ACTA MECHANICA SINICA, Vol.7, No.2, May 1991 Science Press, Beijing, China ALlerton Press, INC., New York, U. S. A.

ISSN 0567- 7718

VARIATIONAL PRINCIPLES OF PERFORATED THIN PLATES AND FINITE ELEMENT METHOD FOR BUCKLING AND

POST-BUCKLING ANALYSIS*

Yang Xiao Cheng Changjun (Department of Mechanics, Lanzhou University, Lanzhau, 730000)

ABSlrRACr: On the basis of the general theory of perforated thin plates under large deflections 11, 2] variational principles with deflection w and stress function F as variables are stated in detail. Based on these princi- pies, finite element method is established for analysing the buckling and post-buckling of perforated thin plates. It is found that the property of element is very complicated, owing to the multiple connexity of the region.

KEY WORDS : variational principle, finite element method, perforated thin plate, buckling and post-buckling analysis

I. INTRODUCTION Perforated thin plate is an important structure in engineering and technology. It is necessa-

ry to analyse its deformation, stress and stability. In this case, generally, the single-valued prop- erty of the stress function can not be ensured, and also the single-valued conditions of displace- ments must be considered as the region occupied by a plate is multiply connected. The general theory for analysing the problems of large deflection and of s tabi l i ty in perforated thin plates has been established by the authors [ 1], [2]. The description of this kind of problems has a rational mathematical model, and the framework for solving this kind of problems is also suggested in [1]. On the basis of the theory established [~'21, variational principles and finite element method for solving large deflection and stability problems in perforated thin plates are established in this paper. Variational principles for thin plate and the corresponding finite element method have been established t3'4. Up to now, to the knowledge of the authors, the variational principles and the finite element method for perforated thin plates are still unknown. As we shall see later that, the formulation of variational principles for perforated plates differs greatly from those of plates without hole owing to the multiple connexity of region and the single-valued requirement of dis- placements. And hence, both the property of element and its solutions are very complicated.

Following the theory established in [1,2] the variational principles with variables w and F are discussed. Here, the single-valued requirement is represented by a: class of generalized natural boundary conditions. Hence, the analysis of the problem is simplified. Secondly, based on these principles, the finite element method analysing the buckling and post-buckling of perforated thin plates is established. The corresponding formulae are derived-and the procedure for solution is given.

II. VARIATIONAL PRINCIPLES For the sake of convenience, we assume that the material of a perforated plate is isotropic

with material constants E and v and that the plate is subjected to a lateral loading q. Let f~ be the region occupied by the mid-plane of the plate before deformation and assume that f~, being a doubly connected region, has F=F0 (outer part )+Fl (inner part )as its boundary. If the forces acting along F i ( i = l , 2 ) a r e in self-equilibrium, it is known ill that a single-valued stress function exists

Received 5 March 1990 * Project supported by National Natural Science Foundation of China.

148 ACTA MECHANICA SINICA 1991

F0. p

r0, u @ / ]:

Fig. 1

with F and the deflection w and F satisfy the following governing equations and conditions I1'21

DAEw=h[w'F]+q "~ in f~ (2.1)

A2F = - --~-E [w,w] J 2

L g ( r ) - N i ( w ) = O (i=1, 2, 3) (2.2) 0F

F = ~ o 0n =Wo on Fo (2.3)

F=@I+AI+A2x+A3y } OF on Fi (2.4) -.~ -- q,Ll + A2nx+ Azny

- - Ow =0-, on ] - - F i w=W -- FI ; (i=0, Mn=M. Q.=Q,, on ,p

1 ) (2.5) E h 3 , h is the thickness of the plate. A1 ,A2,Aa are constants to be determined. where, D= 12(1_v2)

Oi, Wi are known functions Ill determined by the given boundary plane forces on Fi . (nx,ny)iS the unit outer normal vector of Fi, while w, 0. , M. and Q. are the generalized displacements and forces on F. Fi=Fi,u+Fg,p is assumed to be smooth. In addition,

2 f~ ~ 0 L, ( r ) = - g ~r - - -~ -yaFax+-~x AFdy

L 2 ( F ) = - ~ ~ 02-----F dx - 02F dy+ 2 ~ rl Oy 2 ~ T r |

2 ~ O 2 F , . 02F 2 ~r L 3(F)=--~ rl Ox@ a x - r - - d y + Ox 2 -E l N1 (w)= ~ r l Fldx-F2dy

N2(w)=~r YF, dx'yF2dy+(fr, (-~-x ) 2 d x + -

O2W OW 02W OW F I --

OX 2 Oy OxOy Ox

0 - y -~y AFax+y ~ AFdy

+~ A F dy x -~y AFdx-x--~- x

Ow Ow Ox Oy dy

Ow Ow dx + dy Ox Oy

~2W 0W 02W F 2 ~- . . . . 0y 2 0x &0y

Ow ~y

It is worth noticing that (2.2) are the single-valued conditions, which can ensure single-valued solutiOn for displacements of perforated thin plate ill.

Variational Principle I The solutions for the problem ( 2 . 1 ) - - ( 2 . 5 ) are equaivalent to seeking the stationary points of functional H in {F[ satisfying (2.3) and (2.4)} and {w [ satis- fying (2.5a)}

~ ;fo f;o ffo H = -'~ H I (w) dxdy- 2-~-~.. H 2 (F)dxdy + -~ H 3 (F, w )dxdy

- wqdxdy+ (-g. ~ --~,,w)ds d r 0 , p + r l ,p

(2.6)

Vol.7, No. 2

where

Yang & Cheng :Variational Principles of Perforated Thin Plates 149

Ox 2 \ @2 Ox 2 Oy2

(02F) 2+ 1/02F) 27_.2 -2v 02F H 2(F)=~ \ Ox 2 ~ Oy, Ox 2 @z

Ho(F,w)~ __02F (Ow ~ 2 O2F 0w 0w ~y2 \ - 2 OxOy Ox Oy

~){ O2w )~ - - + 2 ( 1 - \ 3xOy

_ _ ) ( 02F )2 02F + 2 ( l + v \ 0x0y

+ 02F ( O w ) 2 - O x :

Noticing that the undetermined constants A~ (i= 1, 2, 3 ) have also to be operated with variations, we are able to prove the variational principle I and to derive the expressions for M~ and Q~. Thus, the expression of M~ is obtained the same as the one of thin plate with small deflections, while Q~ is given as

OMns [~ F c~2F n2- O 2F O 2 F 1 Ow Q,=V,,+---~--- s +h 2 n ~ : n y + - - n ~ [L Oy 2 x ~ " Ox 2 J On

in which, V, and M,s are the shearing force and twisting moment. For the stability problem of a plate, it may be assumed that w =O,=M,=Q,,=O and q = 0

in ( 2.1 ) and ( 2 . 5 ) . Assume also that all the given plane boundary forces are directly proportional to a factor 2 and that the pre-buckling solution of problem ( 2 . 1 ) - - (2.5) is ( F , A , w ) = ( F * , A * , w * ) = ( F * ( x , y ) , A * , 0 ) when 2=1 . Here A = ( A j , A 2,A3) T and A *= (A*l , A*3, A*3)T. Obviously, this solution is a trivial one.

If the following transformations of variables are introduced

W=W then (2 .1) - - (2.5) become

F = 2 F , + ff A=2A,+ A A=(AI, A2,A 3 )T (2.7)

DA2w =h[w, "F] +2h[ w, F * ]

A 2 ~ ' = _ E [ w , w ] } inf~

Li( ' f f ) -Ni (w ) = 0 (i= 1, 2, 3)

~'= OF =0 on r0 On

F=AI+A2x+A3y } _ on Fi

Off _ A2 nx + A3 ny On

Ow w - On =0 on I"i, u

(2.8)

(2.9)

(2.10)

(2.11)

( i=0, 1) (2.12) Mn---Qn=0 on Fi, p

Obviously, for any 2, there is one zero-solution for ( 2 . 8 ) - - (2.12), that is, (w , F , A', 2 ) = ( 0 , 0 , 0 , 2 ) . It corresponds to the unbuckled state of the perforated plate. In order to analyse the buckling and post-buckling behaviour of the plate, we have to seek bifurcation points (or critical loads) and bifurcation solutions of (2 .8)-- (2 .12) . From the variational princi- ple I, we can further deduce the following variational principle for bifurcation analysis.

Variational Principle II The bifurcation solutions of the trivial solution of (2 .8 ) - - (2 .12 ) at bifurcation points are equivalent to seeking the non-zero stationary points of a functional Hb

150 ACTA MECHANICA SINICA 1991

D ffnH,(w)dxdy- h f f H2("F)dxdy n~=T 52-

h ;IoH3(ff', w)dxdy + h-~-~ f fH3(F' ,w)dxdy (2.13) +T in {F I satisfying (2.10) and (2.11)} and { w I satisfying (2 .12a)}.

The proof is also not difficult. Deflection w and stress function F (or F ) as well as A~, A 2 , A 3 are the variables in these principles. Both the variational principles 1 and II may be regarded as generalized variational principles.

III. DISCRETIZED EQUATIONS It has been mathematically proved that there exists one unique bifurcation solution for

( 2 . 8 ) - - (2 .12 ) near a simple bifurcation point 2 *I51 . Here, we employ the variational principle II to derive discretized equations of bifurcation problem (2 .8 ) (2.12). For convenience, we introduce non-dimensional variables and parameters as follows:

W w _ x - Y ,~" ~ * - T _ h

= h -~ -- ~=--~c "rq--~c z--'~--) - f f ( F ' F * ) T T h }

a l - - ~ Ai (a2,a3) = D ( A 2 ' A 3 ) q~i=--ff ~i (3.1)

hR~ 2 t~= D W~ ( i = 0 , 1 ) /~=12(1-v )

in which, R~ is a characteristic length of the mid-plane. We further use the symbols x, y and w in the place of non-dimensional variables ~, r/ and W and use n for region occupied by (~, q). Thus, the variational principle II is equivalent to the following proposition.

Proposition (Variational Principle II ') The bifurcations governed by (2 .8 ) (2 .12) near a simple bifurcation point are equivalent to seeking non-zero stationary points of a functional

I f [ f f H ~I= -~ Hl tw )dxdy- 1 J. n ~ 2 ('S )dxdy

+"~- H3(S', w )dxdy + --~ 3 ( S*, w )dxdy (3.2)

where, S must satisfy

and w must satisfy

=0 on F0

S=al+a2x+aaY -~n =aznx+a3ny on Fl

(3.3)

w = 0W0n = 0 on F 0 , u § (3.4)

028 * 028 * 028 * Let T'x- dy~ , T%.- Ox 2 , T%. dxdy ' then i t is easy to see that ( T * , T3*,

T'y ) is the stress state corresponding to 2=1 . That is to say, the plate is in an unbuckled state.

Now sub-divide the region I) into N subregions f ~ e ( e = l , 2 , . . . , N ) . Assume that the total number of nodeS"is K. Let the nodal parameter vectors {6~,} and {6~} at each node correspond tO the: deflection and stress function ( r= 1, 2, ..., K ) , respectively. For an element e, we can ob- tain the deflection w and stress function S 'by making interpolations from the nodal parameters

we = [N~] {6~} s ' e = [ N ; ] {6~} (3.5)

where, {65} and {6~} are the nodal parameter vectors of the deflection w and stress function

Vol.7, No. 2 Yang & Cheng : Variational Principles of Perforated Thin

S of the element e, respectively. Let

Plates 151

ax 2 @~ axay =[ Me] {a~ } aY 2 ax ~ "] T = [ M;] {a; }

axay )

i , v o ] [ [ D ] = v 1 0 [ E] = --~- - v

0 0 2 ( l - v ) 0

[ T*] = x ,T*, [ T] = 8Y2

" ~ x ~ y

-v~ 1 1 0

0 2( l+v)

ax@ a= ~. - H,(~ ')

c~x 2

{o}= [ ~w

-~x = [ a ] {aS} [ A ] = o aw

@

as 1 {a;} T 1 ~le= T [Kebl {a~b }-- T {a;}r[K;l {a ~}

1 2 + T {aep )T [Kae~. {a/e } _ T {ap }T [K e ] {a; }

[ K;] =fro [ Mb e ] T[ DI [Mb e ] dxdy e

[ K~] =fro [ G]T[ T ] [ G ] ax@ e

then in element e we have the functional H e

in which

o] Ow ~ H 5 (w ) ay aw &

(3.6)

Thus, ~ = E fie, therefore, al~= E a fie = O. e e

non-linear algebraic equations can be deduced for the unknown nodal parameters {ab } and {ap}. Next, let us discuss the properties of the element in two cases.

(1) If there is no point of F~ on the nodes of element e, then, we have

[ K~] =IIOe [ M~] T[ El {M~I dxdy

[K~] =flo~[GlT[ T*l[G]dx@

From the variational equation a~=O, a set of

a~e={d{aa }T, d{a; }T }[Kel { {a;{a; }} } (3.7)

where [K~b]= [G]T[A]T[Me,]dxdy

e

[K~]-=

1 1 e I [ K ~ , ] - 2 [ K ~ ] + ~ [K~o] -~- [Kpb]

J l e e [Kpb ] T -- [ Kp ]

152 ACTA MECHANtCA SINICA 1991

Let the generalized nodal force and displacement of element e be {fe}T={{f/ }T,{f; }T }T and {6 e }T={{6~}T, {6; }T }T, respectively. Then, it follows from (3.7)that

{ f e } = [ g e] {6e} (3.8)

(2) If there are points of F~ on the nodes of element e, then owing to (3.3) and the arbitrariness of ai, the nodal parameters { 6p } on Ft are dependent on ai. Assume that in ele- ment { 6"p } is a vector including ai and nodal stress parameter vectors not attached to points of Fl , it follows from (3.3b) that

{6~ }=[ ~] [ 5"~] (3.9)

Therefore, the variation of { 6"; } is arbitrary, and (3.7) may be expressed as

6IIe={d{6~} T , d{ 6~ }T }[~.e I { sT, ~ } (3.1o)

in which,

[ g e ] =

Similarly, we have

1 [ K S ] - 2 [ K ] ]+-~- [K~o]

1 ]T -]- [7]T[K~b

I [K;,] [7]

- [ 71 T[ K~I [71

{ye }=[ ~-e] {~'e} (3.11)

Thus, we obtain the discretization equations

E {fe}-I-e~, {f~'}~-- E [ge]{6e}"l-eE[ge']{ '~ 'e '}={O} (3.12) e g '

For convenience, (3.12) will be written into the following form

{ f }~ [K] {6}= {0} (3.13) This is the set of discretized equations for solving (2.8)--(2.12) under bifurcation, which is

a set of non-linear algebraic equations. Obviously, for any 2, there exists a trivial solution {5 }={0 } for (3.13).

IV. LINEARIZED PROBLEM AND CRITICAL LOADS Following the general theory of bifurcation TM, in order to obtain non-trivial solution

for (3.13), the Jacobi matrix 8 { f }/8{6 } {~}={0} on the left side of (3.13) must be singular.

From the definition of[K e] in (3,7), this is equivalent to say that the following eigenvalue problem has non-zero solutions. That is

I[K~]-2[Ke] 0 ] { {6~,} ) ={0} e 0 - [ K ; ] { 6 ; }

(4.1)

It is not difficult to see that {6p }={0} as [K~] is positive definite. Hence, (4.1) becomes

( [ K ~ ] - 2 [ K e] ) {6~}={0} (4.2) e

which can be equivalently written as

{ [ K b l - 2 [ G*] }{6b}={0} (4.3)

There are many methods for calculating the eigenvalues of (4.3). We shall employ the iterative method of subspace to calculate the eigenvalues of (4.3) and to check their multiplicity.

V. BUCKLED STATES NEAR A CRITICAL LOAD Next, let us discuss the finite element method for the buckled states of a perforated thin

plate near bifurcation point.

Vol.7; No. 2

Assume that )~* is ,a simple eigenvalue of (4 .1) is the corresponding eigenvector

{6" }={ {6*b} T , {6 ~ }T }T={{(~*b }T, {0 }T} T

which satisfies the condition { 6* 6 }T [ Kb ] { 6 *b] = l. Assume that a non-zero solution of (3.13) is {6 } and that

{~}=,{6.}+8(~}~=~.+~(,1Y}2{{~'b } T , ) {~VV} in which, 8 is a small parameter defined by

8---{6b }T [ Kb] {6*b } Hence

Substituting (5.2) into [ K e] in (3.7), we obtain

[Kpb]=8 [--e ]+~ [--, e g'pb gpb] [ g eba] =8[ geba]

in which,

= ] I [ 61T [ ~-1T [ [-E~I M~] dxdy J ,) u e

[ K;b] = [ [ [ G] T[ ZIT [ M~] dxdy e

[ .E;,I = ( ( [ 61T [ T'I [ el dxdy e

[ T'] = / /4 ( [ N ; ] { 6"~ } )

['-A]=H~([N~]{6*b e} ) [X]=Hs([N~]{6"~}) Thus, we have

Yang & Cheng:Variational Principles of Perforated Thin Plates 153

and has the smallest absolute value. { 5 * }

[ g e] ~ [ B e ] - ~ [ C e] +8[ F e]

(5.1)

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

~-[ K~,] - 2 * [ K,~]

L 0 :1 -[K~I o

[ E'~o] y ( [ p~l +[ K;bl +8 (5.7)

L+ ([ "K";bl "1- [ ~";bl )T 0

Then (3.13) may be expressed as

[ ~e] [ ~"e]_~"~ [ C e ] {~ 'e } + 8 2 [ F e ] {~""~e } e e e

+'~-~ ( [ B e ] - Y [ c e ] + e , [ F `1 ) { 6 * e } = { 0 } e

Obviously, when 8=0 and under the condition (5.4), (5.8) always has a zero-solution

{g'}={0} ~'=0 The linearized problem of (5,8) about the zero-solution (5.9) is given as

(5.8)

(5.9)

154 ACTA MECHANICA SINICA 1991

Z ([ .Be] { ~"e }_~" [ C e] { 6 , e } ) = { 0 } (5.10) e

It is easy to prove that the unique solution o f ( 5. 10) is the zero-solution under the condition (5.4). This means that there is one unique solution for (5.8) with (5.4) and { 6"} = { 0 } and 2=0 when e=0. Therefore, we have proved the existance and uniqueness of bifurcation solution for (3.13), namely, {5"}# {0 }.

Expanding { 6"} and ~" into series of/; { ~'}=/; { L,) }+/;2{ ~2) }+'" 2%/; )"1 +/;2 ~2_t; ... ~ (5.11)

then, we have

[ i ] = Z/; i [ X,;)] [ ~] = Z/; ' [ ~'<,>] i=1 i=1

[ g ;b ] = Z /;i[ gipb ] [ g ~ a ] "-- Z /;i[ rkeba] i=1 i=1

[ F e] ~ [ F~]+/;[ F~I]+/;2[ Fe2]+... (5.12)

0 T Kea] 1 [/~'~b,,] T

= Jrl; [--eKp.blr 0 - - [ g[pb] T

where

+/;2 I ~ Efi~bo] [ K~p,~] T

[ A'(,)] =H 5 ([ N~] {6~'e(i)b }

12 [ /~leb]] Jr ...

o j

[ T'(i)] =H4( [ N ; ] {6"~e(i)p } )

[ K~ipb] = [G]T[A( i ) ]T[M;]dxdy f~e

[ --e II [ T~,)] [ a] Kib o ] = [ G] T dxdy dd fte

Substituting (5.11) into (5.8) and (5.4) and comparing the coefficients of/; in same powers, we deduce the following sets of linear algebraic equations

t Z ( [Be ] {6~"e ( l ) } -~ l [Ce ] {~*e }+ [F~ ] {6*e } )= { O} ( ~ 1 ) e (5.13)

{g'(,)~, }T [ X~,] {6%}=0

t ~e ([ge]i'~e(2)}-~2[ fe]{(~*e}-~l I C e ] {6~1)}

+[ e0] { g'[,) }+ [ F;] {6 ,e } )={0 }

{6(2)b}T[Ka ]{6,b}=O

(,s 2 ) (5.14)

Vol.7, No. 2 Yang & Cheng : Variational Principles of Perforated Thin Plates 155

Taking notice of (5.7), we can obtain a unique solution of (5.13)

{a(,~}={0 } 2,=0 (5.15) The first expression of (5.14) then becomes

([ K ~ ] _ 2 , [ K e ] ) { "~'e ~ a(.~ }-2~ E [ KS] {a*~} e e

+ T ~e ([ K--;b ] { (~ {I)p '} + [ Kerbs] { 6 *g } )={ 0 } (5.16)

"~e 1 T ~ }+ [ K{pbl T { a , g } ) = { 0 } EIK;l{amp}- T Y~([E;b] {a{,)~ e e

From the second expression of (5.16), it follows tha t { a'~{2)p } = { 0 }. Multiplying { 5 *b }r on the both sides of the first expression of (5 .16) and employing the second expression of (5 .14) , we have

and

e e

Under the condition (5.14b), (5.18) has a unique post'buckling response of the perforated plate near the critical load 2 *

{a~ }=~{ a*~ }+d { g~2)~ }+o(~' ) } {(~p }=~ 2 { ~( l )p }-{'- O (~ 4 ) (5.19)

2=2"+~2 ~ + O (e 3 )

It is worth to point out that the three undetermined constants ai( i = 1, 2, 3 ) can also be approximately obtained during the above process and their degree of precision is the same as {ap}.

( 2 ~ 2 ) [ K ~ I - [ K ~ b ~] ) { 6 " ~ } (5.18)

solution {~'(2)b }. Thus, we have obtained the

VI. RESPONSE OF LARGE REGION ON THE POST-BUCKLING BEHAVIOUR In order to understand fully the post-buckling behaviour of a perforated thin plate, it is

necessary to extend the solution of ( 2 . 8 ) - - ( 2 . 1 2 ) near 2* to a point far from the bifurcation point 2*. This is called the continuation of solutions. The continuation of bifurcation solutions, the determination of singular points (secondary bifurcation points ) along the branch of bifurcated route, the computation method through singular points as well as the jump along branches are all given in [6] in detail. We here only present the computation formulae of Newton-Raphson iterative scheme for analysing the post-buckling behaviour of a perforated thin plate in a large region of loads.

From (3.8), we have d { f e }= [ K]-] d {h e } (6.1)

where

Letting

d { f ~ } d2

_ _ - [ C e l { 6 e }

[KeT]~ I[Keb]-2[Ke]-~[Keba] [K;b] 1 [ Kepb] T __[ Kep]

d { f e } {f~ }==- ~ d2 - - ~ [ Ce]{6e} e e

(6.2)

{ f } ~ Z ( f e } [ K r l = Y , [ K ~ l e e

156 ACTA MECHANICA SINICA 1991

then, it is easy to see that [ Kr] is the corresponding matrix of tangent stiffness. Assume that { 20, {6 }(0)} is a post-buckling solution determined by ( 5 . 19) near 2 " , then Newton-Raphson iterative procedure can be given as follows:

(1) Assume that A2i(i = 0, 1, 2, ... ) are the increments of 2,- and if 2i+i=)~i+A2i, then we may obtain an approximate solution { 50} (i+1) from the formula { 60} (i+t) ={ 5,,i} (i) - [ Kr]-I{ f ;v}A2i when 2=2i+1 . Here, m i is the number of iteration within the second step (2) computing 2 = 2 i from 2 = 2 i- l �9

(2) Taking { 60 }ti+ll to be an initial value, we make the following iteration: Firstly, compute {f } and { K r }from { 6j} (i+l), j = 0 , 1, 2 , ' ' ' , mi+ 1 , Secondly, compute modified quantity A{ 6j } ( i + ~ ) = - [ K r ] - t { f } : a n d hence it yields

that { 6j+~ }{i+~)= { 5j }u+~)+A{6j }u+~) until convergence is obtained. Return to the step (1) and repeat the iteration mentioned above. Here, j is an iteration index of the step (2).

To sum up, we can compute not only critical loads of a perforated thin plate sul~ected to self-equilibrium forces along each boundary but also both the post-buckling behaviour near 2* and far from 2* by using the finite element method given in this paper. It is not difficult to generalize this method to the corresponding problem of a perforated thin plate sul~ected to forces not' in self-equilibrium along each boundary following [2].

Critical loads and the post-buckling behaviour of an annular plate sul~ected to a force distribution of ei + / / i cos ni 0 along its boundary are studied using the method stated here. When /~=0, the results agree very well with some existing solutions.

REFERENCES [1] Zhu Zhengyou, Cheng Chan~un, Acta Mechanica Sinica, 2, 3(1986), 278-- 288. [2] Cheng Chan~un, Lui Xiaoan, Acta Mechanica Sinica (in Chinese), 21, 2(1989), 193--203. 13] Chien Weizang, Generalized Variational Principles (in Chinese), Knowledge Press (1985). [4] Zienkiewicz, O.C.,The Finite Element Method (third edition), McGraw-Hill (1977). [5] Cheng Chan~un, Yang Xiao, Modern Mathematics and Mechanics( in Chinese), Chien Weizang, Guo

Youzhong (eds), Science Press (1989), 257-- 267.

[6] Zhu Zhengyou, Cheng Chani~un, Numerical Methods on Bifurcation Problems (in Chinese), Lanzhou University Press, Lanzhou, China (1989).

[7] Bathe, K . L . , Wilson, E.L., Numerical Methods in Finite Element Analysis, Prentice-Hall, Inc. (1976).