Static Condensation

Chapter 12 Static Condensation. Incompatible Elements___________________________

120

CHAPTER 12 INCOMPATIBLE FINITE ELEMENTS 12.1 THE STATIC CONDENSATION The concept of static condensation is used in order to reduce the number of degrees of freedom (DOF) at element level. The quadrilateral element with the external nodes 1 to 4 shown in figure 12.1 can be considered as being replaced by 4 triangular elements connected in the internal node 5. Assuming all elements linear with 2 DOF per node, the total number of degrees of freedom is 10. Thus the assembled stiffness matrix yields a 10 10 table. In order to reduce the number of algebraic equations when merging the elements into a mesh, the internal DOF of node 5 should be eliminated, or condensed. Hence, only the degrees of freedom associated with the external nodes enter the equations and the stiffness matrix dimension yields 8 8. The stiffness matrices of the four triangular elements should be superimposed in order to create the stiffness matrix of the quadrilateral element.

Fig. 12.1 The condensation of internal node DOF

The static condensation procedure begins by partitioning the algebraic equation system as

y,v

x,u

3

2

4 1

5

1

2

3

4

______________________Basics of the Finite Element Method Applied in Civil Engineering

121

=

i

e

i

e

rr

kkkk

2221

1211 (12.1)

where i is the vector of internal displacements corresponding to node 5 and ri the load vector applied to the same node. The (12.1) relationship is only a fragment of the global equation system defining the equilibrium of the structure. Solving for i [ ]eii krk 21122 = (12.2) and substituting in (12.1), the condensed equation system yields [ ] iee rkkrkkkk 12212211221211 = . (12.3) The coefficient of e is the condensed stiffness matrix kc and

iec rkkrr1

2212= is the condensed load vector. Usually no loads are

associated to the internal node, such that rc = re. Equation 12.3 can be solved for the actual corner node displacements in the usual manner:

cec rk = (12.4) The stress field over the quadrilateral element is now calculated according to the known relationship:

AEB == (12.5) with the nodal displacements. Some of these are internal DOF, eliminated in the condensation process, thus unknown. As consequence, in the condensation process the matrix A should be also transformed. Partitioning the equation 12.5 according to the internal and external DOF

[ ]

=i

eie

AA (12.6)


122

and substituting i out of 12.2, the stress vector relationship yields:

[ ] iieie rkAkkAA 12221122 += (12.7) 12.2 INCOMPATIBLE ELEMENTS (EXTRA DISPLACEMENT SHAPES) The linear isoparametric element presented in chapter 10 provides a displacement pattern which brings in shear strains even in pure bending (see figure 12.2). Due to the elements shear strain xy the deformation energy augments, corresponding to the shear stresses xy which do not exist in pure bending. Therefore, the linear element is too stiff when withstanding the bending load. In order to avoid this inconsistence, a quadratic element with 8 nodes and 16 DOF can be adopted.

Fig. 12.2 Incompatible finite element displacements

Even though, a similar result can be obtained by improving the linear element with 4 nodes and 8 DOF. The solution is to complete the standard displacement field by adding two more terms, capable to reproduce the elements side curvature. Comparing the exact solution of the elements displacement components

stEIMu = and )1(

8)1(

82

22

2

tEI

MhsEI

MLv += (12.8) with the displacement field of the standard linear element (with 0=v ), the error in the displacement definition along the y axis has the following form:

)u(s,x)u(s,x

)v(t,y )v(t,yL

hM M M M

xy

______________________Basics of the Finite Element Method Applied in Civil Engineering

123

( ) ( )22 11 tsv += (12.9)

Adding these terms, the displacement field in normalized coordinates becomes:

( ) ( ) ++= 22 11 tsii Nd ; )1)(1(41 iii ttssN ++= (12.10) where and are called non-associated parameters to elements nodes (belonging to element only). Thus, the deformation is described independently for each element by the new parameters and . Although the shape functions terms ( )21 s and ( )21 t provide a correct bending behavior, they are incompatible modes of deformation. The displacement magnitude of a certain point on the elements side is not longer defined only by its nodes. Moreover, the corresponding point belonging to the adjacent element may have a different displacement. Gaps or overlapping between adjacent elements may occur, determining the incompatibility. However, all the numerical tests and current applications have shown an excellent performance of the incompatible element in bending situations, if certain constrains concerning the geometry are fulfilled. The approximation of the unknown displacement function is made in terms of normalized nodal displacements and the parameters [ ]=Ta . As usual, to guarantee the elements coupling in the mesh, its matrices must be expressed in terms of nodal values only. Thus, the unknowns a should be eliminated. The procedure is similar with the one used in the static condensation development. In the energy minimization relationship, the displacement vector is partitioned as follows:

0=E rK

=

E

=a

e (12.11)


124

with e the nodal DOF of the element and a the elements non-associated parameters. The partitioned form of the functional derivatives yields:

=

a

0r

a

KKKK

e

e

e

e

aa

a

E

E

(12.12)

Note that there are no nodal forces corresponding to non-associated parameters. From the second set of equations, dropping the displacements index, the non-associated parameters are withdrawn:

0aKK =+ aa (12.13) KKa Taa

1= (12.14) and by substituting

( ) r KKKK

= T

aaaeE

1 (12.15)

Consequently, the stiffness matrix of the element yields:

Ta

-aa KKKKK 1= (12.16)

By solving the algebraic equation system rK = the nodal displacements are found. For computer programming, the procedure is to eliminate successively and from the last two rows and columns of the original matrix, using the Gauss elimination algorithm. Similar procedures are applied for a 3D element. The augmented displacement field has in this case the general form: ( ) ( ) ( )222 111 rts(s,t,r) ii +++= Nd (12.17)

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Static Condensation