ENGR216 2022 Statics Q A1 (ENGR5004)

ENGR216 2022 Statics Q A1Past paperOld spec ENGR2162:125 marks30 min

Consider the cantilevered structure of Figure A1, loaded by a parabolic distributed normal load $w(x)$ , given by

$w(x) = w_0\left[-\left(\frac{x}{L}\right)^2 + 2\left(\frac{x}{L}\right)\right] \qquad (1)$

where $x$ is the coordinate along the mean line of the horizontal member BC (origin at B), which has length $L$ .

Formulas you may need

Equilibrium of a rigid body: $\sum F_x = 0$ , $\sum F_y = 0$ , $\sum M_A = 0$ (learn this)
Resultant of a distributed load $P = \int w\,dx$ and its position $x_P = \dfrac{\int x\,w\,dx}{\int w\,dx}$ (on the formula sheet)
Load, shear and moment: $\dfrac{dV}{dx} = -w(x)$ , $\dfrac{dM}{dx} = V(x)$ , positive shear and bending as pictured (on the formula sheet)
Elastic curve $EI\,\dfrac{d^2y}{dx^2} = M(x)$ , so $\theta_C - \theta_B = \dfrac{1}{EI}\displaystyle\int_B^C M\,dx$ (learn this; on the 2025 sheet but not the 2026 one)
Combined axial force and bending: $\sigma_x = \dfrac{N}{A} - \dfrac{M y}{I}$ , $|\sigma|_{max} = \dfrac{|N|}{A} + \dfrac{|M| c}{I}$ (on the formula sheet, as eccentric axial loading)
Solid circle: $A = \pi r^2$ , $I = \dfrac{\pi r^4}{4}$ (on the formula sheet)

(a)
Draw the free-body-diagram of the entire structure, indicating all nonzero reactions.
A_y (up +) =w0*L
A_x =w0*L
[7]2:2
(b)
Demonstrate that the bending moment exerted by the fixed support at A has magnitude $M_A = \frac{5}{12} w_0 L^2 \qquad (2)$ and state if this moment is clockwise or counter-clockwise.
M_A (CCW +) =w0*L^2
[6]2:2
(c)
Knowing that the bending moment $M(x)$ along the horizontal member BC is $M(x) = w_0 L^2\left[\frac{1}{12}\left(\frac{x}{L}\right)^4 - \frac{1}{3}\left(\frac{x}{L}\right)^3 + \frac{2}{3}\left(\frac{x}{L}\right) - \frac{5}{12}\right] \qquad (3)$ determine the rotation of section C relative to section B as a function of $w_0$ , $L$ , and the flexural rigidity $EI$ .
theta_C - theta_B (CCW +) =w0*L^3/EI
[6]
(d)
The cross section of members AB and BC are circular with radius of 50 mm, $L = 1$ m and the allowable normal stress of the material is 150 MPa. Briefly explain which cross section(s) of the given structure have maximum magnitude of the normal stress, and calculate the maximum value of $w_0$ in kN/m such that the normal stress in all cross sections of AB and BC never exceeds the allowable normal stress.
w0_max =kN/m
[6]