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good morning and welcome
are to lecture 3 a machine design this'll be
a chapter 4 ARV are
per machine design textbook Mohr's circle analysis
Mohr's circle
is named after our Christian are more it's a graphic representation
have the state of stress at a point we've been talking about stress elements
various assist stress states
10 soul sheer coercion
bending arm were interested in examining
at a particular point what is the level of stress
and it is not frequently in the conventionally thought I love
X why perhaps Z of
cartesian coordinate systems sometimes our maximum stress
occurs at a plane a completely different
somewhat arbitrary angle to those conventional
XYZ coordinates and so is important at
a particular location to examine what is the truth stress
being seen by the material at hand so no more circle we can graphically get an
idea of where
these Macs mom planes a stress
occur and so we do this by representing the
are tensile stress normal stress
on the issue abscissa or X axis and the shear stress on the ordinate or why
access and so are each stress state represents a point
the locus or combination of all of these
a centered on to Sir karma in the center of a circle
create some more circle graphic representation of stress
we use this I to determine the stresses
at this given point as I said and
the angle have orientation that gives the maximum stress
that that particular point in space that point apart we'll see
arm the before possible variations
how that will examine pure
uniaxial tension pure uni actually a compression Muni axial means
only in the axe or only in the Y pure torsion
and then uniaxial tension and torsional shear
to draw
more circle will have to have knowledge
at a particular point in space on our part
armoire are the tensile stress
in the axe in the Y and was the shear stress at that point
with that I'll with that information we can construct
more circle it's helpful to have some
are sign conventions when creating the circles convention holds that potential
stresses
are positive positive tensile stresses
are are to the right compressive
negative test stresses are to the left clockwise sheer
shear stresses that would turn that would tend to Roque AP element
clockwise are plotted up and counterclockwise
sheer shear stresses that word rotator tend to rotate
a stress element counterclockwise plot it down and so we
identify the axe as the
potential stress axis and the Y
as the shear stress axis and we're gonna plot
arm the stressed pairs the
Sigma X Sigma X Y Sigma why
Sigma X Y those two points and then we'll connect those two points
and that is the diameter the circle and so we now can crawl
a circle about those two points the center this look circle
is located at the average up those two
normal stresses and so
here we see the finish more circle
you come back to this as we progress in our conversation
so the average stresses the average between the sick maxine the signal why
and we arithmetic Lee at those
divide by two plying
passing from point 122 and intersecting with the x-axis
is the center of our more circle
and where that circle crosses
the normal axis for the x-axis we will have that nor
max normal stress Sigma max and the place where it crosses the
at axis but left side
would be the map and a minimum normal strikes or Sigma minimum
we can arithmetic Lee
compute what the a sigma max's
using this equation
you can see both Sigma max is sick man very similar the only difference being
a plus in the minus sign prior to that second term the square return
on each side
the vertical diameter which passes service on the circle or Sigma
average arm can be found
in Equation form by this equation
at the bottom the slide we notice that this equation
represents the square root
turn in the previous Signum acted sick man
equation
next up is to develop to determine the angle that the normal stresses acting
arm this is a
twice the actual angle and sold at
are is occurring by
that is computed by to fade equals the arcane ginned
of that equation there and so we would
for we would Perry computer using our
stress state values and then divide by two to get the angle
and then the angle at them extra Mac stresses acting
arm Mac shear stresses acting
his the two prime
and sold the last part is we draw our normal
stress angle at the angle theta
and their Mac shear stress element angle had a bagel think a prime
and so what were happening here is as we rotate
the stress element to the angle theta
we will have the max potential stresses that the
that element will see and if we then rotate the element
to say to prime
which generate that the
max sheer alum shear stress that that out on the quad
see here is an animation
ARV this year
stressed elements and more circle
you see here as the stress element rotates
counterclockwise the stress arrows
the magnitude of them are changing when we're
intersecting the Sigma where the x-axis
the sheer arrows or stresses disappear
so the shear stress at this
Cigna has no shear so that is a principal
stress was the general case
where we have a stress element that has
a signal X a signal why and shear stresses in
in the ex-wife points as well and he rear
the equations that we introduced earlier
this is AB special case
the corn organ is located at the center of more circle
meaning we have uniaxial tansel stress
you re axial stress one direction
so we start to their
and as we rotate we go from max
normal to max sheer
by waving example let's say
on a particular point we have the following or the shown stress state
at that point we have a dual stress state we have a potential stress
in the axe 6 kasi a compressive stress in the Y
-2 kasi in a sheer
in of three kasi so here is our stress element
we're going to now plot more circle we're going to look at
the X face here
we have stress on that's is a
6 case I positive and a shear stress
I've just this blue element which is tending to cause this to rotate
clockwise this would be a clockwise rotation
we will plot that
nicely positive tensile stresses
to the right clockwise rotating
shear stresses upward so that point one
likewise for point2
the negative or compressive signal why
and the sheer which causes a
counter clockwise rotation the negative or compress a potential stressful plot
down and the counter clockwise rotating
sheer will reply all plot to the left and so we have
point2 plotted so we plotted both
sup faces a bar stress element
week connect damn
that is the diameter a far more circle
the intersection his where it crosses the Sigma access
access and that circle there is the locus stop all possible combinations
as we rotate the stress element
through a variety of angles using the equations we can compute
with the average shear stresses recognize that the
max and the median shear stress occurs
on a vertical line passing through Sigma average
we can compute
using for factory in Theorem the radius
and use that expression to you determine what
the magnitude
I love sick not one or Sigma max
Sigma 2 more sick man so here we have
the max Sigma to the right
and the mean Sigma to the left and
the max and mean shear are equal to the radius
occurring again on I'll vertical line passing through the center more circle
here's the max sheer here's the mean shear
the angle that one must wrote kate
from .1 down to the
Sigma access to get power principal
shears I'm sorry principal normal stresses
he's gotten by our to failure expression
and so if we
rotate our element the given element here
18.43 five degrees
we will transform this stress state
into a principal stress state
principal stress state a higher stresses at a given location
and the plains upon which those occur are the principal planes
so here we see we have a three-dimensional
harm all
pencil no shear stress here we have a
three-dimensional stressed
potential stress with some sheer and so on
various stress states different rotations
when we haven't wanted to mention the body relative the other
short such as a thin plate arm
one of the primary direction the state's are zeros
so if you think I love like up the wall other to attack
a scuba tank I'm sorry in which we have
pressure the pressure acting
on the ends at the tank are causing the tank to be
com tension stretching out and the
pressure internal is causing the
diameter radius the tank to white span so we're we're getting a
a bi directional
stress but there is no tensile stress
on the element in the sea
so we have in that situation to normal stresses an issue stress
here's a picture that particular kind of stressed a
going back to our example
we rotate our element 80.4 35 degrees
get down to the principal
stresses and we see this is the max
tensile stress that is generated this is the max
compressive stress that is generated Sigma max
Sigma man do
we're OK to principal element we're okay our stress element to the principal
plane
thats the stresses that will be seen at that part and this is what we want to
base our computations on
likewise if we were broke eight
our element twice that
to a different angle we will generate
the max sheer
and so we rotate from our stress state 20 6.5 65 degrees
we will have generating the max
sheer that that stress element might see
simply bio orienting different
by considering it through a different said accordance
we're interested in is we're interested in these maximin values cuz that's what
we'll want to use his
proceed with our designs
and salt on the left here we have our original stress state
if we take that element and rotated
18.43 five degrees in this case down
week generate the principal stresses you'll notice that there's no shear on
this principle
stresses so this is a max tansel stressed
element no shear stresses and this is the max value
of such a potential stress that the element this eloquent see
simply by change the the access or into action
this is the minimum if we're OK
that very same element not moving anywhere in part
in a different direction in this case twenty 6.5 65 degrees
up we will generate the Mac sheer
notice that we do have tansel stresses
but the Mac shear is generated at this
access and so going back to our part
that element will not see we shouldn't base our design on these numbers
that we have here but on knees number some not six but we should be
bass are due by non 7 ksl we shouldn't basic design on a3 kasi sheer
we should base it on a five kasi sheer because
simply by changing the reference act up axes
we can get different numbers higher numbers so this analysis is to determine
what are the highest numbers possible by
changing our axis orientation and those highest values is what we'll do our
Design Inc com
we can ask alright at a particular angle wonder what are the stress states
typically were interested in %uh maxine the mean values
but you can pick some other arbitrary angle
we can do that by applying
the formulas and the geometry involved in our analysis
and we can determine then what are
he 10 soul and shear stresses
at that arbitrary element location angle
so here we have some arbitrary
axis system Venu and we can figure out for that element
this is how and this is this the Sigma
at that at that angle
are some special cases you have both
stresses have the same sign
I was talking about a cylindrical pressure vessel a scuba tank
again we have no stress
in those easy but we have positive
potential stresses in are
axe and why
that special case will plot thusly
again they're plotting over here because
their positive and the max year will occur
at the average
in this special case signal on
is equal Sigma axes Eagle signal one segment wise Eagle Sigma 2
in this case just because we don't have
a sigma Z we must consider
the other
more circle emanating from that and soul
when this three-dimensional just because we don't have a sickness she doesn't
mean
we can't and we do we can ignore that stress
circle so here we have arm
applauding for the XZ play
combining them week get this representation
and so here was our
X why plane more circle
this is our axe Z plane and see here
we have a much higher sheer
value on
the XZ plane and so we would want to design
our part based on that max here you know
another special case pure Union axial tension
it plots
thusly
and we we have in the past estimated our
yield strength in the sheer by dividing our yield strength
in are 10 so your strength and to when we can see why we do that
here we have this union actually attention
and so are yield strength
in sheer is equal to one-half our yield strength
intention compression
plus likewise on the left side
I love the axis pure torsion
no polling
no pressing no tension no compression
he going to plot so here's our stress state
the
locust a ball point possible or in tations
would plot like this and in fact
signal one is equal to the stress state
of our element we have
uniaxial tension and torsional shear
mean pulling in one direction plus sheer we would get
a more circle plotting something like this
again here's our stress stayed
because there's zero
normal stress on this face we plot
on 0 with the sheer
and we draw our line the intersection on the sig maxis gives us our the center of
the circle
so if we have a rod subjected to tensile force and torque
we can use more circle potential force
2000 Nunes the torque 10 Newton meters
radius
I love point zero zero five meters we have the expression for
the polar moment of inertia cross-sectional area
and we can compute Sigma X and the sheer
occurring at this stress element uniaxial
one direction which
with sheer
to reply are stressed aid on both faces
a NB we construct our circle
and the angle at which that
max tansel stress signal one more current 38 degrees
Sigma one is equal to 66 make a Pascal's
Sigma 2 is equal to -40
make a pass cows' so we would want to do our Design
based on this principal stress
not the 26 that we see here but the sixty-six that actually you
occurs
here is
the principal stress element if we have a 3d stress state
it will plot
something like this you're going to have hit this
Sigma 3 indicating a value other than 0
what offset that entire circle and so we have the 12 or or discontent
commits conventionally the XY
this 13 is XZ
and this 23 is YZ so the design stresses that were most interested in RR this Mac
Sigma
and the max tap and so we can not disregard this third
direction so if we have a 3d stress element
and worsen the principal normal shear stresses and the max normal and shear
stresses
we would use our expressions to calculate these values
and so 130 3.1 make a Pascal's would be the design signal that we will want to
look at
and 30 113 .1 would be the max
sheer that we would want to conduct
our analysis on this concludes
the lecture on Chapter four in machine design
thank you