Engineer's Notebook §2.2

Bending moment

The bending moment at a section is the sum of the moments of every force on one side of it. For a simply supported beam with a central point load, it peaks under the load:

Mmax =P·L4
M
bending moment
N·m
P
point load
N
w
uniform load
N/m
L
span
m
x
distance from left support
m
a
load distance from left support
m
b
load distance from right support
m
R
support reaction
N
A simply supported beam with a central point load and a uniform load along it. Below, the bending moment diagram curves down to its maximum at mid-span. w = 2 kN/m P = 10 kN BENDING MOMENT M M = 24 kN·m
Fig. 2.2-1Sagging moment is drawn below the axis, on the tension side.
MP = PL/415 kN·m
Mw = wL²/89 kN·m
Mmax = MP + Mw24 kN·m

Worked example

Uses the values in Fig. 2.2-1. Change them and the working updates.

Given
  • Ppoint load at mid-span10 kN
  • wuniform load2 kN/m
  • Lspan6 m
Find

The bending moment at mid-span.

Solution
MP = PL/4 = 10 kN × 6 m / 4 = 15 kN·m Mw = wL²/8 = 2 kN/m × (6 m)² / 8 = 9 kN·m M = MP + Mw = 15 + 9 = 24 kN·m
Answer
M = 24 kN·m

Each support reaction is P/2 + wL/2 = 11 kN.

Related cases

Maximum bending moment for other common beams.

  • Simply supported, point load at a

    M = PabL

  • Simply supported, uniform load

    M = wL²8

  • Cantilever, end load

    M = PL

  • Cantilever, uniform load

    M = wL²2

Assumptions
  • Loads add by superposition: the moment from each load is found alone and the results summed. This holds for linear-elastic beams with small deflections.
  • A sagging moment, which bends the beam concave upwards, is positive.
  • The supports are simple, so the moment is zero at each end.
  • The beam's own weight is ignored unless it is included in w.

Derivation

Each support carries R = P/2 + wL/2. Cut the beam at distance x from the left support, with x ≤ L/2, and take moments of the forces to the left of the cut.

  1. M(x) = Rx − wx²/2, since the uniform load on the left part, wx, acts at x/2.
  2. Substitute R: M(x) = Px/2 + wx(L − x)/2.
  3. At mid-span, x = L/2: M = PL/4 + wL²/8.

The point-load term grows linearly to mid-span and the uniform-load term is a parabola. Both peak at the centre, so their maxima add.