
MECHANICS  THEORY



Normal Strain

Prismatic Bar Undergoing Elongation


Strain, represented by the Greek letter ε,
is a term used to measure the deformation or extension of a body that
is subjected to a force or set of forces. The strain of a body is generally
defined as the change in length divided by the initial length.
Often, the change in length of the bar, ΔL, is simply
referred to as the total displacement, or δ. In that
case, strain is then
The elongation of the bar is assumed normal, or perpendicular, to the cross
section. Therefore, like stress, the strain is called a normal strain.
Similar to stress, a tensile strain is generally considered positive and a
compressive strain is considered negative.






StressStrain Relationship

Spring Analogy for
Stress/Strain Relationship


For linearly elastic materials, Hooke's Law relates the stress of a
body to the strain in the elastic range. Through experiments, it can be shown
that most materials act like springs in that the force is
proportional to the displacement (F = kδ). However,
unlike a spring, a bar has a crosssectional area that effects the displacement.
Therefore, it is simpler to relate the stress (F/A) to the strain (δ/L).
This gives the relationship,
where E is the material
property that represents the stiffness of the material (called Young's Modulus).
Simply stated, the stress is directly proportional to the strain. Young's modulus
is determined through experiments and are commonly listed in engineering handbooks
(appendix has
a partial list for many common structural materials).
This stressstrain relationship is called Hooke's Law. In the next section,
this relationship will be expanded to two dimensions.






StressStrain Curve

Prismatic Bar Undergoing Elongation
General StressStrain Curves
for Various Materials


The previous paragraphs developed the basic linear elastic stressstrain
relationship assuming there was no plastic deformation. However, if most materials
are strained enough, they will permanently deform. When a material deforms, the
stressstrain relationship is no longer linear.
It is common to plot the stress as a function of the strain. This
curve is called the 'StressStrain Curve'. Each material has a unique curve, but
for most materials, the initial curve is a straight line reflecting the linear
relationship between the stress and strain. This is called the 'Elastic' range.
Hooke's Law only applies in this range.
If the material is stressed past its elastic limit, the material will be permanently
deformed and plastic deformation will occur. This means the material has yielded
and will stretch easily with little additional load. Next, the material crosssection
will become smaller which is called "necking".
Since the crosssectional area
is reducing, the effective, or true stress goes up. However, many times the
stress calculations will use the original cross sectional area (called the nominal
area) and this makes the stressstrain curve appear to go down.
While most material have an elastic range, some do not. Rubber has no linear
range. Most glasses at room temperature have only an elastic range. Some examples
of basic stressstrain curves are given at the left.






Uniaxial Deflection  Constant Load, Area and Stiffness

Total Deformation
δ
= PL/AE


For a simple homogenous bar with a constant cross section and a constant applied
load, the total deflection of the bar can be determined in terms of P, L, A,
and E. Starting with the one dimensional Hooke's Law,
σ = Eε
and substituting P/A for stress and δ/L for strain gives,
P/A = E (δ/L)
This can be rearranged to give,




Total Deformation
δ = PL_{1}/A_{1}E_{1} + PL_{2}/A_{2}E_{2} +
PL_{3}/A_{3}E_{3} 

If
there are a series of bars, then the deflection of each section can be determined
and then all deflections summed. This can be written in equation form as






Uniaxial Deflection  Variable Load, Area and Stiffness

Integraion Required when
Load, Area, or/and Stiffness Vary


There are cases when the loading, cross section area, and/or the
stiffness varies along the bar. In this case, the bar can be split into small
infinitesimal lengths, dx, and integrated along its length as
To solve, each of the nonconstant parameters (A,E, and P) need to be a function
in terms of its length (x in this case). Generally, only one or two of the parameters
will be nonconstant. An example would be the tapered stake driven into the ground
where the area varies and the loading varies due to friction on its surface.




