Isoparametric Formulation

1. Isoparametric Formulation

2. linear-strain triangular element • the development of element matrices and equations expressed in terms of a global coordinate system becomes an enormously dificult task. • To solve this problem, the isoparametric formulation was developed. • The isoparametric method will lead to a simple computer program formulation. • The isoparametric formulation allows elements to be created that are nonrectangular and have curved sides.

3. Isoparametric Formulation of the Bar Element Stiffness Matrix • Isoparametric element equations are formulated using a natural (or intrinsic) coordinate system s. • This is defined by element geometry and not by the element orientation in the global-coordinate system.

5. Developing the isoparametric formulation of the stiffness matrix of a simple linear bar element Step 1 Select Element Type

6. Step 2 Select a Displacement Function Since u and x are defined by the same shape functions at the same nodes, the element is called isoparametric.

7. Step 3 Define the Strain/Displacement and Stress/Strain Relationships The displacement u, is now a function of s. Therefore, we must apply the chain rule of differentiation and The stress matrix is

8. Step 4 Derive the Element Stiffness Matrix and Equations In general, we must transform the coordinate x to s because [B] is, in general, a function of s. For the one-dimensional case,

9. Rectangular Plane Stress Element Step 1 Select Element Type

10. Step 2 Select Displacement Functions

12. Step 3 Define the Strain/ Displacement and Stress/Strain Relationships

13. Step 4 Derive the Element Stiffness Matrix and Equations Because the [B] matrix is a function of x and y, integration must be performed. where

14. Isoparametric Formulation of the Plane Element Stiffness Matrix The term isoparametric means that when then we use

15. Step 1 Select Element Type The s and t axes need not be orthogonal, and neither has to be parallel to the x or y axis. The orientation of s-t coordinates is such that the four corner nodes and the edges of the quadrilateral are bounded by +1 or -1.

16. For the special case

19. Step 2 Select Displacement Functions The displacement functions within an element are defined by the same shape functions used to define the element shape:

20. Step 3 Define the Strain/Displacement and Stress/Strain Relationships

24. where

25. Step 4 Derive the Element Stiffness Matrix and Equations However, [B] is now a function of s and t. Then we use This integration to determine the element stiffness matrix is usually done numerically.

26. Body Forces

27. Surface Forces The surface-force matrix, say, along edge t = 1 with overall length L: For the case of uniform (constant) ps and pt

28. Numerical Integration Gaussian Quadrature: Sampling points are located symmetrically with respect to the center of the interval. one-point Gaussian quadrature

29. In general, Gaussian quadrature using n points (Gauss points) is exact if the integrand is a polynomial of degree 2n-1 or less.

30. Two-Point Formula There are four unknown parameters to determine: Assuming a cubic function for y: we will assume

31. the error , is to vanish for any C0 and C2

32. In two dimensions: we need not use the same number of Gauss points in each direction, but this is usually done.

33. for a 2x2 Gauss rule the double summation, reduces to a single summation over the four points for the rectangle.

34. In three dimensions:

35. Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature Evaluation of the Stiffness Matrix

36. Evaluation of Element Stresses The stresses are not constant within the quadrilateral element. In practice, the stresses are evaluated at the same Gauss points used to evaluate the sti¤ness matrix [k]. To reduce the data, it is often practical to evaluate stress at s=0, t = 0 instead.