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ENGG2400 – Mechanics of Solids 1
T3 2019 Block Test 3
Question 1: (3 Marks)
A pressure vessel is loaded with torsional and axial loadings. A point P on the inner surface of the pressure vessel experiences a stress state in the x-y plane as shown in the figure. There is also an normal stress due to the internal pressure σzz = − 100 MPa .
a) Draw the Mohr’s Circle for the stress state in the x-y plane. Use Mohr’s circle to determine:
- The principal stresses σ1 , σ2 and the angle of rotation θp to the principal axes
- The maximum shear stressτxy ,max in the x-y plane and the angle of rotation θs to the maximum in-plane shear axes.
Label all calculated points, intersections, diameters and angles.
b) Draw the orientation of the principal axes and maximum in-plane shear axes relative to the x-y axes. Draw the stress state of rotated material elements orientated with the principal axes and maximum in-plane shear axes. Clearly label stress magnitudes and directions.
c) Calculate the absolute maximum shear stress τ abs,max .
d) Which plane does the absolute shear stress occur in?
1-2 plane
1-3 plane
2-3 plane
1, 2 and 3 refer to the principal axes such that σ1 > σ2 > σ3
Question 2: (3 Marks)
Abeam is pin supported at A and is roller supported atB. It is loaded as shown in the figure. The beam has a constant second moment of area I = 170 × 10−6 m and a Young’s modulus of E = 200 GPa .
a) Complete the free body diagram of the beam and calculate all support reactions
e) Which of the following curves best represent the deflection of the beam? The distance between vertical grid lines represents 1m.
Question 3: (3 Marks)
The cantilevered beam with length L has a force F applied to it at an angle θ= 30o above the horizontal. The beam has constant cross-sectional and material properties.
a) Calculate the normal force N , shear force V and bending moment M as functions of position on the beam.
b) Calculate the axial ( Ua )i , shear ( Us )i and bending strain ( Ub )i energies for the beam.
c) Calculate the displacement ∆ of the right end of the beam in the direction of the applied force.