# Draw the shear and moment diagrams for the cantilevered beam

Draw the shear and moment diagrams for the cantilevered
beam.

## General guidance

Concepts and reason
When loads are applied on a beam, it develops an internal shear force and bending moment which varies from point to point along the axis of the beam. The shear and bending moment functions can be plotted on a graph paper, taking the beam as a reference, called shear force and bending moment diagrams.

These diagrams help engineers to decide where to reinforce the beam and were to provide greater areas of the cross section to the beam to withstand the loading.

Fundamentals

The steps required to plot the shear force and bending moment diagrams are:

1. Determine all the reactive forces and couple moments acting on the beam and resolve all of them into components acting perpendicular and parallel to the beam's axis.

2. Take the beam's left end as the origin and extend to regions of the beam between concentrated forces and couple moments, or where there is no discontinuity of distributed loading.

3. Take different beam sections and draw the free-body diagram of one of the segments. The shear force V and bending moment M should be shown acting in their positive sense, in accordance with the sign convention.

4. Calculate the shear force by summing forces perpendicular to the beam's axis.

5. Calculate the moment by summing moments about the sectioned end of the segment.

6. Plot the shear force diagram (V versus x) and the bending moment diagram (M versus x). If V and M in the diagram are positive, their values are plotted above the x-axis, whereas negative values are plotted below the axis.

7. To enhance readability, show the shear force and moment diagrams below the free-body diagram of the beam.

Sign convention:

The positive direction is considered when the distributed load acts upward on the beam; the internal shear force causes a clockwise rotation of the beam segment on which it acts. The internal moment causes compression of the top fibers of the beam such that that it causes the beam to sag downwards.

## Step-by-step

### Step 1 of 3

Consider the free body diagram as follows:

Calculate the horizontal reaction at A as follows:

Apply horizontal equilibrium equation.

Calculate the vertical reaction at A as follows:

Apply vertical equilibrium equation.

Calculate the moment at A as follows:

The cantilever support has 3 reactions. The upward force is considered a positive and downward force is considered as negative. The clockwise moment is considered a positive and anticlockwise moment is considered as negative.

### Step 2 of 3

Calculate the shear force at various lengths.

Calculate the shear force at point A.

Calculate the shear force at point B.

Calculate the shear force at point C.

Draw the shear force diagram as follows:

The shear force diagram is

The Shear force is the summation of the forces either to the left or right of a section.

### Step 3 of 3

Calculate the bending moment at point A as follows:

Calculate the bending moment at the left of point B as follows:

Calculate the bending moment at the right of point B as follows:

‎Calculate the bending moment at point C as follows:

Draw the bending moment diagram as follows:

The bending moment diagram is

Bending moment is the summation of the moment due to the force acting at a section.