How to calculate bearing forces of a beam?

Two rules of statics are used. First, the upward support forces equal the downward loads so that the sum of the support forces equals the total load. Second, moments about any point balance each other so that the load can be distributed according to the distance between supports. For a simply supported beam with a concentrated load, the support closer to the load carries more of the load. The above formula for calculating support forces is shown for every case this calculator covers.

What is the difference between Thrust and Bending Moment?

The shear force is the magnitude of a force trying to separate a point in a beam horizontally and is calculated by summing all vertical forces on one side of that point. The bending moment is the magnitude of a force trying to bend the beam at that point, and is calculated by adding up the moments those forces create at that point. Shear force diagrams typically have peaks at support points while bending moment diagrams usually peak in the middle of the span or at the fixed end. The maximum bending moment is normally critical for design purposes.

Do results change based on where the concentrated load is?

Yes. When a concentrated load is moved the way in which the two supports share the load changes as does the location of the maximum bending moment. The closer the load is to one support the greater that support's reaction force will be. The maximum bending moment, which occurs when the load is placed at midspan, is equal to P times a multiplied by b divided by L. This calculator allows you to place the load anywhere along the span to check out the effects.

Why is the bending moment of a beam with one end significantly greater than that of a simply supported beam?

A cantilever beam is supported on one end only so all of the loads hang off of that fixed end and there are no other supports to help. The bending moment at this position is equal to the product of the load and the distance from the support with no division by the span required. A simply supported beam with the same span and the same loading has two supports sharing the load so the maximum bending moment is many times less.

How can you tell if a support beam is holding up?

Compare the bending stress caused by the load with the allowable stress of the material. The stress is equal to the maximum bending moment of the beam divided by the section modulus. If you enable the "Bending Stress" option and enter the section modulus and the allowable stress, then the calculation tool will give the degree of utilization in percentage. If the degree of utilization is less than 100, the cross-section passes the bending test; however, for a real structure, deformation, shear force, and buckling must also be considered.

Beam Load Calculator

Beam and load type

Beam support type

Simply supported

Cantilever

Load type

Point load

Distributed load

Beam and load details

Span length (L)

Point load (P)

Load position from the near support (a)

Check bending stress

Add the section modulus and an allowable stress to turn the bending moment into a bending stress, a utilization, and the section modulus you need.

Maximum bending moment (kN.m)

Maximum shear force (kN)

Maximum bending moment

kN·m

Maximum shear force: kN
Reaction at support A: kN
Reaction at support B: kN

Governing case: a simply-supported beam with a point load. Reactions R_A = P·b / L and R_B = P·a / L, maximum moment M = P·a·b / L under the load.

The two reactions add up to the total load on the beam. Support A carries 5 kN and support B carries 5 kN.

Shear force and bending moment diagrams

The Beam Load Calculator calculates how a beam transfers the loads to its supports and what shear and bending moments these loads cause in the beam. By selecting the type of beam and loading, as well as entering the span length and load, it will show you the support reactions, maximum shear force, maximum bending moment, and diagrams for shear force and bending moment.

This version supports simply supported and cantilevered beams and takes into account both concentrated loads at any point along the span as well as uniformly distributed loads over the entire length of the beam. If the "Bending Stress" option is enabled, in addition to calculating the loads, it also calculates the corresponding bending stresses for the selected cross-section.

What can be found out through an analysis of carriers?

Any analysis of a beam begins with statics. Downward loads must be balanced by equal upward reaction forces and the beam must be in equilibrium. From this balance three results emerge that engineers need to consider when designing.

The reaction forces are the forces that the supports transfer to the columns or walls below them. The shear force gives the magnitude of the horizontal force acting at any point on the beam and trying to cut it, while the bending moment gives the magnitude of the force acting at any point on the beam and trying to bend it. From the maximum shear force, maximum bending moment and their locations, the required dimensions of the beam can be determined.

Here's how to use this calculator.

First select the type of support for the beam. A simply supported beam rests on supports at both ends and can rotate freely at the support locations. A cantilevered beam has one end fixed while the other end is free.

Next select the type of load. For a concentrated load enter the magnitude of the load and its distance from the left support or fixed end of the cantilever beam. For a uniformly distributed load, enter the load per unit length. The reaction forces, maximum shear force and maximum bending moment are immediately displayed, and the two bottom diagrams show the distribution of shear force and bending moment along the beam.

Support forces

The reaction forces are derived from two equilibrium conditions: the sum of all vertical forces must be zero and the sum of all moments about any point must also be zero. For a simply supported beam with a concentrated load, by considering the moments at either support one can see that the load is distributed to both ends according to their lever arms such that the nearer support carries more load.

Beam and load	Support reactions
Simply supported, point load at a (b = L - a)	$R_A = \dfrac{Pb}{L}, \quad R_B = \dfrac{Pa}{L}$
Simply supported, uniform load	$R_A = R_B = \dfrac{wL}{2}$
Cantilever, point load	$R = P$
Cantilever, uniform load	$R = wL$

For easy reference: if a uniformly distributed load of 5 kN/m is applied to a simply supported beam with a length of 10 m, the total load will be 50 kN so each support carries half, or 25 kN. If the concentrated load is moved from the center, the distribution changes depending on the distance and is no longer uniform.

Thrust and Thrust Diagram

The shear forces in a cross section are the sum of all vertical forces acting on one side of the cross section. One starts at an end support and its reaction force, then follows the loads along the beam subtracting any load that is crossed. The shear diagram represents this stepwise accumulated sum.

Under a concentrated load the shear remains horizontal and drops by an amount equal to the load at the point where it is applied. Under a uniformly distributed load, the shear changes continuously and passes through a value of zero at the point where the bending moment is a maximum. The maximum shear occurs almost always immediately next to a support.

Beam and load	Maximum shear force
Simply supported, point load	$\max\left(\dfrac{Pa}{L}, \dfrac{Pb}{L}\right)$
Simply supported, uniform load	$\dfrac{wL}{2}$
Cantilever, point load	$P$
Cantilever, uniform load	$wL$

Bending moments and bending moment diagrams

The bending moment at a cross-section is the sum of all moments caused by forces acting on that section. It indicates how much a particular point in the beam is bent. The bending moment diagram shows the variation of the bending moment along the span, with the peak value often being the critical design value.

Since a simply supported beam bends downward the maximum bending moment is typically near the center. For a cantilevered beam it bends upward so that the maximum bending moment occurs at the fixed end, i.e., where all of the loads hang outside the wall.

Beam and load	Maximum bending moment
Simply supported, point load at a	$\dfrac{Pab}{L}$
Simply supported, uniform load	$\dfrac{wL^2}{8}$
Cantilever, point load	$Pa$
Cantilever, uniform load	$\dfrac{wL^2}{2}$

Example calculation

A simple beam with one end supported and a span of 6 meters carries a concentrated load of 12 kN located 2 meters from the left support. So, a = 2 m and b = 4 m. The reaction forces can be calculated using the moment arm as follows.

R_A = \frac{Pb}{L} = \frac{12 \times 4}{6} = 8\ \text{kN}, \qquad R_B = \frac{Pa}{L} = \frac{12 \times 2}{6} = 4\ \text{kN}

These resulting forces exactly match the load of 12 kN. This is a necessary result. The maximum shear force is equal to the larger support reaction, so it's 8 kN. The maximum bending moment occurs just below the load.

M = \frac{Pab}{L} = \frac{12 \times 2 \times 4}{6} = 16\ \text{kN}\cdot\text{m}

If the load is moved to the center, both support forces change to 6 kN and the bending moment rises to a maximum value calculated by multiplying P with L and then dividing it by 4. Both the position and size of the load affect the result.

From bending moment to bending stress:

The maximum bending moment determines the stress that a material will experience when bent. By dividing the bending moment by the section modulus (a number that reflects the shape and height of the cross-section), you can find the maximum bending stress.

\sigma = \frac{M}{S} \qquad S_{\text{required}} = \frac{M}{\sigma_{\text{allow}}}

If you enable the 'bending stress' option and enter the section modulus and allowable stress then it will show the actual stress, the degree of utilisation relative to the allowable stress and the required section modulus to remain within the allowable range. This is a tool for quickly assessing whether or not a particular beam can carry a given load.

This calculation tool is for general educational purposes and initial sizing only. It does not replace a full structural design by a qualified professional, nor does it check deflection, buckling, shear stresses, connections or code compliance. Load-bearing components must always be designed and checked by an expert.

Frequently asked questions

How to calculate bearing forces of a beam?: Two rules of statics are used. First, the upward support forces equal the downward loads so that the sum of the support forces equals the total load. Second, moments about any point balance each other so that the load can be distributed according to the distance between supports. For a simply supported beam with a concentrated load, the support closer to the load carries more of the load. The above formula for calculating support forces is shown for every case this calculator covers.
What is the difference between Thrust and Bending Moment?: The shear force is the magnitude of a force trying to separate a point in a beam horizontally and is calculated by summing all vertical forces on one side of that point. The bending moment is the magnitude of a force trying to bend the beam at that point, and is calculated by adding up the moments those forces create at that point. Shear force diagrams typically have peaks at support points while bending moment diagrams usually peak in the middle of the span or at the fixed end. The maximum bending moment is normally critical for design purposes.
Do results change based on where the concentrated load is?: Yes. When a concentrated load is moved the way in which the two supports share the load changes as does the location of the maximum bending moment. The closer the load is to one support the greater that support's reaction force will be. The maximum bending moment, which occurs when the load is placed at midspan, is equal to P times a multiplied by b divided by L. This calculator allows you to place the load anywhere along the span to check out the effects.
Why is the bending moment of a beam with one end significantly greater than that of a simply supported beam?: A cantilever beam is supported on one end only so all of the loads hang off of that fixed end and there are no other supports to help. The bending moment at this position is equal to the product of the load and the distance from the support with no division by the span required. A simply supported beam with the same span and the same loading has two supports sharing the load so the maximum bending moment is many times less.
How can you tell if a support beam is holding up?: Compare the bending stress caused by the load with the allowable stress of the material. The stress is equal to the maximum bending moment of the beam divided by the section modulus. If you enable the "Bending Stress" option and enter the section modulus and the allowable stress, then the calculation tool will give the degree of utilization in percentage. If the degree of utilization is less than 100, the cross-section passes the bending test; however, for a real structure, deformation, shear force, and buckling must also be considered.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Hibbeler, Structural Analysis (statics of beams, reactions and internal loads)
Shear-force and bending-moment diagrams and how they are built from equilibrium.
Wikipedia: Statically indeterminate and determinate structures
Why simply supported and cantilever beams can be solved by statics alone.
Wikipedia: Section modulus
Definition of the elastic section modulus used in the bending-stress check.