Computer-Aided Mechanical Design

Question 1:

Figure 1 shows a 2D mesh for a structural plan stress analysis problem. We assume that an element with No 15 is such a quadrilateral element with 4 corner nodes in the mesh, and its 4 node coordinates in the global coordinate system (in millimetres) are given in Figure 2. Let Young's modulus E = 200GPa and Poisson's ratio ν = 0.3.

As indicated in Figure 3, using the isoparametric Q4 element and 4-node Gaussian quadrature (numerical integration to approximate the element stiffness matrix), calculate:

1 The stiffness matrix of the element 15 by using hand calculations
2 The stress of the element at the local location (ξ=0, η=0) of the element, if we know the displacements for the element No 15 are {d}={10, 5; 20, 1; 15, 5; 10, 1}.

Figure 1: A structural mesh for a 2-D plan stress analysis problem

Figure 2: The quadrilateral element in two dimensions

Figure 3: The isoparametric element using 2-by-2 rule for Gaussian quadrature