ENGR216 2022 Statics Q A2 (ENGR5004)

ENGR216 2022 Statics Q A2Past paperOld spec ENGR2162:225 marks30 min

ENGR216 Summer 2022 Statics Q A2[VALID] official answers

Consider the statically determinate system given by the cantilevered beam with flexural rigidity $EI$ and length $L$ in Figure A2, subjected to a triangular distributed normal load varying from 0 at end A to $w_0$ at end B. The abscissa $x$ is the coordinate along the horizontal axis depicted in the figure with origin at end A of the beam.

Figure A2: cantilever fixed at A, free at B, triangular load from 0 at A to w0 at B. Figure A3: rectangular section w x h.

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 $x_P$ (triangle: $P = w_0L/2$ at $2L/3$ from the zero end) (on the formula sheet)
Load, shear and moment: $\dfrac{dV}{dx} = -w(x)$ , $\dfrac{dM}{dx} = V(x)$ (on the formula sheet)
Elastic curve $EI\,\dfrac{d^2y}{dx^2} = M(x)$ , so $M = EI\,y''$ ; clamped end: $y = 0$ , $y' = 0$ (learn this; on the 2025 sheet but not the 2026 one)
Rectangle bending about the axis parallel to the width $w$ : $I = \dfrac{w h^3}{12}$ (on the formula sheet)
Statically indeterminate beams by compatibility/superposition; cantilever with end load $P$ : $\delta = \dfrac{PL^3}{3EI}$ (learn this)

(a)
For the system in Figure A2, sketch the deformed shape of the beam longitudinal axis as accurately as possible, accounting for the action of the constraint at the end A.
[6]
(b)
Sketch the free body diagram of the loaded beam in Figure A2, clearly indicating all reactions and loads.
A_y (up +) =w0*L
M_A (CCW +) =w0*L^2
[6]Third
(c)
Knowing that the equation of the deformed longitudinal axis of the beam is $y(x) = \frac{w_0}{EI}\left(-\frac{L^2 x^2}{6} + \frac{L x^3}{12} - \frac{x^5}{120L}\right) \qquad (4)$ determine the magnitude of the bending moment at the fixed support as a function of $w_0$ and $L$ .
|M_A| =w0*L^2
[4]Third
(d)
The value of the maximum allowable vertical displacement of the beam free end is 1 mm, and the beam has a rectangular cross section of width $w$ and height $h$ , as shown in Figure A3. Determine the minimum value of $h$ to fulfill the aforementioned rigidity constraint. Use $w_0 = 50$ kN/m, $w = 150$ mm, $L = 1.5$ m and $E = 200$ GPa.
h_min =mm
[4]
(e)
State one constraint which could be applied to the end B to make the beam statically indeterminate, and, for the system thus obtained, list all nonzero reactions acting on the beam.
[5]2:1