Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.

com, ISSN 1743-355X

Non-linear analysis of plates by the analog

equation method
J.T. Katsikadelis, M.S. Nerantzaki
Department of Civil Engineering, National
Technical University, Zografou Campus,
GR-15773 Athens, Greece
ABSTRACT

The Analog Equation Method is applied to large deflection analysis of thin elastic
plates. The von-K2rm£n plate theory is adopted. The deflection and the stress
function of the non-linear problem are established by solving two linear uncoupled
plate bending problems under the same boundary conditions subjected to
"appropriate" (equivalent) fictitious loads. Numerical examples are presented which
illustrate the efficiency and the accuracy of the proposed method.

INTRODUCTION

Many methods have been developed to study large deflections of plates. Analytic
as well approximate methods have been presented over the years [1]. However, all
these methods are restricted to plates of specific simple geometry and boundary
conditions. Realistic engineering problems are solved only by numerical methods
such as FDM, FEM and BEM. The latter has been proven an efficient alternative to
the domain type methods and has been used to study large deflection of plates in the
last decade [2-8]. In this paper a novel solution approach to large deflection analysis
of plates is presented. In what it follows the proposed method will be referred to as
Analog Equation Method [9]. According to this method the non-linear problem
governed by the coupled von-Kirmdn equations is substituted by two linear
uncoupled plate bending problems subjected to "appropriate" equivalent fictitious
load distributions under the same boundary conditions. Subsequently, using BEM
for the non-homogeneous biharmonic equation, the deflection and the stress function
as well as their second derivatives are expressed in terms of the unknown domain
fictitious loads. Substituting the foregoing quantities into the von-Kdrm&i equations
yields a set of non-linear algeabraic equations which permit the determination of the
fictitious equivalent loads. The resulting non-linear equations are solved numerically
by step increasing the actual load. The method is utilized to analyze certain example
problems. The obtained numerical results are in excellent agreement when compared
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166 Boundary Elements

with those obtained from other computational techniques. The proposed method can
be categorized into BEM type methods as its implementation is based on BEM.

GOVERNING EQUATIONS

The non-linear bending of thin plates is described by the von-K£rm£n equations [1].
Thus,

V'w = A + Ai(H,,F) (1)

V*F = - — L(w,w) in Q (2)

in which O is a two dimensional region with boundary F occupied by the plate;

w = w(jt,;y), (z,)0 e O, is the transverse deflection; F = F(x,y) is the
Airy's stress function; D= Eh* /12(l-v*) theflexuralrigidity of the plate,
having thickness h and elastic constants E,v ; V* is the biharmonic operator and L(
, ) is a non-linear operator applied to the functions w and F and represents

L(w,F) = w,« F,» +w,^ F,« -2w,^ F,^ (3)

L(w,w) is obtained by replacing F with w in equation (3), i.e.

, w) = 2(w,« w, -w\ ) (4)

The functions L(w,F) and L(w,w) define the non-linearity of the problem which is
due to the coupling of the transverse deflection with the membrane deformation.
The stress resultants at a point x,y € Q are given in terms of w and F as

My =-
(5)
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Boundary Elements 167

In equations (5) Af^,M^ are the bending moments, M^ is the twisting
moment, Q, , Qy are the transverse shearing forces and TV, , ^ , N^ are the
membrane forces.

The plate is subjected to the following boundary conditions on the boundary F

a^ (6a)
03 (6b)

F=Y\ (7a)
F,. = 72 (7b)

where a, = a,.(j), 0, = 0,(j), (i = 1,2,3) and y* = 7*0*) (* = 1,2) are

functions specified on F. V*w and Afw are the reactive equivalent transverse
force and the bending moment along the boundary given as [6]

V*vv = Vw 4- #„%/,,, +Nnt™n (8a)

Mw = -D(w,^, +vw^ ) (8b)

where

Vw = -Z>[(W),, -(v - !)(%,,„ ),. ] (9)

is the equivalent reactive force of the linear theory. The remaining terms in
equation (8a) are due to the contribution of the membrane force components
NX , N^ in the transverse direction.

Using intrinsic co-ordinates, that is the distance along the normal n to the
boundary and the arc length s, and taking into account that

dt ds (10a)

- «

dndt dsdn ds
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168 Boundary Elements

in which K (s) is the curvature of the boundary, equations (8a,b) are written as
-rfl,,),J + N.X + N.Q,. (lla)

Mw = -D[$ + (v - !)(«,„ +xX)] (lib)

where the following notation has been used

O =w X = w,^ <D = V*w ¥ = (V*w),, (12)

It is apparent from equations (6a,b) that all types of boundary conditions with
respect to the transverse deflection w can be treated by specifying appropriately the
functions #, , 0,. On the other hand stress boundary conditions are considered for
the membrane stresses which are rather more easily expressible in terms of the
stress function (as it is the case of movable edges or edges subjected to prescribed
inplane edge forces).

THE ANALOG EQUATION METHOD

The boundary value problem described by equations (l),(2),(6a,b) and (7a,b) is

solved using the Analog Equation Method (AEM) developed by Katsikadelis [9]. In
the problem at hand this method is applied as following:

Let w and F be the sought solution to equations (1) and (2). These functions are
four times continuously differentiate with respect to the co-ordinates x,y in Q and
three times on its boundary P. If the biharmonic operator is applied to these
functions we have

V*w = 0(jc,y) (13a)

W = 4(*,y) (13b)

Equations (13a,b) indicate that the solution to the original problem (1), (2) can
be obtained as the solution of two uncoupled linear plate bending problems with unit
stiffness and subjected to the equivalent fictitious loads q and q under the given
boundary conditions.

According to the AEM the unknown load distributions q and q can be

established using BEM which is applied as following:

For any function w satisfying the non-homogeneous biharmonic equation (13a)

the following integral representations are obtained [10].
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Boundary Elements 169

ew(P) = J A - J (A,O + A^X + A^ (14)

= J A^^n-j (15)

where £ = 1, 1/2, 0 depending on whether the point P is inside the domain fi,
on the boundary F or outside Q, respectively. Note that the boundary has been
assumed to be smooth at the point P. The kernels A, = A,(r), r = \P - q\,
P € fi, q € F, are given as

coscp
A,(r) = -
r (16a)
i = Inr + 1 (16b)

= —(2ifnr 4-r)cos<p (16c)

4

(16d)

The boundary conditions (6a,b) and the integral representations (14) and (15) for
f e r constitute a set of four boundary equations with respect to the boundary
quantities Q, X, <D, Y. Two of these equations are boundary integral equations
and the remaining boundary differential. These equations are solved numerically.
Thus, approximating the boundary integrals using the boundary element technique,
the domain integrals using domain nodal points (e.g. FEM discretization or Gauss
integration) and the boundary derivatives using finite differences, the following set
of linear equations is obtained:

[0] 0
[0] {X} 0
MM! {*} {0} [Q]
[0] [0] {0} (17)

where [/4,y] (ij= 1,2,3,4) are A^dV known coefficient matrices, {5-} (/=1,2)
known constant column matrices and [CJ (i=3,4) N\M known coefficient
matrices. {#},{X},{O},{Y} are A^cl vectors including the nodal values of the
unknown boundary quantities, while {q} is an Afxl vector incuding the nodal values
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170 Boundary Elements

of the unknown fictitious loading. N is the number of the boundary nodal points,
whereas M is the number of the domain nodal points.

Equation (17) permits the elimination of the boundary quantities O,X,<&,Y

from the discretized counterpart of equation (14), which after collocation at the M
domain nodal Gauss points yields

{"} = [G]{q] (18)

where {w} is an MX 1 vector including the values of the function w at the M Gauss
integrations points and [G] is an MxM known coefficient matrix.

Subsequent differentiation of equation (14) twice with respect to x and y yields

0 + (A «)„

The derivatives of the kernels are given in the Appendix.

Elimination of the boundary quantities from the discretized counterparts of

equations (19a,b,c) using equations (17) and collocation at the M Gauss integration
points inside A yield
. (20)

(2D

(22)

where [G^ ], [G^,], [G^] are known MxM coefficient matrices. Equations
(18),(20)-(22) are valid for homogeneous boundary conditions. For non-
homogeneous boundary conditions an additive constant vector will appear.
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Boundary Elements 171

Analogous relations can be derived for the function F. They differ only in the
boundary conditions. However, equations (7a,b) is a special case of (6a,b), that is
they are obtained by specifying a\ =1, (%% = 0, a, = y,, 0, = 1,
02 - 0, $3 = /2. Consequently, only the upper half part of equation (17) is
affected. Thus, we have

(23)

(24)

(25)

(26)

The matrices [G],[G^],[Gj,[Gj are defined similar to

The final step of AEM is to apply equations (1) and (2) to the M nodal
integration points inside the domain Q . This yields

which by means of equations (13a,b), (18) and (20)-(26) become

D{q}-h{L(q,q} = {g} (27)

)} (28)

Equations (27) and (28) are solved numerically by step increasing the load.

NUMERICAL RESULTS

On the basis of analysis presented in previous sections a computer program has been
written and some example plates have been studied to illustrate the efficiency of the
developed method and investigate its accuracy.
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172 Boundary Elements

Example 1
A circular uniformly-loaded plate with clamped stress-free edge (i.e.,w=0,
%/,„ = 0, F = 0, F,,, = 0) has been studied. The data of the plate are
fl / /* = 50, v = 0.30, g = constant. The obtained numerical results are presented
in Table 1. They are compared with those obtained from a BEM solution as well as
with those from an analytical solution and are found in excellent agreement. In
Figures 1 and 2 the distributions of the fictitious loads q = q(r) and q = q(r)
along the diameter of the plate are shown. From Figure 1 it can be concluded that
for small values of the actual load, the fictitious load q is constant and equal to g
(linear behaviour). However, as g increases q is no more constant. It exhibits peaks
which are shifted away from the center. The distribution takes place so that

2n [ q(r)dr = nafg
Jo
The same is valid for the fictitious load q. The total load in this case is equal to
zero

This was anticipated since L(w, w) is the second invariant of the curvature tensor
of the deflection surface w and it can be proven that in this case it satisfies the
relation

=0
L

Example 2
A circular plate with simply-supported stress-free edge (w = 0,M^ = 0,
F = 0,F,,, = 0) has been studied. The same data as in Example 1 have been used.
Numerical results have been obtained for two values of the load g. They are
presented in Table 2 as compared with those obtained from a BEM solution.

Example 3
A square uniformly loaded plate with clamped stress-free edges has been analyzed.
The obtained numerical results are presented in Figure 3 and in Table 3. The data of
the plate are a/A = 100, v = 0.30. They are in full agreement with those
obtained using BEM [6].

Example 4
A square uniformly loaded plate with simply supported stress-free edge has been
analyzed. The same data as in Example 3 have been used. The obtained numerical
results are given in Figure 4 and in Table 4. They are in full agreement with those
obtains using BEM [6].
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Boundary Elements 173

In Table 3 and 4 the values obtained by BEM [6] are not shown as they differ
neglibly.

CONCLUSIONS

A novel BEM solution, the Analog Equation Method, has been presented for non-
linear analysis of thin elastic plates. According to this formulation the coupled non-
linear governing equations are substituted by two linear plate bending equations
subjected to appropriate fictitious equivalent loads, which are established using
BEM. Its efficiency and accuracy are demonstrated by analyzing several examples.

Table L Deflections, membrane and bending stresses along the radius of a uniformly
loaded circular plate with clamped stress-free edge fV =0.30, #//z = 50,
g*= ga' I Eh' = 10).
w = w /h a; = <7;a* IEh* a; = crX lEtf
r/a Anal. BEM AEM Anal. BEM AEM Anal. BEM AEM
Ref.LH Ref.[6] Ref.[ll Ref.[6] Ref.fll Ref.[61
0 1.310 1.308 1.307 0.782 0.780 0.780 3.250 3.253 3.253
0.098 1.290 1.286 1.286 0.770 0.768 0.768 3.214 3.208 3.207
0.305 1.105 1.101 1.102 0.666 0.664 0.664 2.821 2.812 2.810
0.562 0.660 0.658 0.659 0.420 0.415 0.415 1.214 1.218 -1.208
0.802 0.190 0.188 0.189 0.159 0.171 0.172 -2.285 -2.278 -2.266
0.960 0.010 0.009 0.009 0.026 0.026 0.026 -5.750 -5.777 -5.802

Table 2. Deflections, membrane and bending stresses along the radius of a uniformly
loaded circular plate with simply-supported stress-free edge(v = 0.30, a I h = 50)
w = W tih 0-* = crV lEtf a*=cr*flr/E/%:
g* r/a BEM AEM BEM AEM BEM AlEM
Ref.[lll Ref.mi R ef.flll
0 1.805 1 .805 0.855 0.854 2.599 2.599
0.098 1.788 1 .788 0.847 0.846 2.610 2.610
5 0.305 1.635 1 ,635 0.777 0.776 2.701 2.700
0.562 1.217 1,,217 0.582 0.582 2.700 2.696
0.802 0.606 0,,607 0.287 0.286 1.934 1 .932
0.960 0.124 0,,124 0.070 0.069 0.492 0.477
0 3.662 3..660 2.977 2.969 3.811 3.819
0.098 3.636 3..634 2.957 2.951 3.897 3.904
25 0.305 3.403 3.401 2.802 2.793 4.636 4.642
0.562 2.696 2.691 2.304 2.290 6.530 6.516
0.802 1.455 1.457 1.222 1.213 7.318 7.241
0.960 0.305 0.310 0.286 0.290 2.295 .164
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174 Boundary Elements

Load q

-0.2 0.2
Radial position r/a

Fig.l Distribution of the equivalent load q along the diameter of a clamped

circular plate with movable edge for various values of the load
(v = 0.30, a I h = 50, g* =ga* I Eh* = 10).

81-4

4.E-4-

Load q

Of+O

-4.E-4
.1.0 -0.2
Radial position r/a

.Fig.2 Distribution of the equivalent load q along the diameter of a clamped

circular plate with movable edge for various values of the load
(v =0.30, o/A=50, g*=ga* I Eh* = 10).
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Boundary Elements 175

Table 3. Deflectionu, bending and membrane stresses alemg the halj'linex=(1 in a
square uniformly loaded clamped squcire plate with stniss-free edges
(V=0.3< D,a/A = 100;
f* yIa w °: V a;
',"
0.049 1.355 11.574 11.697 2.695 2.762
0.152 1.185 10.183 11.042 1.755 2.365
128 0.281 0.745 5.687 5.831 -0.369 1.454
0.401 0.232 -2.373 -9.762 -2.224 0.504
0.480 0.013 •10.036 -29.661 -3.696 -0.258
0.049 1.779 13.976 14.210 4.363 4.451
0.152 1.569 12.491 14.2301 2.984 3.824
192 0.281 1.007 7.188 8.694 -0.427 2.370
0.401 0.320 -3.452 -12.701 -3.736 0.824
0.480 0.018 -13.793 -41.063 -6.490 -0.462

Table 4. Deflections, bending and membtan stresses along the half line x=0 in a
square uniformly loaded simply supported square plate with stress-free edges
(V= 0.30,d/A = 100;
g* yIa W 6? a? °: cr;
0.049 2.652 11.842 12.140 5.214 5.195
0.152 2.469 11.488 14.119 4.416 4.404
128 0.281 1.913 9.997 16.924 1.112 2.524
0.401 0.996 5.621 12.874 -5.467 0.721
0.480 0.209 0.394 2.883 -11.662 0.061
0.049 3.262 13.099 13.515 7.243 7.162
0.152 3.056 12.857 16.597 6.395 5.967
192 0.281 2.403 11.645 21.891 1.943 3.224
0.401 1.268 6.685 17.676 -7.933 0.868
0.480 0.267 0.340 3.987 -17.071 0.087
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176 Boundary Elements

4O I . I
3O —• present work
binding stress
r_ Ret. [6]

-1O -• membrane stress!

-2O —
-3O —•
-4O —
-50
o.o o.i 0.2 0.3 O.5
position x/a

Fig.3. Variation of the membrane stresses a"=a"a*/Eh* and bending stress

a? = <T,V I Eh* along the central half line y =0, in a clamped square plate with
stress-free edges (v =0.30, a I h = 100, g* = ga* I Eh* = 192).

O.O 0.1 0.2 0.3 O4 O.5

position x/a

Fig.4. Variation of the membrane stresses o" =o"a* I Eh* and bending stress
G* = G*a* I Eh* along the central half line y = 0, in a simply supported square
plate with stress-free edges (v = 0.30, a I h = 100, g* = ga* I Eh* = 192j.
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Boundary Elements 177

REFERENCES

1. Chia, C.Y. 1980. Analysis of plates. New York: McGraw-Hill.

2. Tanaka, M. 1982. Integral equation approach to small and large displacements of
thin elastic plates. Boundary element methods in engineering, p.526-539. Berlin.
Springer-Verlag.
3. Kamiya, N. & Sawaki, Y. 1982. Integral formulation for non-linear bending of
plates. ZAMM 62:651-655.
4. O'Donoghue, P.E. & Atluri, S.N. 1987. Field/ Boundary element approach to
the large deflection of thinflatplates. Comp. & Struc. 27,3:427-435.
5. Nerantzaki, M.S. & Katsikadelis, J.T. 1988. A Green's function method for
large deflection analysis of plates, Acta Mech. 25:211-225.
6. Katsikadelis, J.T. & Nerantzaki, M.S. 1988. Large deflections of thin plates by
the boundary element method. In C.A Brebbia (ed.), Boundary Elements X, 3,
p.435-456. Berlin, Springer-Verlag.
7. Kamiya N. 1988. Structural non-linear analysis by boundary element methods. In
C.A. Brebbia (ed.), Boundary elements X, 3: p. 17-27, Berlin, Springer-Verlag.
8. Elzein, A. & Syngellakis S. 1989. High-order elements for the BEM stability
analysis of imperfect plates. In C.A. Brebbia & J.J. Connor (eds.), Advances in
boundary elements 3, p.269-284. Berlin, Springer-Verlag.
9. Katsikadelis, J.T. 1993. The analog equation method, to be published.
10. Katsikadelis, J.T. & Armenakas, A.E. 1989. A new boundary equation solution
to the plate problem. ASME J. Appl. Mech. 56:364-374.
11. Nerantzaki, M.S. 1992. Non-linear analysis of plates by the boundary element
method, Ph.D. Dissertation, National Technical University, Athens.

APPENDIX

a. Derivatives of the kernels A,(r)

-r-r- = - —cos(2w -<p)

Ar r (Ala)

^- = -yCOS(2# -<JO) (Alb)

d *A 9
——J- = -4sin(2<o-<p) (Ale)
dxdy r
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178 Boundary Elements

= —r- (sin* Q) - cos* Q) ) (A2a)

r

p- = -7 (cos* a) - sin* co) (A2b)

^--5HL*L (A2c)
dxcty r*

* sin<jO cososinfi) cosy

~= %—
r 2r

siny cos6) sino) cosy

% (A3b)
r 2r

sin<p cos2cu

- r^- = -lnr+-4-- cos^ o> (A4a)

dx* 2 42

(A4b)
<?y 2 42

(A4c)

in which co = JC,r is the angle between the x-axis and the vector r and (p = r,n is
the angle between the vector r and the outward normal n.