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Two-Dimensional Isoparametric Elements and Numerical Integration

Chapter Seven. Two-Dimensional Isoparametric Elements and Numerical Integration. THE FOUR-NODE QUADRILATERAL. Coordinates of nod i. The element displacement vector. The local nodes are numbered as 1,2,3,4 in a counterclockwise fashion. SHAPE FUNCTIONS.

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Two-Dimensional Isoparametric Elements and Numerical Integration

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  1. Chapter Seven Two-Dimensional Isoparametric Elements and Numerical Integration

  2. THE FOUR-NODE QUADRILATERAL Coordinates of nod i The element displacement vector The local nodes are numbered as 1,2,3,4 in a counterclockwise fashion

  3. SHAPE FUNCTIONS In ξ,η coordinates (natural coordinates) at nod i. i=1,2,3,4 = 0 at other nodes at nod 1 C=1/4

  4. Displacement field within the element In the isoparametric formulation , we use the same function N to express the coordinates of a point within the element in terms of nodal coordinates

  5. B MATRIX (I)

  6. Equation (I) can be inverted as : II

  7. Another Method : iii By considering f=u in (II) we have : iv

  8. Equation (iii) and (iv) yield : ** Equation * yield

  9. Element Stiffness Matrix :

  10. Element Body Force Vectors :

  11. Traction force :

  12. Gaussian quadrature : Consider the n-point approximation weights Gauss points n : Number of gauss points m : degree of Polynomial

  13. One point formula : If then

  14. Approximate Area=2f(0) Exact area = -1 0 1

  15. Two- point formula Error =0 if

  16. Example 7.1 evaluate for n=1 , we have for n=2 , we have for n=3 , we have

  17. Two dimensional integrals The extension of gaussian quadrature to two –dimensional integrals

  18. Stiffness integration Let φ represent the ijth element in the integrand If we use a 2×2 rule then:

  19. gaussian quadrature to two –dimensions using the 2×2 rule

  20. Stress calculations Example 7.1 Consider a rectangular element as shown in fig .assume plane stress condition , E= 30×10^6 psi ,v=0.3, and Evaluate J,B and σ at ξ=0,η=0

  21. SOLUTION From Eqs. ** Evaluating G in Eq. *** at ζ=η=0 and using B=AG

  22. Comment on degenerate quadrilaterals In some situations, quadrilaterals elements degenerate into triangles. Numerical integration will permit the use of such elements But the errors are higher than regular elements.

  23. Quadrilateral Higher-order elements For 4-node quadrilateral element For 9-node quadrilateral element

  24. 9-node quadrilateral element For example at nod 1:

  25. 16-node quadrilateral element(4th order) 7 (1/3,-1/3)

  26. 8-node quadrilateral element (serendipity element) +

  27. Triangular Higher-order elements For 3-node triangle element For 6-node triangle element

  28. Six-node triangle

  29. Gauss quadrature formulas for a triangle

  30. Midside node The midside node should be as near as possible to the center of the side The node should not be outside of ¼< s/l <3/4 . l

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