Tangent stiffness properties of finite elements

Pergamon

TANGENT

Compurms & Srructures Vol. 58. No. 2. pp. U-365. 1996 Copyright f; 1995 Elsewer Scmce Ltd Printed in Great Brimn. All rights reserved @.X5-7949/96 $9.50 + 0.00

0045-7949(95)00130-l

STIFFNESS

PROPERTIES

OF FINITE

ELEMENTS

Z. Sindelt and S. S. Tezcant tSTFA Temel Kaziklari AS, Kosuyolu, Istanbul, Turkey SBogazici University, Bebek, Istanbul, Turkey (Received 30 June 1994) Abstract-Original tangent stiffness matrices have been developed explicitly for a triangular and rectangular finite element, by which both the speed of computations and the degree of accuracy are increased. The new tangent stiffness matrices developed in this study represent the final conditions of the material, such as the most recent deformed geometry, the current total stress level, the amount of plastic strains present, as well as the current material constants in the form of either the tangent or secant modulus

of elasticity. The method of development is so general that the comprehensive tangent stiffness matrices may be developed for any other finite element type, such as arbitrary quadrilaterals, prisms, curvilinear elements. etc.

1. INTRODUCTION For over a few centuries structural mechanics and soil mechanics have concerned themselves primarily with linear problems. All structural behaviour in solid and soil mechanics is actually nonlinear. No structures or soil formations behave linearly. In fact, for a proper conventional aseismic design in earthquake engineering, it is the primary responsibility of the engineer to let the materials go into the plastic range and allow the joints to become plastic hinges. All deformations need not be small, nor need they be elastic and recoverable. A tension structure, a cable stayed or a suspension bridge, a dome or an arch structure, a dynamic analysis under earthquake ground motions, the post yielding and large deflection of structures, possibility of finite deformations, post buckling and stability of beams, frames, plates and shells, nearly all problems in soil and rock mechanics require nonlinear, inelastic solutions. The complexities of nonlinear analysis have been treated only in the last 30 years through the efficient use of modern high speed computers. In structural and soil mechanics, the finite element method is successfully used in solving nonlinear problems. The finite element method is ideally suitable for conducting nonlinear analysis under static or time dependent loading conditions. Methods of incremental loading or Newton-Raphson iterative schemes require that in each loading or time step, the stiffness matrices should be modified to be as closely tangent as possible to the load-deformation curve. It is only by the tangent stiffness approach that both the accuracy and speed of computations may be increased in nonlinear analyses. The development of improved and high order element characteristics and the availability of very

efficient nonlinear solution algorithms and the experience gained in their application to engineering problems have indicated that the nonlinear finite element analysis can be performed with success. The limitations to the general solution of nonlinear problems by finite elements have been removed considerably and the process is successfully applied to nonlinear problems in structural and soil mechanics. A special effort is invested in this study to formulate a rigorous mathematical technique for the derivation of the instantaneous stiffness matrices. One of the main difficulties in describing the nonlinear material behaviour is to be able to accommodate the influence of large strains and to modify the stiffness matrix in each cycle of analysis. In fact, most investigators prefer short cuts, by ignoring the quadratic nonlinear terms in the strain-displacement relations. Further, the modified Newton-Raphson iterative scheme is resorted to at the expense of increasing the volume of numerical computations, by employing a constant stiffness matrix at each cycle of analysis, thus avoiding the necessity of inverting a new stiffness at each cycle. When the new methodology presented in this study is employed however, there will not be any need for such an excuse, because there will not be any need for a modified iterative technique, nor will there be any need for inversions in order to arrive at a new stiffness matrix. The comprehensive tangent stiffness matrices are presented herein in explicit forms, for a triangular and rectangular element. The nature of the tangent stiffness matrices introduced in this study is not the so-called “corrective” type as available in the literature, but rather they are very “comprehensive” corresponding to the latest state of stresses and strains in the material and 351

352

Z. Sindel and S. S. Tezcan

containing the finite strain components. These original comprehensive tangent stiffness matrices are essential to increase the speed of computations and also to assure the stability of convergence to the true solution. The second most useful and original aspect of the study is to be able to conduct nonlinear analysis of structures and soils with finite strains by considering the nonlinear quadratic components of the strain-displacement relations as well as the latest nodal displacements.

2. CONCEPT OF TANGENT STIFFNESS

In almost all computational techniques the stiffness matrix of each finite element has to be modified and updated at the beginning of a new step of loading. If no modification takes place, as it is the case in the modified Newton-Raphson method, then the number of iteration cycles may be excessively large. Therefore, it is extremely time saving and accurate to modify the stiffness matrices at the end of each cycle, using the state of stress and deformations obtained at the end of the previous cycle. The parameters that should be taken into account in calculating a new tangent stiffness matrix at the end of each cycle, are: (a) the new coordinates of the nodes, (b) the state of strain at the centroid of each element, (c) the state of stress at the centroid of each element, (d) the modified material properties, or the plasticity matrix, (e) the condition of loading or reloading, (f) the condition of work hardening or work softening. Any of these parameters may be taken into consideration, depending on the availability of data and the requirements of the specific computational technique utilized. If the element remains in the elastic range, only the coordinates change and the initial stresses need to be considerd. If the material yields

increments and knowledge about the elasto-plastic constitutive law. The speed of convergence as well as the assurance for convergence and the elimination of the possibility of an oscillatory iteration all depend on the accurate calculation of the tangent stiffness matrix. In the following paragraphs, a general method of derivation for the tangent stiffness matrices has been described employing the nonlinear higher order terms of the strain-displacement relations. The concept of the tangent stiffness matrix introduced herein is basically similar to the formulations presented earlier by Martin [I], Argyris [2, 31, Przemieniecki [4], Wissmann [5], Oden and Sato [6-S], Tezcan ef al. [9] and others. The emphasis herein however, is in the importance of including the higher order components of the strain energy expressions. Further, the tangent stiffness matrices are presented in explicit forms for easy programming purposes and also for eliminating any matrix inversion or numerical integration.

3. STRAIN-DISPLACEMENT

RELATIONS

The strain vector, at any intermediate stage of analysis, may be expressed as the superposition of the current values of the strains at step i and the additional strains L,, or strain increments, which are developed because of the applications of a new load increment. That is, ie]=ltJI+{~)“.

(1)

The additional strains may be expressed in terms of the generic displacements U, G and w for a plate bending and three-dimensional stress problems as follows [IO]:

(3)

the elastoplastic material matrix should be employed, requiring the values of total stresses, plastic strain

Instantaneous strains ci, are considered to be present and numerically available before the application

Tangent stiffness properties of finite elements

of a new set of external loads. For instance, the state of strain of the preceding cycle of analysis, thermal stresses, prestressing forces, yield stresses, lack of fit, etc., constitute the initial stress vector. It is interesting to note that the instantaneous strain vector L,, due to the loads of the preceding cycle is also calculated from eqn (3) in the same way as the additional strain vector E, using all the nonlinear strain-displacement relations. There is one difference, however, that in the case of c,, the generic displacements U, v and w as well as their partial derivatives, are all numerically available, while in the case of c,, these displacements are variable functions of the nodal displacements. Therefore, from the view of loaddeflection history, it is more appropriate to call ci, the “accumulated” and c,, the “incremental” strain vectors.

353

The strain energy terms which contribute to the tangent stiffness matrix are listed in Table 1. For each type of stiffness matrix listed in Table 1, there is a systematic method of formulation as will be discussed in the following paragraphs. The stiffness matrix formulations involving rotational degrees of freedom are not included herein because they represent linear components and are readily available elsewhere [l-3].

5. LINEAR STIFFNESS MATRIX (TYPE I)

The strain energy component for the linear stiffness matrix is {~,}~[Dl(q,} dv,

U, =; 4. STRAIN ENERGY FOR FINITE STRAINS

where cO= linear strain components given in eqns (2) and (3) and [D] = the elastic material matrix. When the shape function matrix [N] is introduced for the generic displacements u, v and w as

Strain energy U, by definition is

U =;

{t}‘{o} dV =; {E}T[D]{E}dt’, sY sY

(4)

where D = elastic or elasto-plastic material matrix. The integration should be carried over the undeformed volume of the finite element as already discussed by Wissmann [S]. When the finite strain components are introduced, the strain energy becomes u =

s i’

{t, + L”+ L, + L*+ L, + LqJT

x [D] {L,+ ~0+ c, + e2+ ej +

~4}

dV.

(5)

the strain components are obtained by taking the partial derivatives of [N] in the form of

(9) which is substituted in the strain energy expressions and differentiated twice with respect to the nodal displacements {d} to yield the regular linear stiffness matrix K,, which is normally available in all finite element text books [ 11, 131.

After carrying out the products inside the integration, the total strain energy is expressed as the algebraic sum of 36 different strain energy coponents

s

[&I=;{G, ,‘[D]{Go)

as follows:

dV.

(10)

Y

“Z; sY

+

CTDC, Second

+

e;Dc, Second

+

+

+

e:Dc, Third

+

$DC, Second

+

LTDQ Third

L:DCi Second

+

C%DC, First

+

c ;DL, Zonstant

+

C;D$ First

+

L:DC; Second

@G,

CTDC, Second

L;DQ Second

+

ciDe, Third ., CfDC, Fourth

CDQ Third ,._.._.,...._... + efD6, Fourth

+

+

E:Dc, Fourth

+

c;Dt, ; ~. Third :

+

&lDC, Fourth

+

c;Dcz Fourth

+

CTDC, Fourth

+

c;Dc, Third

c;Dc, Third

+

CTDC, Fourth

+

c:De, Fourth

+

C:De, Fourth

+

e:Dc, Second

+

CTDL, Third . .

+

e:De, Third . ..“.........._.

+

c:DC, Third

+

First

+

C,TDC, + Third

L;Dt4 First eiDe, Second

j I

c;Dt, f Third : . . c:De, Second

dV. (6)

354

Z. Sindel and S. S. Tezcan Table I. List of contributing strain energy component Type

Order

Strain energy components

Stiffness matrix

Remarks Linear

I

Instantaneous

II

Linear (bending)

III

Zero, if -no

IV

rotations

-symmetry

about the xy-plane

KO’ Nonlinear (function of u and v) V

3rd

K02 1

6. TANGENT STIFFNESS DUE TO INITIAL STRESSES (TYPE II)

For reason of convenience in formulation, only the first component of the strain energy U, will be discussed involving only the higher order derivatives of the u-displacements. The expressions for iJ2 and U, are exactly the same as for U, , except the indices will be 2 and 3, respectively. Therefore, @}:(a}idV,

(11)

Usually, the state of stress and strain in a finite element are evaluated at the centroid of that element. In order to eliminate the quadratic terms in the strain vector, the following matrix identity will be utilized:

The product vectors of stresses and strains may then be simplified as

where u, = stresses accumulated at the end of the last cycle i and available as numerical quantities, L, = higher order strain components, expressed as partial derivatives of the u-displacements eqns (2) and (3).

(14)

Tangent

stiffness

properties of

in which the column vectors of quadratic terms are eliminated by means of introducing a triple matrix operation containing a rectangular block in the middle. Normally, the middle block is the accumulated stress tensor at the centroid of the finite element. By taking partial derivatives of the u-displacements in eqn (8) and also by introducing the above matrix identities mentioned above the strain energy U, of eqn (11) becomes

finite

355

elements

7. THE TANGENT STIFFNESS DUE TO FINITE DISPLACEMENTS (TYPE V)

The linear tangent stiffness in plate bending (Type III) and its higher order components (Types IV and VI) are not discussed herein for reasons of space limitations. Their formulations and methods of derivation however, are exactly the same as those of Types II and V included herein. A typical strain energy expression involving the third order terms, for the u-displacements for instance, is given by

@I = bw4 tr,,, =;

s,TDr,dI’+;

c:Dc,dI’.

(20)

Replacing the products of D, c,,, by a fictitious single vector {f } u, = f

~4T[~l:bl,[~l, (4 dV sY

+ $ j b4[Gl, bli[Gl:V)T df,‘. ”

(15) b-1 = PW)o =

Realizing that the interior product in the above equation represents a congruent transformation which does not destroy the symmetric nature of the square matrix, we can combine the two separate integrals of the strain energy into one term as follows:

u, = f

~4’[‘4fbl,~Gl, (4 dV.

and using it in eqn (20)

U,,=; WT{W’+; {c}:{f)dV

(22)

(16) and taking advantage of the identity of eqn (12), the strain energy becomes

Differentiating twice with respect to nodal displacements, the tangent stiffness matrix Ki, is obtained as

K, =

lGl:bl;[Gl, d v

sY

which corresponds to the u-displacements only. Identical procedure is applied for the Kn and K,3 tangent stiffness matrices, except the u-displacement is replaced by u and w in obtaining G, and G, matrices, respectively as

Kil =

Kc3=

sV

sv

(1x2)

(17)

(2x2)

(2x1).

The individual terms f,, fi and f, of the middle block of eqn (23) are analogues to Q,~, cry and t,r, of the instantaneous stress block of eqn (13). It is important to note that this middle block, given by

(24)

Fl:bl,[Gl, dV

(18) contains

the unknown nodal displacements Substituting eqns (15) and (24) into eqn (23)

[GlT[~lt[Gl3 dV.

(19)

The important remark in connection with these matrices is that they contain only the state of stress of the last step of loading. The explicit contents of these matrices are supplied in the subsequent chapters, for a triangular and rectangular finite element.

{d}.

In order to obtain the tangent stiffness matrix, the strain energy should be differentiated twice with respect to the nodal displacements {d}. But, it should be remembered that the F-matrix contains the derivatives of the shape functions multiplied by the nodal displacements. In partial differentiating, this fact

Z. Sindel and S. S. Tezcan

356

should be taken into account. We may write the F-matrix as the product of derivatives of shape function and the nodal displacements as follows:

I

in which the typical auxiliary matrices u-displacement case are as follows:

for the

(26) (29)

in which, e,,e2,e3 = the row vectors containing the coefficients of the derivatives of the shape functions as follows:

The components of the K,, and K,,,-matrices A,., C, and A,., C,,., respectively are exactly the same as A,,, and C,, except the G, -matrix is replaced by G, and G3, respectively. The G-matrices are obtained from the first order partial derivatives of the shape functions matrix N, as follows:

(30)

and

@l)=(ell

el2

%

e14

elS

e16

o

o

O)

(31) h)

=

hl

e22

e23

e24

e25

e26

o

o

O)

tell

=

te31

e32

e33

e34

e35

e36

o

o

O)

=

[GlzM

(32)

or

D,2a3

0

0

It is an interesting point to notice that the K,, , K,, and Ko3matrices all contain the nodal displacements of the last loading step. For this reason, sometimes these matrices are called the “displacement str@iess

O)

matrices”. Especially, it is these matrix components that reject the nonlinearity of the finite elements.

D22a3 0 0 0)

(e3)=

& (D13+ D33b3

0

D13a2 D13a3 D3d+ DA 0

8. THE TOTAL TANGENT

0).

The tangent stiffness matrix is obtained by taking the second partial derivatives of the strain energy U,, in accordance with

The total tangent stiffness matrix to be employed in the next step of loading is obtained as the superposition of all linear and nonlinear components as follows:

K

a%

STIFFNESS MATRIX

=

K,

+

W,,

+

K,

+

K,,) + (K,, + Ko2+ K,,) (34)

k,, = ___

ad, ad,

The resulting stiffness matrix becomes Ko,=A,+A:+C,, Ko2= A,. + A: + C, K,,=A,,.+A;+C,,

(27)

in which K, = the regular linear stiffness matrix, which is modified only if the modulus of elasticity changes. Ki, ,K,2and K,3= the tangent stiffness matrix components arising from the accumulated state of stresses at the end of the ith loading step and finally, K,,,, Koz and &, = the tangent stiffness components containing in them the nodal displacements of the ith loading step.

Tangent stiffness properties of finite elements 9. TANGENT

STIFFNESS OF A TRIANGLE

ness the explicit Appendix I.

9.1. The’ linear component (Type I) The local coordinate axes and the nine degrees of freedom of a general three dimensional triangle are shown in Fig. 1. Corners are numbered counterclockwise. The assumed shape functions are

{u) = P34

II I

=ooogq

V

0 5

w u

‘I0

0i

0

0

The strains components matrix are given as

[

0

0

0

00

05

40

i0

1 {d}.

and the linear

357

form of the K,-matrix

is given in

9.2. The initial stress component (Type II) The tangent stiffness matrix due to the accumulated state of stress o,, for the u-displacement is

Kli, =

sY

KAfbl,[Gl, d K

(40)

(35) where

stiffness Gi=&

(36)

Klo = ;

sc

[GliX~l[Gl,dV

(37)

in which the GO-matrix is given by

Flo =

(42)

Gl=;

&

0

0

0

0

0

0

b,

b,

6,

0

0

0

0

0

0

a,

a,

a3 (43)

(38) where A = area of the triangle, plate and a, = xi - x, =

b,= Y, -

YA =

t = thickness

of the

.xk,

Y,k

(39)

The resulting

tangent

stiffness matrix

The resulting stiffness matrix K, is readily available in the literature [I I-131. But, for reasons of complete-

[a,, + m

+ [ai3 =

which is explicitly

defined

is

[Cl

0

0

0

[Cl

0

0

0

[Cl

FHil in Appendix

(45)

I.

9.3. Displacement stiffness matrices (Type V) If the partial derivatives of u and u are taken in accordance with eqn (21), the elements of the middle block matrix-F become

fi = 41 u,,+ Duv,. = (cl 114

fz = Q2u,, + Dz2v,.= (e,){d) Fig. 1. A general triangle in three dimensions.

fi = &u,. + b3vy, =

(d(d),

(46)

Z. Sindel and S. S. Tezcan

358

where (e)L=&(h,D,, a,&,

(eh =

QJL

& (b,D,, alb2

@I3=

&&I

b,D,,

a,&

0

0)

b,D,, b,D,,

a2D2, ad%

& (aID3, b,D33

0

0 0 0)

1

a2D3, a36,

b2D33

hD33

0

0

Fig. 2. Rectangular membrane element [I2 degrees of freedom (DOF)].

(47)

0).

The components of the displacement tangent stiffness matrices are

m3, = Ml,,+ [A 1:+ K%

m3

(48)

= [A I,?+ 1-4IT+ [Cl,. .

The explicit forms of these matrices are given in Appendix 1. The total tangent stiffness then becomes K

=

K,

+

6,

+

K,

+

K,,)

+

(K,,

+

K,,

+

Ko3).

10. TANGENT STIFFNESS OF A RECTANGULAR

(49)

g,, = bU + rl)

gzl =

a(1 + 5)

g,, = b(l - 9)

g2, =

-4

g,, = --HI + ‘I)

g23=4l

gl, = -Ml

g24 =

-q)

--au

+ t) -5) -

5).

The explicit linear stiffness matrix K,of a rectangular finite element is given in Appendix 2.

ELEMENT

10.1. The linear component (Type I)

10.2. The initial stress component (Type II)

The local coordinate axes, the corner numbering and the 12 degrees of freedom for a rectangular finite element in membrane action are shown in Fig. 2. The linear shaoe functions assumed are

The G, -matrix of eqn (15), in connection with the u-displacement is _ . (54)

(50) where =

0

0

0 N,

0 N2

0

0

0 N3

0 N4

N, N2 N, N4 0 00

0 0

N, = (1 + &)(l

0 0

0 0

N, 0

+ wi)/4.

0

0

N2 0

N, 0

1

0 {d}

(51)

N4 0

(52)

l,q, = nondimensional coordinates of the ith corner, 5 = x/a, 0 = y/b, a, b = half size dimensions of the sides of the rectangle. The linear stiffness matrix is WI, =

where

[Gl;PI[Gl, dV> JY

(53)

Tangent

stiffness properties

of finite elements

359

5-12 (1 -s)/a

(1 +q)/a

[F]=f

(1 +5)/b

(-1

-5)/b

The G, and G, matrices are exactly the same as G, , except the nonzero terms are located at columns five to eight in the G, -matrix and at columns nine to 12 in the G,-matrix, respectively. The initial stress tangent stiffness matrix, for the u-diplacements, then becomes

[Cl, =

[Vud vt

(56)

Y

where, u, = the accumulated state of stress x_r-plane at the end of the ith loading step.

in the

The Ki2 and K,, matrices are exactly the same as K,,, except the G, matrix is replaced by G, and G, matrices, respectively. matrix is

The resulting

I

I

[Cl

[Kl,,+ [Kl,,+ [Kl,,=

tangent

I

0

stiffness

1

0

IIikl 0

[Cl

0

>

(57)

I OI OI[c1I I

I

I

1

which is explicitly defined in Appendix 2. Due to the recursive nature of the [S], matrices, the contents of the [Cl,, [Cl, and [Cl, matrices are all the same [Cl, except the locations inside the [K], matrix gradually shift downwards along the diagonal. 11. CONCLUSIONS AND RECOMMENDATIONS

In light of the overview of the various aspects of the nonlinear finite element analysis and also considering the impact of the refined tangent stiffness matrices presented herein, a few concluding remarks and recommendations may be stated as follows:

(1) The refined tangent stiffness matrices developed herein are original matrices and they provide an added assurance in performing a linear analysis as closely tangent as possible to the load-deflection curve.

(2) With the availability of the explicit forms of the refined tangent stiffness matrices for a triangle and rectangle, the speed of convergence as well as the degree of accuracy of nonlinear analyses are significantly increased. Further, these matrices are especially suitable for the purpose of conducting stability analysis for thin plates, since they contain the out-of-plane degrees of freedom. CAS w--1

-Cl +tlM (1 --O/b

(1 -?)/a -(l-M

0 0

(3) The method of generation proposed herein to obtain the refined tangent stiffness matrices is so general and straightforward that the refined tangent stiffness matrices to other types of finite elements including bending action may be easily obtained, following an identical flow of calculations. In fact, the explicit expressions of the nonlinear stiffness matrices of a tetrahedron and other elements are available elsewhere [ 141. (4) It is recommended that, in order to assess the relative advantages and merits of the proposed refined tangent stiffness matrices in other fields of applications, such as stability problems, heat conduction, seepage, fluid dynamics, etc., extensive testing and numerical analyses need to be conducted in future studies. REFERENCES

1. H. C. Martin, On the derivation of stiffness matrices for the analysis of large deflection and stability problem. In: Proc. ConjI on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, OH, pp. 647-716. AFFDL-TR-66-80 (1965). 2. J. H. Argyris, Continua and discontinua. In: Proc. Conf. on Matrix Methods in Structural Mechanics. WrightPatterson Air Force Base, OH, pp. 1I-189. AFFDLTR-66-80 (1965). 3. J. H. Argyris, Matrix analysis of three-dimensional elastic media, small and large displacements. J. Am. Inst. Aeronaut. Astronaut. 3, 45-51 (1965). 4. J. S. Przemieniecki, Discrete element methods for stability analysis of complex structures. J. R. aeronaur. Sot. 72, 107771086 (1968). 5. J. W. Wissmann, Nonlinear structural analysis; tensor formulation. In: Proc. Conf on Matrix Methods in Structural Mechanics. Wright-Patterson Air Force Base, OH, pp. 679696. AFFDL-TR-66-80 (1965). 6. J. T. Oden, Calculation of geometric stiffness matrices for complex structures. J. Am. Inst. Aeronaut. Astronaut. 4, 1480-1482 (1966). 7. J. T. Oden and T. Sato, Finite strains and displacements of elastic membranes by the finite element method. Int. J. Solids Struct. 3, 471488 (1967). 8. J. T. Oden, Finite element applications in nonlinear structural analysis. In: Proc. Symp. on Application of Finite Element Methods m &ii Engineering, Vanderbilt Universitv. Nashville, TN. DD. . . 419456 (1969). 9. S. S. Tezcan, B. C. Mahapatra and C. I. Mathews, Tangent Sttjmss Matrices for Finite elements, pp. 2 17-246. International Association for Bridge and Structural Engineering. Leemann, Zurich (1970). 10. V. V. Novozhilov, Foundations of the Nonlinear Theory of Elasficity. Graylock Press, Rochester, New York (1953). Il. 0. C. Zienkiewicz, The Finite Elemeni Method in Structural and Continuum Mechanics. McGraw-Hill, Maidenhead (1967).

Z. Sindel and S. S. Tezcan

360

12. M. J. Turner, R. W. Clough, H. C. Martin and L. J. Topp, Stiffness and deflection analysis of complex structures. .I. Aerospace Sci. 27, 97-102 (1956). 13. J. S. Prezemieniecki, Theory of Marrix Structural Analysis. McGraw-Hill, New York (1968).

14. Z. Sindel, Tangent stiffness approach in nonlinear analysis by finite elements. Ph.D. Dissertation, Department of Civil Engineering, Bogazici University, Istanbul, (1993).

APPENDIX I: TANGENT STIFFNESS OF A GENERAL TRIANGLE

(a) The K,-matrix of eqn (37) is derived as follows: 1

2

D,,Y:,

1

D33.42

D,,y3y*, -D33~3~3*

D,,Y,Y,,

2

--4,x,x,,

3

-D,,Y~Y*, D33x3x32

KMl= 4

4

-D,,y,yz

D,,Y:

-D,,Y,Y,

D33-4

-D,,x*x,

5

D3,x32y23

D33x2x3Z

D33 x32y3

D,2x3*y23

-

-D,,y*y,

4,~: D33x:

D33x3y23

D33x2y23 -

6

7-9

-D,,x,*Y*

-D,*x,Y*,

42x32~3

-D,,x*x,

D,2x2y23

-43x3~3

D,,x,Y*

- D,2x3y3

D,2x3y3

D33x2~3

--D,,xzY,

D,2x32y2

D,*x,Y*

-D,*x,Y,

4,y:3

D,,Y,Y*,

-

0 D,*x~*Y*, D,,x,*Y*,

5

3

-D,zx,Y*,

6 -

D432~3 -

-D,*x~*Y*

5’33x3~23

D,,x,Y*,

-D,*x,Y,

42x3~2

D,,x,*Y,

- D,,x,Y,

D33x2y3

42x2~2,

42x2~3

-42x2~2

43x3~2

-D,,x*Y,

D33 ~32~2

- D**~3 x32

D2242

D33~3~23 -D*2x3x,*

-D,,Y*Y*, D22 ~2x32

7-9

D,3y2y23 D22 ~2 x32

D,,Y:

--D,,Y~Y,

D**x:

-4*x*x,

- D33y2y3

D33.4

-D22x*x3

D**x:

0

(b) The [Cl-matrix of eqn (45) is

ICI=& i

Cl,

Cl2

Cl3

c*,

c**

c*,

c3,

c32

c33

1 ’

where, in index and explicit notations the C values are: C,,=b,b,u,+(a,b,+a,b,)r,,.+a,a,o,.

C,, = b;u, + 2a, b,q,

(i=1,2,3)Ci=l,2,3)

+ u:u,

C22 = b:u, + 2a,bzr,, + a:u, C,, = b:a, + 2a3b3r,,. + a:u,

C,z=b,b,a,+(a,b,+a,b,)r,,+a,a,u,. C,, = b, b,u, + (a,b, + a, b,)r,, + a, a,u, Cz3 = b,b,a,

+ (a,b, + a,b,h,,

+

a*a,u,..

&.

Tangent stiffness properties of finite elements

361

(c) The Au-matrix of eqn (48) is

where a,,=(b,e,,+c~,e,,)R,+(b,e,~+a,e,~)R,

alI =

(i=1,2,3)(j=1,2

@+ellf~~e~~)R,+(b,e31+ ule2,)R2

uzl

=(b2e,, +u2e3,)R,+(b,e,, + w2,)&

alI

=(b,e,, +a3e3,)4 +(b3e3,+a3e2,W2

aI2 =

(4 e12+

a*2 =

WI2 + u2e3,M,+ (b2e3, + a2ez2)R2

u3* =

(b3q2+ a3e3,)Rl+ (b3e3* + w2,)R2

aI3 =

(heI +ale3,)4 + (he13+ UI~Z#Z

uz3 =

(b2e13 + a2e3dRt+ (b2e3, + a2ez3)R2

aI

q2)R, + @+e32+ atez2)4

% = WI3 + a3e3,)4f (b3e33 + v2dR*

q4

=

az4 =

(b, cl4

+

a, e,,M, + (4 ej4 +

(bze,, + a,e,)R,

u34 = (b,e,, + 03edR,

aIs =

(4 eIs +

aI

+ (b,e,

a~e2P2

+ a2e2dRl

+ (b,e,, + wh)Rz

e3#, + (4 e3s +

01 e2dR2

uzs= (b2eIS + a2e3Ml + (b2qs+ vzsW2 uls =

(b3eIS + a3e3M, + (b3e3s + a3e2dR2

aI6 =

(6 e16+ aI e3JR,+ (4 e16+ a,e,dR,

az6=

(b2e16 + a2e3,)4+ (b2e3, + u2e2dR2

036=

(b3e16 + w%)R, + he,, + u3e2dR2

,...

,6)

Z. Sindel and S. S. Tezcan

362

all elements a,, for the A,,, A, and A, matrices

are identical,

except the R, and Rz values are modified

R,,=xb,d,

Cj= 1,233)

R,, = 1 a,4

(j = I, 2, 3)

For A,

(k = I, 2,3)

For A,

(k = 4, 5,6)

as follows:

For A,$ (k = 7,8,9) R,=b,d,fb,dz+b,d, for [A],-matrix R,=a,d,+a,d,+a,d, R, = b,d,i

I-

b,d, f b3d6 for [A], -matrix

Rz=a,d4+a,d5+a,d,

I-

R, = b,d, + bzd, + b,$ for [A],,.-matrix R,=a,d,ta,d,fa,d,

(d) The C,-matrix of eqn (48) is

f,,

The elements of the C,-matrix are the same as in the [Cl-matrix f2 and &, respectively. (e) The locations of various tangent stiffness matrix components

above.

but the c,,

O, and r,,. are to be replaced

by

are as follows:

K, = K, + (K,, + % + K,,) + (K,, + K,, + K,,)

EliIFzl EliIBl!iEl 0

[Al,=

+[A], =

0

Each subblock

is 3 x 3 in size

+ [A I,, =

0

0

0

0

0

0

AS>

0

0

0

0

0

Af

Al

[AIf =

0

0

0

0

0

A,

0

0

0

0

0

0

0

+[A]:=

0

A;r

0

0

0

0

+[A]:= :o

0

0

0

0

t

Tangent

stiffness

properties

of finite elements

363

APPENDIX 2. THE TANGENT STIFFNESS MATRIX OF A RECTANGLE

(a) The linear stiffness matrix

component

K, of eqn (53) is derived

1

2

3

4

as

5

6

7

4 [Kl, = 5 6 7 8

(9-12)

I

0

in which

k,, = W,,B +4,//Q/3 k>, = W,,8/2

+ k/P)!3

k,, = ~(-~,,P

+ k/28)/3

k,, = [(-u/2 k,, = W,? k,, = t(-D,,

- ~2p)i3 + &,)I4 + W/4

k,, = -k,, k,, = -k,,

(b) The [S], and [Cl matrices

of eqns (55) and eqn (56) are

[Kl,, =

sI

[Gl:bl,[Gl, dV =

”

I.Y,dY= [Cl,

8

(9-12)

Z. Sindel and S. S. Tezcan

364

s,, = o,(l + v)V + 27,(1 + r)(l + II) + ~~(1 + 028 S2, = a,(1 - q*)/p - 27,,.v(l + 5) - o,(l + 028

rj’)/fi - h,,.(l

- bf) - u,.(l -

t*)8

S,, = -u,(l

-

S,, = a,(1 +

r/)*/b - &,(I

S3* = -a,(1

-

tj2)/p + ‘h,.(l + ttl) - o,.(l - t*)8

S,, = -u,(l

+

q)*/b + 2r,,t(l

s,, = a,(1

+ 5)(1 - rl) + a,.(1 + 8*8

- ‘7)+ u,(l -

t*)fi

+ rl)2/s - 25,.(1 - 5)U + v) + q(l - 028

s43=

4

-VW + 27,.rlU + 5) - u,(I - O’S

h=

4

-d/P

+ 27,.(l - 5)(1 - 4) + u,.(l - 5)*P

Some of the integral solutions are

s Y

dV= 4abr

q* dV = 4abt/3

(by analogy to 0.

Y

When these integral solutions are introduced into the S-matrix, the C-matrix is obtained as

l-h Cl,

c2,

c22

c4,

c42

Symmetrical

c4,

c44

where C,, = ‘J,/3fl + %/2 + 0,$/3 G, = uJ6P -Q/3 c,, = -u,/3p

+ u,p/3

C,, = -u,/6fl

- T,$ - 0,4/6

Tangent stiffness properties of finite elements

G, =0,/38 -r,,P+a,P/3 C,, = -a,/68

+ r,,.l2 -q/3/6

G = G,

c,=c,,. Therefore, the total instantaneous tangent stiffness is

Kl;, + WI, + Kl,J =

c

0

0

0

c

0

0

0

c

El

365