Problem 10.24

Obtain the reaction R at the roller using the unit load method.

The problem statement should include something about the member geometric and material properties.  We will assume that the members are uniform and have the same EI, and that axial and shear deformations may be neglected.

Choose R as statical redundant, and form the compatibility equation setting the displacement D at C in the direction of R to be zero.

 D = S∫M'Mdx/EI = 0

EI being constant and the same for both members, the compatibility condition becomes

EID = S∫M'Mdx = 0

M and M' are plotted in the figures on the compression side of the members.

Member AB:  ∫M'Mdx = a(b)(bR-M_o) = ab²R - abM_o

Member BC: ∫M'Mdx = (b/3)(b)(bR) = b³R/3  (using formula for linear moment diagrams)

S∫M'Mdx = ab²R - abM_o + b³R/3 = 0

R = aM_o/b(a + b/3)