Calculate the integral using Gauss Quadrature where fs s2

Calculate the integral using Gauss Quadrature, [where f(s) = s^2 + 2s + 3] and using a 3-point function rule. \"Gaussian Quadrature can only be exact for basic elements, never for higher order elements.\" True or false ? Explain.

Solution

a)Partial derivatives of shape functions with respect to the Cartesian coordinates x and y are required for the strain and stress calculations. Because shape functions are not directly functions of x and y but of the natural coordinates and , the determination of Cartesian partial derivatives is not trivial. The derivative calculation procedure is presented below for the case of an arbitrary iso parametric quadrilateral element with n nodes.

Higher order Gauss rules are tabulated in standard manuals for numerical computation. For example, the widely used Handbook of Mathematical Functions rules with up to 96 points. For p > 6 the abscissas and weights of sample points are not expressible as rational numbers or radicals, and can only be given as floating-point numbers.

b)Computations of shape function derivatives to form the strain-displacement matrix.

Numerical integration is essential for practical evaluation of integrals over isoparametric element domains. The standard practice has been to use Gauss integration because such rules use a minimal number of sample points to achieve a desired level of accuracy. This economy is important for efficient element calculations, since a matrix product is evaluated at each sample point. The fact that the location of the sample points in Gauss rules is usually given by non-rational numbers is of no concern in digital computation.

Verify that for integration rules p=2,3,4 the stiffness matrix does not change and has three zero eigenvalues, which correspond to the three two-dimensional rigid body modes. On the other hand, for p = 1 the stiffness matrix is different and displays five zero eigenvalues, which is physically incorrect.