For each of the following beams, i. (1 point per beam) Sketch the described beam, including supports and external loads.
ii. (2 points per beam) Derive expressions for the internal shear and moment resultant functions, V(x) and M(x).
iii. (2 points per beam) Sketch plots of V(x) and M(x). assuming that L > 0 is a constant with dimensions of length and P > 0 is a constant with dimensions of force over length.
(a) A beam of length L with a pinned support at x = 0, a roller support at x = L/2, a free end at x = L, and a linear external transverse load, p(x) = Px/L. Hint: To compute the reaction forces at the supports using statics, recall that the external transverse loading's net y-direction force and (counterclockwise) moment (about x = 0) are given by the integrals Fy -= S* pvdt and MP xp(x) dx . Reminder: Do not forget to add the concentrated reaction force of the roller support to the external load p(x) before integrating to get V(x).
(b) A beam of length L with a free end at x = 0, a roller support at x = L/3, a pinned support at x = L, and a uniform external transverse loading p(x) = P(1 – H(x – L/3)) applied to its overhang (where H is the Heaviside function).