ILD: shear at section x-x, 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.

Reading of the figure

Two equal spans $AB = BC = 8$ m, hinged at A, rollers at B and C. Section x-x is in BC, 4 m from B (12 m from A).

Method

For a section in span BC the shear is that of the simple span BC corrected for the support moment $M_B$ (which is hogging, so it changes the end reaction at C by $M_B/L_2$):

$$V_{x} = V_{\text{simple}} - \frac{M_B}{L_2}$$

where $V_{\text{simple}} = 0$ for a load in AB, $+(L_2 - z)/L_2$ for a load right of the section ($z$ measured from B, z > 4), and $-z/L_2$ for a load left of the section in BC. With $L_1 = L_2 = 8$: $M_B = -\dfrac{x(64-x^2)}{256}$ for a load in AB, $M_B = -\dfrac{y(64-y^2)}{256}$ for a load in BC.

Sample: load at x = 4 (in AB): $M_B = -0.75$, $V = -(-0.75)/8 = +0.094$. Sample: load at the section from the left ($z = 4$): $V_{simple} = -0.5$, $M_B = -0.75$, $V = -0.5 + 0.094 = -0.406$; from the right: $+0.5 + 0.094 = +0.594$.

Ordinates (2 m interval)

ILD shape

                                    |*
                                    | ***
                                    |    ***
         **********                 |       ***
*********----------*********--------|----------**
                            ****    |
                                ****|
                                    |

^                       ^                       ^

Jump of 1.0 at the section ($-0.406$ to $+0.594$). The ILD is positive but small in AB (maximum about 0.096) and negative in BC between B and the section.