VECTOR FORM INTRINSIC FINITE ELEMENT (VFIFE).pptx

VECTOR FORM INTRINSIC FINITE ELEMENT (VFIFE)

1. VECTOR FORM INTRINSIC FINITE ELEMENT (VFIFE) Theory of 2D Solid Element (How to determine Internal nodal force)

2. 1. INTRODUCTION

3. 4. INTERNAL NODAL FORCE 4.1. Deformation coordinates The transformation between the coordinates and the global coordinates: (7) We choose another point D and impose following conditions on the shape functions for the deformation: at C, (8) at D,

4. 4.2 SHAPE FUNCTIONS FOR THE DEFORMATION The deformation shape functions should satisfy: , i = 1, 2, 3,… , n 1. (11) 2. 3. 4. (12) Continuity conditions in the element and across the boundaries. (10)

5. 4.3 THREE-NODE TRIANGULAR ELEMENT For a three-node constant strain triangle, shape functions have the form: (13) in which:

6. 4.3 THREE-NODE TRIANGULAR ELEMENT To satisfy Eq. (11), it requires (14) Using Eq. (6) and solving for the displacement at C, we get ( 6 ) To satisfy Eq. (16), it requires (16) (11) (15)

7. 4.3 THREE-NODE TRIANGULAR ELEMENT Equations (14), (15) and (16) are the three conditions to simplify Eq. (13). The choice of (C, D) is arbitrary. However, if we use the nodal points, say (1, 2), as the reference, the formulations are simpler. We get

8. 4.3 THREE-NODE TRIANGULAR ELEMENT The shape functions become (17) (18) in which, , i = 2, 3 The coordinate transformation matrix can be found from the condition . The condition implies that the axis is parallel to the deformation vector at node 2.

9. 4.3 THREE-NODE TRIANGULAR ELEMENT For a linearly elastic material with modulus matrix E , we get To define a work equivalent nodal force vector, we use the virtual work: ( 22 ) In which, t e , A e , V e are the thickness, area and volume of the element. Substitution of (20) into (21) yields the internal nodal force vector : (20)

10. 4.3 THREE-NODE TRIANGULAR ELEMENT In Eq. (22), three components of the internal forces are obtained. The other three dependent force components are calculated from the equilibrium conditions: And, we get (23)

11. 4.3 THREE-NODE TRIANGULAR ELEMENT For the assemblage of forces at the nodes, force components are transformed to global coordinates as (24) The total internal nodal force vector at t +  t is (25)

12. 4.3 FOUR-NODE ISOPARAMETRIC ELEMENT To formulate shape functions, a set of natural coordinates (s, t) is defined such that (26) In which, Shape functions for the deformation: (27)

13. 4.3 FOUR-NODE ISOPARAMETRIC ELEMENT Let ( x C , y C ) be the location of C corresponding to its position in natural coordinates ( s C , t C ). The deformation displacements at C are ( u C , v C ). Since C is the origin and its deformation is zero, we have (28) (29)

14. 4.3 FOUR-NODE ISOPARAMETRIC ELEMENT From the coordinate transformation relationship, we get (30) From the definition of deformation, we get (31)

15. 4.3 FOUR-NODE ISOPARAMETRIC ELEMENT We choose an element node (for example, node 1) as the reference point C, the above calculations are simplified. At node 1, N 1 = 1, N 2 = N 3 = N 4 = 0. (32) We choose to orient the x axis parallel to the deformation vector of another node. This gives (33)

16. 4.3 FOUR-NODE ISOPARAMETRIC ELEMENT Using Eqs . (32) and (33), the geometry and deformation shape functions of the isoprarametric element are: (34)

17. 4.3 FOUR-NODE ISOPARAMETRIC ELEMENT Using Eqs . (32) and (33), the geometry and deformation shape functions of the isoprarametric element are: (35) (36)

18. 4.3 FOUR-NODE ISOPARAMETRIC ELEMENT The procedure to calculate internal forces is similar to the traditional finite element. They are summarized in the following: At an arbitrary point (s, t), the strain components are The Jacobian is is an abbreviation of the derivative (37)

19. 4.3 FOUR-NODE ISOPARAMETRIC ELEMENT Substitutions of Eqs . (34), (35), and (36) into Eq. (37) yield In which,

20. 4.3 FOUR-NODE ISOPARAMETRIC ELEMENT Using the virtual work, the nodal internal forces are (38) The nodal forces f lx , f 1y and f 2y at nodes 1 and 2 are found from the equilibrium equations. (39)

21. 4.3 FOUR-NODE ISOPARAMETRIC ELEMENT For assemblage, the internal forces are transformed to global coordinates: (40) For an incremental deformation formulation, the internal forces become at is the stress and E the tangent modulus matrix of the element material at time t. (41)

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