Final exam 2012, questions PDF

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Solid Mechanics...

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Question 1 (20/100) Determine the forces supported by the pins at A, B and C for the frame in Figure 1 loaded by the two 200 N forces. The weight of the members can be neglected. Sketch the shear force and bending moment diagrams of the AB segment of the frame.

N m

4m

4m

N 3m

m 1.5 m

Figure 1

Question 2 (5/100) Consider a cylinder of height h=200mm and top and bottom area A=7850mm2, subject to an axial load applied following a quasi-static load-unload cycle. The applied load P and the relative displacement between the top and bottom of the cylinder are recorded during the test. The maximum measured force Pmax=50,000N and the maximum displacement umax=0.02mm. Plot the expected  curve in case: 1) The material of the cylinder is linear elastic; 2) The material is non-linear elastic; 3) The material is linear elastic up to =5MPa (correspondent displacement u=0.008mm) and it is then plastic. What is the value of the Young Modulus for case 1 and 3? Question 3 (5/100) Which are the three sets of equations used to solve a statically indeterminate structure?

Question 4 (30/100) For the structure shown in Figure 2, the bar on the left is made of bronze (cross section area Ab=1500mm2, Young Modulus Eb=105 GPa, temperature coefficient b=21.6x10-6/°C) and the bar on the right is made of aluminium (cross section area Aa=1800mm2, Young Modulus Ea=73 GPa, temperature coefficient a=23.2x10-6/°C). Knowing that a 0.5mm exists when the temperature is 20°C determine (a) the temperature at which the normal stress in the aluminium bar will be equal to -90MPa, (b) the corresponding exact length of the aluminium bar.

0.5 mm 0.45 m

0.35 m

Figure 2 Question 5 (20/100) Column ABC in Figure 3 has a uniform rectangular cross section with b=13 mm and d=24 mm. The column is braced in the xz plane at its midpoint C and carries a centric load P of magnitude 3.6 kN. Knowing that a factor of safety of 3.4 is required, determine the largest allowable length L. Use E=200GPa

Figure 3

Question 6 (20/100) A beam with the I-shaped cross section showed in Figure 4 supports a bending moment equal to 33625.8 Nm and a shear force equal to 110310.4 N. Calculate (a) the maximum value of the normal stress in the beam and (b) the maximum value of the principal stress at the junction of the flange and the web. The moment of inertia Ix is equal to 118625956.3 mm4.

13

8 mm 318 mm

x 13

167 mm y Figure 4

Formulas Question 4: Question 5:

  T  2 EI Pcr 

L2e

2

Question 6:



 VQ   ;  max       2 It 2 2...