Approximation of minimal maximal deflection of nonlinear beams

Int. J. Engng Sci. Vol. 32, No. 7, pp. 1117-1123,1994 Copyright 0 1994Elsevier Science Ltd Printed in Great Britain. All rights reserved 00207225/94 S7.M)+ 0.00

Pergamon

APPROXIMATION OF MINIMAL MAXIMAL DEFLECTION OF NONLINEAR BEAMS G. T. MCALLISTER Department of Mathematics, Lehigh University, Christmas Saucon Hall, 14 E. Packer Avenue, Bethlehem, PA 18015, U.S.A. (Communicated

by D. G. B. EDELEN)

Abstrsct-A geometrically nonlinear beam of fixed volume is subjected to both bending and stretching. A method is presented for approximating the distribution of mass of the beam so as to get the smallest maximum displacement for a given load. The method allowsfor easy error estimates.

1. INTRODUCTION A geometrically nonlinear beam subjected displacement W(X) satisfying the equation

to both bending and stretching

(/P(x)w”(x))”

has a transverse

- /..&zW(X)= t.(x)

(1)

with x E I = (0,6), p is a constant over I which is equal zero or one, and r-C,,[w,h]:={2~~-‘(t)dr~‘jlw’2(l)dt

(2)

The prime ’ will always denote differentiation with respect to x and p is used for ease of presentation. For given positive constants 6, and & the set of admissible coefficients or shape functions or mass distributions is W = {h(x) E C(Z): 6, S h(x) zs &, and U(h; I) = y} where C(Z) is the set of continuous functions on Z, U(h; K):=

1 h(x)

dx

(3)

K

with K c Z, and y E (b&, &)

is given. If z E W, let

Z*(z) = {x:x E Z and z(x) E (S,, 6,))

and

Z,(z) = {x:x E Z and h(x) = Sk}

with k = 1 or 3. Knowing the value of I- E W over Z2(t) uniquely determines z over Z and a solution of (1) corresponding to that z will sometimes be written as wz. For any v E C(Z) we set JIvlI, = maxOv(t)l:t E [O, b]). The following problem is studied: find an element ho E W such that ~1~~~11~ S lI%L

(4)

for all h E W; i.e. find a mass distribution ho(x) E W which gives the smallest maximum deflection to the beam. The function k(x) is the cross-sectional area of the beam and h”(x) is its moment of inertia. The following assumptions are made as they simplify our computations: the transverse load r(t) applied to the beam is non-positive over Z, the modulus of elasticity is one, and the beam is 1117

G. T. MCALLISTER

1118

simply supported; i.e. w(O) = w(b) = w”(0) = w”(b) = 0. Simply supported write (1) in the form

data allows us to

h”(x)w”(x) - /JZW(X) = Q(x)

(I’)

where Q”(x) = T(X) and Q(0) = Q(b) = 0. W e view (1) and (1’) as the same equation. The maximum principle in [l; p. 61 shows that w(x) 5 0 over I. The effect of stretching on the equation of a beam may be found in [2; p. 1761 and the equations (1) and (2) are found in [3; p. 2621. The study in [3] seeks a minimum of the integral of the pth power of the displacement and uses multipliers to obtain an optimality condition [3; p. 2641. Our methods are different and error estimates are obtained here. In practice the number CY= 1,2, or 3. Let p = (Y+ 1. Let IS] be the measure of the set S, and let (Y(Z) (or C;(Z)) the set of 12times continuously differentiable functions over the set I (and vanishing at the end points). The set Hi(Z) is the completion of C:(Z) with llwll22= lIwl12+ IIw’l12 + llw”l12 where ]jw 11’is the integral of w’(x) over I. In Section 2 we establish sufficient conditions that (1) has a solution for each h E W when Z_L = 1. In Section 3 we find the function ho E W which solves the problem in (4) when Z_L is zero. This case was also studied in [5] under the same conditions except for the constraint on the size of h(x). In Section 4 we do the general case of p = 1. We write the general solution of (1’) as a series in r and integrals of h-‘(t). We then give a method of finding the mass distribution which minimizes the maximum displacement up to the nth order in r.

2. EXISTENCE

OF A SOLUTION

TO (1) WHEN /.L= 1

Let h(x) E W be given. For any number r 2 0 the functional P[z] = 2-l Lb {h”(x)f2(x)

+ d*(x)

- 2r(x)z(x)} dw

(5)

has a unique minimizer w, E Ad with Ad = {w(x): w E Hi(Z) and w(0) = w(b) = w”(0) = w”(b) = O}.

(6)

We will obtain bounds on the norms of the functions w,. By Poincare’s inequality, for any w E A d, ~~w~~~~~*Tc-*IJw’l12.Moreover, [w’(x) - w’(O)15x’/* IIw”]I; i.e. w’ has finite values at the end-points of I. Using (w’w)’ = ww” + w” and Cauchy’s inequality yields lIw’(l*I b2z-* (lw”\12.Thus 1)w I(* 5 b4nw4 IIw”I(~and any minimizer of (5) must satisfy I~w’:I)~~SFm llrll IIw,II or Therefore,

IIw’:l12~b4a-4i3;2a

llrj12.

for any r 2 0 and for any h E Ad, IJw,(x)(I,S

If r and Y are nonnegative the inequality

b1’2 Ilw:ll 5 b3’2z-’ Ilw’:ll5 b7’*ne36;”

Ilrll.

numbers, then the associated minimizers w, and w, of (5) satisfy l](w, - w,)“l( 5 b4 llrll a-3S;a Iz - YJ.

Thus w, and its derivatives with respect to x are continuous functions of r.

Minimal maximal deflection of nonlinear beams

1119

Since Co(r) := (1will’ + 2U(h-‘; I) and b* 5 U(h-‘; I), we obtain, for each fixed h E W the estimates

and I&(r) - C,(Y)1 5 yb’ llr112w-7s;2p )r - VI. Thus C,(r) is a Lipschitz map from

{r : r 2

0) into the bounded closed interval [0, M] where

M = $I jlrll +2x367

(7)

and we can find at least one element r for which C,(r) = r. Since, for a given r, the first variation of (5) has a unique solution, we have our first result. THEOREM 1.

For each h E W there is a unique W, E Ad which minimizes (5) and has the number r determined by z = Ilw:ll’ f 2U(h-',

3. THE

CASE

I).

p = 0

If ZJ = 0, then (1) becomes

with Q(x) 2 0. This has the solution

w&)

=

I

G(x, t)Q(t)h-“(f)

dt,

Q(t)

I

and G(x, t) = t(x - b)b-’

=j-GO,h)rOd

dt,

I

if t IX or x(t - b)b-’

if t 2x. Over Z, wh(x) 5 0 as Q(f) 10. Thus

I%(x)l= -%l(x). For each x E Z and h E W let T,[h] := j- IG(x, t)lQ(Qh-“(t)

dt.

(8)

I

At any x E I the inf{T,[h]:h E W} is achieved if and only if (Cauchy-Schwarz h(t) = h,(t):=

max{min{[lG(x, t)l Q
inequality)

%], 6,)

(9)

with c”~(x):= (y - 83 IZ,(h,)l - 61IWxN-’

j-h,, [IW, t)l Q
making U(h,; I) = y; i.e. h, E W for each x E I. The subscript x in (9) is used as a parameter. Let

F,(x):= T,[hxl = w,,,&> for each x E I. The absolute minimum of T,[h] over W is F;,(x). Note that w(~,J~) need not have

1120

G. T. MCALLISTER

a critical point at x. As (G(x, t)] Q(f) is small for x close to zero or to b it is clear that c(x) goes to zero as x approaches zero or 6. Also, for x E I,

In the next result ’ denoted differentiation

with respect to the parameter x.

THEOREM 2. The

following are true: (a) For each x E I the end points of the intervals Z,(h,), Z,(h,), and I,(&), as determined by h,(t), are continuously differentiable functions of x E I. Moreover, for each x E Z and for each hew

(b) For each x E I, U’(h,, I) = U(h:, Z,(h,)) = {[(b -

x)c-‘(x)]“~} j-,,

[?I’” dt

+ {[xc-‘(x)]“P}’ 1 [@ - b)8@)1”” dt = 0 k!K where ZzA(or ZzL) is that portion of Z,(h,) to the right (or left) of x. Consequently, [xc-‘(x)] + [(b - x)c~‘(x)]’ 9 0 and this implies that c(x)-’

1 c’(x) 2 -c(x)(b

- x))’

over I. Here h:(t) denotes differentiation with respect to the parameter x. (c) Fk(x,) = 0 if and only if w&(x”) = 0 with h = h,,,. (d) F,(x) 5 0 over [0, b] and is continuously differentiable over Z with F,(O) = F,(b) = 0. Therefore F,(x) has a maximum over I. If x0 E Z and &(x0) = max{FO(x) :x E I}, then %h,,,(xd

=

mini

11wh 11m : h

E

WI.

If x, E Z is such that Fb(x,) = 0, then &(x1) = FO(xO)and, for all x E [x0, x,], F;,(x) = F;,(x,J. PROOF.

(a) and (b) follow easily. For (c) note that

I,

F;(x) = lG(x, t>lxQ(f>h;“(t) dt - a

1

h:(t) dt

W,)

and the latter integral is zero by (b); the term IG(x, t)lX is the partial derivative of (G(x, t)l with respect to x. To obtain (d) observe that F&J = 0 and, by (c), F,(%) =

b%h,,,(-%)l

=

b%h,,,lir

4

b”&d

s

/I wOh 11%

for all h E W; the first equality is a consequence of the definition of F,(x), the second follows from convexity, the first inequality follows from the Cauchy-Schwarz inequality, and the last inequality is the definition of a maximum. In particular, the last inequality is true when h(t) = h,(t) for any x E I; i.e. lW”h /hJl s IlW”h,ll=.

Minimal maximal deflection of nonlinear beams

1121

Thus

for all h E W. The first part of (d) is proved. To see the last part note that &(x1) = lwPI,,(Xl)l = IlW”h,,II= 5 Iwdx1)l for all h E W; the second equality is due to convexity and the inequality follows from the definition of h,. Therefore FO(xO)= F,(x,). Thus there is a point x2 E (x0, x1) at which the derivative of F,(x) at x =x2 is zero. Continuing this process shows that the derivative of F;,(x) with respect to x is zero over [x0, x1]. Numerical experiments suggest that F,(x) is strongly convex over I. If we replace W by the set W, = {h : U(h; I) = y, h 2 0, and U(lG(x, t)] Q(r)h-“; Z) < m}, then the minimum of T, over W, exists for each x E I and it is given by = [lG(x, t)l QW-‘(x)1’@.

h,(t)

Letting G,(x) = U(lG(x, t)jQ(t)h&“; I), gives G,(x) 5 F,(x) for each x E I. Thus we have the lower bound F,(x) 2 yd(x) where d(x) = [y-l J: [IG(x, ?)I Q(r)ll’Pdr]‘. Note that when G,(x) = max{G(x):x

E I}, then d(x,) 2 d(x) for all x E I.

4.

THE CASE p =

1

Let z be a given positive number and A = {w : w E Ad, w(x) 5 0, and Q(x) + m(x) > 0 over I} with Ad given in (6). For h E W set

K,(x) := j G(x, r){Q(t) + zw(t)}h-“(t)

dt.

I

Clearly K,(t) I 0 over I. Also {Q(t) + r&(t))”

< [r(t) + rQ(t)]s;”

(IO)

with zero values at t = 0 and

t = b. Thus

r(t) + d;“Q(t) gives a sufficient condition that &:A +A. 1. For any w, u E A there is the estimate llK,,,(x) - K,(x)l(,l

<0

(11)

We will assume that Q(r) + rK,,,(t) is positive over

zb24-‘S;* Ilw(x) - u(x)llco.

If rb2S;” < 4,

(12)

then, for each h E W, K, has a fixed point in A and for each x E Z we have the representation

--w&I

= Iw&)l = lim Iw&)I n--r=

with

Iw&)l

= $ IW,

91 h-“(t)

z. (-l)‘%(t;

h) dt

G. T. MCALLISTER

1122

where M,,(t; h) = Q(t) and * * 6’ IG(t, t,) * . . G(t;-,,

Mi(t; h) = [:.

=

’ IG(t, t,)l Mi-,(t,)h-Yt,) I0

here t,,.. . , tj is a set of i independent w0 = 0, w,(t) = K,,(t), w2 = K,,(t), etc. The inequalities

t,)l Q(ti)h-“(t,)

. . . h-“(ti) dt, * . . dt;

dt,;

(13)

variables. To compute the fixed point of K, we take

IM;(t; h)J i tib2’4-‘6;“’

I(Q(t) 11%

and Ilw&) - w,&)IIs~

IlQ(~>llcc Gn+‘(l - W’,

are true with G = zb2 + 48:. For fixed n we will give a method for finding an h, E W and an X, E I such that

Iwnh,,~xn)l= inf~llw+llm:h

E WI.

(14)

First let z E [0, M] be given. We will do the first order case as the zeroth order was done in Section 3. The functional to be studied is

Iw/,@)I = [; IGk

t)l hpa(t)[Q(U

- M(t;

h)l dt

(15)

with x E I. For each h E W, Iw,,,(x)I is a strongly convex function of x and has a unique critical point. Using [6; Theorem 1.31 we see that a minimum of (15) exists over the set of bounded measureable functions with range in [S,, a,]. Any stationarizer of (15) must be of the form h(t) = h,,(t) = max{min{c(x)v,,(t),

&J, S,]

with

(16) I’,&, x) = [IW,

t)l {Q(l) - TM,@; &)I

- ~Q(tL(f;

hx)l,‘P

and L,(t; h,,) the integral of IG(x, t,)G(t,, t)l f hp,(t,) with respect to t, over 1. The positive constant c(x) is determined by the volume constraint U(h,,; Z) = y. We will assume that

IW,

t)lfQ - TWO; h)I - rQ(t)L(t,

x; h) >O

for each h E W. This is a constraint on the size of r. Note that V,(O, x) = V,(b, x) = 0. Therefore, Z,(h) must contain intervals of the form [0, a,] and [a,, b] with u,, a2 E I. The second variation of (15) is given by p times the expression b

I 0

[IW> t)l > - &Ax;

t)Q(t) - k-4q4x;

t)Q(t) + IW,

[)I zpWlW”(r)q2W

dr

where K = p + 1, I_L= (Yt p, u,(x; t) is the integral with respect to t, over [0, 61 of the expression IG(x, t,)G(t,, t)l f hT,(t,), and z,(t) is that of IG(x, t),lQ(t) f h&(t,). If T > 0 and so small that the numerator is positive, then we have a minimum of (15) once we obtain a solution of (16). Let F,(x) be the right-hand side of (15) with h,(t) replacing h(t). Then F;(x) = j-’ IG(x, t)lx (Q(t) - zM,(t; h,))h;*(t)dt

- crc-O(x) i; h:(t) dt

(17)

0

and the last integral is zero as U(h,(t); I) = y. Thus x0 is a critical point of F(x) if and only if x0 is a critical point of lwlh(t)J where h = h,,(t).

1123

Minimal maximal deflection of nonlinear beams

equation defining h,,(t) in (15) allows for an easy solution up to any order in r. The right-hand side of the equation h,,(t) = c(x)V,,,(t, x) can be expanded in a Taylor series about r = 0 over I,&) for each x E I. In particular, The

k(t)

= &Jr) + r(c/Jv,,(x; t) + c”(x)v*(x; t)) + 2%X,(x;

t) + cb(x)V,(x; t) + c&)v2(x;

r)) + 00-7

(18)

where v,(x; t) has r = 0 and h,, in (16) is replaced by h,(t)

IG(x, [)I Q(f)l”P,

= max{min{[c&)

%I, &I,

v,(x; t) = %7(x; t)[lG(x, 01 M(t; hd + Q(t)b(x; CO(X)= -co(x)U(V,(x;

t); &,x))

+ wJ(x;

t; Ml,

t); WkJ)~

H,(x, f) = co(x)v,(x, t) + co(x)vo(x; 97 V,(x; t) = --(Y/Tv(p+p) (x; t)[lG(x, t)l M,(t; hx) + Q(W,(x,

+ +,VOp(x; t)[lG(x,[>IM,,(x,t) - Q(W,,(x, M,,(x, t) = I” IW, 0 L,(x, G(X)

= [-2U(V,(x,

t; k,x) = lb IW, t), Z2(h,))

[)I Q(~)h?(W,(x,

t,)W,,

+ cO(x)U(vZ(x,

f; &x)1,

f) dt,

[)I Q(Oh?(W,(x, 4,

t; ~IJ*

t) dt,,

~2(W)l + WXx7 0, ~2(hJ)~ etc.

Now F,(x) > 0 over Z with F,(O) = F,(b) = 0 for all h E W and F,(x) is continuous over [0, b] and is C’(Z). Therefore F,(x) has a maximum over I. Moreover, all points of Z where F,(x) has zero derivative must give the same value to F,(x); see the proof of Theorem 2(d). Since 0 < W,,,(X)< W&X) for all h E W, 0 F,(x) - rg(x) with

O
IG(x,x)( [IQ/la.

Similarly, using the definitions of F, and F, and the result in (18) we may easily derive an estimate of the form IF;(x) -F;(x)/ = O(z); thus lx, -x01 = O(r). Any point xl E Z where Fi(x,) = 0 gives a rule h,,,(t) for distributing the mass of the beam so as to give the least maximum displacement up to first order in r; i.e. IIw,~,,II~5 llwhII2 over W. Computing h,,,,(t) with x0 any critical point of F,(x) may be done as in the preceeding section. With minor modifications the above results may be continued to higher order in r. We maximize the function F,(x) = Iwnh,,Jx)(. However, as n gets large, the number of computations becomes formidable. Now consider the general case that r is given by (2) and that the hypothesis of Theorem 1 are satisfied. Let h,(t) = h,,(t) and let ~~ be determined by (2) using ho and we(x) = wohJx) with hg(x)wg(x) = Q(x). Plug r, into (15) and compute the associated maximum of F,(x). Repeat until the desired accuracy is achieved.

REFERENCES

111M. H. PROTTER

and H. F. WEINBERGER, Maximum Principles in Diflerentiuf Equations. Prentice-Hall, N. J. (1967). 121J. T. ODEN and E. A. RIPPERGER, Mechanics of Elastic Structures. Second Edition. McGraw-Hill. New York (1981). 131Z. MR6Z, M. P. KAMAT and R. H. PLAUT, Sensitivity analysis and optimal design of nonlinear beams and plates. J. Strut. Me& 13, 245 (1985). G. H. HARDY, J. E. LITTLEWOOD and G. Pt)LYA, Inequalities. Cambridge (1964). [:I I. G. TADJBAKHSH. Beams of minimum deflection. Iranian J. Sci. & Technol. 1. 1 (19711. [61 B. DACOROGNA, direct Methods in rhe Culcufus of Variations, Vol. 78. Applied M&h. sci. Springer, Berlin. (Received and accepted I July 1993)