Shear and Moment Diagram

Fundamentals

Shear and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear force and bending moment at a given point of a structural element such as a beam. These diagrams can be used to easily determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.

We need to find an equation that estimates the shear force and bending moment on a position along the beam that we can define as x.

Let’s imagine making a cut at a distance x along the beam. Both internal shear force and bending moment revealed by the cut are functions of x. Now, assuming clockwise moments are positive, we’ll determine an expression for M(x).

Where:

w = applied loading factor on the beam position x

x = cut position

L = full length of the beam

With this equation we can determine the value of the internal bending moment for any value of x along the beam.

The maximum bending moment for simply supported beams will happen at the mid-spam x = L/2.

Before you’ll go through this article to learn how to draw the shear and bending moment diagrams for beams, we want to remember that Civils.ai has a free tool where you can easily make it. Visit the page later for discovering more!

Drawing the shear force diagram

In the following example (see the image below), we’re going to ‘trace the impact of the loads’ across a simply supported beam from left to right.

The first load on the structure is the support reaction VA = 67kN acting upwards, this raises the shear force diagram from zero to +67kN at point A. The shear force then remains constant as we move from left to right until we hit the external load of 90kN acting down at D. This will cause the shear force diagram to ‘drop’ down by 90kN at D to -23kN (+67 – 90).

When we reach the point E a 10kN linearly varying load is applied. The shear force diagram starts curving at E with a linearly reducing slope as we move towards F (10 – 10/9*x), ultimately finishing at F with a slope of zero (horizontal).

Finally, we apply an upward load at the point B of 68kN and the diagram goes up to 8.9kN (-59.1 + 68kN).

Following the draw of the diagram for this basic example:

Drawing the bending moment diagram

Considering the example above, the bending moment diagram becomes much easier to determine. First, we can consider the relationship between the shear force V and the slope of the bending moment diagram:

The expression helps us to calculate a qualitative shape for the bending moment diagram, based on our shear force diagram (see the image below).

Considering that the shear force between A and D is constant in our example, the bending moment will be constant too. The same between D and E but the sign changes according to the shear force. The bending moment diagram reaches a local negative peak at D.

As we know, Between E and F the shear force is a curve and this also tells us the bending moment’s slope is a curve too, continuously changing. Since the shear shear force changes sign at point B, the bending moment diagram reaches a peak at that point.

We can combine all this information together to sketch out a qualitative bending moment diagram, based purely on the information encoded in the shear force diagram.

Now we simply have to cut the structure at discrete locations, like the red lines in the draw, to establish the various key values required to quantitatively define the bending moment diagram. In this case three cuts are sufficient:

• at D to determine the local peak (1)
• at E to determine the value between the straight and curved sections (2)
• at B to determine the local peak (3)

Once all internal bending moments have been calculated, we can complete our draw of the shear and moment diagram as following:

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