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## Q. 4.12

A wall bracket with a rectangular cross-section is shown in Fig. 4.39. The depth of the cross-section is twice of the width. The force P acting on the bracket at 600 to the vertical is 5 kN. The material of the bracket is grey cast iron FG 200 and the factor of safety is 3.5. Determine the dimensions of the cross-section of the bracket.
Assume maximum normal stress theory of failure.

## Verified Solution

Given P = 5 kN

$S_{u t}=200 N / mm ^{2} \quad(f s)=3.5 \quad d / w=2$.

Step I Calculation of permissible stress

$\sigma_{\max }=\frac{S_{u t}}{(f s)}=\frac{200}{3.5}=57.14 N / mm ^{2}$         (i).

Step II Calculation of direct and bending tensile stresses
The stress is maximum at the point A in the section XX. The point is subjected to combined bending and direct tensile stresses. The force P is resolved into two components—horizontal component $P_h$ and vertical component $P_v$ .

$P_{h}=P \sin 60^{\circ}=5000 \sin 60^{\circ}=4330.13 N$.

$P_{v}=P \cos 60^{\circ}=5000 \cos 60^{\circ}=2500 N$.

The bending moment at the section XX is given by

$M_{b}=P_{h} \times 150+P_{v} \times 300$.

$=4330.13 \times 150+2500 \times 300$.

$=1399.52 \times 10^{3} N – mm$.

$\sigma_{b}=\frac{M_{b} y}{I}$.

$=\frac{\left(1399.52 \times 10^{3}\right)(t)}{\left[\frac{1}{12}(t)(2 t)^{3}\right]}=\frac{2099.28 \times 10^{3}}{t^{3}} N / mm ^{2}$.

The direct tensile stress due to component $P_h$ is given by,

$\sigma_{t}=\frac{P_{h}}{A}=\frac{4330.13}{2 t^{2}}=\frac{2165.07}{t^{2}} N / mm ^{2}$.

The vertical component $P_v$ induces shear stress at the section XX. It is however small and neglected.

Step III Calculation of dimensions of cross-section
The resultant tensile stress $\sigma_{\max }$. at the point A is given by,

$\sigma_{\max }=\sigma_{b}+\sigma_{t}=\frac{2099.28 \times 10^{3}}{t^{3}}+\frac{2165.07}{t^{2}}$              (ii).

Equating (i) and (ii),

$\frac{2099.28 \times 10^{3}}{t^{3}}+\frac{2165.07}{t^{2}}=57.14$.

or

$t^{3}-37.89 t-36739.24=0$.

Solving the above cubic equation by trial and error method,

$t=33.65 mm \cong 35 mm$.

The dimensions of the cross-section are 35 × 70 mm.