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

displacement in the horizontal direction is denoted
by u together with a subscript identifying the joint
(e.g., u A ). A positive displacement u means that the
joint moves to the right.
SOLUTION
Equilibrium
Draw an FBD that exposes the internal axial force
in member (1). Assume tension in member (1).
The equilibrium equation for this FBD is
∑ Fx = F1
− 4kN = 0
∴ F = 4kN = 4 kN ( T)
1
Draw an FBD for member (2) and assume tension
in member (2).
The equilibrium equation for this FBD is
∑ Fx = F2
+ 2(10 kN) − 4 kN = 0
∴ F2 = − 16 kN = 16 kN ( C)
Similarly, draw an FBD for member (3) and assume
tension in the member. Although two different
FBDs are possible, the simpler one is shown.
The equilibrium equation for this FBD is
∑ Fx =− F3
+ 8kN = 0
∴ F = 8kN = 8 kN ( T)
3
Before proceeding, plot the internal forces F 1 , F 2 ,
and F 3 acting in the compound member. It is the
internal forces that create deformations in the axial
members, not the external forces applied at
joints A, B, C, and D.
y
10 kN
12 kN
4 kN F 1
8 kN
(1)
(2)
(3)
x
A 10 kN B C 12 kN D
2.0 m 2.5 m 3.0 m
y
10 kN
12 kN
4 kN
F 2
8 kN
(1)
(2)
(3)
x
A 10 kN B C 12 kN D
2.0 m 2.5 m 3.0 m
y
10 kN
12 kN
4 kN
F 3
8 kN
(1)
(2)
(3)
x
A 10 kN B C 12 kN D
2.0 m 2.5 m 3.0 m
y
10 kN
12 kN
4 kN
(1) (2) (3)
8 kN
A 10 kN B C 12 kN D
Tension
4 kN
8 kN
Compression
–16 kN
Axial force diagram for compound member.
x
Problem-Solving Tip: When drawing an FBD that cuts through an axial member,
assume that the internal force is tension and draw the force arrow so that it is directed
away from the cut surface. If the computed internal force value turns out to be a
positive number, then the assumption of tension is confirmed. If the computed value
turns out to be a negative number, then the internal force is actually compression.
Force–Deformation Relationships
The relationship between the deformation of an axial member and its internal force is
expressed by Equation (5.2):
δ =
Since the internal force is assumed to be a tensile force, the axial deformation is assumed
to be an elongation. If the internal force is compressive, then the use of a negative value
for the internal force F in the preceding equation will produce a negative deformation—in
other words, a contraction.
FL
AE
89