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

Successful application of the five-step solution method depends in no small part on
the ability to understand how axial deformations are related in a structure. The table which
follows highlights three common categories of statically indeterminate structures that are
made up of axial members. For each general category, possible geometry-of-deformation
equations are discussed.
107
STATICALLy INdETERMINATE
AxIALLy LOAdEd MEMbERS
Geometry of Deformations for Typical Statically Indeterminate Axial Structures
Equation Form Comments Typical Problems
δ
= δ
1 2
δ1 + gap = δ2
δ = δ + gap
1 2
Problems in this category include
side-by-side plates, a tube with a
filled core, a concrete column with
embedded reinforcing steel, and
three parallel rods symmetrically
connected to a rigid bar.
The deformation of each axial
member must be the same, unless
there is a gap or clearance in the
connections.
If there is a gap, then the
deformation of one member
equals the deformation of the
other member plus the length of
the gap.
1. Coaxial or parallel axial members.
B
(1)
A
(2)
360 kN
(1)
(1)
1.2 m
1.8 m
(2)
6 in.
Brass shell (1)
(1)
2.4 m
(2) Ceramic core
2 mm
600 mm
P
(2)
B
(1)
A
A B C
P
δ + δ = 0
1 2
δ + δ = constant
1 2
2. Axial members connected end to end in series.
Problems in this category include
two or more members connected
end to end.
If there are no gaps or clearances
in the configuration, then the
member deformations must sum
to zero; in other words, an
elongation in member (1) is
accompanied by an equal
contraction in member (2).
If there is a gap or clearance
between the two members or if
the supports move as the load is
applied, then the sum of the
deformations of the members
equals the specified distance.
(2)
900 mm
A
(1)
C
B
500 mm P
(1) A
3 in. 2 in.
B
(2)
32 in. 44 in.
0.04 in. gap
C
30 kips
(1) (2)
A B C
30 kips
120 in. 144 in.
12 ft
2 ft
B
A
C
(1)
P
2
— P
2
—
(2)
0.08 in.