MECHANICS RESEARCH COMMUNICATIONS Vol. 17 (2), 111- 116, 1990. Printed in the USA. 0093-6413/90 $3.00 + .00 Copyright (c) 1990 Pergamon Press plc

AXISYMMETRIC BENDING OF THICK CIRCULAR PLATES

K.H.Lee, N.R.Senthilnathan and S.P.Lim Department of Mechanical and Production Engineering National University of Singapore, Singapore 0511.

(Received 15 August 1989; accepted for print 6 December 1989)

Introduction

This paper describes the axisymmetric bending of clamped thick circular plates under a central point load using a simple higher-order shear deformation theory of plates presented in [i] which assumes that the in-plane rotation tensor does not vary across the plate thickness. The theory has one variable ~ess than that given by Reddy [2] yet accounts for a cubic variation of the in-plane stresses and a parabolic variation of the transverse shear stresses with zero values at the free surfaces. It may be shown that the assumption in the theory is exact for cylindrical bending of rectangular plates and axisymmetric bending of circular plates. The governing equations can be uncoupled in the unknown variables and are consequently remarkably simple for obtaining exact solutions. In [i], the authors have shown that the present theory gives results comparable to those from Reissner's, Mindlin's and Reddy's theories for the bending of thick rectangular plates for various boundary conditions. In this study, maximum deflections and stresses from the present theory for the axlsymmetric bending of clamped circular plates subjected to a central point load are compared with those from Reissner's theory. It is shown, in particular, that the present theory leads to a finite central deflection for this problem in contrast to the unrealistic infinite values from Reissner's and Mindlin's theories.

Exact solution for a clamped plate under a central point load

The governing equations of the simple higher-order shear deformation

theory derived in [i] for the bending of isotropic plates are shown to

be given by the following uncoupled form

III

112 K.H. LEE, N.R. SENTHILNATHAN and S.P. LIM

V4w k ~ qa4 /D

I 4(l-u) - - V4w s V2w s = qa4/D (i)

105 (h/a) z

where V 2 = [az/op z + (I/p)a/Op] with p = r/a, D = Eh3/12(l-v2), h is the

plate thickness, a is the plate radius and q is the transverse load. It

k wb+wS/5 where w b is that part of the is understood here that w =

transverse displacement w the derivative of which is numerically equal

to the rotation of the cross-section at the midplane, and w s is the

remaining part of the transverse displacement w due to the effects of

transverse shear of the cross-section.

The equations (i) can be solved for the axisymmetric bending of

thick circular plates subjected to a central point load by setting q - 0

k W s and taking w and to be of the form

k B p21np 2 w = B11n p + + B3P + B 4

w s = CiIo(~p) + CzKo(~p) + C31np'+ C 4 (2)

2 where ~ = - - -- - - -[420(l-v)(a/h) 2] and I and K are the modified Bessel

0 0

functions of the f i r s t and second k ind , r e s p e c t i v e l y .

Boundary conditions

dw k dw s k w s At p=l : w = 0

dp dp

dw k dw" d i Pa z At p~0 : 0 , --(V2w k)

dp dp dp 2~p D

d 105 Pa 2 and--(V z- ~Z)w"

dp 2~p D (3)

where P is the central point load. It is evident that dwb/dp-O at p - 1 in

the above boundary conditions is equivalent to specifying the rotation

of the normal to the midplane (at z = O) to be zero at the clamped edge.

It should be noted that the last two conditions at p~O have been

obtained by setting the effective shear forces ~ and V s (as defined in

[I]) to be equal to -P/2~r and expressing them in terms of w k and w s.

Using equations (2) in equations (3), it is obtained that

I Pa 2 w ~ 2 p 2 1 n p + ( 1 - p 2) - -

16~ D

BENDING OF THICK PLATES 113

] w" - (~p) + Inp -- (4) 2 s 3 i 1 ( ~ ) 2 s z 0 2 s ~ 3 D

w h e r e t e r m s i n v o l v i n g Io(m)/Ii(m), K 0 ( m ) a n d K l ( m ) f o r l a r g e v a l u e s o f

have been appropriately omitted for the sake of brevity.

The total deflection at any point in the plate is then given by k

w -- w +(4/5)w' so that the maximum deflection can be written as

1 in(2/~)-7 Pa 2

max 16s 10s(1-v) D

where 7 is the Euler's constant, 0.577 2157 .... The maximum radial

stress at z - h/2 can be obtained using

-Eh 1 d2w k v dw k 1 dZw" v dw" ' [;[ z] [ ]] o a + -- + - - - - + - - - ( 6 )

r (l.v2) dp 2 p 15 dp 2 p dp

S~stituting for ~ and w" gives the non-dimensionalized radial stress

[ar hz] 3 0"6523(h/a) (h/a)2 (7)

P r -a 2s ~ (1 -v ) lOs

The transverse shear stress distribution at the edge is obtained using

the relevant equilibrium equation of elasticity as

-4 -2 ah ~2 1 6 z z 1 630

P 2 8 s 3 I0 240 s (8)

where z - z/h and terms of order (h/a) and higher are neglected. It is

worth noting that the maximum value of the non-dimensionalized

transverse shear stress does not occur at the midplane but occurs near

the free surface. This qualitative observation is in agreement with the

clamped end shear stress distribution obtained from an elasticity

solution [3] for the bending of clamped circular plates under uniform

load. By comparing with the solutions from CPT, it is interesting to

note that the present shear correction for the maximum

non-dimensionalized radial stress at the edge is of the order of h/a and

that for the non-dlmensionalized maximum transverse shear stress is

independent of the h/a ratio!. This means that the transverse shear

stress from the CPT at the edge of a clamped plate in error both

qualitatively and quantitatively even for thin plates with this type of

clamping condition. A similar observation can also be made from the work

of lyengar et al [4] on clamped circular plates under uniformly

distributed load where it was shown that the distribution of the

1 14 K.H. LEE, N.R. SENTHILNATHAN and S.P. LIM

transverse shear stress at the clamped edge using a higher-order theory

departs considerably from the classical parabolic distribution for a

range of h/a values. The transverse shear stress at the clamped edge was

also found to be a maximum near the free surface with a value nearly

independent of the h/a ratio.

It can further be noted that the integration through the thickness

of the first term in the right hand side of equation (8) gives the edge

reaction due to the transverse load whereas the integration of the

second term through the thickness gives a zero value as expected. The

self-equilibrating traction given by the second term is due to the

effect of the warping of the normal to the mid-plane. It should,

however, be noted 'that the 'clamping' conditions (3) while satisfying

the clamping conditions required by elasticity theory (u - w - 0 for all

z) have also implicitly satisfied aw/ar - 0 at the clamped end. This can

be avoided by assuming that the 'clamping' conditions are given by

@u/az - 0 at z - 0 with an a priori assumption of a parabolic

distribution of the transverse shear stress at the clamped end. The

boundary conditions for such a clamping condition can be shown to be

given by

At p=l : wk -- W s = 0,

dw k (h/a) 2 pa 2 dw s (h/a) 2 pa 2

and ( 9 ) dp 40~(l-v) D dp 8~(l-v) D

The conditions at p=0 are the same as those given in equations (3). Here

again, it is clear that dwb/dp-O at p=l. Substitution of equations (2)

in the boundary conditions (9) gives

I (h/a) 2 Pa 2

I] w = -- 2palnp + I + --- (l-p a) -- 16~ 5(l-v) D

2~ 2 0(~p) + inp -- (i0) D

where terms involving Io(~)/Ii(~), Ko(~ ) and KI(~ ) have also been

appropriately omitted for the sake of brevity.

The maximum deflection can then be sho~ to be

I (h/a) 2 In(2/~)-7 Pa z

max 16~ 80~(l-w) 10~(l-v) D

where 7 is the Euler's constant, 0.577 2157 .... The non-dimensionalized

maximum radial stress at the clamped edge is given by

BENDING OF THICK PLATES 115

a2 ] 3 (l+5w)(h/a) 2 _ __ +

P Jr-a 2~ 20~(l-v) (12)

and the non-dimensionalized maximum transverse shear stress remains as

-0.23873 as implied in the boundary conditions at the clamped edge.

Numerical results and discussion

The maximum deflection and stresses from the present theory are compared

with those from the CPT, Mindlin's and Reissner's plate theories in

Table i for an isotropic (v-0.3) thick (a/h-5) circular plate. It is

observed that the effect of shear deformation is equally pronounced for

both the maximum deflection and stresses. It is further noted that the

type of clamping condition has a negligible effect on the maximum

deflection but has a significant effect on the maximum stresses. The

clamped end stresses from the first set of boundary conditions (3) are

significantly higher than those from the second set of boundary

conditions (9). It is interesting to note that while the present theory

gives a finite deflection at the centre of the plate for both types of

clamping conditions, the deflection expression found in [5] shows

clearly that Reissner's theory predicts an infinite value at the centre.

One possible reason for this is that the present theory allows the

deflection gradient to be specified as zero whereas both Reissner's and

Mindlln's theories have no such provision. Furthermore, the finite

deflection behaviour at the centre of the plate from the present theory

is certainly more realistic and is in agreement with that obtained by an

exact elasticity solution [6]. For the first set of clamping

conditions (3), the CPT, Mindlin's and Reissner's theories give values

for the maximum radial stress at the clamped edge which are lower by

about 25% in comparison to that from the present theory. The

corresponding discrepancy in the numerical value of the transverse shear

stress is about 69%. The maximum value of the transverse shear stress

occurs at z - ±0.38h in the present theory whereas it occurs at the

mid-plane for the other theories. The stress values from the second set

of clamping conditions (9) are nearly the same as those given by the

Reissner-Mindlin theories due to the a priori assumption of a parabolic

distribution of the transverse shear stress at the clamped end.

Finally, it should be noted that for axisymmetric bending, the

number of variables is the same for both the present higher-order theory

and Reddy's [2] higher-order theory and indeed, the results for

1 16 K.H. LEE, N.R. SENTHILNATHAN and S.P. LIM

axisymmetric problems would be identical for both theories. However, it

is a more difficult task to obtain exact solutions to the equilibrium

equations of Reddy's theory.

TABLE I

Comparison of maximum deflection and stresses for v-0.3 and a/h-5.

wD/Pa 2 a hz/P ~ ah/P r ~:z

r~O r-a r-a

CPT 0.01989 -0.47746 -0.23873

Reissner I ~ -0.47746 -0.23873

Mindlin z ~ -0.47746 -0.23873

Present 3 0.0.2778 -0.63338 -0.767195

Present 4 0.02801 -0.47518 -0.23873

ICalculated from the expressions given in [5]

2Calculated from the expressions given in [7]

3Results for the first set of clamping conditions (3).

4Results for the second set of clamping conditions (9).

5Calculated using the equilibrium equation of elasticity and

the maximum value occurs at z = ±0.38h (maximum value occurs

at z = 0 for the other theories).

Conclusions

The governing equations of the simple higher-order theory presented here

are shown to be one of the simplest in obtaining exact solutions for the

axisymmetric bending of thick circular plates. It is found that the

results from the present theory for the bending of clamped circular

plates under a central point load agree better with the exact elasticity

solution than those from the Reissner-Mindlin theories.

References

i.S.P.Lim, K.H.Lee and N.R.Senthilnathan, Comput. Struct. 30, 945 (1988)

2. J.N.Reddy, J. Appl. Mech. 51, 745 (1984) 3. K.T.Sundara Raja lyengar, K.Chandrashekhara and V.K.Sebastian, Nucl.

Engng. Des. 36, 341 (1976) 4. K.T.Sundara Raja lyengar, K.Chandrashekhara and V.K.Sebastian, Proc.

2nd Int. Conf. on Struct. Mech. in React. Tech., Berlin V-Part M, Paper M5/3, i (1973)

5. T.J.R.Hughes, R.L.Taylor and W.Kanoknukulachai, Int. J. Num. Meth. Engrg. Ii, 1529 (1977)

6. S.Timoshenko and S.Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, Tokyo (1959)

7. G.J.Perakatte and T.F.Lehnhoff, J. Appl. Mech. 38, 1036 (1971)