Determine the shape of the beam:

A simply supported beam of span l carries a concentrated load P at midspan. If the beam is to be a beam of consistent strength, determine the shape of the beam by keeping the width constant & varying the depth towards the supports.

Solution

Consider the beam has uniform width b during the span and let d be the maximum depth at midspan.

Consider dx be the depth of beam at any section XX, distance x from the support A.

Bending moment at section XX =  (P/2) x .

Moment of resistance of beam at section XX = (1/6) × σ × b × (d _x)²

Moment of resistance = Bending moment at that section

 (1/6) × σ × b × (d _x)²  = (P/2)x

Therefore, the depth at any section at a distance x from the left support A might be found. The variation of the depth is illustrated in Figure .

Therefore, the depth at midspan, ( at x =  (½) ) , we get

 d =  ((3P/ σ b) ×  (l /2))^{(1/ 2)}  =   ((3/2) ×  (Pl / σ  ))^(1/2) .

Figure