A B q a dx Figure 1: Equilibrium of a simply supported beam Eliminating V, we obtain d 2 M +qo=0 Recall from Unified Engineering or 16.20(we'll cover this later in the course also) that the bending moment is related to the deflection of the beam w(a) by the equation d 2 M=El where e is the young's modulus and i is the moment of inertia of the beam Combining 5 and 6, we obtain d 2 dx2(ar2)+=0.0<x<L The boundary conditions of the beam are: The solution of equations 7 and 8 is given by w(a) 24EI (L-c)(L+Lc-a� � x L q0 z x dx M M+dM V V+dV q0 A B A B Figure 1: Equilibrium of a simply supported beam Eliminating V , we obtain: d2M + q0 = 0 (5) dx2 Recall from Unified Engineering or 16.20 (we’ll cover this later in the course also) that the bending moment is related to the deflection of the beam w(x) by the equation: d2w M = EI (6) dx2 where E is the Young’s modulus and I is the moment of inertia of the beam. Combining 5 and 6, we obtain: d2 d2w dx2 EI dx2 + q0 = 0, 0 < x < L (7) The boundary conditions of the beam are: w(0) = w(L) = 0, M(0) = M(L) = 0 (8) The solution of equations 7 and 8 is given by: q0 � 2 � w(x) = − x (L − x) L2 + Lx − x (9) 24EI 2