Bending Moment Diagrams – Explanations and Drawing

Bending moments are very important aspects of mechanics. Mechanical engineers are generally dealing with these kinds of problems on a theoretical basis. Definitions of correct forces and moments are very important to obtain the required results. 

Drawing bending moment diagrams for different beam sections is also a very important part of the strength of material problems. Here, we explain how to draw bending moment diagrams effectively for all problems.

Bending diagrams are represented with a straight horizontal line right below the beam. This line represents zero. Beneath the line, the bending moment is negative, above the line bending moment is positive. To understand the sign conventions of bending moment and shear forces, consider this article. 

How to Draw Bending Moment Diagrams?

Strength of Materials, Third Edition

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First of all, we need to understand this fact; Bending moment in beam sections is the result of the shear forces acting on the beam. If we know the shear force situation and application areas, we can make comments about possible bending moment diagrams. 

The moment is the multiplication of the distance of a force where the moment that we calculate, with the force itself. With the increasing distance(x), the bending moment is also increasing. So; 

dM(x)/dx = F(Shear Force)

According to the derivation calculus, when the shear force is zero or changes sign from maximum to minimum, the bending moment will have the maximum or minimum on this point. 

Some Graphing Techniques from Math for Bending Moment Diagram

  • If the shear force application is singular on a beam, the shear force diagram will be constant just as shown below. So, the bending moment diagram will be inclined linear. 
  • If the shear force application is a uniformly distributed load on a beam, from left(where the beam is considered as cantilevered) to right, the shear force will decrease as inclined linear. So, the bending moment will decrease as a second-order curve. 

So, you can understand the decrement and increment relation between shear force and bending moment diagrams. 

A Bending Diagram Drawing Method

At the free side of the cantilevered beam, the bending moment will be zero. So, you need to make your calculations from the free side of the beam. 

For Singular Shear Forces

Bending moment diagrams of simple beams.
Singular shear force acting on the free side(Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg.149).

Say that the length of the beam is ‘l’. And say the distance from the free side of the beam is ‘x’. This ‘x’ must be at both sides of the shear force application to see the complete situation of the bending moment.

First, you need to calculate the left side bending moment situation of the singular force, by placing ‘x’ on the left side of the singular shear force. Then calculate the bending moment at that point, from the left side of the point.

Then place the ‘x’ on the right side, and make your calculation again.

Drawing Bending Moment Diagrams for Uniformly Distributed Shear Forces

Shear force bending moment diagram.
Uniformly distributed load and bending/shear diagrams(Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg.150).

If the force is uniformly distributed on a random side of the beam, you need to place your ‘x’ on the place where that distributed load is not applied(left side of distributed load). Then you need to place your ‘x’ on the place where the distributed load is applied.

If the right side of the distributed force is empty, place your ‘x’ on this side.

Also, for each placement, make your bending moment calculations according to these places by considering the left side(free side actually).

For Shear Forces That Distributed as Inclined Linear Curve

Distributed load condition.
Inclined distributed shear force and shear force/bending moment diagrams(Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg.157).

In this case, you must have the second or third order of the equation of shear force. If you derive this equation with ‘x’, you will have the equation of bending moment. 

We want to obtain the graphs of equations in bending moment diagrams. The mathematical methods above for obtaining the equations.

For Complex Problems

You can deal with complex problems such as different shear force forms acting on beams. In this case, you need to make separate calculations for each force as they are only acting forces on beams. Then you need to add these bending moment diagrams to obtain the complete system’s bending moment diagram. 

Conclusion on Bending Moments

Above all, if you understand the general logic of obtaining bending moment and shear force diagrams, you will be able to solve all the problems related to this topic. 

So, do not forget to leave your comments and questions below about the bending moment diagrams and shear force diagrams.

Your precious feedbacks are very important to us.

FAQs About Bending Moment Diagrams

What is a bending moment?

This type of moment takes place because of the perpendicular forces acting on the horizontal elements. Around these horizontal elements, built-in beams are the most important examples. Because of the bending moments, compression and tension take place on the mechanical elements.

What is a bending formula?

Bending is a type of torque and the unit of the torque is N.m. So, you can calculate by multiplying the bending force with the distance of the force to where you want to calculate the bending moment.

How to draw bending moment diagrams?

It is very simple to draw these diagrams. You need to obtain the general bending moment equation of all sections of beams. So, you can draw and calculate the total diagram. You can check the all details in the post above.

Why bending moments are important?

Because the most common engineering failures take place because of the bending moments. We need to calculate the bending moments effectively.

What is the unit of the bending moment?

The unit of the torque or moments is N.m or lbf.ft in English units. So, the unit of the bending moment is the same as the torque. It will be better if you take care of the units of the moment diagrams.

What is the bending of a beam? 

Bending take place because of the perpendicular forces acting above the beams. So, bending takes place on beams because of these forces.


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