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How To Calculate Shear And Bending Moment

By The Calcumatix Team Reviewed by Calcumatix Editorial Review 4 min read

Quick Answer

To calculate shear and bending moment, you must first calculate the reaction forces at the physical supports holding the beam up. Next, cut the beam into mathematical sections between every applied load. Calculate the internal shear force (V) in each section by adding up all the vertical forces to the left of your cut. Then, calculate the internal bending moment (M) by summing the torques (Force × Distance) created by those same forces. Finally, plot the calculated V and M values on a graph to create a Shear Force Diagram (SFD) and Bending Moment Diagram (BMD).

Before a structural engineer approves a steel I-beam for construction, they must prove the beam will not snap in half under the weight of the building. To do this, they analyze two specific internal forces: shear force (the tendency of the beam to slice vertically) and bending moment (the tendency of the beam to physically bow or snap). By mapping these forces along the entire length of the beam, engineers can pinpoint the exact millimeter where the steel will experience the maximum stress. This guide explains how to calculate shear and bending moment, and how to graph them into standard structural diagrams.

What Are Shear And Bending Moment Diagrams?

A Shear Force Diagram (SFD) and a Bending Moment Diagram (BMD) are graphical representations of the internal forces acting inside a structural beam. The X-axis represents the physical length of the beam, while the Y-axis represents the magnitude of the force.

The mathematical relationship: The bending moment is the mathematical integral of the shear force. This means that the area under the curve on the Shear Force Diagram exactly equals the change in the Bending Moment Diagram. Furthermore, the point on the beam where the shear force crosses zero is almost always the exact point where the bending moment reaches its absolute maximum.

Standard sign convention:

  • Upward external forces create positive shear.
  • Downward external forces create negative shear.
  • Forces causing the beam to “smile” (sag downward) create a positive bending moment.
  • Forces causing the beam to “frown” (hog upward) create a negative bending moment.

The Step-By-Step SFD And BMD Diagram Maths Procedure

This procedure covers a simply supported beam (resting on two supports at the extreme left and right ends) with a single point load pushing down in the exact center. You must follow the standard sign convention precisely, or your diagrams will be inverted.

Step by step:

  1. Draw a Free Body Diagram (FBD) showing the beam length and the applied downward point load.
  2. Calculate the upward reaction forces at the left support (R1) and right support (R2). Because the load is centered, simply divide the total downward load by 2.
  3. Begin at the far left edge of the beam (x = 0). The shear force immediately jumps up to match the R1 upward reaction force.
  4. Move across the beam to the right. The shear force remains constant until you hit the downward point load in the center.
  5. At the center, subtract the downward point load from your current shear force. The shear force drops into the negative.
  6. Continue to the right edge. The shear force returns to zero exactly when it meets the upward R2 reaction force. Plot these values to create the SFD.
  7. To find the maximum bending moment in the center, multiply the R1 reaction force by the distance from the left edge to the center. Plot this triangular peak to create the BMD.

Worked example: Inputs: A 10-meter simply supported beam has a 100 kilo-Newton (kN) point load applied exactly in the center (at 5 meters). Step 1 (FBD): Length = 10m. Load = 100 kN downward at x = 5m.

Step 2 (reactions): 100 kN / 2 = 50 kN. R1 = 50 kN upward. R2 = 50 kN upward.

Step 3 (left edge shear): At x = 0, the shear force is +50 kN. Step 4 (midpoint shear): From x = 0 to x = 5, the shear remains at a flat +50 kN line. Step 5 (load drop): At x = 5, the 100 kN load pushes down. 50 - 100 = -50 kN. The shear instantly drops to -50 kN.

Step 6 (right edge shear): From x = 5 to x = 10, the shear remains flat at -50 kN. At x = 10, the R2 upward force of 50 kN brings the shear exactly back to zero (-50 + 50 = 0). Step 7 (max moment): The maximum moment occurs at the center (x = 5). Multiply the R1 force (50 kN) by the distance (5m). 50 × 5 = 250 kNm.

Result: Max Shear 50 kN, Max Moment 250 kNm. Your SFD consists of a positive rectangular block on the left and a negative rectangular block on the right. Your BMD is a triangle starting at zero on the left, peaking at 250 kNm in the center, and sloping back to zero on the right. Use the shear and moment diagram calculator to build these diagrams automatically, and see the engineering calculators hub for related structural tools.

Sources and References

Frequently asked questions

What happens if the load is not perfectly centered?

If the point load is offset to the left or right, the reaction forces at the supports will not be equal. You cannot simply divide the load by two. You must use the sum of moments equilibrium equation ($\sum M = 0$) to calculate exactly how much weight is being supported by each individual pillar before you can start drawing the shear diagram.

How does a distributed load change the diagrams?

A uniform distributed load (like a concrete slab resting across the entire beam) changes the shape of the diagrams entirely. Instead of a flat, rectangular shear force, the distributed load causes the shear force to slope diagonally downward. Because the bending moment is the integral of the shear, this diagonal shear line creates a curved, parabolic Bending Moment Diagram instead of a straight-line triangle.

What is a cantilever beam?

A cantilever beam is completely anchored into a wall on one side, but entirely unsupported on the other side (like a diving board). The mathematics for a cantilever are drastically different. The maximum shear force and the maximum bending moment always occur exactly at the fixed wall anchor, and they both drop to zero at the free-floating edge.

Why does the bending moment have to be zero at the supports?

On a standard simply supported beam, the ends are resting freely on rollers or simple pins. Because they are free to rotate slightly like a door hinge, they cannot mathematically resist a bending force. Therefore, the internal bending moment at those exact unanchored points must physically be zero.

What units are used for bending moment?

Because a bending moment is a rotational torque (Force × Distance), the units must combine a weight measurement with a length measurement. In the metric system, engineers use kilo-Newton-meters (kNm). In the American imperial system, engineers use foot-pounds (ft-lb) or kip-feet (k-ft). One "kip" is equal to exactly 1,000 pounds of force.