Acta Appficandae Mathematicae 17: 269-286. 1989. 269 © 1989 Kluwer Academic Publishers. Printed #t the Netherlands.

Buckling of Randomly Imperfect Beams

W. D A Y Department of Mathematics, Auburn University, AL 36849, U.S.A.

A. J. K A R W O W S K I Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A.

and

G. C. P A P A N I C O L A O U Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A.

(Received: 9 January 1989; accepted: 27 September 1989)

Abstract. The qualitative behavior of buckled states of two different models of elastic beams is studied. It is assumed that random imperfections affect the governing nonlinear equations. It is shown that near the first critical value of the buckling load the stochastic bifurcation is described asymptotically by an algebraic equation whose coeffficients are Gaussian random variables. The corresponding asymptotic expansion for the displacement is to lowest order a Gaussian stochastic process.

AMS subject classifications (1980). 58F14, 35C20, 35F20.

Key words. Stochastic bifurcation, buckling, asymptotic expansion, nonlinear equation.

I. Introduction

In this paper we study the qualitative behavior of buckled states of two different models of elastic beams under compression by axial forces. In both examples, we assume that random imperfections affect the governing nonlinear equations. The imperfections are described by stationary random functions of the form f(x/e) with zero mean and with e a small parameter. This means that the stochastic perturb- ations are rapidly varying relative to the dimensions of the beam but are not necessarily small. Our object is to analyze the asymptotic bifurcation picture in a suitable scaling.

We show that near the first critical value of the buckling load the stochastic bifurcation is described asymptotically by an algebraic equation of the second or third order whose coefficients are Gaussian random variables. The corresponding

* Work supported by NSF Grant No. DCR81-14726. ** Work supported by NSF Grant No. DMS87-01895.

270 w. DAY ET AL.

asymptotic expansion for the displacement is to lowest order a Gaussian stochastic processes. The Gaussian nature of the stochastic terms in the solutions is indepen- dent of the statistical properties of the imperfections except for some general assumptions which are necessary for the validity of the central limit theorem. We apply the method used in [4, 10] to handle similar problems. However, here we do not go into the detailed mathematical analysis. We focus attention on the results and their interpretation.

The first problem we consider is that of a beam resting on a nonlinear foun- dation. The second example concerns an elastic rod, whose axis has small but spatially rapidly varying imperfections. Each problem is taken up in two parts. The first contains a detailed formulation. The second section deals with the stochastic bifurcation analysis and contains the solution for each problem. The bifurcation analysis is carrried out by methods described in [2, 3, 9, 10, 13]. For the sake of simplicity, some rather lengthy mathematical arguments, similar to those in [4, 10, 16], are omitted here.

2. Formulation of the Problem for the Von K~rm~n Beam

We consider an elastic beam resting on a nonlinear elastic foundation. The re- sponse of the foundation is described by the stationary random function

f(oJ, x/e, w) = {k, + o-q(w, x&)}w(x) + k3w(x) 3 (2.1)

which depends on the vertical displacement of the beam w and where o- is the amplitude of the imperfection q. Here w labels the realization of the random function and will be usually omitted. Stationarity means that the joint distribution of q(z~) . . . . . q(z~) and q(zi + h) . . . . . q (z , + h) for any zt . . . . . z,, and h is the same. We also assume that the mean of q(z) is zero and that k~ is a positive constant.

If the beam is subjected to compression at one end then the dimensionless equations of the problem are (see Appendix (A))

DIDul + ½{Duz} 2} = 0,

ID4u2 - D{Du t Du2 + ½{Du2} 3} + f(x/•, u2) = 0 (2.2)

where

d D = ~ , 0~<x~<l,

and

u~(0) = 0, u1(1) = -A, u2(0) = u2(1) = Du2(0) = Du2(1) = 0.

Here u~(x), U2(X ) are dimensionless horizontal and vertical displacement of the beam, I is a bending stiffness and A is the horizontal displacement at the end of the beam. The bifurcation parameter for Equation (2.2) is A.

BUCKLING OF RANDOMLY IMPERFECT BEAMS 271

It is more convenient to transform Equations (2.2) by letting

Wl(X) = Ul(X) + ax , we(x) = u2(x)

so that w~ and We satisfy

D{DWl + ½{Dw2} 2} = 0,

ID4w2 + A D 2 w 2 - D{DwxDw2 + l { D w 2 } 3 } + {kl + erq(x/O}wa + k3w 3 = O,

Wl(0) = wl(1) = 0, w2(0) = w2(1) = Dwe(0) = Dw2(1) = 0. (2.3)

If o-= 0 then Equation (2.3) undergoes bifurcation at the first critical value A0. Our aim is to find out what happens to the solution of (2.3) as • goes to zero when A is close to Ao and er 4= 0 but small.

3. Stochastic Bifurcation for the Yon K~rm~n Beam

The trivial solution (wb w2) = (0, 0), er = 0 satisfies Equation (2.3) for any A. We linearize (2.3) and obtain the following eigenvalue problem

D2wt = O,

ID4w2 + AD2w2 + klw2 = 0, (3.1)

Wl(0) = W~(1) = 0, W(0) = W(1) = Dw2(0) = Dw2(1) = 0.

Equation (3.1) has a countable number of solutions A,,, (wt., = O, w2., :~ 0) for n = 0, 1 . . . . . 0 < A0 < A~ < • • .. The differential Equation (3.1) is selfadjoint in the space of pairs of square integrable functions w = (w~, w2) with the inner product

£ <wlv) = {wL(w)v,(x) + we(x)ve(x)} (3 .2) )

Our object is to find all solutions of Equation (2.3) for small er and A close to Ao. The problem can be solved by the Liapounov-Schmidt method (see [2, 3, 9]).

We write Equation (2.3) in the operator form

G{w, o-2, o-3} = A{er2, er3}w + N{w} = 0,

(0" 3 ~ 0", /~ = Ao(1 + er2)), (3.3)

where

A{er2, er~}w = { D2w' } " 1D4w2 + A,,(1 + o'2)D2w2 + (kt + er3q)w2

= { A , w , ~ [Ae{er-,, er3}wzJ' (3.4)

N{w} : { I'2D{Dwz}Z } : ~N'{w}I. (3.5) t -D{DwIDwz + 1/2{Dw2} 3} + k3w~ {N2{w}J

272 W. DAY ET AL.

The linear eigenvalue problem (3.1) takes the form

DwG(O, O, 0)w = A(0, 0)w = 0, (3.6)

with a unique, nontrivial solution Wo(X) = (0, Wz,o(X)) satisfying the condition

f2 (wo, wo) = W2,o(X)W2,o(X) dx = 1. (3.7)

Since the solution of Equation (3.3) can be represented in the form

W ( 0 - ) = 0-1W0 + l t~ (O ' ) , ~ = ( 0 - 1 , 0"2, 0"3) ,

l t l l (O ' , X) : (~11(O' , X ) , {]t2(nr , X ) ) ( 3 . 8 )

where the function 0 satisfies the orthogonality condition

(Wo I*(~r)) = We.o(X) ~02(~r, x) dx = 0. (3.9)

Equation (3.3) can be reduced to the system of two equations (see [2, 3, 13]).

QG{001Wo + ~(0-), 002,003} = 0,

PG{001~o + 0(~r), 002,003} = 0,

where Q and P' are the projections

Qw = w - (wl Wo)Wo, P 'w = (WlWo)Wo.

(3.10)

(3.11)

(3.12)

We write Pw for (P'wnwo). For small ~r there is a unique solution for 0 = 0(,r) of (3.10) such that 0(0) = 0 and (Wo]0(,r)) ~ 0. Thus, Equation (3.3) has a solution for small cr if and only if Equation (3.11) is satisfied. That is,

P(cr) =--P(G(0-~Wo + 6(00), 002,003) = 0. (3.13)

The original bifurcation problem becomes now a finite-dimensional one. The next step is the analysis of (3.10) and (3.11) u.sing the Taylor's expansions

of 0(~r) and e0r ) (see [3, 13]).

0 0 r ) = ~ or" - - a " P ( 0 ) , (3 .14) - - 0 ~ 0 ( 0 ) , P(cr) = ~ cr~ I,,l=k a! I,~l=k 0~!

where

- - ot 1 o~ 2 or- 4 Ot = (O/1 , 0¢2, O'3) , ~t'! = O~!10~]20~!3, O ' ~ - - O"1 0°2 003" ,

lal -[- 0¢ 2 q- 4 3 , O a = °tl a ot:~ = 01 0 2 2 0 3 - ,

O Oi = - , i = 1 , 2 , 3 .

00-i

In order to obtain these expansions we differentiate Equation (3.10) with respect

,BUCKLING OF RANDOMLY IMPERFECT BEAMS 273

to o'i and obtain

O~'QG{o'lwo + 0(o)}lo=0 = 0. (3.15)

These are linear differential equations for 0"0(0) which must be solved subject to the orthogonality condition (Wo[ 0"0(0))= 0. Once Equation (3.15) are solved we calculate 0~P(0) by simple differentiation of Equation (3.13). We list the results of our calculations in Appendix C.

To continue further with the analysis, we need the following fact concerning the central limit theorem (see [6]). When q is stationary and mixing with exponen- tial rate, then as E goes to zero, the integral

E--1/2 q(xlE)f(x) dx (3.16) .30

tends in distribution to a Gaussian random variable ~ with zero mean and variance given by

o-~ = E{~} = l ime -1 E{q(xle)q(y/e)}f(x)f(y) dx dy E~,O

2 fo o-} = R(z) dz f (x ) 2 dx, R(z ) = E{q(x + z)q(x)}. (3.17)

Using this theorem, we proceed with the asymptotic solution of (3.13) in two stages. First we put o-3 = 0 and we use the Newton diagram method (see [2, 13]) to find the scaling necessary to solve the equation

P(o-~, o-z, 0) = 0. (3.18)

In our case, ~rl = ~5~1, o'z = 62~2. Next we let 6 depend on E, we scale all of the variables

o-t = 6(e)#1, 0-2 = 62(E)0~2, 0-3 = ~ 3 , 6(e) = eP (3.19)

and assume that the scale of or3 = e's'~3 is known in advance from the formulation of the problem.

Now we calculate the reduced bifurcation equation for Equation (3.13) by taking the limit (see [13], Appendix [C])

P ( ~ , ~z, ~3) = lim 6-3(E)P{6(e)o~l, 82(E)o~2, Ekt~3} al , 0

= 0,02P(0)5,~2 + ~ O~P(O)o=~ +

+l ira Ol(r3e k 2p q(xle)w~.,~(x)dx + ~ 0 )

+ lira r(e, ~, , o=2, ~3, P, k). (3.20) ~+0

274

If k - 2p = - 1/2, then from the central limit theorem we have that

s c= lim e -1/2 q(x&)w2o(x) dx (3.21) ~ ; o

is a Gaussian random variable with mean zero variance

2 f2 o-~ = R(z) dz W4o(X) dx. (3.22)

Hence for the given scale e k of o'3 the unknown scale 6(e) - Ep must be chosen so

that 1 p = lk + s (3.23)

with p > 0. One can check that for any k/> 0 (see [10])

lim r(e, 071, 072, 073, k, p ) = 0 (3.24)

so the reduced bifurcation Equation (3.20) has the form

P(071, 072, 073) = --x~t071072 "~ /~073 ..~ ~:071073 = 0, (3.25)

where

fit = Ao Dw~,o(X) dx, B = Dw~.o dx + k3 W4o dx. (3.26)

Equation (3.25) defines asymptotically the scaled amplitude 071 of the displacement as a function of the scaled bifurcation pa ramete r 072.

In this particular problem we could assume that k = 0 as well. This would not produce a significant change in the final results. However , as we shall see later,

this is not always the case. In the second example, different values of k produce

different results. Therefore, to make the method clear, we have retained k in (3.19).

Once the scaling for the problem is established we are able to obtain an asymp-

totic expansion for w = O-lW0 + 0(or). From the Taylor 's expansion we have (see Appendix C).

w(x) = ( )071Wo(X) + +

"]- ~3(ff){O~lOzZ01O211l(X ) + 07310311tll(X ) ~- ¢Yao=z{lim • 1/201031tll(x)}}-I- ~1, o

+ 0(63), (3.27)

where

ID4010302 + AoD2010302 "}- k1010302 h- Qz{q(x/e)w2,0} = O,

0 1 0 3 0 2 0 ) = 010302(1) = D010302(0) = DO103t~2(1) = 0, ( 3 . 2 8 )

Q2w = W -- W2, 0 WW2, 0 dx, 010301 ~ 0.

W. DAY ET AL

BUCKLING OF RANDOMLY IMPERFECT BEAMS 275

If K(x, y) is the Green's function for the problem (3.28) then by the central limit theorem

f2 W2(x) = lim E-ln0102qJ2 = lim -x/2 K(x, y)q(ylOw2.o(y) dy (3.29)

is a Gaussian process with independent increments, zero mean and variance given by the integral

a2w2(X) = {K(x, y)w2.0(y)} 2 dy R(z) dz. (3.30)

Thus the lowest terms in the asymptotic expansion for w(x) are

w ( x ) : a(0 lw0(x) + a(E) 1½a20(x) +

-~- ~3(~){~1~20102~(X ) -~- ~lO~3W(x)}, (3.31) where

W(x)= W (x)

We now can summarize the results of the bifurcation analysis. Equation (3.25) has two solution branches

(i) oh = 0 which is the beam remaining unbuckled, (ii) ~2 = BI.4~ + ~3/.4 which is the buckled state.

The buckled state of the beam appears when ~2 is bigger than

(~2)~-- ~o~3/A, (3.32)

which is a Gaussian random variable with mean zero and variance

E{(d2)~r} = o~A -2 R(z) dz W4.o(X) dx. (3.33)

Therefore the critical value of A for k/> 0 is

(A)cr = Ao{1 + ek+ u2s~o~3-A-1} + higher order terms (3.34)

and the displacement of the buckled beam is described by the function

W(X) = W(X)16"3-0 + e'3k/2+3/4~2(f3W(x) nt'- higher-order terms. (3.35)

The results described above modify the usual bifurcation picture of the problem. The bifurcation diagram (see Figure 1) is still a pitchfork but its focus shifts randomly along the A axis, producing a correction to the value of the critical force (see Appendix C)

(N)c, -- YSAo(1 + e~/z~3/A), k = O. (3.36)

2 7 6 w . DAY ET AL

/

A: Bifurcation diagram when ~3 = 0 B: Bifurcation diagram when o'3 :# 0

Fig. 1.

The critical force Ncr is a Gaussian random variable with the mean equal to its deterministic value No = YSAo and variance giVen by the formula (see (2.1))

O'2N• = , ( y s ) 2 0 "2 ~+_~R(z)dz f£ w42,o(X)dx{ f£ Dw2o(X)dx} -2. (3.37)

Therefore, the size of the uncertainty in the axial force responsible for the buckling is proportional to

ON,- ,l/2{ f ~ R(z) dz}l/2.const. (3.38)

and may be appreciable if the power spectrum at zero frequency of the initial perturbation q(x/~) is large.

BUCKLING OF RANDOMLY IMPERFECT BEAMS 277

4. Formulation of the Imperfect Rod Problem

Let us consider a thin, incompressible elastic rod subjected to an axial compression of magnitude P. In the presence of the initial imperfection 0, the governing differential equation for the slope of the rod is (see Appendix B)

D20 + )t sin 0 = D20,, 0(0) = 0,(0),

(4.1) 0(1)=0,(1) , ( D = d 0 ~ < x ~ < l ) ' d x '

where the bending stiffness E1 and the length L are incorporated into the dimen- sionless parameter h = pL2/EI. Let Oo(Z) be a stationary random function with mean zero and finite moments. We assume that

O,(x) = o-Oo(x/e), (4.2)

where o- is an amplitude parameter. The small parameter • makes precise the fact that imperfections vary rapidly on the scale of the rod. If we substitute w = 0 - 0,,

then (4.1) leads to

DZw + h sin(w + o-0,) = O, w(O) = w(1) = O. (4.3)

If o- = 0 then the rod buckles at h = )to = ~ . Our objective is to find the limiting behavior of the solution of (4.3) as • + 0 for small o- and h close to )to.

5. Stochastic Bifurcation for the Imperfect Rod Problem

We notice that the triple (w = 0, o- = 0, A) satisfies Equation (4.3) and linearization- leads to the spectral problem

D2w + hw = 0, w(0) = w(1) = 0. (5.1)

The smallest eigenvalue of Equation (5.1) is ho = ~ with Wo = ~ 2 s i n 7rx the corresponding eigenvector. We write (4.3) in the form

G{w, o-2, o'3} = DZw + ho(1 + o-2) sin(w + o-300) = 0, (5.2)

where the operator G is defined for square integrable functions w. Since G(0, 0, 0) = 0 and the selfadjoint spectral problem (5.1) corresponds to

OwG(O, O, 0)w = 0, (5.3)

we can use again the Liapounov-Schmidt method. Let P ' , Q be the orthogonal projections defined by the solution of Equation (5.3)

P'w = (w I Wo)Wo, Qw = w - (WlWo)Wo, (5.4)

f2 (w I wo) = w(x)wo(x) ax.

278 w. DAY ET AL.

We write P w for (P'wl Wo). As in paragraph 3 we write w in the form w = 0-1Wo + O and reduce Equation (5.2) to the system of equations

QG{o-xw 0 + i~(~), 0-2, 03} = 0 ,

PG{0-1wo + 4'(or), o'2, 0-3} = 0, (5.5)

<wol~> = wo(x) O(,,, x) dx-- O,

O" = (0-1, 0"2, 0"3)"

The first Equation (5.5) has a unique solution 4' = 4'(¢r) for small values of ¢r. This solution satisfies the condition 4'(0) = 0. We insert 4,(¢r) into the second Equation (5.5) and obtain the algebraic equation

P{(r} = PG{0"1Wo + 0(or), o'z, 0-3} = 0 (5.6)

for ~. We repeat all the steps in the analysis described in Section 3, expanding 4' and P in Taylor's series. The results of our calculations are in Appendix D. Next, we use Newton's diagram method to obtain from the unperturbed bifurcation Equation (5.6) (0-3 = 0) the scaling

13"1 = ~G1, 0"2 = 6252 •

We can scale Equation (5.6) by setting

0-I = 6(~ ' )OZl , 0-2 = 8 2 ( f f ) O ~ 2 , 0-3 = EkO=3, 6 = E p, p, k > 0 (5.7)

and calculate the limit

/~(#~, ~2, 53) = lim 6 3(e)P(a(E)o~,, a2(~)~2, EkO~3) E~O

where

= l im {L ¢k-3p+ 1/2/~0~3~3~ -[- hOO~l 0~2 --[- ~-&-P+ 1/2,~013~20~3~E q- ~ 0

_ ho~3A _ ~k p + l / 2 h o ~ 3 r f _ E2k-Zpho~B,_

-- ~ '3k -3P l~o~3C~ } q- lim r(~f2, 0~2, 0~3, e, k , p ) , e ; 0

= Oo(x/e)wo dx,

1__ ! r ' w~(o~ + Oo(x/~)) ,ix, n'=x/~,zj o

B" = W2o(034' + Oo(xle)) 2 dx,

(5.8)

(5.9)

BUCKLING OF RANDOMLY IMPERFECT BEAMS

c" = 6 wo(O3q,+ Oo(x/~)) ~ dx,

lfl We note that

03~b= -Ao K(x, y)Oo(y/e) dy,

where K(x, y) is the generalized Green function for the equation

f2 A w = Qf---~ w(x) = K(x, y ) f ( y ) dy

described in Appendix C, and it is given by

K(x, y) = ~ Wk(X)Wk(y) where wk = V ~ sin(k + 1) 7rx. k=~ ~ ( 1 - (k + 1)2) '

Thus, by the ergodic theorem

fl 1 lim B" =/} = 1 E{O~} w 2 dx = 2 E{02}' ,~o 2

lim C ~ = C = _1 E{03 } wo dx. ~to 6

By the central limit theorem (el. (3.19))

lim ¢~ = ~:, lim , / ' = r/, e ~ O e J, 0

where both ~:, 77 are Gaussian random variables. There are three possible results for the limit (5.8)

(i) O < k < ¼ p = k

P(~3~,, O~2, 13~3) = AO{O~, O72 - - A o ~3 - - BO~,O7 2 - - 1~0~3},

1 I ( i i ) k = ~ p = z

P ( O ~ I , O~2, iT3) = ')t0{O73~ at- O'IO'2 ---z~O~O~3 -- CO~33},

where

f+ E{s c2} = R(Z) dz, R(z ) = E{Oo(x + z)Oo(x)},

(iii) ' k > Z P = 1/3k + "~

279

(5.10)

(5.11)

(5.12)

(5.13)

(5.14)

(5.15)

(5.16)

(5.17)

280 w. DAY ET AL,

P(O~I, ~3~2, 073) = /~0(O~3~ + O~10~2 --Ao~3), (5.18)

where ~ is the same as in (5.16).

For the corresponding asymptot ic expansion, for w we obtain the expression (see Appendix D)

w = 8( efiYl W° + ek- 3P+ l/z ~311im ~ O3tb} 63( ") , ; o

-~- 63(e )lEk_p+l/2l~2(~ ~t l im 1 02030} + L - L,~,Lo V e

2k 2 -~[" 1 2 ] I,~,!, o 2 J

+ek+p+t /2( l i m _ l O~330}o.20.3+e._td~llimsOStb~t+. - e - 3k ,~ ~ f 1 _ ]3 le,L o X f e l e l O O JJ

+ O( , ) : 6 = eP. (5.19)

The re fo re to lowest o rder

(i) i f 0 < k < ¼ p = k , then

1 fl } w = ePdlwo + e 3p cf 3 ~ O + e3 ~ )toE{ 03} K(x, y) dy , (5.20)

1 1 (ii) if k = z p = ~, then

1 , 3 hoE{O O}foK(x,y ) W _~ Ell4~lWo qt_ ff314 { ~316O31~lt..~ 1 dy} + e,314{d.3g},

(5.21)

and W is a Gaussian process with zero mean and

f2 E{W2(x)} = A2E{02} K2(x, z) dz, (5.22)

(iii) if k > 1 1 1 p = ~k + g then

: p -3- l 3 W t~p(~lWo ..~ if3 {or1601~j.. [_ 073W } (5.23)

where W is the same as in (5.21).

The asymptotic results can be recapi tula ted as follows. In each case the bifur- cation equat ion as the form

P : 6"~ - ate1 + b = c, (5.24)

(i) a = A-~{~Y2 - /3i f2}, b = / [ - ~ C o =3 , (5.25)

BUCKLING OF RANDOMLY IMPERFECT BEAMS 281

(ii) a = A - 1{~2 - /~ f2} , b = A-1{(9#33 - ~3~}, (5.26)

(iii) a = A-~o~2, b = - A - t o 7 3 ~ . (5.27)

The solution of (5.24) changes its multiplicity when

a 3 b 2 - (5.28)

27 4"

Therefore Equation (5.28) defines the following critical value of ~r2

(i) if 0 < k < 1, then

(~2)cr : {3(A~'2/4) 1/3 + / ~ } ~ , (5 .29)

1 (ii) if k = z, then

(o72)c r : 3(X/4)1/31 ~cf3 _ ~3~:1z/3 + / ~ 2 , (5.30)

(iii) if k > ¼, then

(~f2)cl = 3(e{/4) 1/3 Icf3~l 2'3. (5.31)

We notice that if the amplitude of the imperfections is large enough, no random terms appears in (~2)cr.

6. Discussion of the Results

We saw in Section 5 that the character of the bifurcation changes as the defining equation is subjected to perturbations with varying amplitude ~r3 = ~k~3. Contrary to what happens in problem (2.3), the amplitude of the imperfections must always be kept small, of order ~ , with k strictly greater than zero. Otherwise our analysis is not valid. We discuss here the most interesting case of large amplitude (0 < k < ¼) when no random variable appears in the bifurcation Equation (5.24).

In Figure 2, we present two possible bifurcation diagrams for this case. The first one is the simple pitchfork with focus shifted in proportion to the magnitude of the second moment of the initial perturbation Oo(x/E). The second diagram is a perturbed pitchfork which appears when Oo(x&) has nonvanishing third moment.

The formula for the critical force of the rod (see (5.8), (5.13), (5.29)) is

(A)cr : ho{1 + E2k{3{AC2/4}1/3 +/1}~3}

= ~{1 + e-3k{3{A~'2[4}l/3 + 1E{0~0}}~3}, (6.1)

It contains both second and third moments. It implies that in the presence of random perturbations the critical force of the rod is always strictly greater than the corresponding force of an ideal rod. Within the range of validity of the present model, we conclude that the more irregular the rod is the larger the force applied to it must be to buckle it.

282 w. DAY ET AL

o~

A: Bifurcation diagram when E{03}4: 0 B: Bifurcation diagram when E{0~]}= 0 C: Bifurcation diagram when V~ = 0

All Fig. 2.

A particularly simple picture of bifurcation emerges when E{O 3} = 0. The bifur- cation diagram and the buckled shape of the rod are distinguishable from the unperturbed solution of the problem (see (5.15); (5.20)) except for the higher value of the critical force

()t)cr = h0{1 + 1/2~2k~E{~}) (6.2)

1 which for 0 < k < z may be appreciably greater than its ideal counterpart. The remaining two cases (5.16) and (5.17) complete the general buckling picture

as follows. For small amplitude of the imperfections (k > ¼) the bifurcation diagram is randomly perturbed about the classical pitchfork. The random critical force is always slightly greater than its deterministic counterpart. When k decreases to its

BUCKLING OF RANDOMLY IMPERFECT BEAMS 283 !

intermediate value k = Z, the whole diagram moves up to fit the perturbed pitchfork 1

shown on Figure 2. Firially when k decreases below z the whole bifurcation diagram ceases to depend on particular realization of the random function 00(o), x/e) and it falls into one of the two patterns shown on Figure 2.

Appendix

A. Dimensional form and scaling of the yon K~rmhn equation. The dimensional form of the von Khrm~n equation is (see [14], p. 143)

d---~N=0, N = YS~ dUt -t- l-{dU212"~ dz [ d z 2 [ d z J J '

d2M t_ d IN dU2"~_ dz z -dzz[ Tz J F[z'Ue]=O' M=-yjd2U2 dz 2 '

where U1Ue are the horizontal and vertical displacements of the beam and M and N are the bending moment and normal force, respectively. S and J are the area and the moment of inertia of the cross-section of the beam. Y is the Young modulus

We assume that the beam rests on a nonlinear elastic foundation which response can be described by a function,

F[z, U 2 ] : {KI + Q(z)}U2 n t- K3U~2, Ks = const, K 3 ---- const.

If L is the length of the beam then the scaling

Ut = U1L -1, U 2 = U2L -1, x = z L -1

leads to Equation (2.2)

I D { D u I + I { D u 2 } 2 } = O , (D = d )

ID4u2 - D{DUlDU2 + ½{Du2} 3} + {kl + o'q(x/~.)}Uz + k3u32 = O,

where

J = SLZ[, K1 = Y S L - 2 k l , K3 = Y S L - a k 3 ,

Q(z) = YSL-Zoq(z/eL)

B. The imperfect rod equation. The equation of equilibrium for an imperfect rod of the length L is (see [14],

p. 70) M-Py=O. If s is the arc length of the deformed rod and 0 the slope of its axis then

approximately

284 w. DAY ET AL.

where 0, is the s lope of the u n d e f o r m e d rod . I f we pu t now s/L = x and use the

re la t ion dy/ds = sin 0, then we ob t a in

EJ. 2 dMds - P sin 0 = - L-S {D 0 - D 2 0 ~ } - P sin 0 = 0

H e n c e

{D20 - O20~} + A sin 0 = 0,

where A = pL2/Ej.

C.

and

(o=d)

The nonvan ish ing der iva t ives , up to the th i rd o rde r , of P ( , r ) and ~(~r) a re

fo Ol02P(O) = - A o Dw2o dx,

f2 o~o~p(o) = q(x/~)w~,o dx,

o3p(o) = 3 {f~ Dw~,o dx}2 + 61,3 f j w~,o aX,

fo O~02p(o) = 2Ao D2w2.oO~0202 dx,

010203P(0) = Ao O2w2.00,0302 dx + q(x[~)w2.0OlO2t~2 dx,

OIO~P(O) = q(xle)w2,00~0302 dx,

02~(0) = {0";~0~(x)};

0 , 031~J(O)= {O3~2(X)}" H e r e

A2(0, 0)w = ID4w + AoDZw + klW,

02~01 = x Dw2.0 dx - Dw2.0 dx,

A2(0, 0)0,02qJ2 + Q2{AoDw2.0} = O,

A2(0, 0)0103tP2 + Q2{qw2.0} = O,

A2(O,O)a3~b2 + Q2{6k3w3,o- 3 f~ Dw~,odxD2w2.o}=O

w(O) = w(1) = Dw(O) = Dw(1) = O,

BUCKLING OF RANDOMLY IMPERFECT BEAMS

f2 Q2w = w - W2,o ww2,o dx.

D. The nonvanishing derivatives, up to the third order of P(or) and ~¢r) are

03P(O) = Ao Oo(x/~)Wo dx,

0102P(0) = ho,

fo 02cq3P(0) = )to Oo(x/E)Wo dx,

O~lP(O) = - x o w~ dx,

O~lo~e(o) = - ~o w~{o~4, + Oo(x/~)} dx,

o,o~e(o) = - ~ o w~o{o~ + Oo(x/~)} ~ dx,

o~e(o) = - ~ o Wo{O~4,+ Oo(x/~)} ~ dx,

A O ~ + Aoe{oo(x/~)} = O,

A0~034, + XoQ{Oo(x/e) + o34,t = O,

AO3,P- AoQ{w})} = O,

AO2O3qJ- AoQ{{w2{Oo(x/e) + 03~b}} = O,

aoxo2qt- AoQ{wo{Oo(x/~) + 03qJ} 2} -~ O,

AO203~b + 2AoQ{O203qJ} = O,

Ab3qJ- AoQ{Oo(x/e) + 03~b} 3 = O,

where

A w = DZw + AoW, w(O) = w(1) = O,

Qw = w - Wo WWo dx.

285

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286 w. DAY ET AL.

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