What Are Beams?

Beams are structural members primarily loaded transverse to their long axis, causing them to bend rather than compress or stretch along their length. They transfer applied loads to supports through a combination of shear forces and bending moments distributed along the member, and their analysis is among the most fundamental topics in structural and mechanical engineering. Beams appear in virtually every engineered structure, from bridges and building frames to printed circuit board assemblies and microelectromechanical systems (MEMS).

The theoretical treatment of beams draws from the mechanics of materials, solid mechanics, and applied mathematics. The classical Euler-Bernoulli beam theory, developed in the 18th century, relates the curvature of a slender beam to the internal bending moment and the material's flexural stiffness, expressed as the product of the elastic modulus E and the second moment of area I. Timoshenko beam theory, introduced in the early 20th century, extends this by accounting for shear deformation and rotational inertia, providing more accurate results for short, deep beams and dynamic loading situations.

Bending and Stress Distribution

When a beam bends, fibers on the concave side are compressed while fibers on the convex side are stretched, with a neutral axis separating tension and compression zones where longitudinal stress is zero. The bending stress at any point in the cross section is proportional to its distance from the neutral axis and to the internal bending moment at that location, following the flexure formula σ = Mc/I. Shear stress is also present, distributed across the cross section according to the Jourawski formula. University of Washington course materials on beam stress and deflection cover the derivation of these distributions for common cross-sectional geometries including rectangular, I-section, and circular profiles.

Deflection and Support Conditions

Beam deflection, the transverse displacement of points along the member under load, is governed by the relationship between bending moment and curvature. Integrating the moment-curvature relation twice, with boundary conditions imposed by the support type, yields the elastic curve. Simply supported beams, cantilevers, and fixed-fixed beams each produce distinct deflection profiles under the same applied load, and support conditions profoundly affect both maximum deflection and reaction forces. The MechaniCalc reference on beam stress and deflection provides a systematic treatment of these calculations for uniform, concentrated, and distributed loading patterns. Maximum deflection must remain within serviceability limits; in steel structures, deflection limits are often set at span/360 under live load.

Cross-Sectional Geometry and Material Selection

The efficiency of a beam at resisting bending depends on how its cross-sectional area is distributed relative to the neutral axis. Wide-flange I-sections concentrate material far from the neutral axis, maximizing the second moment of area per unit weight. Hollow rectangular and circular tubes offer good torsional stiffness in addition to bending resistance. Composite beams, combining materials such as steel and concrete or carbon fiber and epoxy, achieve stiffness and strength profiles unattainable with any single material. SkyCiv's guide to beam deflection discusses how modulus of elasticity values, ranging from roughly 70 GPa for aluminum to 210 GPa for structural steel, govern deflection under identical loading.

Applications

Beams have applications in a wide range of engineering fields, including:

• Building and bridge structures, as primary load-carrying elements in frames and decks
• Mechanical machinery, as shafts and support members in equipment frames
• Microelectromechanical systems (MEMS), as cantilever sensing elements in accelerometers and pressure sensors
• Aerospace structures, as wing spars and fuselage frames
• Offshore platforms and marine structures subjected to wave and current loading