Applied Mathematics and Mechanics English Edition, Vol. 5 No. 3, Jun. 1984) Published by }~ST Press, .Wuhan, China
ENDING
OF CORNERSUPPORTED RECTANGULAR PLATE UNDER CONCENTRATED LOAD
Lin Pengcheng ~r )
(Fuzhou University, Fujian)
Received Oct. 17, 1982)
ABSTRACT
In this paper the solution for the bending of cornersup ported rectangular plate under concentrated load at any point (~/2, 7) of the middle line of the plate is given by means of a conception called modified simply supported edges and the me thod of superposition. Some numerical example is presented. The solution obtained by this method checks very nicely with what was obtained by G.T. Shih [3] by means of spline finite element method when ~ ~ d/2 This shows that this method of solution is satisfactory~
I. Introduction
In [I], Prof. Chang Fovan considered the bending of cornersupported rec tangular plate under distributed load, while in this paper, the solution for the bending of cornersupported rectangular plate under concentrated load at any point
(a/2,rl)
of the middle line
x=a/2
of the plate is discussed. For this reason the problem is tackled as follows: Within the bounds of rectangular plate, the derlection w must satisfy the partial differential equation
a,m a.w
a w
Pc) .xa/2,1Jn)
a~' + ~ a~'r~u +~u c=~ ~
in which O is flexural rigidity of the plate,
<} xa/z, u,)=<} xa/2)<~ u~)
6 is Dirac function. Furthermore, it must satisfy the boundary condition
o,o ,, o,:, .
1
Oy #,.a LOCi u~ay J,'. (1.2)
8iw . 8 w , ^
[ a va O~w ]
~u v~ .,., a~+ 2~)~ l:l=o 1.3)
The solution of this problem is given by means of a conception called modified simply supported edges and the method of superposition.
II.
The
ConSit1 tuent Parts of Superposition
ethod
* Communicated by Chang Fovan.
1410 Lin Pengcheng
(A) A rectangalar plate with four simply supported edges is subject to con centrated load at .any point (a/2,~) of the middle line
x=a 2
of plate. In this case, the bending surface is/5]:
Pc
v ~ smh ?77 sinh ~Y'(I + a~.cotha,.

~t~:/'coth a/~
y' w,
= /)zr s ,~.,,s.~ . mSsinhu,~
in which If
,cc:v
in the exp.
b b t a 2
mzb
.,=5 i and .tT'~ then the quantity y~ must be replaced by y , and the quantity ~ by b~ (1.1). Therefore,
'~ + (1
p)a,~cotha=
(1#)
ct~(b~7)
coth
a.,(bT1)
V~Jv'~ u ~.,,~,...  b sinh
.a,,, ( b rl)
b m~ gt~/T,~
9 sinha.~ sin ,7~ sin (2.2) (V,) vb = ~
o+(11~)a~cotha~_(l_~t ) rl
mI 3 ...
sinh a,.r/
b l/l:I
DZ?I'X
;inhaT sin 2 sin a (2.3)
Also the bending surface of the plate can be expressed by
sin
i~rr/
u Pb2
,, i%oshbqC b 2 2 b b __ . b ( 2~
o
which
fl+=i:za/b.
This equation of ber.,ding surface is satisfied for
x~a/2.
By
symmetry., we can compute the deflections and internal component as bending mom ents of the other half plate with this equation
2+ (1U)~'tanh~ _ ~V ...... 2 2
sin
ira? icty
(V.) ._.

2blZ~ cosh/4 b
sin
b (2.4)
,)
(b) Suppose the edges
u=O,
u=b
5f
the rectangular plate are simply sup ~orted edges, while the edges x=0, xa are modified sLmply supported edges, the deflections along two edges x=0 , x=a are:
(w). _ 0
 ~'
b, sin
irry
xa Ii b  in
this case, the bending surface of plate is:
2 i~rx Drx iz~y
(Vv) ~o = D l(lll) n2 V~ X mSb~ ~rtx
b 3 .~ a2 2 sin + (2.6) (,n' ) a m ~,... ~ i i ~+b:
D4(1  tL)~tr ~ \~ V~
mSb, mzx
(I%) ,0 = 
a ., ,..., a z z c~  (2.7)
+~ ~
m.I ~1..
.1
[ b~
Bending of Corner Supported Rectangular Plate 1411
coshfl~ 1 3 +
p
(I~)
.=:y:
,~b,
(~ ,.) ~i,,'~
2.8)
V,)=.,
:
D 2 
b .
sinhfl lu b
(C) Suppose the edge y=O of rectangular plate is the modified simply sup ported edge and the other three edges are simply supported edges, the deflections along edge
y=O
are:
t~),. 0
= ~' C.
sin 1 m.l~St...
In this case, the bending surface of plate is:
w~= c. cotha., sinh m~y 1t~ m~YsinhmaY
=,,
,.. 2 sinhZa= a ~ 2 a a
(2.9)
(V,) ,.0 =D (1 )'~ ~ m ~ 3*. a.. ) ,,,~
....... c,n(~fi~ ~zotham~si~ahzh~
sin
(2 10)
03
m.[ S . ~/
(V,),.~ =D (I#)'~
~, m%..l_.(
3+# +a.cotha.)sin m~x (2 ll)
2a
..~,s,...
stnna,~x ~~
a
(V.)..o:
2D
(1 _p)~z
Pc,.cosma i~y
s
~ ~ /b z
~z \. sin ~ (2.12)
,,,,  ,
m
L ~
+ ~  J
(D) Suppose the edge
y=b
of rectangular p]~ is the modified simply sup ported edge and the other three edges are simply :T~orted edges, the deflects along edge
y=~
are
:
In this case, the bending su~face of plate is:
III.
w~:' 1,u Y~'
sinha.L~ 1
Zp +
a~cotha.
) sinh mzy
rrury
coshrr~y ~
sin
mzx
2 m~ ...

a a a A
~
(2.13) (V,)~.0=D (1t~)2w
.m3a,. I
3+u .
mzx
mll$~, (]
(V')'=~
2
..... s ...
) a (2.15) ~,,s,... sinhZa= + otha.
sin
rnzx
2~_ V
a~ v~ iScosin izry
(V.)=..=D(1#)'
a ~ m.l s. m ~_~ CZ
i2)2
sin ~
2.16)
The Solution of the Problem of CornerSupported Rectangular Plate Bending by Means of Superposition Method
In order to satisfy the condition that the transverse forces along the free edge y=Oi must be equal to zero, by adding up the exps. (2.2), (2.6), (2.10), (2.14) and equating thblr sums to zero, we therefore obtain: _am_ {_3+~ +a.cotha,~ \ /
3+~
.. _ a. \_
8
a' V
b,
sinuam
1g \
1/~
sJnn
am
u o
~.~ 4a
_rex
,Lb 2
t]:2y
Pa z r
~(1~) ~,~ L 2 + (1 /~)a.cotha=
1412 Lin Peng cheng
(1~) a=(bTr/) r ]sinh a ~(b~) b

sinha= sin
rt~__
2
(m=l, 3,
5,
...)
3.11
In order to satisfy the condition that the transverse forces along the free edgey=b must be equal to zero, adding up the exps. (2.3), (2.7), (2,11), (2.15) and equating their sums to zero, we thus obtain:
a=( a,. q_ 3+~ cotha '~__c~( 35F/z_+a.cotha~,. ) sinhZa=
lt~ / smna=~ l~
8 a s
b~coshr
2Pa~
[2
+~b' c~1 (;: +'inzz)/= D(1#)2=Sn; +
(1
~ ~ ~ ~c~t h~
sinh a,.~,
 lp)olcoth b j slnha,,, sin ,~ m=1,3,5,...) 3.2)
In order to satisfy the transverse forces along the free edge
x a
to be equal to zero, by adding up the exps. 2.4), 2.8), 2.12), 2.16) and equating their sums to zero, we then obtain:
COS
~ \ 'n
am
f b z
i, \=
'll3t''* )T/
]2
+ m2
ml n ...
t ~
+,;,
a'a s coshfl,I
3+I' ,
x +4b s b, sinhfi,
( ..... fi Pa 1
li~
sinh
fie) = Db
4(1~)~'i ,
t4
fl,
+ (1 tz)t~ e anh
2
coshfl

sin b ~ (i=1,2,3,...)
(3.3)
2 By symmetry we can obtain just the same set of equations for edge x=0 as those for edge x=a . Hence, solving the above three sets of infinite sLmultaneous eqs. (3.1) (3.3), we can find a=, b,, c, and thus the deflections, internal force component as bending moments, etc. at any point of plate and it will be possible to compute con centrated reactive forces R at the corners of the plate according to the following
expressions: (R),.,2])(1a),.~( 8'w, { a ~ imTtanhfl'2
.~ exau
)~:~= (1 p) P._zb ,.,. si b :osh fl, ~os i=
2
mzu., D[ 1 +/~
(1/~)~r* .~Y
Pa z
~1~ c~ sin~~=a
)
rn. ,~ 9
,ZbjD
.
coshfle ~ l+i, fie /
+ (1.)='~ p~cos,~
~inh);
~i.  s~nh~ l
+(1a)n: V'
mZc D ( I+P
+a,cotha= )] (3.4)
,,.l''~,.. Pa=sinha,, ] v
tanh~t
{ ~b.
o 1 ~osh _p
R) =~o =  (1  u) P  sin~ ,~ + ( 1 
,,) ~' ~.
mtsv*.
2
m c~DX
1
+
Po' t ] ,u. cotho=