in the simpler form x = rx cos
j = rx sin #>! = r2 sin #>2.
(14.29)
We have already found the stress function F at the generic point E on the wedge due to a single force P at ft. Now the function Fx = r^F
(14.30)
is a new stress function for the transformed point S, having the property of being singular at O and ft in the new domain. However, despite the singularity at the vertices, the boundary of the transformed domain is not free, because it is subject to normal traction. This traction has the same value at all points on the transformed boundary and may be eliminated by superposition of another additional uniform state of stress that can create uniform tractions on the boundary, but of opposite sign to the first.24 Under the change of variables (14.29), the stress function Fx for the transformed domain becomes
This peculiar property of the transformation by inversion was pointed out by Michell.
116
14. The Influence of Curvature F\ = r2 (C2cpp sin cp + C3(pp cos
T
= 4K
/ r"
C2(py + C 3 a9| x — —-
= 4ic2(
(14.31)
The associated stresses are expressed as follows:
2KC3
sin 2(
2KC}
sin
^
dxdy
— LT\ r21
=4 , C
2
I rx
+ 2KC3 sin(^! +
/
^ - ^2 r
r
\
sin^i s i n ^ ^ 1 I sin(^i + cp2) — 2(
\
r2 J
r\
r1sin^Y^l-^)l
+
\
r\
r\ ) \
•]
sin2
2
(14.32)
Since we are particularly interested in finding the stresses along the section BA, characterized by the properties r{ = r2 = r and
(14.33)
Let us now consider the particular, but important case in which the force P acting at the vertex £2 of the wedge forms an angle \jr = 7T with the edge
Rod Theories: Three-dimensional Approach Q
O
B
Q
Q
117
B ax=0.4525
av =0.0739
ax=-0.0757
Fig. 14.5 (n — a) + sin a cos a (TT
— a) — sin a
sin2 a (7r — a) 2 — sin2 a In order to evaluate Q, we use the equilibrium condition n —a A
hxdyf=
J
Z
f axK-^- z =
J
cos cp
o
B
whereas we have the condition n — ot
faxy'dy'= f From the first of these equations we obtain P = Q/(4K2), and thus the constants C2 and C3 are completely determined. Figure 14.5 shows the distribution of the stresses ax and ay in the middle section AB for Q = 1.
15. The Influence of Initial Twist
When a prismatic bar possesses an initial twist, its torsional rigidity will be greater than that of the same bar with no initial twist. The relative increase in torsional rigidity depends on the initial twist, the shape of the cross-section, and the elastic modulus of the material. This property was first studied by Chu (1951), who calculated the increase in torsional rigidity and confirmed the result with a series of experiments on initially twisted rectangular steel strips. The improvement in the stiffness of a bar by pretwisting is used in aircraft propellers and turbine blades.
118
15. The Influence of Initial Twist
However, although in good agreement with experimental data, Chu's theory was incomplete, because he used a nonorthogonal system of curvilinear coordinates, which complicated the derivation of the result considerably, with the consequence of introducing certain terms that, in effect, should disappear. Another criticism of the theory is that it does not include warping (Rosen 1980). However, notwithstanding these two criticisms, Chu's idea is interesting. A correct version of the theory can be simply derived (Rosen 1980), and the improved analysis shows why, in certain cases, Chu's theoretical results are almost exactly corroborated by experiment. Let us consider a prismatic bar of length I and constant cross-section A. The longitudinal axis connecting the centroids of the cross-sections is denoted by x and y, z these are orthogonal axes placed in the plane of the cross-section. In addition to the y and z axes, we introduce a second axis system x, rj, and f, where r] and £ are the principal axes of the cross-section that may rotate with the cross-section along x. The initial twist is represented by a rotation /3 of the cross-section about the x axis. The y and z axes are fixed and the rj and f axes are movable, and ft is the angle between axes rj and y. The angle p is a function of x, but we consider the special case in which ft is linear in x, so that
is a constant. It is important to emphasize that the initial configuration, including the initial twist, is free from stress. In other words, the initial twist is a pure geometrical modification of a prismatic beam, ideally obtained by cutting the latter into an infinite number of small transverse slices and then superimposing these slices in such a manner that each of them is rotated by a small angle with respect to the preceding one. Let us now suppose that, at the ends of the beam, two equal and opposite torques of magnitude Mt are applied by means of tangential tractions exerted on the terminal sections and distributed over these according to specific laws that must be determined a posteriori, the only restriction being that they are equivalent to a couple, say, Mt at the section x = £, and a couple -Mt at the section x = 0. If we denote the displacement components along the fixed axes by w, v, and w, respectively, we assume that they have the form
v = -0xz, w = Oxy,
(15.2)
where i/r = i/r(x, y, z) is a function, called the warping function, Ox is the angle of elastic rotation of a section relative to the section x = 0, and u = u\(x) is another function, unknown at this stage. If the bar is prismatic, then ux is, identically, zero and (15.2) coincides with Saint-Venant's solution, provided that ^ depends on y and z but not on x. The function V is harmonic and satisfies the condition. dxlr — = zcos(y, v) — ycos(z, v), av
Rod Theories: Three-dimensional Approach
119
at all points on the bounding curve of each cross-section. Alternatively, as a consequence of (15.2), the strain components that do not vanish are du
du\
du
dv
d\jf
du dw +
The strain energy per unit volume has the form
where £" and G are (constant) elastic moduli. The total potential energy of the beam is thus
-Mtfodx,
(15.4)
which may be written in the simpler form
W-L = J{^ where
(156)
A
K= / / ( —) dydz, It can be seen immediately that GJ$ is the torsional rigidity of Saint-Venant's theory, while S and K are integrals associated with the dependence of yfr on x. If the beam is in equilibrium, the functional W — L attains a stationary point, and the Euler equations that correspond to the vanishing of the first variation of the functional with respect to u\ and 6 are ox GJSO + E( S — + KG) = Mt,
(15.7) (15.8)
120 so that elimination of
15. The Influence of Initial Twist 3M 1 /3X
yields (15.9)
This means that the initial twist produces a variation in the torsional rigidity GJS, calculated according to Saint-Venant's theory, to give a new value (15.10)
GJ = G
and it is clear that we have / ^ / s as a consequence of Schwarz's inequality. A useful application of the theory is offered by the case, very common in practice, in which the cross-section is a slender profile that is symmetrical with respect to its principal moving axes rj and £ (Figure 15.1) (Rosen 1980). Since all cross-sections are identical except for the initial rotation ft, xjr will not be a function of x, if represented by the coordinates rj, f. Therefore, dx/s _ dx/s drj ~dx~~drjlhc
dxjr d£ ^ax'
(15.11)
with dxjs/dr] and dx/f/dt; independent of x. On the other hand, by differentiating the relations (see Figure 15.1) rj = y cos $ + z sin fr f = — y sin $ + zcos ft,
Fig. 15.1
Rod Theories: Three-dimensional Approach
121
we obtain
— = (-ycos£ - zsin£) — = -rjk. ox ox Substitution of these equations in (15.11) and then in (15.6) gives
A
Since now the function \/r9 considered as a function of r\ and f, is the torsion function according to Saint-Venant's theory, it may be written as df
a^
= ?+
90
9TA
=
I?
90
^' at " " V
(15 14)
*
where 0 is another harmonic function, called Prandl's function. It is known that 0 satisfies the equation
and vanishes on the boundary. For a thin-walled cross-section the stress function 0 may be approximated as
where h is the thickness of the cross-section measured perpendicular to the rj axis. If we substitute this expression for 0 in (15.14) and then in (15.12) and (15.13), we obtain
It is clear that the term f2/rj2 does not contribute very much to the values of S and K, and it is thus sometimes neglected. The influence of bending effects in a pretwisted section is a very important problem in the analysis of turbine blades, because, when the beam is bent, the bending stiffnesses of a pretwisted cross-section are very different from those of the same section in an untwisted beam. An approximate analysis of the problem can be made under certain strong assumptions about the magnitude of the pretwist (Downs 1979), which must be small.
122
15. The Influence of Initial Twist i Z
Fig. 15-2 Let us consider a pretwisted beam the cross-section of which (see Figure 15.2) undergoes a uniform initial twist in the direction of the x axis, perpendicular to the plane of the figure and directed inwards. The initial cross-section, at x = 0, is referred to a system of fixed Cartesian axes y and z that coincide with the principal axes of the section. Another section, at a distance x from the former is identified by the angle a0, representing the rigid rotation of the section about the x axis with respect to thefixedaxes y and z. A point of coordinates y, z in thefixedsection is at a distance
from the origin. A corresponding point in a section at unit distance from the fixed section, is connected to the first point by a helical fiber of length (15.15) By imposing a further elastic twist 0 per unit length (0 <& a0) in the direction of the pretwist, while allowing no axial movement or bending, we produce a strain
ee = {[1 + (ao + efcPj - (1 + ald2f}-
1
(15.16)
where, in deriving the last formula, we have neglected the terms involving 02 and higher powers of 0, and the terms involving al and higher powers of a0. The corresponding stress acting along the helical filament is ae ~ E0(a0d2 -
a30d4).
(15.17)
Rod Theories: Three-dimensional Approach
123
We now impose an axial extension a per unit length, without allowing any other movement, and produce the strain ea = {[(1 + "? + *U2i - (1 + *ld2i}
1 2 2
—~- ~ a(\ - a\d\
(15.18)
(l d f
where we have ignored the terms in a2 and higher powers of a, and the terms in a^ and higher powers of a0. The corresponding filament stress is oa~Ea{\-a\d2\
(15.19)
Through its length the filament is inclined with respect the x axis at a helical angle P = arctan(ao
dF~o\\-
-aid2 )dA,
(15.20)
and the axial component of this elementary force is dFa = dF cos p - a{\ - a2od2)dA,
(15.21)
and the tangential component is dFx = dF sin p ~ a(aod - a3od3)dA.
(15.22)
The examination of the terms discarded indicates that the accuracy remains within one percent for a maximum helix angle of 16°, and within two percent for a maximum helix angle of 23°. Let us now apply a unit relative translation, along the x axis, between two sections unit distance apart; and let us apply three unit relative rotations between the same sections, about the x, y9 and z axes, respectively. The displacement generates strains and stresses along the helical fibers, from which we can find the stress resultants TV, Mx, My, and Mz on the basis of elementary beam theory. Let us call s the extension, K and K' the curvatures, and r the torsion. We then obtain four constitutive equations of the type N = KUS + KyiK + KUK' + /C14T, Mz = K2\e + K22K + K23K M
y
Mx
= / C £+ K K + K K
f
+ K T
= K416 + K42K + K43K + /C44T,
75. The Influence of Initial Twist
124 where
Kn=EJj{\-<xld2fdA, a
Kn=K2X=-EJjy{\-ald2fdA,
K14 = K4l = Efj(a0d2
- asgrf*Xl - <*2od2)dA,
A
K22 = EJjy2{\-ald2fdA, A
K23=K32
(15.24)
EJjyz(l-a20d2)2dA,
= A
"24 = K42 = -Ejjy(aod2
- ald4)(l - a20d2) dA,
A
= *43 = -EJjz(a d 0
2
- a2od2)dA,
-
A
= E jj(a0d2 - a30d4)2 dA + GT0, GT0 being the torsional rigidity of the section according to Saint-Venant's theory. The formulae for the stress resultant show that there is an intimate coupling between them, in the sense that each stress resultant depends on all the strain parameters. There is the immediate question of whether (15.23) can be simplified; this can be done by introducing the notion of a neutral axis. Let us consider the axial force and the moment induced by imposing a small rotation of a pretwisted section about a line parallel to the neutral axis. Then, if n is the distance of the neutral axis from the line, we must have the equation (15.25) where dFa is given by (15.19) and (15.21), and s is the distance of the element of force dFa from the given line. Thus from the last equation we obtain n=
ff s(l - a20d2)2 dA x - A
Rod Theories: Three-dimensional Approach
125
and, calculating the coefficients of equations (15.23) with y and z measured from the neutral axis, we find that KU, KU, and K23 vanish, thereby permitting some simplification.
16. The Energy of an Elastic Rod
When an elastic rod is viewed as a three-dimensional body it is always necessary, at least in principle, to solve a system of partial differential equations in order to determine the state of strain and stress in the body. Solving this system is very difficult even when using the assumptions of the linear theory of elasticity, but practically impossible when large strains or even large displacements are involved. In the absence of a systematic method for tackling the nonlinear problem, efforts have been made towards reducing the original problem to one of a single dimension through the averaging procedure, which we saw in Section 8. However, sometimes these methods are also difficult, so the procedure for solving the averaged equations needs further reduction. Instead of finding a complete solution, we content ourselves with characterizing certain properties of the solutions, or evaluating some useful quantities related to them. The techniques involved in this are almost always of the variational type. The simplest application of such methods is illustrated by the case of a prismatic beam, of rectangular cross-section and sufficiently thin for plane stress theory to apply (Rychter 1988a, 1988b). The beam is bent in the xz plane, which coincides with the middle plane of the rectangular prism. The x axis coincides with the center line of the prism, the y axis passes across the width, and the z axis points downards; y and z are the principal axes of inertia of the section. The end surfaces of the beam are defined by x = ±1, the side surfaces are described by y = ±b, and the upper and lower surfaces are described by z = ±h. The beam is bent in the xz plane by a surface load p(x) distributed over the upper face z = —h\ the lower face and both side faces are free. The boundary conditions for the end faces are specified in the following form: the left face x = — I experiences tractions ax = cr* and rxz = r*z, and the right face x = t experiences displacements u = u* and w = w*. The body forces are zero. In order to solve the problem we must know the constitutive equations of the material. In the present case, we assume that the material is homogeneous, linearly elastic, and orthotropic, and is characterized by a stress-strain relation of the type ax = Eu,z +Daz,
rxz = G(w,z +w, 2 ), ) ^ ,
(16.1)
where E and E' are the Young's moduli in the x and z directions, respectively, G is the shear modulus, and D = a{E/Ef). As it is difficult to determine an exact solution to the problem, we consider two approximate fields. The first is a kinematically admissible displacement field, that is to say a field of displacements (u, v, w) that is sufficiently regular, in particular with piecewise continuous first partial derivatives, and satisfies the kinematic boundary
126
16. The Energy of an Elastic Rod
conditions on the face x = I. For reasons that will be clarified later, it is convenient to split the field into two parts (fi, v, w) = (u\ v\ wl) + {u1, v\ w%
(16.2)
where (u\ v\ wl), the interior field, need not necessarily satisfy the boundary conditions on the face x = l, whereas (ul,vl, wl), the exterior field, must satisfy the boundary conditions ue = u*- u\
wl = w*- w\ at x = L
(16.3)
Once we have the displacements, we can derive the associated stresses (ax, az, xxz) through the constitutive equations, and decompose them into the sum (£*. <*z, *xz) = (<%> %> ?xz) + (<& °t *iz)-
(16.4)
The second field is a statically admissible stress field (<xx, oz, axz) which satisfies the equilibrium equations in the interior and all the boundary conditions of statical type. Again, it is useful to write (ax, &z, rxz) = (alx, alz1 rlj + (ox, a£Zi ixz),
(16.5)
where (cfXi alz, rlxz) satisfy the equilibrium equations and the traction boundary conditions on the lateral surface, while (ax, crz, xxz) satisfy the equilibrium equations, the boundary condition on the lateral surface, but with p(x) = 0, and the end conditions
5i = °Z- Si,
4 =
(16.6)
Let us now suppose that the two fields (w, v, w) and (cfx, az, xxz) have been determined. In terms of these fields we may define
-h h
1 / -
1
-h
where w is a mean lateral deflection, and b is a rotation represented by the difference between the axial displacement of the upper and lower edges. At the same time we introduce the conventional bending moment M and shear force Q, defined as follows: h
h
M(x) = j zax(x, z) dz, Q(x) - j rxz(x, z) dz. -h
(16.8)
-h
From the field equations we derive the equilibrium condition M\x) = Q, =0
(16.9a) (16.9b)
Rod Theories: Three-dimensional Approach
127
and the constitutive relations
f,
(16.10a) (16.10b)
At the ends we must have M = M*,
Q = Q* for x = £; b = b\
w = iv\ for x = -I,
(16.11)
where the starred quantities are prescribed. To account for the elongation or contraction of the beam, we introduce the average axial displacement h
W(JC) = ^
f u(x,z)dz,
(16.12)
-h
and the axial force h
N(x)=
Jax{x,z)dz. -h
For the equilibrium we also need the conditions N'(x) = 0,
(16.13)
and N = N* at x = £,
v = v* at x = -£,
where, again, the starred quantities are known. In addition we have the constitutive relation N = 2hEu' + hDp.
(16.14)
The equations just derived represent the averaged version of the original twodimensional problem. Our task is to exploit the one-dimensional solution in order to obtain estimates of the two-dimensional solution. This may be done by constructing a displacement field, satisfying only the interior conditions, having the form ff = u(x) + thb(x) + (t3 2
i)f(x\
w< = w(x) + ts(x) + Qt - l)g(x) + (t4 - It1 + ^)r(x) - (t4 - 6t2 - 8t + §)
128
16. The Energy of an Elastic Rod
where = l,
(16.16a)
= -hDu,
(16.16b)
= -h2D^,
(16.16c)
6
= -hD^-,
(16.16d) ^
(16.16e)
The corresponding stress field becomes Si = E[u' + thb' + (t3 - 0 / ] + Dal xxz = 3(1 - t2)%- + G[tsf + (t4 - It2 + £)/] - G(t4 - 6t2 - St + §)#',
(16.17)
4 The interior statically admissible stress field is given by
2D hp
l " ) '' aiUii
+i t t ^ + a
-if
+
In these equations, we may express/' in terms of the load/?, by combining (16.16f) with (16.9b) and (16.16c) with (16.10a). The result is
Similarly, from (16.16b) and the two equations written for N, we derive s' = -^P'.
(16.20)
Since we already have formulae (16.17) and (16.18) for the two states of stress, we may write
o*x-6*x = O(h2p"\ 4 - 4 = O(hp'), oi - Si = O(h2/).
(16.21)
A first conclusion to be drawn from (16.21) is that the two stress states coincide for a constant load p on the upper surface and, consequently, the exact two-dimensional solution is then reached, provided that the end conditions are just that of the state
(cflx, alz, xlxz) 2Xx = l and that of the state (plx, olz, xlxz) at x = -t. In the general case of
Rod Theories: Three-dimensional Approach
129
a load varying with x we may impose an energy bound in the linear elasticity,24 which allows the strain energy of an elastic body to be bounded by means of a kinematically and statically admissible state such as those constructed above. Let (ax, az, xxz) be the exact solution, and let U be a quadratic norm associated with this solution having the form h
I
U(ax, crz, rxz) = I I \a2x + (cr2z - 2aaxcrz)— + x\y - I dxdz. -h
-t
Then Synge and Prager's theorem states that the following equality holds: = u(alx - alx, a\ - alz, ilxz - x l x z \
(16.22)
We can now use (16.21) to obtain an asymptotic estimate of the right-hand side of (16.22). So far we have treated the case in which the boundary data at the terminal faces have two special distributions such that the other externalfieldsvanish. If, however, the boundary conditions at the ends are different, having forms that allow the states (ax,az,rxz) and (ax,az,rxz) to be constructed without excessive difficulty, then a formula like (16.22) still holds, and we obtain an estimate for the complete solution. The level of accuracy of an estimate like that derivable from (16.22) can be improved further through a process of successive generation of approximate twodimensional fields of any order of accuracy, provided that, at the ends of the beam, we demand no more detail than the shear stress resulting from the average vertical displacement and the stress couple or the gross rotation (Duva and Simmonds 1990). In order to get an idea of the method, let us consider a rectangular beam which, when undeformed, occupies the region 0 ^ x ^ L, \y\ ^ 2?, \z\ ^ H. We assume that the beam is clamped at x = 0 and is stress free at x = L, that there are no body forces, and that the faces of the beam are under the tractions crz(x, ±H) = ± ^pop(x/L) and rxy(x, ±H) = 0, where p0 is a dimensionless normal load. On assuming that the material is linearly elastic, orthotropic, and that the thickness B is small, the stress in the beam may be considered as a generalized plane state. A statically admissible stress field may be derived from a stress function F(x, z) through the relations ax = FjZZ,
az = Fxx,
ixz = -FtXZ,
(16.23)
where F(x, z) must also satisfy the face traction conditions F,xx (x, ±H) = ±\P<*P(T),
F,XZ (X,
±H) = 0,
(16.24)
and the condition that the normal and shear stresses at the end x = L have a zero resultant and zero couple with respect to the y axis. A kinematically admissible state is simply generated by assigning the axial and normal displacements U and W, with 24
Known as Synge and Prager's method (Prager and Synge 1947).
130
16. The Energy of an Elastic Rod
the only restriction that the averaged vertical deflection and the averaged vertical rotation are zero at x = 0. Once we have the two admissible states we can calculate the differences between the strain components
texz = exz - exz = UtZ +WX + ^ FXZi
Aex = ex-ex=Ux--
(16.25)
(Fzz - oFxx),
where E, E\ and G are elastic moduli, and a is Poisson's ratio. The actual strain field is kinematically and statically admissible and the actual stress function must satisfy the compatibility equation Fzzzz + 2EFXXZZ + EFfXXXX = 0,
(16.26)
where
In order to construct a statically admissible field, we introduce the dimensionless variables ^
(16 27)
^
-
and the parameter e = H/L. It thus follows that (16.26) may be rewritten as
and the face traction conditions (16.24) as 0.
(16.29)
Because the stress resultant and the couple at the right end of the beam vanish, we may integrate (16.29) with respect to § and replace them by the conditions25 \
,
0
(16.30)
where
is the dimensionless bending moment along the longitudinal axis of the beam. We assume that the prescribed stresses and displacements at the ends are such that ox and U are odd in z, while xxy and W are even in z. It thus follows t h a t / must also be odd in f. Hence, on integrating (16.28) four times with respect to f, we obtain 25
This modification of the boundary conditions in plane elasticity is known as Michell's transformation.
Rod Theories: Three-dimensional Approach
131
J
Lo
M
,t)dt
(16.31)
.0
where A and B are unknown functions of §. Imposing the boundary conditions (16.30) yields (16.32) where (16.33a)
t)dt-2
, t) dt,
(16.33b)
o(16.33c)
-*/*
The integrodifferential equation (16.32) may be solved by using a perturbation method, on putting / = / ° ) + e 2 / 1 ) + ...,
(16.34)
and equating the terms of equal order in s. We already have f®\ and thus / ^ becomes /
— ~^j^ w "(£)f(l ~ K )
=
4fiEp(£)i>(\ ~~ K ) »
(16.35)
which may then be followed by the calculation off®. The interesting fact is that the approximation/^ + £ 2 /^ leads to the statically admissible stress field obtained by Rychter. To approximate the displacements we put EH2 '
E'H3
(16.36)
Then (16.25) takes the form
AeX2=:^ Ae*=^<
(16.37)
132
16. The Energy of an Elastic Rod
In these equations the function/ is supposed known in the expansion (16.34). We now assume similar expansions for u and w (0)
<16-38)
2 (1)
w - w{0> + s2wm + ..., and choose the coefficients in the expansion so that
„,(»-') = -aE$-2) + Efjg-3*, «J-») =/%-» - oEf%2\
(16.39a) (16.39c)
with the convention that the terms of order less than zero vanish. Equations (16.39) may then be solved in sequence. On integrating the first with respect to f we obtain (16.40) where un_i(£) is an arbitrary function. On inserting (16.40) in (16.39b) and integrating with respect to £, we have
oE - 1 ^ - 2 ) -E
- ry<-3> dt
(16.41)
Lo without any function of integration, as u must be odd in f. The vn terms are determined by substituting (16.41) in (16.39b). In particular, we find from (16.33a) the relation
that is, the value vfm = -\m^\
(16.42)
which is just the moment-curvature relationship of the elementary theory. The two methods presented above allow us to estimate the energy of the strains and stresses in a thin rectangular beam, by means of constructing two or more admissible fields. Both methods lead to the conclusion that, if we restrict ourselves to satisfying static or geometric boundary conditions on the ends solely in the mean, then the approximation can be progressively improved, whatever the pressure distribution over the upper face of the rectangular beam may be. However, it often happens that even this procedure presents difficulties in general circumstances, when, for instance, the cross-section of the beam is not a thin rectangle, or the longitudinal axis is naturally curved. In these situations we can still apply a variational argument, but using a different technique (Berdichevskii 1980) and with a different purpose. Instead of trying to compare the strains and stresses calculated using two-dimensional theory with those obtained using elementary beam theory, we content ourselves with evaluating certain averaged elastic moduli of the
Rod Theories: Three-dimensional Approach
133
stiffness of an equivalent one-dimensional continuum, which approximates a given three-dimensional tubular elastic body. In other words, the problem consists of determining the bending and torsional stiffnesses of the first continuum by averaging in some way the elastic moduli of the second. The one-dimensional continuum is regarded as a curve F endowed with an orthonormal triad di,d 2 ,d 3 , the last of which is tangential to F. The points of F are characterized by a radius vector ro(S), where S is a parameter in F, and the derivative
is a vector tangential to F, but not, in general, of unit length. In the unstrained state the curve F occupies the position F o , determined by a radius vector Ro(S) and by an orthonormal triad e1? e2, e3, the last of these being tangential to F o at the point RoCS), and the first two oriented, for instance, along the principal axes of inertia of the cross-section, although this choice is not necessary. As a result of the deformation, the vector Ro(S) moves to the vector ro(S), and the ratio Y
2\
\dR0\2
)
is a measure (one among many) of the elongation of the rod. If the parameter S is the arc length in F o , then we have \dR0\ — dS and the elongation assumes the form
Y = W*\2-V'
( 16 - 43 )
In addition, let us introduce the quantities K, K, T, and /c0, K0, r0 defined by (14.6) and (14.8) and assume, as measures of the change in curvature, the differences
(16.44)
Our purpose then is to represent the total energy of the three-dimensional elastic body by means of a functional of the form W-L
=[<S>(Y,
Va,n)ds-
L(S)
(a - 1 , 2 ) ,
(16.45)
r0 where O and L are obtained by averaging the elastic moduli of the material over the cross-section. The three-dimensional undeformed configuration of the rod is a domain generated by the motion along the space curve F o of a plane figure A, intersecting F o at its centroid and perpendicular to the tangent to this curve at each point of F o . Let us introduce a system of curvilinear coordinates £", S by means of the formula e a f. 1
(16.46)
The point with coordinates if = 0 is the centroid of the cross-section. We assume that A is centrally symmetric; this means that, in addition to every point with
134
16. The Energy of an Elastic Rod
coordinates %* there is the point — §". The section is, in general, nonhomogeneous, but the inhomogeneities are also assumed symmetric, that is to say the Young's modulus E(%*) and the shear modulus /x(§") are even functions of the coordinates §", and Poisson's ratio a is constant. As a consequence of the deformation, the new position of a point becomes !•(§", S), and the components of the strain tensor are determined by the formula ( 16 - 47 )
2£ab=*X*-|ab
where xl and xt are the contravariant and covariant components of r, the new position, and gab are the components of the metric tensor in the unstrained configuration. The strain energy of the three-dimensional solid assumes the following form:
W= f ds ffA^dfd^, r0
(16.48)
A
with A = I [A(f b s a b ) 2 + 2//| a b | c d e a c £ b d ],
(16.49)
g = detail; and the external work L may be written as
= f ds f[FixiJ§d?xd^+ J
r0
JJ
L
f
J
(16.50)
dA
A
where Ft are the body forces and Pt are the surface forces. In these formulae, the components of the metric tensor in the reference configuration can be derived from (16.46) and their detailed expressions are o2uxux
Tfr Os-2
o33
g
/ i . o«.l • o/«.2\—2
=(1+/C? +K$)
O
= ^ 5 , #23 = ol3 °^-2/i | oul - o/«.2\—2
g =-r$(l+<£
+Jcf
) ,
(lO.Jl)
The strains ^ may thus be written in explicit form when we recall that the position , S) admits the two-fold representation
Now, from (16.47) and (16.51), we can derive the complete expression for the strain energy density A^/f. We now take into account the fact that terms in ij? are of the order of h, the diameter of the cross-section. This is small compared with the length L of the beam, making terms of the order of (h/L)2 negligible, and so the strain energy can be written in the form
Rod Theories: Three-dimensional Approach
135
W= where
2d> = y21 If Ed? dg j + III E?t d? dt J nBfy + CO2,
(16.52)
E being Young's modulus of the section and C its torsional rigidity. This result rests on the assumption that the triad d!,d 2 ,d 3 is orthogonal, as assumed in the so-called Kirchhoff hypothesis for beams. However, the method may be generalized further to the Timoshenko theory, in which the vectors di and d2 are of unit length and mutually orthogonal, but not orthogonal to the vector d3 (Berdichevskii and Starosel'skii 1983) (see Section 14). Yet another extension of the method lies in the treatment of naturally twisted rods (Berdichevskii and Starosel'skii 1985), a particular class of curvilinear rods generated as follows. We consider a segment 0 ^ S ^ L located on the x3 axis of a Cartesian system of coordinates and take a two-dimensional domain A in the plane x 3 = 0 and displace it along the x 3 axis, while simultaneously rotating it through an angle cp = cox3, where co is a constant, about the x3 axis. For naturally twisted rods it is also possible to define an averaged strain energy like (16.52). A problem that can be dealt with by means of a variational argument is that of the so-called generalized torsion of a cylinder (proposed by Truesdell 1959, 1978). This problem may be formulated as follows. Saint-Venant's theory of torsion of an isotropic linear elastic cylinder shows that between the torque M and the twist r the relation M = fiRr
(16.53)
holds, where R is a geometric quantity. The product fiR is the torsional rigidity. However, (16.53) holds only for the exact Saint-Venant's solution corresponding to a very special distribution of tractions on the plane ends of the cylinder, and the question thus arises of deciding whether (16.53) is valid in general for solutions that are not exact. The answer is that (16.53) holds if the twist r is defined as the constant associated with Saint-Venant's torsion field which approximates a meansquare solution in the strain (Day 1981) instead of the exact solution. In order to prove the result, let us consider a cylinder V of height 2/z, the crosssection A of which is a domain in the xy plane and the generators of which are parallel to the z axis, where x, y, and z are rectangular Cartesian coordinates. The vector field w0 = -zy, v0 = zx,
(16.54)
w0 =
where cp(x, y) is a harmonic function in A and satisfies the Neumann condition — = jcos(x, v) — xcos(y, v), ov
(16.55)
136
16. The Energy of an Elastic Rod
is called a torsion field and corresponds to unit twist. If (w, v, w) is any continuous differentiable field, we define the generalized twist r, associated with (w, v, w), as the unique number that minimizes the function
/(<*) = fjf I J^[ey(u, v, w) - aey(uo, v0, w0)]2 dxdy dz
(16.56)
where etj denote the strains. On squaring the integrand and carrying out a short calculation we obtain x = IJI \y v
eij(u0, v0, wo)eij(u, v, w) \dxdydz
=///(£•
3
eij(u0, v0, W0) J dxdydz > .
(16.57)
In this expression we put / / / I 5 ^ 4(«o. ^o» w0) Irfxdy dz = 2h // [Up x - yf + Up v + x)2]rfxJv = 2/zi^, yyy v ^ J I JJ * *y V
A
(16.58) and we transform the numerator in (16.57) in the following manner:
/7/fv* , ///
N/
J^^ ^
2 ^ e#(w» v» w)e!y(wo» ^o» w o)
dxdydz
^z + w ^ X ^ + ^)]rfxrfyrfz = III \\ — (w(^? x — v) + v(^> v + x)) H JJJ [Idz
'
J
[w(^? ^ — ^)] H
[w(^ v + x)] \ dx dy dz
dx
ty
\
\- x)] \ dx dy dz
+ / dz | / [w((piX - y) cos(x, v) + w(
V. oA
}
to obtain the result — Illl—[u((p x—y) + v((p,y+x)]\ dxdydz, JJJ [dz * J T =
(16.59) +v(
'
\
-
(1660)
Rod Theories: Three-dimensional Approach
137
Let us now suppose that (w, v, w) is a solution of the torsion problem, in the sense that the stresses associated with these displacements satisfy the equilibrium equations in V, the tractions on the lateral surface vanish, and the tractions on the plane ends of the cylinder are purely tangential. It is clear that the field (w0, t>o> Wo) is a solution in this sense. Another consequence of the definition is that the torque is given by
M=
II [crzy(x, y , h)x - crzx(w, y , h)y] dx dy A
= - jj [
where azx and crzy are the tangential tractions on the end faces. If we have two solutions of the torsion problem, (w, v, w) and (w, v, w), then, by Betti's reciprocal term, we may write II kzx(w, y, h)u + crzy(x, y, h)v\ dx dy + 11 [azx(x, y, -h)u + crzy(x, y, -h)v\ dx dy A
A
= / / [d'zxix, y, h)u + azy(x, y, h)v] dx dy A
+ / / [&zx(xi y> —h)u + cfzy(x, y, —h)v] dxdy.
(16.61)
A
In particular, if we chose (w, v, w) = (u0, v0, w0), the tractions over the plane ends are Zzx(x, y, h) = -azx(x, y, -h) = IM(
+ M- II (V,x ~ y)u(x, y , -h) + (
V
On the other hand, the left-hand side of (16.61), with (w, 5, w) = (u0, v0, w0), becomes h II [-crzx(x, y, h)y + crzy(x, y, h)x] dxdy-h A
/ / [-crzx(x, y, -h)y A
+ azy(x, y, -h)x\ dx dy = 2hM. Thus, by comparing the last two equations, we arrive at the equality M — any solution of the torsion problem.
(16.63) /JLRT,
for
138
17. The Influence of Large Strains 17. The Influence of Large Strains
The systematic application of three-dimensional theory to large deformations leads to equations of considerable complexity, the solutions of which exist for only a few special cases. For practical applications, it is useful to write the equilibrium equations relating the displacements to certain averages of the exterior loads taken over the cross-section. However, these equations must be such that they allow the generation of a sequence of models at increasing orders of approximation. Classical theories should be included in the early stages (Parker 1979) as, for example, in the case of small strains and displacements, Saint-Venant's solution should result as a first approximation. On the other hand, if we allow for small strains but large displacements and rotations, Kirchhoff s theory of rods should naturally emerge. In order to elaborate the approach, let us suppose, for simplicity, that the rod is straight in its unstressed reference configuration. This will be a cylindrical domain V, with cross-section A, referred to a Cartesian orthogonal system, such that the X3 axis coincides with the center line and the X\ and X2 axes are parallel to the principal axes of inertia of the cross-section. The length of the rod is I, such that 0 ^ X3 ^ £, and the diameter of A is assumed to be small compared with L We recall that we are concerned with deformations in which each cross-section remains nearly plane, although its orientation and displacement may vary appreciably, and so introduce the new variable X — eX3, where s is a parameter, and let L = si be the length of the beam. We now associate with each X a triad of vectors &i(X), d2(X), d3(X) the components of which along the Xx, X2, and X3 axes are the columns of an orthogonal matrix H = H(X). Thus each position r ^ , X2, X?) of the deformed configuration may be represented as r(Xa, X3) = ro(X3) + uada + u3d3 = ro(X3) + Hu
(a = 1, 2),
(17.1)
where ro(X3) denotes the deformed position of the line of the centroids and (wa, u3) = u denotes the position of each point of the cross-section relative to its centroid. The unit vector d3 is tangential to the deformed center line, and we choose di and d2 such that
u3=su3. Equation (17.1) then becomes r(Xa, Z3) = ro(X3) + X,dx + X2d2 + s(uada + u3d3).
(17.2)
This means that the deformation of the cross-section can be split into a term representing the motion of the centroid, a rotation which aligns the Xx and X2 axes along d! and d2, and a superimposed small displacement s(ui,u2,u3) with components parallel to da and d3. The rotation of this triad is associated with a skew-symmetric matrix fl, for which T
= o),
(17.3)
Rod Theories: Three-dimensional Approach
139
where ft has the form ft=
-r
V k
f
0 -K
KV
(17.4)
o/
K and /c' being the curvatures and x being the twist. Formula (17.3) is exactly equivalent to equations (14.7), which give the derivatives of (da, d3) in terms of the curvatures. The extension of the central line is defined in terms of 1^(^)1 = a(X), and, as a{X) is close to unity, we write The position of each point on the rod is determined from (17.2), and we know the terms composing its right-hand side, or, at least, an approximation of it. To determine these terms we set (ui, w2, w3) = u = u + ev + ...
(17.5)
where u has the form u = w{l\Xa) (1)
+ K?iP\Xa) + x
(17.6)
(2)
The displacements w and w correspond to pure bending about axes parallel to d{ and d2, respectively;
(17.7)
where Ho is a constant vector and ft is given by (17.4) but with K = K = 0. Using the characteristic equation ft3 + r2ft = 0, we can write exp(ftTZ) = [1 + x-lilT sin xX + r"2(ftT)2(l - cos xX)] =
/cosrZ -sinrZ 0\ sin xX cos xX 0 ,
V 0
0
1/
and we approximate l o ^ ) by taking
with a displacement u = [0,0, x
,
(17.8)
140
17. The Influence of Large Strains
which, in the special case r0 = 0 and H o = 1 reduces to x2 = Xi sm(srX3) + X2 cos(srX3),
(17.9)
x3=X3+enp(Xa). These formulae generalize, for arbitrary X3, the standard solution for torsion X\ = X\ — szX2X3, x2 = X2 + sxXlX3,
(17.10)
x3=X3+ertfXa). Let us now consider the case in which K = x = 0, with d2 fixed. We choose 0 sinic'X' 1 0 0 cos K'X ,
COSK'X
0 . — sin K'X and
(
1<X2 •
XXX2
o
,,
so that i COS(SK'X3) \
r = ro(X3) + H o |
/ 1 - COS(SK'X3) \ 1
X2
|+(e*T Ho|
xxx2
IS(EK'X3)
0
\
In the special case for which r0 = 0 and H o = 1, we derive {eic')-\\ -
COS(SK'X3)]
+ \eic'o{Xl - X$)cos(eit!X3), (17.11)
x3 =
[(BKT
1
- Xi - \eic'o(Xl - Xi)] sin(sK'X3),
and the standard solution for small displacements can be recovered simply by linearizing the functions of sin(£K-'Z3) and COS^K'A^):
x, = Xi + \&tX\ + \eic'o{X\ - Xl), x2 = X2 + SKaXxX2, x3 = X3 -
(17.12)
?XX
Another interesting solution may be obtained by putting K — 0, K = bic, x = bx, with {ic)2 + f2 = 1, so that the position vector r can be approximated as
(
ic'[cos(bX) - 1] \ K'x[bX - sin(feX)] + H o exp(ftTZ)u,
?bX + K'2 sm{bX))
Rod Theories: Three-dimensional Approach where
(
and
u=
cos bX
x sin bX
x sin bX ic2 + f2 cos bX -ic'sin bX ic'x[l-cos(bX)]
141
—ic sin bX icx[\ - cos{bX)] \, x2+ic'2 cos bX
'Xx+\eic'G(Xl-Xiy
The orthogonal matrix H is periodic in X, and the fibers Xa = constant are coaxial helices with axis parallel to HK 3 = H O K 3 , where K3 = (0, ic\ f)T is the unit null vector of SI. These solutions are known as Ericksen's helical deformations of prisms (Ericksen 1977). In some cases the purpose of the theory is that of obtaining a set of equations which, while still preserving the relevant nonlinear terms, are sufficiently simple that they can be treated mathematically or even be solved in some particular cases. A theory of slender straight beams undergoing small strains and moderate rotations, which is useful for describing the behavior of rotor and turbine blades, can be formulated as follows (Rosen and Friedmann 1979). The, initially straight, longitudinal axis of the beam is the line connecting the centroids of the cross-sections. This line is taken as the Z axis of a Cartesian system of coordinates, with the origin O placed at the centroid of the left face of the cylinder; the X and Y axes coincide with the principal axes of inertia of this face and are also the axes of symmetry of the cross-section. Now, we introduce a triad of orthonormal unit vectors e!,e 2 ,e 3 placed along the coordinate axes so that the position vector of any material point before the deformation is represented by R(X, Y, Z) = Xex + Ye2 + Ze3.
(17.13)
The vector equation of the central axis is then the straight line R0(Z) = R(0,0,Z) = Ze3. After the deformation, the curve Ro(^) is displaced to a new curved line ro(Z), and the tangential vector to this line is the, nonunit, vector g3=ii(Z).
(17.14)
The unit vectors drawn from the centroid of the generic cross-section parallel to the X and Y axes are changed to a pair of nonunit vectors gx and g2. Knowledge of the triad gi,g 2 ,g 3 characterizes the deformation of the cross-section. However, in order to explain the process of deformation, it is often convenient to define another orthonormal triad d1? d2, d3, such that d3 coincides with g3, but is of unit length, di is a unit vector perpendicular to d3 and contained in d3g! plane, and d2 is a unit vector perpendicular to di and d3. The triad d1? d2, d3 defines the principal torsion-flexure axes. The introduction of the principal torsion-flexure axes helps in understanding how the deformation of the cross-section can be split into successive stages. First, the triad e1?e2,e3 is rigidly translated and rotated so as to coincide with the triad di, d2, d3; next, this triad is stretched and sheared so that it is superimposed on the
142
17. The Influence of Large Strains
triad gl9 g2, g 3 . In linear theory it is commonly assumed that the two deformations are both infinitesimal, but sometimes it is necessary to remove the hypothesis that the rigid rotation is small. This means that the position of the principal torsion-flexure axes cannot be described by a skew-symmetric matrix like £2, but must be treated by means of at least three finite angles. The canonical choice of these angles is the Euler angles of rigid body dynamics. According to Euler's method, the passage from e l 5 e 2 , e3 to di,d 2 ,d 3 may be done by means of successive rotations cp, 0, and ty, which yield (Love 1927, Art. 253) di = (— sin \jr sin
Fig. 17.1 26
Rosen and Friedmann (1979); however, some terms of second order are retained.
Rod Theories: Three-dimensional Approach
143
d! = e 1 +
(17.16)
d3 = uex + t/e 2 + e 3 . These expressions for the rigid rotation of the triad ej, e2, e3 are useful for describing the case of small strains accompanied by moderate rotations. Curvature and twist follow from (17.16) through (14.6): K
= d2 d3 = - d 2 • d3 = v + ucp,
K? = 4b-&l=i/' + t/'
(17.17)
If the strains within the cross-section are neglected, the relevant strains can be derived from (14.17), and they assume the simplified form e33 = -KX + KY,
e3l = -xY,
e32 = xX.
(17.18)
The bending moments about the d2 and d3 axes and the torque about the d3 axis can thus be written as
M2 = ffcr33XdA = -EI2K\
MI
= ffa33YdA = EI{K,
A
A
(17.19)
M 3 = / / {anX -
These couples are related to one another by the equilibrium equations and the resulting couple is a vector m, which may be expressed as the sum m = MXAX + M 2 d 2 + M 3 d 3 ,
(17.20)
or, alternatively, written in terms of the fixed triad, m = Mt(ex +(pe2-
w'e3) + M2(-(pe{ + e2 - t/e 3 ) + M3(uex + ve2 + e 3 ).
(17.21)
The vector couple m must satisfy the equilibrium equation m , z + r o x n = O,
(17.22)
where n is the resultant of the shear forces and the axial force acting on the crosssection. In the absence of a load distributed along the longitudinal axis the equilibrium equation for n becomes n, z = 0, whence we immediately derive the result n = constant vector = n(L),
(17.23)
where L is the value of Z at the right-hand end of the beam. By using the last result we can integrate (17.22): m + r 0 x n(L) = constant vector.
(17.24)
In this vector equation n(L) is known, m is an explicit function of cp, u, and v as a result of (17.21) and (17.19). The vector r 0 is also representable, as the sum
144
17. The Influence of Large Strains r o ^ R o + we! +^e 2 + we3,
or also by its approximation r0 ~ Ro + uex + ve2.
(17.25)
It turns out that (17.24) is equivalent to a nonlinear system of three ordinary differential equations in u, v, and cp. Despite the difficulty involved in effecting a solution, a remarkable simplification often occurs for beams having a fairly thin cross-section, so that the rigidity factor, say EI2, is much greater than EI\ and GJ, and the terms containing EIX and GJ may thus be neglected without appreciable error (Rosen and Friedmann 1979). In some other circumstances, as, for example, in the design of tailored helicopter blades, the nonlinear effects occur in naturally curved and twisted beams. However, their treatment may be simplified if we assume that the cross-section is undeformable in its own plane and that the initial curvatures of the beam are small (Bauchau and Hong 1988). In its undeformed state, the beam is defined by it axis, which is the line of centroids of the cross-sections, and by a plane perpendicular at each point to the central line, called the plane of the section. The position vector of a particle before the deformation is a vector function of the form (17.26) where Xa and X3 are curvilinear coordinates chosen in such a way that (17.27) is the equation of the central line. The direction of the central line and the position of the cross-section for each X3 are defined by an orthonormal triad e l5 e2, e3, where the unit vector e3 coincides with the tangent, and the unit vectors ex and e2 coincide with the principal axes of inertia of the cross-section. The derivates of this triad with respect to X3, denoted by ei, e2, e3, are related to the initial curvatures and twist by a formula like (14.9). If, in particular, the coordinates Xi and X2 are Cartesian coordinates in the plane of the cross-section, the position vector can be expressed as R(XlfX2, X3) = R0(X3) + exXx + e2X2,
(17.28)
and the base vectors, still recalling (14.10), become G^R^ej,
(17.29a)
G2 = R 2 = e 2 , G 3 = R 3 = e 3 = (R'o - Kf0Xxe3 + K0X2e3) - TQX2ex + r0Xxe2.
(17.29b) (17.29C)
The vector Rf0 is parallel to e3, but not of unit length. However, by using the relation G33
_ — e3,
(17.29c) assumes the form G3 =
(JGT3
-
K'0XX
+ K0X2)e3 - To^e, + r^Xxe2.
(17.30)
Rod Theories: Three-dimensional Approach
145
The geometric structure of the beam after the deformation is defined by a position vector T = T(X1,X2,X3),
(17.31)
the values of which for Xx = X2 = 0 define the central line ro = r(O,O,Z3) = ro(Z3).
(17.32)
In addition, for the strained central line we may introduce an orthonormal triad di, d2, d3, which usually coincides with the principal torsion-flexure axes, such that the derivatives with respect to X3, denoted by di, df2, d3, are related to the triad by formulae like (14.7), where /c, K, and T are the curvatures and the twist of the deformed axis. If we assume that the cross-section does not deform in its own plane, the position vector in the strained configuration can be written as (Bauchau and Hong 1988) r(Xu X2, X3) = vo(X3) + d!*! + &2X2 + [83(X3)w3(XuX2) e23w2(Xu X2)]d3,
(17.33)
where wu w2, and w3 are three functions representing the warping of the crosssection, the forms of which are considered as given;27 83(X3) is an unknown function characterizing the magnitude of the torsional warping w3; and eX3 and e23 are two unknown averaged shearing strains characterizing the magnitude of the two other bending components of warping. From (17.33) we can derive the base vectors in the strained configuration: gx = r ! = di + [83w3A + el3wul + e23w2A]&3, g2 = r 2 = d2 + [83w3a + el3wh2 + e23w2a]&3, g3 = r 3 = ro,3 + di * i + d ^ ^ + [8'3w3 + e[3wx + e'23w2]&3, [83w3 + eX3wx + 523w2]d3, where again di, d2, d3 may be expressed in terms of d b d2, d3 through the curvatures K and K and the twist r. The vector r03 is still parallel to d3, but not of unit length. It may be written as
where s is the elongation of the central line. In order to reduce the number of unknowns, let us introduce the displacement u(X3) of a particle of the central line, so that By differentiating this equation with respect to X3, and recalling the connection 2 8
we obtain 27 28
I n fact W\ is the torsion function
146
17. The Influence of Large Strains (17.35)
This equation can be exploited in two ways. First, we may express the elongation s in terms of u'; then we may find a relationship between £13, e23, an( * u ' by observing that, by definition, we have the values
(0,0 f Jr 3 ).d 3 .
(17.36)
On applying the general formulae (14.11), we thus obtain the strains in terms of the seven unknowns (w1? u2, w3) = u, K, K, T, and the warping amplitude <53. The resulting expressions for the strains are still very complicated, but they can be simplified by neglecting terms that are presumably small with respect to the others, a process of simplification which is not rigorous in general. It remains for us to discuss the solution of the equations, because there are seven unknowns, whereas the balance equations contain only six unknowns. This indeterminacy can be overcome by recourse to a direct variational method. The full nonlinear treatment of a prismatic slender beam may be performed using the method outlined in Section 9 (Dokmeci 1972). The beam is referred to a system of Cartesian coordinates, such that the X$ axis is the locus of centroids of the crosssections and the X\ and X2 axes are the principal axes of the cross-section. In order to ensure that the solutions are sufficiently smooth, say continuous with their partial derivatives up to second order, we assume that the cross-section is bounded by a simply connected Jordan curve C, that is to say, a non-self-intersecting and continuously differentiable curve. The displacement components of a generic point are represented by uk(XuX2,X3,t)~
£
^m\Xi)Q^{X2)u^n\x,j\
(17.37)
n+m—O
where F^\X\) and Q^(X2) are given functions and u^n) are functions to be determined. There is a certain freedom of choice in the form of the functions F$(X{) and Qm\x2), provided that they make a complete system in the sense of the convergence in the mean, and satisfy the conditions
For example, we may take \
«\
X"2,
(17.39)
or some other system of functions, such as Legendre or Jacobi polynomials. In terms of the displacement components, the strain is defined by Yu = Wk,i + ultk + urtkurtl), or by Yu = £u + 3 (srk + a>rk)(eri + &ri)>
(17.40)
Rod Theories: Three-dimensional Approach
\A1
where Gkt =£ik=j
(ujc,e + ul%k\
cbH = -cbtk = \ (uktt - uttk).
(17.41)
On using (17.37) and (17.39) the strains take the form YK = (XUX2, X3, t) ~ JT XTX^n)(X3,
0,
(17.42)
n+m=0
where 1 m+n
E # 4 } \\ p+q=0
)
],
(17.43)
with
£
(m,n) _ (m,n)
33
— M 3,3
'
= 3
(17.44) The stresses are derived from the momentum balance equation *#P + PoVGf = P0VG —j,
(17.45)
where p0 is the density of the undeformed body, G is the determinant of the metric tensor in the reference configuration, and G~1xp is the Piola-Kirchhoff stress vector. This is related to the Cauchy stress vector tp by the equation (Antman 1972, Sec. 3) xp = y/gtp,
(17.46)
<Jg being the determinant of the metric tensor in the deformed configuration. The vector xp can be expressed in terms of a stress tensor G~T, with components Tpq (Tpq =
(Y1AT)
For the problem of the cylinder, the reference coordinates X\, X2, X3 are orthogonal Cartesian coordinates, and hence the metric tensor in the reference state is a unit tensor whereas the base vectors in the actual state may be written as %q = *,q = (R + *),q = % + U q. It then follows that the balance equations, written in terms of components, become [T"r(Srq + uqj)\p
+ pofq = A, -$• •
(17.48)
In general, surface tractions or displacements on the surface of the cylinder are known, as are the displacements and velocities at the instant t = 0. However, in
148
17. The Influence of Large Strains
order to eliminate unessential terms, let us assume that the lateral surface and the end faces are free from tractions, so that only body forces pof and inertial forces po(32u/3?2) act on the cylinder. For each section we may define a resultant body force of order (m, n)
ft* = f Pof X?X dX dX , k
n
2
x
(17.49)
2
and a stress resultant of order (m, n) T(m,n) =
f
^
^
Similarly, we may introduce an average displacement of order (m, ri)
p+q=0 m+
n+(
where j( P> fi represents a moment of inertia of order =
(m+p,n
f X™
As Xi and X2 are the principal axes of inertia, the above definition yields
Moreover, the following relations hold for a symmetric cross-section with respect to r(m,n)
P
where n0 stands for any even integer and 7(mw) is a centrifugal moment of order (m, n). It is clear that for a symmetric section with respect to X2 we have
with m0 an even integer. For a section that is symmetric with respect to Xx and X2, 7 (w ' n) becomes j{m,ri)
1
c —
XT °mm(i0nno1(m,n)'
Stresses and strains must be related by constitutive equations. In the case of a hyperelastic material, the stress-strain law has the form
(£ + £).
(.7.53)
where W is the strain energy function measured per unit volume of the undeformed body. By using (17.42) and (17.50) we can derive the constitutive relations for the stress resultants:
Rod Theories: Three-dimensional Approach T(m,n)
_ 1/
c/2,
c)2,
149
\
(17.54)
where E= / ^ ^ J Z
2
represents the strain energy function measured per unit length of the undeformed beam. We now proceed to construct the balance equations involving the stress resultants of order (m, n) with the purpose of transforming the original system of partial differential equations into a sequence of systems involving the partial derivatives with respect to X3 and those with respect to time. We start from (17.48), where the displacement components uq are replaced by their formal expansions \X2U i+k=0
The balance equations thus become
W=0
01
\i+k=0
)
Let us now multiply both sides of these equations by X^X^ and integrate the resulting expressions over stf. The term T^XfX^ can be integrated by parts, and becomes
f T]
dX2 =
l ds - f mTl
l
X^ dXx dX2,
and a similar expression holds for the term with p = 2. The term / \i+k=0
after integration by parts becomes
\i+k=o
dX2 where the second term can again be integrated by parts when r = 1, 2. A similar expression may be derived for p = 2.
150
17. The Influence of Large Strains
Now, for each pair (m, n), the sum of integrals over si gives an equation of the form T(,m,n)
T(m-l,n)
T(m,n-l)
U/n.n) _
a2
°
rr{m,n)
/i?
r^
where N^"'^ is the collection of all remaining integrals over J / . On the other hand, the sum of the integrals through dstf imposes a further condition, which may be written symbolically as ff'n)
+ Rf>n) = 0.
(17.57)
We have supposed that the boundary conditions on the end faces are free, although this is not an essential restriction. The free state, however, requires, say for X3 = 0 and X$ = L, the equation rApi,ri)
2
3q
n
-
s\
U
*
n)
The integrals N^ \ Rf> in (17.56) and (17.57) have been isolated because they contain all the nonlinear terms of the problem, and so may be neglected in a linearized theory. For this case, in addition to neglecting iv£m'^ and R^n\ the constitutive equations may also be linearized using the Lame equations, for example. The known theories of beams can be recovered by particular choices of the functions i/™'"* in (17.37). For instance, let us take
U2
where ,.(2,0) _ i « - l f i (U)
_ p-l_
(0,2) _ f .(l,0)
_
i p-1
(0,0) _ i r>-l
y2
p-lY
In this case, the displacements are exactly those corresponding to Saint-Venant's theory of a bar bent by two terminal couples about the X2 axis (Love 1927, Art. 87). The method of expanding the solutions in powers of the distances of a point on the cross-section from the centroid may be extended to spatially curved beams (Gordienko 1979). The beam is defined geometrically by a curve in three-dimensional Euclidean space, the points of the curve being specified by the arc length S from a given point on the axis, and a cross-section J / ( S ) , which in general varies with S(0 < S < 1). The material is supposed linearly elastic, homogeneous, isotropic and the loading conditions are assumed known. Let PQ be a point on the axis in the unstrained state and let si(S) be the crosssection drawn through Po perpendicular to the axis. If t is the unit tangent vector, and n and b are the unit vectors of the normal and binormal to the axial line, we may choose a system of reference axes coinciding with these three unit vectors, and denote the coordinates of a point P of J / ( S ) with respect n and b by r\ and f, respectively. If Ro is the radius vector that identifies P o , the point P is specified by the vector
Rod Theories: Three-dimensional Approach
151
R = Ro + *7ii + £b.
(17.58)
After the deformation, the point P moves to another point P\ and occupies a new position r = R + u,
(17.59)
where u denotes the displacement, also representable as the sum of its projections along the directions t, n, and b: u = tu{ + nw2 + bw3.
(17.60)
On differentiating (17.59) and utilizing the Frenet-Serret formulae,29 we have 3R
fo
3R
f
8R ,
8u
= [(1 + du2 n + du3 b + D^/con - "2(^0* + To*>) + "3^on] ^K.
(17.61)
In the unstrained state the base vectors are Gi = — = (1 - KOrj)t + ror]n - T07?b,
G 2 = n,
G 3 = b.
Let us denote the derivatives of u with respect to S, rj, and f by u,z (/ = 1,2, 3) and write, still using the Frenet-Serret formulae « 1 = ("1,1 ~ K0"2)t + ("2,1 + KoWl - T()W3)n + (W3,l u
,2 = "1,2* + "2,2 n + W3,2l>>
« 3 = "1,3* + W2,3ll + "3,3 b -
It is also convenient to put +
^ =
+
^13 = "3,1
-
(1 = 2, 3; 7 = 1,2, 3). My-,
As we already know the metric tensor in the unstrained state, the metric tensor in the deformed body becomes
gij = it ' ij = (G/ + 11,/) • (Gj +U,y) = Gtj + Isfj, where the quantities etj are expressed as 2etj = etj + 29
eji
+ eisejs - (1 + ^ ( r y , f),
The Frenet-Serret formulae are: 3RQ
9t
9b
where /t0 and TQ are the curvature and the torsion of the unstrained axis, respectively.
(17.62)
152
17. The Influence of Large Strains
and 01/ = fro^/i - *oefl)ri + ^0^-2? (/ = 1. 2, 3), «21 = 012.
031 = «13»
022 = 033 = 023 = 032 = 0.
We now expand the functions uj, eg, and etj in power series in t] and f, and obtain (up ef Sij) = (ujtPq\ eijtPq; eij%pq)rf?,
(17.63)
having used the following relations e
U,pq = u\,pq -
e
U
K u
e
l3,pq = U3,pq
0 2,pq>
K u
T u
\2,pq — 2,pq + 0 l,pq ~ 0 3,pq>
e3jtPq = (q+ l)«/,^+i) 0//>^ = (KOeji,(p-l)q 021,p^f = 012,/?^'
0" = 1.2, 3),
r e
0 j3,(p-l)q)
0 3 1 , ^ ~ 013,^»
022,/?^ = 033,/?^ = a23,pq = a32,pq = 0-
Here the indices /? and q denote the summation from zero to infinity, and k and I denote the summation from zero to p and q. Since the material obeys the Lame constitutive equations, with moduli E and a, the Cauchy stresses associated with the strains etj have the form
where 0 = ess and B = a/{\ — 2a). The quantities|^. have the physical interpretation of being nondimensional stresses, and their derivatives with respect to rj and f are given by
)
(1764)
b-
'
The resultant stress vector in dimensionless form has components of the type
N°ik = N°kl = 1 J °eikdA =§k,p/2pq
(k = 1, 2, 3),
and the components of the couple resultant vector are Mn=Mt
=-
1
1V±22 — M-n — "~7
f J ($\3r)-e\2t)dA = (e\3,(p-\)q-e\2,p{q-\))lpq, I
A J
6\\^aJi
—S\\ p(q—l)^pqi
sf(S)
1 C ^33 = Mb = --£ / e\\r\dA —
-en,(p-\)qlpq,
where summation with respect p and q is understood, and we have put
Rod Theories: Three-dimensional Approach
t00 =
1,
^ i o = '/0>
153
^01 — SO-
In these expressions Ipq denotes the moment of inertia of order (p, q), and r/0 a n d ?o the coordinates of the centroid of the cross-section in the system (t, n, b). If the axis connects the centroids of the cross-sections, then rj0 = f0 = 0; and if the axes n and b are the principal axes of inertia, £n = 0 and lpq — 0, provided that at least one of the numbers p or q is odd. Having defined the stress resultants and the stress-couple resultants, the formal structure of the balance equations may be obtained from the Hamilton principle, which states that the integral (T-W)dS)
(17.65)
Lo must be stationary for all synchronous motions, that is to say, the candidate functions ut must have the same values for t = t0 and t = tx. In (17.65) W represents the (dimensionless) strain energy stored in the cross-section: W = (1 + a) j
^Sy dA = (\+
a)Al2p^jMsil(p.k)(q_l),
and T is the (dimensionless) kinetic energy J where the dot denotes the derivative with respect a parameter t = (cT0/L), To is the physical time, and c2 = 1/(1 + a) is the speed of sound corresponding to a material of unit density. In the functional T —W the variables are dcp
dcp
and the Euler equations, which follow from the condition that the functional (17.65) should be stationary, are — (—) - — ( — ) + — = 0, (17.66) at \dr J aS \op J dip and they must be supplemented by certain initial and boundary conditions, which depend on the specific problem that we have formulated. The advantage of this method is that we are in a position to recover all the known theories of curved beams by simply truncating the expansions (17.63) up to a given finite order. For example, let us assume that the displacements have the form «i = («i)o + («i)i V + («,-)2?>
(17.67)
154
17. The Influence of Large Strains
where (w;)0, («/)i> and {ut)2 are functions of the only variable S. If the beam has small transverse dimensions with respect to the length, then in the formulae for the strains we may neglect the terms quadratic in rj and f. In particular, it is possible to put
9«3 , , - ^ T = ("3)2
For a first theory we assume that the stresses | 2 2 , £33 > a n d £23 = £32 are absent. The assumption that | 2 3 = 0 implies that | 2 3 = 0, and hence the result ("2)2 = - (
Furthermore, the assumption that | 2 2 = £33 = 0 yields the connections
and, consequently, by disregarding the terms in r) and f, we have the formulae
In this manner we have found that the displacements (17.67) take the form "1 =(wi)o + (wi)i*7 + (Mi)2f> "2 = ( " 2 ) 0 - ^ - ^
(17.68)
"3
where cp and T/T have the values
In a second theory we put £22 = £33 = £23 = 0, so that (w2)i = (w3)2 = 0,
(M 2 ) 3 = -(w 3 ) 2 .
Then the displacements become
w2 = ("2)0 - P?.
W
3 = ("3)0 + W-
(17.69)
The version (17.68) corresponds to a Saint-Venant's model of curved beams; the version (17.69) describes a beam which is perfectly rigid in each cross-section. The effect of nonlinear terms can also be taken into account in the problem of the torsion of an initially twisted bar (analyzed in Section 15). Here, however, it is also important to consider the influence of a tensile force acting at the same time as the torsional moment (Rosen 1983), because the strains induced by the two kinds of force cannot be studied separately. We wish to study an initially twisted bar like the one shown in Figure 17.2, of length I and constant cross-section. Each section is rotated about the longitudinal x
Rod Theories: Three-dimensional Approach
155
Fig. 17.2
axis of an angle ft with respect a fixed section x = 0, and the measure of pretwisting is the quantity K
=
dp dx'
(17.70)
which is assumed to be a constant. We also limit ourselves to the study of twisted bars the cross-sections of which admit two axes of symmetry, in order to decouple the torsion and extension from the bending. We choose a system of Cartesian axes, such that the origin O is placed at the centroid of the bottom section, the x axis is drawn vertically upwards, and the y and x axes coincide with the axes of symmetry of the section x = 0. A generic crosssection, at a distance x from the basis, is still plane in the undeformed state and its axes of symmetry, denoted by rj and £, are rigidly rotated with respect the y and z axes. The angle between rj and y is set equal to /?. Upon the action of a torque Mt and an axial force T applied at the upper end of the bar, a state of displacement is created at each point, and its vector form is u = uex
vev
wez,
(17.71)
where ex, ey, and ez are unit vectors in the x, y and z directions. Since we now wish to account for large displacements, we assume the following representations for u, v, and w: u = u{(x) + 0(x)\l/(x, y, z), v = y(cos
(17.72)
156
17. The Influence of Large Strains
where ty is a warping function, (p = cp(x) is the angle of rotation of the cross-section due to the elastic deformation, and 0 is the twist associated with
. dx In order to simplify the treatment, we suppose that the bar is free to warp at the ends, so that the torsion is uniform along the bar, with the consequence that 0 is a constant and not a function of x. The nonlinear strains associated with the state of displacement (17.72) have the form
2 ,x
(tXty
fXty
+ 0V + z2)l +
WfXWj)
-z + fr/
(17.73)
*xz = U,z + W,x + (u,xU,z + V,XV,
= Qltz+y +
t
z
(
)
where
is the elongation, which is assumed to be constant along the bar. In general, the terms occurring in the definition of the strains are of different orders of magnitude, because pure strains are often much smaller than rotations. This implies that, in a simplified theory, the underlined terms in (17.73) may be neglected. The consequence is that the equilibrium equations are derivable as the Euler equations of the variational problem i
W-L = fdxjj^ 0
e\x + 1 ( 4 + 4)1 dydz - Tux{t) - MMf) = min, (17.74)
j/
in the class of functions ux and
W -L=
f dx\^(As\ o ! ^
2
+ 2Sex0 + KO2 + Ipex# + DO3 + \F04)
- Tex - Mto\ = min,
(17.75)
Rod Theories: Three-dimensional Approach
157
where we have put
D = ff ^x(y2 + z2) dy dz, F = ff (y2 + z2)2 dy dz, Ip = ff(y2 + z2)dydz, Js = JJ[(-z + V^)2 + 0 + tzf] dy dz, K = ff tfxdydz, S = fj ir,xdydz. si
s#
The condition for the first variation of W — L to vanish with respect to ex and 0 yields the pair of algebraic equations
ESsl + (GJS + EK)0 + EIPGX9 + \EDG2 +±EF03 = Mt.
(17.76)
Elimination of s\ from these two equations yields
(17.77) The equation so obtained deserves some comment. The most surprising fact is that the presence of a tensile force T increases the torsional rigidity by the amount (Ip/A)T, whereas the effective torsional moment is no longer Mt but [Mt — (S/A)T\. These two resulting effects are well known by designers of blades propellers and helicopter rotors. The formulae found so far can be further simplified when the cross-section is very thin, as occurs in a rectangle with sides 2b and 2h with h <£ b. A generic section at a distance x from the origin is defined by the inequalities
where rj and £ are the local coordinate related to x and y by the formulae rj = c
so that, by repeating the argument given in Section 15, we have
158
18. Justification of the One-dimensional Model
It thus turns out that the various constants D and F have the expressions:
s= where, sometimes, the terms containing powers of the ratio f/?/ may be disregarded.
18. Justification of the One-dimensional Model The three-dimensional theory of rods is essentially founded on the idea of representing the position of a point in the deformed configuration of a rod by a formula like (9.1) or, more generally, (8.1), which formally separates the dependence of the unknown functions on the S variable, which denotes arc length along the central line, from the other two variables, which represent the distance of a point in the cross-section from the axis. The second step of the theory is that of choosing a specific form for the function by which the unknowns depend on the variables representing the distance of a point from the axis, with the hope that, provided that the rod is slender, some form of the function of these "transverse" variables might constitute an approximation to the exact law of dependence. In the usual theories, we assume that these functions are polynomials. The third step in the procedure is that of averaging the three-dimensional equations of elasticity over the cross-section in order to obtain a system of ordinary differential equations in only one variable, S. However, this procedure, purely heuristic, in the sense that the choice of the above-mentioned polynomial functions is justified a posteriori, conditioned by the requirement that the classical theories of rods emerge as corollaries of particular choices of the polynomial functions. It is thus understandable that the need has arisen to justify all these attempts in a more rigorous way, and to explain how the results might be interpreted as steps in an asymptotic approach to the study of slender rods. The method applies to a cylindrical domain Q = co x (0,1), where co is a bounded regular two-dimensional domain in the plane x3 = 0 of a Cartesian system of coordinates, with the origin at the centroid of co, and with the x\ and x2 axes coinciding with the principal axes of inertia of co. Without loss of generality, we assume that the area co is unity, and the length of the cylinder is unitary (0 < x3 < 1). Now, for e > 0, we define co8 to be a domain of the x\Xi plane, such that
Rod Theories: Three-dimensional Approach
159
(0
Fig.18.1 and, accordingly, we introduce the cylinder
As e varies the sets Q£ describe a family of cylindrical domains the cross-sections of of which have a diameter of order s (Figure 18.1). We consider the sets Q£ as reference configurations of a family of slender elastic rods the constitutive equations for which are Lame's equation, with constant moduli X and /x. The beam may be loaded and constrained in several ways. For simplicity, however, we suppose that the lower face CO£Q at x3 = 0 is rigidly clamped, and only surface tractions h\ are applied on the upper face co\ at x3 = 1. It is well known that, if we denote by Vs the set of (smooth) kinematically admissible displacements (in the present case, the field of displacement v£ vanishing on col), the solution us and the associated stresses
where Yij(uk) = \(uh + "/,*)>
are
solution of the functional equation
= j h\v\ dxx dx2,
(18.2)
for all vE in Vs. However, in order to take into account the smallness of co£, we formulate a problem equivalent to problem (18.2), but now posed over a domain that does not depend on s. Accordingly, we associate each point (jq, x2, x3) of Q with each point of & through the correspondence:30 First used by Ciarlet and Destuynder (1979).
160
18. Justification of the One-dimensional Model (x\, x2, x3) e Q - • (sxi, sx2, x3) e &.
Then, for all v£ in Vs we set va(xu x2, x3) = e~lvsa(exl,£x2,
x3),
v3(xu x2, x3) = £~2v£3(£xusx2,
x3),
*2, x3) = 8~4aeap(sxusx2,
x3),
x2, x3) = e~3asa3(£xuex2,
x3),
2
U
£ X
2
,
(18.3)
x3).
In addition, in order to control the way in which the assigned loads depend on e, we require that ha(X\,X2, X3) = £~3h£a(£XU£X2, X3),
h3(xi,x2,
x3) = e~2hea(exi,ex2, x3),
(18.4)
where the functions ha and h3 are independent of £. An immediate consequence of (18.3) is that the strains have the form , X2, X3) = y£a£(£XU £X2, X3),
i,x2, x3) = e^yitiexuexfr Y33(xux2,
x3),
x3) = £-2ye33(£XU£X2, x3).
It might appear that formulae (18.4) are the result of an unjustified choice, but the explanation lies in the invariance, up to a certain multiplicative power of e, of the integrals in (18.2). We have, in fact, / °ijYfj(vk) dxx dx2 dx3 =£2 I <TijYy(vk) dxx dx2 dx3, I hsvs dx\ dx2 = £2 I hfVi dx{ dx2.
Replacing the domains Qe and co\ by Q and coi, the original boundary value problem (18.2) allows the following alternative formulation. We wish to find a state <jg, ue such that the equation / °ijYij(vk) dx\ dx2 dx3 = I h£Vidxx dx2,
(18.5)
holds for any vt belonging to the set V of functions vf which is kinematically admissible. If we apply (18.3) and (18.4), and we associate the terms of the same order in e, we arrive at the conclusion that (18.5) is equivalent to the problem
Rod Theories: Three-dimensional Approach
(t
161
t
I / cr33Y33(vk) dx\ dx2 dx3 — I h3v3 dx\ dx2 /
\
(18.6)
for every vt e V. In order to solve this problem, we make the assumption that ut and cry can be expanded in the formal series Ui
= w? + eu\ + £2M? + . . . ,
with atj related to uf through the constitutive equations (18.1) and the straindisplacement relation. By substituting (18.7) in (18.6) and equating the terms of the same order in e to zero, we get a sequence of linear problems in the unknowns uj, u], u2t,..., and, of course, in the stress components o^, crjj, ajj,..., which depend on the partial derivatives of the displacements. The method of asymptotic expansion is remarkable for two reasons. The first is that we can derive all Saint-Venant's solutions in sequence. This means that the terms of zero order in (18.3) permit us to evaluate the influence of the loads h3 parallel to the longitudinal axes, the terms of order unity take account of loads ha parallel to the terminal sections, and the terms of higher order are solutions of homogeneous equations. The second argument for which the procedure is important is that it applies also to nonlinear strains and constitutive relations. For instance, it may be extended to the case in which the strains maintain the nonlinear terms and have the form Yij = \ (uij + UJJ + «r,i«rj)»
(
18
*8)
where y^ is now the Green-Saint-Venant strain tensor, and the stress, or, more specifically, the second Piola-Kirchhoff stress tensor, has the form ^AB an
are
=
XO^AB
+ YIYAB + YI(YAB)\
(18.9)
functions of the principal invariants of the strain tensor where yo> Y\> d Yi (Cimetiere et al. 1988). The application of the asymptotic expansion of the solutions is now more difficult, but the result is that we can recover, in increasing order of approximation, the nonlinear versions of Saint-Venant's problem in all its six cases. Naturally, the question of the convergence of the formal expansion of solutions, and the question of solving the sequence of nonlinear problems is still unanswered.
Chapter III Rod Theories: Director Approach
19. Rods as Oriented Bodies
Rods or, more generally, thin bodies are conceptual abstractions of physical bodies having one or two dimensions much smaller than the third. The prototype of a thin body is a cylinder in which the diameter of the cross-section is much smaller than the length of its generators. If we have to determine the stresses and strains in an elastic cylinder under the action of given loads, the solution of equations of threedimensional elasticity appears as the canonical procedure. Sometimes, however, an alternative method of solution may be suggested by the slenderness of the cylinder. This leads us to attempt a solution which will take the form of a limit of the threedimensional solution when the diameter of the cylinder approaches zero, while the resultant of loads and the global stiffness of the body tend to finite limits. Obviously, the cylinder must not be conceived as a straight line which can only be lengthened, but account must also be taken of the bending and twisting of this line. This is achieved by regarding the cylinder in the limit as a line endowed with directions that may rotate independently of the deformation of the line to which they are attached. The directions, associated with the axis, are called directors and, a material curve, together with a collection of directors assigned to each particle, that is able to deform independently constitutes a Cosserat rod (Cosserat and Cosserat 1907). In general, the present configuration of a Cosserat rod is specified by a vector function of the variable S (Sx ^ S ^ S2) and the time t: r = Bo(5,O.
(19.1)
This gives the position vectors of a point of a curve in three-dimensional Euclidean space at S, the arc length of the curve. The directors are a set of vectors dk(k = 1,..., K), which are also functions of (S, t). We require that, under the change of frame (19.2) the vector p0 transforms according to the rule p o (S,O^c(O + Q(OPo(S>O,
(19.3)
and the directors transform according to the rule (19.4) 162
Rod Theories: Director Approach
163
This means that, while p0 behaves like a position vector under a change of frame, the dfc terms behave like differences between position vectors. In other words, directors are not influenced by pure translations of the frame. In addition, we require that the vectors p0 s, d1? and d2 to be independent: (dixd2)>O,
(19.5)
which is called the continuity condition (Antman 1972, Sec. 14). The most important case is that with K = 2, because it embodies the simplest director theory of rods that is able to account for bending, torsion, extension, and also for shears and transverse extensions. The configuration of the rod at time t = 0 is conventionally taken as the reference configuration. This is defined by the vector R = P 0 (S),
(19.6)
and by a set of vectors Dk(S)(k = 1 , . . . , AT), called the directors of the reference state. In general, both the position vector, and the directors of the reference configuration are assumed to be known and, in the most interesting case (K = 2), the initial configuration is characterized by the three vectors R, D1? and D 3 , with the continuity constraint P 0 , s • (D! x D2) > 0.
(19.7)
An important preliminary question is that of constructing a differential description of the undeformed rod, starting from a knowledge of the position vector and the two directors. As a first choice, Dx is placed along the principal normal and D 2 along the binormal, so that the triad D l5 D 2 , P O s = D 3 , is oriented as a right-handed system, coinciding with the principal triad of the curve. The vectors D^ (k = 1, 2, 3) are not necessarily of unit length, but can be taken to be so, without loss of generality. In this case the variation of D^ with respect to S is defined by the Frenet-Serret formulae T0D2,
(19.8) where KQ and r0 are the principal curvature and the torsion of the curve. We must observe, however, that such a result holds only upon the assumption that the directors remain coincident with the unit normal, binormal, and tangent to the curve as S varies. However, we are also free to choose a set of directors attached to the curve but turning independently of it. For another special choice of directors, let us consider a single unit vector D1? normal to the curve, but subtending an angle cp(S) with the principal normal N, and an angle [(TV/2) —
(19.9)
The quantity
(19.10)
164
19. Rods as Oriented Bodies
or, alternatively, by (N' = - ^ D 3 + r0B),
(19.9)
and
the
Frenet-Serret
formulae
D[ -N + T O sin^= -smcpip'.
for
N'
(19.11)
Provided that cp^O, this equation and its variants (Truesdell and Toupin 1960, Sec. 63) permit us to determine the twist in terms of the torsion and the derivative ofDi. If, instead of one unit vector D1? we consider two unit vectors D t and D 2 , there is, in general, no connection between their twists. However, often D t and D 3 are normal to one another, so that the twist of the unstrained rod may be defined as the twist of either of these vectors relative to the axis. Let us now assume that the rod undergoes a deformation. The deformed configuration is thus defined, not only by (19.1), but also by a relation between the new directors d^ with the original directors D^: 4 = d,(D1,D2,D3,^0.
09.12)
In general, the directors d^ are no longer orthogonal or of unit length. However, the common director theories of rods put more restrictions on the law of dependence (19.12). It is usually assumed that dj and d2 remain at right angles and preserve their initial unit length. The plane with the normal d3 = di x d3, is called the section of the rod at S. However, unlike the situation in the reference configuration, in which D 3 was tangential to the curve, in the present configuration d3 is not coaxial with the vector r' = p0 s. In order to exclude the case in which a section may be sheared so severely so as to contain the tangent to the axis, we require that the continuity condition (19.5) should hold, whatever the magnitude of loads acting on the rod may be. The continuity condition is the counterpart of the condition / > 0 of the three-dimensional theory. We may ask whether it might have been better to choose a triad of directors such that d3 = r'/|r'l and with d* and d2 perpendicular to d3 and perpendicular to one another. The reason for this is that the first triad is more natural both physically and mathematically (Antman 1974). We will explain the advantages of the first basis below, when we define the stress resultants and introduce the constitutive equations. For the kinematical description of the rod in its deformed state, with the triad d^ orthonormal, d^ can be localized with respect to a fixed orthonormal triad e1? e2, e3 by means of just three parameters; for example, we can use the Euler angles \//, 0, and cp employed in Section 17, with the differences that the triad el9 e2, e3 is not necessarily a triad coinciding with the direction of the (initially straight) central line and the two principal axes of the cross-section in Figure 17.1, and d1? d2, and d3 are not chosen as the so-called principal torsion-flexure axes of the three-dimensional theory. In the fixed triad e1? e2, e3 we set r = xk(S)ek, r'(S) = 4(S)e* = yk(S)dk(S) = ya(S)da(S) + z(S)d3(S)
(y3 = z),
(19.13)
Rod Theories: Director Approach
165
and substitute (17.15) in (19.13) to obtain x[ = yx (— sin x/r sin cp + cos \ff cos ^ cos 0) — J 2 (sin ^ cos (p + cos i/r sin cp cos 0) + zcosi/'"sin0, x'2 = y\ (cos ^ sin cp + sin ^ cos cp cos 0) + j>2(cos xj/ cos
= &2 • d3 = 0r sin
K' = d3 d! = ^ / c o s ^ + Vr/sin^sin6>, /
(19.15)
/
r = d{ -d 2 = <^ + ^ cos6>. The functions yk, K, K\ and r constitute the strains of the director theory. These quantities are almost universally adopted as appropriate measures of strain in a rod, and they were accepted as far back as the second half of nineteenth century after the publication of Clebsch's (1862) and Love's (1927) treatises. However, it seems that Saint-Venant and Binet preferred the definition of twist as given by (19.11) (Truesdell and Toupin 1960, Sec. 63A). Another unexpected extension of director theory is that we may also introduce directors to characterize certain privileged directions of heat propagation. These directors are now scalar fields of the type Xk(S, t), and if we take k = 0 the heat flow in only one direction is taken into account, which could possibly be along the axis.31 The balance equations connecting the mechanical variables must now be added to the system of kinematic variables. However, since Cosserat rods are not classical three-dimensional bodies, they cannot be regarded as trivial applications deducible from three-dimensional theory. The fundamental assumption of the mechanical theory is a stress principle for rods, which states that, across any section ideally splitting the rod into two parts, the action of the material on one side upon the material of the other is equipollent to that of a resultant stress vector n and a resultant couple m. Let us denote the forces distributed along the axis by f and the couples by I per unit length. The balance equations at the equilibrium then read v
and
[n]£+ f fdS = 0, u
(19.16)
v
[m]£ + [r x n]£ + f(t + r x f) dS = 0, u
(19.17)
where [U, V\ is an arbitrary subinterval of [S\, S2]. The first equation describes the balance of linear momentum, and the second describes the balance of angular 31
This observation is due to Antman (1972, Sec. 14).
166
19. Rods as Oriented Bodies
momentum. We need not postulate conservation of mass, as we have employed a strictly material description. In local form, the balance equations become n + f = 0, f = 0.
(19.18)
Traction, displacement and mixed boundary value problems may be formulated by prescribing n, and m, and r, and r', respectively, or a combination of these at S = S\, S2, or when the rod is a ring, simply by prescribing the periodic conditions. To render determinate the rod theories based on the two vector equations (19.18), which are equivalent to six scalar equations, it is necessary to reduce the number of geometric variables to six or less. This reduction may be effected in various ways. The commonest procedure is that of adopting Kirchhoffs hypotheses, which require that planar cross-sections normal to the axis in the reference configuration should remain planar, undeformed, and normal to the same strained axis. Other physically meaningful methods for reducing the kinematical quantities through additional constraints also exist. The commonly used component version of (19.18) is obtained by setting n(S) = nk(S)dk(S) m(S) = mk(S)dk(S),
(19.19) (19.20)
from which, again by using (14.7), we obtain n[ + KYI3 -
xn2
«i + wi! - ra3+/2
o, = 0, o, o,
(19.21a) (19.21b) (19.21c)
and m[ + Kfm3 - xm2 + y2n3 - zn2 m2 + xmx -Km3 f
+znx
m3 + Km2 - K mx + yxn2 -
-yxn3
+ (r x f) • di = 0, + (r x f) • d2 = 0,
y2nx
+ (r x f) • d3 = 0.
(19.22a) (19.22b) (19.22c)
The choice of n with components along d^ will illustrate the advantage of having dk as a local basis of strain in the deformed rod. In fact, it is the components nx and n2 that are responsible for the shearing, whereas the components along the d? and d2 directions, perpendicular to d!$ = r'/lr'l, have no direct influence on the shearing. This is illustrated for the two-dimensional case in Figure 19.1 (Antman 1974). We take n2 = 0 and r ; , d1? and d3 coplanar. It is reasonable to expect that an increase in nx will produce a decrease in the angle between di and r', whereas an increase in «*, the component of n along d*, will have no such effect. As for the constitutive relations, the most natural form of these equations, which is valid in the purely elastic case, is
nk = nk(ya(S), z(S), K(S), K'(S), r(5), S), mk=mk(ya(S),z(S),K(S),,c'(S),T(S),S),
Rod Theories: Director Approach
167
d[
Fig. 19.1
where nk and mk are continuous differentiable functions of their arguments. These can assume any real values, except z which is restricted by the continuity condition r' • d3 > 0, and S which varies between Sx and S2- Equations (19.23) are invariant under rigid motions, and it is possible to show that this condition necessarily implies the form (19.23).32 If we substitute the constitutive equations in the balance equations (19.22), we obtain a system of six nonlinear ordinary differential equations in six unknowns. The solution of the system constitutes a difficult problem, which is solvable only in special cases, but at least we have derived a system in which the number of equations equals the number of unknowns.
20. Kirchhoff s Theory The first complete theory of large displacement but small strains in initially straight elastic rods was proposed by Kirchhoff (1859), who extended an earlier result of Euler (1744). Kirchhoffs theory is an attempt to solve system (19.22), with the following constitutive assumptions: (1) the axis of the rod is inextensible; (2) the stress couple depends linearly on the curvature and the twist; (3) there is no shear between the cross-section and the axis; and (4) the cross-section is undeformable in its plane. The analytical consequences of Kirchhoffs assumptions are that, whatever the deformation, d3 must remain coaxial with r', and, therefore, in (19.13) we must put ya = 0 and z = |r'|. The postulate of inextensibility implies that |r'| must be unity. Again, Kirchhoffs theory assumes that constitutive relations are not only linear in the corresponding kinematical variables, but also have the special form mi = Ate, 32
m2 = BK\
This result is due to Green and Laws (1966).
m3 = Cr,
(20.1)
20. Kirchhoffs Theory
168
where A, B, and C are constants depending on the elastic quality of the material and the shape and dimensions of the cross-section. The indeformability of the crosssection is characterized by the property that after deformation the directors Di and D2 of the unstrained state maintain their orthogonality and their unit length. Several remarkable solutions for Kirchhoffs rods have been derived in the particular case in which the axis is straight in the reference configuration; in the deformed state, d3 coincides with r', and d^ and d2 are constant functions of S. The values of the quantities ya and z and the curvatures in the reference configuration are then = 0i z = l ,
=KQ
= r 0 = 0.
(20.2)
The first application of the theory is to the problem of determining the forms in which a thin rod, which is straight in the unstressed state, can be held by forces and couples applied at its ends only, when the rod is bent in a principal plane, so that the axis becomes a plane curve and there is no twist. This is the problem of the elastica, which was first solved by Euler in 1744. We take as the plane of bending the plane for which the flexural rigidity is B, so that K and T vanish, and the initially straight axis becomes a curve in the plane d ^ (Figure 20.1) having curvature K — K'{S). The load is a single vertical force R, directed downwards, applied at the S = S2 end of the axis. The stress resultants at S, resolved along the directions of dt and d3, are then rii = RsinO,
n3 = RcosQ,
where 0 is the angle that d3 makes with the line of action of the force R. The flexural couple at the section S is then m2 =
BK,
where K = dO/dS. We note that, according to the convention for the signs of moments, m2 is positive.
Si=O
Fig. 20.1
Rod Theories: Director Approach
169
Equation (19.22b), adapted to the present situation, yields
m2+nx= 0, or, in another form, d2f)
B—^ + Rsm0 = 0,
(20.3)
with a first integral 1 /d9\2 +
R
2{ds) B
(C S6
° ° ~ C ° S6) = °'
(2
°- 4)
where 0O is the value of 0 at S = S2 = L. A second integration gives the total length
L=
1
T
d
*
=
K
<®
(205)
where A:sin0 = sin(0/2) and k = sin(0o/2). Then K(k) is a complete elliptic normal integral of modulus k. Using tables of elliptic integrals, the value of R can then be obtained from a knowledge of k for any value of 00. For a point (jq, x2) in the fixed system of axes shown in Figure 20.1, we have
Hence, from the above relation, the values of xu and x3 can be expressed in terms of elliptic functions. The elastica takes different forms according to whether or not there are inflexions. At an inflection dO/dS, and hence the bending moment vanishes, so that a rod can be held in the form of an inflexional elastica by a terminal force without couples. The ends are points of inflexion, and all intermediate points of inflexion must lie on the line of action of the terminal force. When there are no inflexions the strained axes are themselves referred to as noninflexional elastica. In his original theory, Euler classified all the possible forms of the elastica into nine types. The discussion of all the types of deflection of a rod under the action of terminal forces is particularly important. It is particularly relevant in the study of the stability of an initially straight rod under the action of a compressive force R directed along the undeformed axis (Figure 20.2). When the straight configuration becomes unstable under an increasing load R, there may exist a neighboring bent state (0 small) of curved form, as shown in Figure 20.2. In this case, sin 6 is approximated by 0 and (20.3) has the approximate solution
where Cx and C2 are constants, which must be determined by the boundary conditions 0(0) = 0,
^ ( i ) = 0. aS
(20.6)
20. Kirchhoffs Theory
170
Fig. 20.2 It turns out that nonzero values of the constants C\ and C2, compatible with the end conditions, are possible, provided that (20.7)
This minimum value of R, for which a possible stable bent configuration arises, is called the Euler buckling load. It should be noted that, in the approximate theory, it is the ratio C\/C2 that is determined, and not the individual values of these constants, and the approximate theory can only give the critical load. In order to determine the deflection of the terminal point of the axis in the new configuration under the buckling load, we must return to (20.4). When we remember that dxx = dS sin 0 and that k and 0 are defined above for (20.5), a simple integration yields Xl(L)
=
7T/2
2k
JR/B
J
2k ^
(20.8)
An approximate version of this formula can be found on remembering that, for small k, we have the approximation (Jahnke, Emde, and Losch 1966, p. 62) -K(k)~l+^-k2
+ ...,
(20.9)
where K = L^/R/B. Hence, on solving (20.9) with respect to k and substituting in (20.8), we obtain
xx{L)~A
2L
'©•
(20.10)
Rod Theories: Director Approach
171
which is in agreement with a formula registered in Love's (1927, Art. 264) treatise. A slight generalization of the result is achieved by removing the hypothesis of inextensibility of the axis under longitudinal thrust. In this case the length of the deformed axis is L[\ - (R/EA)] and the Euler buckling load of (20.7) must be modified as
However, the correction is not of importance in practice. The question of finding the conditions of the buckling of a rod by linearizing (20.3) involves some consideration. We have determined the buckling load by investigating the possible equilibrium configurations close to the rectilinear configuration, but we do not know whether the critical load of the linearized theory is related to the critical load of the nonlinear theory. In addition, because the constants C\ and C2 are only partially determined, the impossibility of determining the precise magnitude of the deformation in the buckled configurations is clearly a deficiency of the linearization process. In order to clarify this situation, it is necessary to analyze the structure of the solutions of (20.3) in more detail. For simplicity, we denote the ratio R/B by X. The linearized problem has the solutions
0(S) = An sin(H S) n = 0,1, 3, 5,...,
(20.12)
when X = kn = ( H ) ,
n = 0,1, 3, 5,....
(20.13)
For the value Ao = 0, it has only 0 = Ao = 0 as a solution, and so Xo is not an eigenvalue. The nonlinear problem has the first integral (20.4), that is
GH)~
c o s
(2 l4)
°°^
°*
where eOi
o^eo^7t.
(20.15)
Since 0 is continuous and satisfies (20.15), and the right-hand side of (20.14) must be nonnegative, we must have
Now, from the identity dS,
172
20. Kirchhoffs Theory
and the result OsinO^ \6\ |sin<9| s£ 02, we have the inequality L
V
2
"1
(20.16) This inequality becomes a strict equality only for k = ki = (7t/2L)2, with 0 = Ax sin(7tS/2L), that is, when k and 0 are the first eigenvalue and the first eigenfunction of the linear problem, respectively. This is a consequence of the minimum principle giving a minimum for k = k\, which implies that the integral in (20.16) is positive for 0 < k < kx for all admissible 0 ^ constant. Hence, for 0 < k < ku the only solutions of the nonlinear problem are 0 = 0 or 0 = n. To study the solutions for k > A1? we return to (20.4) and write it in the form
fs = 2Vly/sin2 | - sin2 \ = 2
^ - sin2 \
= 2Vkky/l-sn2(u,k) = 2kcn(u, k),
(20.17)
where u = S^/R/B, k = sin 2 (0 o /2), and sn(w, k) and cn(w, k) are Jacobian elliptic functions (Jahnke et al. 1966, p. 73). The function cn(w, k), regarded as a function of w, vanishes from u = nK(n = 1, 3, 5,...), where K(k) is the elliptic integral as defined before. For S = L, we have the boundary condition cn[w(L), k] = cn( This condition is verified, provided that Lk = Lkn = LJ^- = nK(k)
(n = 1, 3, 5,...),
(20.18)
or
R
/nK\2
k\ = - = l — \
(#i = 1, 3, 5,...).
(20.19)
Thus, for each n we have obtained the response of the rod, that is to say, a relationship between the load parameter R/B and the measure of deformation k = sin(0o/2). The curves kn(k) are indicated in Figure 20.3. Since K(0) = n/2, each curve branches from k = 0, k = kn, which are just the eigenvalues of the linearized problem. It thus follows that the linear eigenvalue problem yields the points of the bifurcations for the nonlinear problem. From (20.18) we find that dkn/dk > 0 for k > 0, and hence the curves kn = kn{k) are monotonically increasing. In addition, the curves do not intersect and have as a common asymptote the straight line k = 1 as kn tends to infinity. The behavior of the rod for k > A1? for instance for kn < k < kn+2 (n— 1,3,...), must be clarified, because we must decide whether the rod will buckle and, if so,
Rod Theories: Director Approach
173
Fig. 20.3
which branch is to be preferred. The answer can be obtained by examining the total potential energy associated with the possible states. The energy of the unbuckled state 0 = 0 is clearly zero, provided that we continue to maintain the hypothesis of inextensibility of the axis. The energy of the buckled state is given by the difference
-&=
l\B(jt)
dS-R
- cosO)dS,
(20.20)
where the first term represents the stored strain energy and the second the work done by the thrust R. On using (20.17) and the properties of elliptic functions, we transform (20.20) into the expression (20.21) where E(k) is the elliptic integral of the second kind. Now the functions K(k) and E{k) have the following expansions, which converge for k2 < 1 (Jahnke et al. 1966, p. 62):
On substituting these expressions for K(k) and E(k) into (20.21), and considering the total energy {iV — 5£\ corresponding to a buckled state which branches from AM, as given by (20.19), we obtain
174
20. Kirchhoffs Theory k2
(20.22)
This means that every buckled state has less energy than the straight configuration. We can also show that, if we fix X such that Xn < X < Xn+2, the energies are ordered in the sequence '-&)n>(iT-
&)n_2 > . . . > (TT - &\.
(20.23)
After having determined X (Figure 20.3), let kn(X) denote the unique root of (20.18): (20.24)
Since K(k) is increasing monotonically we conclude from (20.24) that, for m > n, kn < km. We now differentiate (20.21) with respect to k, and obtain
±
o,
(20.25)
since, as a consequence of the differentiation formulae for elliptic integrals (Jahnke et al. 1966, p. 64), we have E
K
The equality in (20.5) holds only when k = 0; otherwise, [E — (1 — 1^)K\ is strictly positive. It thus follows that the energy (TST — J£?) decreases strictly monotonically for k > 0, and increases strictly monotonically as a function of n, so that (20.23) is established. Under a given X(> Xi), the rod tends to deform in the buckled state of least energy, which is the state branching from A1? and will remain in this state for all X> kx. The existence of a bifurcation in the curve k = k(X) in the kX plane when X(0) exceeds the critical value Xx was recognized and discussed by Poincare, who described such a point as a point of bifurcation, at which an exchange of stability takes place because the points on the X axis with X > Xx are unstable and stability only occurs for points on the curve branching from X = Xi. The presence of the other curves branching from the points X = X3, X5,..., is not in conflict with Poincare's theory, because the curves do not issue from the stable branch but from the straight X axis, where this has become unstable.
Rod Theories: Director Approach
175
21. Solutions for More General Loads When the rod is bent and twisted by a more complex system of loads, the integration of Kirchhoff s equations is possible only in some special cases. If the load consists of a force R and a couple K, applied at one end, and balanced by another force and couple at the other end, so that the total system of loads is statically equivalent to zero, the stress resultants in a section of the deformed rod at a distance S from the end at which R and K are applied, can be expressed by means of the Euler angles defined by (17.15). The stress resultants in this section are given by
The equations of the problem are again (19.22), but they lack the terms representing the influence of the distributed loads and torques f and £. In these equations the couples mu ra2, andra3are related to the curvatures by constitutive relations like (20.1). However, it is very difficult to obtain their solution, unless we introduce the hypothesis that the strains ya of the deformed state are negligible and z is approximately equal to unity. Equations (19.22) thus become dK
A—--(BdS
C)K X = n2,
(21.2a)
B^-(C-A)xK=-nu C^-(AdS
(21.2b)
B)KK = 0.
(21.2c)
On eliminating n\ and n2 from (19.21), with the help of (21.2) we find dm uo
A
die uo
n
, dK
, A t — B)XKK . = 0,
(21.3)
uo
or, by (21.2c), -BK'2-
This equation may be integrated directly, and by using (21.1) we obtain a first integral of (21.2): \(AK2 + BK2 + Cx2) + R d3 = constant.
(21.4)
A second integral may be obtained by observing that, if we consider the moment m = (m\, m2,ra3)at any point S on the strained axis, its projection along the fixed direction e3 must remain constant. Since the components of the unit vector e3 along the directions d1? d2, and d3 are e3 • dt = — sinflcos^,
e3 • d2 = sin0sin#>,
e3 • d3 = cos0,
the above condition yields m-e 3 = — AK sin 0 cos (p + Btc' sin 0 sin (p+ Cx cos 0 = constant.
(21.5)
Having determined two integrals of the equilibrium equations, we might hope to find a third integral of the same equations in order to obtain three finite relations
21. Solutions for More General Loads
176
between the Euler angles 0, i//, and (p. In general, no other integral is known; however, if the flexural rigidities A and B are equal, then from (21.2c) we have at once (21.6)
T = constant,
and therefore the system of equations (21.2) can be integrated in finite form. An example of this result is given by a rod which is bent into a helical form,33 with a constant inclination 0 = (TV/2) — a to the upward vertical, by a longitudinal force of magnitude R and a twisting couple of magnitude K. Since dO/dS = 0, (19.15) give K
dilr
,
dxlr
dw
d\lr
= —— cos a cos cp, K = —— cos a sin cp, r = — 4- sin a ——, dS dS dS dS
and from (21.4)-(21.6) we obtain dx// U
in
Y (21.7) — = constant. dS This means that the strained axis is a curve of constant curvature and torsion; that is, a helix traced on a right circular cylinder. Once we have (21.7), we can determine the constant a, the radius r of the cylinder on which the helix lies, and the constants on the right-hand sides of (21.7) by exploiting the connection between K, K\ and r and the stress and moment resultants. An explicit solution for the problem of the elastica, which has some importance in the analysis of mechanical springs, can be obtained for the case of a slender rod of length L, weighted at one end by a vertical load P, while the other end is hinged so that a couple M is applied (Figure 21.1) (Wang 1981a). The form of the elastica can assume various configurations as the couple M is changed, and our aim is to find a connection between M and the angle a, which represents the slope of the axis at the origin. If 0 denotes the slope at a point S on the abscissa, the equilibrium equation for the elastica is
x = constant,
K + K — constant,
Fig. 21.1 Love (1927, Art. 270), but the solution is due to Kirchhoff.
Rod Theories: Director Approach
111
B ^ = PXl-M,
(21.8)
where B is the flexural rigidity, and xx the distance of the current section from the JC3 axis. After differentiation with respect to S, (21.8) becomes
with the boundary conditions 0(0) = a,
^-(L) = 0.
(21.10)
The first integral of (21.9) is
where ft is the terminal slope at S = L. The formal solution of the above equation is thus
where F is an elliptic function of the first kind. The angle f$ has to be found as one of the roots of the transcendental equation
P+A ^yyi+cos^
2
The law of dependence of M on a is given by (21.8), written at the origin, M=-B^(0).
(21.13)
Analysis of the curve M == M(a) for 0 ^ a ^ In shows that, if F/B is small, then M is a single-valued function of a as illustrated by curve I in Figure 21.2; if F/B is large, then the curve behaves like curve II in Figure 21.2, and snapping may occur. The analysis may be extended to the case in which the weight of the rod itself is taken into account (Bickley 1934; Wang 1981b). The qualitative behavior of the curve M = M(a) is similar to that already described. The equation describing deformed configurations of heavy rods can be treated mathematically without recourse to an explicit solution (Hsu and Wang 1988). We
178
21. Solutions for More General Loads
Fig. 21.2
consider again the case shown in Figure 21.1, with no load at the end S = L, but under the influence of a vertical body force pg applied at the points of the strained axis and directed downwards. Having taken the origin coincident with the hinged end, the equilibrium equation is
d20
(21.14)
with the boundary conditions 0(0) = a,
^ ( L ) = 0.
(21.15)
On putting s = S/L and K = [(pgL3)/B]l/\ (21.14) and (21.15) become —j = K\l-s)sinO, K>0, 0<s<\, as 0(0) = a, 0'(1) = 0, -7T ^ a ^ 7T,
(21.16) (21.17)
and the problem is one of analyzing the solutions of (21.16) under the conditions K > 0 and — TT < a ^ n. In order to simplify the notation, let i/,(s) = 0(1 - s) 0 < s < 1, and reformulate the problem in the form —y z= K3ssimls, ds ^'(0) = 0,
K>0,
0<s
(21.18) (21.19)
In addition, since a solution \//(s) corresponding to a given a is associated with a solution -i/s(s) when a is replaced by -a, we only consider the problem with 0 ^ a ^ n.
Rod Theories: Director Approach
179
Now (21.18) and (21.19) admit one solution at least, since the right-hand side of (21.18), namely K3s sin \jr, is a bounded function for 0 ^ s < 1. Moreover, the solution is unique, provided that we have K3 < A/45. In order to prove this result, let \//(s) be a solution and write it in the form l
= a - f K3^ si
,s
(21.20)
where G(s, f) = 1 - max(s, £ Let
and
be two solutions, then we have l
K* / G(s, J 0 1
-li
Lo or
As a consequence of the result 11
00 6
it follows that, if K /45 < 1 or AT3 < >/45, we must have the identity Vi = ^2In order to understand the bifurcation phenomena that may occur in general, we consider the important case where a— it, which corresponds to a vertical slope, upwardly directed, at the initial section. Let us make the substitution v(s) = ir(s/K) - 7T, so that (21.18) and (21.19), written with a = 7t, take the form t/(0) = 0,
^ '
}
^ '
J
and consider the following initial value problem v/;(j) + . t/(0)
=
where a is a real constant. If we denote the solution of this problem by v(s, a), then from the uniqueness of solutions of ordinary differential equations with Cauchy data we have
180
21. Solutions for More General Loads v(s, lit + a) = v(s, a) + 2TT, v(s, lit — d) = —v(s, a) + 2TT, v(s, a) — —v(s, —a), v(s, 0) = 0, v(s, TT) — it,
which means that here again we may study v(s, a) only in the interval 0 < a < it. Now multiply (21.22) by v\s) and integrate from zero to s, so as to derive s
^[v\s)f
= scosv(s) - f cosv($d% ^ 0.
-f
(21.23)
0
Now, if a lies in the interval 0 < a < TT/2, then we have cos a = cos v(0) > 0, and this implies the positive-definiteness of cosi>(s) > 0 for all s^0. If not, then there is a s0 > 0 such that cos v(s) > 0 for all s in the interval 0 ^ s < s0 with cos v(s0) = 0. But this contradicts (21.23) when we put s = s0, which proves the results -TT/2 < v(s, a) < TT/2 for 0 < a < n/2, s ^ 0. If a lies in the interval n/2 ^ a < n, then cos a = cos v(0) satisfies the inequality — 1 ^ cos a < 0, and this implies the result cos v(s) / 1 for all s ^ 0. If not, then there is an s0 > 0 such that cost^o) = — 1 and cosv(s) > — 1 for 0 ^ s < s0. Again from (21.23), we obtain a contradiction, and hence we must have — n < v(s, a) < n for 7r/2 < a < 7t and s^0. But we can prove that v(s, a) is oscillatory for any a in the interval 0 < a < n. The function
satisfies the inequality
Then we have the results 1 - cos v(s) ^ V(s) ^ F(0) = 1 - cos a. As v(s) lies in the interval — ix < v(s) < 7t, then \v(s)\ ^ a for s ^ 0. On observing now that (21.22) can be written as v (s) + s\ —-^— |v(^) = 0,
(21.24)
we put 0 < 8 < min
v J
and we consider the equation 0.
(21.25)
This equation is oscillatory for s ^ 0, and hence, by virtue of Sturm's comparison theorem for ordinary differential equations, the solution of (21.24) is also oscillatory.
Rod Theories: Director Approach
181
In order to characterize the multiplicity of solutions of (21.21), we must consider the family of solutions of (21.22) for 0 < a ^ In and find the value of v(K) corresponding to each a. The values of a for which the equation v(K) = 0
(21.26)
is satisfied, represent the possible solutions of our problem. If K is small the solution is unique, but as K becomes larger than k\ ~ 1.98635, where k\ is the first eigenvalue of (21.25) written for 8 = 1, then the solution is no longer unique. Another useful extension of the theory of the elastica is necessary when a cantilever beam is subjected to a concentrated load at its tip, the direction of which is not fixed, but can change with the rotation of the terminal section (Figure 21.3). Loads of this type are said to be nonconservative. A simple exact solution can be found for a simple beam clamped at its left end, and loaded at the other end by an initially vertical load R, which rotates through an angle /fa as the tip A rotates through an angle a. This angle a is a consequence of the elastic deformation of the beam. Since ft is a constant, we have implicitly assumed that the rotation of the load is proportional to the rotation of the terminal section (Nageswara and Venkateswara 1986). On taking the origin of the curvilinear abscissa S along the deformed axis at A, the equilibrium equation, a modification of (20.3) becomes d20
(21.27)
where 0 is the slope of the strained axis at S, and B is the flexural rigidity. The boundary conditions are
— (o) = o, e(L) = o.
(21.28)
Once we know the function 0 = 0(s), the coordinates x\ and JC3 of the strained section at S are given by = I sin 6 dS,
x 3 = / cos OdS.
Fig. 21.3
(21.29)
182
21. Solutions for More General Loads
In order to integrate (21.27), we put i- = S/L, boundary conditions assume the form
so that the field equation and
Q" + X cos(j8a - 0) = 0, 0 < § < 1,
(21.30a) (21.30b)
0'(O) = O, 0(1) = 0,
(21.30c) (21.30d)
xx=L = L ff sin si 0d%,
(21.30e)
l
x3=L =L [cos9d%, [
(21.30f)
where A = (RL2)/B. A first integral of the equation can be obtained by multiplying (21.30a) by 9' and integrating with respect to £: 0'2 = {2A.[sm(/Ja - (9) - sin(^of - a)]},
(21.31)
which satisfies the boundary conditions whenever the slope at the tip is a. From (21.31) we can obtain 9' but, since we expect 9f to be a decreasing function of 9, we take the solution 9f = -{2k[sin(Pa - 6) - sin(£ - a)]}K
(21.32)
On writing sin(^or - 6) = 1 - 2k2 sin2 cp and sinO&x - a) = 1 - 2£ 2 , (21.32) is transformed into
(21.33)
and the boundary conditions for cp become |,
(21.34a)
= arcsinl"^ sm (?r " ^ l .
(21.34b)
[A: 4 J By integrating this equation together with the condition ^?(0) = TT/2, we obtain the connection between A and the tip angle a: A = {F[K *<0)] - F[fc, (^(l)]}2,
(21.35)
where F(A:, ^) is the elliptic integral of the first kind, given by
After having determined A. as a function of a, we can also obtain the expression for x\ and x2 in terms of the variable
l
Rod Theories: Director Approach
183
where G((p) and H(cp) have the form 2k G = —j=z [cos cp(X) — cos
2k
and
E(k,
- k2 sin 2
is the elliptic integral of the second kind, with
184
22. Nonclassical Problems for Rods d26 B—T + (P + wS)cos0 = O,
0^0
^2a,
(21.37)
where 0(S) is the slope and 2a is the slope at S = 0. We assume that the end S = L is clamped. Let us change the asymmetric interval of variation of 0 into a symmetric one by putting and rewrite (21.37) in the form B —-j + (P + w*S)(cos a cos ^ — sin a sin cp) = 0, — a ^
sin? = 2Jx(ct)Tx (£) - 2J3(a)T3 Q + ...,
(21.39)
cos
(21.40)
where J((a) is an ordinary Bessel function and T^cp/a) is the Tchebychev polynomial of /th order, that is To = 1, T{ = cp/a, T2 = 2(cp/a)2 - 1,.... On substituting (21.39) and (21.40) into (21.38), and retaining only the terms of zeroth and first order, we obtain the linear equation
On integrating this equation and using the boundary conditions (d0/dS)(0) — 0, (p(0) = a, and cp(L) = —a, we determine the approximate solution.
22. Nonclassical Problems for Rods In the classical theory of the elastica, external loads are always regarded as given quantities, as are the boundary conditions, of either geometric or mechanical type. However, in some cases, the quantities customarily assumed as the data of the problem are also unknown and must be determined as part of the solution. A problem of this kind arises in contact mechanics when we wish to find the equilibrium configuration of a rod initially having the shape of an arc of a ring clamped at one end and impinging on a rigid barrier at the other (Figure 22.1). The initial length of the circular arc (dashed line in Figure 22.1) is L, and the arc AB passes through the sliding clamp A, smoothly touching the rigid barrier at the point B. Let us now suppose that an additional length AL is forced through the clamp at A, but this motion is hindered by the reaction exerted by the rigid barrier against the other end B. The barrier, which for simplicity is considered vertical, is not perfectly smooth, but offers a small reaction AF inclined at an angle T/T, which has the effect of holding the edge of the ring in place. Whether the ring sticks or slips depends on the value of the angle of friction tan -1 (/x) between the end B and the
Rod Theories: Director Approach
185
N /
tanV)
Fig. 22.1 vertical wall. If i/s > tan *(//,), the point B will slide up; if f ^ tan ^/x), the point will stick. An analysis of the type of contact at B is useful in some practical applications, such as in the making of plain paper copies, in self-threading tapes, and in sewing machines. A simple solution can be obtained by means of perturbation procedure (Benson 1981). By neglecting gravity and transient effects, the differential equation and the boundary conditions for the deformed beam are (22.1)
±.
(22.2)
where S is the distance along the deformed axis measured from the partial clamp, and 0(S) is the slope with respect to the horizontal. Before and after the deformation the coordinates of the beam tip remain unchanged: L+AZ,
•-Ms)-
cos OdS,
(22.3)
L+AL
h = R-Rcos[-\
= / sinOdS.
(22.4)
o
Let us now introduce the dimensionless quantities S
g— y ,
K
_L
_AL
— ~H> A L — — - ,
AF =
AF
(22.5)
186
22. Nonclassical Problems for Rods
and let us call A0 the change in slope from the undeformed circular state, which has the value S/R = K%. Hence the final slope is given by &
(22.6)
Thus equations (22.1)-(22.4) become:
jp (A0) = AF sin(tfr - *§ - A0),
(22.7)
A0(O) = 0,
(22.8)
-^- A0(1 + AL) = 0, d%
I+AZ
I+AZ 1
f
==
si
(22.9)
However, if the deformed configuration is very close to the circular form, we may linearize the above equations and write ,—x
-™ . , .
l+AL
sin K K
f = / J
^ 1
f cos(/c£) d%-
AL sin(*§)
I+AI
f
=
*
(22.11)
J
0
1—cos/c§
(22.10)
I
.
/
f
sin(/c£W£ + / A0 cos(/c£)
J
(22.12)
J
o o On integrating the first term on the right-hand side of the last two equations and retaining only the terms of first order in AL, we obtain I
AL cos K = I A0 sin(/c£) d%, o l
- AL sin K — I ~A0 cos(/c£) d%. o Thus, by eliminating AL from these equations we have 0 = f A0 cos(/c - K$) di=. (22.13) o Next, (22.10) may be integrated, making use of its boundary conditions, to give AF A0 = — Y [ s i n & ~ sin(i^ — ict-) — K^COS{\//
— /c)].
It thus follows that, by substituting this value of A0 in (22.13) and integrating the resulting expression with respect to £, we arrive at the equation sin T/T(2 sin K COS K — K COS K — sin K) + cos \jf(2 cos2 K H- K sin K — 2 cos K) = 0,
Rod Theories: Director Approach
187
which can be solved with respect to V" to give cos K + sin /c(l — 2 cos K) J' In this manner we have found the reaction angle of the beam in terms of the dimensionless curvature K. Under the condition \jr < t a n " 1 ^ ) the beam sticks at B, otherwise it slips under a small advancement AL at the left-hand end. We also observe that, for small /c, (22.14) can be replaced by its Taylor expansion ^ ~ § * + ...,
(22.15)
which implies that ^ is a linear function of /c, for small K. A problem which is, in some respects, the converse of the one just described is represented by the peeling of tough adhesive tape from a plane surface to which the tape was initially glued. This is a typical example of a free boundary problem in mechanics, since the boundary of the tape is not prescribed in advance, but must be found as part of the solution. Problems of this kind are encountered in fracture mechanics, when we must determine the extension of a rectilinear crack, or in seismology, when we must appreciate the boundary of a fault plane over which slipping occurs. The surprising feature is that these different problems, if restricted to a onedimensional version, require the same mathematical treatment (Sawyers and Rivlin 1974; Burridge and Keller 1978). We shall consider an elastica formed during the process of peeling a tape from a horizontal plane under the effect of a tension To applied at a point of the tape at a very large distance from the point of detachment (Figure 22.2). Let S denote the arc length along the tape measured from a point S = 0, and let us assume that we are considering the process of peeling at a given instant t, at which S = a(i) represents the point of separation between the part of the tape already detached and that still glued to the plane surface. If r(5, i) is the position of the particle S, then r s = d3 is the unit tangent vector to the axis, and r ss is related to the curvature K by the Frenet-Serret formula r ss = K'A\, d{ being the unit normal directed to the left as one advances in the direction of d3. The bending moment is of magnitude BK in the
Fig. 22.2
188
22. Nonclassical Problems for Rods
direction d3 x d\ and the shear force is —BKSd\. Since there is a tension T(S, t) along the axis, the total force acting at a section at S is n = Td^-BK[sdx.
(22.16)
Let p be the density of the material and A the area of the cross-section. The equation of motion of the part S < a(t) of the tape is X
(22.17)
At the point of separation S = a{t) we have the conditions )
(22.18)
rsWO, 0 = 0,0).
(22.19)
In addition, we have at o(f) a condition of rupture, because the detachment of the beam at a(t) occurs just when the bending energy \BK2 per unit length is equal to the energy y per unit length required to create a local fracture. On putting K = (ly/B)1 this condition becomes K' = \r,ss(a(t),t)\=K.
(22.20)
It is clear that the problem has more boundary conditions than necessary, but in the present case o(i) is also an unknown in addition to r. In order to find r(S, t) and a(t) we seek a solution of the form r(S, 0 = R(S') + (vt, 0),
S' = S- vt,
a(t) = vt,
(22.21)
which has the property of being stationary if viewed from the point of separation. We now substitute (22.21) in (22.17) and in the boundary conditions (22.18)(22.20), noting that r Jt = v2RfSS = v2Krd{ and {d/ds)dx = -ic'd3, according to (14.7) with r = 0. We then drop the primes and write (22.17) in the form pAv2K'dx = (BK'K'J + Ts)d3 +{TK' - BK[ss)dx,
S < 0,
(22.22)
R(0) = (0,0),
(22.23a)
R 5 (0) = (l,0),
(22.23b)
\Rss(P)\ = K.
(22.23c)
After separation of the normal and tangential components in the differential equation, we get
BK'K[S + Ts
= 0.
These are just the equations of the elastica for the static deflection of a rod when its tension is not T, but T — pAv2. A simple solution to these equations has the form (22.24) (22.25)
Rod Theories: Director Approach
189
where To is a constant representing the tension when K = 0, and b is given by b2=^(T0-PAv2).
(22.26)
To find 0(S), the angle between d3 and the horizontal axis, we integrate the equation (dO/dS) = K\ with 0(0) = 0, and use (22.25) for K . This yields 0(S) = 4 tan"1 L n h ^ ~ 5 o ) 1 + 4 tan"1 ( t a n h ^ ) . In addition, we can determine R(5) in terms of 0(S): s R(S) = /"(cos 6>(£), sin
5 < 0,
(22.27)
(22.28)
which is adapted so that the first two boundary conditions (22.23) are satisfied. By using (22.28) in (22.23c), we get ^ ( 0 ) = \K'\=K.
dS
Thus from (22.25) we obtain b sech -^—-—— = K,
(22.29)
which gives So and completes the solution. From (22.27) we find that, as S tends to —oo, 0(S) tends to \\r — it, where \\r is given by
_, r Vr = 4tan
/z>s o \l
tanh -— .
This means that the tape is asymptotic to the direction xjr. To express v in terms of \J/, we note that from the above equation we can derive
then sech(Z>S0/2) can be written as follows:
sech ( ^ ) =
^-4- =
H f = coS(t).
(22.30)
Thus by squaring (22.29) and using (22.26), (22.30), and the condition of detachment |/c'| = K, we get p^=l-|-(l+cos^r1.
(22.31)
This relation determines the velocity v of the point of separation tip in terms of \jr and the other data. The formula shows that there is a solution only if
190
22. Nonclassical Problems for Rods
To > y{\ + cos i/r)~l, and the velocity is greatest when \\r — 0, in which case the tape is being pulled forward parallel to the horizontal axis, that is to say, through an angle of 180° from its original direction. Furthermore, v is zero when \\r assumes the value X/TQ such that cos i/ro = ^ - - 1. There is no solution with v real if it ^ xjr > \lr0. This kind of argument may be applied to study the interesting problem of the longitudinal cleavage of a beam when a longitudinal crack, initially created at one end, is gradually extended along the axis of the beam by two equal and opposite transverse forces F pulling apart the two halves of the terminal section (Figure 22.3). Sometimes, the same effect of the propagation of a crack may be obtained by prescribing the displacement between the two halves of the terminal section. This problem arises when woodmen split a piece of wood longitudinally. The opposing forces F are supplied practically through the insertion of a rigid wedge. Suppose that an initially rectilinear beam is split from x = 0 to x = cr0 and its two ends are pulled apart gradually. At first the crack length will remain constant while the two sides of the crack separate, but after the curvature at the crack tip has become large enough the crack starts to elongate. We wish to determine the law of dependence between the prescribed forces F and the displacements U and establish whether a small increase in each of these parameters will result in a small increase in the length of the crack, or result in a catastrophically fast propagation (Burridge and Keller 1978). We suppose that the end of the beam at x = 0 is displaced by an amount U and that the crack length is cr. Since we are interested in predicting the response of specimens made of a slightly deformable material, and in the phase preceding the catastrophic rupture, we apply the static linear beam theory. On denoting the transverse displacement of the upper half-beam by u(x), elementary beam theory permits us to determine u(x) as the solution of the equations
Fig. 22.3
Rod Theories: Director Approach 0 < x < a,
(22.32)
«(0) = U,
uxx(0) = 0,
(22.33)
u(a) = 0,
ux(a) = 0.
(22.34)
*,xxxx
= 0,
191
In addition, we require that the curvature at the tip does not exceed a given value K: uxx(a) < K.
(22.35)
Then the solution of (22.32) with its boundary conditions (22.33) and (22.34) is 0 < x :% a,
(22.36)
from which we calculate (22.37)
Urr =
We have thus found the distribution of the transverse displacement u(x) between the end at which a displacement U is prescribed and the tip of separation. In practice, however, the process is slightly different. The tape is initially separated along an interval of extent a0, starting from the end x = 0, and increasing displacement U is impressed at this end. As long as we have the inequality
O)
the fracture condition at the tip is not violated, and therefore the extent of the separated portion remains equal to cr0. When U exceeds the value Uo = (KOQ/3), the separation begins to propagate, and the instantaneous value of a is obtained by setting 3U/a2 = K. This yields if a — <
^r)
if U >
3 '
(22.38)
3 '
and the graph of a as a function of U, shown in Figure 22.4(a), shows that a is a continuous increasing function of U. The force required to maintain the displacement U is F = Buxxx(0), and can be calculated from (22.36) and (22.38). It is given by 3UB
if (22.39)
F= if
and the graph of (22.39) is shown in Figure 22.4(b). We can thus see that the prescription of a displacement U or a force F have radically different effects on the propagation of the crack. If the force F is prescribed at a value greater than Fo = KB/a0, which represents the maximum of the function
192
22. Nonclassical Problems for Rods G
(a)
O
O
U
Uo
U
Fig. 22.4
Fig. 22.5
(22.39), there is no static solution, and catastrophic failure will ensue. However, if U is prescribed, there is always a static solution. The theory of the elastica may also be applied to explain the apparently anomalous behavior of a high-flexure manipulator tube element, the cross-section of which is a semi-circular ring composed of a thin corrugated bellows for the curved part, and a flat reinforced strip along the rectilinear side (Figure 22.5). Rigid caps are provided at each end. The tube is pressurized from the inside, and the pressure load
Rod Theories: Director Approach
193
is applied at a point of the cross-section situated on the axis of symmetry at a distance dd from the neutral axis, which is located in the flat strip parallel to the rectilinear edge. When the tube is pressurized, the noncoincidence of the center of the pressure and the centroid of the cross-section causes a deflection of the, initially straight, central line. The resulting pressure rotates to follow the tangent to the deformed axis. By regulating the internal pressure it is possible to lift a payload Wd applied at the movable end. The manipulator arm has many practical applications, especially in medicine. To study this problem mathematically (Wilson and Snyder 1988), we regard the tube as a slender cantilever beam with an end load composed of P d , Qd, and M d as shown in Figure 22.6; the subscript "d" denotes quantities that have physical dimensions. We fix a system of Cartesian axes x and y with their origin at the clamped end of the beam; the x axis is placed along the unstrained longitudinal axis, and the y axis is perpendicular to it. Pd and Qd are the components of the end load along the reference axes. On adopting the customary assumptions of the theory of the elastica, the differential equation for the inclination 6 is (22.40)
where sd is the arc length evaluated from the origin, xd and yd are the coordinates of the section at sd, and Xd and Yd are the coordinates of the terminal section at sd = L. The boundary condition is 0(0) = 0. In order to solve (22.40), we introduce dimensionless parameters _sjL ~ V
S
_xd ~ L'
X
p=
B
y
~
_ j d L'
Q=
_JQ L'
v
_ Yd L'
MAL B '
194
22. Nonclassical Problems for Rods
and write the equation in the nondimensional form ^ = P(Y - y) + Q(X -x) + M,
(22.41)
with the same boundary condition 0(0) = 0. On differentiating (22.41) with respect to s and using the relations dx — = cos 0, ds
dy — = sin 0, ds
(22.41) becomes d20 = -P sin 0 - Q cos 0. ds2
(22.42)
We must now add the boundary condition (22.43)
because we know the bending moment at s = 1. The first integral of (22.42) is - ( — } = - M 2 + P(cos 0 - cos a) + g(sin a - sin 0), 2 \dsj 2
(22.44)
where a is the slope of the elastic curve at s — 1. If we now make the substitutions a = M2-b
cos(a + 00),
b = 2yjP2 + g 2 ,
(22.45)
with 0O arctan(g/P) for — n < 00 ^ it, we can take the positive square root of the right-hand side of (22.44) and integrate. This yields
-7
y/a + b cos
(22.46)
0o
where = 0 + 00. The nondimensional x, >> coordinates at s are cos(
/ _
(22.47)
0+0oo
/• sin(
+ b cos (^
and the angle a is given by (22.46) on setting s = 1 and 6 = a. We thus have the solution in terms of the forces P and Q and the couple M. These quantities must now be related to the loading parameters of our problem. Let us define W& and Nd as the force and moment due to the weight, respectively, of the payload at the tip of the element. As shown in Figures 22.5 and 22.6, gravity acts at an angle ft to the x axis. Fd is the pressure load at the tip acting at a distance dd from the neutral axis. The magnitude and direction of W& remain unchanged as the
Rod Theories: Director Approach
195
element rotates, while Fd follows the slope at the tip. The couple N& varies with both a and the configuration of the payload. On introducing the nondimensional parameters
*£ B
Z**
M
B
B
d=
*
LJ L
(22.48)
the relations between these nondimensional loads and the nondimensional loads given by equations (22.45) are P = - F c o s a - Wcosfi,
Q = Fsina+ Wsinfi,
M = N + Fd.
(22.49)
The solution can be computed by fixing W and f$ and letting F vary.35 The slope a of the tip increases gradually as F increases, as does the curvature of the element. However, shortly after the tip rotates past 90°, the vertical component of F opposes W so that a relatively small increment of pressure load causes a "flip-over," or a large change in the shape of the elastica. Another, apparently abnormal, phenomenon that can be explained using the theory of the elastica is the kinking of submarine cables when the initial tension is temporarily reduced (Yabuta, Yoshizawa, and Kojima 1982). It has frequently been noted that loops occur in oceanic cables, which are perfectly straight under a given tensile force, as soon as the tension is relaxed, and when the cable is retensioned the loops decrease in diameter, kink, and may cause damage. The explanation of this kinking is relatively simple if we assume that a state of initial twist accompanied by tension preexists in the cable. The twist is due to the fact that the wires composing the cable are helically stranded so that torsional strains are induced under tension. Then, if the tensile load is decreased, perhaps as a result of wave motion, a reversed torsional load is induced, which may also cause loops. This results from a transfer of torsional strain energy to bending strain energy. However, when the cable is retensioned, the loop gradually decreases in diameter, and eventually reopens, transferring back the bending strain energy to torsional strain energy. The initial straight twisted stated is shown in Figure 22.7(a), the cable with a loop is shown in Figure 22.7(b), and in Figure 22.7(c) the loop has become very small under increasing load and may eventually disappear. To study the problem, we neglect the cable weight and the elongation of the cable, and assume that the cable is linearly elastic, with a bending stiffness B and a torsional rigidity C. The cable loop is regarded as a closed curve branching from the line of action of the tensile force, as represented in Figure 22.7(b). The points of the central line of the loop are identified by Cartesian coordinates x and y with respect to a system having its origin at the center and the x axis parallel to the line of action of P. Alternatively, polar coordinates r and 0 can be specified, so that the loop crossing point is at x = 0, 0 = —(TT/2). If the loop is perfectly planar, the strain energy is purely flexural. However, if the loop leaves the xy plane and becomes displaced so as to form a cylindrical helix with the z axis perpendicular to the xy plane, part of the 35
Wilson and Snyder (1988) described the deformation of the elastica numerically, but it would be interesting to treat the problem in terms of bifurcation theory.
196
22. Nonclassical Problems for Rods P (a) P
A
(b)
AL
-I / v (c)
Fig. 22.7
strain energy is transferred to torsional strain energy. In this configuration, the coordinates of the central line of the loop have the expressions x = r cos 0, y = r sin 0, z = z.
(22.50)
The curvature K and torsion cp can be calculated from (22.50) according to the formula for the differential geometry (Eisenhart 1947) ic2 = [(r2 + ra + zt2)(r"2 + 4r'2 + r2 - 2rr" + z//2) - (r'r" + rrf + z'zf')2}
x[r2 + / 2 +z / 2 r 3 , 2
(22.51) 2
2
//
/
f
(22.53)
where r0 indicates the residual twist of the straight configuration which cannot be converted to bending energy of the loop. In the intermediate helical configuration we thus have the bending strain energy WB and the torsional energy WT: L
L
.2. 2
WB=^I
BK ds, 0
Cxlds,
WT=-
(22.54)
0
where L is the length of cable under consideration. The mutual displacement AL between the ends A and A' of the cable when it passes from the configuration shown
Rod Theories: Director Approach
197
in Figure 22.7(a) to the one shown in Figure 22.7(b) is the perimeter of the loop. As a consequence, the work done by the external forces has the value ££ = PAL,
(22.55)
provided that we assume P to remain constant during the formation of the loop. We observe that this work is negative when the cable passes from state (a) to state (b), and positive when it passes from state (b) to state (c) in Figure 22.7. The total potential energy of the piece of cable of length L is thus given by E = WB + WT - &,
(22.56)
and the stability of a loop is determined by the critical points of this energy. The analysis of stability is complicated by the fact that r = r(6), the equation of the loop, is not known, but needs to be determined. However, progress could be made if the form of the loop were known, and so we tentatively assume that it takes the shape of an arc of a circular helix having the parametric equations z = R08,
0 ^ 0 ^ 7T, 7T
r = R = constant { z = i),
(22.57)
where 6, the displacement in the z direction, is assumed to be very small. Outside the loop, the shape of the cable is regarded as perfectly straight. The curvature K is then easily derived from (22.51) and (22.57) as /C 2 =-~ 2
R
r^,
0<6<7t,
2 2
(1 + 8 )
**=* -5<0
0
(22.59)
TC<0.
The arc element has the expression ds = dO(R2 + z'H =l*dW+ [RdO,
8
^
O^O^jt, 0
The bending strain energy and the torsional strain energy are evaluated by putting the curvature equal to zero in the straight part of the cable and assuming the torsion to be r0 everywhere except along the arc 0 ^ 0 ^ n of the circular loop:
^ B = 2 -? ( 1 + 5 2 ) 4 + 2 : 7 r i' -^-^
(22 60)
-
2
+ r0j (1 + & f + \ C^LX,
(22.61)
198
22. Nonclassical Problems for Rods
where n
LX=L-
(
y/\+82Rd0.
o The work <£ is considered to be divided into two parts <£ \ and if2: the first represents the work done by the forces P applied at the ends of a portion of cable of length L during the formation of a loop from the state shown in Figure 22.7(a) to the state shown in Figure 22.7(b); the second part is the work done in a small reopening of the cable during the passage from the state shown in Figure 22.7(b) to that in Figure 22.7(c). The work ££x is clearly <£x = -P(27tR); the work <$?2
ma
(22.62)
y be expressed in terms of the angle of the helix P = - z',
(22.63)
and is given by
g 2 = P I (I or, more simply, on putting 1 — cos/J ~ \(z'jrf 7t
by
2
= P [1 (j\ o
sin (9 ds = PR82(l + 82)K
(22.64)
The total potential energy E is then given by the sum E = WB + WT - &x - S£2. If we exploit the condition that 8 is much smaller than unity, the energy assumes the following simpler form
E = ^ (1 - Is2) + it £- 82 + \ R \
4 )
2/v
2
TTCXQ8{\
- 82) + \ CxlL + 2nPR - PR82. (22.65) 2
The equilibrium states of the cable are defined by the pairs 8, R, such that dE dS dE
3nB 2R it.
s +
xCs R
and the stability of the equilibrium is determined by the nature of the Hessian form associated with (22.65).36 36
The stability criterion used by Yabuta et al. (1982) is wrong.
Rod Theories: Director Approach
199
In order to have a very simple criterion of stability, let us consider the case in which r0 = 0, and the cable has no residual twist. Then the equilibrium is determined by the solution (22.66) After calculating the second derivatives of E in the equilibrium state, the Hessian form becomes
H=
o
R3
The result is that the Hessian form is negative definite whenever the first term of the matrix H is negative, and a loop is unstable. Kinking is thus difficult when bending stiffness is large and torsional rigidity is small. This confirms the common experience that wires that are stiff in bending do not generate loops. The elastica theory has several practical applications, even in circumstances of everyday experience which seem to have escaped any attempt at a mathematical approach. One of these unusual problems is the "spaghetti problem," named and studied by Carrier (1949), which endeavors to determine the motion of a dangling inextensible string when it is drawn upwards through a hole in a rigid wall. If the string initially has a transverse motion, the kinetic energy of the string does not change while it is being drawn through the hole and being squeezed into an evershrinking length. As a result, on reaching the hole, the tail of the string must have infinite transverse velocity; this explains why sucking spaghetti improperly leads to a spot of sauce on one's blouse or tie (Mansfield and Simmonds 1987)! The spaghetti problem has an equally interesting reverse problem, which consists of finding out what happens when one spits out a piece of spaghetti which, if undercooked, is not totally limp. The problem arises not only for spaghetti but also in the study of the motion of a sheet of paper issuing from a rigid horizontal guide into a uniform gravitational field. The same dynamic effect as in the spaghetti problem explains why the ends of a sheet of paper extruded by a copier are often curved. To study this last problem, we model the paper as forming an elastica of length L, bending stiffness B, and mass per unit area m. The paper is expelled by a guide and its free part is subject to a weight mg per unit area, g being the acceleration due to gravity (Figure 22.8). Let us call V the constant velocity at which the paper issues from the guide. With reference to thefixedCartesian axes X and Y (Figure 22.8), the sheet initially occupies the interval 0 < X ^ L of the horizontal X axis. At an instant T after the onset of motion, the coordinates of the points of the deformed paper elastica, denoted by X and F, satisfy the conditions X = X+VT,
Y = 0 0<X
(22.67)
200
22. Nonclassical Problems for Rods
mg
Fig. 22.8
inside the guide, and the balance equations d2X
dH
dV dX
m dM dX
„
n
yytrr —
mg i
-i sin
n
d2Y
Wl
dT2'
(22.68) (22.69)
t
for L — VT < X < L, outside the guide. As indicated in Figure 22.8, H, V, and M denote the resultants generated by the material to the right of a particle initially at (X, 0) in the material to the left; the angle ^ represents the inclination of the deformed axis with respect to the X axis. In addition, we have the constitutive relation M = B%,
(22.70)
and the conditions of inextensibility dXL_ J7 . (22.71) SP dX~°° ' dX~~ In order to integrate equations (22.68) and (22.69), we introduce the functions x x P(X, T)= f JT(§, T) d%, Q(X, T)= f #(£, T) d$. (22.12) L
L
Because the end X = L is free, (22.68) can be integrated once with respect to X to yield H =m
dT2'
V = m g{X-L)
+
dT1
and, on substituting these relations along with (22.70) in (22.69), we get B^-mUL-
(d2P . = ml
(22.73)
Rod Theories: Director Approach
201
But, after combining (22.71) and (22.72), we also have the equations
0
^
(22.74)
which, together with (22.73), constitute the field equations outside the guide. The associated boundary conditions are that, at the wall, the velocity is purely horizontal and p is zero, while at the free end the bending moment is zero. In terms of P, Q, and P these conditions assume the form ^ = 0, dX
0 = 0 at X = L - FT,
(22.75)
4 ^ = 0,
P = g = 0 a t X = L.
(22.76)
aX We still have to state the appropriate initial conditions. However, these are not completely obvious because all the paper is initially contained within the guide. The initial conditions therefore apply only at the point (L, 0), and may be obtained by requiring that the conditions at the end of the guide (X — L — VT) match those at the free end (X = L) as T tends to zero. Let us introduce dimensionless quantities by putting P = L2p,
Q = L2q,
X = Lx = L(\-s),
T =
yt.
On substituting these expressions in (22.73) and (22.74) and setting mgL3
Im
we obtain the dimensionless field equations
4 ds
2 dft
(dn
?-4-liscosP = v2[—?-smp L \dr
da \ £COSJ8|, dr I
(22.77) (22.78)
which must be solved in the triangular domain 0 < s < t and 0 < t < 1, subject to the boundary conditions
* co—. As a result of direct computation, the solution can be found as a formal expansion in powers of s and t9 of the form
202
23. Initially Curved Rods
p = l ^ ( s 3 - t 3 ) + f^- (ft - 6s312 + 5t5) + ...,
(22.79)
p = -s + - - s t + ...,
(22.80)
o
24
q = j^(s5-l0s2t3
+ 15st4) + ....
(22.81)
The dimensionless coordinates at the end of the elastica, relative to the end of the guide, defined as
become xe = / + . . . , t4
(22.82)
* = — + ....
Of course, this solution is crude, but might give a rough idea of the form of the bent sheet for short times / and at short distances from the end of the guide.
23. Initially Curved Rods When a rod is naturally bent and twisted, the general equations are still as given in (19.22), but now the constitutive equations, although linear, must be modified. In its unstressed state the center line of the rod is a tortuous curve, the shape of which may be defined by considering an orthogonal triad of directors D,- chosen, for instance, in such a way that D 3 is tangential to the axis of the curve and the other two directors coincide with the principal axes of the cross-sections at their centroids. As the origin of this triad moves along the curve with unit velocity, the components of the angular velocity of the moving triad, referred to the instantaneous position of the axes, are denoted by /c0, KQ, and r0, where r0, the component along the tangent, is the initial twist, and /c0 and KQ, the components along the other axes, are the initial curvatures. When the rod is bent and twisted further, we can construct a second triad d,-, such that d3 is directed along the strained central line, the plane dxd3 contains the unit vector D1? and d2 is perpendicular to the d ^ plane and directed in such a way that the new triad is right-handed. As before, the components of the angular velocity of this triad, as its origin moves along the strained axis, define the twist x\ and the curvatures KX and K[. If we introduce the bending stiffness with respect the d! and d2 axes, and denote them by A and B, and we call C the torsional rigidity, the relations between stress couples and the changes in curvature assume the form mx = A{KX - /c0),
m2 = B(K[ - KQ),
m3 = C(xx - r 0 ).
(23.1)
These formulae are known as Clebsch's constitutive relations (Love 1927, Art. 259). Now, after these preliminaries, let us return to Kirchhoff s equations for the case in which the sections of the rod have equal moments of inertia, so that A = B, and
Rod Theories: Director Approach
203
the rod is a circular spiral traced in its unstressed configuration. This is a spiral on a circular cylinder of radius r having a constant inclination a with respect to the planes perpendicular to the generators of the cylinder. As a consequence of the assumption A = B, all pairs of orthogonal axes in the plane of the cross-section issuing from its centroid are principal axes. We may then choose the triad D z , such that D 3 is the unit vector tangential to the center line, Dx coincides with the principal normal to this curve pointing towards the center of curvature, and D 2 coincides with the binormal and is oriented so that the triad 0 ^ 0 2 , 0 3 is right-handed. In this particular reference system the curvatures and the twist have the expressions
,0 = 0,
4 =^ ,
ro =
^ ^ .
(23.2)
These quantities define the initial unstressed state. Let us now apply terminal forces and couples to the rod. The new state of strain is defined by the formulae (23 .3)
where rx and ax are the radius and the inclination of the new helix. It then follows that the stress couples are given by the equations _
/cos oi\ cos a\ \ rx r )
_/sinai cosai \ rx
sin a cos a\ r )
and the stress-resultants have the form
= C
cos «! /sin <x\ cos a\ 1
—B These are the basic formulae governing the theory of spiral springs. From them we can calculate the resulting force R parallel to the axis of the cylinder of magnitude R = « 3 sin a?! + n2 cos ax,
(23.4)
and the couple K parallel to the same axis
K = m3 sin ctx + m2 cos ax.
(23.5)
If R and K are given, we can determine r\ and c^; alternatively, if we know rx and a\9 the calculation of R and K is immediate. Very often, the deformation is small so that we can write r + 8r, a + 8a for r{ and a1? and linearize the equations. The equations are simpler in the case of a planar deformation, that is to say, when a rod, the axis of which was initially contained in a plane, is loaded by forces and couples that maintain the strained configuration in the same plane. In this case it is even possible to find exact solutions when forces are applied along the length of the beam.
204
23. Initially Curved Rods
An example of a case of this type is the equilibrium of a heavy wire suspended from two fixed points at the same level.37 Let us call 0 the inclination of the tangent to the axis with respect to the horizonal, and let y be the weight of the wire per unit length. As the strained axis is contained in a vertical plane passing through the ends of the wire, the curvatures are K = 0,
K' = ^ , (23.6) as and the twist x is zero. The self-weight of the wire acts vertically, and its components along the tangent and the normal to the strained axis are / 3 = -ysin<9,
fx = -ycosO.
(23.7)
The equilibrium equations become mf2 + nx = 0, n[ -\-n3Kf — ycosO = 0, = 0,
where m2 = BK = B(dO/ds). Elimination of nx and n3 from the last equations yields B
' d_ (d3O/ds3\ ds \ dO/ds
dO
\
{dO/dsf
(23.8)
\
This equation admits a first integral after multiplying both sides by cos 0: 3
n((d
*(
0/ds3)cos0
d20 . +
s
cos20
\ m
d
)
+
y
To
'
(23 9)
-
where the constant of integration To is the tension «3 at a point 0 = 0. A subsequent integration of (23.9) yields d20 B~~^SQc0-\-ys=T 0 dsL
tan 0,
(23.10)
where s is measured from the point 0 = 0. For the case in which the wire is stretched taut under high tension so that any additional strains are small everywhere, (23.10) can be simplified by putting sec# ~ 1 and tan# ~ 0. The resulting differential equation is linear, with constant coefficients. The solution just found may be extended further to take account of extensibility in the axis. The theory for fixed end supports is particularly important in the case of an initially rectilinear bar subjected to transverse loads. Rods with constraints at the ends are called tie rods and their equilibrium configurations under transverse loads may be determined by applying Kirchhoff s theory, but removing the hypothesis of inextensibility of the central line. The problem is well illustrated by the case of a tie rod pinned at its unyielding supports and subjected to a single vertical load P (Figure 23.1) (Huddleston and Dowd 1979).
37
According to Love (1927, Art. 273A), this solution is due to A. E. Young (1915).
Rod Theories: Director Approach
205
Let us consider a tie rod AB, which is straight in its unstrained state, AB being a segment of length L, loaded by a single transverse force at C, at a distance a from A and b from B. After the deformation the rod becomes an arc connecting the fixed points A and B, and C, the point of application of the force P, will shift to a new point C\. On taking a system of axes x and y, as shown in Figure 23.1, the position of a point D l5 which initially had coordinates (x, 0), is now (x + w, v), where u and v are the displacement components, while the position of C\ is determined by the coordinates [a + u(a), v(a)]. As before, let us call B theflexuralrigidity of the rod in the xy plane, and let EA be its extensional rigidity. At a generic point D{ on the deformed axis, the slope of the curve is denoted by Q(s), where s is the arc length measured from A. The bending moment at D! is then (23.11) where M is expressed by Rv
P
Rv
P
M =
~b - u(a)' [x + u(x)] L
'a + u(aj L
{L-[x
+ u(x)]}
0
0 (23.12) a^*
The axial force at D t is denoted by N, where N has the form RcosO + N= < RcosO-
'b - u{d)
L a + u(aj
PsinO (23.13) PsinO
The ratio N represents the elongation of the strained axis, and the coordinates of Pi are related to the slope by the geometric conditions
Fig. 23.1
206
23. Initially Curved Rods = (
,, . ^ =( ds
(23.14)
As the ends A and B are fixed, we must have the boundary conditions w(0)
= v(0) = 0,
u(L) = v(L) = 0;
(23.15)
in addition, the bending moment must also vanish at the ends, so that we have
The unknowns of the problem are the displacements u and v, the rotation 0, the stress resultant N, the stress couple M, and the shear force Q. The stress resultants are related by the equilibrium equations
ds ds dM Q
The full system can be integrated by a shooting method in which the unknown quantities 0(0), R, and u(a) are given tentative values. This converts the problem into an initial-value type one, which gives us u(L) and v(L) and the recomputed u(a), so that we can adjust the initial data to improve the values of u(L), v(L), and u(a).3S The specific influence of extensibility in a naturally curved rod can be well explained in terms of the deformation of a circular ring of radius a, when it is bent and stretched in its plane. In this case it is convenient to introduce the angle 0 between the radius drawn from the center of the circle to any point, and a chosen radius. The initial curvature KQ is then given by dO , 1 - = KO=-.
(23.17)
We also note that, if the plane of the circle is a principal plane of the rod at any point, then K0 and r0 vanish. Let us call u the displacement component directed along the radius inwards and w the component directed along the tangent to the circle in the sense in which 0 increases; the v component, perpendicular to the plane of the circle, is now zero. If we assume that the central line is not extended, we must add the condition
Under the assumption that if the rod is bent by forces having components fr and/# directed along the radius and the tangent at any point, the equilibrium equations are 38
This method has been applied by Huddleston and Dowd (1979), but with a slight modification to (23.11), where they put M = B(dO/dx).
Rod Theories: Director Approach ^
207
=0,
(23.19a)
= 0,
(23.19b)
=0,
(23.19c)
where a\ = a\(0) is the radius of the central line at any point after the deformation. In (23.19c) the stress couple m2 is related to the change in curvature by the equation (23.20)
where B is the flexural rigidity for bending in the plane of the rod. The above equations have been written without making any assumption about the magnitude of strains, but if we suppose these to be small, we can put ,
1
1 d f\ du w
where a is the radius of the undeformed ring. From this equation, using (23.18), we derive m =
2 ~^2\ "XT + ~Ia )•
(23.21)
Hence we find that nx and n3 are expressible in terms of w through the equations B(d4w
d2w\
_
B(£w
d\\
and that w satisfies the equation39 B (d6* ,
*W , d2w\ _ Jdfr (23
-23)
Let us now remove the assumption (23.18) and try to reformulate the problem, still maintaining the hypothesis that strains are small. Then the extension of the central line is (23.24)
and the tension «3 is n3 = EAB.
(23.25)
The expression for the couple m2 and the shearing force n\ depends on the flexural rigidity B, while the expression for n3 depends on the extensional rigidity EA. However, the contribution of the terms containing B may be omitted, to a first
39
According to Love (1927, Art. 292), the result is due to Lamb.
23. Initially Curved Rods
208
approximation, and the equations to be satisfied by u and w are (23.19a) and (23.19b), that is EA
(23 26)
"
The result is that the assumption of inextensibility in a curved rod is one that introduces considerable sensitivity, because it leads to a mathematical description and results that may be far from those furnished by the set of equations involving extensibility. The effect of the self-weight in a circular elastic ring hung at its highest point cannot be evaluated by direct integration of the equation of the elastica, but an approximate solution can be found by using a perturbation method (Wang and Watson 1980). Let us consider a thin circular ring suspended at the top A and subjected to the action of its own weight (Figure 23.2), defined by a constant value pg per unit length. Let L be the perimeter of half the ring, El be the flexural rigidity, and F the lateral force exerted across a section at O. On choosing a system of axes x' and y' with their origin at O and directed as shown in Figure 23.2, the equation of the heavy elastica can be written as 2 dd0 0 El —z — -F sin 0 - gps cos 0, ds
dx
(23.27)
dy'
(23.28)
where 6 is the local angle of inclination and s' is the arc length measured from O. The boundary conditions are (23.29) 0(L) = 7t, x\L) = 0.
(23.30)
r
o Fig. 23.2
Rod Theories: Director Approach
209
In order to treat these equations, we normalize all lengths by putting x'
/ y
and the governing equations become d20 —~= -AsinO-BscosO as1
(23.31)
^ = cos 0, % = sin 0, as as where A and B are nondimensional quantities defined by A=
, El '
(23.32)
* = El
The boundary conditions are
0(1) = 7T,
JC(1) = 0.
(23.33)
No solution of (23.31) and (23.32) in closed form is available, because A is also unknown. However, an approximate solution can be found for small values of the constant B, which measures the relative importance of density and length to rigidity. A small value of B means relatively stiff rings with almost circular shape. Since the effect of gravity is small, we expect A also to be small. Now, let B = e«,l,
A=ae,
where a is of the order of unity, and consider the formal expansions 0 = 00(s) + eel(s) + ...,
(23.34a)
x = jco(s) + exi(s) + . . . , y = yo(s) + eyi(s) + ..., a = ao + sal+....
(23.34b) (23.34c) (23.34d)
When these expansions are substituted in the field equations and boundary conditions, we find that the zero-order terms must satisfy the equations >=
0.
^ = cos0o, as
^ = sin % as
(23.35) = 0,
( 23 - 36 )
and thus the solution is 0 O = TCS, *0 = ^
(23.37a) ,
(23.37b)
TV 1
yo=
-COS7TS
71
.
(23.37c)
210
23. Initially Curved Rods
The equations of first order are = a0 sin 0Q— s cos 0Q,
—^- = — Q\ sin 0O,
as 0{ (0) = *!(()) = ^(0) = 0,
(23.38)
0,(1) = x{(\) = 0.
(23.39)
The corresponding solution is (23 40)
-
3
^sin27T^
cos 2ns
s sin ns
«o = - ^ -
cosns (23.43)
An idea of the final configuration of the ring is given by its maximum extended vertical length
l
(^£)>
(23-44)
and by its maximum width at 0 = n/2. To calculate the latter, we use (23.34a), (23.37a), and (23.40) to obtain the corresponding arc length
On substituing this value of s in (23.37b) and (23.41) we obtain the maximum width 2x(s*) =
l-e(2--^+
0(e2).
(23.45)
One of the most important applications of the theory of naturally curved rods is in the analysis of wire ropes. Over the years, many types of wire rope have been designed and tested for various applications, but the theoretical analysis of the problem is very complex. It is thus not surprising that, until 1950, the only reliable criteria for designing wire ropes were based on extensive programs involving testing under both static and fatigue loading conditions. According to Phillips and Costello (1985), the first assessment of useful design criteria taken from existing experimental data was done by Drucker and Tachau (1945), and the first attempts at theoretical models predicting the stresses in the individual wires of a strand were made by Hall (1951) and Hruska (1951). Wire ropes are complex structures formed by helical wires of circular cross-section wound round a central rectilinear wire called the core (Figure 23.3). The wires and core form a strand, and complex cables are formed by the union of strands which are also formed into helices. Sometimes the strands have no core and the single wires are wound directly around one another. The cross-section of a strand with a core is
Rod Theories: Director Approach
211
Fig. 23.3
sketched in Figure 23.4 where, for clarity, the section of the core has also been drawn separately. Since the wires are aligned along the helices, their horizontal crosssections are ellipses; the horizontal section of the core is a circle. Wire ropes have been employed since the beginning of the nineteenth century, but calculations of their behavior have been extremely crude. As ropes are essentially destined to bear axial tensile forces, the immediate initial assumption is that both the wires and the core simply carry an axial tensile force, parallel to the central line of each element. The
212
23. Initially Curved Rods
sum of the tensile force in the core and of the vertical components of the tensile forces in the wires must balance the total axial force in the strand. All the formulae proposed in manuals of mechanical engineering rest on this assumption and the difference between one formula and another depends only on the criterion used to assign the fraction of load absorbed by the wires and that absorbed by the core. Traditional methods of calculating the stresses in wire ropes have been resolved by a large sequence of experiments accompanying the technological progress in structural design. However, these tests have in fact confirmed that there are some effects in the deformation and the generation of stresses in a wire rope that are not negligible. First, the flexural rigidity of each wire is often very high, especially in wires made of high tensile steel. The flexural rigidity of the wires is also important in very flexible ropes, such as those employed by rock climbers, in connection with knots, when these are under severe tension. Secondly, the wires and the core, being in contact with one other, transmit transverse forces between them. This has the result of altering, sometimes radically, the state of the stress calculated under the assumption that each of these components is independent. The situation is even more complicated if we take into account the fact that the transverse forces are purely compressive, and may disappear as soon as loads are removed and contact is lost. In the limit, the separation between the wires and the core in some parts of the cord is so high that a strange local inflation occurs, creating a form like a hollow bird cage. This effect is particularly dangerous in cables used in mobile situations, because, if a bird-cage condition is generated in the vicinity of a pulley, the cable is at risk of immediate damage. The influence of flexural rigidity and lateral forces, resulting from the packing of wires or cords in a strand without a core, can be evaluated approximately as follows (Phillips and Costello 1973). A cable is regarded as a collection of m smooth wires the unstrained configurations of which are helices, defined by the radius r of the cylinder on which they are wound and their helical angle a. If we take a system of directors coinciding with the principal normals of each helix, the curvatures of the unstrained helix are (see (23.2)) , K0 = 0 ,
cos2 a
K0=
,
and the twist is sin a cos a When the cable is loaded, the new configuration of each wire, provided that it is still a helix coaxial with the initial helix, is characterized by two curvatures and a twist, typically ,
cos2Qf!
sin c^ cos c^
where r{ and a\ are the radius and angle of the new helix. Let us call N the shear force along the normal to the helix, N' the shear force along the binormal, and T the tension along the tangent. These forces have previously been denoted by n\, n2,
Rod Theories: Director Approach
213
and «3, respectively. The stress couples are represented by a couple G' round the binormal, a couple G round the normal, and a twisting couple H round the tangent. In the notation used for the general theory these couples were denoted by m1? m2, and ra3, respectively. Here, N and G vanish, since K0 and KX vanish. Let us now assume that the exterior body forces and body couples per unit length are zero. We now wish to describe the effect of the contact force per unit length on each wire as a constant body force X directed along the principal normal. In these circumstances we suppose that stress resultants and stress couples are independent of the variable S\, the arc length along the central line of each strained wire. It thus follows that the equilibrium equations, relative to the strained configuration, reduce to -G'TX
0, + HK[ - N = 0. f
(23.46) (23.47)
As, by virtue of the constitutive equations, we know the values Gf and H, G' = B(K[ - *J),
H=
C(rl-z0)9
f
and by eliminating N from (23.46) and (23.47) we can express X in the form X = [Cfa - to)*{ - BQci - i ^ n t o - TK[.
(23.48)
Hence the contact force is determined by the tensile force T and the curvatures and twists. The total axial force and the total axial twisting moment on the cable are given by F = m{Tsmax +N'cos ai), M = m(H sina x +G' cos ax + Trx cos otx - N'rx sin ax).
(23.49) (23.50)
The force F and the moment M form a wrench the axis of which coincides with the axis of the helix. An improvement in the theory may be achieved by specifying the magnitude of the contact forces mutually exerted between the single wires and the positions of these forces (Costello and Phillips 1973, 1974). Let us consider the cross-section of a wire, traced perpendicularly to its helicoidal longitudinal axis in the strained configuration. This cross-section is a circle of radius R, the center Ox of which is at a distance rx from the axis of the helix and describes a curve, which, in a Cartesian x, y, z system having the z axis coinciding with the axis of the helix, has the parametric equations x — rx cos (p,
y — r\ sin cp,
z — kcp,
where A: is a constant related to the pitch h of the helix by the equation h = Ink. The points of the circle of radius R where it touches the contiguous wires are placed at an angle p symmetrically with respect to the binormal b, along which the force per unit length X is directed (Figure 23.5). On denoting the magnitude of the effective contact forces per unit length at the contact points by Q, the relation ^
( 23 - 51 )
must hold, and hence once we know X and f$ we can find Q. Since X is given by (23.48), it remains for us to determine p. For this purpose, let us consider the section
214
23. Initially Curved Rods
Fig. 23.5
of the helical tubular wire with a plane perpendicular to the axis of the helix. This section is approximately an ellipse with half-axes (^/sino^) and R, where a{ is the helical angle (Figure 23.6). As there are m wires, the angle made by the tangents to the ellipse drawn from O is lir/m. The distance of Ox from each of the points of contact of the tangents B and B' is
Rod Theories: Director Approach
215 -i!
cos' 2
m)
sin2
whence we are able to express r{ as a function of a\\ OB/
/n
.
n\
n
2
sin2
1 cos l\2— m
m
Having now obtained r1? the angle /* in Figure 23.5 can be evaluated approximately by assuming that the point of application of each force Q is not far from the point of contact of each of the tangents drawn from O to the circle of radius R and the center at Ox. It thus follows that we have
4
tan"
R cos ft = — =
(23.52) \2 m, 2 sin ci\ This formula is simple, but not exact. It can be improved further, but the result is much more complicated (Costello and Phillips 1974; Kunoh and Leech 1985). So far we have not discussed possible loss of contact due to a local separation of the wires. When the wires are separated in the deformed configuration, the inequality tan'(---'15 V2 m sin2 a.\ must hold and X must vanish. We use (23.48), and recall that in a circular section of radius R, like that of wires, the bending stiffness B and the torsional rigidity C are given by B=
nER4
C=
nER4 4(1+or)'
where E is Young's modulus and a is Poisson's ratio. The resultant stress T is either prescribed, or, provided that the wires are extensible, related to the elongation e by an equation of the form T = 7tER2e. In the second case, on substituting the values of B, C, and T in (23.48) and using the relations = 0,
K[
=
cos 2 ot\
cose*!
we find that rx satisfies a quadratic equation for which the only physically meaningful root is (Phillips and Costello 1979)
±,
216
23. Initially Curved Rods
where A
f / •
L
sine*!,
+ °") J
L
a
• 2
c= —
2
sin a\ cos ofj.
In these expressions, s and a.x are still unknown because they are part of the solution. But, if s and ax are given, we can express all the other relevant quantities (rls JV', X, etc.) in terms of e and c^. As, in general, the data for the problem are F and M, as expressed by equations (23.49) and (23.50), respectively, the unknowns s and otx must be determined, at least in principle, by solving these two equations with respect to s and otx. Owing to the nonlinear nature of the governing equations, they are often solved numerically (Phillips and Costello 1979). None of the above results takes into consideration the presence of an elastic core, which could absorb a certain part of the applied axial force F and twisting moment M. In order to treat this situation as well, it is necessary to divide each of the quantities into two parts (Huang 1979): M = M^ + MQ,
(23.53)
where Fw and Mw are carried by the helical wires, and Fc and Mc are carried by the central core. Therefore, the core is under the action of Fc and Mc and lateral forces X along the line of contact with the wires. In order to evaluate the deformation of the core, we replace the line forces X on the mantle of the core by a statically equivalent uniformly distributed lateral normal pressure of intensity mX ~ 2n(rx - R) sinc^ '
q
where (rx — R) is the radius of the core (see Figure 23.4) and a{ is the slope of the central axis of the helix of each wire. The axial strain of the core is therefore mX and the transverse strain in the core is
k £ }
<2355>
Solving these equations we obtain the results
2*^-^1 -A =
2£7r(r 1 JQsina 1 7j—;—TTj
,
; T T " ( £ C - crsc).
(23.57)
m{\ +CT)(1— 2a)
It should be noted that the transverse strain e'c is related to r^ and R by e'c =
r — Ko
r
±^-h
(23.58)
Rod Theories: Director Approach
217
where r is the radius of the axes of the helices of the wires in the unstrained state, and Ro is the radius of the cross-section of each undeformed wire. The angle of twist per unit length of the core is related to the twisting moment Mc by the formula
£Jfi±2> ±2 „..
(23.59,
— R) The axial force due to the wires is given by (23.49), which now reads F w = m(T sinai + A^'cosc^), and the associated twisting moment, according to (23.50), is Mw = m(Hsinax +G ! 'cosa 1 + Trx cosax - N'rx sinai). Let us consider a wire of M o turns in the undeformed state, and assume that, after the deformation, it has M turns. Hence the undeformed and deformed lengths of the wire measured in the direction of the axis of the strand are, respectively, Lo = 2nM0k0,
L = 2nMk,
where k0 and k are two constants related to the pitches of the two helices by h0 = 2nk0y
h — Ink.
The undeformed and deformed lengths of the wire are then where
p = Jr2 + R20,
Pl =
RQ and R being the radii of the cross-sections of a wire before and after the deformation. The axial strain of the strand is therefore Lo
^ i , Moko
(23.60)
1 ^ 1 -
(23-61)
and the axial strain of the wire is
The lateral contraction is then l—l
= -<xew.
(23.62)
After elimination of £w and M/Mo from the last three equations, we have
In the case where there is no rotational slip between the central core and the wires, the angle of twist of the strand per unit length of the strand is ^(MM0)
=^ r ( ^ l ) .
(23.64)
218
24. The Stability of Rods
After elimination of £w and M/Mo from (23.61), (23.62), and (23.64), we obtain (23.65) Thus, if the ends of the strand are unconstrained, elongation of the strand will be coupled with torsion. On the other hand, if the ends of the strand are fixed, Q vanishes and we have the following condition of constraint Pi = ,
In both cases, the procedure for solving the problem is as follows. For given values of the inclination of the unstrained helix a, the number of wires m, Poisson's ratio a, the total twisting moment M, and the ratio of the radius of the cross-section of each wire R/Ro, we assume a tentative value of A:. We then find a solution and, eventually, we start the procedure again, using a correct value of k. The method also allows us to ascertain if there is any separation between wires. It is found that, when the central core and surrounding wires are made of the same material, the extension of the strain always gives rise to a separation between helical wires.
24. The Stability of Rods One of the merits of Kirchhoff s theory is that it allows an investigation of the stability of a rod under combined states of stress. It uses a systematic method, in which all approximations that are applicable can be put into practice. The simplest of the cases concerned with a loss of stability under combined loads occurs when a rod, assumed straight in the unstrained state, is subjected to a thrust R and a couple K. Let us consider the case for which the cross-section is doubly symmetric so that the principal moments of inertia A and B are equal, and assume a system of fixed unit vectors such that e3 coincides with the axis of the rod and e t and e2 are placed in the perpendicular plane containing the lower end of the rod (Figure 24.1). In a strained state, a generic cross-section, initially placed at a distance S from the origin, is moved to a new position defined by an arc of length S\ and by a triad of directors dl5 d2, d3 coinciding with the principal torsion-flexure axes of the rod at the section. The directions of these axes are defined by Euler's angles 6, \fr, and (p, of the new triad with respect the fixed axes. The direction cosines of the axes x, y9 and z are expressible in terms of Euler's angles, and their detailed form is obtainable from equations (17.15) by observing that we have lx — dx • e l s
m\ = &i • e2,
, n3 = d3 • e 3 .
(24.1)
By using these direction cosines in the equations we can express the resultant couples acting on the section in terms of the force R and the couple K: AKIX + Btc't2 + CU3 = f
-Ry,
+ BK m2 + Crm3 = Rx, IX 4- BK'H2 + Crn3 = K,
AKYYIX
(24.2)
Rod Theories: Director Approach
219
Fig. 24.1
where x and y are the coordinates of any point on the strained axis. When we replace K, K\ and r using (19.15) and £1? ..., «3 using (24.1), we obtain a system of three ordinary differential equations in 0, T/T, cp, and x, y. A relationship between JC, y and Euler's angles can be found using the formulae of the differential geometry, but their precise expressions are not necessary. Since the treatment of system (24.2) is very complicated, we approximate the equations by considering small deviations from the unstrained rectilinear state, and study the stability of the solutions by applying the linearized criterion for stability. In an approximate analysis, the coefficients lu m1? ..., are given by the equations (Love 1927, Art. 258) dx
dx
dy_ dS'
dy
(24.3)
where the quantity f$ is, to first order, equal to (i/r +
K' =
dS2'
X =
dp dS'
(24.4)
On substituting (24.3) and (24.4) in (24.2), and neglecting the terms of higher order, we obtain the simpler system
220
24. The Stability of Rods
(b)
(a) Fig. 24.2
d2x
dy
(24.5)
or, alternatively, the pair of equations
7 ax
(24.6)
dy
If the beam is hinged at its end, the boundary conditions are x(0) = x(L) = y(0) = y(L).
(24.7)
The combinations of K and R for which (24.6) and (24.7) allow nontrivial solutions determine the conditions of instability of the structure. In practice, as we have undertaken a linearized analysis, only the combinations giving the lowest eigenvalue of the problem are of interest regarding the onset of instability. The situation is well illustrated by considering two limiting cases. If A' is zero, then system (24.6) splits into two separate equations, from which we obtain the result that the critical value of R is exactly Euler's load (jt^B)!!}. Alternatively, if R is zero, then the first eigenvalue of the problem is K = (2TTB)/L9 and a behavior well known to designers of wires having a moderate flexibility occurs in which, when the torsional couple exceeds a certain critical value, the wire suddenly becomes a spiral. Let us now consider the case of a cantilevered rod the cross-section of which has a flexural rigidity B with respect to one principal axis, which is much larger than the flexural rigidity with respect to the other principal axis, and much larger than the torsional rigidity. This would be the case for a rectangular cross-section in which one pair of sides is much longer than the other pair (Figure 24.2(a)). The rod is built in at
Rod Theories: Director Approach
221
one end and bent by a vertical transverse load R applied at the centroid of the other end, and placed in the plane of highest flexural rigidity. Let us choose a system of Cartesian axes such that the origin O coincides with the centroid of the clamped section, the X axis is the central line in the unstrained state, and the Y and Z axes are the principal axes of inertia of the initial section. If the length L of the beam is not too great and the load R is sufficiently small, the rod will simply bend in the YZ plane. However, if the magnitude of the load exceeds a certain limit, the plane bent state will become unstable, and the rod will be distorted and twisted out of its plane. This kind of buckling is known as Michell-Prandtl instability40 (Figure 24.2(b)). The equations that govern the problem are again Kirchhoff s equations (24.2), where the right-hand sides must now be modified. Let us call S the distance of a point on the central axis from the origin in the unstressed state; after the deformation, the initially rectilinear central axis is displaced into a curved arc, and a point of the deformed axis will have coordinates x, y, z with respect to the fixed axes, and its arc length measured from the origin will be S\. Let us call x\, y\, and zx the coordinates of the centroid of the end of the rod where the load R is applied. The equilibrium equations, written for an intermediate deformed section, become AKIX + BK'12 + CU3
= R(yx - y),
f
BK m2 + Crm3 = -R(xx - x),
(24.8)
f
+ BK n2 + Crn3 = 0, which again can be transformed into three differential equations in Euler's angles 6, iff, and
*~
ds9
K
~ds'
x
T
~ds'
On recalling that A and C are small compared to B, (24.8) can be replaced by
B^=-R(L-S), dS
(24.10)
As dy/dS = m3 = sin 0 sin \jr = x/r, nearly, we derive from the first two equations the new equation C^ 40
+ ^ [R(L - S)ir] = -Rx/r,
(24.11)
Love (1927, Art. 272). Note that, in contrast to the customary notation, the central line of the unstrained rod is denoted by X.
222
24. The Stability of Rods
and from the last two equations we deduce the result A-—= R(L — S)(p,
(24.12)
and, eliminating \\r from (24.11) and (24.12), we obtain ~(L-S)2(p = 0.
(24.13)
This is an ordinary linear differential equation in (p, which is reducible to a Bessel equation. In addition, we know that (p vanishes for S = 0, where the cross-section is clamped, and that d(p/dS vanishes at the free end S = L. The combination of the coefficients in (24.13), which corresponds to the first eigenvalue of the problem, is 4
R2L4
where y is a number very nearly equal to 2. It then follows that for a load R such that R>y
1T
the rod, bent in the plane of greatest flexural rigidity, becomes unstable. Note that the critical value of R does not depend on B. Both Michell's and Prandtl's theories use as unknown functions Euler's angles of the strained axis and, consequently, are unable to embrace all the types of boundary condition that may occur in beam theory when these conditions also involve the displacements of the central line. To overcome this lack of completeness it is necessary to find the connection between Euler's angles and the displacements u, v, and w along the fixed axes X, Y, and Z. 41 These relations are again obtainable from the scheme (19.24), by observing that, if S\ is the arc length of the deformed axis, we have
that is to say, since x = X + u, y = Y + v, and z = Z + w = S + w, it follows that du dv dS dw i 0 V , - j - - = sin0sinV, -JTT + -JTT = cos 0. (24.15) Since the elongation e is small, differentiation with respect to S\ may be replaced by differentiation with respect to S. Thus the second derivatives of u and v with respect to S become u" = —j = cos0 fl' d2v
v" =—2 = 41
Celigoj (1979), with a slight, but unessential, change in sign.
Rod Theories: Director Approach
223
where the primes denote differentiation with respect to S. From these equations we immediately have the relation The following equalities are also easily verified u" cos(#? + VO + v" sin(#? + T/0 = 9f cos 0 cos (p + \frf sin 0 sin
n 2 e = ^ ' + ^'cos6> - M^r/ + w V . rcos<9 = x/rf + cp'cosO- \jr' sin If we use the approximations of the linearized theory (24.4), and put we obtain an explicit formula for the first- and second-order terms in the curvatures and the twist: = u" + v"fi = v"p + KP\
(24.17)
= p>- u'v» + u"v = -u'v" + fi V If we accept the ordinary linear theory, according to which the stress couples are connected (as before, but with a change of superscripts) with the curvatures and the twist by equations of the form (24.18) we can write (24.18) in the more explicit form
= -Avn + Bi = Bu"--Gmp, A C 0'
+ £:G(V A
(24.19)
C JJ
which show that, in contrast to the first-order theory, each stress couple in the second-order theory is connected with all the kinematical functions u\ v\ and p and their derivatives. In other words, in the second-order theory, a twist ft about the Z axis generates stress couples round the other axes, and so on. An illustrative application of the theory is the combined application of a longitudinal compressive force T = —P along the Z axis, two transverse distributions of uniform loads along the longitudinal axis, and possible bending moments at the ends. If the transverse loading is represented by fx &ndfy per unit length along the
224
24. The Stability of Rods
X and Y axes, respectively, the field equations, derived from Kirchhoff s theory with the application of (24.19) become
Buiv + Pu" + (l- ^ ) ( G ( 1 W =/*,
(24.20)
" -(l- §V'V - (l - §V(1V = 0. It is easy to see that these equations are not independent but coupled, and so their solution, although simple, may be cumbersome. But, to illustrate the consequences of (24.20), let us study the stability of the solutions under the action of a pure bending couple Mo about the X axis (fy=fx = G'(1) = P = 0 and G(1) = M o ), and assume that A is much larger than B or C If the lateral displacements vanish at the ends, in addition to the stress couples (except Mo), a simple computation yields the condition of critical equilibrium
f)(S)ia formula which was first found by Chwalla (1939, 1943). If we work only with the linear terms, we obtain (Vielsack 1975) ^ .
(24.22)
which, for A » B, C, can be approximated by %
(24.23)
which is the classical formula obtained by Prandtl (1899). The variational method is the most powerful one for treating the linear stability of elastic structures. The method is based on the fact that the critical compressive thrust P on a simple rod clamped at its ends minimizes the ratio L
JBv"2 dz (24.24)
vf2dz where L is the length and B is the flexural rigidity. The minimum is contained in the class of functions v(z), which have piecewise continuous second derivatives, such that v(0) = vf(0) = v(L) = v\L) = 0. This ratio is known as Rayleigh's quotient.
Rod Theories: Director Approach
225
The variational formulation of problems of buckling is not restricted to beams, but can also be extended to cases of structural stability involving frames, plates, or shells, provided that the Rayleigh quotient (24.24) is defined as the ratio W k =j -
(24.25)
where W is the strain energy arising from the buckling and L is the change in the potential energy of the loads due to buckling. If the loads acting on the structure are proportionally increased by a factor A., the minimum of the ratio (24.25) defines the value of the multiplicative parameter of the loads at which the elastic equilibrium becomes unstable. Once the condition defining the onset of structural instability has been formulated in the form of a minimum principle, it is possible to apply the general theory of minima to determine the dependence of the solutions on the data. For instance, on examining (24.24), it is clear that an increase in the flexural rigidity B will not decrease the numerator of the right-hand side of (24.24), and so P does not decrease. Alternatively, a relaxation of the boundary conditions, such as, for example, the suppression of one of the end constraints on v(z), will enlarge the class of admissible functions while having the effect of not increasing the minimum of P. All these results have been codified by Courant (Courant and Hilbert 1953, Vol. 1, Chap. VI) in the form of a list of theorems on the variational characterization of eigenvalues of self-adjoint problems, such as the case of elastic stability under conservative loads. The consequence has been that there is now a widely held conjecture that reinforcing a structure, either by increasing its stiffness or by providing additional constraints, increases (not decreases) the critical load. However, although accepted almost as dogma, the principle has been proved false, at least when the restrictions are not chosen carefully (Tarnai 1980). In order to elucidate how the additional constraints modify the critical load, it is necessary to distinguish two kinds of constraint: those that impede the displacement components arising during buckling and acting in the buckling directions; and those that prevent the other displacement components, and not occurring in the equations of buckling. The former are constraints on solutions of the eigenvalue problem and called proper constraints; the latter are constraints on displacement, not found in the eigenvalue problem, and are called improper constraints. For example, in the case of in-plane buckling of a compressed straight bar, the lateral components of displacement of points on the axis will be eigen-displacements; but the axial component of the displacement will not. Provided that the structure is of a linearly elastic material admitting a stored energy function and the external loads are conservative, only bifurcation problems need to be considered, so that the critical load factor means the load factor associated with a point of