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

88
AxIAL dEFORMATION
Equation (5.5) applies only to linear-elastic material (since Hooke’s law was assumed).
Also, Equation (5.5) was derived under the assumption that the stress distribution was uniformly
distributed over every cross section [i.e., s = F( x)/ Ax ( )]. While this is true for prismatic
bars, it is not true for tapered bars. However, Equation (5.5) gives acceptable results if
the angle between the sides of the bar is small. For example, if the angle between the sides of
the bar does not exceed 20°, there is less than a 3 percent difference between the results obtained
from Equation (5.5) and the results obtained from more advanced elasticity methods.
mecmovies
ExAmpLE
m5.3 A load of P = 50 kN is applied to a compound axial
member. Segment (1) is a solid brass [E = 100 GPa] rod
20 mm in diameter. Segment (2) is a solid aluminum [E =
70 GPa] rod. Determine the minimum diameter of the aluminum
segment if the axial displacement of C relative to support
A must not exceed 5 mm.
ExAmpLE 5.1
4 kN
y
10 kN
12 kN
(1) (2) (3)
A 10 kN B C 12 kN D
2.0 m 2.5 m 3.0 m
8 kN
x
The compound axial member shown consists of a
solid aluminum [E = 70 GPa] segment (1) 20 mm in
diameter, a solid aluminum segment (2) 24 mm in
diameter, and a solid steel [E = 200 GPa] segment (3)
16 mm in diameter. Determine the displacements of
points B, C, and D relative to end A.
Plan the Solution
Free-body diagrams (FBDs) will be drawn to expose the internal axial forces in each segment.
With the use of the internal force and the cross-sectional area, the normal stress can
be computed. The deformation of each segment can be computed from Equation (5.2),
and Equation (5.3) will be used to compute the displacements of points B, C, and D relative
to end A.
Nomenclature
Before we begin the solution, we will define the terms used to discuss problems of this
type. Segments (1), (2), and (3) will be referred to as axial members or simply members.
Members are deformable: They either elongate or contract in response to their internal
axial force. As a rule, the internal axial force in a member will be assumed to be tension.
While this convention is not essential, it is often helpful to establish a repetitive solution
procedure that can be applied as a matter of course in a variety of situations. Members are
labeled by a number in parentheses, such as member (1), and deformations in a member
are denoted δ 1 , δ 2 , etc.
Points A, B, C, and D refer to joints. A joint is the connection point between components
(adjacent members in this example), or a joint may simply denote a specific location
(such as joints A and D). Joints do not elongate or contract—they move, either in translation
or in rotation. Therefore, a joint may be said to undergo displacement. (In other
contexts, a joint might also rotate or deflect.) Joints are denoted by a capital letter. A joint