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(b) determine all nodal reaction forces and moments due to this moment, and represent these reactions on a sketch of the frame. Show that both force and moment equilibrium is satisfied.

(c) If, due to a manufacturing defect, the joint at the lower end of the vertical member undergoes an angular displacement of ML/4EI, whilst all other properties remain unchanged, obtain a new expression for the angular displacement at the common junction.

[0, —3M/16L, M/8, —3M/4L, 0, 0, 3M/16L, M/8, 3M/4L, 0, M/4, ML/16EI]

9.9 Using the displacement based finite element method and a three-node triangular membrane element representation, determine the nodal displacements in global coordinates for the continuum shown in Fig. 9.58. Take advantage of any symmetry, assume plane stress conditions and use only two elements in the discretisation. For the material assume Young's modulus, E = 200 GN/m2 and Poisson's ratio, v = 0.3.

[-3.00 x 10"6, 10.01 x 10"6, -3.00 x I0"6, 10.01 x 10~6 m]

9.10 A crude lifting device is fabricated from a triangular sheet of steel, 6 mm thick, as shown in Fig. 9.59. Assume for the material Young's modulus, E = 200 GN/m2 and Poisson's ratio, v = 0.3, and that plane stress conditions are appropriate.

(a) Taking advantage of any symmetry, ignoring any instability and using only a single three-node triangular membrane element representation, use the displacement based finite element method to predict the nodal displacements in global coordinates.

(b) Determine the corresponding element principal stresses and their directions, and show these on a sketch of the element. [-0.05, -0.17, -0.60 mm, 134.85 MN/m2 (T) at 31.7° from ^-direction, 51.50 MN/m2 (C)]

9.11 The web of a support structure, fabricated from steel sheet 1 mm thick, is shown in Fig. 9.60. Assume for the material Young's modulus, E = 207 GN/m2 and Poisson's ratio, v = 0.3, and that plane stress conditions are appropriate.

(a) Neglecting any stiffening effects of adjoining members and any instability and using only a single three-node triangular membrane element representation, use the displacement based finite element method to predict the nodal displacements with respect to global coordinates.

Mechanics of Materials 2 4 kN Total load

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