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ENGR5004 / Week 9
ENGR216 2019 Q A1Past paperOld spec ENGR2162:225 marks30 min

ENGR216 Summer 2019 Q A1[VALID] official answers

Answer ALL parts (a) - (e)

Section A Appendix (as printed with the paper):

The generalised Hooke's laws for multi-axial load of an isotropic homogeneous material are:

ϵx=+σxE−νσyE−νσzE\epsilon_x = +\frac{\sigma_x}{E} - \frac{\nu\sigma_y}{E} - \frac{\nu\sigma_z}{E} ϵy=−νσxE+σyE−νσzE\epsilon_y = -\frac{\nu\sigma_x}{E} + \frac{\sigma_y}{E} - \frac{\nu\sigma_z}{E} ϵz=−νσxE−νσyE+σzE\epsilon_z = -\frac{\nu\sigma_x}{E} - \frac{\nu\sigma_y}{E} + \frac{\sigma_z}{E}

where σ\sigma denotes a normal stress component, ϵ\epsilon denotes a normal strain component, xx, yy, and zz denote the axis of a three-dimensional Cartesian system, EE denotes Young's modulus and ν\nu denotes Poisson's ratio.

The expression of the maximum and minimum normal stresses in a plane stress state are:

σmax/min=σx+σy2±(σx−σy2)2+τxy2\sigma_{max/min} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}

Formulas you may need
  • Generalised Hooke's law ϵx=(σx−νσy−νσz)/E\epsilon_x = (\sigma_x - \nu\sigma_y - \nu\sigma_z)/E etc. (given in the question; learn this)
  • Principal stresses σmax/min=σx+σy2±(σx−σy2)2+τxy2\sigma_{max/min} = \frac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2} (given in the question; learn this)
  • Maximum in-plane shear stress τmax=(σx−σy2)2+τxy2=σmax−σmin2\tau_{max} = \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2} = \frac{\sigma_{max}-\sigma_{min}}{2} (radius of Mohr's circle) (learn this)
  • Normal strain ϵ=δ/L\epsilon = \delta / L (learn this)
  • Plane stress: σz=τxz=τyz=0\sigma_z = \tau_{xz} = \tau_{yz} = 0 (learn this)
  1. (a)
    Making use of the generalised Hooke's laws for isotropic homogeneous materials subject to multi-axial load provided in the Appendix explain why a three-dimensional structure subject to a single axial load can deform also in directions different from that along which above said axial load acts.
    [5]Third
  2. (b)
    Provide the definition of plane stress state and one example of this multi-dimensional loading condition.
    [5]Third
  3. (c)
    A rectangular thin plate of homogeneous isotropic material is subject to a biaxial loading state made up of a tensile stress σx=120 MPa\sigma_x = 120\ \mathrm{MPa} in the xx direction and a tensile stress σy=171 MPa\sigma_y = 171\ \mathrm{MPa} in the yy direction. Determine the magnitude of maximum shear stress in the plate (HINT: the expression of the maximum shear stress can be obtained from that of the maximum and minimum normal stresses provided in the Appendix noting that the difference between maximum and minimum normal stresses is the diameter of Mohr's circle).
    [5]Third
  4. (d)
    Before being loaded, the plate has length lx=1571 mml_x = 1571\ \mathrm{mm} along the xx axis and ly=533 mml_y = 533\ \mathrm{mm} along the yy axis, and after being loaded it elongates by 0.6 mm0.6\ \mathrm{mm} in the xx direction and 0.4 mm0.4\ \mathrm{mm} in the yy direction. Determine Young's modulus and Poisson's ratio of the plate material.
    [5]
  5. (e)
    Knowing that the plate has thickness lz=20 mml_z = 20\ \mathrm{mm}, determine the variation of its thickness in mm due to the applied loads, indicating whether this is a contraction or an elongation.
    [5]