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ENGR5004 / Week 10
ENGR216 2017 Q A3Past paperOld spec ENGR2162:125 marks30 min

ENGR216 Summer 2017 Q A3[VALID] official answers

Answer ALL parts (a) - (e)

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

Figure A3: symmetric two-bar frame. Rods AB and CB, each of length L, are pinned to the ground at A (left) and C (right) and pinned together at the apex B. Each rod makes angle theta with the horizontal (marked at A). A vertical load P acts downwards at B; a dashed vertical line through B marks the axis of symmetry.
Figure A3: symmetric two-bar frame. Rods AB and CB, each of length L, are pinned to the ground at A (left) and C (right) and pinned together at the apex B. Each rod makes angle theta with the horizontal (marked at A). A vertical load P acts downwards at B; a dashed vertical line through B marks the axis of symmetry.
Formulas you may need
  • Euler load for a pinned-pinned column Pcr=π2EIminL2P_{cr} = \dfrac{\pi^2 E I_{min}}{L^2} (Le=LL_e = L) (given in the question; also on the formula sheet)
  • Solid circle I=πr4/4I = \pi r^4/4, A=πr2A = \pi r^2 (given in the question; also on the formula sheet); hollow circle I=π(r24−r14)/4I = \pi(r_2^4 - r_1^4)/4, A=π(r22−r12)A = \pi(r_2^2 - r_1^2) (on the formula sheet)
  • Strength requirement σ=F/A≤σall\sigma = F/A \le \sigma_{all}; buckling with a safety factor Pcr≥FS×FP_{cr} \ge FS \times F (learn this)
  • Axial deformation δ=FLAE\delta = \dfrac{FL}{AE} (learn this)
  • Equilibrium of a pin joint, ∑Fx=0\sum F_x = 0, ∑Fy=0\sum F_y = 0, with two-force members carrying force along their axes (learn this)
  1. (a)
    Provide a brief general definition of buckling in the context of the three main structural design requirements.
    [3]Third
  2. (b)
    Consider the statically determinate symmetric structure depicted in Figure A3, made up of 2 pinned rods subject to a vertical load PP of 50 kN50\ \mathrm{kN} acting on the joint B, as indicated in the figure. Both rods have length LL of 500 mm500\ \mathrm{mm} and θ=60∘\theta = 60^\circ. Their material has Young's modulus E=200 GPaE = 200\ \mathrm{GPa}, and allowable normal stress of 150 MPa150\ \mathrm{MPa}. Assume a buckling safety factor of 2, and also that the Euler critical load (axial force) of the rods is that of doubly pinned columns. Determine the internal force in each rod.
    [3]Third
  3. (c)
    Assuming that both rods have circular cross section, determine the minimum radius of the cross section required for the structure to safely support the load PP, fulfilling buckling and strength requirements. Euler's critical load for doubly pinned rods and the second area moment of solid circular cross section about any diameter are given in the Appendix.
    [5]2:2
  4. (d)
    Assuming that both rods have hollow circular cross section with inner radius of 10 mm10\ \mathrm{mm}, determine the minimum outer diameter of the cross section to safely support the load PP.
    [7]
  5. (e)
    Starting from the load specifications and final geometry as given in question (d) above, determine the outer diameter of the rods' cross section if the length of each rod must not decrease by more than 400 μm400\ \mu\mathrm{m}.
    [7]