Old spec Statics Session 9 Problem 2 (ENGR5004)

Old spec Statics Session 9 Problem 2TutorialOld spec ENGR2162:212 min

ENGR216 (old spec) Statics session 9 worked example 2 (Problem 2, slides 6-9) official answers

A simply supported timber beam AB of length $L$ and rectangular cross section (width $b$ , height $h$ ) carries a single concentrated load $P$ at its midpoint C.

Simply supported timber beam AB with load P at midpoint C (AC = CB = L/2); rectangular section of width b and height h.

Formulas you may need

Maximum transverse shear stress, rectangle: $\tau_{max} = \dfrac{3V}{2A}$ (learn this)
Maximum bending stress $\sigma_{max} = \dfrac{|M|c}{I}$ , rectangle $I = \dfrac{bh^3}{12}$ (on the formula sheet)
Simply supported beam with central load: $V_{max} = P/2$ , $M_{max} = PL/4$ (learn this: from the shear and moment diagrams)

(a)
Show that the ratio $\tau_{max}/\sigma_{max}$ of the maximum shear and normal stresses in the beam is equal to $h/(2L)$ .
(b)
Determine the cross section height $h$ and width $b$ , knowing that $L = 3.5$ m, $P = 60$ kN, $\tau_{max} = 1100$ kPa and $\sigma_{max} = 31.0$ MPa.
h =m
b =m
(c)
For the beam just designed, determine the maximum load $P$ that can be applied if the maximum allowable shear stress is 1600 kPa.
P_max =kN