3. Figure 2 shows a cantilever beam (dark black box) of length L and subjected to a linearly varying load, q(z) — go (1)

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3 Figure 2 Shows A Cantilever Beam Dark Black Box Of Length L And Subjected To A Linearly Varying Load Q Z Go 1 1 (242.4 KiB) Viewed 80 times

3. Figure 2 shows a cantilever beam (dark black box) of length L and subjected to a linearly varying load, q(z) — go (1) that is acting upwards (green triangle). The beam is also restrained by a spring of stiffness, k = 24 at end B. •y, v 4(z) = 4(1-²) A EI.L Figure 2: A Cantilever beam subjected to a linearly varying load load g(x) in the upward direction and restrained by a spring at end B • Draw the shear force, Sy(2), and bending moment, M₂(2), dia- gram of the beam. Scale (non-dimensionalize) the shear force as Sy(z)/qoL, the bending moment as M₂(z)/qoL², and the distance as z/L. (20 points) Hint: This is a statically indeterminate problem. One cannot draw the shear force and bending moment diagrams from simple statics, as we do not know the reaction provided by the spring, RB. We can solve this problem by using the linear supposition approach as described in the previous problem, namely by solving two statically determinate problems: (a) Determine the deflection of the free end, B (say u) by making the end B free, i.e., it is free from any loads, and (b) Determine the tip deflection at the tip of a cantilever beam under a unit tip load, applied in the same direction as that as- sumed for Rg. Let this deflection be vg. From the fact that the beam and the spring must have the same displacement, you have to first derive the following compatibility condition: +(¹+)-0 VB = (2) The reaction Rg is assumed to be acting in the same direction as considered positive for the beam deflection. - You can obtain, v, by determining the deflection profile of a cantilever beam subjected to a linearly varying load by solving the following fourth-order boundary value problem: d² 2 (B) = (¹-1) 90 0 <z <L dz² v(0) = 0 dv = 0 dzlz=0 d²v = 0 dz²|z=L d = 0 dz dz² |z=L You can obtain, v, by using the well-known relation for the tip deflection of a uniform cantilever beam of bending rigidity, PL³ EI, subjected to a tip load, P, as v = 3EI EI