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

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87
dEFORMATIONS IN AxIALLy
LOAdEd bARS
L 1
F 1
(1)
B
(1)
B
(1)
B
(1)
B
P B
P B
P B
F 2
P B
L 2
(2)
PC
(2)
P
C
(2)
P C
(2)
PC
C
C
C
C
F 3
L 3
(3)
D
(3)
D
(3)
D
(3)
D
P D
(a) Three-segment
axial member
P D
(b) FBD for
segment (1)
P D
(c) FBD for
segment (2)
FIGURE 5.7 Compound axial member and associated free-body diagrams (FBDs).
P D
(d) FBD for
segment (3)
are first calculated individually. Then, the three deformation values are added together to
give the overall deformation. The internal force F i in each segment is determined from the
free-body diagrams shown in Figure 5.7b–d.
For those cases in which the axial force or the cross-sectional area varies continuously
along the length of the bar (Figure 5.8a), Equations (5.1), (5.2), and (5.3) are not
valid. In Section 2.2, the normal strain at a point for the case of nonuniform deformation
was defined as ε = dδ
/ dL . Thus, the increment of deformation associated with a
differential element of length dL = dx may be expressed as dδ = ε dx. If Hooke’s law
applies, the strain may again be expressed as ε = s / E , where s = F( x)/ Ax ( ) and both
the internal force F and the cross-sectional area A may be functions of position x along
the bar (Figure 5.8b). Thus,
Fx ( )
dδ =
AxE ( )
dx
(5.4)
Integrating Equation (5.4) yields the following expression for the total deformation of
the bar:
L L Fx ( )
δ = ∫ dδ
=
AxE ( )
dx
0 ∫
(5.5)
0
x
dx
F(x)
F(x)
P A
P B
L
δ
dx
dδ
(a)
FIGURE 5.8 Axial member with varying internal force and cross-sectional area.
(b)