DISCUSSION THREAD
Hints:
Note that this is an indeterminate beam. Because of this, you are unable to determine the reactions on the beam strictly from equilibrium considerations. Through the use of integration to determine the rotation θ(x) and displacement v(x) of the beam, you will need to leave the reactions on the beam as unknowns until the end. The final equation needed to solve for the reactions will be found from enforcing the geometric boundary conditions on your solutions.
The process for the integration is the same as earlier problems that we have seen with determinate boundary conditions. It is recommended that you use the second-order approach for the integration process for finding rotations and displacements. For this, consider the two sections of the beam BC and CD.
- Make a mathematical cut through the beam between B and C, and from equilibrium considerations, determine the bending moment M(x). Integrate once and twice to get the rotation θ(x) and displacement v(x):
θ(x) = θ(0) + (1/EI) ∫M(x)dx
v(x) = v(0) + ∫θ(x)dx
where θ(0) and v(0) are integration constants. - Make a mathematical cut through the beam between C and D, and from equilibrium considerations, determine the bending moment M(x). Integrate once and twice to get the rotation θ(x) and displacement v(x):
θ(x) = θ(a) + (1/EI) ∫M(x)dx
v(x) = v(a) + ∫θ(x)dx
where θ(a) and v(a) are integration constants. θ(a) and v(a) are found from the results of integrating from B to C. - Enforce the displacement boundary condition at x = a to determine the unknown integration constant. This enforcement will lead to the additional equation needed to solve for the reactions.
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