Simple beams are straight, single-span members that can generally be analysed using the equations of static equilibrium. They include simply supported beams and cantilever beams and form the basis of many standard beam calculations.
This page provides an overview of the principal support arrangements, loading types and beam sections used in single-span beam analysis. For bending stress, deflection and the underlying beam equations, see Beam Theory.
Three basic support arrangements are commonly encountered. More complex beam systems can be formed by combining or extending these arrangements.
A simply supported beam is normally represented by a pin or hinge support at one end and a roller support at the other. The supports prevent the required translations while allowing rotation of the beam ends.
For the usual two-dimensional loading case, the support reactions can be determined directly from the equations of static equilibrium. The beam is therefore statically determinate.
A cantilever is fixed at one end and free at the other. The fixed support provides force reactions and a reaction moment.
A single cantilever under plane loading is also statically determinate because the support reactions can be obtained from the equilibrium equations.
A beam fixed at both ends is restrained against translation and rotation at both supports. Unlike the simple cases above, a fixed-fixed beam is statically indeterminate and requires compatibility or deformation relationships in addition to the equations of static equilibrium.
Beams may be subjected to several different forms of loading. The principal loading types used in beam calculations are:
A practical beam may be subjected to several of these loading types simultaneously. The applied loads and support reactions are first established from a free-body diagram before the internal shear forces and bending moments are determined.
For the construction of shear-force and bending-moment diagrams, see Shear Force and Bending Moment Diagrams.
For a linear-elastic beam undergoing small deflections, the principle of superposition provides a convenient method of analysing several loads acting on the same beam.
The effect of each load may be calculated separately and the resulting reactions, shear forces, bending moments, slopes and deflections combined algebraically.
Beams commonly have a constant cross-section along their length. Structural beam sections include rectangular sections, circular sections, channels, angles, I-sections, hollow sections and many other standard profiles. Section properties for a range of engineering profiles are included in the RoyMech Sections reference.
In mechanical engineering, circular sections such as shafts and rectangular sections are frequently encountered. Other machine components may behave as beams even though they are not normally described as structural beams. Examples include shafts under transverse loading, levers, springs and gear teeth.
Beam sections may also vary along their length. In these cases the section properties, particularly the second moment of area I, may vary with position and must be treated accordingly in the bending and deflection analysis.
For a statically determinate beam, a typical analysis proceeds by first identifying the applied loads and support conditions and then calculating the support reactions from static equilibrium.
The internal shear force V and bending moment M can then be determined along the beam. These provide the basis for calculating bending stress, shear stress, slope and deflection.
For elastic bending of a straight beam, the fundamental bending relationship is:
where M is the bending moment, I is the second moment of area, E is Young's modulus, R is the radius of curvature, σ is bending stress and y is the distance from the neutral axis.
The maximum elastic bending stress is commonly expressed using the section modulus Z:
Detailed derivations and deflection relationships are given on the Beam Theory page.
Standard beam cases provide a convenient way of obtaining reactions, maximum bending moments, slopes and deflections for commonly encountered support and loading arrangements.
Where several independent loads act on a linear-elastic beam, the appropriate standard cases can often be combined using superposition.
RoyMech beam and stress calculation formulae are available in the Stress and Strain Formulae reference.