Title 21 Beam Deflection (Algebraic Methods).pdf ✅

In the double integration method, a downward deflection is considered negative. Thus, if the downward direction is considered positive instead, EIy = Pbx 6L (L 2 − x 2 −b 2 ) For the maximum deflection, set the slope equal to zero, EIy ′ = Pbx 2 2L − P 2 〈x −a〉 2 + Pb 3 6L − PbL 6 Assume that the maximum deflection is before the concentrated load, x < a and 〈x − a〉 = 0, EI(0) = Pbx 2 2L + Pb 3 6L − PbL 6 0 = x 2 + b 2 3 − L 2 3 x = √ L 2 − b2 3 For the maximum deflection, EIymax = Pb 6L √ L 2 − b 2 3 (L 2 − L 2 − b 2 3 −b 2) EIymax = Pb(L 2 − b 2 ) 3 2 9√3L If the concentrated load is at midspan, a = b = L 2 , EIymax = Pb (L 2 − L 2 4 ) 3 2 9√3L EIymax = P ( L 2 ) ( 3 4 ) 3 2 (L 3 ) 9√3L EIymax = PL 3 48 The principle of superposition method states that if there are multiple loads acting on a beam, the total deflection is equal to the sum of the deflections of the individual loads. For a beam with a uniformly distributed load, EIy = ∫ (w dl)(l)(x) 6L (L 2 − x 2 − l 2 ) x 0 + ∫ (w dl)(L − l)(x) 6L [L 2 − x 2 − (L − l) 2 ] L x EIy = wx 24 (L 3 − 2Lx 2 +x 3 )