ENGR216 2018 Q A1 (ENGR5004)

ENGR216 2018 Q A1Past paperOld spec ENGR2162:125 marks30 min

ENGR216 Summer 2018 Q A1[VALID] official answers

A pin-ended metal post of length $L = 2\ \mathrm{m}$ with circular cross section of radius $R$ supports a compressive axial load $P = 400\ \mathrm{kN}$ applied at distance $e$ from the section centre, as indicated in Figure A1. Use a Young's modulus $E$ of $70\ \mathrm{GPa}$ .

Figure A1: circular cross section with axes x and y through the centre 0; A at the top (y = R), B at the left (x = -R), C at the bottom (y = -R); the load acts on the y axis at distance e below 0.

Formulas you may need

Euler critical load, pin-ended: $P_{cr} = \dfrac{\pi^2 E I_{min}}{L_e^2}$ with $L_e = L$ (on the formula sheet; also printed in this paper's appendix)
Circle: $A = \pi R^2$ , $I = \dfrac{\pi R^4}{4}$ , $\rho = \sqrt{I/A} = R/2$ (on the formula sheet)
Eccentric axial loading: $F = P$ , $M = Pd$ , $\sigma = \dfrac{N}{A} - \dfrac{M y}{I}$ , $M > 0$ compresses fibres at $y > 0$ (on the formula sheet; also printed in this paper's appendix)
Safety factor: $P_{cr} = FS \cdot P$ ; allowable stress $|\sigma| \le \sigma_{all}$ (learn this)
Uniaxial stress: maximum shear $\tau_{max} = |\sigma|/2$ on planes at $45^\circ$ (learn this)
Principal axes $\tan 2\theta_p = \dfrac{2\tau_{xy}}{\sigma_x - \sigma_y}$ , max shear axes $\tan 2\theta_s = -\dfrac{\sigma_x - \sigma_y}{2\tau_{xy}}$ (learn this; printed in this paper's appendix but not on the 2026 sheet)

(a)
Assuming $e = 0$ , an allowable normal stress of $80\ \mathrm{MPa}$ and a buckling safety factor of 2.5, determine the minimum radius of the cross section required for yield- and buckling-free operation.
R_min =mm
[3]2:2
(b)
Assuming $e = 0.25R$ and using the value of $R$ obtained at (a): determine the distance (with sign) of the neutral axis from the $x$ -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 $y$ axis of the section clearly indicating the neutral axis;
y_NA (up +) =mm
[4]2:2
(c)
at positions A, B and C determine the normal stress acting on the post cross section and the maximum shear stress on the post surface.
sigma_A (tension +) =MPa
sigma_B (tension +) =MPa
sigma_C (tension +) =MPa
tau_max at A =MPa
tau_max at B =MPa
tau_max at C =MPa
[6]2:2
(d)
Consider three small squared surface elements with two sides parallel to the axis of the post (axis $z$ ) and centered at points A, B and C. At these three points, determine the orientation of the principal axes and the maximum shear axes with respect to the axis $z$ .
theta_p at B and C (from z) =deg
theta_s at B and C (from z) =deg
[7]
(e)
Briefly comment on how the stress variations due to the axial load eccentricity $e = 0.25R$ may impact on the strength and the buckling stability of the post using the value of $R$ determined at (a).
[5]First