The beam AB is used in a railroad yard for loading and unloading cars. The hoist travel along the bottom flange of the I section beam AB (fig. a). Assume the beam AB is pinned to the column at B and roller supported at A. At what position of the hoist load, the Bending Moment in beam AB will be maximum? The C/S details of beam AB are shown. If the allowable bending stress in the beam material is 170 MPa. What will be the maximum value of hoist load that can be allowed to be lifted? Calculated maximum bending moment corresponding to maximum hoist load?

1. Calculation of $\bar y$ and $I_{NA}$:

$\bar y_{top}=\bar y_t=\frac{120*20*\frac{20}{2}+200*20*(20+\frac{200}{2})+20*120*(20+200+\frac{20}{2})}{120*20+200*20+120*20}$

$y_t=120mm$

$y_t= y_{bottom}=120mm$ (symmetry about x-axis}

$I_{N-A}=I_{big rectangle}-I_{small rectangle}$

$=\frac{120*240^3}{12}-\frac{(120-20)*200^3}{12}$

$I_{N-A}=71.573*10^6 mm^4$

1. Calculation of maximum bending moment:

$\frac{M}{I}=\frac{f}{y}$

Allowable bending stress=$170MPa=f=170 N/mm^2$

$\bar y=\bar y_t=\bar y_b=120mm$

(Here $\bar y_t=\bar y_b$ values are same but if value of $\bar y_t=\bar y_b$ are different then consider only minimum value, after maximum bending moment $M\propto \frac{1}{\bar y}$

$M=\frac{f*I_{NA}}{y_t}=\frac{170*71.573*10^6}{120}=101.395*10^6 Nmm$

i.e M=101.4 KNm

On I beam, the hoist carrying the load acts a point load.Let the point load be w (KN) which is maximum.

Maximum Bending Moment=M=101.4 KNm and also $M=\frac{wL}{4}$

$101.4=\frac{w*9}{4}$

i.e $w=45.067 KN$ which is maximum