ENGR216 2018 Q A3 (ENGR5004)

ENGR216 2018 Q A3Past paperOld spec ENGR2162:225 marks30 min

ENGR216 Summer 2018 Q A3[VALID] official answers

Consider the cantilever beam of length $L$ depicted in the left plot of Figure A3. The beam is subject to a linear distributed load which is zero at $x = 0$ (point A) and is $w_0$ N/m at $x = L$ (point B), has Young's modulus $E$ , and the second area moment of its cross section about the neutral axis is $I$ .

Figure A3: left, cantilever AB of length L built in at A (x = 0), free at B, with a downward linearly varying load from zero at A to w0 at B; right, rectangular cross section of width w and height h.

Formulas you may need

Distributed load resultant $P = \int w\,dx$ and its position $x_P$ (on the formula sheet)
Sign convention for positive shear and bending, $dV/dx = -w(x)$ , $dM/dx = V(x)$ (on the formula sheet; also printed in this paper's appendix)
Transverse shear stress $\tau = \dfrac{VQ}{It}$ , $\tau_{max} = \dfrac{3V}{2A}$ for a rectangle (learn this; on the 2025 sheet but not the 2026 one; $VQ/It$ also printed in this paper's appendix)
Rectangle $I = \dfrac{w h^3}{12}$ (on the formula sheet)
Elastic curve $EI\,\dfrac{d^2y}{dx^2} = M(x)$ , integrated twice with boundary conditions (learn this; on the 2025 sheet but not the 2026 one; also printed in this paper's appendix)

(a)
Determine the magnitude and the orientation of the reactions acting on the beam.
R_A (up +) =w0*L
M_A (CCW +) =w0*L^2
[6]
(b)
Demonstrate that the shear load $V(x)$ along the beam is given by: $V(x) = \frac{w_0 L}{2} - \frac{w_0 x^2}{2L}$ and sketch as accurately as possible the function $V(x)$ along the beam axis.
[4]
(c)
Assuming the beam has a rectangular cross section with sides of width $w$ and height $h$ , shown in the right plot of Figure A3, and considering the shear stress variability in the cross section, indicate the position along the beam ( $x$ value) where the maximum shear stress occurs, and, at such position, determine the maximum shear stress and the shear stress at $y = h/4$ as functions of $w_0$ , $L$ , $w$ and $h$ .
tau_max (at x = 0, y = 0) =w0*L/(w*h)
tau at y = h/4 =w0*L/(w*h)
[7]
(d)
Knowing that the bending moment $M(x)$ along the beam is: $M(x) = -\frac{w_0 L^2}{3} + \frac{w_0 L x}{2} - \frac{w_0 x^3}{6L}$ determine the equation of the deformed mean line of the beam.
[5]
(e)
Determine the rotation (magnitude and direction) of the end B of the beam as a function of $w_0$ , $E$ , $I$ and $L$ .
theta_B (CCW +) =w0*L^3/(E*I)
[3]