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ENGR5004 / Week 9
ENGR216 2021 Statics Q A1Past paperOld spec ENGR2162:123 marks28 min

ENGR216 Summer 2021 Statics Q A1[VALID] official answers

Consider the cantilevered beam of length LL in Figure A1, subject to the transverse load FF. The beam has rectangular cross section of width ww and height hh, and the second area moment past the neutral axis of this cross section is II. The beam's material has Young's modulus EE. The line of action of FF is parallel to the side of height hh, as indicated in Figure A1.

Figure A1: cantilever AB of length L fixed at A with transverse load F at B; rectangular section w x h with point S at mid-height of a side and T at the top.
Figure A1: cantilever AB of length L fixed at A with transverse load F at B; rectangular section w x h with point S at mid-height of a side and T at the top.
Formulas you may need
  • Pure bending: σx=−MyI\sigma_x = -\dfrac{M y}{I}, σmax=∣M∣cI\sigma_{max} = \dfrac{|M| c}{I} (on the formula sheet)
  • Rectangle: I=bh312I = \dfrac{b h^3}{12}, A=bhA = bh, ρ=I/A\rho = \sqrt{I/A} (on the formula sheet)
  • Transverse shear stress: τ=VQIt\tau = \dfrac{VQ}{It}, τmax=3V2A\tau_{max} = \dfrac{3V}{2A} for a rectangle (learn this; on the 2025 sheet but not the 2026 one)
  • 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)
  • Euler buckling: Pcr=π2EIminLe2P_{cr} = \dfrac{\pi^2 E I_{min}}{L_e^2} with Le=2LL_e = 2L for a cantilever (on the formula sheet; Le=2LL_e = 2L is also given in the question)
  • Safety factor and strength limit: Pallow=Pcr/FSP_{allow} = P_{cr}/FS, P≤σallAP \le \sigma_{all} A (learn this)
  1. (a)
    Considering the beam cross section at the fixed support A and making use of the plane stress theory, show that the maximum normal stress at point S of the beam surface is given by Equation 1 σS,max=3F2wh(1)\sigma_{S,max} = \frac{3F}{2wh} \qquad (1) and state if this normal stress is parallel to the beam axis, justifying your answer.
    [7]2:2
  2. (b)
    Assuming L=1L = 1 m, F=40F = 40 kN, w=50w = 50 mm, determine the height hh of the beam cross section if the maximum allowable normal stress of the material of 120 MPa is not to be exceeded in any part of the beam.
    [9]
  3. (c)
    The beam designed in part (b) is now subject only to a centric compressive axial load, and E=200E = 200 GPa. Assuming maximum allowable normal stress of 120 MPa and buckling safety factor of 2, determine the maximum centric compressible axial load applicable to the beam fulfilling the strength and buckling stability requirements. Please note that the equivalent length LeqL_{eq} of a cantilevered beam subject to axial load is twice its geometric length.
    [7]2:2