Final - MAE 131A/SE110A – FALL 2009 Final Exam Monday...

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Unformatted text preview: MAE 131A/SE110A – FALL 2009 Final Exam Monday December 7, 2009 Time: 3:00 to 6:00PM Problem 1 (6 points) A 2‐inch wide and 0.125‐inch thick bar is connected to a support by a 0.375‐inch diameter pin, as shown in the figure at right. An axial force of 4,000lbs is to be carried by the bar. The ultimate strength (i.e., failure stress) of the bar in tension is 30,000psi and the ultimate strength of the pin in shear is 24,000psi. (1.5 points each): 1. Draw the free‐body diagram for the pin and for the bar. 2. Determine the normal stress in the bar. 3. Determine the shear stress in the pin. 4. Determine the factors of safety for both the bar and the pin. Problem 2 (6 points) The shaft shown in the figure at right consists of a 30mm diameter solid section CD and a tube AC that has an outside diameter of 45mm and an inside diameter of 35mm. The shaft is subjected to concentrated torques at B, C, and D. The elastic shear modulus is G = 75GPa. Compute: 1. the maximum shear stress in the shaft, 2. the angle of rotation of C relative to A, and 3. the angle of rotation of D relative to A. Make sure to draw the free‐body diagram for each section of the shaft. Problem 3 (10 points) Consider the cantilever beam AB, fix at A into a rigid support, and carrying a uniform load of 1kN/m as well as a concentrated load of 2kN at a point 0.2m from its free end, as shown. The beam is fabricated from a steel plate of 10mm uniform thickness, welded to form a wide flange with the cross‐section, as shown on the right. 2kN 1. (1.5 point): Find the area of 60mm 1.0kN/m the cross section, the first moment of the area with A B 50mm respect to the bottom of 0.2m 0.6m the beam (dashed line), identify the neutral axis and 2. 3. 4. 5. 6. calculate the moment of inertia and the corresponding section modulus. (1 point): Identify and calculate the reactions at the rigid support. (2 points): Draw the shear and bending moment diagrams. (1.5 points): Find the maximum tensile and the maximum compressive stresses in the beam. (1 point): Find the shear stress at the upper weld. (3 points): Find the slope, dy/dx, and the deflection, y, as functions of x, and evaluate them at x = 0.6m. The elastic modulus is 200GPa. Problem 4 (12 points) Consider the cantilever beam AB, fix at A into a rigid support and simply supported at point B, as shown (note: this is a statically indeterminate beam). The beam has a moment of inertia I and an elastic modulus E. It is carrying a uniform load of 1kN/m. (Express your results in terms of EI.) 1.0kN/m 1.0kN/m 1. (2 point): Identify all the reactions and B assuming that the reaction at the support A x B is RB, use the two equilibrium equations 2.0m to express MA and RA (the reactions at the fix end A) in terms of RB and the applied load. 2. (2 points): Sketch the shear and bending moment diagrams. 3. (2 points): Express the bending moment as a function of x (that is find M = M(x), assuming you know the reaction RB. 4. (2 points): Write down the corresponding differential equation for y (x) (the lateral deflection of the beam) and the boundary conditions (that is the conditions that the slope and deflection, dy/dx and y, must satisfy at point A, and the condition that y must satisfy at B). 5. (4 points): Integrate the equation EI d2y/dx2 = M(x) (assume EI is known) to find dy/dx and y as functions of x. Problem 5 (6 points) Consider a column with a circular cross section (E = 200GPa), fixed to a rigid foundation and free at the other end (the fixed‐free case). The column is 400mm long and has a 20mm diameter. 1. (2 points): Calculate the Euler buckling load for this column. 2. (2 points): If the yield stress of the material is 700MPa, calculate the Euler buckling stress for the column. 3. (2 points): If instead of fixed‐free, the column was pinned at both ends (the pinned‐pinned case), what would be the Euler buckling load. ...
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