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## Q. 4.6

A wall-rack, used to store round steel bars, consists of two I-section cantilever beams fixed in the wall. The bars are stacked in a triangular fashion as shown in Fig. 4.25(a). The total weight of the bars is 75 kN. The permissible bending stress for the cantilevers is 165 N/mm².

## Verified Solution

$\text { Given } \quad W=75 kN \quad \sigma_{b}=165 N / mm ^{2}$.

Step I Calculation of bending moment
There are two cantilever beams and the load

Select a standard rolled I-section beam from the following table:

 $I_{x x}\left(m m^{4}\right)$ h (mm) b (mm) Designation $688.2 \times 10^{4}$ 150 80 ISLB 150 $1096.2 \times 10^{4}$ 175 90 ISLB 175 $1696.6 \times 10^{4}$ 200 100 ISLB 200 $2501.9 \times 10^{4}$ 225 100 ISLB 225 $3717.8 \times 10^{4}$ 250 125 ISLB 250

supported by each beam is (75/2) or 37.5 kN. For a triangular load distribution, the centre of gravity of the resultant load is at a distance of (2000/3) mm from the wall. Therefore,

$M_{b}=\left(37.5 \times 10^{3}\right)\left(\frac{2000}{3}\right)=25 \times 10^{6} N – mm$.

$\text { Step II Calculation of }\left(I_{x x} / y\right)$.

From Eq. (4.12),

$\sigma_{b}=\frac{M_{b} y}{I}$            (4.12).

$\frac{I_{x x}}{y}=\frac{M_{b}}{\sigma_{b}}=\frac{25 \times 10^{6}}{165}=151.51 \times 10^{3} mm ^{3}$.

Step III Selection of beam
The cross-section of the beam is shown in Fig. 4.25
(b), (y = h/2)

Trial I  Suppose beam (ISLB 175) is suitable for the application. For this beam,

$\frac{I_{x x}}{y}=\frac{1096.2 \times 10^{4}}{(175 / 2)}=125.28 \times 10^{3} mm ^{3}$.

$\text { Since the required }\left(I_{x x} / y\right) \text { is }\left(151.51 \times 10^{3}\right) mm ^{2}$,

beam (ISLB 175) is not suitable.
Trial II Suppose beam (ISLB 200) is suitable for the application. For this beam,

$\frac{I_{x x}}{y}=\frac{1696.6 \times 10^{4}}{(200 / 2)}=169.66 \times 10^{3} mm ^{3}$.

$\left(I_{x x} / y\right)>\left(151.51 \times 10^{3}\right) mm ^{3}$.

Therefore, the cantilever beams of standard cross-section ISLB 200 are suitable for this application.