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ENGR5004 / Week 9
Tutorial 3 Problem 2Exercise sheetCurrent spec2:125 min

ENGR5004 Statics Tutorial Sheet 3, Problem 2 official answers

Consider the free-body diagram of the axle AD depicted in the left plot below, with dimensions AB = CD = 500 mm and BM = MC = 1000 mm. The axle cross section, depicted in the right plot below, has radius R=150R = 150 mm. The axle is subject to an engine torque T0=350T_0 = 350 kNm applied at position M, two resistive torques T0/2T_0/2 applied at A and D, two vertical loads P0=700P_0 = 700 kN applied at B and C, and two vertical reactions P0P_0 applied at A and D.

Left: axle AD along x with z up; P0 down at B and C, P0 up at A and D, torque T0 at M, torques T0/2 at A and D. Right: circular cross section of radius R in the yz plane with point K on the surface at the top (on the z axis).
Left: axle AD along x with z up; P0 down at B and C, P0 up at A and D, torque T0 at M, torques T0/2 at A and D. Right: circular cross section of radius R in the yz plane with point K on the surface at the top (on the z axis).
Formulas you may need
  • Load-shear-moment relations and sign convention: dVdx=−w\dfrac{dV}{dx} = -w, dMdx=V\dfrac{dM}{dx} = V (on the formula sheet)
  • Bending stress σx=−MzI\sigma_x = -\dfrac{Mz}{I} (formula sheet: σx=−My/I\sigma_x = -My/I with the vertical coordinate), solid circle I=πR44I = \dfrac{\pi R^4}{4} (on the formula sheet)
  • Torsion: τ=TcJ\tau = \dfrac{Tc}{J}, J=πR42J = \dfrac{\pi R^4}{2} (learn this)
  • Principal stresses and maximum in-plane shear: σ1,2=σx2±(σx2)2+τ2\sigma_{1,2} = \dfrac{\sigma_x}{2} \pm \sqrt{\left(\dfrac{\sigma_x}{2}\right)^2 + \tau^2}, τmax=(σx2)2+τ2\tau_{max} = \sqrt{\left(\dfrac{\sigma_x}{2}\right)^2 + \tau^2} (learn this)
  1. (a)
    Determine the expression of the bending moment M(x)M(x), shearing load V(x)V(x) and torsional load T(x)T(x) along the entire axle.
  2. (b)
    Sketch M(x)M(x), V(x)V(x) and T(x)T(x) along the entire axle.
  3. (c)
    State at which portions (xx interval) of the axle, one has a plane stress state on the axle surface and justify your answer.
  4. (d)
    Determine the magnitude of the maximum normal stress at point K indicated in the right plot below acting on the axle surface at any axial position between B and M, and state if this stress is tensile or compressive.
  5. (e)
    Determine the magnitude of the maximum shear stress at the point K indicated in the right plot below acting on the axle surface at any axial position between B and M.