Statics Progress Test 2024 Q1 (ENGR5004)

Statics Progress Test 2024 Q1Past paperOld spec ENGR2162:135 min

ENGR216 Statics Progress Test 2024, Question 1 (solved in Statics session 21)[VALID] official answers

Consider the structure of Figure 1: a vertical member PQ of length $L$ rigidly joined at Q to a horizontal member QR of length $2L$ . P rests on a roller support (vertical reaction only) and R is pinned. Member PQ is loaded by the harmonic distributed normal load

$w(x) = w_0 \sin\left(\frac{\pi x}{L}\right)$

acting horizontally to the right, with position $x$ measured along PQ with origin at P, and $w_0$ in N/m.

Figure 1: frame PQR; roller at P, pin at R; half-sine load w(x) acting to the right on the vertical member PQ (length L); QR has length 2L.

Formulas you may need

Resultant of a distributed load $P = \int_0^{L} w(x)\,dx$ and its position $x_P - x_0 = \dfrac{\int_0^{L} w(x)(x - x_0)\,dx}{P}$ (on the formula sheet)
$\dfrac{dV}{dx} = -w(x)$ , $\dfrac{dM}{dx} = V(x)$ , $V_D - V_C = -\int_{x_C}^{x_D} w\,dx$ , $M_D - M_C = \int_{x_C}^{x_D} V\,dx$ (on the formula sheet)
Positive shear and positive bending sign convention (pictures) (on the formula sheet)
Combined axial force and bending (eccentric loading form) $\sigma_x = \dfrac{N}{A} - \dfrac{My}{I}$ (on the formula sheet)
Solid circle $I = \dfrac{\pi r^4}{4}$ (on the formula sheet)
$\int x \sin(ax)\,dx = \dfrac{\sin(ax)}{a^2} - \dfrac{x\cos(ax)}{a}$ (learn this; integration by parts)
Static equilibrium $\sum F_x = 0$ , $\sum F_y = 0$ , $\sum M = 0$ ; roller gives one force, pin gives two (learn this)

(a)
Draw the free-body diagram of the entire structure, indicating all nonzero reactions.
2:2
(b)
Determine all reactions acting on the structure as functions of $w_0$ and $L$ .
R_P (upward) =w0*L
R_RH (to the left) =w0*L
R_RV (downward) =w0*L
2:2
(c)
Calculate the axial load distribution $N(x)$ , the shearing load distribution $V(x)$ and the bending moment distribution $M(x)$ along members PQ and QR.
M at Q (both members) =w0*L^2
V in PQ at Q =w0*L
(d)
Draw the $N(x)$ , $V(x)$ and $M(x)$ diagrams along members PQ and QR.
2:2
(e)
Draw the free-body diagram of the vertical member PQ, indicating clearly all nonzero external and internal loads, and give the expressions of all internal and external loads acting on it as functions of $w_0$ and $L$ .
F_QH on PQ (to the left) =w0*L
F_QV on PQ (downward) =w0*L
M_Q on PQ (clockwise) =w0*L^2
2:2
(f)
Calculate the magnitude of the maximum compressive stress, in kPa, in the cross section of member PQ at $0.75L$ from end P. Assume a circular cross section of radius $R = 200$ mm, and use $L = 1$ m and $w_0 = 30$ kN/m.
M at x = 0.75L =kNm
max compressive stress =kPa