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ENGR5004 / Week 10
ENGR216 2019 Q A3Past paperOld spec ENGR2162:225 marks30 min

ENGR216 Summer 2019 Q A3[VALID] official answers

Answer ALL parts (a) - (d)

Section A Appendix (as printed with the paper):

Euler's critical load PcrP_{cr} for doubly pinned rods is:

Pcr=π2EIL2P_{cr} = \frac{\pi^2 E I}{L^2}

where II is the smallest second area moment of the rod cross section, EE is the elastic or Young's modulus and LL is the rod length.

The second area moment II of a solid circular cross section of radius rr about any diameter is:

I=14πr4I = \frac14 \pi r^4

Formulas you may need
  • Euler load Pcr=π2EIminLe2P_{cr} = \dfrac{\pi^2 E I_{min}}{L_e^2}; pinned-pinned Le=LL_e = L (given in the question; also on the formula sheet)
  • Solid circle I=πr4/4I = \pi r^4/4 (given in the question; also on the formula sheet); hollow circle I=π(r24−r14)/4I = \pi(r_2^4 - r_1^4)/4 (on the formula sheet)
  • Radius of gyration ρ=I/A\rho = \sqrt{I/A}, slenderness ratio Le/ρL_e/\rho, critical stress σcr=π2E(Le/ρ)2\sigma_{cr} = \dfrac{\pi^2 E}{(L_e/\rho)^2} (on the formula sheet)
  • Normal stress σ=P/A\sigma = P/A and allowable load with a safety factor Pall=Pcr/FSP_{all} = P_{cr}/FS (learn this)
  1. (a)
    Provide a brief general definition of buckling in the context of the three main structural design requirements.
    [5]Third
  2. (b)
    Define the slenderness ratio of a doubly pinned beam, and briefly explain how the likelihood of buckling occurrence varies with the value of this parameter.
    [6]Third
  3. (c)
    A beam pinned at both ends has length of 1.3 m1.3\ \mathrm{m} and consists of a solid steel rod with Young's modulus E=120 GPaE = 120\ \mathrm{GPa}. The beam has a circular cross section with diameter of 40 mm40\ \mathrm{mm}. To reduce the weight of the member, the solid rod is replaced by a hollow rod, obtained by cutting a concentric circular hole with diameter of 24 mm24\ \mathrm{mm}. Assume that the effective length of the beam is the same in all planes through the beam axis. Determine the value of Euler's critical load of both the solid and the hollow rods, and calculate the percentage reduction of the critical load of the hollow beam with respect to the solid beam. The general expression of Euler's critical load and the second area moment of a solid circular section past any diameter are provided in the Appendix.
    [6]
  4. (d)
    Determine the maximum compressive load applicable to the hollow beam if the maximum allowable normal stress is 60 MPa60\ \mathrm{MPa} and a safety factor of 2 is used for computing the actual critical buckling load.
    [8]