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.
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) .
Simply Supported Beam