the application of a two-dimensional higher-order theory for the analysis of a thick elastic plate

compters &Structures Vol. 45, No. 4, PP. 633-648, 1992 Printed in Great Britein.

O&45-7949192 15.00 + 0.00 Q W92PCQlUUOllRCglLtd

THE APPLICATION OF A TWO-DIMENSIONAL HIGHER-ORDER THEORY FOR THE ANALYSIS

OF A THICK ELASTIC PLATE

H. MATSUNAGA

Department of Architecture, Setsunan University, Osaka, Japan

(Receiued 3 October 1991)

Ah&act--The application of approximate equations of a two-dimensional theory for the analysis of stress and displacement distributions of a thick elastic plate with lixed boundaries is presented. By using the method of power series expansion of displacement components, a set of fundamental equations of a two-dimensional higher-order plate theory is derived through the principle of virtual work. In order to assure the accuracy of the present theory, the governing equations of several sets of tnmcated approximate theories are solved numerically by the tinite ditference method. Higher-order finite difference operators with error terms of o(h’) are applied in order to validate the numerical accmacy of results. The convergence properties of the present numerical solutions have been shown to be accurate for the expanded displacement components with respect not only to the number of linite difference mesh intervals but also to the order of approximate theories. Stress components are determined by satisfying the equilibrium equations of a three-dimensional elastic continuum and the stress boundary conditions on the upper and lower surfaces of a plate. A direct comparison is also made with the numerical results of the integral method obtained by using the solutions of a three-dimensional elastic continuum.

1. INTRODUCT'ION

In the two-dimensional theory of a thin plate, a great number of investigations have been developed on the basis of the well-known Rirchhoff-Love hypothesis. Any plate theory is necessarily of an approximate character to provide a two-dimensional represen- tation of an intrinsically three-dimensional phenom- enon. On the basis of intuition and experience, an expectation is that the thinner the plate is, the more accurately the actual three-dimensional stress and displacement fields can be predicted by an approximate two-dimensional solution except near edges or regions of highly concentrated loadings. Although the classical plate theory is refined and well established, the applicability range of such a simplified theory would naturally be limited to a thin plate. In order to analyze a thick plate, various two- dimensional theories have also been investigated by taking account of the three-dimensional characteristics of stress and displacement fields.

Under the same assumption of displacement distributions in the classical theory, Reissner [l] has developed a generalized theory of the bending of elastic plates by taking into account the effects of transverse shear deformation. By introducing a set of stress distributions which satisfy the boundary conditions on the surfaces of a plate, the fundamental equations have been derived through the variational principle of the three-dimensional elastic body. Mindlin [2] has also derived a set of governing equations under the same assumption of displacement distributions. In Mindlin’s theory, the distri-

butions of stress components are not specified in the thickness direction of a plate and, therefore, stress boundary conditions on the surfaces of a plate cannot be satisfied. Although the first-order effects of transverse shear deformation have been taken into consideration, the effects of thickness change are neglected in both investigations. Introducing the same displacement components with those of the second-order approximation theory in the present paper, Reissner [3] has developed a two-dimensional bending theory of a plate including the effects of transversely shear deformation. The solution for a numerical example of the bending of a plate with a circular hole has been obtained and compared with the solution from the three-dimensional theory of elasticity. This theory is a minimum revision of the classical plate theory by taking into account the effects of both bending and transverse shear deformations and does not contain the effects of in-plane deformation. Stress boundary conditions on the upper and lower surfaces of a plate are not generally satisfied in the theory. Lo et al. [4] have presented a higher-order theory by taking into account the effects of in-plane deformation. The selected combination of displacement components corresponds to the second order approximation of the present theory. Stress components which satisfy the boundary conditions on the surfaces of a plate have been obtained. On the basis of the in-plane stress components obtained in

[4], Lo et al. [Sl have proposed an approach to determine other stress components by integrating the equilibrium equations of a three-dimensional elastic continuum that satisfies the stress boundary

633

canditions an the upper and lower surfaces ofa plate. Against the ~fo~ention~ theories which were based on the assumed ~st~butia~s of displa~rn~~~ components, Ambartsu~yan [6] has developed an a~~~x~rnate tw~d~rn~~on~ theory based on the power series expansions of stress ~orn~ne~~. Although a little revision may be noticed in the ~st~bution of in-plane stresses, the displacement solutions make no great difference with those in 111.

Various sets of tw~dime~s~~nal higher-order approximate theories using a finite number of terms in the power series expansions of d~~~a~rnen# corn- ponents have been investigate to refine the dassicai plate theory (for instance, [4], 173). Within the seape of the two-~me~sional ~~er~o~der theory, a gend nonlinear theory of elastic shells has been developed by Vokoo and ~a~unaga fg] without ~nt~du~j~g the K~r~hho~-Love ~~t~~~_ A set of fund~enta~ equations has been derived by expanding the disp~a~m~t eomponents into power series in the she11 thickness coordinate and by including the terms representing the higher-order stretohing and bending On the basis of &e resuhs in fg], a set of f~~rne~t~~ equations for a plate can be heavy dire&y and has been apphed to the Linear analysis of an extremely thick plate in [9]. The fundamental set of equations is completely divided into the sets of equations governing the m-plane and out~f-place defo~ati~~ modes. Fourier series solutions for approximate ~ua~ons of various orders of the theory have been obtained for a simply supported thick plate of an infinite Iength in one dire&ion subjected to sinu- soidally or unif~~ly dis~ibut~ pressure only on the upper surface of a plate. A close agreement between the so&ions of tw~~rn~sio~~ other-order theories and the exact solutions of the Dimensions elastic theory flO] has verified the ae~uraey and reliability of the approximate two-dimensional theories at least with respect to the specific problems in 191.

This paper presents the ap~~i~~~o~ of a~rox~rn~t~ equations of a two-~rn~s~ona~ theory for the anafy- sis of stress and ~s~~a~ent ~s~~bu~ons of a thick elastic pIate. As an extension of the classical thin plate theory, appheabifity and reiaibility of the two-dimensional higher~~~der theory are elarified in detail through the numerical examples of bound~y-vales probiems of an extreme& thick plate. On the basis of the power series expansions of ~~~a~rnent components, a f~damenta~ set of approximate equations of a two-dimensional higher-order plate theory has been derived through the p~~~~~e of virtual work. Linear constitutive relations for an elastic plate of isotropic mate~~ls have also been derived in terms of the expanded ~s~~~ent ~rn~~~~~ Severa sets of truncated a~prox~ate tuition of the present theory are solved for a thick plate with fixed boundaries subjected to u~fo~ly d~s~~buted pressure on the upper and/or lower surfaces. The equatians of ~n~~b~~ and boundary locutions sprinted in

terms of the displacement camponents are solved nurne~c~ly by finite difference approxim~tiQns. Higher-order finite di~eren~ operators with error terms of ofh’) are applied in order to validate the nurn~~~ accuracy within small nmber of unknowns fg I]. Stress ~orn~~~n~ are determined that satisfy the equilibrium equations of a three- dimensioned elastic continuum and the stress boundary ~ndit~~~s on the upper and lower surfaces of a plate. The convergence properties of the present numericaf solutions have been shown to be accurate for the expanded dis~~a~rn~~t component with respect not oniy to the number of finite difference mesh intervals but also to the order of approximate threories. Another proof on the accuracy of the numerical results has been made by comparing the rna~j~de of the internal work in the @ate with that of the external work. A dire@ ~rn~~so~ is also made with the numerical results of the integral method obtained by using the solutions of a three- dimensia~~l elastic continuum [ 121.

~echan~~~ characteristics of a thick plate with fixed boundaries are &armed in detail through the stress and d~sp~a~ent ~~tr~bn~ons under uniformly ~st~buted pressure on the upper and/or lower surfaces of the piate.

~n~~~~~~~ the Cartesian coordinates (x3 yf z) on the middle plane of a plate of uniform thickness 8, the displacement components may be expressed in power series ex~~~s~on of the thickness coordinate z by

On the basis of this expression of the dis~~a~~~t components, a set of the linear funda~~~ta~ equations of a two-dirnens~~~al higher-order plate theory can be s~mma~zed in the fo~~o~n~ (see, for instance, [8] and [9]).

Analysis of an elastic plate using higher-order theory 635

and strain-displacement relations can be written as

7(x”! =f{(n + l)u(*+‘)+ w$‘}

y$) = i{(fI + l)v(“+‘) + w~$‘}

$1’) = (n + I)@‘, IZ (3)

where a comma denotes partial differentiation with respect to the coordinates that follow.

With the use of the strain-displacement relations, the equations of equilibrium and natural boundary conditions can be derived through the principle of virtual work.

2.2. Equilibrium equations

&(“): @“, + N$J& - &‘-” +p$!” = 0

6v’“‘:N~~~+N~~-nQ(“-1)+p~)=0 Y

&,,,(“): Q(J; + Q,” _ nT’“-” +p4”’ = 0, (4)

where, introducing stress components s,, , syy, sxy , s,, , syz and s,,, stress resultants are defined as follows:

s

+ r/2 + r/2 NC”) =

xx s,,z” dz, N:;! = syy z” dz - r/z s -r/2

s

+ r/2 + r/2 NC”) =

XY s,,z”dz, Q$“’ = s

s,,z” dz -r/2 -l/2

s

+ r/2 + r/2 Q(n) =

Y syz z” dz, T’“’ = s,z” dz. (5)

-r/z s -r/2

Load terms measured per unit area of the middle plane are expressed as

pl”’ = [s:,z”l!$:, p?’ = [sy~zn]-cg:,

p!“‘= [s*zq’“z 2.2 r/2 3 (6)

where the stress components marked with an asterisk denote the prescribed quantities on the upper and lower surfaces of a plate.

2.3. Constitutive relations

For elastic and isotropic materials, the constitutive relations can be written as

s,, = (&, + E, )Y,, + E, (yyy + Y~Z )

syu = 0’0, + E, hyy + El (yzz + yxx )

sxy = D, ~xy 3 sxz = D, yxz 9 syz = D, ?/;r

szz = (Dw + El )~zr + E, (~xx + yyy 1, (7)

where Lame’s constants Dw and E, are defined by using Young’s modulus E and Poisson’s ratio v

D,S E vE

-’ E1=(1+v)(l-2v). 1+v (8) where qx and qy denote tangential loads and p denotes normal load on the surfaces of a plate.

2.4. Boundary conditions

The equations of the boundary conditions on the upper and lower surfaces are expressed as

* SE = SAX9 syr = sy* f * szz = szz (9)

and along the boundaries on the middle plane at x = const as follows:

&O = u(“)* or N$!!! = N!J”+

v(“) = v(“)* or N!$ = N!$*

w(“) = @O* or Q(n) = Q(n)* x x (10)

and at y = const

where

U(n) = &O* or Ng = NC*

@O = v(“)* or N$ = N$*

W(n) = w(“)* or Q”’ = Ql”‘* Y 9 (11)

NC”)* = XX s

+1/z

s” z” dz XX 9 - r/2

+ r/2 NC”)* =

YY 5

s* z” dz YY - r/2

+ l/2

N$* =

I

s* z” dz XY -t/z

+ r/2 Qt’* =

I

s*znd.z zz 7 -r/2

I

+ r/2 Q(n)* =

Y s*zndz YZ ’ (12) -r/2

where the quantities marked an asterisk are those prescribed along the boundaries on the middle plane of a plate.

Within the range of linear problems, the governing equations can be divided into two types of in-plane and out-of-plane problems according to the symmetry or antisymmetry conditions of loads with respect to the middle plane of a plate. The prescribed stress components on the upper and lower surfaces of a plate are expressed for these types of problems, respectively as follows:

(I) In -plane problem

s,: = fqJ2, s,: = fqy/2, s,: = -p/2

at z = *t/2. (13)

(II) Out-of-plane problem

s:x= +4x/2, s;= +qy/ 2, s,:= fPl2

at z = &t/2, (14)

636 H. MATSUNAGA

2.5. Stress resultants expressed in terms of the expanded ~p~a~e~nt components

Stress resultants can be derived from eqns (5) and (7) in terms of the expanded displacement components separately for in-plane and out-of-plane problems.


Ng = g [(D,, + E,)c$) -t E, uy, j-0

+(2j+l)E,w”i+‘l.2i+;j+l~ ; 2i*2j+1 8 0

Q$?+‘) = 5 [(2j + 2)*‘Uf 2) + q+ “1 j-0

2 t 2i+2j+3

‘2i+2j+3’ 2 0

Qt$+h t [(2j+2)u’Y+2’+w~~+l’]

j-0

2 t 2i+2jt3

‘2i+2j+3’ ?! 0

To”= i [(2j+ 1)(D,+E*)w’~+‘) I-0

+~,(~~~)+u~))]. 2 t 2if2j.i.1

2i+2j+l’ Z 0

(II) Out -of -plane problem

@f+ I) = xx f [(D,+E,)u~~+‘)+E,u~+‘) I-0

+(2j+2)Elw(Y+r)]. 2i+;j+3 9 ; 0

zi+zj+3

QF = i [(2j + l)#+Q + w;g j-0

2 t 2i+2j+l

*2i+2j+l’ 1 0

Qi2Q = t [(Q’ + I)@+ 1) + w,ff] j-0

2 t 2it2j+I

‘2i+zj+1’ z 0

T@i+l) = 5 [(Zj + 2)(0, + E,)w(4’+2) j-0

+ E, (a;:* 0 + 79; + I’)]

2 t .2+2j+3

‘2i$2j+3’ ? ’ 0 (16)

where i, j = 0, 1,2, . . . , co.

2.6. Equations of equilibrium expressed in terms of the expanded dtiplacement components

Equilibrium equations (4) can also be expressed in terms of the expanded displacement components seaparately for in-plane and out-of-plane problems.

(I) In-plane problem

+ (fD(@ + E,)u$ + (2j + l)E, w;F+‘)]

(20 -- 2 Dw[(2j + 2)~(~+~) + w$” ‘1

+ (4 Do0 + E,)z@ + (2j + l)E, w$‘+ U 1

-$,.!&,[(2j +QJ(U+~I + ,$j+I,

2 t 2j+l

‘2i+2j+l’ Z 0 +q,=o

~o~D,&p”‘+ w;gtl)

t aI+2 . - 0 2

-(2i+1)[(2j+l)(Ds0+E,)

xw(U+l)+~,(u~)+o~))l. 2

2i + 2j -t- 1

. f” 01 2 -p=o. (17)


(2i + 1) --&)[(2j 4 1)FPQ

2

+ My] ‘ 2 tY

Zi+2j+1’ Z 01 +q,=o

2 * a+2

‘2i+2j+3’ ? 0

(2i + 1) --D@J[{2j t_ I)vfa+It

2

+ wfyJ* +q,=o

sw*: 2 f&JwJ$~+ wg j-0

+(2j + ~)(u(~+‘)+o~+~))] 9

- (2i)[(2j + 2)(& -+ .&“l)w~~iz~

+~~(~~*I~+~~*‘))] 2

2i + 2j -I- 1

+p=D,

wherei,j=0,1,2 ,..., a3.

2.7. Stress components

Stress components can be derived in terms of the expanded displacement components by introducing the st~~spla~ment relations (2) and (3) into the cunstitutive relations (7). These stress ~rn~nent~ however, cannot satisfy the stress boundary conditions on the upper and lower surfaces of a plate with the ex~tion of the in-plane stress ~rnpon~~ which have no reference to the boundary conditions. With the use of the in-piane stress ~rnpon~~~ therefore, other stress comportants are determined by integration of equilibrium equations of a three- dimensional continuum with satisfying the stress boundary conditions on the uppar and lower surfaces of a plate.

I ’ z”+--(2jf3) 2 [

t 2i+2

’ (2j+2)(2j+3) 0 1 ‘2

638 H. MATSUNAGA

where

+ (2j + 2)E, w(8 + 2,

i- (2j + 2)E, W@ + 2)

S15+ I) XY

= fDm(Uy+l) + y$j+Of. (22)

In the above results, it is not clear whether the stress boundary conditions are satisfied for s,, and sYz of the in-plane problem and g of the out-of-plane problem. However, these stress components can satisfy the stress boundary conditions on the surfaces of a plate through the corresponding equations of equilibrium (17) or (18) for i = 0.

3. lwtb ORDER APPROX~~~ THEOffIES

Since the fundamental equations mentioned above are complex, approximate theories of various orders may be considered for the present problem. A set of the following combiantions of the Mth (M 2 1) order approximate equations is proposed here. This combination of the selected terms of displacement components is suggested from the forms of shear strain components (3).


m=O

M-2

w= 1 w (ZrnC IizZm+ I

1 (23) m=O

where M 3 2 for w.

(II) &d-of-plane probh

M-l u= c U(tn+l)ZZm+l

1 ISI=

M-l

M-l

w= c w(tn)zh, (24) l?l=O

where m =0, 1,2,3,. . . . The number of the unknown displacement components is (3M - 1) for in-plane problems and 3M for out-of-plane problems.

In the above cases of h& = 1, an approximation of plane strains is imposed. For the case of M = 1 of out-of-plane problems, the set of governing equations

can be reduced to those of the Kirchhoff-Love theory with the use of an approximation of plane stresses and the normal strain in the thickness direction is obtained as

4. INTERNAL AND EXTRRNAL WORKS

In order to evaluate the applicability range of the present theory and to clarify the numerical accuracy of the solutions, the magnitudes of the internal work and external work are calculated.

(A) Internal work

+ Q$o + T(“)y$)) dS

= z. lSOINll:Ui:) “+ tN$!$(u!;” + v!;‘) + iv$$’

+ fQ$“{(n + 1)~” + 1, + w$‘)

+ $Q$?{(n + l)@+ ‘) + w!;‘)

+ (n + 1)T’“‘w’” + ‘1 dS, (26)

where S, denotes the middle plane of a plate.

(B) External work

R,= s (s&u + s,:v + st,w) dS s*

+ I

(N$d“)* + N$f”)* f Q$%v~“)*) dL/,_,, L,

+ s

(N$*,(“) + N$?*u’“’ -I- Q :“)*w9dLl,=cons~r L,

(27)

where S, denotes the upper and lower surfaces of a plate and L, the boundary of the middle plane of a plate. The subscripts u and s shows the prescribed quantities of displacement and stress, respectively.

Analysis of an elastic plate tig higher-order theory

Due to Clapeyron’s theorem, the equilibrium state of a plate may be characterized by

5. DIMENSION= QU- IN THE FUNDAMENTAL EQUATIONS

The dimensionless quantities used in this paper are summarized as follows:

(1) Coordinates

x = I& y = F$, 2 = tr, (29)

where I is a half-span length of a plate in the x-direction and rc denotes the ratio of side length of a rectangular plate. The dimensionless coordinates ~~~n~e~~tsof -lgt,~G+l and -l/2 G r d + l/2.

(2) Displacement components

u=tu, v= tv, w = tw u(“)= t’-qJ(“),

“m = t’-“V”, 7 wfn) = ti-“W’“’ ’ (30)

(3) Stress resultants

N’;1? =Et”+z Ntj, 1

N($ _ Et”+’ N(n) 1 YY’

Q,“’ = Et” + IQ!), fl@ _ Et”+ ITb).

(4) Stress components

(31)

sxx=E ; *s,,, 0

syy=E f ‘s,, 0

s,,=E ; ‘s, 0

s,, = E 0

f ‘sr,, sYz = E(;pyz, s,, = E(&.

(32)

(5) Internal and external works

R, = Et%&, C& = Et%,. (33)

6. ANALYSIS OF A SQUARE PLWE (x -1) WITH FIXED BOUNDARIES

6.1. Load and boundary conditions

A square plate with fixed boundaries subjected to uniformly distributed loads on the upper and lower surfaces of the plate as shown in Fig. 1 is analyzed, Boundary conditions are summarized as follows:

(1) Along the boundaries of the middle plane of a plate (e = fl,q = &l)

u(*) iii 0, a(“) = 0, w(“) I 0. (34)

pw////////~///. / ! I

/ i i i i

oi _------------_ --_-.-.-_. / /

I-- x=IE

iy=lI

/ I

/ I I I

/m////////f///////:

t 1 1

t

s&=ap, 1

tt?ttfttt’Ttf?ttttt I

639

t

I

1

+ 1 1

I

Fig. 1. !$uare plate with fixed boundaries subjected to uniformly distributed load.

(2) On the upper and lower surfaces of a plate

(r = f l/2)

B sx = 0, s,: = 0, s,: = a II

p, (35)

where in correspondence to symmetry and/or anti- s~et~ conditions of loads, a and fl are given as follows:


a = -l/2, fl B -l/2. (36)


0: = -l/2, fi = +1/2. (37)

6.2. Finite dgference expressions of Mth order approximate equations

With the use of the finite difference operators, the equations of equilibrium and boundary conditions expressed in terms of the dimensionless displacement components are approximately transformed into a set of finite difference equations. In order to obtain the reasonably accurate numerical solutions with the use of the small number of unknowns, the slightly complicated higher-order finite difference operators with error terms of o(h’) (where h denotes a finite mesh length) are applied both to the equations of equilibrium and to the boundary conditions [l 11.

640 H. MAT~UNAGA

Table 1.

f(i,j - 3) f(i,j - 2) j((i. j - 1) f(i,j) f(i, j + I) f(i,j + 2) f(i,j f 3) f(i, j + 4) j(i, j f 5) f(i,j + 3)

1 -8 0 8 -1 QY&,j) -3 -10 18 -6 I

-25 48 -36 16 -3

-1 16 -30 16 -I 12hY.&j) 10 -15 -4 14 -6 1

45 -154 214 -156 61 -10

1 -8 13 0 -13 8 -1 8h3j& f&j) -1 -8 35 -48 29 -8 1

-15 56 -83 64 -29 8 -1 -49 232 -461 496 - 307 104 -15

When the central difference operators cannot be defined near or at the boundary points, appropriate forward or backward difference operators of the same order are applied.

Let f denote an arbitrary function which is continu- ously differentiable up to the third order. The higher- order finite difference operators on f for a square mesh which are applied to the present analysis may be expressed, for instance, in the 5 -direction as shown in Table I. Operators in the n-direction can be derived by interchanging the independent variable r with 9 and by considering that the integers i and j are corresponding to n with C, respectively. The mixed derivatives off, i.e. ir;,, , xtc,, and xer4 are obtained by multiply- ing appropriate operators in the 5 - and v-directions.

By intr~u~ng the finite difference operators, a set of differential equations for the boundary-value problem can be transformed into a set of algebraic equations expressed in terms of the displacements at the finite number of mesh points. A set of finite difference meshes of the middle plane of a plate are shown in Fig. 2. In Fig. 2, N is the number of mesh intervals over half spans in the 5- and q-coordinate directions and h is the mesh length, i.e. h = l/N. Due to the symmetric condition of the geometric configur- ation and loading of a plate, only one-eight of the plate (- 1 < q < c $ 0), as shown in Fig. 2, needs to be analyzed. The unknown displa~ment components of U(“) 9 V(“) and W(“) are to be determined at the interior mesh, point. The displacement components for the boundary point are prescribed by the boundary conditions at the point. Then the number of the final equilibrium equations for the interior field points is (3M - l)N(N + I)/2 for in-plane problems and 3~N(N + I)/2 for out-of-plane problems and coincide with the number of unknowns in each case, respectively.

With the use. of the displacement components, stress resultants and stress components can be obtained by eqns (15) and (16) and by eqns (19)-(22), respectively. The magnitudes of the internal and external works can also be calculated from eqns (26) and (27). The integration is carried out numerically by using Simpson’s formula. For this purpose, the mesh points in Fig. 2 and finite difference operators in Table 1 are also used.

7. NUMERICAL EXAMPLES AND RESULTS

7.1. Numerical examples

A thick elastic plate as shown in Fig. 1 is analyzed for five numerical examples with the thickness parameter

r/t =0.5, 1.0, 2.0, 5.0, 10.0. (38)

Poisson’s ratio is fixed to be v = 0.3. All the numerical results are shown in the dimensionless quantities and at the load-parameter

P 1 4 ( 10 z ; = 1.0.

Since the fundamental equations in the present problem are separated into two sets of equations of (I) in-plane and (II) out-of-plane problems, numerical results are also shown in each case separately.

In order to verify the accuracy of the present results the convergence properties of the numerical solutions according to the number of finite difference mesh intervals, N and to the order of approximate theories, M are examined.

Fig. 2. Finite difference mesh.

Analysis of an elastic plate using higher-order theory

Table 2.

(I) In-plane problem

MIN N=4 N=6 N=8

WfU -5.908 + I -0.5902 + t -0.5900 + 1 M=2 n, 0.6490 + t 0.6431+ t 0.6413 + I

n, 0.6384 + 1 0.6400+1 0.6402 + 1

WC’) -0.6247 + t -0.6243 + t -0.6238 + 1 M=3 Q 0.7040 + 1 0.7023 + t 0.7002 + 1

a, 0.6859 + I 0.6937 + t 0.6958 + t

W”’ -0.6020 f 1 -0.6040 + 1 -0.6037 + 1 M=4 Qi 0.7129 + 1 0.7139 + t 0.7130 + 1

a, 0.6930 + t 0.7034 + 1 0.7067 + 1

WC’) -0.5959 + I -0.5995 + t -0.5998 + t M=5 lzi 0.7141 + I 0.7166 + I 0.7165 + I

n. 0.6940 + 1 0.7055 + 1 0.7096 + t


MIN N=4 N=6 N=8

W(O) 0.3263 + I M=l ni 0.6124 + 1

a* 0.6092 + 1

WC*’ 0.2993 + t M=2 ni 0.9614 + 1

4 0.9412 + t

W(O) 0.2945 + 1 M=3 q 0.9999 + 1

a, 0.9738 + 1

W”) 0.2965 + 1 M=4 CL, 0.1006+2

a, 0.9787 + 1

W” 0.2969 + I M=5 fJi 0.1002+2

a, 0.9791 + t

0.3263 + 1 0.6103 + 1 0.6096 + t

0.2995 + I 0.9538 + 1 0.9462 + 1

0.3263 + 1 0.6098 + 1 0.6096 + t

0.2994 + I 0.9503 + I 0.9471 + 1

0.2957 + 1 0.2960 + t 0.9978 -I- t 0.9953 $1 0.9845 + t 0.9877 + 1

0.2982 + t 0.1008 + 2 0.9923 + 1

0.2987 + 1 0.1006+2 0.9969 + 1

0.2986 + 1 0.1009+2 0.9935 + 1

0.2993 + 1 0.1009+2 0.9988 + 1

Although the present sets of approximate theories can easily he applied to a moderately thick plate, higher orders of the expanded two-dimensional theories may be necessary to obtain reasonably accurate solutions for an extremely thick plate. As the most disadvantageous case, an extremely thick plate with the thickness parameter I/t = 0.5 is analyzed in

detail. In Table 2, the normal displacement at the central point on the middle plane, WC” (in-plane problem) or W(O) (out-of-plane problem) and the internal work Sz, and the external work & are shown in each case under several combinations of N and M. A numerical value 0.1234 x IO’, for example, is expressed as 0.1234 + 1 in this numerical table. For the present range of the thickness parameter, reasonably accurate displa~ment solutions are obtained by N = 4-8 and M = 2-5. It is also indicated that the magnitudes of the internal and external works are numerically equal with each other.

In the out-of-plane problem, Fig. 3 shows the normal displacement WC’) along t] = 0 on the middle plane for the specific value of M with respect to N = 4,6,8. It is noticed that the finite difference solutions are converged accurately enough within the present number of the mesh intervals. In the following, discussion is made only on the numerical results

Fig. 3. Distribution of W@) (q = 0) (convergence property due to N).

for iV = 8 which are considered to be sufficient with respect to the accuracy of the solutions.

Figure 4 shows the normal displacement W(O) along q = 0 on the middle plane for the specific value of the thickness parameter with respect to M = 1,2,3,4. It is noticed that the proper order of appro~mate theories may be estimated according to the level of the thickness parameter.

The convergence properties of the stress distributions in the thickness direction at the middle point of the fixed boundary (r = - 1, q = 0) are shown in Fig. 5 for the thickness parameter 1/t = 0.5. At this point, the distribution of the stresses is very

Fig. 4. Distribution of W(O) ((t = 0) (convergence property due to M).

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44% H. MA~UNAGA

complicated due to the influences of fixed boundaries and it is noticed that the convergence property of stress components is very poor. At points a short distance from the fixed boundaries, stress components are rapidly converged. It is also pointed out that the convergence property of solutions is rapid enough for a thin plate with larger values of the thickness parameter. Stress boundary conditions on the upper and lower surfaces along the fixed boundries are not necessarily satisfied, especially for S,, in the in-plane problem and for S, in the out-of- plane problem. This is because all the displa~m~nt components are prescribed to be zero along the fixed boundaries. So the stress components along the fixed boundaries are determined with the use of the equilibrium equations of a three-dimensional continuum in the present analysis. At interior points far from fixed boundaries, stress boundary conditions on the upper and lower surfaces can be satisfied exactly through the equilibrium equations (17) and (18).

7.3. Comparison with other solutions

An alternative verification of the present results can be made by a direct comparison with the three- dimensional elastic solutions in [ 121. For this purpose, stress and displacement distributions of a thick plate with fixed boundaries subjected to uniformly distributed load only on the upper surface are analyzed. The present solutions for this probIem are obtained by the sum of solutions of two cases of in-plane and out-of-plane problems of the same order of approximation.

Figure 6 shows a comparison of stress and displacement distributions at the specific points for the thickest plate with the thickness parameter l/t = 0.5. Poisson’s ratio is 0.2 only in the results of this figure. The solutions expressed by the open circles are due to the results of [12] and the solid lines show the present results for N = 8 and M = 5. It is clear that the distributions of stress and displacement components agree well with each other. The result corresponds to the thickest plate in the numerical

__ 41. 2. 3. 4, 5. 10. t 20.

Fig 7 W(O) ,-(r/t) relation. * . max

examples of the present analysis where the numerical accuracy is considered to be poorer for the two- dimensional plate theories. Therefore when thinner plates are analyzed, the lower-order approximate theories may be used with sufficient accuracy. The applicability and reliability of the present approximate theories are then confirmed.

7.4. Mechanical characteristics qf a thick plate with fixed boundaries

Some remarkable features in the mechanical characteristics of a thick plate with fixed boundaries may be summarized as follows:

(1) The maximum dejection on the middle plane Wzx. For out-of-plane problems, the maximum deflection on the middle plane is plotted in Fig. 7 with respect to the thickness parameter in comparison with the theoretical solutions of a thin plate. The present numerical solutions are obtained for N = 8 and M = 5 which are considered to be sufficient with respect to the accuracy of solutions. The thin plate solutions show the results of the Kirchhoff-Love theory and an approximate theory for M = 1 in the present paper which corresponds to the classical thin plate theory with the approximation of plane strains.

(2) Distribution of stress resultants in out-of-plane problems. The stress resultants N$?, N’,‘), N$, QF) and Qr) correspond to those in the classical plate theory M,, h4,, M_YY, QX and Q,, respectively. Other stress resultants in the present theories may be defined as higher-order stress resultants in eqn (5). Figure 8 shows the distributions of the dimensionless stress resultants along q = 0. The order of approximate theories will be determined according to the thickness parameter.

(3) Distribution of stress and d&placement components. The distribution of stress and displacement components of a thick plate with I/t = 0.5 along 9 = 0( - 1 ,< 5 < 0) is shown in Fig. 9 for two cases of in-plane and out-of-plane problems, separately. Since this is the case of the thickest plate treated in the present paper, a higher-order approximate theory for M = 5 may be required in order to assure the sufficient accuracy of the solutions. The numerical convergence of stress components is not so good in comparison with the other numerical examples of ComparativeIy thin plates, especially at the points on the fixed boundaries as shown in Fig. 5. However, at the interior domain of the plate far from the fixed boundaries, the numerical convergence property is exceedingly good for the present case. In compara- tively thin plates with l/t = 1.0, for instance the convergence property of solutions is sufficient for M = 5 and/or less. For displacement components and stress resultants, accurate solutions can be obtained beyond comparison with those for stress components under the same order of approximate theories.


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1.07 -.02so -.293 z.332 -331 -331 _.__-.327 -324 -3221 t: r

Fig. 9(I) Distribution of stress and displacement components (v = 0; l/t = 0.5). In-plane problem.

8. DJSCUSSIONS AND CONCLUSIONS correspond to transverse shear deformations and/or

Beyond the limits of applicability of the existing thickness changes.

thick plate theories, various orders of the expanded On the other hand, many investigations on a

approximate theories of a thick plate have been thick plate by numerical analyses, such as finite

applied to analyze a thick plate with fixed boundaries element methods and boundary element methods

subjected to uniformly distributed load on the have been done. It is very difficult to estimate the

surfaces. Within the range of linear problems, the error of solutions obtained by these numerical

governing equations can be divided into two types of methods and, therefore, the n~e~~l solutions must

in-plane and out-of-plane problems. Three-dimen- be verified by comparing with other reliable solutions.

sional stress and displacement distributions of a The present results for boundary-value problems of a

compratively thick plate have been obtained with a thick plate have been obtained with high accuracy

considerably high accuracy by two-dimensional and can be regarded as the benchmark data of the

higher-order approximate theories. problem.

The following conclusions may be drawn from the 2. An actuate solution for the distributions of

present analysis: displacements of an extremely thick plate has been obtained by solving the governing equations of

1. Reissner’s and other approximate theories of equilibrium and boundary conditions. Since stress

almost the same order may be applied to the analysis resultants can be easily calculated from the displace- of not so thick plates. However, for an extremely ments through a differential operation, the accuracy thick plate with complex distributions of stresses and of the results has been shown to be sufficient. displacements, two-dimensional plate theories are Although the stress components can be obtained required to introduce the higher-order effects which directly from the constitutive equations (7), these


Fig. 9.(H) Distribution of stress and displacement components (q = 0; l/t = 0.5). Out-of-plane problem.

stresses may not satisfy the stress boundary conditions on the upper and lower surfaces of a plate. In the present paper, therefore, the stress components have been determined by integrating the ~~lib~urn equations of a three-dimensional elastic continuum and satisfying the stress boundary conditions on the surfaces of a plate.

Higher-order finite difference operators with error terms of o(h’) are used to obtain accurate solutions within a smaller number of unknowns. An estimation on the approximate order of the governing equations may be concluded according to the thickness parameter. Within the range of the thickness parameters of the numerical examples in the present paper, following approximate orders of the governing equations may be required in order to obtain the solutions with sticient accuracies

i/r = 0.5 N 1 .o, M-3-4

r/t = 2.0 * 5.0, M = 2-3

r/t > 10.0, M = l-2,

where the lowest order of approximate theory for M = 1 is a modified theory with the approximation of plane stresses and it corresponds to the Kirchhoff-Love theory.

3. There is almost no data to be compared directly with the present results. Only the results in [12], which are obtained by the integral method for the governing equations of a three-dimensional theory, are useful for this purpose. The present two-dimensional approximate theories require a large value of M for thick plates with smaller values of the thickness parameter to ensure the numerical accuracy of the results. On the other hand, three-dimensional treat- ments of the problem may not be succesful in the cases of moderately thick plates with large values of the thickness parameter due to convergence difficulti~ in the numerical analyses. For an extremely thick plate with the thickness parameter l/t = 0.5, the stress distributions obtained indepen- dently by the different methods have agreed well with each other. It can be said that two-dimensional higher-order plate theories in the present paper are very useful for the analysis of a thick plate as extended theories of the classical thin plate theory.

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REFERENCES

E. Reissner, The effects of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, 69-77 (1945). R. D. Mindlin, Influences of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech. 18, 31-38 (1951).

g,

E. Reissner, On transverse bending of plates, including the effects of transverse shear deformation. Int. J. Solids Sfructures 11, 569-573 (1975). 10. K. H. Lo. R. M. Christensen and E. M. Wu, A high-order ‘theory of plate deformation, Part I: homo- 11, geneous plates. J. Appl. Mech. 44, 663-668 (1977). K. H. Lo, R. M Christensen and E. M. Wu. Stress solution determination for high order plate theory. Int. J. Soliak Structures 14, 655-662 (1978). 12. S. A. Ambartsumiyan, Theory of Anisotropic Plates (translated from Russian into Japanese by N. Kamiya and Ed. Y. Ohashi). Morikita, Tokyo (1975).

M. Lcvinson, An accurate, simple theory of the static and dynamics of elastic plates, Mech. Res. Commun. 7, 343-350 (1980). Y. Yokoo and H. Matsunaga, A general nonlinear theory of elastic shells. Int. J. Solids Structures 10, 261-274 (1974). H. Matsunaga, On the analysis of displacement and stress distributions of a thick elastic plate by two-dimensional higher order theory. Tram A.I.J. 367, 48-58 (1986) (in Japanese). R. W. Little, Elasticity, pp. 1099112. Prentice-Hall, Englewood Cliffs, NJ (1973). Y. Yokoo and H. Matsunaga, Higher order finite difference solutions to non-shallow hyperbolic paraboloidal shells. Comput. Struct. 4, 593-614 (1974). I. Shimada and H. Okamura, Stresses and deformations in thick rectangular slabs. Proc. Japan Sot. Civil Engineers 233, 13-23 (1975) (in Japanese).

the application of a two-dimensional higher-order theory for the analysis of a thick elastic plate

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