# Beam Structures Classical and Advanced Theories

## Beam Structures Classical and Advanced Theories

Beam theories are extensively used to analyze the structural behavior of slender
bodies, such as columns, arches, blades, aircraft wings, and bridges. The main

advantage of beam models is that they reduce the 3D problem to a set of vari-
ables that only depends on the beam-axis coordinate. The 1D structural elements

obtained are simpler and computationally more efficient than 2D (plate/shell) and
3D (solid) elements. This feature makes beam theories very attractive for the static
and dynamic analysis of structures.
Many methods have been proposed to overcome the limitations of classical
theories and to allow the application of 1D models to any geometry or boundary
condition. Many examples of these models can be found in many well-known

books on the theory of elasticity.

A brief description of the book’s layout is given here to provide a brief overview

of what will be discussed. Chapter 1 presents the basic equations that the struc-
tural analysis is based on: equilibrium equations, strain–displacement geometrical

relations, and constitutive equations. The principle of virtual displacements is also
introduced in strong and weak forms.
Chapter 2 focuses on the description of classical beam theories: namely, the
Euler–Bernoulli and Timoshenko models. The kinematics model of these theories
is introduced and then strains, stresses, stress resultants, and elastica equations
are derived. Numerical examples are given in order to highlight the differences
between these two models.
The first refined model of this book is given in Chapter 3, where the complete
linear expansion case is presented. Particular attention is given to the importance of
the in-plane stretching terms that characterize this model. Examples are provided
in order to underline the importance of these terms and the ineffectiveness of
classical models to deal with in-plane stretching.