Bending Moment at a section of beam - Definition, equation, calculation and diagram

Bending moment and Shearing force are very important factors to be considered for design of any structural component.

Bending moment at a section of a structural element is defined as the reaction developed in a structural element when an external force or moment is applied to the element that causes the element to be bent.

Bending moment at a section of beam is calculated as the algebraic sum of the moment of all the forces acting on one side of the section. For example consider the following figure which shows the free body diagram of a simply supported beam having F₁, F₂, F₃ and F₄ as the applied loads on the beam and R_A, R_B as the support reactions.

It is very important to define the sign convention for bending moment before calculating the values. The possible bending of a beam is shown in the following figure.

When the beam bends in such a way that it forms concavity downwards (cup-shaped) it is called as sagging. Whereas the bending which results in convexity upwards (like a hump); is known as hogging.

Sign convention:

Sagging bending moment is taken as +ve,  it results in developing tension in the bottom fibers and compression in the top fibers of the beam.

Hogging bending moment is taken as -ve, and it develops compression in the bottom fibers and tension in the top fibers.

Based on these sign conventions we can write the equation of bending moment at section x-x of the beam given above;

Considering forces on the left of the section x-x

M_x  = R_AL_A - F₁x₁ - F₂x₂

We can also get the values of M_x by considering the forces on the right of section x-x.

M_x  = R_BL_B - F₃x₃ - F₄x₄

The above expression is called as bending moment equation which can be written depending upon the loading on the beam. A graphical representation of the bending moment equation along the span of the beam is known as bending moment diagram (BMD).

Bending moment is required for design of beam and also for the calculation of slope and deflection of beam

Get more information about bending moment diagram shape and curvature due to different loading cases.