# What is Beam Calculation? (with pictures)

D. Poupon
D. Poupon

Beam calculation is the measurement of the stress and deflection of a structural beam when a given load is applied to it. Many factors contribute to a beam’s capacity to resist bending, such as characteristics of the beam, the load and the supports. Calculating the load displacement of a single beam using the Euler-Bernoulli beam equation is straightforward, but in most practical applications, beam software is used. Beam calculations are used to ensure safety and avoid overbuilding in a variety of disciplines such as construction and aeronautics.

It is necessary to calculate beam load capacity in order to construct structures with the lightest and most inexpensive materials while fulfilling safety requirements and maintaining the structure’s aesthetic quality. The entire discipline of structural engineering is devoted to this analysis and design, ensuring that roofs do not collapse under the weight of snow, that underground parking garages are safe when traffic drives overhead and that skyscrapers built along fault lines meet earthquake safety requirements. Beam calculation also has its applications in mechanical engineering, when testing the load resistance of individual parts of a machine, such as the load that an airplane wing can withstand before developing potentially dangerous stresses. Finally, architects must consider beam deformation when building and renovating houses with post and beam construction and when considering the visual impact of sagging floors, roofs and balconies.

One of the most important factors when calculating a beam’s load bearing capacity is the choice of materials. Typically, beams are made out of wood, steel, reinforced concrete or aluminum. Each material has a different tendency to deform elastically, called the modulus of elasticity, which refers to the material’s ability to spring back into place. At its yield point, the material will deform plastically, maintaining the deformation after the applied force is removed.

The cross-sectional shape of the beam is the second characteristic that is considered in beam calculation. Beams might be rectangular, round or hollow, as well as having many types of flanking, such as I-beams, Z-beams or T-beams. Each shape has a different moment of inertia, otherwise known as second moment of area, which predicts a beam’s stiffness.

The force per unit length is another parameter used in beam calculation, and it is dependent on the load type. Dead loads simply are the weight of the structure, and imposed or live loads are the forces that the structure will be exposed to intermittently, such as snow, traffic or wind. Most loads are static, but particular attention must be paid to dynamic loads, earthquakes, waves and hurricanes, which repetitively apply force for an extended duration. A load might be distributed, typically uniformly or asymmetrically, such as snowfall or a pile of dirt. It also might be concentrated at a point, centrally or at various intervals.

The boundary conditions for beam calculation depend on the beam support type. A beam might simply be supported on both ends, like a floor joist between two load bearing walls. It might be cantilevered, or supported on one end, like a balcony or airplane wing. The boundary conditions apply to all points along the beam's length.

The relationship between a beam’s deflection and a static load is described by the Euler-Bernoulli beam equation. Another equation, the Euler-Lagrange beam equation, describes this relationship for a dynamic load, but because of the complexity of its application, static approximations typically are used. The deflection, bending moments and shear force of a beam given an applied load can be derived. In a practical setting, load charts are used to summarize this information, and they list common materials that fulfill the safety requirements for a known load. For more complicated applications, beam calculators are readily available on company websites and as add-ons for computer aided design (CAD) software.