1) The triangular plane stress problem connected to a spring, is subjected to a uniformly distributed load and two point forces. Calculate the nodal displacements and the spring force using two triangular finite elements. k = 3x105 kN/m (spring constant) h = 0.20 m (thickness) E = 3x107 kN/m² (modulus of elasticity) v = 0.20 (Poisson's ratio) 1.5 m 40 kN 30 kN/m 50 kN 2 m 2 m

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A cylindrical bar made of brittle material has been subjected to pure torsion by a torque M and tension from the force F (Figure Q6.c). The torque gives rise to the shear stresses T = T = 32 MPa and the force F gives rise to the tensile stress ху yx 0, = 65 MPa. The stress along the y-axis is zero (o, = 0). Determine whether there will be a failure due to the maximum tensile stress if the tensile strength of the material is 90 MPa. F X Figure Q6.c

Two concentric plain helical springs of the same length are wound out of the same wire circular in cross section and supports a compressive load P. The inner spring consists of 20 turns of mean dimeter 160 mm and the outer spring has 18 turns of mean diameter 200 mm. Calculate the maximum stress induced in the spring if the diameter of wire is equal to 10 mm and Pis equal to 1000 N. (Take K,-1)

The bar shown in the figure is made of cold-drawn steel, having an yield strength of 280 MPa. Loads applied are F = 1 kN, P = 5 kN, and torque T = 20 kN. For each element shown in the figure (element A and element B, separately, • Determine the principal stresses at A and B (eigenvalues) • Using DET, determine whether the bar will yield. What is the factor of safety for the loading situation shown, based on DET? 15-mm D. -100 mm

2. A composite spring has two closed coiled springs connected in series, each spring has 12 coils at a mean diameter of 25mm. Find the wire diameter for one if the other is of 2.5mm diameter and the stiffness of the composite spring is 700N/m. Estimate the greatest load that can be carried by the composite spring, and the corresponding extension, for a maximum shear stress of 180MPa. Take G=80GPa.

2-A wheel of diameter d and width w carrying a load F rolls on a flat rail. Estimate the maximum contact pressure for these materials. Por vedit -3D d 5 in 150 mm 3 in W 2 in 40 mm 1.25 in F 600 lbf 2 kN 250 lbf Wheel Material Steel Steel Cast iron Rail Material Steel Cast iron Cast iron.

Consider the loaded shaft illustrated in Figure Q4 (i). The shaft is simply supported at A and D. The shoulder fillet radius is 0.3 cm at B and C. If the shaft is subjected to a force of F-800 N, determine the following: (a) The reaction forces at the two ends of the shaft A and D. (b) The bending moments at shoulder fillets B andC. (c) The nominal stresses at shoulder fillets B and C. (d) The stress concentration factor at shoulder fillet B. (e) The maximum stress in the shaft taking into account stress concentrations. 800 N T-0.3 can A d, - 3 cm d, = 5 cm d, - 3 cm 5 cm 8 cm 8 cm 12 cm Figure Q4 (i) The reference parameters are shown in Figure Q4 (ii) in the page that follows. For 0.002 srlds 0.3 and 1.01 s Dlds 6.0, the stress concentration factor for bar bending is given as: kn -4.4 = 0.632 + 0.377 -0.14 – 0.363(P/a)" + 0.503(P/a) 1- 2.39(P/a) + 3.368(2la)* -0.5 The nominal stress is: 32M %3D nd3

For the rod in the figure, an allowable stress of 10 ksi, calculate the maximum torque T which can be applied. d= 0.5 in, D= 0.835 in, r= 0.045 in .

Consider a one-element triangular mesh shown in Figure 2. The edge AC is fixed. The edge BC is subject to lin- early varying traction normal to the edge as shown in Figure 2. Assume plane-stress conditions with thickness 1 = 1 cm and Young's modulus E = 70 GPa and Poisson's ratio v = 0.3. 150 kN/m 50 kN/m C (0,0.3) B (0.4,0.3) a) Specify the boundary condi- tions. b) Construct the stiffness matrix. A (0,0) c) Calculate the global force ma- Figure 2: Triangular element. All coordinates are in meters. trix. d) Solve for the unknown displace- ments and compute the stress at (0.1, 0.2).