Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

5.3 Deformations in Axially Loaded Bars
L
FIGURE 5.5 Elongation of a
prismatic axial member.
δ
F
A member that is subjected to no
moments and has forces applied
at only two points is called a
two-force member. For
equilibrium, the line of action of
both forces must pass through
the two points where the forces
are applied.
A material of uniform
composi tion is called a homogeneous
material. The term
prismatic describes a structural
member that has a straight
longitudinal axis and
a constant cross section.
When a bar of uniform cross section is axially loaded by forces applied at the ends (such a
bar is called a two-force member), the normal strain along the length of the bar is assumed to
have a constant value. By definition, the deformation (Figure 5.5) of the bar resulting from
the axial force F may be expressed as δ = εL. The axial stress in the bar is given by s = F/A,
where A is the cross-sectional area. If the axial stress does not exceed the proportional limit
of the material, Hooke’s law may be applied to relate stress and strain: s = Eε. Thus, the
axial deformation may be expressed in terms of stress or load as:
or
s L
δ = εL
= (5.1)
E
FL
δ = (5.2)
AE
Equation (5.2) is termed the force–deformation relationship for axial members and is the
more general of these two equations. Equation (5.1) frequently proves convenient when a
situation involves a limiting axial stress, such as the design of a bar so that both its deformation
and its axial stress are within prescribed limits. In addition, Equation (5.1) often
proves to be convenient when a problem involves the comparison of stresses in assemblies
consisting of multiple axial members.
Equations (5.1) and (5.2) may be used only if the axial member
• is homogeneous (i.e., E is constant),
• is prismatic (i.e., has a uniform cross-sectional area A), and
• has a constant internal force (i.e., is loaded only by forces at its ends).
If the member is subjected to axial loads at intermediate points (i.e., points other than
the ends) or if it consists of various cross-sectional areas or materials, then the axial member
must be divided into segments that satisfy the three requirements just listed. For compound
axial members consisting of two or more segments, the overall deformation of the axial
member can be determined by algebraically adding the deformations of all the segments:
FL i i
δ = (5.3)
AE
∑
i
Here, F i , L i , A i , and E i are the internal force, length, cross-sectional area, and elastic modulus,
respectively, for individual segments i of the compound axial member.
i
i
F
δ
δ
F
In Equation (5.3), a consistent sign convention is necessary to calculate the deformation
δ produced by an internal force F. The sign convention (Figure 5.6) for deformation is
defined as follows:
• A positive value of δ indicates that the axial member gets longer; accordingly,
a positive internal force F produces tensile normal stress.
• A negative value of δ indicates that the axial member gets shorter (termed
contraction). A negative internal force F produces compressive normal stress.
FIGURE 5.6 Positive sign
convention for internal force F
and deformation δ.
A three-segment compound axial member is shown in Figure 5.7a. To determine the
overall deformation of this axial member, the deformations for each of the three segments
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