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ENGR5004 / Week 10
Tutorial 4 Problem 2Exercise sheetCurrent specFirst30 min

ENGR5004 Statics Tutorial Sheet 4, Problem 2 official answers

A cantilevered metal beam has length L=0.6L = 0.6 m, circular cross section of radius RR, and is subject to a compressive axial load P=650P = 650 kN applied at distance ee from the section centre, as indicated in the figure below. The material of the beam has Young's modulus E=200E = 200 GPa and allowable normal stress of 150 MPa.

Parts (b) to (e): assuming e=0.40Re = 0.40R and using the value of RR obtained at (a).

Circular cross section with centre 0, x axis horizontal and y axis vertical; the load acts at distance e below 0 on the y axis; A at the top (y = R), B at the left end of the x axis, C at the bottom (y = -R).
Circular cross section with centre 0, x axis horizontal and y axis vertical; the load acts at distance e below 0 on the y axis; A at the top (y = R), B at the left end of the x axis, C at the bottom (y = -R).
Formulas you may need
  • Euler critical load Pcr=π2EIminLe2P_{cr} = \dfrac{\pi^2 E I_{min}}{L_e^2}; fixed-free Le=2LL_e = 2L (on the formula sheet)
  • Solid circle A=πR2A = \pi R^2, I=πR44I = \dfrac{\pi R^4}{4} (on the formula sheet)
  • Eccentric axial loading: F=PF = P, M=PdM = Pd, σ=PA−MyI\sigma = \dfrac{P}{A} - \dfrac{My}{I} (on the formula sheet)
  • Maximum in-plane shear for a uniaxial surface element: τmax=∣σ∣2\tau_{max} = \dfrac{|\sigma|}{2}; principal axes along the stress directions, maximum-shear axes at 45∘45^\circ (learn this)
  1. (a)
    Assuming e=0e = 0, and a buckling safety factor of 2.0, determine the minimum radius RR of the cross section required for yield- and buckling-free operation.
  2. (b)
    determine distance (with sign) of the neutral axis from the xx-axis. Briefly comment on this result with regard to the sign of the normal stress in the section, and sketch the diagram of the resulting normal stress along the yy axis of the section clearly indicating the neutral axis;
  3. (c)
    at positions A, B and C determine the normal stress acting on the beam cross section, and the maximum shear stress on beam surface.
  4. (d)
    Briefly comment on results of (c) with regard to capability of the designed beam to withstand the considered eccentric axial load.
  5. (e)
    At points A, B and C determine the orientation of the principal axes and the maximum shear axes with respect to the axis zz coinciding with the axis of the beam.