Generalized Hooke's Law Video Lecture - Civil Engineering (CE)

unit 8.4 generalized Hookes law in this
unit we focused on the following course
outcome demonstrate the ability to
transform stress and strain and find
principal normal and shear stresses in
this lesson we'll focus on the outcome
understand material property
relationships for three-dimensional
stresses and strains let's first review
some important material property
relationships we've been using during
scores first let's review Hookes law
Hookes law is commonly shown in this
form which is Hookes law for normal
stress and strain in a single dimension
another convenient form that we'll be
using in this lesson is also shown
Hookes law shows us that there is a
linear relationship between stress and
strain Hookes law has been very useful
to us throughout this course all of the
structural deformation equations that we
have been using and we'll be using in
this course are based on this
fundamental property by definition
Hookes law only applies when a material
is linearly elastic the second
relationship I'd like to review is in
regards to Poisson through ratio
Poisson ratio is a material property and
shows us that there is a linear
relationship between longitudinal strain
and lateral strain the form that we'll
be using in our discussion is shown here
let's begin with a stress element this
is a volumetric stress element it
represents an infinitesimal point in a
material that is subjected to loading
and let's say we have stresses in all
three dimensions Sigma X Sigma Y Sigma Z
as a result of these stresses we will
expect the element to increase in size
there will be strain in all three
dimensions and we can consider the
strain in all three dimensions as strain
occurring independently
in each of the three dimensions X
dimension Y dimension n Z dimension
let's consider the strain that will be
occurring in the x-direction when we
have stresses in all three dimensions XY
and z let's begin with our x-direction
stress is a normal stress in tension
when it is applied to the element we
will expect there to be strain also in
the direction of the stress so that will
be a positive strain and that strain is
going to be equal to the stress in the X
Direction divided by the modulus of
elasticity that's Hookes law in one
dimension so now let's consider the
strain in the X direction when our
stress is in the Y direction when we put
a normal stress in tension on the
element the element will get longer in
the Y direction but the strain in the X
direction will be negative this is a
this is a lateral strain and it's
negative we can see from the poisons
ratio equation this is what we expect
and we can write the strain in the X
direction from a stress in the Y
direction is minus Poisson ratio times
the longitudinal strain which is simply
Sigma Y divided by Young's modulus
according to Hookes law and when we
consider the strain in the X direction
from our stress in the Z direction it
will look like this once again this is a
lateral strain it is a negative strain
and it will be equal to poisons ratio
times the longitudinal strain in the Z
direction which we could rewrite here
with Hookes law now if we can combine
these three equations we'll get an
expression for the strain in the X
direction in consideration of stresses
in the x y&z direction and simplifying
that we get this expression here which
is the generalized Hookes law equation
for strain in the X dimension when we
have stresses in all three dimensions x
y and z using a similar process we can
develop generalized Hookes law equations
for
in the y-direction and z direction for
stresses in the XY and z direction now
let's consider shear strains we have
equations for Hookes law for shear
stress and strains and here's an example
the shear strain is equal to 1 over the
shear modulus times the shear stress and
what we found is that shear strains
cause deformations only or changes in
the element shape and shear stress shear
strains in one plane will not cause
shear strains to develop in other planes
therefore we do not have generalized
Hookes law equations force your strain a
couple more material property
relationships to discuss first of all we
saw shear modulus it shows up in Hookes
law equation for shear stress and strain
using the generalized Hookes law
equations we can derive this expression
here which is that the shear modulus of
a material is related to the modulus of
elasticity and poisons ratio of that
material with this expression shown this
is very useful this is a very useful
relationship because we can find the
shear modulus of material knowing just
the elastic modulus and M poisons ratio
another relationship is the bulk modulus
which is a measure of the stiffness of a
volume of material it can be derived
also using generalized Hookes law
equations and we see here from this
expression that there is a limit on
poisons ratio in fact poisons ratio
cannot be greater than 0.5 for most
metals that we use in engineering
poisons ratio has a value of about 0.3
and we're done