Question $2(30$ marks$)$

An $8$ m long l$-$beam made of mild steel is used to support an overhead crane in a workshop, as shown in Figure $2($a$).$ Figure $2($b$)$ gives the dimensions of the $1-$beam. When answering the following questions, include the weight of the beam in your calculations.

a$.$ At what horizontal position on the beam would the load cause the maximum bending moment?

$(8$ marks$)$

b$.$ What is the minimum load that can initiate plastic deformation on the beam? What will be its location on the beam?

$(22$ marks$)$

For the mild steel, take the elastic modulus as $210$ GPa $,$ yield stress as $350$ MPa $,$ and density as $7850kg{m}^{-3}.$

$($D$)$

Figure $2($a$)$ I$-$beam supporting an overhead crane; $($b$)$ eross section of the libeam

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The figure consists of two parts. Part a illustrates an l$-$beam supporting an overhead crane in a workshop as part of a question. A frame consists of two vertical columns connected by a roof at the top. Below the roof, a long I$-$beam is shown with its left and right ends sitting on roller joints on horizontal platforms affixed to the left and right columns of the frame, respectively. A string connects the left end of the beam to a crane depicted as a vertical rod with a hook at the bottom. The crane hangs from the centre of the beam. A weight depicted as a box is hooked onto the crane. Part b illustrates the cross section of an l$-$beam showing its dimensions as part of a question. The $1-$beam is set against the $yz$ coordinate axes with the $z-$axis horizontal to the right and the $y-$axis upward. The I$-$beam consists of a horizontal flange above and below a vertical web, each depicted as rectangles. The overall height of the beam is $400$ millimetres. The horizontal width of each flange is $300$ millimetres. The height of the web is $200$ millimetres. The distance between the right edge of the web and the right edge of the flange is $100$ millimetres.