Load–Shear–Moment Relations

\frac{d V}{d x} = - w \frac{d M}{d x} = V

Shear V is the negative integral of distributed load w. Moment M is the integral of shear. Jump in V equals a concentrated force; jump in M equals a concentrated couple. Useful for drawing V and M diagrams.

$w$ (kN/m): Distributed load
$V$ (kN): Shear
$M$ (kN·m): Moment

Step-by-step solved example

Simple beam L=4 m, w=2 kN/m uniform. V(x) from left and Mmax?

Load → shear → moment by integration. dV/dx = −w, dM/dx = V.

1. Reactions
RA=RB=4 kN.
2. V and M
V(x)=4−2x kN. V=0 at x=2 m. Mmax=4×2−2×2×1=4 kN·m = wL²/8.
$M_{ma x} = 4 kN \cdot m$

Answer: V=4−2x kN; Mmax=4 kN·m

10 practice questions

0/10 correct

1.dM/dx equals

2.dV/dx equals

3.A downward point load causes V to

4.Mmax on a simple uniform beam is

5.Where V=0 (no point moment), M is

6.Cantilever with end load P: V is

7.A couple M0 on the beam makes M

8.Uniform w=3 kN/m, L=6 m simple. RA=

9.If w=0 between two points, V is

10.M diagram slope is