Euler–Bernoulli beam theory

Euler–Bernoulli beam theory models a beam as a slender, line-like body whose cross-sections remain plane and perpendicular to the deformed neutral axis. ¹ The assumption suppresses transverse shear deformation, so the slope of the deflected centerline equals the section’s rotation. The beam is usually treated as prismatic or piecewise prismatic, linearly elastic, and subject to small strains and small rotations. Its material may be homogeneous and isotropic in the simplest form, although the mathematical structure can be adapted to varying stiffness. These assumptions make the theory especially effective for long, narrow beams under static or low-frequency bending, including floor joists, rails, shafts, and many idealized members in frames.

The model is one-dimensional: loads and reactions are represented along a reference axis, while cross-sectional geometry enters through the second moment of area, I. Local effects near supports, load introduction points, holes, and abrupt changes in section are therefore outside its ideal resolution.

The central equation connects distributed transverse load to beam deflection through bending stiffness. For a uniform beam with constant elastic modulus E and second moment of area I, the static relation is commonly written as EI d⁴w/dx⁴ = q(x), with sign depending on convention. ¹ Here w(x) is transverse displacement and q(x) is distributed load. Integrating the equation introduces the internal bending moment and shear force, while four integration constants are determined by boundary conditions.

Curvature is proportional to bending moment: M = EIκ, with the sign again convention-dependent. A larger E makes a material harder to bend; a larger I makes a cross-section more resistant by placing more area farther from its neutral axis. ² Boundary conditions may prescribe displacement, rotation, moment, or shear. A clamped end fixes displacement and rotation; a simple support fixes displacement while allowing rotation; and a free end has prescribed moment and shear, often both zero.

The theory is valuable because it converts many beam problems into tractable boundary-value problems with closed-form or readily computed solutions. It predicts deflection, slope, bending moment, shear force, and support reactions for common cases such as point loads, uniform loads, cantilevers, simply supported beams, and continuous members. ² It also supplies the classical basis for finite-element beam elements, where nodal translations and rotations approximate the displacement field.

Its main limitation is the neglect of shear deformation and rotary inertia. For deep or short beams, sandwich panels, timber members with low shear stiffness, and high-frequency vibration, Timoshenko beam theory is generally more appropriate. ³ Large deflections require geometrically nonlinear equations; plasticity, cracking, contact, and damage require constitutive or specialized structural models beyond linear Euler–Bernoulli theory. Three-dimensional stress concentrations also cannot be inferred reliably from the one-dimensional solution alone.

The theory’s most consequential approximation is not simply “small deflection,” but the kinematic statement that cross-sections remain perpendicular to the centerline. That statement makes the model exact only in idealized slender-beam limits; it can remain useful at moderate deflections in carefully linearized engineering calculations, but the underlying small-rotation equations then cease to describe the geometry accurately. ¹

Cross-sectional warping is another overlooked boundary of the model. Ordinary bending about a principal axis is represented well, whereas torsion, restrained warping, open thin-walled sections, and coupled bending-torsion behavior require additional theories. The neutral axis also need not coincide with the geometric centroid in unsymmetrical or composite sections. In dynamics, Euler–Bernoulli theory predicts a frequency-dependent error because it omits rotary inertia and shear flexibility; the error generally grows for higher modes and shorter wavelengths. ³ These edge cases explain why the classical model remains a benchmark rather than a universal beam description.

Bending stiffness

The product EI, combining material stiffness with cross-sectional resistance to bending.

Second moment of area

A geometric quantity measuring how cross-sectional area is distributed about a bending axis.

Neutral axis

The longitudinal line in a bending member whose normal fibers experience approximately zero axial strain in the classical model.

Timoshenko beam theory

A beam model that includes transverse shear deformation and rotary inertia.