The Finite Element Method for the Analysis of Linear Systems

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1 Swiss Federal Institute of Technolog Page The Finite Element Method for the Analsis of Linear Sstems Prof. Dr. Michael Havbro Faber Swiss Federal Institute of Technolog ETH Zurich, Switzerland

2 Swiss Federal Institute of Technolog Page Contents t of Toda's Lecture - Bi-linear four node element - Singularit of Jacobian matrix Element matrices in global coordinate sstem

3 Swiss Federal Institute of Technolog Page 3 Bi-linear four node element: In the following we will for matters of convenience consider the iso-parametric representation: Displacement fields as well as the geometrical representation of the finite elements are approximated using the same approximating functions shape functions uˆ, vˆ -, s, u ˆ, v ˆ 3 uˆ, vˆ 3 3 r uˆ, vˆ x -, -, -

4 Swiss Federal Institute of Technolog Page Bi-linear four node element: For the bi-linear four node element the shape functions in this coordinate sstem stem become: h s = ( r) ( s) -, h = 3 ( + r) ( s) h3 = ( + r) ( + s) -, - h = ( r) ( + s),, - r

5 Swiss Federal Institute of Technolog Page 5 Bi-linear four node element: s Numerical integration: Gauss rule (x) (, ) 3 (, ) r Stiffness matrix: K = V T B CB dv dv = det J dr ds dt (, ) (, ) T K = B CBdet J V dr ds dt 5-Apr-08

6 Swiss Federal Institute of Technolog Page 6 Singularit it of Jacobian matrix: Considering the general three-dimensional case there is: φ x z φ r r r r x φ x z φ = s s s s φ x z φ t t t t z The inverse of Jacobian matrix is = J = J r x x r required! 5-Apr-08

7 Swiss Federal Institute of Technolog Page 7 Singularit it of Jacobian matrix: The existence of the inverse of Jacobian matrix provides the unique correspondence between een natural and local coordinates. Let us consider a quadrilateral element -, s, 3 x xrs (, ) = ( r)( sx ) + ( + r)( sx ) + ( + r)( + sx ) + ( r)( + sx ) rs (, ) = ( r )( s ) + ( + r )( s ) + ( + r )( + s ) + ( r)( + s ) 5-Apr-08 ( ) ( ) -, -, r

8 Swiss Federal Institute of Technolog Page 8 Singularit it of Jacobian matrix: Let us consider a quadrilateral element xrs (, ) = ( r)( sx ) + ( + r)( sx ) + ( + r)( + sx ) + ( r)( + sx ) ( rs, ) = ( r)( s ) + ( + r)( s) + ( + r)( + s) + ( r)( + s) ( ) 3 ( ) 3 x -, -, - s, 3, - r det J x r r = x s s x+ x + x3 x + s( x x + x3 x) s( + 3 ) = x x + x + x + r ( x x + x x ) r ( + ) Apr-08

9 Swiss Federal Institute of Technolog Page 9 Singularit it of Jacobian matrix: Let us consider a quadrilateral element -, -, - s, 3 det J is a linear function of the coordinates r and s. Therefore, det J 0 in the element onl if its values in all nodes are positive or negative. x, - r We have to inspect the values at the nodes. node : ( )( ) ( )( ) det J = x x x x 5-Apr-08

10 Swiss Federal Institute of Technolog Page 0 Singularit it of Jacobian matrix: Let us consider a quadrilateral element ( )( ) ( )( ) det J = x x x x x -, -, - s, 3, - r It is connected with the value of the product of the vectors: b α 3 a b = a b sinα a a b is positive if α < π x 5-Apr-08

11 Swiss Federal Institute of Technolog Page Singularit it of Jacobian matrix: Let us consider a quadrilateral element If all angles of the quadrilateral element are smaller than 80 degrees, det J will not be 0. If this is not the case, there will be singularit somewhere in the element and it will not be possible to establish unique relation between natural and local coordinates. α 3 5-Apr-08 x

12 Swiss Federal Institute of Technolog Page Following the same principle one ma define isoparametric shape functions for three-dimensional quadrilateral elements (see Bathe pp ) 35 z t r s x

13 Swiss Federal Institute of Technolog Page 3 We can also construct the triangular element directl from the quadrilateral element b so-called collapsing: 3 3

14 Swiss Federal Institute of Technolog Page We can also construct the triangular element directl from the quadrilateral element b so-called collapsing: x = hxˆ + hxˆ + hxˆ + hxˆ = h ˆ ˆ ˆ ˆ ˆ ˆ + h + h33 + h 3 = xˆ = xˆ x = hxˆ + ( h + h) xˆ + hxˆ 3 = h ˆ + ( h + h ) ˆ + h ˆ 3 3

15 Swiss Federal Institute of Technolog Page 5 Element matrices in global l coordinate sstem Local to global coordinate transformations: It is often more convenient to define the element stiffness relations and to calculate their contributions to the load vector in a local coordinate sstem (e.g. displacements ũ) this is often specific for each individual element. In this case we need to transform the element matrixes into global coordinates (e.g. displacements û) before we can assemble the global stiffness relation. Transformation relationship can be written as: u = Tuˆ T being a transformation matrix.

16 Swiss Federal Institute of Technolog Page 6 Element matrices in global l coordinate sstem Y v u v u α X u = Tuˆ u cosα sinα 0 0 u v sinα cosα 0 0 v = u 0 0 cosα sinαu v 0 0 sinα cosα v

17 Swiss Federal Institute of Technolog Page 7 Element matrices in global l coordinate sstem Let us tr to establish the stiffness matrix of the truss element using directl global nodal point displacements. (see example 5., Bathe pp ) 388) K = V T B CB dv For the truss element considered we have [ α ] u=cos ( ru ) + ( + ru ) sinα ( r ) V + ( + r) V

18 Swiss Federal Institute of Technolog Page 8 Element matrices in global l coordinate sstem Let us tr to establish the stiffness matrix of the truss element using directl global nodal point displacements. (see example 5., Bathe pp ) 388) u u Using ε =, ε = in the natural coordinate sstem, x L r B= [ cosα sinα cosα sinα] L

Swiss Federal Institute of Technolog Page 9 Element matrices in global l coordinate sstem Let us tr to establish the stiffness matrix of the truss element using directl global nodal point

19 Swiss Federal Institute of Technolog Page 9 Element matrices in global l coordinate sstem Let us tr to establish the stiffness matrix of the truss element using directl global nodal point displacements. (see example 5., Bathe pp ) 388) we have AL dv = dr, and C = E Finall, we obtain cos α cosαsinα cos α cosαsinα AE sin α cosαsinα sin α K= L cos α cos α sin α sm sin α

Lecture 21: Isoparametric Formulation of Plane Elements.

Lecture 21: Isoparametric Formulation of Plane Elements. 6.6. Rectangular Plane Stress/Strain Element. he CS element uses a triangular shape. he 3 nodes of the CS element allow us to employ linear multivariate approximations to the u and v displacements. he