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2

Bars and Trusses

2.1 Introduction
This chapter introduces you to the simplest one-dimensional (1-D) structural element,
namely the bar element, and the FEA of truss structures using such element. Trusses
are commonly used in the design of buildings, bridges, and towers (Figure 2.1). They are
triangulated frameworks composed of slender bars whose ends are connected through
bolts, pins, rivets, and so on. Truss structures create large, open, and uninterrupted space,
and offer lightweight and economical solutions to many engineering situations. If a truss,
along with the applied load, lies in a single plane, it is called a planar truss. If it has mem-
bers and joints extending into the three-dimensional (3-D) space, it is then a space truss.
Most structural analysis problems such as stress and strain analysis can be treated as
linear static problems, based on the following assumptions:

1. Small deformations (loading pattern is not changed due to the deformed shape)
2. Elastic materials (no plasticity or failures)
3. Static loads (the load is applied to the structure in a slow or steady fashion)

Linear analysis can provide most of the information about the behavior of a structure,
and can be a good approximation for many analyses. It is also the basis of nonlinear FEA
in most of the cases. In Chapters 2 through 7, only linear static responses of structures are
considered.

2.2 Review of the 1-D Elasticity Theory
We begin by examining the problem of an axially loaded bar based on 1-D linear elasticity.
Consider a uniform prismatic bar shown in Figure 2.2. The parameters L, A, and E are the
length, cross-sectional area, and elastic modulus of the bar, respectively.
Let u, ε, and σ be the displacement, strain, and stress, respectively (all in the axial direc-
tion and functions of x only), we have the following basic relations:

Strain–displacement relation:

du ()
x
x
ε() = (2.1)
dx
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