Applied Mathematics and Mechanics (English Edition, Vol.7, No.l, Jan. 1986) Published by SUT, Shanghai, China APPROXIMATE SOLUTION FOR BENDING OF RECTANGULAR PLATES Kantorovich-Galerkin's Method WangLei(ZE i~) LiJia-bao(~) (Hunan University, Changsha) (Received Jan. 3, 1981 Communicated by Chien Wei-zang) Abstract This paper derives the cubic spline beam function from the generalized beam differential equation and obtains the solution of the discontinuous polynomial under concentrated loads, concentrated moment and uniform distributed by using delta function. By means of Kantorovich method of the partial differential equation of elastic plates which is transformed by the generalized function ( 6 function and ~r function), whether concentrated load, concentratedmoment, uniform distributed load or smail-square Ioad can be shown as the discontinuous polynomial deformed curve in the x-direction and the y- direction. We change the partial differential equation into the ordinary equation by using Kantorovich method and then obtain a good approximate solution by using Glerkin's method. In this paper there 'are more calculation examples involving elastic plates with various boundary-conditions, various loads and various section plates, and the classical differential problems such as cantilever plates are shown. I. Introduction Kantorovich. Kralove presented Kantorovich approximate variational approach to deal with the functional variation of multivariable functions. In our country Chien, W. Z. has noticed very much Kantorovich-method. In his work ~ the theory is described in detail and rich calculation examples are contained. The calculation examples of elastic plates include such boundary conditions as four clamped edges, three clamped edges and one edge free, or three edges free and another clamped. But the loads are limited to uniform distributed load and the beam function simply satisfies such conditions as two clamped ends or one end clamped and the other free. Hence, the application range is smaller. This paper derives the cubic spline function starting with the generalized beam differential equation and it is possible to obtain the solution of the discontinuous polynomial under uniform distributed load, abrupt load, concentrated load and concentrated bending moment by using 6 function and ~ function. This spline function is used as the beam function in the directions of x and y to deal with the functional variation problems of two varibles functions, which changes a partial differential equation into an ordinary differential equation, next, the approximate solution can be obtained by using Galerkin's method. In order to deal with the term of load of a partial differential equation, this paper derives the intergral formulas of product of ~ function and ~ function times any continuous functionJ(x), therefore, the 87
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Applied Mathematics and Mechanics (English Edition, Vol.7, No.l, Jan. 1986)
Published by SUT, Shanghai, China
A P P R O X I M A T E S O L U T I O N FOR B E N D I N G OF R E C T A N G U L A R P L A T E S
Kantorovich-Galerkin 's M e t h o d
WangLei(ZE i~) L i J i a - b a o ( ~ )
(Hunan University, Changsha)
(Received Jan. 3, 1981 Communicated by Chien Wei-zang)
Abstract This paper derives the cubic spline beam function from the generalized beam
differential equation and obtains the solution of the discontinuous polynomial under
concentrated loads, concentrated moment and uniform distributed by using delta function.
By means of Kantorovich method of the partial differential equation of elastic plates which
is transformed by the generalized function ( 6 function and ~r function), whether
concentrated load, concentratedmoment, uniform distributed load or smail-square Ioad can
be shown as the discontinuous polynomial deformed curve in the x-direction and the y-
direction. We change the partial differential equation into the ordinary equation by using
Kantorovich method and then obtain a good approximate solution by using Glerkin's
method. In this paper there 'are more calculation examples involving elastic plates with
various boundary-conditions, various loads and various section plates, and the classical
differential problems such as cantilever plates are shown.
I. Introduct ion
Kantorovich. Kralove presented Kantorovich approximate variational approach to deal with the functional variation of multivariable functions. In our country Chien, W. Z. has noticed very much Kantorovich-method. In his work ~ the theory is described in detail and rich calculation examples are contained. The calculation examples of elastic plates include such boundary
conditions as four clamped edges, three clamped edges and one edge free, or three edges free and another clamped. But the loads are limited to uniform distributed load and the beam function simply satisfies such conditions as two clamped ends or one end clamped and the other free. Hence, the application range is smaller. This paper derives the cubic spline function starting with the generalized beam differential equation and it is possible to obtain the solution of the discontinuous
polynomial under uniform distributed load, abrupt load, concentrated load and concentrated bending moment by using 6 function and ~ function. This spline function is used as the
beam function in the directions of x and y to deal with the functional variation problems of two varibles functions, which changes a partial differential equation into an ordinary differential equation, next, the approximate solution can be obtained by using Galerkin's method. In order to deal with the term of load of a partial differential equation, this paper derives the intergral formulas
of product of ~ function and ~ function times any continuous functionJ(x), therefore, the
87
88 Wang Lei and Li Jia-bao
approximate solution of bending of rectangular plates with any boundary condition under uniform distributed load, small-square load, concentrated load, concentrated moment and line load can be
obtained.
Hu Hal-chart t21 has introduced Kantorovich method and defined the ordinary differential equation system of the finite strip to solve the plane-stress problems by means of Kantorovich
method. His work is new as compared with the solution of Y. K. Cheung's finite strip method. Because of the importance of Kantorovich method in mechanics, scholars of the world have
been paying attention to it's development and various practice applications. So it is significant for this paper to make further development. The relatively rich calculation examples in this paper can
be taken as the supplement to Kantorovich method.
II . D e r i v a t i o n o f S p l i n e F u n c t i o n
Let us start with the generalized beam differential equation.
N - I
E l - d4w d x ~ - ~ P , 6 ( x - - x , ) (2.1)
i - I
the difference of this statement from ordinary beams lies in the term of right-hand side. Where P~ is the concentrated load exerted on the beam (for i= 1, 2 ..... N-l), 6(x--x~) is delta function.
Integrating (2.1) once gives:
N - 1
E I d3w -~ d x 3 = c 3 + 5~ P , c s ( x - - x , ) (2.2) i - I
statement (2.2) has a definite mechanical meaning, that is, the shear chart has sudden change.
cr(:c--x~) is sigma function or step function. Integrating (2.1) twice gives:
N - I
F.I d2w.l_ --c:~+cj + V' p,(x--x~) t (2 3)
Integrating (2.1) three times gives:
E I ~ = c t 4-c~.x +c3x ~ (9 t)
Integrating (2.1) four times gives:
�9 N . . 1 x ~ x ~ . \ , ( x - x ~ )
L l w . = C o + C ~ X + C ~ - - + c 3 - - ~ i~ " " 2! 3! ~ 3T i - I
(2.5)
For the integral of the fight hand side of eq. (2.1), we introduce the concept of "discontinuous
polynomial." For any positive integral number k defines.
j ( x - - x , ) ' ( for xL~O) ( x - x , ) ~_
0 ( for x < 0 )
For k = O, we define:
Approximate Solution for Bending of Rectangular Plates 89
0 ( for x<.v , )
l (X--Xl)O=cr(x--xl) = f ( for x = x , )
t ( for x'>.'.~ )
For simplisity we only give the figure for xi = xl shown as Fig. 1
(x-xO~_ ]tO
i [~/2
(x-x,);
Y . X l " I
t----x,_--__4
Fig. I Fig. 2
obviously the discontinuous polynomial is a step function.
For k = I define
.f 0 ( for x~xt) (x-x~)l+
(x--xt) 1 ( for x~>xl)
Fig. 3
shown as Fig. 2
For k = 2 define
0 ( for x<~xl) (.v-x~)-' = {
(x--xl) ~ ( for x~>xl)
shown as Fig. 3
For k = 3 define
0 ( for x~xl ) (
( x - x l ) ~ = k (x--x,)S ( for x>xl)
Where, " + " is called discontinuous sign and above four statements are known as half-
discontinuous polynomial or interrupt polynomial. The figures of (x- - x~ ) h. (for k = 0, 1, 2, 3) are shown as Fig. 1, 2, 3, 4. This kind of discontinuous polynomial is the important component of the
spline function, which can be represented by the unit step function, namely
tO (x-x,) ?
- ~.___:/e . . . . . ~_. t/e__ -4 -~r �9 .{
Fig. 4 Fig. 5
( x - x O ; = x~(x-xl)~
For concentrated moment, the generalized differential equation is to be written:
90 Wang Lei and Li Jia-bao
E I m N - 1
d'w -- \~ M~d'(x--x~) d x 4 z..,.a i - I
(2.6)
the first integration gives:
N - I
E I daw X_~ d-3-U-- = c ~ + M,c3(x-x,)
i - I
(2.7)
the second integration gives: N - 1
d:"w t 2 I ~ = c : + c j + \,..~ M , a ( x - - x , ) i - I
where a(x--x~) is just (x--x~) ~ it has definite mechanical meaning, that is, the moment chart has a sudden change. the third integration of eq. (2.6) gives:
( 2 . 8 )
E l =c~+c~x+c3- - -~+ ~ M,(x- -a ' , ) ' INI
(2.9)
and the fourth integration of eq. (2.6) gives:
x 2 ,r ' " ( x - x , ) : lilW=Co + q X + C 2 ~ c 3 \ ' - -~C+ ~ M, (2.10) �9 ,-t 21
Compare (2.6) with (2.7), we observe that the right term in statement (2.7) decreases one power.
Let us take an example to illustrate the generalized beam differential equation.
Try to derive the deform curve of beam. Suppose a simply supported beam (Fig. 5) is supported by a concentrated bending moment M
at the central point. From statement (2.5), using the boundary conditions
dZw dZw x = 0 w=O dx z =0 ; x = l w = 0 dx z = 0
We can find four integral constants as follows
-•4 Mo co=O cz=O c1= Mol e3-- I
thus obtain the deflection curve:
E I w - - - - - ~ ( lZx--4x 3) ( O~ x ~ l )
It is the same as the result given in ~ Strench of Materials ~ (P. 640) by Timoshenko.
Here is another case. A beam with the left end simply supported and the right end clamped beam a concentrated
bending moment M and a concentrated load P. The generalized beam differential equation is
written as
Approximate Solution for Bending of Rectangular Plates 91
% 1 ~ _ _ ..t/3_____f . . . . t/3 _ .. _f._.
Fig. 6
- f f = l J
t / ~- ----d ~ L ~ U T ' T T ~ ~ _ _ _ - ~ A - relative displacement
Fig. 7 ~g. 8
El- d'Wdx, = M ( x - 3/-- ) +P3. ( x - 2 I ) ( 2 . 1 1 )
After integration yields:
~? x~ +__~( ( I = P .3
For beam with two free ends the statement of deflection is to be derived by means of fifth power
spline function as well as by ordinary means:
Coasidering the following polynomial: x6/l 0 - 3xS/ l~+ 5x'/214 which has satisfied the
conditions that both the bending moment and the shear are equal to zero at two ends.
In the calculation of elastic plates, the beam function must satisfy the condition that the vertical
relative displacement is one i.e. A = 1 and the rotating angle is zero at the central point shown as Fig. 7.
Hence, the beam function is written in this case:
X e X 5 5 ~4 - ~ + B ~=F- 3V+-~ F -+ A
with
a l to 1 I X s X 4 ~3 --d~ = T t 6 ~ - - 1 5 F + 1 o~- + - / / )
For x = 1]2, w = 1 (relative displacement at central point)
For x = i/2, dw/dx = 0, A = - i/2, B = 75/64.
For such condition as one end simply supported and the other free, shown as Fig. 8 the beam
function is written as:
x s 10 x ' 10 -~:~ x w = l~ 3 I' + 3 13 + l
which satisfies the conditions that both the deflection and the bending moment at the left end are
zero and the bending moment and shear at the right end are zero. In addition the condition that the displacement at the right end is two is satisfied.
Above two sorts of static-beam are unequilibrum. Hence, it is impossible to find the absolute
displacement. Thus, we have to call the displacement to equal any constant which is called the
relative displacement. However, the concept of relative displaoement is different from that in the
structure mechanics. In the calculation of bending plates, we need only to learn the beam function
(which is relative displac~aent) in the x-direction and the y-direction, and then we can obtain the
absolute displacement by means of the variational approach. Integrals of generalized function $ (x - -x , ) and integrating by Parts.
92 Wang Lei and Li Jia-bao
thus, integrating 5( .~ ..... -, ) following equations
Integrating (3(x--.r~) once we obtain:
[a(x--x,)dx= ( x - x , ) 0
Integrating once again and not considering the integral constant we obtain:
i ~ ( x - - x , ) d x d x = ( x - - x , ) }
k + 1 times and ignoring all integral constants, we obtain the
If 0~.r~,(a , we can prove:
[~ o
,~(-v--.':l)dx=~r(:,--xl) = c r ( a ) - - a ( 0 ) =1 �9 0 0
The formula of integral by parts is
(~ b (b f ( x ) g ' ( x ) d x = f ( x ) g(x) -- g ( x ) f ' ( . v ) d x (2 13)
Suppose fix) to be a smooth enough continuous function and g(x) a generalized function,
above equation becomes:
bf (x )~V(x- -x , )dx=fc~(x- -x , ) -- cr(x--xz)f ' ( .v)dx a a
According to the properties of cr function, above equation becomes:
f ( x ) a ' ( x - - x , ) d x = / ( b ) a ( b - - x , ) - - /'(x)dx=/(b)--/(x) r X l X 1
=f (b ) --~- f (b) - - f (Xl) ] = f ( x , )
Hence, we can define the generalized derivative $ ( x - - x I ) of a ( x - - x t) as such a function
which satisfies the following integral.
(o<.,<b) (,..i,)
where, fix) is continuous at place x = xlThis statement has definite mechanical meaning.
Ordinary use of Concentrated Load. The fight hand side of equation (2.14) can be understood as the work done by unit concentrated
load. The generalized function 8(x--x1) can be taken derivative again and function
~'(x--xt ) is also a generatized function.
From statement (2.14) we obtain:
~bS' (x- -x , ) f ( xld.,:= -- f ' ( x) (2.15) t l
Approximate Solution for Bending of Rectangular Plates 93
similarly proved if ,.Cx~<~b ,
then
(3 ~ ( x - - x ~ ) f ( x ) d x = ( - - 1 ) " ?~(x--.v~)f " ( x ) d x = ( - - 1 ) ~ f '~ (x , ) (2.16) . a . a
where f " ( x ) is continuous at x=x~, otherwise, the right hand side of statement (2.16) has no
meaning. Thus, we can define the function ~, " ' (x- -x~ ) by means of ~quation (2.16). In this way
8<"(x- -xL) is also the generalized function.
What follows is a theorem. Assume J ( x - - x , ) to be the step function defined in the interval [a, b] with discontinuity
points x~ (i= 1, 2 .. . . n) with ( z = x o ~ x , . C x . : ~ . . . . - ~ x , = b , In addition, assumeflx) to be any
continuous function in [a, b], the derivative of which f ' ( x ) can be integrated, then, the following
formula
I b a ( x - - x ~ ) f ' ( x ) d a = c r f (2 17) \ ' [cr(x~)]f(x) ~ o
C l t - I
is true.
Where
Proof: Since in [a, b]
function i.e.
Ecr(x~)]=cy(x,+ 0 ) - , r ( x , - 0 ) is jumpmg value offlx,) at place x=x , or(x-x~ ) can be represented as the linear combination of the unit step
n
, : ~ ( x - x , ) = ~ Er
Taking derivative to ~7(x--x~) and noting that
,T' ( x - x, ) =,~( , : - x, ) =
Thus, using integrating by parts we obtain:
n
\:~ Ef(.v,) -!,~(_v-.v, )
Since
(2.18)
E )" f c c ( x - - x , ) f ' ( x ) d x = c r f -- ~ T ' ( x - - x , ) f ( x tx (2.19) ) a , a
" I b .v) ,t.v= ~ [,~(.x', ) ] J(_,.)<'~I x-- v d .v �9 r
n
= E [c~(:<,)~/(x,) (2.2o) i - I
Substitution of above statement into equation (2.19) yields formula (2.17). The theorem is proved. Formula (2. l 5) has definite mechanical meaning, that is, the right hand side of formula (2. I 5) is
the integral of concentrated bending moment times the generalized function <'5' ( x-- x, } timesflx) and the right-hand side of formula (2.17) can be understood as the work done by unit bending
moment. The integral of the step function times the continuous functionflx) in formula (2.17) is often
used in where the section of plates varies suddenly.
94 Wang Lei and Li Jia-bao
III. Kantorovich-Galerkin's Approximate Solution
Kantorovich-Kralove presented Kantorovich approximate variational approach to deal with
the variational problems of multiple variables function.
T h e m e t h o d i s t o c h o o s e t h e f u n c t i o n s e q u e n c e ,/'~ ( x~ x . . . . . x , , ) ( f o r k = l , 2 .. . . m)
which satisfies the boundary conditions and write the approximate solution of variation problems
as follows: tr~
I I ( x ~ . x : , . . x , ) = '> ' i I . , ( x , ) , / . ( x , x., ... x , ,) - I
where Ak(x,) is undetermined functions of the x,. Substituting I I ( x , . x : - . . . x , ) into the
functional, the origional functional I I ( t t ) of function 11 (-~, . -L: , . . . .x , ) becomes the
functional of functions d~ (x,,) st:. ( x , ) .... , ~t,, ( x , ) which is written ~s
Now, the question becomes to choose such ..t, . .4 . . . . . ,Am(x , ) that let
I[*(..I, ..4........-1,,~ reach the extreme value.
The procedure of finding the extreme value of 17 ~ (A ~ . . . . . A,,) is to find Euler equations of
.-t~ ( x,, ). .-t: ( x,, ) . . . . . . 4 ~, C .v, ) and the concerned boundary conditions by means of variation.
These Euler equations are generally ordinary differential equations. In this way, the origional
partial differential equation containing multiple variables becomes the ordinary equation
containing single variable, which is the essentiality of Kantorovich method. Setting m~r and
taking lira?t, we can obtain the exact solution under certain conditions. Let m be a limited number,
then we obtain the approximate solution by such method.
The functional of rectangul',r elastic plates with clamped or simply supported edges is:
Now we solve the ordinary differential equation by Galerkin's method. Suppose
X4 -V :~ X
where ct, c, . . . . . is undetermined coefficient.
Taking the first term cl and substituting u(x) into Galerkin's equation gives:
I [ (u ' - -22"74344"~zu"+238"80597b 'u--4 9 7 5 1 2 ~ - - ) ( - ~ ~4 "xs ~ ) d x = �9 - z - - C - + 0
After integration arranging yields:
cl 4.8 (~1'~ +11.0t59.1 _ +11.75059 =0.995024" D
If a = b, then cl = 0.0360561qadD Thus the deflection equation of square plates is:
w = c l ~ a: ~ a l \ a --5 a3 + 3
At p l a~ x = a/2, y = a]2. the deflection of middle point is
aq~ 5 i q a 4 w=0.0360361 • 1-~-• D
The error of the result is very small compared with the classical solution. As we choose the beem function with one clamped end and the other simply supported or two
simply supported ends, we will encounter the follo~ving types of integrals.
which need not calculating and can be found out in the attached Table 1 at the end of this paper. E x * m p l e l The conditions are the same as above example but uniform distributed load is
changed into line uniform distributed load. (See Fig. 14) Solution: The functional equation is the same as (3.8) but the term of load should be changed as
f l I 2 P S ( . x - - 2 ) . V Y d x d ~ , = P I I Y d y X ~ ( 10)
Still using equation (3.8) but must changing the beam function so that it satisfies the load and
boundary conditions i.e. 1,> . ( ,~3 \
,,(.~) = t o T - - 4 7 ) ( c , +c, .x+ c;x~ 4 - ... ) (3 .11)
consider the following ~ni,::~ra 1~:
Approximate Solution for Bending of Rectangular Plates 97
. Q I l
"0
and use the value of the term of load at place x = 1/2
( |
u:'18 , 2 [T.-V ".\ 'dx-- I ,8 - 0 a
( x _ . , d P x . 3 u 1 aS- ) x = 2
2
Thus integrating after variation calculus yields:
= P x 1
b @ ~ u P I).030156• a~ ; 2: (~.:',12857I• 1.8 + 7 . 2 • . . . . t ) . 1 ? ; 6 - ~ -
For square plate, a = b. et-0.01706p,t '/D. The deflection of the middle point is
I ) a :~ l 1 ~ a :~ ~1.f)1766--73- • 1 x - ~ - = 0 . 0 0 , 1 4 1 5 ~
]Example II A rectangular plate with two adjacent edges simply supported and another two
adjacent edges clamped bears of uniform distributed load. Try to find the deflection of the middle
point. (See Fig. 15)
Solution" Suppose _ !/a ,/z .~
Similarly to Example I we obtain Euler equation, then solve the ordinary differ~,ntial equation by
: > . . . . . : , a c t solut ion 0.11~C.,, L,. the error it 17 8% . By m,zar: " ,,~ Kantorov ich-
Approximate Solution for Bending of Rectangular Plates 101
~ , ~ ~ t ~- ~
o
[ - -
~ m
t .--
~ ~ ~ ~ ' ~ o ~ c~ ~ ,--- I
! c ~ I ~ "
t . . J - " ~ ,
�9 ~ ~ ~:~ ~ ~
,.....-. o ~ ~, ,=,
0 ~ ~ - c.o
o . I
4-
I
o0
oo
~l~. ~1 ~
I V /
:"t
I
0
O4
c,.o oo.
.J.
LO "~
4-
I
4-
?
co
0
2., oo
4-
4-
-I
E
0
E # ~ . s U ~ ~ ,
-i
o
U
102 Wang Lei and Li Jia-bao'
Galerkin's method, the error is dropped under 5.0%. E x a m p l e 7 An elastic plate with three simply supproted edges and one free edge.Under
uniform distributed load q. Poisson ratio u = (1.3 . Try to find the deflection of the middle point. The beam function of beams with one simple supported end and the other free can be taken as:
.\" I 0 _v ~ I ( l _v : .v , \ - - + + - - t l " , ) d 4 ,, ') , ) t l ' (I
The concerned five types of integrals can be all looked up in the attached Table 1. Substituting each value of integrals and performing variation we obtain:
c O.Olf12056"<:; .80.q6 - - - t l
b' i '~'..,ql~ : ; ' . ' ( - -I / 185676)>,'0.9811321--a l__t,v
+ I . . . . 8,< 1 0503-I" aO-~7-> =-1.I>~ 0 185711:< 1:3 '2~6 l~a b - =o.8333:o3xO.2i~-~b
For square plates, a = b, c=O.0210976qa*/D.
The deflection of the middle point of the free edge of the square plates is:
qd ~ 7) qd ~ rt,,. = O. 0210976 - -77 • ;< '2 = O. 013186 [-~-
Compared wi th the exact so lu t ion O.O1286qa'/D. the error is on ly 2.47%.
R e f e r e n c e s
[ 1 ] Chien, W. Z., Variational Methods and Finite Elements, Science Press (1980). (in Chinese) [ 2 ] Hu, H. C., Variational Principles in Elasticity and Their Applications, Science Press ( 1981 ). (in
Chinese) [3 ] Beijing Mechanics Institute, The Science Academy of China: The Buckling Stability of
Sandwich and its Vibration, Science Press (1977). (in Chinese) [ 4 ] Timoshenko, S. P., Theory of Plates and Shells, McGraw-Hill Book Company, New York
(1960). [ 5 ] Timoshen k o. S. P., Strength of Materials, Science Press (1978. 3). (in Chinese)
[6 ] Cheung Y. K., Finite Strip Method of Structure Analysis (1976). [ 7 ] Xu, Z. L., Elastic Theory, People's Education Press (1979.8) (in Chinese) [ 8 ] Hu Hai-Chang, Suggestions on the application of finite element method with examples of
plane stress problems in elasticity, Acta Mechanica Solida Sinica (1981. 1). [ 9 ] Wang. Lei, Trial function and weighted residuals method, Journal ofHunan UniversiO' (1981.
1). [10] Wang. Lei, On middle-thick plates and try-functions, Engineering Mechanics, (1984. 1). [11] Wang. Lei. Analysis of boundary integrate method for middle tliick plates, Computational