ENGR216 2019 Q A2 (ENGR5004)

ENGR216 2019 Q A2Past paperOld spec ENGR2162:125 marks30 min

ENGR216 Summer 2019 Q A2[VALID] official answers

Answer ALL parts (a) - (d)

Consider the statically determinate beam with flexural rigidity $EI$ and length $L$ depicted in Figure A2, subject to the distributed load $w(x)$ , where $x$ denotes the coordinate along the horizontal axis depicted in the figure with origin at end A of the beam.

Figure A2: beam AB of length L, roller support at A and pin support at B, downward distributed load w(x) decreasing from w0 at A to zero at B; x axis along the beam from A, y axis upwards at A.

Formulas you may need

Elastic curve $\dfrac{d^2y}{dx^2} = \dfrac{M}{EI}$ (given in the question; learn this; on the 2025 sheet but not the 2026 one)
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)
$dV/dx = -w(x)$ , $dM/dx = V(x)$ and the positive shear / positive bending sign convention (on the formula sheet)
Equilibrium $\sum F_x = 0$ , $\sum F_y = 0$ , $\sum M = 0$ ; roller gives one reaction normal to its surface, pin gives two (learn this)
Boundary conditions for a simply supported beam: $y = 0$ at each support (learn this)

(a)
Knowing that the bending moment $M(x)$ is positive over the entire beam, and noting that the relationship between $M(x)$ and the local deflection $y(x)$ is: $\frac{d^2y}{dx^2} = \frac{M}{EI} \qquad (1)$ where the $y$ axis is depicted in Figure A2 and has origin at end A of the beam, sketch the deformed shape of the beam mean line. For your sketch, use the same Cartesian axes $x$ and $y$ defined in Figure A2.
[5]2:2
(b)
Sketch the free body diagram of the loaded beam in Figure A2.
[5]Third
(c)
Knowing that the distributed load $w(x)$ is given by: $w(x) = w_0\left[1 - \frac{x^3}{L^3}\right] \qquad (2)$ where $w_0$ is a constant of dimension N/m, demonstrate that the vertical reactive forces $R_A$ and $R_B$ at the beam ends A and B are respectively: $R_A = \frac{9}{20} w_0 L \quad \text{and} \quad R_B = \frac{3}{10} w_0 L$
R_A (up +) =w0*L
R_B (up +) =w0*L
[10]2:2
(d)
Knowing that the equation of the deformed mean line of the beam is: $y(x) = \frac{w_0}{EI}\left(\frac{x^7}{840L^3} - \frac{x^4}{24} + \frac{3Lx^3}{40}\right) + C_1 x + C_2 \qquad (3)$ determine the constants $C_1$ and $C_2$ by imposing the compatibility condition of the beam deformation expressed by Equation (3) with the constraints acting on the beam depicted in Figure A2.
C1 =w0*L^3/EI
C2 =w0*L^4/EI
[5]2:2