Wu, BAISHENC: Secondary Buckling of an Elastic Strut 74 1 -------

ZAMM . Z. angew. Math. Mech. 75 (1995) 10, 741 -751

WLJ, BAISHENG

Akademie Verlag

Secondary Buckling of an Elastic Strut under Axial Compression

Diese Arbeit beschaytigt sich mit dem Nachbeulverhalten einer undehnbaren, linear elastischeiz, zylindrisch in eiizem Gelenk endenden Stiitze niit rechteckigein Querschnitt unter axialer Kompression. Das Langenverhaltnis der Seiten des Querschnitts sei 1 1 2 ~ . Fur p 4 1 gibt es zwei uizterschiedliche Knicklasten, welchefur p = po = 1 verschnielzen. Unter Verwendung drr Liapunov-Schmidt-Reduktion zeigen wir die sekundare Bifurkation der Stiitze fur jedes p nahe p = po, doch p $; po. Weiier wird die Stabilitat jedes Zustandes durch die Verwendung der reduzierten Potentialenergie analysiert. Es wird gezeigt, daJ der stabile primare Zustand der niedrigsten Knicklast nach der sekundaren Bijiurkation instabil wird, wahrend alle anderen Zustundt instabil sind.

This paper is concerned with the post-buckling behavior of an inextensible, linear elastic, cylindrical hinges-ended strut with rectangular cross-section, under axial compression. Let the ratio of the lengths of the sides of the cross-section be 1 : 2p. For p =k 1 there are two distinct buckling loads which coalesce when p = p o = 1. By employing the Liapunov-Schmidt reduction, we reveal the secondary bifurcation of the strut for each p near p = p o but p =k po . Furthermore, the stability of each stair is analysed by using the reduced potential energy. I t is shown that the primary state from the lowest buckling load becomes unstable after secondary bifurcation, while all the other states are unstable.

MSC (1991): 73H05, 73K05

1. Introduction

Many engineering structures can be modeled as strut-like slender members. This includes, for example, structural components of large structures, tall engineering structures, and piston struts. The understanding of the behavior of the stability of such structures is of primary importance to their proper functioning and usefulness. The identification of the different types of buckling phenomenon allows more accurate and reliable structural and control system designs to be accomplished in the future.

The literature concerning stability problems of struts is voluminous, so only those directly related to this investigation nre cited. In the particular case of a strut that is uniform along its length and in the case of linear constitutive relations, EULER [l] gave a complete analysis of planar equilibria in terms of elliptic functions. In the same special case, a modern bifurcation analysis of the boundary-value problem considered by Euler was given by REISS [2] who conjectured various results on stability that were later proved by COFFMAN [3]. The conclusions were that the primary state from the first buckling load is stable and there is no secondary bifurcation. Here, secondary bifurcation means that, with the change of loading parameter, except a basic state (a trivial equilibrium state corresponding to undeformed configuration), a new state (secondary state) bifurcates from a primary state branching off the basic state. We refer readers to [4] for these ideas. These previous analyses were concerned with planar equilibria, and are restricted to the consideration of stability within the plane of deformation, which may lead to the loss of complex post-buckling behavior of a strut, for example, Secondary bifurcation. Therefore, a full investigation should consider spatial deformation of a strut.

Evidence for secondary bifurcations of a strut has firstly been obtained by KOVARI in [5], by using complex elliptic functions and numerical computations for the strut with the cross-section of kinetic symmetry and with one end held in sleeve and the other end clamped. Using second variations with the constraints of potential energy, MADDOCKS [6] investigated stabilities of planar equilibrium configurations of a nonlinearly elastic strut that is buckled under the action of an axial load. In the cases of three kinds of boundary conditions: 1) ball-and-socket joints at both ends, 2 ) clamps at both ends, and 3) both ends held in sleeves, he found secondary bifurcations on the first primary states. In these studies, however, the secondary bifurcation loads are far from the primary bifurcation loads at the basic state, and are no1 concerned with geometries of the cross-section of the struts.

Early in 1975, BAUER and the other [4] proposed that “multiple eigenvalues lead to secondary bifurcation”. This technique has been coupled with singularities theory [7] in [8] by BUZANO who found secondary bifurcation of a thin non-linear hyperelastic strut with near square cross-section, under axial compression. However, the method in [S] will be ineffective for the linear elastic strut because of its indefinite codimension. For other structures exhibiting secondary bifurcations we refer readers to [9, 10, 113.

In this paper, we consider the problem of secondary bifurcation of an inextensible, linear elastic, cylindrical hinge-ended strut with rectangular cross-section having the ratio of the sides 1 : 2p, under axial compression. Here, the cylindrical hinges mean that a strut is simply supported and clamped respectively in its two principal planes of inertia at both ends, see Fig. 1. From the variation of the total potential energy we derive the equilibrium equations with two Parameters A and p, where 1 represents the axial load. With this model we investigate the interaction between the buckling modes which represent respectively the bucklings in the two principal planes of inertia of the strut. For p $; 1 there are

--h - 142 ZAMM . Z. angew. Math. Mech. 75 (1995) 10

/

Fig. 1. A cylindrical hinges-end( strut

two distinct buckling loads which coalesce when p = po = 1. By employing the Liapunov-Schmidt reduction, we fm that when the values of the parameter p are slightly different from the critical value p = po the multiple primary bifurcatio point splits into two simple primary bifurcation points and two secondary bifurcation points. We also give the asymptot expansions of the primary and secondary states. Moreover, the reduced potential energy of the strut is derived and use to study the stability of post-buckling states. We will show that the stable primary state from the lowest buckling loa becomes unstable after secondary bifurcation, and the primary state bifurcating from the second primary buckling loa and the secondary states are unstable.

2. Equilibrium equations of the strut

We model the strut as a continuum of plane sections with an inextensible curve C passing through the centroid of eac section. Without loss of generality we may assume that the length of the strut is 1. We parametrize C by arc-lengt s E J = [0, 11 and denote differentiation off by f’. Let us set a right-handed orthonormal reference frame [ e , , e,, e,] fixe in space. We assume that the strut is naturally straight with the end s = 0 at the origin configuration and its axis alon the direction e3. The deformed configuration is described by three vector functions r, d,, d , of the variable s E J . As usui r(s) is the position vector jointing the origin t = 0 and the centroid t = s, and d,(s), d2(s ) are two principal axial un vectors fixed flat to the cross-section through the centroids. Define

d3(4 = d l ( 4 x d2(4

and assume that the strut can suffer neither extension nor shear; that is that (Kirchhoff hypothesis)

r’(s) = d 3 ( s ) .

Let u(s) be the unique vector satisfying

dJ = U ( S ) x d j ( s ) , j = 1,2, 3 .

Set 3 3

r(s) = 2 xi($ ei, u(s) = ui(s) di(s). i = 1 i = 1

In our theory u1 and u2 measure flexure about the axes d, and d, while u3 measures twist [12,§ 1.11. From (2.1) we have

x;(s) = [I - (x;2(s) + X;2(S))]l’Z . (2..

Let 0, q, 4 be the Euler angles describing the rotation of { d , (s)} with respect to {ei(s)} (see [13, Article 253]), and let

then it is easy to compute uj in terms of xi, x i and y, to obtain (Cf. [12, 5 1.13):

cos y

0 [ = [ sin y

-sin y cos y

0 (2;

Wu, BAISHENG: Secondary Buckling of an Elastic Strut 743

we remark that the uj are well defined on a neighborhood of the straight configuration (x,(s), x2(s), x3(s)) = (O,O, s). Finally xl, x2 and y are the displacements chosen by us. We note that y’ measures the amount of twist about e3 md introduction of y removes the polar singularity of the Euler angles. We refer readers to [12, Chap. 11 for the

- r,

details. We hold fixed the end s = 0 and apply a terminal compressive load force ,I along the axis e3 of the strut at the end

1. Thus the total energy U of the strut is the deformation energy minus the work done by 1 in moving along its line of action. Assume that the material of the strut is linearly elastic, isotropic, uniform along its length, then we have

where 1 2

f = - [EJ , (p) cos2 y + EJ,(p) sin’ y1

1 2

+ - [ E J , ( p ) sin2 y + E J , ( ~ ) cos2 y ]

+ [EJ,(p) - E J , ( p ) ] cos y sin y 1 + x;

(2.4 b)

and E J , and E J , are the bending rigidities respectively in the two principal planes of inertia of the strut, C ( p ) is the torsional rigidity:

(2 .4~)

(2.4d)

where a is the length of the short side of the cross-section and vo is Poisson’s ratio. We consider a strut with cylindrical hinges at its two ends. Specifically, we suppose that the strut is simply

supported in the principal plane T, (x , = 0) of inertia with the bending rigidity EJ,, and clamped in the principal plane T2(x2 = 0) of inertia with the bending rigidity E J , , at both ends, see Fig. 2. Consequently, the following set of displacement boundary conditions holds :

x,(O) = x,(l) = x2(0) = x2(1) = 0,

Xi(0) = x;(l) = 0 ,

(2.5a)

(2.5b)

y(0) = y(1) = 0 . (2.5~)

Conditions (2.5b) mean that that ends s = 0 and s = 1 can rotate only in the plane Tl. In particular it is easy to prove that the boundary conditions (2.5a)- (2.5b) correspond to (see Fig. 3)

Fig. 2. Two principal planes of inertia of the strut of the strut

Fig. 3. Two coordinate systems at the left end

744 ZAMM . Z. angew. Math. Mech. 75 (19951 10

By the principle of the stationary potential energy, from (2.4) we get the equilibrium equations of the strut:

and the mechanical boundary conditions:

X’;(O) = Xi(1) = 0, ( 2 3 where aflax; etc. are listed in the Appendix. The formulae (2.5 - 6) are the governing equations of equilibrium of the str( under axial compression.

Define the following Banach spaces:

x = x , x x , x x , , Y = Y , x Y , x Y , ,

x , = {x:x c4[0, I), x(o) = x(i) = x’(o) = x’(i) = 01 ; x, = iX : E c4[0, I), x(o) = x(i) = x”(o) = x”(i) = oj ; x3 = {x:x E C2[0, 11, x(0) = x(l)} ;

Y, = {Y : y E C[O, 11) 9

where

and X i ( i = 1,2, 3 ) and Yl are equipped with the maximum norm, for example

lxll = sup max Ixjt)(t)l, x1 E X I , Osis4 O s s s l

and X and Y with the product-norm 1x1, ( Y ( respectively. Let

X o = x : x ~ X , min (1 - xi2 - xi2) > o} t { osss 1

and

R ; = {(A, p) : (A, p) E R2, 2 2 0, p > 0 } ,

Then (2.5 - 6 ) may be rewritten as:

F(x,A,p) = 0 , F : X o x R ; 3 Y , (2.7

where x = (xl, x2, y)‘. In the following discussion we will regard A as a bifurcation parameter and p as an auxiliary one

3. Secondary bifurcation of the strut

Obviously, for any (A, p) E R;, (x,, x,, 7)’ = (0, 0,O)’ (0 5 s 1) is always a solution of (2.7), and corresponds to the bad solution of equilibrium of the strut. The linearized equations followed from (2.7) about the basic solution is F,(O, A, p) x = c which are

(3.18

(3.lb

EJ,(p)xj4’ + nx; = 0 ,

E J , ( p ) x:”’ + nx; = 0 ,

X,(O) = X,(l) = Xi(0) = x;(l) = 0;

x2(0) = x2(1) = xZ(0) = XZ(1) = 0;

-C(p) y” = 0 , y(0) = y(1) = 0 . (3.1~

From the equations above, it is well known that for p is near po = 1, the two lowest buckling loads are

/Zf,(p) = 47C2EJ,-,p, A.$(p) = 4X2EJ,P3 ,

and the corresponding two buckling modes are respectively

G1 = (h, 0,OY and G2 = (0, 4,, 0)t , where

(3.2

(3.38

4, = (8’” (1 - cos 2ns) and 4, = 2’12 sin as, 0 6 s 6 1 . (3.3b

Wu, BAISHENG: Secondary Buckling of an Elastic Strut 745 /

1

Here, we have made the normalization according to 4; ds = 1 (i = 1,2). 0

Notice that when p is near 1 but p $. 1, the lowest buckling load

&A4 = min { X A P ) ? A,’,(P,)

is a simple buckling load; while for p = p o = 1,

& = I c r ( l ) = 4n2EJ0

is a double buckling load and the corresponding two buckling modes are given in (3.3). Define Si E L ( Y ) (i = 1,2) such that

where S , ( i = 1 , 2) is the reflection through yi axis respectively. Then we have

si * I , s; = I(i = 1 ,2 ) ,

where I is the identity operator on Y . Set

X,(S,) = {xeX:S,x = + x ) , Y,(S,) = { y ~ Y : s , y = + y } , i = 1 , 2 .

Then

x = x, (S,) 0 X-(S,), Y = Y+ (S,) 0 Y- (S , ) , I = 1 , 2 .

(3.4)

(3.5)

(3.6)

From (2.4b), it is easy to prove that for any (x, I , p) E X 0 x R;, f ( S i x , I , p) = f ( x , 1, v), i = 1,2, that is the potential function (2.4a, b) is equivalent under the two reflections in (3.5). By using (2.6), we obtain

F(S ix , 1, p) = SiF(x, A, p), V(x, A, p) E X, x R; , i = 1, 2 . (3.7)

Furthermore, the following relations hold:

S IQl = G 1 ,

S2@] = --a1 ,

S1@, = -@,,

S2@, = @, .

Let q’], q; be continuous linear functionals on Y defined by ( y = (yl, y 2 , y J )

I 1

( Y , ~ i ) = J 41.~1 ds := (Y, @I) ( Y , &> = J 4 2 ~ 2 ds := ( y , @ J , y E Y . 0 0

Noticing that the equation FJO, I , , po) x = 0 is self-adjoint (see (3.1)), we get

W , ( O , 10, Po) ) = i u = (Yl, Y,, Y J E y : (Y, @ l ) = 0, (Y , @,I = 0) t (3.8)

where I? represents the range of the operator F,(O, lo, p o ) (we note that F,(O, Ao, p o ) := F: et al.). Then F: is a Fredholm operator with index 0 and a two-dimensional null space.

Now the Liapunov-Schmidt reduction procedure [7, 141 can be applied to obtain the solution of (2.7) near the double buckling load. Let Q be a complementary space of X to N = ker F: and P : Y -+ R”(F:) a projection operator with kernel N . The implict function theorem guarantees the existence of a unique mapping W : N , x D, + Q such that

PQ@l + B @ 2 + W(a@, + P@,,A, PI, 1, p) = 0 I (3.9)

where N 6 = {(a, /3) E R 2 : la1 < 6,181 < S } and D, = { ( A , p) E R 2 : [ A - Iol < S, p - pol < S} for some positive number 6. Let

(3.10) G(a, /3, 1, PI = (I - 4 F(a@, + B @ 2 + W ( a @ , + D@,,A, p), 1, p) ‘

We may write (3.10) in terms of its components again as

(3.1 1)

746 ZAMM . Z. angew. Math. Mech. 75 (1995) 10

Then G(a, P, A, p) = 0 parametrizes the solutions to (2.7). If we define

V ( a , P I A, p) = U(M@, + /!?a, + W(a@, + P @ 2 , A , A, 2, p) 7

then (cf. [15]) (3.

(3.

Let Z : N x N -+ Q be a bilinear mapping defined by Z(u, u ) = W,,(O, Lo, po ) uu, or equivalently, differentiating (:

(3.

we have

F:Z(u, U) + P F : p = 0 .

Under the “double” symmetries (3.7), the bifurcation equations (3.11) are of the following form [16]:

Gl = ala(A - A,) + b,a(p - pol + pa3 + rap’ + ah,(a, p, A, p), G, = azP(A - A,) + bZP(P - Po) + qa2P + sp3 + M a , PI A, p) ,

(3.1

(3.1:

where the coefficients

ai = (F:,@, @ J , bi = (F:,,Qi, ai), i = 1,2 ; (3.1

P = (Co@? + Qo(@i, z(@i, @i)L @A; (3.1:

(3.1

(3.1

4 = (3C0@% + 2Q0(@,, z(@17 @ J ) + Qo(@,, Z(@,, @A G2);

s = (Co@ + Qo(@z, z(@z* @z)), @2) ;

r = ( ~ C O @ I @ ~ + ~ Q o ( @ z , Z(@19 @A) + Q o ( @ ~ , Z(@Z, @z)X @I) ;

(3.1:

and Co = F:,,/6, Qo = F%/2, h , and h, represent higher order terms in (a, /?, A - A,, p - p0), and are even in botl and for all (A, p) E D,.

Let us compute the coefficients a,, a2, b,, and b, firstly. By utilizing the commutativity of the derivative operation, we 1

F,O,@l = (4:, 0,O)’ , F,O,@, = (EJo4?’, 0,O)’ ,

F:,@, = (0, 4;, 0)‘ , F,O,,@, = (0, 12EJ04$4), 0)‘ . Then from (3.15~) we get

1 1

0 0

1 1

0 0

a, = S 4,4: ds = - & n 2 3 , 02 = S dz4;ds = - n 2 ,

b1 = S E J 0 4 , 4 $ ~ ’ d s = yEEJon4,

It remains to calculate the coefficients p , q, r, s. Note that

b, = 12EJ04,4$4)ds = 12EJ0n4.

F,”,@: = 0 , i = 1,2 ;

F:x@,@, = (0,0,40(s))‘,

where

Using the formula above and (3.14) we obtain

Z(Qi,@J = 0 , i = 1 , 2 ;

Z(@l, @,I = (090, $1) 3

where

As

Wu, BAKHENG: Secondary Buckling of an Elastic Strut 147 /

We obtain the ith component of the vector above:

- d [ - ; C(1) &+;I. ds

Then by using (3.15d-g) one has

16EJon6 ds = ~ p = s = / ~ , j , ds = EJ,n6, J 9 ’

0 0

We now analyse the solutions of the bifurcation equations (3.15a, b). As a,a , = 47?/3 4 0, one may determine the solution 1 = X(a,p) of (3.15a) with /3 = 0, and the solution 1 = 1(/3,p) of (3.15b) with ci = 0 (cf. [16]). The solvabilities of X and 1 are guaranteed by adapting the result of CRANDALL and RABINOWITZ [17] to the restriction: F : X + ( S i ) x R i -+ Y+ (Si) for i = 1,2; and they correspond respectively to (2.7)’s primary states Ts; and fs;. Here, Tf; is symmetric about Si (cf. [16]) and represents the buckling in the principal plane T3-i of inertia for i = 1,2; and for fixed p

(3.16 a)

(3.1 6 b)

where a,, and Po are some positive numbers. In addition, as c(1) < 4 (see (2.4d)), it is easy to check that

a l b , - a,bl = -32EJ0n6/3 < 0 ,

a,q - “ z p = 16EJon8[40(1 + v O ) - 5~(1)]/[9C(l)] > 0 ,

a,s - a2r = -4EJ0n8[40(1 + v o ) - 5c(1)]/[3c(l)] < 0 ,

PS - qr = 1 6 E Z J ~ ~ 1 2 [ 4 0 ( 1 + YO) - 5c(l)] [ 7 ~ ( 1 ) - 40(1 + ~ 0 ) ] / [ 3 ~ ( l ) ] ~ < 0 .

By introducing the following rescaling functions:

1 - 1, = v ’ A , p - po = V ’ T , CI = v t , /3 = vv (3.17) and using the results of [16] we see that there are exactly two secondary bifurcation points for each p =k po. These two secondary branches of solution of (2.7) intersect

Ts; if p > 1 i rs; if p < l

respectively and are written as T$;(p > 1) and T%(p < 1). Moreover, for fixed and small v 4 0, set p = 1 + v 2 , then

T$;(p > 1) = {x = vt(v, q) @ 1 + V F p , + W(Vt-(V, q ) @ 1 + VV@,, 10 + v2A(v, q), po + v2) , 1 = 1,

+ v 2 N J , q ) : Ivl < q o ) Y (3.18)

where t(v, q)and A(v, q)are the solution ofequations (3.19)near (t, A , q, v ) = (to, A,, 0,O) where to = k {6c(1)/[40(1 + \lo)

- k(l)]}1’2/n, A0 = 4EJon2[40(1 + vo) - 3C(1)]/[40(1 + Yo) - k( l ) ] :

a l A + b , + pt2 + rqz + el({, A , q, V) = o + &(t, A , q, V ) = 0 ’ i a,A + b , + q t 2 +

Where ei(t, A , -q , V) = ei(t, A, q, v) , ei(t, A, q, 0) = 0, i = i ,2 .

(3.19)

748 ZAMM . Z. angew. Math. Mech. 75 (1995) 10

In the following we will give the asymptotic expansions of Tf; and T:;. Firstly, by using the Taylor expansion Q

get (see (3.16a))

r;; = {X = a@, + 0 ( a 3 ) , 1 = 4Zmop + $ ~ ~ ~ n ~ a ~ + o ( ( ~ - a2, ~ t ~ ) } . (3.N; As for secondary branches Ts;, since A(v, yl) = A(v, -yl) and ((v, yl) = ((v, -yl), from (3.18) we have

The derivatives appearing in the right hand of (3.21) can be derived from (3.19), then (3.21) may be written in terms cif components

a = n O v 2 + {EJon4[7c(l) - 4o(i + vo)l/c(i)) v2y12 + 0 ( ~ 3 , v2V4) . (3.2214

Similarly we can get the asymptotic expansions of the primary and secondary states T:; and Ys;, which are omitted.

T;; is subcritical. From (3.20) and (3.22) we know that Tf; is supercritical while Ts; is subcritical. Similarly Tf; is supercritical whilig

4. Stability of each state

In this section, we determine the stability of each equilibrium state of the strut near the double buckling load. Using thg expansions of the eigenfunctions of (3.1) with p = 1, one may prove that there is some positive constant such that th8 following inequality on the second variation of potential energy (2.4a, b) holds for all w E Q:

1 V,,(O, 0, 0, Ao, p o ) w 2 L co 1 (w;’ + w;’ + wi2) d s ,

0

where w = (w,, w2, w3) E Q. Then the solution W(a@, + pQ2, 1, p ) of the equation (3.9) makes the potential energy (2.4a, b) take the local minimal value with respect to w [15]:

and we obtain the reduced energy function (3.12). One can get two bifurcation equations (3.15a, b) by giving V in (3.121, an extremal value with respect to a and p (see (3.13)).

In order to investigate the stability of each state, we need to examine the extremal value of I/, which requires to‘ compute the eigenvalues of the 2 x 2-stability matrix S corresponding to all points on the considered equilibrium state. A state (parametrized by M , p, I , and p) satisfying (2.7) is stable, if all the eigenvalues of S are positive, unstable if at lee] one of the eigenvalues of S is negative.

Taking the first variations of the reduced energy function I/, and differentiating a second time with respect to a aqz p yields the following stability matrix:

1 S = [ a l j + b,fi + 3pa2 + rp’ + h , , 2rap + h,, 2qaB + h12 u2A + b2ji + qa2 + 3sp2 + h, , ’ (4.4

where j = 1 - ,lo, fi = p - po, h,, h,, and h,, are higher order terms in (a, p, 1 - lo, p - po). Under the rescaling transformation of (3.17), S becomes

a1A + b,r + 3pt2 + ry12 + O(v) 2qryl + O(v)

2r1;yl + W V )

U 2 / l + b2T + qr2 + 3Sy12 + [ s = vz

in addition, one can get the reduced primary and secondary states (in parameter space)

rsi 1P : A = - (p t2 + blT) /a , + o ( ~ ) , rl = 0 ;

r f p = - (sq2 + b 2 ~ ) / ~ 2 + O(V), ( = 0 ;

a,A + b,z + p t 2 + rq2 + O(v) = 0 a2A + b 2 ~ + q(’ + sq2 + O(V) = 0

r;;(r;;):

Wu. BAISAENG: Secondary Buckling of an Elastic Strut 749

inserting the reduced rf;, Yf;, and Ys;(Ys;) into (4.3) yields S for all points on these branches respectively as

0 1

a2 1 a2 0 2sq2 + O(v)

I. [ 2qtq + O(v) 2sq2 + O(v) 2 p r 2 + O(V) 2rrq + O(v) S(rs;) (I-$): v2

(4.8)

(4.9)

BY using the coefficients in (3.15) we can determine the stability of each state. Finally, the bifurcation and stability diagrams of the strut based on @ 3 and 4 are summarized in Figure 4, in which the solid line and dotted line represent stable and unstable states respectively.

O k A-A0

> 1

= 1

< I

\ Fig. 4. The bifurcation and stability diagrams of the strut

We remark that at the double eigenvalue or after the secondary bifurcation no stable state exists. The local bifurcation analysis above does not answer where the strut is going, i.e. which stable configuration finally will be attained.

5. Conclusions

With three generalized displacement coordinates, we have derived the equilibrium equations of an inextensible, linearly elastic, cylindrical hinge-ended strut with rectangular cross-section, under axial compression. Then the Liapunov-Schmidt reduction has been used to investigate the bifurcation behavior of the strut. Our results take into account all the geometric non-linearities in the differential equations of the system. By varying the aspect ratios of the cross-section, a coincident Primary bifurcation point is split, giving rise to secondary bifurcation points for each p < po, p > p o occurring on the Primary state from the lowest buckling load.

By examining the extremal values of the reduced energy function, we have found that the stable primary state from the lowest buckling load (corresponds to bending in one of two principal planes of inertia of the strut), after secondary bifurcation, becomes unstable; while the corresponding secondary states are also unstable. This means that secondary buckling occurs for the strut, which is completely different from the planar post-buckling behavior of the strut. Moreover, the secondary buckling load can arbitrarily approach the lowest buckling load as p approaches po.

For other struts with cross-sections having two orthogonal principal axes of inertia, for example, H-type ones, both ends held in cylindrical hinges and nearly simultaneous buckling loads in the two principal planes of inertia, we may Predict similar post-buckling behavior. The importance of this study is that structural optimization cannot be over- emphasized as pointed out in [18, 191.

-- 750 ZAMM . Z. angew. Math. Mech. 75 (1995) 10

Acknowledgement

This work has been supported financially by the National Science Foundation of China. The author would like. to thank Pro1 T. KUPPER and the Mathematisches Institut der Universitat zu Koln for their hospitality. This paper was finished whlle the author v Koln, when he was supported by Volkswagen-Stiftung, and he would like to thank it for its support. Finally, the comments of the re1 were helpful to the revision of this paper.

References

I EULER, L.: Methodus inveniendi lineas curvas maximal minimivi proprietate gdudentes. (1744). Opera Omnia I, V O ~ . 24, F

2 REISS, E. L.: Column buckling - An elementary example of bifurcation. In: KELLER, J. B. and ANTMAN, S. S. (eds.): Bifurc

3 COWMAN, C. V. : The nonhomogeneous classical elastica. Technical Report, Department of Mathematics, Carnegie-Mellon Unwe

4 BAUER, L.; KELLER, H. B.; REISS, E. L.: Multiple eigenvalues lead to secondary bifurcation. SIAM Rev. 17 (1975), 101 - 122. 5 KOVARI, K. : Raumliche Verzweigungsprobleme des diinnen elastischen Stabes mit endlichen Verfomungen. 1ng.-Arch. 37 ( I '

6 MADDOCKS, J. H.: Stability of nonlinear elastic rod. Arch. Rational Mech. Anal. 85 (1984), 31 1 - 354. 7 GOLUBITSKY, M.; SCHAEFFER, D.: Singularities and groups in bifurcation theory. Vol. 1 . Spring-Verlag 1985. 8 BUZANO, E.: Secondary bifurcation of a thin rod under axial compression. SIAM J. Math. Anal. 17 (1986), 312-321. 9 MATKOWSKY, B. J.; PUTNICK, L. J.; REISS, E. L.: Secondary states of rectangular plates. SIAM J. Appl. Math. 38 (1980), 38-

I0 OYESANYA, M. 0.: Secondary bifurcation and imperfection sensitivitiy of cylindrical shells of finite length. Int. J . Non-linear M

1 I WU, BAISHENG: Secondary buckling of a strut on elastic foundation under axial compression. Acta Mechanica Sinica 25 (1993), 443 - 12 BUZANO, E.; GEYMONAT, G.; POSTON, T.: Post-buckling behavior of nonlinearly hyperelastic thin rod with cross-section invai

13 LOVE, A. E.: A treatise on the mathematical theory of elasticity. Fourth edition. Cambridge Univ. Press, Cambridge 1927 (D

14 ZEIDLER, E. : Nonlinear functional analysis and its application. I : Fixed-point theorems. Springer-Verlag 1986. 15 POTIER-FERRY, M.: Bifurcatipn et stabilite pour des systtmes derivant d'un potential. J. Mecanique 17 (1978), 576-608. 16 SHEARER, M.: Secondary bifurcation near a double eigenvalue. SIAM J. Math. Anal. 11 (1980), 365-389. 17 CRANDALL, M. G.; RABINOWITZ, R. H.: Bifurcation from simple eigenvalue. J. Funct. Anal. 8 (1971), 321 -340. 18 THOMPSON, J. M. T.; SUPPLE, W. J.: Erosion of optimum designs by compound branching phenomena. J. Mech. Phys Solid

19 BUDIANSKY, B.; HUTCHINSON, J. W.: Buckling progress and challenge, trends in solid mechanics. In: BESSELING, J P.; VAN

Zurich 1960, pp. 231 -287.

theory and nonlinear eigenvalue problem. Benjamin, New York 1969, pp. 1 - 16.

1976.

393 -416.

25 (1990), 405-415.

under the dihedral group D,. Arch Rational Mech. Anal. 89 (1985), 307-388.

reprint, 1944).

(1973), 135-144.

HEIJDEN, A. M. A. (eds.): Proc. Symp. dedicated to the 65th Birthday of W. T. Koiter 1979, pp. 93- 116.

Received January 31, 1994, revised and accepted version August 4, 1994

Address: Dr. WU, BAISHENG, Mathematics Department, Jilin University, Changchun 130023, P. R. China

Appendix

Set j = 3 - i for i = 1,2, then

1 (

x'.x'I ) [ x;x; ]

{( , x'.x'' ) [ 1 +

x;x; - x'xq( -x; )

Xi2

(1 + x;) x; [ E J , ( p ) cos' y + EJ, (p ) sin' y ]

+ [EJi(p) sin2 y + E J , ( ~ ) cosz yl x; - 13 1 + x;

+ ( - l Y - ' [ ~ ~ j ( p ) - E J ~ ( ~ ) ] cos y sin y x'! -

a f ax:, - =

( 1 + x;)x;

(1 + x;) x; 1 + x;

1 + x ; 1 + x ; '

( ::;> A , + [ E J , ( p ) sin2 y + EJ, (p ) cosz yl x i - 13 a f - = [ E J , ( p ) cos2 y + EJ, (p ) sin' y ] x: - - ax; ( "li;i;> Bi

+ (- 1Y-I [ E J , ( ~ ) - E J , ( ~ ) ] cos y sin y

1 + x;

Wu, BAISHENC: Secondary Buckling of an Elastic Strut 75 1 -.--

where

- (2x:x; + xjx;, x;(l + x;) + x;(x;’x:’ f x;xjx;) (2 + l/X>)

x;q1 + xi)’ -

B, = ”( - -) axi

- x;,w;x;(l + x i ) + x;(xjx;x;‘ + xjzx;, (2 + l/x;) x;’(l + xi)’

-

x;x;(l + x;) + x;(x;x; - x:’xJ)

x;q1 + xi)’ - -