CEE 220
1D Elasticity
Notes
This handout presents the basic concepts and analytical framework associated with engineering solid
mechanics considered in the simple context of 1Dimensional systems. Although this is not a particularly
useful class of models from a practical point of view, it is very useful from a conceptual point of view because
the big picture can be presented with a minimum of complicating factors.
1
Fundamental Quantities
As discussed in class, the two fundamental quantities of interest in solid mechanics are stress and strain. In
their 1Dimensional manifestations we have the following conceptual definitions:
Stress
Intensity of force experienced by a material, i.e., amount of force carried normalized by the area of
material available to carry it. For an axiallyloaded (1D) rod we have:
σ
=
P
A
Strain
Intensity of deformation, i.e., amount of displacement that must be accommodated normalized by
the length of material loang which the displacement is distributed. For a 1Dimensional rod with an
initial length,
L
0
, and an overall elongation,
δ
, we have
=
δ
L
0
These conceptual descriptions both define the units of stress and strain, and provide a basic foundation we
can build on.
2
Fundamental Relations
To move from basic definitions to relations we can use to generate quantitative results, we need to cast our
notions of stress and strain into a more mathematical context. To this end, we will develop mathematical
relations governing the behavior of stresses/loads and strains/displacements in the 1Dimensional case.
x
A
b
(
x
)
Figure 1: A 1D elastic rod.
Figure 1 shows the basic setup for our derivations. We have a 1D rod of material with the location of
each point given by a coordinate,
x
, and the displacement of each point from its original location denoted
u
(
x
). In general, we can have distributed applied axial loading, denoted here as
b
(
x
).
2.1
Kinematics: StrainDisplacement Relation
To characterize the strain in this context, we need to determine the elongation intensity at each point in the
body as function of the overall displacements. Applying the concept of strain as length change normalized
1
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CEE 220
2
x
Δ
x
x
u
(
x
)
u
(
x
+ Δ
x
)
before deformation
after deformation
Figure 2: A 1D elastic rod.
by original length of a particular segment as shown in Figure 2, we can write
=
u
(
x
+ Δ
x
)

u
(
x
)
Δ
x
(1)
To get the strain at the point
x
itself, we take the limit as Δ
x
→
0, and this becomes our 1D strain
displacement relation:
(
x
) =
lim
Δ
x
→
0
u
(
x
+ Δ
x
)

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 Spring '08
 Mackenzie
 Force, Boundary conditions

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