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A perturbation solution to a plate postbuckling problem
A PERTURBATION SOLUTION TO A PLATE POSTBUCKLING PROBLEM E. M. D~MBOURIAN, C. V. SMITH, JR. and R. L. CARLSON School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia, U.S.A. (Received 26 November 1974) Abstract--The modal coupling behavior of a compressed, imperfect plate is examined as a six degree of freedom structural system. A direct variational solution procedure is developed by the use of the functional for the total potential energy of the system. The non-linear algebraic equations generated are solved by use of a continued perturbation technique. The segmented deflection history is then developed for a variety of imperfection compositions and aspect ratios.
INTRODUCTION
The capacity of plates to carry loads after buckling has motivated many investigators to study plate postbuckling behavior[l-121. Unlike some other structural members, initial imperfections do not reduce the ability of a plate to carry continuously increasing loads. In 1970 Supple[12] analyzed a rectangular plate with constant in-plane compressive forces on opposite edges. He used Karman’s large deflection equations with the out-of-plane deflection and the Airy stress function being the dependent variables. He also included the effect of an initial out-of-plane imperfection. Using two degrees of freedom for the out-ofplane deflections, he developed an approximate solution based on the use of a Galerkin procedure. Two non-linear algebraic equations were ultimately obtained, and these were solved by elimination techniques. In the present paper a plate with an initial imperfection wO(w,y) is loaded with a prescribed in-plane edge-displacement. Upon the application of load, the plate deflects elastically with displacements represented by u, u, and w. An approximate solution is initiated by selecting functions u, v, and w which satisfy all of the prescribed boundary conditions. These functions are used as a basis for developing a direct variational solution; i.e. the total potential energy of the plate, which is the appropriate functional for this problem, is minimized in the Ritz procedure, and a system of non-linear algebraic equations is obtained. A perturbation method is then developed and used to obtain solutions for various imperfections and aspect ratios; i.e. the length to width ratios of plates. FORMULATION
OF
THE
PROBLEM
Figure 1 shows the simply supported rectangular plate with sides 2a and 2b. The displacements are ii, 6 and Z in the X, jj and Z directions respectively. The boundary conditions are as follows.
j!=
in:w=O,ATyd(~+“~)=O fi=o,fi=+pl
(2)
where p= 6*/u can be shown to be a constant compressive strain in the j direction. The total potential energy fi is composed of &,, the strain energy due to the in-plane membrane stresses, and 0, the strain entrgy due to the internal bending and twisting moments. The external potential, however, is zero because the edge displacements rather than the edge tractions are specified for this problem. NLM Vol. 11, No. 1-D
49
E. M. DOMBOURIAN, C. V. SMITH, JR. and R. L. CARLSON
, i,i
Fig. 1. The rectangular
plate.
0, may be expressed in terms of the middle surface strain components & EGO, and $& in the following manner :
(3) The strain energy due to bending and twisting of the plate is Eh3 ir, = 12(1-v*)
j;bi;.
{(~+~)‘-2(l_v)~~~-(~~]~d~d~
(4)
Combining, the total potential energy is then ii = u,,,+& For convenience the problem is non-dimensionalized .T xc- b’ n,!! Then the non-dimensional
a’
J
L: v = -, a
a,
as follows: ,=
z=i
-i WC1 h’
wo=h
\E’o
(5)
total potential energy is given by [ 143
where
Note that TI = (2(1 -v)/Eh)i’% For a direct variational and w can be chosen as follows U(x9Y)
=
solution procedure functions u, v
aiXik Y)
1
i
V(X.3 Y) = C bi YdXv Y)-PY i w(X9Y) = C Ci4dxv Y) i WO(&
Y) =
C 44i(x,
i
Y)
(10) (11) (12)
(13)
A perturbation solution to a plate postbuckling problem
51
An approximate solution may now be developed by use of the Ritz procedure. This is accomplished by rendering the functional II stationary with respect to the coefficients of the expression for u, Dand u’ as follows arI_O ,al--’
an_0
an
ab,_’
z=’
APPLICATION
TO AN
o
r = 1,2,...
IMPERFECT
(14)
PLATE
In the present work the plate is allowed to have six degrees of freedom; i.e. two each for u, u, W.The functions chosen are as follows u(x, y) = ai sin 7r.xcos y2 + a2 sin nx sin ny
(13
2
u(x, y) = bl cos 2 smny+b2cos~sin2ny-/rL. ’
w(x, I’) =
7cx
Cl
cos 22cos ~+c2cos1smny 2
(16)
.
(171
WIJX,y) = d, cos 22cos y+d,cosysintry
(18)
2
The Ritz procedure produces a system of non-linear algebraic equations. For the six degrees of freedom the unknowns are ulr a2, bl, b2, cl and c2. It may be noted that these are redefined in terms of the quantities x1, x 2, x3, x4, ,x5 and .x6 respectively, for convenience in the computer programming. The governing equations are:
XI 0: c~~x*+c~~X~+C~4_Y~+C~~X~+C~~x~+Ct~5x:+C~6~X~ =0 8x1 an = 0: c~~x~+c~5x~+c~~x~+cq5~-Yjx~ = 0 ax2 an = 0 -= 0: c~,x,+c~~X~+C~~X~+C~5jX: ax3 an - = 0: c~~x~+c~~x~+c*6x6+c~66x~ = 0 ax,
-=
an
+C~~~X~X~+C~~~X~X~+C~~~X:+C~~~X~+C~~~~X~.Y~+C~S~~X:+~(C~~
=
0
(23)
an ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 0 (24, where the Cij, Cijk, Cijkl, css,,, and cs6.vr are known quantities which depend upon the properties of the plate[14]. A method for developing solutions to the non-linear algebraic equations can be based on the use of a perturbation procedure. Application yields a system of simultaneous equations whose solution can be obtained by elementary methods. In the perturbation method the independent variables x1, s2, x3, x4, x5 and x6 are each expressed as power series of a perturbation parameter E; that is, or
2
xi =XiO+EXil+&
Xi2+*-.
+
E5Xi5
(23 Xi
=
kio
EkXjk
Here, the perturbation parameter is chosen to be E = p-po, where pc, is a value of ,U= I:*/a about which a Taylor series expansion is implied. A graphical representation of a first- and second-order expansion is given in Fig. 2.
E. M. DOMBOURIAN. C. V. SMITH, JR. and R. L. CARLSON
: P
PO
Fig. 2. Graphical
representation
of perturbation
expansions.
The system of equations (19)-(24) is expressed in terms of the sum above. Collecting like powers of E’Sgives equations of the form Cijo &O + iyjlE+ clj2E2+ LYj3 E3+ Ctj4 E4+ Ctjs 2 = 0,
where the Ctjk’S are functions of follows that
Xik’S.
Since E, Ed,
E3,
j=l
,...’ 6
etc. are linearly independent,
Cijo= Ctjl= Ctj2= C?j3= Ctj(j4 = Cij5= 0
(26) it (27)
This requirement yields a set of inhomogeneous equations in xik which can be successively solved by linear algebra techniques. In the indicated manner the six non-linear equations are successfully reduced to six groups with six algebraic equations in each group in the form Aji Xik
=
Bjk 9
i,j,=
1,2 ,..., 6,
k=0,1,...,5
The Bjk are functions of the Xi, with I = 1,2,. . . , k-i. We begin the solution with 11= 0.0, from which it follows that Bjo = 0 for all j. Then the equations for xi0 have the form AjiXio = 0 Since the Aji array is non-singular, it follows that xi0 = 0 for all i. Then the .Yil can be determined from the equations AjiXii = Bj, This procedure is continued until all the xik are calculated. The procedure outlined provides a description of the variable Xi in terms of the endshortening variable p. Although the resulting expressions could in theory be used to describe the entire range of the behavior of interest, it early becomes clear that this would require a much larger number of terms than the six indicated in equation (25). To avqid this, results for the xi as determined from expansions about p = 0 (pug= 0) were evaluated at some p > 0, say p = fi. These values were then used as starting values for a new expansion about IL= fi (that is, the new fl o = j). This new solution was extended to another point which then served as the starting point for a third expansion. In this manner, segments of the xi functions were obtained by continuing the solution procedure about successive values of po. To evaluate the continuation scheme as applied to the perturbation procedure a numerical trial was performed. The square plate with the specifications given in Fig. 3 was used for this trial. If the functions of equations (15)-(18) with a2 = b2 = c2 = d2 = 0 are used and an extremum of II is attained with respect to al, b, and cl, non-linear equations
A perturbation solution to a plate postbuckling problem
,’
*--
:
E~ponsion
53
Point
0.01
4( I
d2. d,*
0.0
0 *
b s 2.92 khsr
h n 0.112 inch 2.
E *
0.1
0.2
0.44110~
0.3
psi
0.4
’ xs
Fig. 3. Compressive strain vs transverse deflection.
which can be solved by elimination are obtained. As an alternative, the perturbation procedure was applied to the same problem and the results for solutions expanded about six values of p. (six segments) are presented in Fig. 3. The functions were first expanded about ,u~ = 0.0 and the first portion of the curve was observed to coincide with the true curve for all values of ,u c 0X)005. To provide an accurate continuation the function was expanded about p. = 0.0004 for the second segment. The computer was then allowed to calculate until p = 0.0006 was reached. This expansion procedure was then repeated four more times. Comparison of the curves obtained from the perturbation method with the algebraic solution indicates that there is good agreement. RESULTS
The plates considered are assumed to have known im*rfections dr cos 7 cos 9
and
d2 cos 7
sin xy
whose components are:
(see equation 18)
By including both imperfections simultaneously, a realistic model is attained. One component of the imperfection may be small but it is not likely to be completely absentt The objective of this work is to study the deflection response of such plates under various combinations of the aspect ratio and amplitudes of imperfection. For all the results presented on the cases of dr # 0 and d2 # 0 the perturbation method described previously was used. Note that for all these computations, six term approximations based on a p continuation increment of 5 x 10m6 were used. The deflection behavior was investigated by determining the effect of changes of the aspect ratio of a plate for a given set of initial imperfections as ,u increases. To this end, initial imperfections of dr = 0.01 arrd d2 = O-01 were chosen, and the aspect ratio was varied between 1.0 and 3.0. The results of the calculations are presented in the x5 vs x6 graph shown in Fig. 4.$ t If one component of imperfection is zero, then bifurcation type behavior is possible for certain conditions. These cases have been studied in detail by Dombourian[ 141. If both components of imperfection are zero, the perturbation procedure described here is not applicable because the derivative of a deflection variable with respect to p becomes infinite at a bifurcation point. These cases were, however, solved by an algebraic elimination scheme similar to that used by Supple[ll]. Note that Supple’s work is based on a formulation involving the Airy Stress function and the deflection. Here, the three displacement variables have been used. The curves shown in Fig. 4 are traces in the xs -x6 plane of loading curves in xs, x6, p space. This method of presentation has been used extensively in previous work[8-121.
54
E. M. DOMBOURIAN, C. V. SMITH, JR. and R. L. CARLSON
To illustrate the convergence properties of the continued perturbation x5
=
solution, values of
-2ckX5k
k=O
are presented in Table 1 with increasing p for m = 2, 3,4 and 5. The data presented are for r = l-85 and n, = n, = 0.01. Table Convergence data for x5 with d, = d2 = 0.01 and CIp continuation increment of 5 x 10-G &,‘m
2
3
0+)0010 oGQO15 o%QO20 0~00025 O-00030 o@M35 oGOG4o 00lO45 00lO50
OQO753906 0.0180183 0.0555135 0235483 0.415435 0.546354 0652594 0.744 107 0825658
OGO754676 0.0180563 0.0558705 0.235624 0.415577 0.546475 0652694 0.744191 082573 I
4 OGl754676 0.0180553 0.0558703 0.235620 0415576 0.546475 @652694 0.744191 0.82573 1
5 0.00754676 0*0180563 0.0558696 0.235625 0.415577 0+%476 0.652694 0744192 0.82573 I
A comparison of the results indicates that the difference between m = 2 (three terms) and m = 5 (six terms) could not be discerned on a plot of xg vs 11.The difference between values for m = 4 and m = 5 occurs in the fifth or sixth significant figure. Comparisons for x6 reveal similar convergence behavior.
Fig. 4. Effect of aspect ratio on x6 vs xI diagram
for dl = d2 = 0.01.
It is of special interest to observe that since each of the solutions for m involves a continuation based on a preceding calculation, the error generated should be accumulative. There is, however, no tendency for the results to diverge as p increases. It would appear that for the results presented, both the number of terms taken (m = 5) and the size of the continuation increment (cc = 5 x 10W6)provide good convergence behavior. Note that in all cases examined in Fig. 4 the behavior was coupled; i.e. as endshortening proceeded, the plate deflected with both x5 # 0 and x6 # 0. The variation in behavior as r increases is interesting. For r = 1-O the plate deflects primarily in a one half-sine mode (x5 dominant). Though the two half-sine wave component is present, it is small and ultimately decreases as ,u increases. A plate with an aspect ratio of 2.00 still exhibits a dominant x5 component. However, for r = 2-l there is an abrupt change in the composition of the deflection pattern of the plate. In fact it may be seen that ultimately, loading increases are accompanied by decreases in the xg component while the x6 component increases. Thus, for a given imperfection, a very small change in r can result in drastic changes in the deflected shape as end-shortening proceeds.
A perturbatiob solution to a plate postbuckling problem
55
In all of the preceding presentations of the results obtained, the role of end-shortening is not clearly indicated because the curves shown are traces in the x5 -x6 plane. The effect of the value of end-shortening is illustrated in Fig. 5 where results for d, = 0.01 and dz = O-01are presented. Here the abscissa parameter is r and the relative contribution of the x5 component is represented by use of the ordinate parameter x,/,/(x: +x2). The progression to either an x5 domination or an x6 domination with increasing p is clearly indicated. Further, the abrupt transition in behavior in the vicinity of r = 2 is distinctly
& 1.0 . 0.9 . 0.8 . 0.7 . 0.6. 0.5. 0.4. 0.3 . p.O.00023 0.2. 0.1 . -
0.0 .
L
*
0.5
p~0.00035
.
.
I.0
1.5 2.0 2.5
.
.
*
3.0 35
. 4.0
r’
%
Fig. 5. Deflection parameter vs aspect ratio for d, = d2 = 0.01.
-*6
0.5.
Fig. 6. Effect of aspect ratio on xs vs x5 diagram for dl = 0.01 and d2 = 04Ml.
demonstrated. It is interesting to observe that for r = 2, slight changes in effective lengths due to alterations of boundary conditions during loading could conceivably result in an abrupt transition in modal composition. The example just illustrated was concerned with plates whose imperfections dl and d2 were equal. Generally a given plate may possess imperfections of varying orders of magnitude. To study such cases, calculations were made for plates whose imperfections dl and d2 were unequal. By holding a given set of imperfections constant and varying the aspect ratio, the modal composition was studied and graphs were drawn on the x5 -x6 plane as seen in Figs. 6 and 7.
56
E. M.
D~MBOURIAN,
C. V. SMITH,JR. and R.L. CARLSON
In Fig. 6 the imperfection dl >>d 2. As a consequence, in the initial stages of loading, the plates develop a modal composition such that the x5 mode (associated with dl imperfection) tends to dominate for the range 2 < r < 3. The influence of the imperfection dl is especially appreciated in the light of the modal composition of the plate whose r = 2.55, where the x5 component dominates throughout. For at this aspect ratio, the postbuckling loading path of a corresponding initially flat plate would coincide with the x6 axis. *X6
0.5.
0.4.
II 0.2 I\
PI.95
o$l+ 0.1
0.2
,x, 0.3
0.4
Fig. 7. Effect of aspect ratio on x6 vs x5 diagram for d, = 0301 and d2 = 0.01.
Fig. 8. Effect of imperfection on x6 vs x5 diagram for r = 2.
In Fig. 6 there is a separation of behavior regions depicted by a dashed curve (this curve is not a result of computations) which Supple[l2] has called the locus of imperfections. The curves that lie above the separation ultimately exhibit a dominant x6 component. It was found that numerically unstable results were obtained in the vicinity of the locus of imperfections. This was due to the fact that within the range of values of aspect ratio where transition from dominance of one mode to another occurs, relatively large changes in x5 or x6 were observed. As observed earlier the perturbation parameter used in this work is not suitable for this type of behavior. In Fig. 7, d2 >>d, and the locus of imperfections is biased towards the x6 axis. This indicates that at the initial stages of loading, the plates deflect and attain a modal shape with a more pronounced x6 component. Here again the importance of the imperfection should be emphasized viz-a-viz the modal composition of the plate whose r = l-95. This plate, if perfect, would have had a postbuckling loading path coincident with the x5 axis. In conclusion it would be plausible to expect that, at least in the initial stages of loading, the modal composition of plates is decided by the preponderance of a particular form of imperfection.
A perturbation
solution
to a plate postbuckling
problem
57
A question may also arise as to the direct effect of variations of imperfections on the deflection shapes of a plate with a fixed aspect ratio. To isolate this effect from other considerations, the aspect ratio was held constant at r = 2.00. Variations of imperfection were assigned by letting d, = 0.01 throughout and changing d2 from 0.001 to 0.05. The results of these calculations are presented in the x5 vs x6 graph shown in Fig. 8. Note again that in all of the cases studied, the behavior was coupled. The variation in behavior as d2 increases is clear. For dl = 0.01 and d, = 0901 the plate deflects in a very pronounced one half-sine wave shape as p increases. This is reasonable, since the dl imperfection (a half-sine wave shape) is much larger than the d2 (two half-sine wave shape) imperfection. The plate with d, = 0.01 and dz = 0.01 still exhibits a dominant x5 component. However, for dl = 0.01 and d2 = 0.02 there is a change in the composition of the deflection pattern of the plate. Examination of the curves ford, = 0.01 and d, 2 0.02 reveals that continuing end-shortening is ultimately accompanied by decreases in the x5 component and increases of the x6 component. This behavior indicates that it is possible for a plate with I = 2 to develop a deflection configuration in which the two half-sine mode dominates if d2 is sufficiently large. The above numerical experiments emphasize again the fact that the phenomenon of changes in deflection pattern depends not only on the aspect ratio, but also upon the relative values of the imperfection components. All of the preceding developments have focused on the manner in which the transverse deflection changes with end-shortening. The solution technique used also provides a description of how the in-plane displacements CIand u change. These latter functions can be used to determine the middle surface strain and stress distributions. Since these quantities are of secondary interest in the discussion presented here, no results for these displacement functions are presented in this paper. CONCLUSIONS In the work described, a perturbation procedure was used to examine the effect of imperfection and aspect ratio on the transverse deflection of a plate subjected to in-plane edge displacement. Some of the features of the perturbation technique developed may be summarized as follows :
Expressing each unknown in powers of a perturbation parameter E reduces the non-linear equations to a system of equations which can be successively solved by elementary linear algebra techniques. If the number of unknowns in a system of non-linear equations is increased, algebraic elimination techniques may be expected to become difficult to apply and impractical to use. The perturbation technique of solution does not appear to be limited in this respect. Improved accuracy of solution may be achieved by increasing the number of terms used in the power series expansion. This can, however, lead to algebraic manipulation problems. The continuation scheme adopted here provides a means for achieving accuracy without resorting to the use of an unmanageable number of terms. Difficulties may be anticipated if the first derivative in a power series expansion of an unknown is very large or unbounded. This problem can, however, be obviated by the selection of another perturbation parameter for which this difficulty does not arise. REFERENCES plates under various conditions of end 1. W. T. Koiter, The effective width of infinitely long flat rectangular restraint, National Aeronautical Research Institute, Amsterdam, NLL Report S 287, December (1943). (in Dutch), Delft, Holland (1945). Translated 2. W. T. Koiter, The Stability of Hasric Equilibrium. Dissertation into English as AFFDL-TRIlO-25, Wright-Patterson Air Force Base, Ohio. on the postbuckling 3. W. T. Koiter. Introduction to rhe Postbucklina Behacior of F/at Plates. Colloouium behavior of piates used in metal structures. Umversity of Liege, Belgium (1963). . 4. Manuel Stein, The phenomenon of change in buckle pattern in elastic structures, NASA TR R-39 (1959). of buckled rectangular plates, NASA TR-JO (1959). 5. Manuel Stein, Loads and deformations 6. I. Hlavacek, Acta Tech. CSAV Praha, No. 2 (1962). 7. G. Augusti, Publication No. 172. Ionic. Napoli. Fat. Ing.. Institute Sci. Construe. (1964).
E. M.
58
DOMBOURIAN, C. V. SMITH, JR.
and R. L. CARLSON
8. A. H. Chilver, Coupled modes of elastic buckling, J. IUech. PIP.Solids 15, 15 (1967). 9. W. J. Supple, Coupled branching configurations in the elastic buckling of symmetric structural systems, Int. J.Mech. Sci. 9.97 (1967). IO. W. J. Supple and A. H. Chilver. Elastic postbuckling of compressed rectangular flat plates (Edited by A. H. Chilver). Chatto and Windus (1967). Il. W. J. Supple, On the change in buckle pattern in elastic structures, fnt. J.Mech. Sci. 10, 737 (1968). 12. W. J. Supple, Changes of wave-form of plates in the postbuckling range, Inr. J.Solid Strucr. 6, 1243 (1970). 13. A. S. Vol’mir, Flexible PmtesandShells, Technical Report AFFDL-TR-66-216. p. 14. Wright-Patterson Air Force Base, Ohio (1967). 14. E. M. Dombourian, On non-linear coupling effects in imperfect elastic plates in the postbuckling range, a Ph.D. Thesis submitted to the Georgia Institute of Technology (1974).
R&urn; On examine en tant que systsme structural a six degr& de liberte le comportement en couplage de mode d'une plaque comprimi$e imparfaite. On developpe une methode de rkolution variationnelle directe en utilisant la fonctionnelle de 1'6nergie potentielle totale du syst;me. On resout les equations algebriques non lingaires g&&Ses en utilisant une technique de perturbation continue. On developpe ensuite un historique de la de'flexion pour une variete de compositions d'imperfections et de param;ters d'aspect.
Zussammenfassung: Das modale Kopplungsverhalted einer zusammenged&ckten mangelhaften Platte, dargestellt als Fachwerksystem mit sechs Freiheitsgraden, wird untersucht. Ein direktes Variationsl&ungsverfahren wird entwickelt unter Verh?endung des Funktionals der Gesamtpotentialenergie des Systems. Die so erzeugten nicht-linearen algebraischen Gleichungen werden unter Verwendung eines fortgefiihrten Perturbationsverfahrens gel&t. Der unterteilte Auslenkungsverlauf wird dann f& eine Reihe von Mangelzusammensetzungen und Langenverhh'ltnisse entwickelt.
INTRODUCTION
The capacity of plates to carry loads after buckling has motivated many investigators to study plate postbuckling behavior[l-121. Unlike some other structural members, initial imperfections do not reduce the ability of a plate to carry continuously increasing loads. In 1970 Supple[12] analyzed a rectangular plate with constant in-plane compressive forces on opposite edges. He used Karman’s large deflection equations with the out-of-plane deflection and the Airy stress function being the dependent variables. He also included the effect of an initial out-of-plane imperfection. Using two degrees of freedom for the out-ofplane deflections, he developed an approximate solution based on the use of a Galerkin procedure. Two non-linear algebraic equations were ultimately obtained, and these were solved by elimination techniques. In the present paper a plate with an initial imperfection wO(w,y) is loaded with a prescribed in-plane edge-displacement. Upon the application of load, the plate deflects elastically with displacements represented by u, u, and w. An approximate solution is initiated by selecting functions u, v, and w which satisfy all of the prescribed boundary conditions. These functions are used as a basis for developing a direct variational solution; i.e. the total potential energy of the plate, which is the appropriate functional for this problem, is minimized in the Ritz procedure, and a system of non-linear algebraic equations is obtained. A perturbation method is then developed and used to obtain solutions for various imperfections and aspect ratios; i.e. the length to width ratios of plates. FORMULATION
OF
THE
PROBLEM
Figure 1 shows the simply supported rectangular plate with sides 2a and 2b. The displacements are ii, 6 and Z in the X, jj and Z directions respectively. The boundary conditions are as follows.
j!=
in:w=O,ATyd(~+“~)=O fi=o,fi=+pl
(2)
where p= 6*/u can be shown to be a constant compressive strain in the j direction. The total potential energy fi is composed of &,, the strain energy due to the in-plane membrane stresses, and 0, the strain entrgy due to the internal bending and twisting moments. The external potential, however, is zero because the edge displacements rather than the edge tractions are specified for this problem. NLM Vol. 11, No. 1-D
49
E. M. DOMBOURIAN, C. V. SMITH, JR. and R. L. CARLSON
, i,i
Fig. 1. The rectangular
plate.
0, may be expressed in terms of the middle surface strain components & EGO, and $& in the following manner :
(3) The strain energy due to bending and twisting of the plate is Eh3 ir, = 12(1-v*)
j;bi;.
{(~+~)‘-2(l_v)~~~-(~~]~d~d~
(4)
Combining, the total potential energy is then ii = u,,,+& For convenience the problem is non-dimensionalized .T xc- b’ n,!! Then the non-dimensional
a’
J
L: v = -, a
a,
as follows: ,=
z=i
-i WC1 h’
wo=h
\E’o
(5)
total potential energy is given by [ 143
where
Note that TI = (2(1 -v)/Eh)i’% For a direct variational and w can be chosen as follows U(x9Y)
=
solution procedure functions u, v
aiXik Y)
1
i
V(X.3 Y) = C bi YdXv Y)-PY i w(X9Y) = C Ci4dxv Y) i WO(&
Y) =
C 44i(x,
i
Y)
(10) (11) (12)
(13)
A perturbation solution to a plate postbuckling problem
51
An approximate solution may now be developed by use of the Ritz procedure. This is accomplished by rendering the functional II stationary with respect to the coefficients of the expression for u, Dand u’ as follows arI_O ,al--’
an_0
an
ab,_’
z=’
APPLICATION
TO AN
o
r = 1,2,...
IMPERFECT
(14)
PLATE
In the present work the plate is allowed to have six degrees of freedom; i.e. two each for u, u, W.The functions chosen are as follows u(x, y) = ai sin 7r.xcos y2 + a2 sin nx sin ny
(13
2
u(x, y) = bl cos 2 smny+b2cos~sin2ny-/rL. ’
w(x, I’) =
7cx
Cl
cos 22cos ~+c2cos1smny 2
(16)
.
(171
WIJX,y) = d, cos 22cos y+d,cosysintry
(18)
2
The Ritz procedure produces a system of non-linear algebraic equations. For the six degrees of freedom the unknowns are ulr a2, bl, b2, cl and c2. It may be noted that these are redefined in terms of the quantities x1, x 2, x3, x4, ,x5 and .x6 respectively, for convenience in the computer programming. The governing equations are:
XI 0: c~~x*+c~~X~+C~4_Y~+C~~X~+C~~x~+Ct~5x:+C~6~X~ =0 8x1 an = 0: c~~x~+c~5x~+c~~x~+cq5~-Yjx~ = 0 ax2 an = 0 -= 0: c~,x,+c~~X~+C~~X~+C~5jX: ax3 an - = 0: c~~x~+c~~x~+c*6x6+c~66x~ = 0 ax,
-=
an
+C~~~X~X~+C~~~X~X~+C~~~X:+C~~~X~+C~~~~X~.Y~+C~S~~X:+~(C~~
=
0
(23)
an ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 0 (24, where the Cij, Cijk, Cijkl, css,,, and cs6.vr are known quantities which depend upon the properties of the plate[14]. A method for developing solutions to the non-linear algebraic equations can be based on the use of a perturbation procedure. Application yields a system of simultaneous equations whose solution can be obtained by elementary methods. In the perturbation method the independent variables x1, s2, x3, x4, x5 and x6 are each expressed as power series of a perturbation parameter E; that is, or
2
xi =XiO+EXil+&
Xi2+*-.
+
E5Xi5
(23 Xi
=
kio
EkXjk
Here, the perturbation parameter is chosen to be E = p-po, where pc, is a value of ,U= I:*/a about which a Taylor series expansion is implied. A graphical representation of a first- and second-order expansion is given in Fig. 2.
E. M. DOMBOURIAN. C. V. SMITH, JR. and R. L. CARLSON
: P
PO
Fig. 2. Graphical
representation
of perturbation
expansions.
The system of equations (19)-(24) is expressed in terms of the sum above. Collecting like powers of E’Sgives equations of the form Cijo &O + iyjlE+ clj2E2+ LYj3 E3+ Ctj4 E4+ Ctjs 2 = 0,
where the Ctjk’S are functions of follows that
Xik’S.
Since E, Ed,
E3,
j=l
,...’ 6
etc. are linearly independent,
Cijo= Ctjl= Ctj2= C?j3= Ctj(j4 = Cij5= 0
(26) it (27)
This requirement yields a set of inhomogeneous equations in xik which can be successively solved by linear algebra techniques. In the indicated manner the six non-linear equations are successfully reduced to six groups with six algebraic equations in each group in the form Aji Xik
=
Bjk 9
i,j,=
1,2 ,..., 6,
k=0,1,...,5
The Bjk are functions of the Xi, with I = 1,2,. . . , k-i. We begin the solution with 11= 0.0, from which it follows that Bjo = 0 for all j. Then the equations for xi0 have the form AjiXio = 0 Since the Aji array is non-singular, it follows that xi0 = 0 for all i. Then the .Yil can be determined from the equations AjiXii = Bj, This procedure is continued until all the xik are calculated. The procedure outlined provides a description of the variable Xi in terms of the endshortening variable p. Although the resulting expressions could in theory be used to describe the entire range of the behavior of interest, it early becomes clear that this would require a much larger number of terms than the six indicated in equation (25). To avqid this, results for the xi as determined from expansions about p = 0 (pug= 0) were evaluated at some p > 0, say p = fi. These values were then used as starting values for a new expansion about IL= fi (that is, the new fl o = j). This new solution was extended to another point which then served as the starting point for a third expansion. In this manner, segments of the xi functions were obtained by continuing the solution procedure about successive values of po. To evaluate the continuation scheme as applied to the perturbation procedure a numerical trial was performed. The square plate with the specifications given in Fig. 3 was used for this trial. If the functions of equations (15)-(18) with a2 = b2 = c2 = d2 = 0 are used and an extremum of II is attained with respect to al, b, and cl, non-linear equations
A perturbation solution to a plate postbuckling problem
,’
*--
:
E~ponsion
53
Point
0.01
4( I
d2. d,*
0.0
0 *
b s 2.92 khsr
h n 0.112 inch 2.
E *
0.1
0.2
0.44110~
0.3
psi
0.4
’ xs
Fig. 3. Compressive strain vs transverse deflection.
which can be solved by elimination are obtained. As an alternative, the perturbation procedure was applied to the same problem and the results for solutions expanded about six values of p. (six segments) are presented in Fig. 3. The functions were first expanded about ,u~ = 0.0 and the first portion of the curve was observed to coincide with the true curve for all values of ,u c 0X)005. To provide an accurate continuation the function was expanded about p. = 0.0004 for the second segment. The computer was then allowed to calculate until p = 0.0006 was reached. This expansion procedure was then repeated four more times. Comparison of the curves obtained from the perturbation method with the algebraic solution indicates that there is good agreement. RESULTS
The plates considered are assumed to have known im*rfections dr cos 7 cos 9
and
d2 cos 7
sin xy
whose components are:
(see equation 18)
By including both imperfections simultaneously, a realistic model is attained. One component of the imperfection may be small but it is not likely to be completely absentt The objective of this work is to study the deflection response of such plates under various combinations of the aspect ratio and amplitudes of imperfection. For all the results presented on the cases of dr # 0 and d2 # 0 the perturbation method described previously was used. Note that for all these computations, six term approximations based on a p continuation increment of 5 x 10m6 were used. The deflection behavior was investigated by determining the effect of changes of the aspect ratio of a plate for a given set of initial imperfections as ,u increases. To this end, initial imperfections of dr = 0.01 arrd d2 = O-01 were chosen, and the aspect ratio was varied between 1.0 and 3.0. The results of the calculations are presented in the x5 vs x6 graph shown in Fig. 4.$ t If one component of imperfection is zero, then bifurcation type behavior is possible for certain conditions. These cases have been studied in detail by Dombourian[ 141. If both components of imperfection are zero, the perturbation procedure described here is not applicable because the derivative of a deflection variable with respect to p becomes infinite at a bifurcation point. These cases were, however, solved by an algebraic elimination scheme similar to that used by Supple[ll]. Note that Supple’s work is based on a formulation involving the Airy Stress function and the deflection. Here, the three displacement variables have been used. The curves shown in Fig. 4 are traces in the xs -x6 plane of loading curves in xs, x6, p space. This method of presentation has been used extensively in previous work[8-121.
54
E. M. DOMBOURIAN, C. V. SMITH, JR. and R. L. CARLSON
To illustrate the convergence properties of the continued perturbation x5
=
solution, values of
-2ckX5k
k=O
are presented in Table 1 with increasing p for m = 2, 3,4 and 5. The data presented are for r = l-85 and n, = n, = 0.01. Table Convergence data for x5 with d, = d2 = 0.01 and CIp continuation increment of 5 x 10-G &,‘m
2
3
0+)0010 oGQO15 o%QO20 0~00025 O-00030 o@M35 oGOG4o 00lO45 00lO50
OQO753906 0.0180183 0.0555135 0235483 0.415435 0.546354 0652594 0.744 107 0825658
OGO754676 0.0180563 0.0558705 0.235624 0.415577 0.546475 0652694 0.744191 082573 I
4 OGl754676 0.0180553 0.0558703 0.235620 0415576 0.546475 @652694 0.744191 0.82573 1
5 0.00754676 0*0180563 0.0558696 0.235625 0.415577 0+%476 0.652694 0744192 0.82573 I
A comparison of the results indicates that the difference between m = 2 (three terms) and m = 5 (six terms) could not be discerned on a plot of xg vs 11.The difference between values for m = 4 and m = 5 occurs in the fifth or sixth significant figure. Comparisons for x6 reveal similar convergence behavior.
Fig. 4. Effect of aspect ratio on x6 vs xI diagram
for dl = d2 = 0.01.
It is of special interest to observe that since each of the solutions for m involves a continuation based on a preceding calculation, the error generated should be accumulative. There is, however, no tendency for the results to diverge as p increases. It would appear that for the results presented, both the number of terms taken (m = 5) and the size of the continuation increment (cc = 5 x 10W6)provide good convergence behavior. Note that in all cases examined in Fig. 4 the behavior was coupled; i.e. as endshortening proceeded, the plate deflected with both x5 # 0 and x6 # 0. The variation in behavior as r increases is interesting. For r = 1-O the plate deflects primarily in a one half-sine mode (x5 dominant). Though the two half-sine wave component is present, it is small and ultimately decreases as ,u increases. A plate with an aspect ratio of 2.00 still exhibits a dominant x5 component. However, for r = 2-l there is an abrupt change in the composition of the deflection pattern of the plate. In fact it may be seen that ultimately, loading increases are accompanied by decreases in the xg component while the x6 component increases. Thus, for a given imperfection, a very small change in r can result in drastic changes in the deflected shape as end-shortening proceeds.
A perturbatiob solution to a plate postbuckling problem
55
In all of the preceding presentations of the results obtained, the role of end-shortening is not clearly indicated because the curves shown are traces in the x5 -x6 plane. The effect of the value of end-shortening is illustrated in Fig. 5 where results for d, = 0.01 and dz = O-01are presented. Here the abscissa parameter is r and the relative contribution of the x5 component is represented by use of the ordinate parameter x,/,/(x: +x2). The progression to either an x5 domination or an x6 domination with increasing p is clearly indicated. Further, the abrupt transition in behavior in the vicinity of r = 2 is distinctly
& 1.0 . 0.9 . 0.8 . 0.7 . 0.6. 0.5. 0.4. 0.3 . p.O.00023 0.2. 0.1 . -
0.0 .
L
*
0.5
p~0.00035
.
.
I.0
1.5 2.0 2.5
.
.
*
3.0 35
. 4.0
r’
%
Fig. 5. Deflection parameter vs aspect ratio for d, = d2 = 0.01.
-*6
0.5.
Fig. 6. Effect of aspect ratio on xs vs x5 diagram for dl = 0.01 and d2 = 04Ml.
demonstrated. It is interesting to observe that for r = 2, slight changes in effective lengths due to alterations of boundary conditions during loading could conceivably result in an abrupt transition in modal composition. The example just illustrated was concerned with plates whose imperfections dl and d2 were equal. Generally a given plate may possess imperfections of varying orders of magnitude. To study such cases, calculations were made for plates whose imperfections dl and d2 were unequal. By holding a given set of imperfections constant and varying the aspect ratio, the modal composition was studied and graphs were drawn on the x5 -x6 plane as seen in Figs. 6 and 7.
56
E. M.
D~MBOURIAN,
C. V. SMITH,JR. and R.L. CARLSON
In Fig. 6 the imperfection dl >>d 2. As a consequence, in the initial stages of loading, the plates develop a modal composition such that the x5 mode (associated with dl imperfection) tends to dominate for the range 2 < r < 3. The influence of the imperfection dl is especially appreciated in the light of the modal composition of the plate whose r = 2.55, where the x5 component dominates throughout. For at this aspect ratio, the postbuckling loading path of a corresponding initially flat plate would coincide with the x6 axis. *X6
0.5.
0.4.
II 0.2 I\
PI.95
o$l+ 0.1
0.2
,x, 0.3
0.4
Fig. 7. Effect of aspect ratio on x6 vs x5 diagram for d, = 0301 and d2 = 0.01.
Fig. 8. Effect of imperfection on x6 vs x5 diagram for r = 2.
In Fig. 6 there is a separation of behavior regions depicted by a dashed curve (this curve is not a result of computations) which Supple[l2] has called the locus of imperfections. The curves that lie above the separation ultimately exhibit a dominant x6 component. It was found that numerically unstable results were obtained in the vicinity of the locus of imperfections. This was due to the fact that within the range of values of aspect ratio where transition from dominance of one mode to another occurs, relatively large changes in x5 or x6 were observed. As observed earlier the perturbation parameter used in this work is not suitable for this type of behavior. In Fig. 7, d2 >>d, and the locus of imperfections is biased towards the x6 axis. This indicates that at the initial stages of loading, the plates deflect and attain a modal shape with a more pronounced x6 component. Here again the importance of the imperfection should be emphasized viz-a-viz the modal composition of the plate whose r = l-95. This plate, if perfect, would have had a postbuckling loading path coincident with the x5 axis. In conclusion it would be plausible to expect that, at least in the initial stages of loading, the modal composition of plates is decided by the preponderance of a particular form of imperfection.
A perturbation
solution
to a plate postbuckling
problem
57
A question may also arise as to the direct effect of variations of imperfections on the deflection shapes of a plate with a fixed aspect ratio. To isolate this effect from other considerations, the aspect ratio was held constant at r = 2.00. Variations of imperfection were assigned by letting d, = 0.01 throughout and changing d2 from 0.001 to 0.05. The results of these calculations are presented in the x5 vs x6 graph shown in Fig. 8. Note again that in all of the cases studied, the behavior was coupled. The variation in behavior as d2 increases is clear. For dl = 0.01 and d, = 0901 the plate deflects in a very pronounced one half-sine wave shape as p increases. This is reasonable, since the dl imperfection (a half-sine wave shape) is much larger than the d2 (two half-sine wave shape) imperfection. The plate with d, = 0.01 and dz = 0.01 still exhibits a dominant x5 component. However, for dl = 0.01 and d2 = 0.02 there is a change in the composition of the deflection pattern of the plate. Examination of the curves ford, = 0.01 and d, 2 0.02 reveals that continuing end-shortening is ultimately accompanied by decreases in the x5 component and increases of the x6 component. This behavior indicates that it is possible for a plate with I = 2 to develop a deflection configuration in which the two half-sine mode dominates if d2 is sufficiently large. The above numerical experiments emphasize again the fact that the phenomenon of changes in deflection pattern depends not only on the aspect ratio, but also upon the relative values of the imperfection components. All of the preceding developments have focused on the manner in which the transverse deflection changes with end-shortening. The solution technique used also provides a description of how the in-plane displacements CIand u change. These latter functions can be used to determine the middle surface strain and stress distributions. Since these quantities are of secondary interest in the discussion presented here, no results for these displacement functions are presented in this paper. CONCLUSIONS In the work described, a perturbation procedure was used to examine the effect of imperfection and aspect ratio on the transverse deflection of a plate subjected to in-plane edge displacement. Some of the features of the perturbation technique developed may be summarized as follows :
Expressing each unknown in powers of a perturbation parameter E reduces the non-linear equations to a system of equations which can be successively solved by elementary linear algebra techniques. If the number of unknowns in a system of non-linear equations is increased, algebraic elimination techniques may be expected to become difficult to apply and impractical to use. The perturbation technique of solution does not appear to be limited in this respect. Improved accuracy of solution may be achieved by increasing the number of terms used in the power series expansion. This can, however, lead to algebraic manipulation problems. The continuation scheme adopted here provides a means for achieving accuracy without resorting to the use of an unmanageable number of terms. Difficulties may be anticipated if the first derivative in a power series expansion of an unknown is very large or unbounded. This problem can, however, be obviated by the selection of another perturbation parameter for which this difficulty does not arise. REFERENCES plates under various conditions of end 1. W. T. Koiter, The effective width of infinitely long flat rectangular restraint, National Aeronautical Research Institute, Amsterdam, NLL Report S 287, December (1943). (in Dutch), Delft, Holland (1945). Translated 2. W. T. Koiter, The Stability of Hasric Equilibrium. Dissertation into English as AFFDL-TRIlO-25, Wright-Patterson Air Force Base, Ohio. on the postbuckling 3. W. T. Koiter. Introduction to rhe Postbucklina Behacior of F/at Plates. Colloouium behavior of piates used in metal structures. Umversity of Liege, Belgium (1963). . 4. Manuel Stein, The phenomenon of change in buckle pattern in elastic structures, NASA TR R-39 (1959). of buckled rectangular plates, NASA TR-JO (1959). 5. Manuel Stein, Loads and deformations 6. I. Hlavacek, Acta Tech. CSAV Praha, No. 2 (1962). 7. G. Augusti, Publication No. 172. Ionic. Napoli. Fat. Ing.. Institute Sci. Construe. (1964).
E. M.
58
DOMBOURIAN, C. V. SMITH, JR.
and R. L. CARLSON
8. A. H. Chilver, Coupled modes of elastic buckling, J. IUech. PIP.Solids 15, 15 (1967). 9. W. J. Supple, Coupled branching configurations in the elastic buckling of symmetric structural systems, Int. J.Mech. Sci. 9.97 (1967). IO. W. J. Supple and A. H. Chilver. Elastic postbuckling of compressed rectangular flat plates (Edited by A. H. Chilver). Chatto and Windus (1967). Il. W. J. Supple, On the change in buckle pattern in elastic structures, fnt. J.Mech. Sci. 10, 737 (1968). 12. W. J. Supple, Changes of wave-form of plates in the postbuckling range, Inr. J.Solid Strucr. 6, 1243 (1970). 13. A. S. Vol’mir, Flexible PmtesandShells, Technical Report AFFDL-TR-66-216. p. 14. Wright-Patterson Air Force Base, Ohio (1967). 14. E. M. Dombourian, On non-linear coupling effects in imperfect elastic plates in the postbuckling range, a Ph.D. Thesis submitted to the Georgia Institute of Technology (1974).
R&urn; On examine en tant que systsme structural a six degr& de liberte le comportement en couplage de mode d'une plaque comprimi$e imparfaite. On developpe une methode de rkolution variationnelle directe en utilisant la fonctionnelle de 1'6nergie potentielle totale du syst;me. On resout les equations algebriques non lingaires g&&Ses en utilisant une technique de perturbation continue. On developpe ensuite un historique de la de'flexion pour une variete de compositions d'imperfections et de param;ters d'aspect.
Zussammenfassung: Das modale Kopplungsverhalted einer zusammenged&ckten mangelhaften Platte, dargestellt als Fachwerksystem mit sechs Freiheitsgraden, wird untersucht. Ein direktes Variationsl&ungsverfahren wird entwickelt unter Verh?endung des Funktionals der Gesamtpotentialenergie des Systems. Die so erzeugten nicht-linearen algebraischen Gleichungen werden unter Verwendung eines fortgefiihrten Perturbationsverfahrens gel&t. Der unterteilte Auslenkungsverlauf wird dann f& eine Reihe von Mangelzusammensetzungen und Langenverhh'ltnisse entwickelt.