ENGR216 2024 Statics Q A2 (ENGR5004)

ENGR216 2024 Statics Q A2Past paperOld spec ENGR2162:125 marks30 min

ENGR216 Summer 2024 Statics Q A2[VALID]

Figure A2 shows a beam of negligible mass with flexural rigidity $EI$ and length $L$ . The beam is constrained by a roller at end A and a fixed support at end B, and is subjected to a distributed load $w(x)$ given by:

$w(x) = w_0 \sin\left(\frac{\pi x}{2L}\right) \qquad \text{(Equation 1)}$

where $x$ denotes the distance from end A along the beam.

Figure A2: beam of length L, roller at A, fixed at B, load w(x) rising from zero at A.

Formulas you may need

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)
Equilibrium of a rigid body in the plane: $\sum F_x = 0$ , $\sum F_y = 0$ , $\sum M = 0$ (learn this)
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 $EI\,\dfrac{d^2y}{dx^2} = M(x)$ , slope $\theta = \dfrac{dy}{dx}$ (learn this; on the 2025 sheet but not the 2026 one)
Solid circle $I = \dfrac{\pi r^4}{4}$ (on the formula sheet)
Degrees to radians, $1^\circ = \pi/180\ \mathrm{rad}$ (learn this)

(a)
Draw the free-body-diagram of the entire structure, indicating only nonzero reactions and applied loads.
[4]Third
(b)
Explain why the system is statically indeterminate (3 marks), and write the equations enforcing the static equilibrium of the system (3 marks).
[6]2:2
(c)
Knowing that the elastic curve of the beam is: $y(x) = \frac{w_0 L}{EI}\left(\frac{2}{\pi}\right)^4 \left[-\frac{1}{2}x^3 + \frac{3}{2}L^2 x - L^3 \sin\left(\frac{\pi x}{2L}\right)\right] \qquad \text{(Equation 2)}$ calculate all reactions acting on the beam as functions of $w_0$ and $L$ .
A_y (up +) =w0*L
B_y (up +) =w0*L
M_B (CCW +) =w0*L^2
[6]
(d)
Provide a schematic of the elastic curve, making sure your drawing is consistent with the applied constraints.
[4]2:2
(e)
The beam has a circular cross section. Determine the minimum value of the section radius such that the slope of the elastic curve at end A does not exceed 1 degree. Use $w_0 = 80$ kN/m, $L = 3.0$ m and $E = 200$ GPa.
r_min =mm
[5]2:2