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

d 0
to 2d 0 at the other end. A concentrated axial load P is applied to the
ends of the rod, as shown in Figure P5.14. Assume that the taper of
the cone is slight enough for the assumption of a uniform axial
stress distribution over a cross section to be valid.
(a) Determine an expression for the stress distribution over an
arbitrary cross section at x.
(b) Determine an expression for the elongation of the rod.
P
x
P
2d 0
p5.15 The conical bar shown
in Figure P5.15 is made of aluminum
alloy [E = 10,600 ksi
and γ = 0.100 lb/in. 3 ]. The bar
has a 2 in. radius at its upper end
and a length L = 20 ft. Assume
that the taper of the bar is slight
enough for the assumption of a
uniform axial stress distribution
over a cross section to be valid.
Determine the extension of the
bar due to its own weight.
y
L
L
FIGURE p5.14 FIGURE p5.15
x
5.4 Deformations in a System of Axially Loaded Bars
Many structures consist of more than one axially loaded member, and for these structures,
axial deformations and stresses for a system of pin-connected deformable bars must be
determined. The problem is approached through a study of the geometry of the deformed
system, from which the axial deformations of the various bars in the system are obtained.
In this section, the analysis of statically determinate structures consisting of homogeneous,
prismatic axial members will be considered. In analyzing these types of structures,
we begin with a free-body diagram showing all forces acting on the key elements of the
structure. Then, we investigate how the structure as a whole deflects in response to the deformations
that occur in the axial members.
A homogeneous, prismatic
member (a) is straight, (b) has
a constant cross-sectional area,
and (c) consists of a single
material (so it has exactly one
value of E).
ExAmpLE 5.3
The assembly shown consists of rigid
bar ABC, two fiber-reinforced plastic
(FRP) rods (1) and (3), and FRP post
(2). The modulus of elasticity for the
FRP is E = 18 GPa. Determine the vertical
deflection of joint D relative to its
initial position after the 30 kN load is
applied.
y
2.4 m 1.8 m
Rigid bar B
A
(1) (2) (3)
C
3.0 m
x
Plan the Solution
The deflection of joint D relative to its
initial position must be computed. The
deflection of D relative to joint C is
simply the elongation of member (3).
The challenge in this problem, however,
lies in computing the deflection
at C. The rigid bar will deflect and
rotate because of the elongation and
contraction in members (1) and (2). To
3.6 m
F
A 1= 500 mm 2
A 3 = 500 mm 2
A 2 = 1,500 mm 2
The three axial members are connected to the rigid beam by pins. Assume that
member (1) is pinned to the foundation at F and member (2) is fixed in the
foundation at E.
E
D
P = 30 kN
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