This guide helps you select the appropriate method for analysing a beam and points you to the detailed RoyMech engineering reference for the calculation.
Beam calculations usually involve some combination of support reactions, shear force, bending moment, bending stress, shear stress, slope and deflection. The correct method depends on the beam supports, loading, cross-section and the result you need.
The first step is to identify how the beam is supported. The support conditions determine which reactions can develop and whether the problem can be solved using static equilibrium alone.
A simply supported beam is normally represented by a pin support at one end and a roller support at the other. The beam is free to rotate at its supports and the end bending moments are zero.
For a single span, the support reactions can normally be obtained directly from static equilibrium.
Single-Span Beam ReferenceA cantilever is fixed at one end and free at the other. The fixed support provides both a reaction force and a reaction moment.
Cantilevers occur frequently in brackets, machine components, structural members and projecting supports.
Single-Span Beam ReferenceA fixed beam has rotational restraint at its supports. A continuous beam extends over more than one span.
These beams are generally statically indeterminate, so equilibrium equations alone are insufficient to determine all reactions and moments.
Continuous Beam ReferenceNext identify how the loads are applied. A beam may carry one type of load or several simultaneously.
| Loading | Description | Typical Effect |
|---|---|---|
| Point load | A concentrated force acting at a particular position | Step change in shear force and changing bending moment |
| Uniformly distributed load (UDL) | Constant force per unit length | Linear variation of shear and parabolic variation of moment |
| Variable distributed load | Load intensity changes along the beam | Requires integration or an equivalent analytical method |
| Applied moment | A concentrated couple acting on the beam | Produces a change in the bending-moment diagram |
| Combined loading | Several loads or load types acting simultaneously | Individual elastic effects may often be combined by superposition |
Once the supports and loading are known, select the engineering quantity required.
For a statically determinate beam, start with the equations of static equilibrium:
Determine the reactions before constructing the shear-force and bending-moment diagrams.
Shear & Bending ReferenceShear force and bending moment describe the internal actions within the beam. They are normally determined before calculating stresses.
The fundamental relationships are:
If the bending moment and section properties are known, ordinary straight-beam bending stress is determined from the flexure relationship:
At the extreme fibre this is commonly written using the section modulus Z:
Transverse shear force produces shear stress through the beam cross-section. The general beam shear relationship is:
The shear stress is not generally uniform through the section. For a rectangular beam it varies parabolically and reaches its maximum at the neutral axis.
Beam Shear Stress ReferenceA beam may satisfy the stress requirement but still be unsuitable because its deflection is excessive. Beam stiffness depends strongly on the product EI.
Standard formulas are convenient for common loading arrangements. Direct integration, singularity functions or energy methods may be preferable for other cases.
Beam Deflection TheoryElementary beam deflection calculations often consider bending deformation only. This is normally adequate for sufficiently slender beams, but shear deformation can become significant for short or deep beams and for members with relatively low shear stiffness.
If shear deformation may be important, include it in addition to bending deflection rather than assuming it is negligible.
Shear Deflection Reference| Problem | Useful Method | RoyMech Reference |
|---|---|---|
| Common single-span loading case | Standard beam equations | Single Span Beams |
| Need bending stress or basic deflection theory | Euler-Bernoulli beam theory | Beam Theory |
| Need shear-force or bending-moment diagram | Equilibrium and load/shear/moment relationships | Shear & Bending |
| Several point and distributed loads | Singularity functions / Macaulay notation | Singularity Functions |
| Fixed beam or beam with several spans | Compatibility / three-moment theorem | Continuous Beams |
| Need deflection at one particular point | Strain energy / Castigliano's theorem | Beam Energy Methods |
| Need to calculate stored elastic energy | Strain-energy integration | Strain Energy |
| Short or deep beam / shear may be significant | Shear-stress and shear-deformation analysis | Beam Shear Stress |
| Initially curved member such as a hook or ring | Curved-beam theory | Curved Beams |
For a conventional straight beam, a useful calculation sequence is:
A standard beam formula is normally the quickest approach when the beam, supports and loading correspond directly to a published case. It also provides a useful independent check on a more detailed calculation.
When several loads are present, the principle of superposition may be used for a linear-elastic beam undergoing small deflections. The response to each load is calculated separately and the compatible results are added with their signs retained.
If the loading does not correspond conveniently to standard cases, singularity functions provide a compact way of describing several loads in one beam equation.
For continuous or otherwise statically indeterminate beams, additional compatibility conditions are required. The Continuous Beams reference introduces the three-moment method for this class of problem.
In a slender beam, bending deformation commonly dominates the total deflection. This is why many elementary beam equations consider bending alone.
The assumption becomes less satisfactory as the beam becomes shorter and deeper. Shear deformation can also become important in sections where the web carries most of the shear, and in materials or constructions having relatively low shear stiffness.
The RoyMech Shear Stress and Shear Deflection reference develops the shear-stress distribution and the associated strain-energy method for calculating shear deflection.
Ordinary beam flexure theory is based on a straight member. If the beam is initially curved and its cross-sectional depth is significant compared with the radius of curvature, the resulting stress distribution differs from that predicted by straight-beam theory.
In a curved beam the neutral axis does not generally coincide with the centroidal axis. This is particularly important in components such as crane hooks, rings and curved machine members.
Use the Curved Beam Stress reference for these cases.
Energy methods are particularly useful when a displacement or rotation is required at one specific point in a beam or structure.
The elastic strain energy may contain contributions from bending, axial loading, torsion and shear. Castigliano's theorem relates the derivative of this total strain energy to the displacement corresponding to an applied force or moment.
See Strain Energy for the underlying energy expressions and Beam Energy Methods for applications of Castigliano's theorem.
A distributed loading used in beam equilibrium should be expressed as force per unit length, such as N/m or kN/m. A value specified as kg/m is a mass per unit length and must be converted to weight per unit length when gravitational loading is intended.
Beam bending calculations use the second moment of area I about the appropriate neutral axis. It should not be confused with mass moment of inertia.
Bending moments, shear forces, slopes and deflections can be positive or negative. Choose a sign convention and use it consistently throughout the calculation.
A beam may have acceptable stresses but excessive deflection. Strength and stiffness are separate design requirements and both may govern the section.
Neglecting shear deformation may be reasonable for a slender beam but should not be applied automatically to short, deep or shear-flexible members.
For strongly curved components, the displacement of the neutral axis and the non-linear stress distribution should be considered using curved-beam theory.
Flexure theory, bending stress, beam curvature, slope, deflection and section behaviour.
Beam TheorySupport reactions, shear-force diagrams, bending-moment diagrams and the relationships between load, shear and moment.
Shear & BendingMacaulay notation for representing several point loads, moments and distributed loads in a single beam equation.
Singularity FunctionsReference information for common simply supported and cantilever beam arrangements.
Single-Span BeamsFixed beams, continuous beams and analysis using Clapeyron's three-moment theorem.
Continuous BeamsNeutral-axis position and bending stress in initially curved members, including crane hooks and rings.
Curved BeamsCastigliano's theorem and strain-energy methods for calculating particular beam and structural displacements.
Beam Energy MethodsThe VQ/Ib relationship, shear-stress distributions, combined stress and deflection caused by transverse shear.
Beam Shear Stress