Tutorial 1 Problem 2 (ENGR5004)

Tutorial 1 Problem 2Exercise sheetCurrent spec2:120 min

ENGR5004 Statics Tutorial Sheet 1, Problem 2 official answers

A steel beam has a rectangular cross section of height $l_x = 20$ mm and width $l_y = 30$ mm, and length $l_z = 1$ m (lengths $l_x$ , $l_y$ and $l_z$ are measured respectively along $x$ , $y$ and $z$ axes of a Cartesian system). The material of the beam has Young modulus $E = 200$ GPa, Poisson ratio $\nu = 0.29$ , and maximum allowable normal stress of 175 MPa. The beam is subject to a compressive centric axial load $P_z$ of 80 kN applied at its ends (load acts along $z$ axis).

Formulas you may need

Normal stress under centric axial load: $\sigma_z = P_z / A$ (learn this)
Generalised Hooke's law: $\epsilon_x = \dfrac{\sigma_x}{E} - \dfrac{\nu\sigma_y}{E} - \dfrac{\nu\sigma_z}{E}$ (and cyclic) (learn this)
Maximum shear stress under axial load, on planes at $45^\circ$ : $\tau_{max} = \dfrac{P}{2A}$ (learn this)
Bulk modulus: $k = \dfrac{E}{3(1-2\nu)}$ (learn this)
Dilatation: $e = \epsilon_x + \epsilon_y + \epsilon_z = \dfrac{1-2\nu}{E}(\sigma_x + \sigma_y + \sigma_z)$ (learn this)

(a)
State whether the area of the cross section of the beam will increase or decrease under the effect of the applied centric axial load and explain why.
(b)
Determine the variation $\delta_x$ of the section height $l_x$ in mm, indicating if such variation is a contraction or an elongation.
delta_x (elongation +) =mm
(c)
Determine the maximum axial load $(P_z)_{max}$ applicable to the beam and the maximum mean shear stress in these conditions.
(P_z)max =kN
tau_max =MPa
(d)
In the loading condition (c), state whether the uniformly distributed normal load to be applied on the beam faces normal to the $x$ axis leading to a zero variation of the section height $l_x$ is compressive or tensile and justify your answer.
(e)
In the loading condition (c), determine the magnitude of the uniformly distributed normal load $\sigma_x$ to be applied on the beam faces normal to the $x$ axis resulting in zero variation of the section height $l_x$ .
sigma_x (tension +) =MPa
(f)
Calculate the bulk modulus $k$ of the beam material, and the dilatation $e$ of the beam due to the axial load $(P_z)_{max}$ and the distributed load $\sigma_x$ , stating if the applied loads result in an increment of reduction of the beam volume.
k =GPa
e =- (dimensionless)