Bending Moment Diagram - shape and curvature

Bending moment is required for design of beam and also for the calculation of slope and deflection of beam. The following examples will illustrate how to write bending moment equation for different types of load and then draw bending moment diagrams.

Case I Bending moment due to point load

Bending moment due to  a point load is the product of the load and its perpendicular distance from the point of moment. as shown below.

Consider a cantilever subjected to a point load at its free end.

Bending moment at the fixed end = W x L = WL

Bending moment Mx at a distance x from free end = W x x = Wx

This is equation of a straight line and the plotted bending moment diagram in the above figure shows that the variation of bending moment along the span of a cantilever is a straight line.

Case II Bending moment due to a uniformly distributed load

Bending moment due to a uniformly distributed load (udl) is equal to the intensity of the load  x length of load  x distance of its center from the point of moment as shown in the following examples.

Bending moment at the fixed end =  10 x 2 x 1= 20 kNm

Bending moment Mx at a distance "x" from the free end = 10 x (x) x (x/2)= 0.5 x2

which is a second degree function of "x" and therefore parabolic.

Case III Bending moment due to uniformly varying load

Bending moment due to a varying load is equal to the area of load diagram  x distance of its centroid  from the point of moment.

The shape of bending moment diagram due to a uniformly varying load is a cubic parabola.

Case IV Bending moment due to a couple

Bending moment at a section due to a couple is equal to the magnitude of the couple and in the same sense as the couple.

#### Excellent Calculators

Stress Transformation Calculator
Calculate Principal Stress, Maximum shear stress and the their planes

To determine Absolute Max. B.M. due to moving loads.

Bending Moment Calculator
Calculate bending moment & shear force for simply supported beam

Moment of Inertia Calculator
Calculate moment of inertia of plane sections e.g. channel, angle, tee etc.

Reinforced Concrete Calculator
Calculate the strength of Reinforced concrete beam

Moment Distribution Calculator
Solving indeterminate beams

Deflection & Slope Calculator
Calculate deflection and slope of simply supported beam for many load cases

Fixed Beam Calculator
Calculation tool for bending moment and shear force for Fixed Beam for many load cases

BM & SF Calculator for Cantilever
Calculate SF & BM for Cantilever

Deflection & Slope Calculator for Cantilever
For many load cases of Cantilever

Overhanging beam calculator
For SF & BM of many load cases of overhanging beam

Civil Engineering Quiz
Test your knowledge on different topics of Civil Engineering

Research Papers
Research Papers, Thesis and Dissertation

List of skyscrapers of the world
Containing Tall building worldwide

Forthcoming conferences
Containing the List of civil engineering conferences, seminar and workshops

Profile of Civil Engineers
Get to know about distinguished Civil Engineers

Professional Societies
Worldwide Civil Engineers Professional Societies