Objectives:

• To understand the procedure of discretization and interpolation.

• To learn the algorithm of Gauss quadrature.

1. 1D element shape functions
Consider a 4-node cubic element in 1D. The length of the element is 3 with the first node located at xi = -2. The remaining nodes are equally spaced.

1) Construct the Lagrange shape functions for this element.

2) Interpolate the function ue(x) (e.g. the displacement along the axis of a uniform bar) in the element when the nodal values are given as

with the same unit as the coordinates of the nodes. (Writing out the expression for u°(x) would suffice).

3) Evaluate the derivatives of the shape functions, i.e. the Be matrix, and use it to find the expression for the derivative of u°(x) (i.e. the axial strain e(x) = due /dx).

4) Plot ue(x) and a°(x) obtained in (2) and (3).

5) What are the corresponding displacement field and strain field when the nodal displacements are given by

Why is this result expected?

2. Gauss quadrature

1) Write a short Matlab (or whatever language you prefer) program to perform 2 point Gauss integration of a function f over a given interval (a, b). Inputs: a, b, and the functional values of f at the two Gauss points x1 and x2, f 1) and f (x2). (The program should calculate the actual coordinates of the two Gauss points). Output: the (approximate) value of the integral ∫fdx. (If you are able, write a program with adjustable order of integration.)

2) Check your program with the following integrals by comparing to the exact values:

₀∫⁴ (x³+1)dx,  _-1∫² (x⁴ + 3x²)dx,  _-1∫¹ cos²Πx dx