The estimation of elastic critical loads

J.Mech.Phy~.Soli&, 1967,Vol. lS,pp.3llto317. Psl~onPrcu

THE

ESTIMATION

Ltd. printed&ckat Britain.

OF ELASTIC

CRITICAL

LOADS

By J. M. T. THOMPSON Department of Civil Engineering, University College, London (Receiued 1711rApril 1967)

in the unloaded state of the path derivatives of the stability determinant of a diecrete conservative system yields an ordered form of stability esthnation. A particular advantage of the resulting approximation scheme in the case of a nonlinear fundamental equilibrium path is that analytical effort is appropriately deployed between the equilibrium and stability problems. The proposed method of analysis is presented in general terms, and two Zowcrbound theorems are established. TEE EVALUATION

1.

INTRODUCTION

the loss of stability of an equilibrium path of a conservative structural system it is natural to examine the variation of the stability determinant along the path. One way in which this can be done lies in the intrinsic determination of the derivatives of the determinant with respect to some path parameter. These derivatives can be readily evaluated in the unloaded state using the path derivatives of a perturbation analysis, and their systematic evaluation yields an ordered form of stability estimation. A particular advantage of the resulting ‘ frozen-coordinate ’ approximation scheme in the case of an initially-unknown nonlinear equilibrium path is that analytical effort is appropriately deployed between the equilibrium and stability problems. The proposed method of analysis is here outlined in general terms, and two theorems are established. These show that in certain definable and relevant circumstances the stability estimates represent lower bounds for the first critical load of a system. The discrete representation of continuous structural systems, using for example the finite-element Rayleigh-Ritz procedure, is now an established tool in the analysis of practical solids and structures. Much work has been done in the small-deflection linear field, and recent interest has focus& on large-deflection nonlinear problems with their associated stability questions. It is felt that the ordered stability analysis presented here, together with the two lower bound theorems, may prove useful in this area. IN STUDYING

2.

EQUILIBRIUM

ANALYSIS

In the proposed approximation scheme the equilibrium and stability problems are studied concurrently, and it is thus appropriate to present here the equilibrium analysis which corresponds to the static perturbation technique outlined by SEWELL (1966). 811

312

J. M. T.

THOMPSON

Consider a discrete conservative structural system described by the well-behaved total potential energy function V (Qr, A), where Q4 represents a set of n generalized coordinates and A is a loading parameter (THOMPSON 1968). The n equilibrium equations of the system are I’4

(&A 4 = 0

(1)

where a subscript on V denotes partial differentiation with respect to the corresponding generalized coordinate. These equations define a series of equilibrium paths in the (n + I)-dimensional A-Q4 space, and we shall here focus attention on the path emerging from the unloaded state Q4 = A = 0, writing this path in the parametric form

81 = Qt (4, A = A(s).

(2)

Here s might represent any suitable parameter defining progress along the path, but we shall in fact assume that s is to be equated to one of the basic variables, Q4 or A. This variable must be chosen with some care, since it is essential that progress along the path should be associated with a monotonic change of the variable : that is to say the functions of (2) must be single-valued. The formulation naturally takes a particularly simple and symmetric form if s is equated to the loading parameter A, but this choice is inappropriate if the system exhibits a limit (snapping) point. The equilibrium equations must be satisfied at every point on the equilibrium path, so we can substitute (2) into (1) to give the n equations Vt

[Qj (4,

A (s)] = 0.

PO)

The left-hand side of each equilibrium equation is now an implicit function of s, so we can differentiate each equation with respect to s as many times as we please. Differentiating once we have vrj Q, + 74’ A = 0 (El) where a dot denotes differentiation with respect to s, a prime denotes differentiation with respect to A, and the dummy-suffix summation convention is employed with all summations ranging from one to n. Differentiating a second time we have (Vtjr Qk + vrp’ A) 0, + vr, Q, + (V4j’ Qj + Vt” A) A + V,’ A = 0,

(J32)

etc, giving us the equations El, E2, Es, E4, . . . . These equations can be evaluated at the unloaded state Q4 = A = 0, and we shall refer to the evaluated equation Emlo as the mth order equilibrium equation. In this equation all path derivatives are evaluated at the unloaded state. We now allow one of the basic variables, Q4 or A, to be independent by equating it to s, and we correspondingly replace its first derivative with respect to s by unity, and its higher derivatives with respect to s by zero. The ordered equations Emlo can then be solved sequentially as a system of linear equations. Thus &lo represents a set of linear equations in the remaining first derivatives, and can be readily solved. These first derivatives can be substituted into the secondorder equilibrium equation E212 which now represents a set of linear equations in the remaining second derivatives which are thus readily obtained. The known

313

The estimation of elastic critical loads

and second derivatives can be substituted into the third-order equilibrium equation Es[o which now represents a set of linear equations in the remaining third derivatives, which are likewise readily obtained. Clearly all the path derivatives can be obtained sequentially in this manner. As observed before, the equations take a particularly symmetric form if we equate s to the loading parameter A, and in this case we see that the determinant of each equation Emlo is simply IV*, (0, o)\. The quadratic coordinate form corresponding to VQ (0, 0) is assumed to be positive definite, and it follows that there will be a unique solution for all the path derivatives &lo, &lo, &lo, . . . . There will then be a unique equilibrium path emerging from the unloaded state. Having evaluated the path derivatives we finally construct the parametric equations of the equilibrium path in the series.form first

Qt (4 = ‘&/II s + 4 &jos2

+ $Qtlos

+ . . .

n(S)=rllOS+gnlos2+galoss+...

3.

> *

(3)

STABILITY ANALYSIS

We have seen that we can readily generate a series solution for the equilibrium path emerging from the unloaded state, and we consider now the stability of this path. The path is by assumption initially stable, and any loss of stability can be identified by the vanishing of the stability determinant IV*,\. Such a loss will normally be associated with either a limit or a branching point (THOMPSON1963), but the form of the loss will not concern us here. We simply note that a limit, point will be identifiable in the equilibrium solution itself (provided of course that s has been chosen appropriately) while a branching point will not. The stability analysis is thus essential for the location of branching loads, but is unnecessary if we are simply concerned to locate a limit load, since ordered estimates of a limit load can be obtained from the equilibrium analysis : the stability analysis may nevertheless be useful in providing a second (and normally different) series of estimates of a limit load. Denoting the stability determinant by A (Qa, 4

=

j&r (QK, 41,

(4)

we now start the stability analysis by considering its variation along the equilibrium

path, writing 9 (4 = A [Qz (4, A (41.

(5)

The total derivatives of the determinant with respect to s can be written as & (s) = Aa &p + A’ (1 3 (4 = (A, &j + At’ A) 4?<+ 4 & + (At’ &z + A” A) /i + A’ A,

(6) 1

etc, and these derivatives can be evaluated at. the unloaded state using the path derivatives of the equilibrium analysis. The variation of the stability determinant along the path can then be written in the series form

J. M. T. THOMPSON

314 9 (8) =

52 (0) +

La (0) s +

6 9 (0) 9

+

. . .

(7)

and truncations of this series can be used to yield estimates of the first critical load at which 9 (s) = 0. Thus we have the first-order stability equation 9 (0) + a (0) s = 0, the second-order stability

(Sl)

equation 9 (0) + a (0) s + 4 3 (0) ss = 0,

(Sa)

etc, an ordered series of stability equations that can be solved concurrently with the equilibrium equations in a unified ‘ frozen-coordinate ’ analysis. It is felt that in many problems the above scheme can be used directly with the aid of a digital computer, but when such a direct approach seems inappropriate the foregoing theory can be used to indicate ordered and consistent energy truncations. Thus for a ’ first-order analysis ’ it is consistent to truncate the series expansion 1 Sl v(Q<,&

+

=[JWo) -

+pyw)

+)AqYl(o,o)

SV~~(O,O)QrQr +$fir~(O,o)QtQjQ~+ i

I

+Yr'(o,o)Qt + i!yt~'(o,o)Q~Qg

+

I

....

f....

. ..] 1

3 (8)

+ . . . ...*. according to the line El for the equilibrium for the stability analysis. 4.

analysis, but according to the line Sr

GENERALRESULTS

We consider now the application of this stability analysis to a general class of systems, which we proceed to define. Starting again with the potential energy function V (Qr, A), we suppose that the equilibrium equations Vi = 0 yield a single-valued fundamental solution Qr = QfF (A), and we introduce the ’ sliding ’ set of coordinates qgdefined by the equations Qc = QiF (4

+ qt.

(9)

A new energy function, W, is now defined by the equation F+’(qg, A) = V [QfF (4

+ w 41,

(10)

this function having the property, arising from its derivation, that Wg (0, (1) = 0. In many structural problems W (qt, A) is linear in /1, or can validly be linearized with respect to A, and we focus attention on these problems, writing w (qt, A) = wo (qe) + AW (nt).

(11)

Assuming that W# (0) is positive definite, we can simultaneously diagonalize lV# (0) and WC+ (0) by means of the non-singular linear transformation qg = rxtfuj, this transformation being independent of A and representing a special case of a

815

The estimation of elastic critical loads

more-general dia@;onalizing transformation (THOMPSON 1965). We write finally the transformed energy function

A (w, A)

w (cyj99

E

4

A0 64) + A A’ (ut) E W” (artujl + A W1 (aij uj)

(12)

>

with the properties AtO (0) = A$1 (0) = 0 A(jO(0) = Arjl(o)

(18)

for i # j.

= 0

>

Following the stability analysis of $8 and equating s to R, we now consider the variation of the determinant lV$jl along the fundamental equilibrium path and observe that

[QkB(4 A]1 = p,, to,4 1 I = g IAt,(094

J (4

=

IJ$

where K = 1ct&2.Thus remembering the continuous can write 9 as the simple product 9(A)

=K

(14) 1 diagon8hzation

of A, we

,$C,+ADt)

(15)

where cr E A0

(0)

(16)

Di z Au'(O).

>

Differentiating this product we have

@(A) _ zi3(4) B (A)

B (A)

ci=n i=l

--w)~W=

[9

Dr

Ct + A Dt

_

WI2

giving on evaluation

22 -- (0) = 9 (0) 9 (0)

9 (0) -

i-n

c i=l

.a(0) B (0) =

P WI2

EL

i=n

-

G

c i=l

1 z

_E k]“=_E i=l

i-1

[$]2

I I

(17)

OS)

where At is the critical load associated with the principal coordinate ~6. Two interesting results are immediately seen. Firstly, from equations (Sl) and (18) we see that A*, the first-orderestimate of the lowest critical load, is given by 1 -= II*

i=n

c i-l

1 3’

(19)

316

J. M. T. THOMSON

Secondly, if Z (l//P) and consequently & (0) are zero, the first-order estimate A* becomes infinite and the second-order estimate, A**, is given by

(20) This simplifies further when the eigenvalues of .4 are symmetrically disposed about A = 0, so that for every positive critical load there is a corresponding negative critical load, to give

[+ij2=q+ [$I23

(21)

the summation now extending over the positive critical loads only. Equations (19) and (21) supply two useful conclusions for the class of system under consideration. THEOREM I. If all the critical loads of a system are positive, the first-order stability estimate mill yield a lower bound for the first critical load. This theorem is illustrated by Fig. 1. It may prove useful in structural analysis, since it is frequently physically obvious that all the critical loads of a structure must be positive : the Euler strut forms an example of this.

FIG. 1.

All critical loads positive.

FIG. 2. Critical loads symmetrically disposed about zero.

The estimation of elastic critical loads

317

THEOREM II. If the critical loads of a system are symmetrically disposed about zero, th$rst order stability estimate is in&&e, and the second order stability estimate will then yield a lower bound for the jirst critical load. This theorem is illustrated by Fig. 2. It may also prove useful, since the required symmetry can usually be observed in the physical system : the buckling of a rectangular plate in shear forms an example of this. ACKNOWLEDGMENT The immediate stimulus for this work was a talk given by Professor E. F. MASUR at University College London, baaed on a paper to be published in the Donnell Aniversary Volume.

REFERENCES SEWELL, M. J. THOMPSON,J. M. T.

1966 1963 1965

J. I&&. Phys. Solids 14, 203. J. Mech. Phys. Solids 11, 13. Ibid. 13, 295.