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A three-dimensional solution for simply supported thick rectangular plates
NUCLEAR ENGINEERING AND DESIGN 6 (1967) 155-162. NORTH-HOLLAND PUBLISHING COMP., AMSTERDAM
A THREE-DIMENSIONAL SOLUTION FOR SIMPLY SUPPORTED THICK RECTANGULAR PLATES * C. W. LEE Department o f Engineering Mechanics, University o f Tennessee, Knoxville, Tennessee, USA
Received 22 July 1967
A solution is obtained for simply supported rectangular plates based on a three-dimensional thick plate theory developed earlier by Donnell and the author. Sinusoidal and uniform loads are studied, and in the latter case a Levy type solution is reached. Four boundary conditions are satisfied at each edge. Numerical results for a square plate are compared to those from Reissner and classical plate theories.
1. INTRODUCTION A t h r e e - d i m e n s i o n a l thick plate theory was developed by Donnell [1] in 1954, dealing with n o r m a l loads applied on the top and bottom s u r f a c e s . The solution was given in infinite s e r i e s with the f i r s t t e r m s r e p r e s e n t i n g the c l a s s i c a l thin plate theory. A s i m i l a r solution was obtained in 1958 for t a n g e n tial loads [2], where the loads were divided as s y m m e t r i c and a n t i s y m m e t r i c p a r t s . The f i r s t t e r m s of the s e r i e s solution r e p r e s e n t e d the c l a s s i c a l plate theory for a n t i s y m m e t r i c loading, and the p l a n e s t r e s s solution for s y m m e t r i c loading. The governing d i f f e r e n t i a l equations for both n o r m a l and tangential loads were two f o u r t h - o r d e r equations s i m i l a r to that of the c l a s s i c a l plate theory. This i m p l i e d that four boundary conditions should be s a t i s f i e d at each edge, as in c o n t r a s t to two conditions in the c l a s s i c a l plate theory and t h r e e conditions in R e i s s n e r plate theory. In the p r e s e n t p a p e r the thick plate theory [1,2] is u s e d to investigate r e c t a n g u l a r plates with simply supported edges. Two types of loading a r e c o n s i d e r e d in the study, n a m e l y , the s i n u s o i d a l l y and the u n i f o r m l y d i s t r i b u t e d loads. In the case of u n i f o r m load, a Levy type solution is obtained. N u m e r i c a l r e s u l t s obtained for a s q u a r e plate a r e c o m p a r e d to those b a s e d on R e i s s n e r plate theory [3,4] and c l a s s i c a l plate theory [5].
2. BASICEQUATIONS FOR NORMAL LOADS The equations which a r e n e c e s s a r y for the p r e s e n t i n v e s t i g a t i o n will be r e p r o d u c e d here f r o m ref. [1]; m i n o r change of notation is made to conform m o r e closely with T i m o s h e n k o [5]. Let T(x, y) and B ( x , y) r e p r e s e n t the n o r m a l unit p r e s s u r e at top (Z =-½h) and bottom (Z =½h) s u r f a c e s , r e s p e c t i v e l y , and let D(x,y) = T-B
,
S(x,y) = T+B .
(1)
v4S=S,
(2)
The governing diffe:rential equations a r e : v4D
=D,
where V 2 i s the t w o - d i m e n s i o n a l Laplace operator. The s t r e s s r e s u l t a n t s a r e given as N x =-½vh v2S,yy
+ ...
,
N x y = ½vh v 2 S , x y
+ ...
,
Vx = - V 2 D , x
+ ...
,
* Presented at the 5th U.S. National Congress of Applied Mechanics, Minneapolis, Minnesota, 14-17 June 1966.
(3a)
156
C.W. LEE
Mx = - ( D , x x + v D , yy) + lo
v u,yy +~vh4D,yy
+ ... , (35)
Mxy = - ( 1 - v ) D , x y - ~gh2v2D,xy + 4~ooVh4 D,xy + . . . , where subscripts after comma indicate partial differentiation; the expressions for Ny, Vy, and My are obtained from Nx, Vx, and Mx, respectively, by interchanging x and y. The stress components Tij and displacement components u i are given as eqs. (2) and (4) in ref. [1]. Let u, v, and w represent respectively the values of Ux, Uy, and u z at the middle surface, then
u =-~-l+v (uv2S,x - ~ h 2 S , x + 1
.. .) ,
v =~-~-l+u ( v v 2 S , y - ~ h 2 S , y
8-3vh 2 227-157v i - u 40 V 2 D 1-v
w = ~ , (D
h4
67,20~D+''') '
+ ...)
, (4)
where K is the flexural rigidity, namely K = Eh3/12(1 - u2).
3. RECTANGULAR PLATE UNDER SINUSOIDAL LOAD Assume that the loading condition is given by
T(x,y) = p c o s ~ c o s ~ ,
(5)
B(x,y)=O,
where p is a constant and a, b are lengths of the sides (fig. 1). It is clear then
D(x,y)= S ( x , y ) = p cos ~ cos ~ .
(6)
The edge boundary conditions to be satisfied by the present theory are x =+½a:
w=0,
Mx=O ,
v=O,
Nx=O ;
y = +{b:
w =0,
My = 0 ,
u =0,
Ny = 0 .
-Ip /
~/
/
1~
,t
/
(7)
h
Z
1,__..
_ j L--2~_ ~ ~r-~-_.._.~
I
l ! /
,.i
J
b/2
r]
.1 ]
b/2
Jr-7
F--l
F-
qr-qr-q ~] J
a/2
a/2 Y
Fig. I. Rectangular plate under sinusoidal load.
X
S I M P L Y S U P P O R T E D THICK R E C T A N G U L A R P L A T E S
157
T h e s e b o u n d a r y conditions should be v e r y n e a r l y the a c t u a l ones to o c c u r for a p l a t e r e s t e d on s h o r t r o l l e r s along the e d g e s , a s i n d i c a t e d in fig. 1, The f i r s t two conditions a r e the ones u s e d in the c l a s s i c a l p l a t e t h e o r y for a s i m p l y s u p p o r t e d edge. A solution which s a t i s f i e s the d i f f e r e n t i a l equations (2) and the b o u n d a r y conditions (7) i s obtained a s
D
= S
-
b4p cos~ 7r4(1 + b2/a2) 2
c o s ~ -~
(8)
The s t r e s s e s , d i s p l a c e m e n t s , and s t r e s s r e s u l t a n t s a r e c o m p l e t e l y d e t e r m i n e d by s u b s t i t u t i n g eq. (8) into a p p r o p r i a t e e x p r e s s i o n s . A d i r e c t substitution, when r e d u c e d to a s q u a r e p l a t e , g i v e s the following e x p r e s s i o n s f o r the p r e s e n t t h e o r y :
""~X)x~ a
pa = - ~
try
cos a'
7r2v h 2 [ (Mx)ma x _ 1 + v pa 2 47r2 \1 + 5(1+v) a 2
7r4v
h4
1050(1+v) a 4 + " " j ' (9)
pa4 (1 + ~r2(8- 3v) h 2 7r4(227- 157v) h 4 Wma x - 4zf4K 2 0 ( 1 - v ) -a- ~ - 1 6 , 8 0 0 ( 1 - v ) a 4 + ' " " j ' (rxx)max -
- - p a 2v ) _ _ 3(1+ 27r2 h2
( 1 + 7r2 h 2 15a 2
717r4 h 4 + .) 3 1 5 0 a 4 "" '
w h e r e the f i r s t t e r m of each s e r i e s , except Vx, r e p r e s e n t s the c l a s s i c a l plate t h e o r y , and in the l a t t e r t h e o r y the s h e a r i n g f o r c e at an edge and the c o n c e n t r a t e d f o r c e at a c o r n e r a r e 3-v Try (Vx)x~ a = - ~ pa c o s -~- ,
1-u R = - - pa 2
(10)
2~2
The R e i s s n e r p l a t e t h e o r y h a s two governing d i f f e r e n t i a l equations [3, 5]: h22-vv2 K~4w = T - 1-0 1---:~v
T
V2~b_~-~ I0 q5 = 0
'
.
(11)
The edge b o u n d a r y conditions to be s a t i s f i e d by R e i s s n e r p l a t e t h e o r y a r e
x=+½a:
w=0,
Mx=O ,
t3y = 0 ;
y =+½a:
w =0,
My = 0 ,
fix = 0 .
(12)
T h e s e a r e e q u i v a l e n t to the f i r s t t h r e e conditions l i s t e d in eq. (7). The quantity fix i s defined a s ~w
12 l + v
~x---~+~
-
E---~ Vx '
and By i s defined s i m i l a r l y . The solution to the b o u n d a r y value p r o b l e m i s e a s i l y obtained a s
w
:=
b4p lr4(l+b2/a2)2K
+-10 1 v
1+
cos- h-cos b '
"
A d i r e c t c a l c u l a t i o n for a s q u a r e p l a t e , using a p p r o p r i a t e e x p r e s s i o n s of R e i s s n e r p l a t e t h e o r y [3,5], gives
---~¢V'3x-- a "':~"
=-
,
l+v 2/ 7r2 v h 2 - - ~ pa ~1 + 5 l+v
, (15)
lr2 2 - v h -~2) 2 4z4K \ 1 + 5 1 -v '
= pa4 [ Wmax
3 ( l + v ) p a 2 \[1 +~r2 v ( r x x ) m a x = 21r2 h2 5 l+v
J
158
C.W. LEE
Here again the first t e r m s , except the e x p r e s s i o n Vx, are the s a m e as c l a s s i c a l plate theory. It is interesting to compare the r e s u l t s between the present theory, eqs. (9), and R e i s s n e r plate theory, eqs. (15). The shearing force (Vx)x_--~a is identical in both t h e o r i e s , and i s independent of the plate thickness; it may be e a s i l y checked that there is no concentrated reaction R at the c o r n e r s of the plate, as s u g g e s t e d by the c l a s s i c a l plate theory. All other quantities calculated depend on the t h i c k n e s s ratio h / a . Since the s e r i e s in eqs. (9) converge very rapidly, the third t e r m i s s m a l l comparing to the second t e r m , and s i n c e for (Mx)ma x the first two t e r m s are the same for both t h e o r i e s , the n u m e r i c a l v a l u e s are p r a c t i c a l l y identical. For Wmax and (Txx)max on the other hand, s i n c e the second t e r m s are not the s a m e between the two t h e o r i e s , the differences are appreciable. For the n u m e r i c a l c o m p a r i s o n of a square plate, let pa 2 (~'xx)max = ~ ( 0 . 1 9 7 5 8 ~ ) .
Wmax = a ( 0 . 0 0 2 5 6 6 5 ~ ) ,
(16)
1.025 Present Theory
1.20, 1.020.
Reissner
Theory
Classical
i
Present
Theory
1.015-
I I Reissner Classical
Theory
Theory Theory
1.010
i.i.~
\
1.oo~
1.000
- -
i.i( i0
15
20
25 a/h
Fig. 2b. Sinusoidal load.
-[ b/2 1.05 X
b/2
A
1 ~
1.00
10
15
20 a/h
Fig. 2a. S i n u s o i d a l
load.
a
1-
Y
Fig. 3. R e c t a n g u l a r
plate under uniform
load.
SIMPLY SUPPORTED THICK RECTANGULAR PLATES
159
V a l u e s of a and fl for the p r e s e n t , R e i s s n e r , and c l a s s i c a l plate t h e o r i e s a r e shown in fig. 2. T h e s e v a l u e s a r e functions of the t h i c k n e s s r a t i o h / a , and u = 0.3 has been u s e d in the calculation.
4. R E C T A N G U L A R
PLATE
UNDER
UNIFORM
LOAD
Take the c o o r d i n a t e s y s t e m a s shown in fig. 3. The load condition is T(x,y) : p ,
B(x,y)
(17)
=0 .
It follows that
(18)
D(x , y ) = S(x , y ) = p .
The edge b o u n d a r y conditions to be s a t i s f i e d by the p r e s e n t t h e o r y a r e x=O,a: 1
x = ±~a:
w =0,
M x =0,
v =0,
Nx =0;
(19a)
w =0 ,
My:
u : O,
Ny = O.
(19b)
0 ,
It is noted that in view of eq. (18) a l l the s e r i e s e x p r e s s i o n s [1] b e c o m e c l o s e d f o r m . In e q s . (3) and (4) it i s o b s e r v e d that t:he q u a n t i t i e s w, Mx, and M y depend on D only, while the quantities u, v , N x , a n d N y depend on S only. To solve the b o u n d a r y value p r o b l e m , we c o n s t r u c t a Levy type solution in the f o r m D = ~p(x4-2ax3+a3x)+
~ (A m cosh tony + B m t o n y sinh %nY) sin mnYa m=l a T '
pa 4
(20)
S = ~ p ( x 4 - 2ax 3 +a3x) , A p p l i c a t i o n of F o u r i e r a n a l y s i s g i v e s [5] 4pa 4 ~ 1 S = ~m=l ~
mnx
sin--,a
(21)
w h e r e m = 1 , 3 , 5 , . . . and throughout the p a p e r only odd i n t e g e r s will be u s e d for rn. A s t r a i g h t f o r w a r d c a l c u l a t i o n b a s e d on eqs. (20) and (21) shows that d i f f e r e n t i a l e q u a t i o n s (2) and the four b o u n d a r y conditions (19a) a r e s a t i s f i e d with a r b i t r a r y v a l u e s of A m and B m . When the c a l c u l a t i o n of b o u n d a r y conditions (19b) i s c a r r i e d out, the f i r s t two conditions a r e s a t i s f i e d if we choose Brn - cosh a m
5
3-3,6-~
1+
40
n2m2 (22)
Am
-1 (w-~m 8 - 3u 1 h2 cosh a m 5 + 1 0 ( 1 - u ) 7r3m3 a 2
n2m2 8 - 3__uh~)
2 2 7 - 157u h 4 16,~--u)~)-
B i n ( a m tanh a m
20
1- u
w h e r e the p a r a m e t e r am = mvb/2a
.
(23)
The l a s t two conditions of (19b) a r e not s a t i s f i e d . A solution to the r e s i d u e p r o b l e m is to be d e t e r m i n e d . Let the solution of the p r o b l e m be defined as [S] = [S1] + [$2] ,
(24)
w h e r e the solution [!71] i s given by eqs. (20) and (22) in a c c o r d a n c e with r e f . [1]. The solution [$2] m u s t be d e t e r m i n e d such that when added to [$1] the sum w i l l s a t i s f y c o m p l e t e l y the g o v e r n i n g d i f f e r e n t i a l equations (2) and the b o u n d a r y conditions (19a) and (19b). In view of t h e s e r e q u i r e m e n t s , a solution [$2] m a y be o b t a i n e d f r o m r e f . [2]. Take p r o b l e m (B) of s y m m e t r i c loading [2], and let the loading be in the
160
C.W.
LEE
y - d i r e c t i o n , then eqs. (6') and (8') of ref. [2] become 2 r x x =~ @ ,yxx - tap,yyy) ,
2 -ryy = - ~ [~p,yyy + (2+v)~,yxx ] ,
"rzz ='rzx = r z y = 0 ,
2(l+v) 2 Eh ~k,yx ,
u =
2 rxy =-~ ( - ~ , x x x + taP,yy x) ,
2(l+v) v =- ~ [ ( 1 - v ) ~ , y y + 2 ~ k , x x ]
,
w =0,
(25) (26)
where the s t r e s s function ~ ( x , y ) is taken as a h a r m o n i c function (it is then also a b i h a r m o n i c function), v2~ = 0 .
(27)
The s t r e s s resultants may be obtained f r o m integration of s t r e s s e s given in eqs. (25), thus N x = 2@,yxx- t@,yxx) ,
Ny = -2 [qC,yyy + (2+v)qJ,yxx ] ,
M x = My = 0 .
(28)
To obtain a solution [$2] , we choose the h a r m o n i c function =Pa 4 ~ C m sinh tarry mux m =1 ~ sin
(29)
Then both the last two conditions in eqs. (19b) a r e satisfied if we take Cm =
1 h -. l + v v4m4 c o s h a m a v
(30)
It is readily checked that the solution [S] : [S1] + [$2] , where [S1] is given by eqs. (20) and (22) and [82] by eqs. (29) and (30), satisfies completely the differential equations (2) and the boundary conditions (19). Hence, the solution of the p r o b l e m is complete. A s t r a i g h t f o r w a r d calculation, when reduced to a square plate, gives the following e x p r e s s i o n s for the p r e s e n t theory: (Vx)x= 0 = R.T. +
~r3(227 157v) pa ~ 16,800 m=l
3 cosh ~ - -
1 +-~-
cosh
½1rm a 4 '
(Mx)ma x = R.T. + ,(227-157v)_~,80_O pa2 ~ (_1)½(m-1) 1 (l+40~m+10(l"v)y2m2tanh½m~ h4 m=l cosh ½m~ 40 + (8+v)~2m2(h2/a2) /-a ~ ' 1 Pa 4 ~ (_1)½(m-1) ~ l - s e c h ½ m u 8 - 3 v h 2 Wmax = C . T . + ~ ( l _ v ) K m=l m3 107r2 a 2
I
(1 -
W/ \
1 cosh 1
7)"
1 cosh
(31)
40(1- v)Tr2m2 tanh½rmr~ 227 - 157v h47 40 +(8+,),2m2(h2/a2)'
1- v h2 0"xx)max = C.T. + 7r ~P a2 m=l ~ ( - l ) ½ ( m - l ) I s - ~ m ( 1 - c o s h. . . .½rmr) a 2 227- 157u
2800
m_
cosh
1
4Ovum + 10(1-u)n2m 2 tanh½m~r) h 4
,( +
where the abbreviations R.T. and C.T. have the meaning of R e i s s n e r theory and c l a s s i c a l theory, r e spectively, and m = 1 , 3 , 5 , . . . only. The solution of the s a m e p r o b l e m b a s e d on R e i s s n e r plate theory was obtained by Salerno and Gold-
SIMPLY SUPPORTED THICK RECTANGULAR PLATES
161
b e r g [3] and Koeller and E s s e n b u r g [4]. The e x p r e s s i o n s listed here are taken f r o m ref. [3]. The solutions to the differential equations (11) for a simply supported rectangular plate under uniform load are 4pa 4 ~
w=+--~K,.,.,=1~
-
(
1+
1 [1
+
~ 2 m 2 2 - v h2 retry sinh m ~ y / a + __ 10 1 - v a 2 2a c o s h a m
~2m2 2 - v h 2 am ) coshmTty/a sinmUX --+ - - tanh a m -10 1 - v a 2 2 cosha m a '
(32)
~=0, and the following expressions a r e obtained for a square plate. (Vx)x__O 4pa ~ = ~-
1
(1 - cosh muy/a" I
~=1 ~
cosh ½~
] '
2v ~ 1 ½(m-l) 1 h2 (Mx)ma x = C.T. + ~ pa 2 (-) m =1 m cosh ½rmr a2 ' (33) 4 2pa Wmax = C . T . + l r 5 K
~ (_l)½(m-1)__1 ( 1 - s e c h ½ m r r } y 2 2 " v h 2 m=l m3 10 1 - v a 2 '
1 h2 1 2 v p a2 ~ (-1) ½(re'l) (Zxx)max = C.T. + - - - 51r h 2 m=l m cosh ½mTr a 2 ' where again C.T. stand for c l a s s i c a l theory and m = 1, 3, 5 , . . . only. The c o r r e s p o n d i n g solution f r o m c l a s s i c a l plate theory [5] is well known, but the pertinent e x p r e s sions a r e listed h e r e for the c o m p a r i s o n purpose; these are 4pa ~ 1 (VX)x=O = ~ m=l ~ ~-pa2 (Mx)max =--8-
Wmax
(1 _ c o s h
m~ry/a~ cosh ½mu /
m=l m cosh ½mu
anh
cosh mlry a
a sinh
(1 - 8 ~ (_1)½(m-1) 4+(1-v)m~tanh½mlr~ ~
m=l
m 3 cosh ½m~
/ '
(34)
= -4- pa 4 ~ ( _ l ) ~~ ( m -1) _ 1_ ( 1 - l+¼mutanh½mu~ u 5 K m=l m5 cosh ½mTr -] '
_ 3 pa 2 (1 8 ~ (.1)½(m-1) 4 + ( 1 - v ) m ~ t a n h ½ m ~ (rxx)max - 4 h 2 " ~ m=l m3 cosh ½mTr / where m = 1 , 3 , 5 , . . . only. A c o m p a r i s o n of eqs. (31) for the p r e s e n t theory and eqs. (33) for R e i s s n e r plate theory shows that in this case (Vx)x= 0 told (Mx)ma x a r e p r a c t i c a l l y identical for both theories (different only in t e r m s involving h4/a4), but there a r e appreciable differences in Wmax and (Txx)ma x. It may be shown without difficulty that for both plate theories no concentrated reactions will occur at the c o r n e r s of the plate. In the n u m e r i c a l c o m p a r i s o n for a square plate, v = 0.3 is used and the values of a and ~ are shown in fig. 4. In this cat~e Wmax = a (0.0040628 ~ ) ,
(rxx)max = fl (0.28734 h~--~) .
(35)
162
C.W. LEE i. 2C
-
-
Present
- -
Theory
Reissner Classical
Theory Theory
i.i.=
1.020 i.i{ Present T h e o r y Reissner
Theory
1.015.. Classical
Theor~
1.010 1.05
1.005
1.000-1.00
-
-
i0
15
20
5
i0
15
a/h
20
25 a/h
Fig. 4. Uniform load. It should be m e n t i o n e d that d u r i n g the c o m p u t a t i o n the f o l l o w i n g f o r m u l a e h a v e b e e n u s e d . F o r m = 1 , 3 , 5 , . 1 m = l m~
~-2 =8'
~ (_l)½(m-1)(1 1 m=l -m 'm3
~5) '
(~" ~-3 5 ~ 5 ) = 4'32 ' 1536 / "
(36)
REFERENCES [1] L. H. Donnell, A theory for thick plates, Proc. 2nd U.S. National Congress of Applied Mechanics (ASME, 1955} p. 369. [2] C. W. Lee and L. H. Donnell, A study of thick plates under tangential loads applied on the faces, Proc. 3rd U.S. National Congress of Applied Mechanics (ASME, 1958} p. 401. [3] V. L. Salerno and M. A. Goldberg. Effect of shear deformations on the bending of rectangular plates, J. Appl. Mech. Trans. ASME 27 (1960) 54. [4] R. C. Koeller and F. Essenburg, Shear deformation in rectangular plates, Proc. 4th U.S. National Congress of Applied Mechanics (ASME, 1962} p. 555. [5] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959).
A THREE-DIMENSIONAL SOLUTION FOR SIMPLY SUPPORTED THICK RECTANGULAR PLATES * C. W. LEE Department o f Engineering Mechanics, University o f Tennessee, Knoxville, Tennessee, USA
Received 22 July 1967
A solution is obtained for simply supported rectangular plates based on a three-dimensional thick plate theory developed earlier by Donnell and the author. Sinusoidal and uniform loads are studied, and in the latter case a Levy type solution is reached. Four boundary conditions are satisfied at each edge. Numerical results for a square plate are compared to those from Reissner and classical plate theories.
1. INTRODUCTION A t h r e e - d i m e n s i o n a l thick plate theory was developed by Donnell [1] in 1954, dealing with n o r m a l loads applied on the top and bottom s u r f a c e s . The solution was given in infinite s e r i e s with the f i r s t t e r m s r e p r e s e n t i n g the c l a s s i c a l thin plate theory. A s i m i l a r solution was obtained in 1958 for t a n g e n tial loads [2], where the loads were divided as s y m m e t r i c and a n t i s y m m e t r i c p a r t s . The f i r s t t e r m s of the s e r i e s solution r e p r e s e n t e d the c l a s s i c a l plate theory for a n t i s y m m e t r i c loading, and the p l a n e s t r e s s solution for s y m m e t r i c loading. The governing d i f f e r e n t i a l equations for both n o r m a l and tangential loads were two f o u r t h - o r d e r equations s i m i l a r to that of the c l a s s i c a l plate theory. This i m p l i e d that four boundary conditions should be s a t i s f i e d at each edge, as in c o n t r a s t to two conditions in the c l a s s i c a l plate theory and t h r e e conditions in R e i s s n e r plate theory. In the p r e s e n t p a p e r the thick plate theory [1,2] is u s e d to investigate r e c t a n g u l a r plates with simply supported edges. Two types of loading a r e c o n s i d e r e d in the study, n a m e l y , the s i n u s o i d a l l y and the u n i f o r m l y d i s t r i b u t e d loads. In the case of u n i f o r m load, a Levy type solution is obtained. N u m e r i c a l r e s u l t s obtained for a s q u a r e plate a r e c o m p a r e d to those b a s e d on R e i s s n e r plate theory [3,4] and c l a s s i c a l plate theory [5].
2. BASICEQUATIONS FOR NORMAL LOADS The equations which a r e n e c e s s a r y for the p r e s e n t i n v e s t i g a t i o n will be r e p r o d u c e d here f r o m ref. [1]; m i n o r change of notation is made to conform m o r e closely with T i m o s h e n k o [5]. Let T(x, y) and B ( x , y) r e p r e s e n t the n o r m a l unit p r e s s u r e at top (Z =-½h) and bottom (Z =½h) s u r f a c e s , r e s p e c t i v e l y , and let D(x,y) = T-B
,
S(x,y) = T+B .
(1)
v4S=S,
(2)
The governing diffe:rential equations a r e : v4D
=D,
where V 2 i s the t w o - d i m e n s i o n a l Laplace operator. The s t r e s s r e s u l t a n t s a r e given as N x =-½vh v2S,yy
+ ...
,
N x y = ½vh v 2 S , x y
+ ...
,
Vx = - V 2 D , x
+ ...
,
* Presented at the 5th U.S. National Congress of Applied Mechanics, Minneapolis, Minnesota, 14-17 June 1966.
(3a)
156
C.W. LEE
Mx = - ( D , x x + v D , yy) + lo
v u,yy +~vh4D,yy
+ ... , (35)
Mxy = - ( 1 - v ) D , x y - ~gh2v2D,xy + 4~ooVh4 D,xy + . . . , where subscripts after comma indicate partial differentiation; the expressions for Ny, Vy, and My are obtained from Nx, Vx, and Mx, respectively, by interchanging x and y. The stress components Tij and displacement components u i are given as eqs. (2) and (4) in ref. [1]. Let u, v, and w represent respectively the values of Ux, Uy, and u z at the middle surface, then
u =-~-l+v (uv2S,x - ~ h 2 S , x + 1
.. .) ,
v =~-~-l+u ( v v 2 S , y - ~ h 2 S , y
8-3vh 2 227-157v i - u 40 V 2 D 1-v
w = ~ , (D
h4
67,20~D+''') '
+ ...)
, (4)
where K is the flexural rigidity, namely K = Eh3/12(1 - u2).
3. RECTANGULAR PLATE UNDER SINUSOIDAL LOAD Assume that the loading condition is given by
T(x,y) = p c o s ~ c o s ~ ,
(5)
B(x,y)=O,
where p is a constant and a, b are lengths of the sides (fig. 1). It is clear then
D(x,y)= S ( x , y ) = p cos ~ cos ~ .
(6)
The edge boundary conditions to be satisfied by the present theory are x =+½a:
w=0,
Mx=O ,
v=O,
Nx=O ;
y = +{b:
w =0,
My = 0 ,
u =0,
Ny = 0 .
-Ip /
~/
/
1~
,t
/
(7)
h
Z
1,__..
_ j L--2~_ ~ ~r-~-_.._.~
I
l ! /
,.i
J
b/2
r]
.1 ]
b/2
Jr-7
F--l
F-
qr-qr-q ~] J
a/2
a/2 Y
Fig. I. Rectangular plate under sinusoidal load.
X
S I M P L Y S U P P O R T E D THICK R E C T A N G U L A R P L A T E S
157
T h e s e b o u n d a r y conditions should be v e r y n e a r l y the a c t u a l ones to o c c u r for a p l a t e r e s t e d on s h o r t r o l l e r s along the e d g e s , a s i n d i c a t e d in fig. 1, The f i r s t two conditions a r e the ones u s e d in the c l a s s i c a l p l a t e t h e o r y for a s i m p l y s u p p o r t e d edge. A solution which s a t i s f i e s the d i f f e r e n t i a l equations (2) and the b o u n d a r y conditions (7) i s obtained a s
D
= S
-
b4p cos~ 7r4(1 + b2/a2) 2
c o s ~ -~
(8)
The s t r e s s e s , d i s p l a c e m e n t s , and s t r e s s r e s u l t a n t s a r e c o m p l e t e l y d e t e r m i n e d by s u b s t i t u t i n g eq. (8) into a p p r o p r i a t e e x p r e s s i o n s . A d i r e c t substitution, when r e d u c e d to a s q u a r e p l a t e , g i v e s the following e x p r e s s i o n s f o r the p r e s e n t t h e o r y :
""~X)x~ a
pa = - ~
try
cos a'
7r2v h 2 [ (Mx)ma x _ 1 + v pa 2 47r2 \1 + 5(1+v) a 2
7r4v
h4
1050(1+v) a 4 + " " j ' (9)
pa4 (1 + ~r2(8- 3v) h 2 7r4(227- 157v) h 4 Wma x - 4zf4K 2 0 ( 1 - v ) -a- ~ - 1 6 , 8 0 0 ( 1 - v ) a 4 + ' " " j ' (rxx)max -
- - p a 2v ) _ _ 3(1+ 27r2 h2
( 1 + 7r2 h 2 15a 2
717r4 h 4 + .) 3 1 5 0 a 4 "" '
w h e r e the f i r s t t e r m of each s e r i e s , except Vx, r e p r e s e n t s the c l a s s i c a l plate t h e o r y , and in the l a t t e r t h e o r y the s h e a r i n g f o r c e at an edge and the c o n c e n t r a t e d f o r c e at a c o r n e r a r e 3-v Try (Vx)x~ a = - ~ pa c o s -~- ,
1-u R = - - pa 2
(10)
2~2
The R e i s s n e r p l a t e t h e o r y h a s two governing d i f f e r e n t i a l equations [3, 5]: h22-vv2 K~4w = T - 1-0 1---:~v
T
V2~b_~-~ I0 q5 = 0
'
.
(11)
The edge b o u n d a r y conditions to be s a t i s f i e d by R e i s s n e r p l a t e t h e o r y a r e
x=+½a:
w=0,
Mx=O ,
t3y = 0 ;
y =+½a:
w =0,
My = 0 ,
fix = 0 .
(12)
T h e s e a r e e q u i v a l e n t to the f i r s t t h r e e conditions l i s t e d in eq. (7). The quantity fix i s defined a s ~w
12 l + v
~x---~+~
-
E---~ Vx '
and By i s defined s i m i l a r l y . The solution to the b o u n d a r y value p r o b l e m i s e a s i l y obtained a s
w
:=
b4p lr4(l+b2/a2)2K
+-10 1 v
1+
cos- h-cos b '
"
A d i r e c t c a l c u l a t i o n for a s q u a r e p l a t e , using a p p r o p r i a t e e x p r e s s i o n s of R e i s s n e r p l a t e t h e o r y [3,5], gives
---~¢V'3x-- a "':~"
=-
,
l+v 2/ 7r2 v h 2 - - ~ pa ~1 + 5 l+v
, (15)
lr2 2 - v h -~2) 2 4z4K \ 1 + 5 1 -v '
= pa4 [ Wmax
3 ( l + v ) p a 2 \[1 +~r2 v ( r x x ) m a x = 21r2 h2 5 l+v
J
158
C.W. LEE
Here again the first t e r m s , except the e x p r e s s i o n Vx, are the s a m e as c l a s s i c a l plate theory. It is interesting to compare the r e s u l t s between the present theory, eqs. (9), and R e i s s n e r plate theory, eqs. (15). The shearing force (Vx)x_--~a is identical in both t h e o r i e s , and i s independent of the plate thickness; it may be e a s i l y checked that there is no concentrated reaction R at the c o r n e r s of the plate, as s u g g e s t e d by the c l a s s i c a l plate theory. All other quantities calculated depend on the t h i c k n e s s ratio h / a . Since the s e r i e s in eqs. (9) converge very rapidly, the third t e r m i s s m a l l comparing to the second t e r m , and s i n c e for (Mx)ma x the first two t e r m s are the same for both t h e o r i e s , the n u m e r i c a l v a l u e s are p r a c t i c a l l y identical. For Wmax and (Txx)max on the other hand, s i n c e the second t e r m s are not the s a m e between the two t h e o r i e s , the differences are appreciable. For the n u m e r i c a l c o m p a r i s o n of a square plate, let pa 2 (~'xx)max = ~ ( 0 . 1 9 7 5 8 ~ ) .
Wmax = a ( 0 . 0 0 2 5 6 6 5 ~ ) ,
(16)
1.025 Present Theory
1.20, 1.020.
Reissner
Theory
Classical
i
Present
Theory
1.015-
I I Reissner Classical
Theory
Theory Theory
1.010
i.i.~
\
1.oo~
1.000
- -
i.i( i0
15
20
25 a/h
Fig. 2b. Sinusoidal load.
-[ b/2 1.05 X
b/2
A
1 ~
1.00
10
15
20 a/h
Fig. 2a. S i n u s o i d a l
load.
a
1-
Y
Fig. 3. R e c t a n g u l a r
plate under uniform
load.
SIMPLY SUPPORTED THICK RECTANGULAR PLATES
159
V a l u e s of a and fl for the p r e s e n t , R e i s s n e r , and c l a s s i c a l plate t h e o r i e s a r e shown in fig. 2. T h e s e v a l u e s a r e functions of the t h i c k n e s s r a t i o h / a , and u = 0.3 has been u s e d in the calculation.
4. R E C T A N G U L A R
PLATE
UNDER
UNIFORM
LOAD
Take the c o o r d i n a t e s y s t e m a s shown in fig. 3. The load condition is T(x,y) : p ,
B(x,y)
(17)
=0 .
It follows that
(18)
D(x , y ) = S(x , y ) = p .
The edge b o u n d a r y conditions to be s a t i s f i e d by the p r e s e n t t h e o r y a r e x=O,a: 1
x = ±~a:
w =0,
M x =0,
v =0,
Nx =0;
(19a)
w =0 ,
My:
u : O,
Ny = O.
(19b)
0 ,
It is noted that in view of eq. (18) a l l the s e r i e s e x p r e s s i o n s [1] b e c o m e c l o s e d f o r m . In e q s . (3) and (4) it i s o b s e r v e d that t:he q u a n t i t i e s w, Mx, and M y depend on D only, while the quantities u, v , N x , a n d N y depend on S only. To solve the b o u n d a r y value p r o b l e m , we c o n s t r u c t a Levy type solution in the f o r m D = ~p(x4-2ax3+a3x)+
~ (A m cosh tony + B m t o n y sinh %nY) sin mnYa m=l a T '
pa 4
(20)
S = ~ p ( x 4 - 2ax 3 +a3x) , A p p l i c a t i o n of F o u r i e r a n a l y s i s g i v e s [5] 4pa 4 ~ 1 S = ~m=l ~
mnx
sin--,a
(21)
w h e r e m = 1 , 3 , 5 , . . . and throughout the p a p e r only odd i n t e g e r s will be u s e d for rn. A s t r a i g h t f o r w a r d c a l c u l a t i o n b a s e d on eqs. (20) and (21) shows that d i f f e r e n t i a l e q u a t i o n s (2) and the four b o u n d a r y conditions (19a) a r e s a t i s f i e d with a r b i t r a r y v a l u e s of A m and B m . When the c a l c u l a t i o n of b o u n d a r y conditions (19b) i s c a r r i e d out, the f i r s t two conditions a r e s a t i s f i e d if we choose Brn - cosh a m
5
3-3,6-~
1+
40
n2m2 (22)
Am
-1 (w-~m 8 - 3u 1 h2 cosh a m 5 + 1 0 ( 1 - u ) 7r3m3 a 2
n2m2 8 - 3__uh~)
2 2 7 - 157u h 4 16,~--u)~)-
B i n ( a m tanh a m
20
1- u
w h e r e the p a r a m e t e r am = mvb/2a
.
(23)
The l a s t two conditions of (19b) a r e not s a t i s f i e d . A solution to the r e s i d u e p r o b l e m is to be d e t e r m i n e d . Let the solution of the p r o b l e m be defined as [S] = [S1] + [$2] ,
(24)
w h e r e the solution [!71] i s given by eqs. (20) and (22) in a c c o r d a n c e with r e f . [1]. The solution [$2] m u s t be d e t e r m i n e d such that when added to [$1] the sum w i l l s a t i s f y c o m p l e t e l y the g o v e r n i n g d i f f e r e n t i a l equations (2) and the b o u n d a r y conditions (19a) and (19b). In view of t h e s e r e q u i r e m e n t s , a solution [$2] m a y be o b t a i n e d f r o m r e f . [2]. Take p r o b l e m (B) of s y m m e t r i c loading [2], and let the loading be in the
160
C.W.
LEE
y - d i r e c t i o n , then eqs. (6') and (8') of ref. [2] become 2 r x x =~ @ ,yxx - tap,yyy) ,
2 -ryy = - ~ [~p,yyy + (2+v)~,yxx ] ,
"rzz ='rzx = r z y = 0 ,
2(l+v) 2 Eh ~k,yx ,
u =
2 rxy =-~ ( - ~ , x x x + taP,yy x) ,
2(l+v) v =- ~ [ ( 1 - v ) ~ , y y + 2 ~ k , x x ]
,
w =0,
(25) (26)
where the s t r e s s function ~ ( x , y ) is taken as a h a r m o n i c function (it is then also a b i h a r m o n i c function), v2~ = 0 .
(27)
The s t r e s s resultants may be obtained f r o m integration of s t r e s s e s given in eqs. (25), thus N x = 2@,yxx- t@,yxx) ,
Ny = -2 [qC,yyy + (2+v)qJ,yxx ] ,
M x = My = 0 .
(28)
To obtain a solution [$2] , we choose the h a r m o n i c function =Pa 4 ~ C m sinh tarry mux m =1 ~ sin
(29)
Then both the last two conditions in eqs. (19b) a r e satisfied if we take Cm =
1 h -. l + v v4m4 c o s h a m a v
(30)
It is readily checked that the solution [S] : [S1] + [$2] , where [S1] is given by eqs. (20) and (22) and [82] by eqs. (29) and (30), satisfies completely the differential equations (2) and the boundary conditions (19). Hence, the solution of the p r o b l e m is complete. A s t r a i g h t f o r w a r d calculation, when reduced to a square plate, gives the following e x p r e s s i o n s for the p r e s e n t theory: (Vx)x= 0 = R.T. +
~r3(227 157v) pa ~ 16,800 m=l
3 cosh ~ - -
1 +-~-
cosh
½1rm a 4 '
(Mx)ma x = R.T. + ,(227-157v)_~,80_O pa2 ~ (_1)½(m-1) 1 (l+40~m+10(l"v)y2m2tanh½m~ h4 m=l cosh ½m~ 40 + (8+v)~2m2(h2/a2) /-a ~ ' 1 Pa 4 ~ (_1)½(m-1) ~ l - s e c h ½ m u 8 - 3 v h 2 Wmax = C . T . + ~ ( l _ v ) K m=l m3 107r2 a 2
I
(1 -
W/ \
1 cosh 1
7)"
1 cosh
(31)
40(1- v)Tr2m2 tanh½rmr~ 227 - 157v h47 40 +(8+,),2m2(h2/a2)'
1- v h2 0"xx)max = C.T. + 7r ~P a2 m=l ~ ( - l ) ½ ( m - l ) I s - ~ m ( 1 - c o s h. . . .½rmr) a 2 227- 157u
2800
m_
cosh
1
4Ovum + 10(1-u)n2m 2 tanh½m~r) h 4
,( +
where the abbreviations R.T. and C.T. have the meaning of R e i s s n e r theory and c l a s s i c a l theory, r e spectively, and m = 1 , 3 , 5 , . . . only. The solution of the s a m e p r o b l e m b a s e d on R e i s s n e r plate theory was obtained by Salerno and Gold-
SIMPLY SUPPORTED THICK RECTANGULAR PLATES
161
b e r g [3] and Koeller and E s s e n b u r g [4]. The e x p r e s s i o n s listed here are taken f r o m ref. [3]. The solutions to the differential equations (11) for a simply supported rectangular plate under uniform load are 4pa 4 ~
w=+--~K,.,.,=1~
-
(
1+
1 [1
+
~ 2 m 2 2 - v h2 retry sinh m ~ y / a + __ 10 1 - v a 2 2a c o s h a m
~2m2 2 - v h 2 am ) coshmTty/a sinmUX --+ - - tanh a m -10 1 - v a 2 2 cosha m a '
(32)
~=0, and the following expressions a r e obtained for a square plate. (Vx)x__O 4pa ~ = ~-
1
(1 - cosh muy/a" I
~=1 ~
cosh ½~
] '
2v ~ 1 ½(m-l) 1 h2 (Mx)ma x = C.T. + ~ pa 2 (-) m =1 m cosh ½rmr a2 ' (33) 4 2pa Wmax = C . T . + l r 5 K
~ (_l)½(m-1)__1 ( 1 - s e c h ½ m r r } y 2 2 " v h 2 m=l m3 10 1 - v a 2 '
1 h2 1 2 v p a2 ~ (-1) ½(re'l) (Zxx)max = C.T. + - - - 51r h 2 m=l m cosh ½mTr a 2 ' where again C.T. stand for c l a s s i c a l theory and m = 1, 3, 5 , . . . only. The c o r r e s p o n d i n g solution f r o m c l a s s i c a l plate theory [5] is well known, but the pertinent e x p r e s sions a r e listed h e r e for the c o m p a r i s o n purpose; these are 4pa ~ 1 (VX)x=O = ~ m=l ~ ~-pa2 (Mx)max =--8-
Wmax
(1 _ c o s h
m~ry/a~ cosh ½mu /
m=l m cosh ½mu
anh
cosh mlry a
a sinh
(1 - 8 ~ (_1)½(m-1) 4+(1-v)m~tanh½mlr~ ~
m=l
m 3 cosh ½m~
/ '
(34)
= -4- pa 4 ~ ( _ l ) ~~ ( m -1) _ 1_ ( 1 - l+¼mutanh½mu~ u 5 K m=l m5 cosh ½mTr -] '
_ 3 pa 2 (1 8 ~ (.1)½(m-1) 4 + ( 1 - v ) m ~ t a n h ½ m ~ (rxx)max - 4 h 2 " ~ m=l m3 cosh ½mTr / where m = 1 , 3 , 5 , . . . only. A c o m p a r i s o n of eqs. (31) for the p r e s e n t theory and eqs. (33) for R e i s s n e r plate theory shows that in this case (Vx)x= 0 told (Mx)ma x a r e p r a c t i c a l l y identical for both theories (different only in t e r m s involving h4/a4), but there a r e appreciable differences in Wmax and (Txx)ma x. It may be shown without difficulty that for both plate theories no concentrated reactions will occur at the c o r n e r s of the plate. In the n u m e r i c a l c o m p a r i s o n for a square plate, v = 0.3 is used and the values of a and ~ are shown in fig. 4. In this cat~e Wmax = a (0.0040628 ~ ) ,
(rxx)max = fl (0.28734 h~--~) .
(35)
162
C.W. LEE i. 2C
-
-
Present
- -
Theory
Reissner Classical
Theory Theory
i.i.=
1.020 i.i{ Present T h e o r y Reissner
Theory
1.015.. Classical
Theor~
1.010 1.05
1.005
1.000-1.00
-
-
i0
15
20
5
i0
15
a/h
20
25 a/h
Fig. 4. Uniform load. It should be m e n t i o n e d that d u r i n g the c o m p u t a t i o n the f o l l o w i n g f o r m u l a e h a v e b e e n u s e d . F o r m = 1 , 3 , 5 , . 1 m = l m~
~-2 =8'
~ (_l)½(m-1)(1 1 m=l -m 'm3
~5) '
(~" ~-3 5 ~ 5 ) = 4'32 ' 1536 / "
(36)
REFERENCES [1] L. H. Donnell, A theory for thick plates, Proc. 2nd U.S. National Congress of Applied Mechanics (ASME, 1955} p. 369. [2] C. W. Lee and L. H. Donnell, A study of thick plates under tangential loads applied on the faces, Proc. 3rd U.S. National Congress of Applied Mechanics (ASME, 1958} p. 401. [3] V. L. Salerno and M. A. Goldberg. Effect of shear deformations on the bending of rectangular plates, J. Appl. Mech. Trans. ASME 27 (1960) 54. [4] R. C. Koeller and F. Essenburg, Shear deformation in rectangular plates, Proc. 4th U.S. National Congress of Applied Mechanics (ASME, 1962} p. 555. [5] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959).