Stress Transformation Example Video Lecture - Civil Engineering (CE)

all right well some people will come
back to structure free learning and in
this video we're going to do stress
transformation without more circle and
that means we're going to use stress
transformation equations which I'll
write out over here and hopefully you've
seen equations like this and how they
were derived normally these equations
are derived or at least this set is
based on starting out with the plane
stress element whose horizontal axis is
defined as theta is equal to zero
degrees or some other axis is defined as
theta equals zero degrees and then going
at a new angle theta rotated
counterclockwise so this would define
our X prime and our x axis the Y and the
Y prime axis would be perpendicular to
their respective you know prime or no
prime States then if you take the
derivative of these with respect to
theta and set them equal to zero then
you can find the principal stress
equations and also the maximum in plane
shear stress equations which look
something like this so here are the
stress transformation equations and the
principal stress state equations and the
maximum in plane shear stress equations
these equations can be used to solve any
stress transformation prop and so what
we're going to do is take this problem
over here which I actually solve once
using Mohr circle and use just equations
to calculate principal stresses and show
all the representative volume element
and maximum plane shear stresses and
draw the representative volume element
so given the stress state here the first
thing we need to do is define a
coordinate system or a local axis for
this element and that means selecting
what we consider positive X and positive
Y and here I'll just go with the
standard plus X to the right and plus Y
vertical and by choosing that we are now
able to define our stresses which
is that Sigma X is positive 20 mega
pascals Sigma Y minus 10 mega pascals
and tau XY because it's on the plus X
face in the plus y direction positive 30
mega Pascal's and with these stresses
defined now we can go ahead and solve
for principal stresses and maximum plane
shear stress states and so we'll do the
principal stresses first and the first
thing you want to find in this is the
angle or angles really associated with
the principal stress T and here we're
going to apply this tangent to theta P
and when we plug and chug we get this
tangent to theta P equals 2 solve for
two theta P we get sixty three point
four four degrees and any positive
number is associated with a positive
rotation as these equations were defined
which is counterclockwise so this
indicates counterclockwise and also
another thing to remember is that 180
degrees from this or plus 180 there is
another angle that will may give you a
correct solution which would be 180 from
this is 243 point four four degrees
going counterclockwise angles actually
associate with the principal stress we
got divide these numbers by two is Theta
P the two possible answers are
thirty-one point seven two degrees
and 121 point seven two degrees both
going counterclockwise so these are the
angles associated with my principal
stresses now I have to determine what
the actual principal stresses are and I
can do that using this relationship
right here and here I can just go ahead
and plug and chug my values so my my
principal stresses are five plus or
minus thirty three point five four I
have two of them Sigma one I will call
my major principle stress or my most
positive which in this case would be
five plus thirty three point five four
or thirty eight point five for
megapascals and Sigma two would be my
minor principle stress will be 5 minus
thirty three point five four or negative
twenty-eight point five for megapascals
and these are my principal stresses here
are the angles and really there's only
one last thing to do and that is figure
out which angle goes with which stress
and in order for me to do that I what
I'm going to do is go back to this
equation here and plug in one of the
angles and see what principle stress I
get so if I take this equation just plug
and chug with one of the principal
stresses I'll go with thirty one point
seven two degrees and two times thirty
one point seven two is sixty three point
four four and when I calculate I'll get
thirty eight point five for MPA which
tells me that this thirty one point
seven two degrees is associated with
Sigma one so I can just write here theta
P one thirty one point seven two degrees
counterclockwise and and I know the
other one theta P two or the angle
associated with my second principal or
my minor principle stress is one hundred
and twenty-one point seven two degrees
counterclockwise and now I can go ahead
and draw the principal stress State this
essentially is my answer for principal
stresses and the angles associated with
them and these angles are with respect
to the horizontal as I defined it in my
coordinate
system at the very beginning and if I
wanted to draw that representative
volume element I'm going to start with
the line that represents theta equals
zero degrees which in this case was
horizontal and if I start with Sigma one
I know that's thirty one point seven two
degrees from the horizontal going
counterclockwise and if I draw a square
with respect to this so that means the
first line I'm going to do is draw a
line perpendicular to that whore that's
thirty one point seven two degree line
I'm going to complete the square from
here my normal stress on this face is a
positive thirty eight point five for
indicating that I have a tension or
positive arrow going away from the face
and equal and opposite on the other side
and if I look 121 point seven two
degrees from my horizontal that should
take me to this face over here which
would be a line going
I drew this kind of messed up but it
would be this line right here and this
angle would be 120 one point seven two
degrees which would take me to this
Sigma 2 and that's twenty eight point
five four or negative twenty-eight point
five four mega Pascal's which indicates
compression on this face and I draw the
direction of the arrow and the magnitude
of 28.5 four mega Pascal's and this is
my principal stress state so if we go
back and look really in order to do any
transformation towards the principal
stress I only have to plug and chug
three times and if you've got kind of a
fancy calculator that remembers things
all you have to do is calculate the
angles the two angles that are
associated with the principal stresses
then go to this equation right here plug
in thirty one point seven two degrees
get one strip one principal stress and
then plug in 120 one point seven two
degrees and all in order to plug in 120
one point seven two degrees you probably
scroll up in your calculator replace
values and bam you would get Sigma two
then you'd be done and then you just
have to draw the representative volume
element out no need for more circle now
that you have like fancy schmancy
calculator what's up