Reference no: EM132287979
Question 1 -
(a) A truss is loaded as shown in Figure 1a. The cross-sectional area of all members is 1000 mm2 and made of steel. Modulus of Elasticity (E) is 200 GPa.
(i) Determine the total stiffness matrix for the truss.
(ii) Using the finite element method determine the displacement at node 2.
(b) A beam is loaded as shown in Figure 1b. A downward settlement of 0.02 m occurs at support. (Note: Use the penalty approach to treat the settlement). EI = 4000 kNm2 are used for members.
Using the finite element method,
(i) Determine the total stiffness matrix for the beam.
(ii) Determine the equivalent nodal loads.
(iii) Determine the displacement and rotation at node 3.
Question 2 -
(a) Derive the shape functions in intrinsic (ξ, η) coordinates for a 4-noded quadrilateral element using the interpolation and rigid translation conditions.
(b) The coordinates of the element shown in Figure 2a are in metres and the element has a thickness of 1 m.
The nodal displacements of the element are shown in Table 1 below.
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Table1
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Node 1
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Node 2
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Node 3
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Node 4
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x-disp (u) m
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0.1
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0.4
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0.8
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0.4
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y-disp (v) m
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0.2
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0.3
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0.3
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0.1
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Calculate the displacements at a point (ξ, η) = (0.5, 0.5) in intrinsic space. What is the corresponding point in global coordinates?
(c) A steel plate (dimensions in meters) is loaded as shown in Figure 2b.
Thickness of the plate, t = 0.01 m, E = 200 GPa and v = 0.3.
The element stiffness matrix for the element (units in N/m) is given below.
(i) Determine the displacement at each node.
(ii) Determine the displacement at point P located at (0.3, 0.3) in the element.
(iii) Determine the strains and stresses in the plate.