UNIVERSITY OF GLASGOW
Degrees of MEng, BEng, MSc and BSc in Engineering
Mechanics of Solids 3 (ENG3037)
Thursday 13 December 2018
09:30-11:30
Duration: 2 hours
Attempt any THREE questions
The numbers in square brackets in the right-hand margin indicate the marks allotted to the
part of the question against which the mark is shown. These marks are for guidance only.
A FORMULA SHEET IS PROVIDED AT THE END OF PAPER
An electronic calculator may be used provided that it does not have a facility for either
textual storage or display, or for graphical display.
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Question 1
The 3D state of stress at a particular point Q in a material (defined in a given x,y,z coordinate
system) is given by:
σxx = 77 MPa, σyy = 31 MPa, σzz = -46 MPa, τxy = 0 MPa, τxz = 0 MPa and τyz = 0 MPa
A new orthogonal coordinate system is defined by rotating the original axes system so that the
̂ + 0𝑘̂), but the
new positive y-axis still lies in the same direction as the original y-axis (0𝑖̂ + 1𝑗
new positive x-axis lies in the direction of the vector 2𝑖̂ + 0𝑗̂ + 1𝑘̂ . (𝑖̂, 𝑗̂ and 𝑘̂ , here, are unit
vectors in the original x, y and z directions).
(a) Determine the 3x3 matrix of direction cosines L for the new coordinate axes (with
respect to the original coordinate system).
[10]
(b) Determine the nine-component stress tensor for the same material point, but in the new
orthogonal coordinate system.
[10]
(c) (i) In the new coordinate system, which of the three orthogonal planes (passing through
point Q) has the maximum absolute value of resultant shear stress?
[4]
(ii) Determine the global maximum absolute value of resultant shear stress at point Q.
i.e. the maximum over all planes that pass through the point Q.
[6]
Question 2
The 3D state of stress at a particular point P in a material (defined in a given x,y,z coordinate
system) is given by the matrix of stresses:
57 0
𝜎𝑖,𝑗 = [ 0 50
24 0
24
0 ] MPa
43
(a) The principal stresses for this system are integer values. One of the principal stresses is
exactly 50 MPa. Establish the principal stress cubic equation and determine the
maximum, intermediate and minimum principal stresses at point P.
[15]
(b) Determine the angles (α1, β1, γ1) that the maximum principal stress direction makes
with each of the original x,y,z coordinate axes.
[9]
(c) Determine the octohedral normal stress and the octohedral shear stress at point P. [6]
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Question 3
A rectangular strain gauge rosette bonded at a point O on a free surface of an engineering
component gave the following readings during testing: εa = 1240(10-6), εb = 400(10-6) and εc =
200(10-6). The rosette (with gauges marked a, b and c) is shown in Figure Q3. The Young’s
Modulus and Poisson’s ratio for the material from which the component is made are E = 210
GPa and ν = 0.3, respectively.
(a) Determine the in-plane principal strains and the maximum absolute value of in-plane
engineering shear strain at point O. (Note: in-plane means: “in the x-y plane”)
[12]
(b) Hence, determine the maximum and minimum in-plane principal stresses at point O.
What is the value of the third principal stress at point O? i.e. the one acting
perpendicular to the surface.
[10]
(c) Now, determine the value of the third principal strain (i.e. the principal strain normal to
the surface).
[4]
(d) Is the maximum absolute value of in-plane engineering shear strain calculated in Part
(a) also a maximum for all possible planes passing though Point O? i.e. Calculate the
global maximum absolute value of engineering shear strain at Point O and compare to
the value calculated in Part (a).
[4]
Figure Q3
Question 4
(a) Describe how yield occurs in metals and explain how the Von Mises and Tresca yield criteria
allow engineers to assess whether or not yield is occurring. Include a sketch of the yield
envelopes on the π-plane of principal stress space.
[8]
(b) A computer based finite element analysis of a mild steel engineering component gives the
following principal stresses at the most stressed point in the component: σ1 = 850 MPa, σ2 = 800
MPa and σ3 = 700 MPa. If the yield stress of the steel in simple tension is 360 MPa, determine
if yield will occur using the Von Mises yield criterion.
[6]
(c) A solid shaft is subjected to a combination of torsional and axial loading. The shaft diameter
is 10 mm and it transmits a torque of 32.5 Nm in combination with a compressive axial force of
15 kN. The shaft is made from steel with a yield stress of 300 MPa. Determine if yield will occur
in the shaft according to the Von Mises yield criterion.
[16]
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End of question paper
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