Question

Question 1 (10 marks)

As indicated in Figure 1, we assume that there is one quadrilateral element (No. 15) with 4 corner

nodes (Q4) in the mesh. After meshing, the node coordinates in the global coordinate system

(millimetres) are given in Figure 2. Let Young’s modulus E = 100GPa and Poisson’s ratio ν = 0.3,

and the element thickness is h=1 mm.

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 [K] of the element

2 The stress of the element, if the displacements of the element are given as {d}={10, 5; 20, 1; 15, 5;

10, 1}×10-4 (mm). Note: stress location is at (ξ=0, η=0).

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

2

Figure 2: The quadrilateral element in two dimensions

( ) ( )

( ) ( )

( ) ( )

( ) ( )

1 1

2 1

1 2

2 2

Gauss point 1: , 0.5773, 0.5773

Gauss point 2: , 0.5773, 0.5773

Gauss point 3: , 0.5773,0.5773

Gauss point 4: , 0.5773,0.5773

ξ η

ξ η

ξ η

ξ η

 =− − 

 = −

 = − 

 = 

1

2

1

Weights: 1

W

W

 = 

 =

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


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Answer 1

3

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