ENGR216 2025 Q A2 (ENGR5004)

ENGR216 2025 Q A2Past paperOld spec ENGR2162:125 marks30 min

ENGR216 Summer 2025 Q A2[VALID] official answers

Figure A2 shows a beam of negligible mass with flexural rigidity $EI$ and length $2L$ . The beam is constrained by a fixed support at end A and a roller at end B, and is subjected to a uniform distributed load $w_0$ .

Figure A2: beam of length 2L, fixed at A, roller at B, uniform load w0.

Formulas you may need

Equilibrium $\sum F_x = 0$ , $\sum F_y = 0$ , $\sum M = 0$ ; a fixed support gives two forces and a couple, a roller one force; degree of indeterminacy = unknown reactions minus independent equilibrium equations (learn this)
Elastic curve $\dfrac{d^2y}{dx^2} = \dfrac{M(x)}{EI}$ , slope $\theta = dy/dx$ (learn this; on the 2025 sheet but not the 2026 one)
Load, shear and moment: $\dfrac{dV}{dx} = -w(x)$ , $\dfrac{dM}{dx} = V(x)$ , with the positive shear and positive bending pictures (on the formula sheet)
Slope is extreme where $d\theta/dx = M/EI = 0$ ; deflection is extreme where $\theta = 0$ (learn this)
Pure bending $\sigma_{max} = \dfrac{|M| c}{I}$ (on the formula sheet)
Rectangle $I = \dfrac{bh^3}{12}$ , so a square of side $s$ has $I = s^4/12$ , $c = s/2$ (on the formula sheet)

(a)
Draw the free-body-diagram of the entire structure, indicating ONLY the nonzero reactions and applied loads, and state if the structure is statically determinate or indeterminate, justifying your answer.
[4]2:2
(b)
Determine all nonzero reactions acting on the beam as functions of $w_0$ and $L$ , knowing that the vertical displacement of its cross sections is: $y(x) = \frac{w_0 L^4}{EI}\left[-\frac{1}{4}\left(\frac{x}{L}\right)^2 + \frac{5}{24}\left(\frac{x}{L}\right)^3 - \frac{1}{24}\left(\frac{x}{L}\right)^4\right] \qquad \text{(Equation 3)}$ where $x$ denotes the position along the axis of the beam.
A_y (upward +) =w0*L
B_y (upward +) =w0*L
M_A (anticlockwise +) =w0*L^2
[6]
(c)
Draw as accurately as possible the bending moment diagram of the beam and the deformed shape of its mean line (elastic curve).
M_max sagging =w0*L^2
x at M_max =L
y_max (upward +) =w0*L^4/EI
[4]2:2
(d)
Calculate the positions along the beam at which the elastic curve has maximum slope as functions of $L$ .
x1 (steepest downward slope) =L
x2 (largest slope, at B) =L
[4]
(e)
Knowing that the beam has square cross-section, determine the minimum side length of the cross-section such that the maximum normal stress in the cross-section at end A does not exceed the material's allowable normal stress of 200 MPa. Use $w_0 = 20$ kN/m and $L = 2.0$ m.
s_min =mm
[7]2:2