Dynamic buckling of a thin elastic plate: Non-linear theory

Journal of Sound and Vibration ( 1977) 54( l), 75-82

DYNAMIC

BIJCKLINC

OF A THIN ELASTIC

NON-LINEAR

PLATE

THEORY

B. K. SHIVAMOGGI~ Department of Aerospace Engineering, University of Maryland, College Park, Maryland 20742, U.S.A. (Received 19 August 1976, and in revisedform 13 April 1977)

This paper treats the dynamic buckling of a rectangular thin elastic plate under a uniaxial harmonically varying load (with a compressive stationary part). The linear theory is known not to lead to a valid description of the dynamic buckling phenomenon, particularly, the dynamic post-buckling behavior ofthe plate. Consequently, a non-linear theory is developed in order to remedy the defects of the linear theory. The evolution of the dynamic postbuckling response of the plate is then illustrated as a bifurcation problem associated with the non-linear partial differential equations, and the important influence on the onset of the bifurcation exercised by the periodic part of the load is pointed out. The non-linear theory is further shown to modify the linear stability characteristics, and to give rise to an amplitude-frequency relationship for the non-linear transverse vibrations of the plate.

1. INTRODUCTION The first work in the area of dynamic buckling of elastic systems was due to Beliaev [I], who considered the linear problem of a straight column subjected to a periodic loading, and deduced the corresponding stability characteristics. Almost the same problem was studied by Lubkin and Stoker [2] independently. In the past three decades, an enormous number of papers has appeared in the Soviet Union on various problems of this class (for a general outline, see reference [3]). The classical linear theory, that considers infinitesimal deflections and establishes the: conditions under which a thin plate under the action of a uniaxial dynamic compressive load can take up equilibrium configurations other than the one corresponding to a uniform compression, reveals that the forced vibrations of the elastic plate show a parametric resonance, so that periodic forces acting in the middle plane of the plate can excite intense transverse vibrations for certain relationships between the forcing frequency and the natural frequency of transverse vibrations of the plate. Further, if the applied load P(t) consists of a stationary part and a periodic part, P(t) = PO + P,cos.Qt, and if PO is a compressive load greater than the static critical load P,, for the onset of buckling, the plate in the undeflected configurations is unstable. But if P, and 52 are chosen properly, the small motions in the neighborhood of the undeflected configuration can be stabilized (i.e., the time average of P(t) over a cycle can be made much larger than the static critical load for the onset of buckling). On the other hand, for certain other pairs (P,,sZ) it is quite possible that the undeflected configuration of the plate may be unstable when PO is a compressive load smaller than the static critical load for the onset of buckling or when P,, is a tensile load rather than a compressive one. However, in the case of static instability, the linear theory leads to lateral deflections, whose magnitude remains indeterminate. In the case of dynamic instability, the linear theory t Now at the Department of Aerospace Engineering Sciences, University of Colorado, Boulder, Colorado 80309, U.S.A. -I=

76

B. K. SHIVAMOGGI

predicts vibration amplitudes that increase unboundedly with time. It is found that experimentally observed amplitudes do not increase indefinitely with time under these circumstances, but level off to a stationary value signifying the advent of a supercritical state of equilibrium. In order to determine whether the linearly unstable vibrations become stationary, and the magnitude of the stationary amplitudes, one has to consider the non-linear problem.

2. NON-LINEAR THEORY A few remarks about the general nature of non-linear effects are in order. It is well known that essentially new phenomena occur in non-linear problems, which have no place in the corresponding linear problems. Therefore, in the following, the major purpose is not so much to introduce methods of improving the accuracy obtainable by the linear problem, but to focus attention precisely on those features of the non-linearities that result in distinctively new phenomena (some of which serve to remedy the defects produced by the linear problem). Among the latter are (i) existence of solutions for all values of the frequency rather than only for a set of characteristic values, (ii) dependence of amplitude on frequency, (iii) removal of resonance infinities and (iv) appearance of jump phenomena. Abandonment of one or more assumptions made in developing the linear theory leads to non-linear equations. Two types of non-linearities arise: (i) non-linear inertia; (ii) nonlinear elasticity. Non-linear inertia constitutes additional inertia forces which arise during coplanar displacements u, V; the latter are coupled non-linearly with the transverse deflection of the plate, w(x,y, t; E). The second type arises in considering the dynamic response of thin plates loaded beyond their buckling limits so that one needs to retain in the expressions for strain components the squares and products of the deflections and their derivatives. Besides, the linear theory involves an assumption that the middle plane of the plate is free from stress, so that there are no resultant forces acting in the middle plane of the plate. However, it is well known that the membrane forces arise if(i) there are external loads acting in the middle plane of the plate, and/or (ii) the plate is bent into a non-developable surface that results in strains on the middle plane of the plate. Under these circumstances, it is necessary to take into consideration the effect on bending vibrations of the plate of the stresses acting in the middle plane of the plate. Thus in the case of a plate the elastic non-linearity is not only geometrical but also physical in origin. It can be readily seen that the membrane forces produce an additional resultant transverse load (the forces N,, N,, N,., being measured per unit length) N&J’ w/8x2) - 2NX,(a2w/&c@, + Ny(a2 w/8y2). One can ignore the longitudinal inertia effects (that become significant when the frequencies of the applied loads are close to the natural longitudinal frequencies of the plate-a case which is excluded from the following) and the rotary inertia effects of the mass of the thin plate, and consider only the inertia effect due to the lateral motion of the plate. Then w(x,y, t; E) will satisfy a non-linear differential equation a2 w __ ay

XYax

(1)

where D = E/?/12(1 - v2), E being the modulus of elasticity, and v the Poisson’s ratio. Equation (1) in the static case reduces to von Karman’s non-linear differential equation for the bending of thin plates corresponding to large deflections [4].

7’7

DYNAMIC BUCKLING OF A PLATE

3. A PERTURBATION

ANALYSIS

Mathematical problems associated with non-linearities are so complex that a comprehensive theory of non-linear phenomena is out of the question. Consequently, the trend is to settle for something less than complete generality. Accordingly, one gives up study of global behavior of solutions of a non-linear problem, and seeks solutions in the neighborhood of a known solution-a method called bifurcation theory (see, for instance, references [5] and [6]). Thus the non-linear problem (1) will be analyzed in the following by various perturbation procedures.

o 0

5 x

I

(b) T

Figure 1. Sketch showing the geometry of the loaded plate: (a) plan; (b) section.

Consider the dynamic post-buckling response of a rectangular thin elastic plate of constani thickness h, simply supported along the edges and subjected to a compressive distributed load P(t) per unit length at the edges x = 0, a. Attention is confined to the middle plane (taken to be the x,y plane) so that the displacement components u and u can be ignored. When a uniaxial applied periodic load P(t) = P, + P, cos SZt per unit length at edges x = 0, a, where PO is stationary, and PI is of small magnitude, acts in the middle plane of the plate (see Figure I), then, upon using (see reference [7] for the derivation of strain terms)

78 equation

B. K. SHIVAMOGGI

(1) becomes

where p = h/2, with the boundary

conditions

x=O,a:

w = 0,

a2 w/ax2 = 0,

y=O,b:

w = 0,

i32w/ay2

= 0.

(4)

In deriving equation (3) it has, in effect, been assumed that the forces P(t) applied at the edges of the plate are transmitted throughout the plate to a sufficiently close approximation without any change in their magnitude along the x-direction. It can be readily verified that this assumption is valid when the frequencies of the applied loads are much smaller than the natural longitudinal frequencies of the plate. Consider cases wherein the amplitude of the periodic load P, is much smaller than the static critical load P,, for the onset of buckling. If one seeks solutions of the form w(x,y,t;~)=~

5 m=l

where Is] $ 1, which will be identified PO = POP,,, equation

(3) becomes,

upon neglecting

m7Lx nny sin-, a b

2 J&t;s)sinn=l

(5)

below, and puts T = SZt

Pl = P,IP,r,

the non-linear

(6)

modal coupling,

where, P,, = D(a/mz)‘[(mr/a)” fl = (dl”/Q’) (1 -PO),

+ (n7c/b)2], E = --i4”/Q2)P1,

c = (D/phpQ2)[&(m7t/a)4 - (1 - fv) (mn/a)2(nn/b)2 + +(n7c/b)4]. w,, is the natural frequency of transverse vibration of the plate. It may be noted that it is the particular choice of the boundary conditions as in equations (4) that made possible separating the variables as in equation (5). Let the initial conditions be T=o:

fm.= Am

df,,/dz

= 0.

(7b)

One thus has a non-linear Mathieu’s differential equation (7) so that the non-linear problem of the dynamic buckling of a thin elastic plate under a periodic load P(t) reduces to the investigation of the parametric stability (where the time-dependent load P(t) appears as a parameter) of the solution of the non-linear Mathieu differential equation. In particular, the issue of stability requires that all solutions &.(T) of this equation for m, n = 1, 2, . . . and 0 < z < ~0 remain bounded when arbitrary initial conditions are imposed. The well-known general theory of linear differential equations with periodic coefficients (viz., Floquet theory-see, for instance, reference [8], which becomes useful here since one is constructing non-linear solutions in the neighborhood of the linear solution) shows that the c(, s-plane is divided into regions of stability and instability separated by transition curves; (here stability is taken to imply boundedness as t 3 E). Further along the transition curves, periodic solutions of period 271 or 47~exist (as well as linearly increasing solutions except at the so-called critical points, according to Ince’s theorem [9]). Also, according to Floquet

79

DYNAMIC BUCKLING OF A PLATE

theory, in the unstable regions the growth takes place exponentially. The transition curves intersect the E = 0 line at the critical points ~1,= n2/4, n = 0, 1,2, . . .. In order to find an approximation to the transition curves for E < 1, one looks for LX(E)such that periodic solutions of period 2n or 47c result. One may use the method of multiple scales (see reference [lo]) to find a typical transition curve. Consider the cases for which the stationary part P, of the applied load P(t) is slightly greater than the static critical load P,, for the onset of buckling so that, corresponding to the neighborhood of the critical point a, = 0, one writes c?(E)= -&CL1- I? a2 + O(E3) and seeks solutions

of the form MT;

E) =fntnO(% f) + Efm”l(? ?) + ~2_L*(~, f) + W3),

where 5 = ~5. Then equation

(7) gives

a2fmno/aT2=o, a2fmn1/a22 = -2 a2fm,,laza?+ (a, --OS 7)fmno, a2fmn2/aT2 = -2a2fm.l~a~at +(a1- COST)fm,l - a2fm,0/a~2 i- a2fmn0 In order to have bounded

solutions

one has, from equation

-

Cfi"O.

(lo),

fm.o(T,q = &I(?). Combined

with equation

(13), equation

( 13)

(11) becomes (14)

a2fmnl/aT2 = (al - COST)&(?).

In order to have bounded

solutions,

one requires (15)

aI = 0 so that fmnl(T,?) = B,(T)+

Combined

with equations

a2fmn2 = aT2

The removal

Zd$Sin

(13) and (16), equation

(16)

&(?)COST.

(12) becomes d2Bo

T - COST [B,(f)+

&(?)COST]-

of the secular terms in equation

T

(17) requires

+ a2

(7b), one obtains

from equation

(dB,/d?)’

= - ;(A:.

(17)

that

d2 B,,/dS’ + (-a2 + 3) B. + cBf, = 0. Using conditions

B,(S) - c[B&)]~.

(18)

(18) - B;) (~2- B;),

(19a)

where F= {(-a2 +3)/(--c)} - Ai,. Different cases arise depending on the signs and magnitudes of c and ?. If c < 0, since B,(O) ::= A ,,,“, Bi cannot exceed A,f,, if F > 1 and cannot be smaller than AZ, if E< 1; otherwise, dB,/d? will be imaginary. Thus if c”> 1, &, is bounded and oscillates between A,, and -A,,, and if F < 1, B,, is unbounded. The special case F = 1 separates the stable oscillations from unstable oscillations, and hence the transition curve (there is only one in this case) corresponds to a2 =++ CA;,,. (20a) On the other hand, if c > 0, equation

(19a) can be written

as

(dB0/d?j2 = +c(AZ,, - B;) (B; - F).

(19b)

In this case, Bi is bounded and oscillates between A,, and F if F > 0, and oscillates between ID and A,, if F < 0. In all cases, the solutions of equation (19a) and equation ( I9b) are given by Jacobian elliptic functions.

80

B. K. SHIVAMOGGI

Using equation

(20a) one finds

a = -&Q + CA;“) + O(E3), (2Ob) which determines the amplitude of a periodic vibration corresponding to a given load (see Figure 2). Note the bifurcation phenomenon which corresponds to the branching of a nonlinear solution corresponding to the undetlected configuration at PO = 1 - G&“/2~2)P: and signifying that the load is a monotonically increasing function of the deformation. It is important to note the backward shift of the branching point caused by the periodic load PI which signifies that the state of equilibrium corresponding to PO z P,,can always be destabilized by a periodic load irrespective of its frequency. However, once the bifurcation occurs, the amplitude A,, of the vibration depends only on the stationary part PO of the applied load P(t).

I

0

PO

-4” \

A2 2i-f

Figure 2. Determination

>

e,

pl

of the amplitude of a periodic vibration corresponding

to a given load.

Another viewpoint in regard to buckling is that, corresponding to PO > P,,, a deflected configuration of equilibrium is the one that is more stable, while the undeflected configuration is unstable. This is apparent by taking PI= 0 (i.e., no periodic load), and then equation (7) can be written as dfmnldfmn = (-afmn - c2 cA3,‘imm

fm.= dfmnld?.

One sees that the origin, f,,, = fmn = 0,in the phase plane corresponds to a state of equilibrium. The character of the singularity there determines whether the state fmn =fmn = 0 is stable or not. It can be readily verified that if c( > 0 the singularity is a center, while it is a saddle if IX< 0. Thus the undeflected state of equilibrium of the plate is stable if PO< PC,, and unstable if PO > P,,. Note also the slight modification due to the non-linear effects of the transition curve given by the linear theory, according to equations (20), as shown in Figure 3-a trend that is consistent with Figure 2 and the foregoing remarks. Next, in order to determine the amplitude-frequency relationship, one may take again P,= 0,for a further illustration of the effect of non-linearities, and make use of a perturbation procedure introduced by Lindstedt (1882) and Poincare (1892) in connection with problems in dynamical astronomy. Thus equation (7) becomes d2f,,/dr2

T=o:

+

aE,,,, + E2 cfm”.= 0,

fm. = 4nm

dfm,ldT = 0,

(21)

DYNAMIC

--,

BUCKLING

OF A PLATE

8 11

Figure 3. Stability boundary as given by linear and non-linear theories. The shaded region is unstable. Linear theory (a = -+* + . . .) ; -. -, non-linear theory (a = de’ -2cA,f,, + . . .).

where now E is some appropriate

parameter,

E < 1. One can put

t = S(1 + &Al+ &21, + O(E3) and seek solutions

_A,“(~; E) =_A”&) so that equation

+ Efm”I(S) + ~2f,“z(~) + W),

d2fmnllds2 + afmnl = -24fmno, d2fmnz 7 + ufmn2 = -24fmnl - cfZno - 4% + 2A2>fmno.

(24:)

(25:) cw

(24),

so that equation

The removal

(23)

(21) gives

d2fm,olds2 + afmno =O,

From equation

(22)

of the form

fmno = A,,,,, cos &is,

(27)

d’fm,, = -21, aA,, cos T’&. -p + afmnl

(28:)

(25) becomes

of the secular terms requires I., = 0,

(29)

fmnl = B,,,,cos &s.

(30)

so that

Figure 4. Plot of A,, as a function of w/wO.

82 Combined

B. K. SHIVAMOGGI

with equations ds?

(27) and (28), equation

+ afmnz = A,,[-2&

(26) becomes

a - 3c(A2,,/4)] cos 1/1;zs- c(AQ4)

cos 36.~.

(31)

In order to remove the secular terms one requires A2 = -3c(A$,/8a),

(32)

so that fmnz = (cA;fJ32a)

cos 3 &.s.

(33)

Therefore

&,,(r ; E) = (A,, + ~B,,)cos

WV&

+ E2(cA&/32a) cos 3v’& + O(e3),

(34)

where o = [l - e2(3cAi,/8a)

+ O(E~)]-‘,

or o = 1 + E2(3cAi,/8a) From this, noting

that oO = v’Z, one finds

which is depicted

in Figure 4.

+ O(E~).

(w/w,,) - 1 = (s2 3c/8) A:,,

(35)

(36)

REFERENCES 1. N. M. BELIAEV1924 Collection ofPapers, EngineeringandStructuraI Mechanics. Leningrad: Put’. 2. S. LUBKIN and J. J. STOKER 1943 Quarterly of Applied Mathematics 1, 275-236. Stability of columns and strings under periodically varying forces. 3. V. V. BOLOTIN1964 Dynamic Stability of Elastic Systems. San Francisco: Holden Day, 4. TH. VON KARMAN 1910Enzyklopadie der Mathematischen Wissenschaten 4,31 l-385. Festigkeitsprobleme im Maschinenbau. 5. J. B. KELLER1969 in Bifurcation Theory for Ordinary Differential Equations (Editors: J. B. Keller and S. S. Antman). Bifurcation theory and non-linear eigenvalue problems. New York: W. A. Benjamin and Company. 6. I. STAKGOLD1971 SIAM Review 289-332. Branching of solutions of non-linear ecmations. 7. S. P. TIMOSHENKO and S. WOINOWSKY-KRIEGER1959 Theory of Plates and Shells. New York: McGraw-Hill Book Company, Inc. 8. E. A. CODDINGTONand N. LEVINSON1955 Theory of Ordinary Differential Equations. New York: McGraw-Hill Book Company, Inc. 9. E. L. INCE 1964 Ordinary Difirential Equations. New York: Dover. 10. J. D. COLE 1968 Perturbation Methods in Applied Mathematics. Waltham, Massachusetts: Blaisdell Publishing Company.