Question

Q2 The simply supported beam of length is subjected to a vertical point load at its middle, as shown in Figure Q2. Both young's modulus and second moment of area of this structure are given as and. Please provide your answers in terms of letters. Self-weight of the beam is neglected. Figure Q2 (a) Determine the reactions, bending moment equation along the beam and draw the corresponding bending moment diagram [10] (b) Determine both the slope and deflection at the midspan of the beam. [15)

Answer 1

Q2) IP V c B I L/2 42 PRA=P/ RB=P/ Taking moment about A MA=0 RB X LE рх ч/, => BB = P² E Fx=0 Rat RB =P RA= P/ 2 Bending moment Caliulations MA=0 & MB=0 BM at B & v Bm at С. Me = RB XL% PIL 2 2 PL 4 PL 4. BM (+) Diagram C B. ② slope & deflection at Midspan of beam 4x P 2 I OA 08 dB Yc Elastic Curlle E = younges Area Differential equation fel elastic Custle of a beam M= EI day is modules daz o I= second moment of Boundary condations Deflection at end support is support is zero YA = YB=0 centre of beam, Oc=0 deflection is marimunn at Centre, Ye = Ymax Slope is is Zero at r

Let us considee One section XX at a distance & from end A, dy = Slope M₂=+RA.X pa 2 dhe Now, Px = Eld24 Q daz Wosit x Integrating Px² + c = EI dy an 4 r When x= 1/2 Oc= dy the lo => Substitute in egin (2 PL2 +c=0 G ==PLE 16 Paz - PL? - EI dy 4. 16. an Again Integrate slope equation PL? x + C q = Ely · Pa3 12 16 v when x=0 C2 = 0 Mid Where x=L Now Deflection at x=1 / span substitute in equation Oc=0 ЕТус - Pa3 12 PL2R 16 PL? (1/2) 16 3 • Yc= Imaxe = P63 EI 4c = P(1/2) 48 EI 12