Structural tool

Free Beam Calculator

Calculate bending moment, shear force, deflection & reaction forces for cantilever & simply supported beam.

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Theory & FAQ

Beam length: 10.0m
Second moment of area: 473.0cm⁴ Youngs Modulus: 210.0GPa

Support conditions are unsatisfactory, add another support.

Max BM: - Min BM: -

Max SF: - Min SF: -

Max Deflection: -
Min Deflection: -

Method & theory

How the beam analysis is solved

The beam is solved using the direct stiffness method. Each span is discretised into elements with two degrees of freedom per node, assembled into a global stiffness matrix, and solved for nodal displacements and rotations under the applied load vector. Bending moment, shear force and deflection are then recovered element-by-element.

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What's this calculator used for?

Free online beam calculator to draw bending moment diagrams, shear force diagrams, calculate reaction forces and measure deflection for an indeterminate beam span. This calculator can be used for structural analysis and beam analysis. Maximum bending moment can be calculated along with shear forces, reaction forces and deflection using steel section properties. Specify beam geometry and point loads or distributed loads to get started analysing the beam.

Linear-elastic material behaviour, small deflections

Euler–Bernoulli beam theory — shear deformation neglected

Prismatic section, constant EI along each span

Loads applied statically in the plane of bending

Common questions

When we apply a force somewhere along the span of a beam, we generate internal stresses. There are two types of stress which are generated:

Normal stress: Generated from forces act along the length of the beam in compression or tension.
Shear stress: Generated from forces act parallel to the direction of the load.

The origin of these two stress components can be seperated into two force components we refer to as Bending moment and Shear force.

The simplest loads applied to a beam can be divided into three categories:

Point loads: This is a force applied to a single point on the beam.
Distributed loads: This force is distributed over a certain length and acts as a form of pressure.
Concentrated moment: This is a pure moment acting on a beam at a given position.

The simplest supports provided to a beam can be divided into three categories:

Pinned support: Restrain the beam in both horizontal and vertical directions without providing any restraint against rotation.
Roller support: Support provided to the beam in only a single direction and no restraint against bending or rotation.
Fixed support: These supports provide restraint to the beam in all directions and restrain against rotation.

Free body force diagram

To find the forces acting on a beam we must follow the process of creating a free body force diagram.

We first draw the beam to scale indicating the force components of the applied loads and then calculate the support reaction forces.

To calculate reactions we must use equilibrium equations to resolve the external forces:

Σ Vertical forces applied = 0kN
Σ Horizontal forces applied = 0kN
Σ Moments applied = 0kNm

There are two types of cases when resolving beam external conditions:

Statically determinate: Where the number of equations is equal to the number of unknowns and the calculations can be solved using closed form hand calculations.
Statically indeterminate: Where the number of unknowns exceeds the number of equations and more advanced methods such as Moment Resistribution Theory need to be used.

To create a shear force diagram you work from left to right across the beam summing the applied forces and reactions at each point they are applied. Cantilevers are the most straightforward example of this with a single reaction at one end of the beam span with zero shear force at the unsupported end.

To create a bending moment diagram you again must work from left to right across the beam and calculate the moment left of each point you are interested in.

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