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DEPARTMENT OF MECHANICAL ENGINEERING

EXAMINATION

MECH2413: ENGINEERING MECHANICS

December 13, 2021

Time: 2:30 p.m. — 3:30 p.m.

SECTION A

A1. (a) Calculate the center of mass of the following beam using the dimensions given in the figure. (5 marks)

(b) Calculate the rectangular moment of inertia about y-axis using the dimensions shown in the following figure. Give the value of product of inertia Iyz. (5 marks)

(c) A beam is under a concentrated force and a concentrated moment as shown below.

(i) Draw the free body diagram. (2 marks)

(ii) Calculate the shear force and bending moment based on the equilibrium conditions. (6 marks)

(iii) Plot the shear force and bending moment diagrams. (2 marks)

A2. (a) Give the shear force and bending moment boundary conditions for the following supports. (4 marks)

(b) The beam shown below is under a distributed force. Given that l = 3 m, a = 2m, and q0 = 10 N/m.

(i) Give the boundary and matching conditions. (4 marks)

(ii) Use the integral method to calculate the shear force and bending moment. (6 marks)

(iii) Plot the shear force and bending moment diagrams. (2 marks)

(c) Given the normal stress distribution in the cross-sectional area, σ(z) = cz, how to calculate a statically equivalent bending moment? (4 marks)

A3. (a) Consider a thin-walled spherical pressure vessel with inner radius r and wall thickness t. It is subjected to an internal gage pressure p.

(i) Derive vessel formulas for membrane stresses σφ and σt. (4 marks)

(ii) Determine the maximum shear stresses Tmax for in-plane rotation of the stress element. (2 marks)

(iii) Neglecting stress in radial direction, determine the maximum shear stresses Tmax for out-of-plane rotation of the stress element. (4 marks)

(b) The cantilever beam AB of length L shown in figure below carries a uniformly distributed load of intensity wo, which includes the weight of the beam. Calculate the deflection of the beam. (10 marks)