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ENGR5004 / Week 9
ENGR216 2018 Q A2Past paperOld spec ENGR2162:125 marks30 min

ENGR216 Summer 2018 Q A2[VALID] official answers

Consider the free-body-diagram of the axle AD depicted in the left plot of Figure A2, with dimensions AB=CD=250 mmAB = CD = 250\ \mathrm{mm} and BM=MC=760 mmBM = MC = 760\ \mathrm{mm}. The axle cross section, depicted in the right plot of Figure A2, has radius R=100 mmR = 100\ \mathrm{mm}. The axle is subject to an engine torque T0=150 kNmT_0 = 150\ \mathrm{kNm} applied at M, two resistive torques T0/2T_0/2 applied at A and D, two vertical loads P0=400 kNP_0 = 400\ \mathrm{kN} applied at A and D, and two vertical reactions P0P_0 applied at B and C.

Figure A2: left, axle AD along x with z up: downward loads P0 and resistive torques T0/2 at A and D, upward reactions P0 at B and C, engine torque T0 at the midpoint M. Right, circular cross section of radius R in the y-z plane with surface point K at the top (on the z axis).
Figure A2: left, axle AD along x with z up: downward loads P0 and resistive torques T0/2 at A and D, upward reactions P0 at B and C, engine torque T0 at the midpoint M. Right, circular cross section of radius R in the y-z plane with surface point K at the top (on the z axis).
Formulas you may need
  • Sign convention for positive shear and bending, dV/dx=−wdV/dx = -w, dM/dx=VdM/dx = V (on the formula sheet)
  • Bending stress σx=−MyI\sigma_x = -\dfrac{M y}{I} (yy measured from the neutral axis; here the vertical coordinate is zz), I=πR44I = \dfrac{\pi R^4}{4} (on the formula sheet; also printed in this paper's appendix)
  • Torsion τ=TcJ\tau = \dfrac{T c}{J}, J=πR42J = \dfrac{\pi R^4}{2} (learn this; printed in this paper's appendix but not on the 2026 sheet)
  • Principal stresses σ1,2=σx+σy2±(σx−σy2)2+τxy2\sigma_{1,2} = \dfrac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\dfrac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}, tan⁡2θp=2τxyσx−σy\tan 2\theta_p = \dfrac{2\tau_{xy}}{\sigma_x - \sigma_y} (learn this; printed in this paper's appendix but not on the 2026 sheet)
  • Maximum in-plane shear τmax=(σx−σy2)2+τxy2\tau_{max} = \sqrt{\left(\dfrac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}; Mohr's circle centre (σx+σy2,0)\left(\dfrac{\sigma_x + \sigma_y}{2}, 0\right) and radius τmax\tau_{max} (learn this)
  1. (a)
    Determine the expression of the bending moment M(x)M(x), shear load V(x)V(x) and torsional load T(x)T(x) along the entire axle.
    [4]2:2
  2. (b)
    Sketch M(x)M(x), V(x)V(x) and T(x)T(x) along the entire axle.
    [3]2:2
  3. (c)
    State in which portion (xx interval) of the axle one has a plane stress state on the axle surface and explain why.
    [4]
  4. (d)
    Determine the magnitude of the maximum normal stress at the point K indicated in the right plot of Figure A2 acting on the axle surface at any axial position between B and M, and state if this stress is tensile or compressive.
    [7]
  5. (e)
    Determine the magnitude of the maximum shear stress at the point K indicated in the right plot of Figure A2 acting on the axle surface at any axial position between B and M.
    [3]2:2
  6. (f)
    Calculate the centre coordinates and the radius of Mohr's circle referring to the axle surface point K at any position between B and M, and draw as accurately as possible Mohr's circle in the σ\sigma-τ\tau plane, indicating in such plane the points corresponding to principal stresses and maximum shear stress.
    [4]2:2