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3

Beams and Frames

3.1 Introduction
Beams and frames are frequently used in constructions, in engineering equipment, and in
everyday life, for example, in buildings, lifting equipment, vehicles, and exercise machines
(Figure 3.1).
Beams are slender structural members subjected primarily to transverse loads. A beam
is geometrically similar to a bar in that its longitudinal dimension is significantly larger
than the two transverse dimensions. Unlike bars, the deformation in a beam is predomi-
nantly bending in transverse directions. Such a bending-dominated deformation is the
primary mechanism for a beam to resist transverse loads. In this chapter, we will use
the term “general beam” for a beam that is subjected to both bending and axial forces, and
the term “simple beam” for a beam subjected to only bending forces. The term “frame” is
used for structures constructed of two or more rigidly connected beams.

3.2 Review of the Beam Theory
We start with a brief review of the simple beam theory. Essential features of the two well-
known beam models, the Euler–Bernoulli beam and the Timoshenko beam, will be intro-
duced, followed by discussions on the stress, strain, and deflection relations in simple
beam theory.

3.2.1 Euler–Bernoulli Beam and Timoshenko Beam
Euler–Bernoulli beam and Timoshenko beam, as shown in Figure 3.2, are two common
models that are used in the structural analysis of beams and frames. Both models have at
their core the assumption of small deformation and linear elastic isotropic material behav-
ior. They are applicable to beams with uniform cross sections.
For a Euler–Bernoulli beam, it is assumed that the forces on a beam only cause the beam
to bend. There is no transverse shear deformation occurring in the beam bending. The
neutral axis, an axis that passes through the centroid of each beam cross section, does not
change in length after the deformation. A planar cross section perpendicular to the neu-
tral axis remains plane and perpendicular to the neutral axis after deformation. Due to its
neglect of shear strain effects, the Euler–Bernoulli model tends to slightly underestimate

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