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ENGR5004 / Week 10
ENGR273 2026 Q2Past paperCurrent spec2:225 marks30 min

ENGR273 Summer 2026 Q2[VALID] official answers

Figure Q2-1 shows a rod of negligible mass of length LL subjected to a compressive axial load FF. The rod is constrained by a fixed support at both end A and end B. The rod's cross section is depicted in Figure Q2-2, in which b=10ab = 10a.

Figures Q2-1 and Q2-2: rod along z, fixed at A and B, load F at B; I-section with flanges b wide and a thick, web of thickness a and clear height b between the flanges (overall depth b + 2a).
Figures Q2-1 and Q2-2: rod along z, fixed at A and B, load F at B; I-section with flanges b wide and a thick, web of thickness a and clear height b between the flanges (overall depth b + 2a).
Formulas you may need
  • Euler critical load Pcr=π2EIminLe2P_{cr} = \dfrac{\pi^2 E I_{min}}{L_e^2}, σcr=π2E(Le/ρmin)2\sigma_{cr} = \dfrac{\pi^2 E}{(L_e/\rho_{min})^2}, ρ=I/A\rho = \sqrt{I/A} (on the formula sheet)
  • Effective lengths: fixed-free Le=2LL_e = 2L, pinned-pinned LL, fixed-pinned 0.7L0.7L, fixed-fixed 0.5L0.5L (on the formula sheet)
  • Rectangle Ix=112bh3I_x = \dfrac{1}{12} b h^3, Iy=112hb3I_y = \dfrac{1}{12} h b^3, A=bhA = bh (on the formula sheet)
  • Parallel-axis theorem I=Ic+Ad2I = I_c + A d^2 (or build the I-section by subtracting rectangles) (learn this)
  • Axial stress σ=F/A≤σall\sigma = F/A \le \sigma_{all}; buckling safety factor Pcr≥FS⋅FP_{cr} \ge FS \cdot F (learn this)
  1. (a)
    Assuming that the compressive load FF exceeds by a small amount Euler's critical load, determine and clearly state if the rod will buckle in the xzxz or yzyz plane.
    [6]
  2. (b)
    Draw as accurately as possible the shape of the buckled rod, labelling the two Cartesian axes defining the plane in which the rod buckled, and fulfilling the applied constraints in the drawn schematic.
    [6]
  3. (c)
    Using F=40F = 40 kN, allowable normal stress of 100 MPa, Young's modulus of 200 GPa, L=1.6L = 1.6 m and a safety factor of 2 for the buckling critical load, calculate the minimum value of aa required for the rod not to buckle and not to yield.
    [7]
  4. (d)
    Assuming that the rod designed in part (c) is used with the fixed support at A but no constraint at B, and using a safety factor of 2 in the buckling analysis, determine and state if the rod will buckle or yield under the new constraint set-up.
    [6]