ILD: bending moment at B, two-span beam

Sign convention: sagging moment (+), hogging moment (-); shear positive when the left part tends to go up; reaction positive upward. Ordinates are for a unit load moving along the beam.

Method

Two-span beam ABC (A hinged, B and C rollers), $L_1 = 8$ m, $L_2 = 10$ m, constant $EI$. Take $M_B$ as the redundant; by Muller-Breslau the ILD for $M_B$ is the deflected shape of the beam with a hinge inserted at B and a unit rotation (kink) applied there. Ordinates follow from the three-moment equation with zero moments at A and C.

For a unit load at $x$ from A in span AB:

$$M_B = -\frac{x\,(L_1^2 - x^2)}{2L_1(L_1+L_2)}$$

For a unit load at distance $y$ from C in span BC:

$$M_B = -\frac{y\,(L_2^2 - y^2)}{2L_2(L_1+L_2)}$$

Sample: $x = 4$ m: $M_B = -\dfrac{4(64-16)}{2\times8\times18} = -0.667$. $y = 4$ m (x = 14): $M_B = -\dfrac{4(100-16)}{2\times10\times18} = -0.933$.

Ordinates of $M_B$

ILD shape

**-------------------**------------------------**
  ****             **  *                     **
      *****   *****     **                ***
           ***            ***         ****
                             *********
^                    ^                          ^

The ILD is negative (hogging) throughout and zero at A, B and C. The largest ordinate in AB is about $-0.68$ near $x = 4.6$ m, and in BC about $-1.07$ near $x = 12.2$ m (5.8 m from C); the tabulated value at 12 m is $-1.067$.