What And Why Is Mohr's Circle is an Important Topic?
Mohr's circle explained [In English]
method for planar body so in this video we are going to cover about why is more circle and important topic then derivation of mole circle followed by steps to draw a Mohr circle and also we will cover about Mohr circle properties and Mohr circle for the different cases like uniaxial biaxial stresses or piers here.
Mohr's Circle Method for Planer Body
how the more stress is going to look for all those different cases finally I will we will also discuss about the application of Mohr circle other areas like in this this video we are mainly going to discuss about Mohr circle for plane stress phenomena but the Mohr circle is applicable in different areas those things we will discussed at the end end of this video so let's start so the first thing is that why is Mohr circle such an important topic so first of all the process of changing a stress from one set of coordinate to another set of coordinate is known as stress transformation and that can be done using the algebraic equations like suppose if you want to compute a normal stress at any plane that is inclined at an angle we can directly write down the formula for Sigma N and tau T Sigma n corresponds to the stress that is normal to the plane cutting plane and tau T represents the shear stress that is presented at that cutting plane there are different ways also like to find those stress at any set of coordinates that is rotated by some angle theta using some transformation matrix or by any other transformation equations but for this case like I am just going to write those equations so if you look at these equations maybe these equations look familiar with you if you are already into the strength of material so these equations are simply of normal the ten-year cell stress at any plane that is cutting the component at theta angle with respect to the x-axis like that I will give you the graphical representation for this like suppose if I we have some element like.
this in XY plane and if we are cutting this element by a some plane which is making an angle theta with respect to the x axis this formula can be derived and this formula will give the normal stress that is acting at display and the tangential stress that will be acting at this plate now this formula is sometimes hard to memorize because it also like contains the sign conventions that we need to remember so what happened i that a person named Chris Otto Christian would come up with the graphical representation of this equation so that it will be easy to memorize so he gave the representation for this Sigma and tau T in a graphical format and when he derived those things and represented those things using some graph table h found that these equations can be represented as a circle so that's why this whole thing is known as a
representation of this transformation equation is known as Mohr circle because it gives you a circle and which mainly gives you like at every point on the Mohr circle gives the stress values that will be acting in Sigma and and tau T corresponding to that plane for any value of theta at any point that is located inside the material so this gives the whole picture that is currently happening inside your material
What And Why Is Mohr's Circle is an Important Topic?
Ok so let's start and first discuss how we can derive the Mohr circle so to derive the Mohr circle first thing is that we can go back to the same wedge element so if you look at the wedge element there are different stresses that will be acting on this which ok now you can derive the formula for Sigma n tau T by balancing the forces like if you balance the forces in the X direction so to keep this wage in equilibrium there must be some force px that will be in this direction similarly py in this direction and then we can further dissolve this px force as after converting in into like px if we resolve it in there this theta direction it will give you a pn force and similarly this tau t will also give you some other forces the PT forces and then we can divide it by correspondingly by area to find the Sigma n tau T so I had already made a video on this you can go through the link that I am going to put in the description to know how we can come up with the formula that I am going to write so this is the formula for normal stress that will be acting in this direction and tau T that will be acting in this direction now if we look at these two equations what we can do is that we can write this equation in such a form that on right hand side only sinusoidal and cosine terms will present on left hand side it will be Sigma and the constant term that will be present in these equations can be categorized okay so let's write those so we are going to get these equations now at this point like suppose if I Square this third equation and the fourth equation and then add them up what we are going to get is that Sigma n minus Sigma X plus Sigma Y divided by 2 whole square plus tau T Square and on this side it will give like for first term it will become Sigma X minus Sigma Y by 2 cos square theta and we are going to when we are going to add it with the fourth after squaring it up it will simply become Sigma X minus Sigma Y by 2 whole square because cos square theta plus sine squared 2 theta will become 1 similarly this term will also give you tau square XY and because there is a plus minus sign in plus plus sign so the other equations that if a plus B Square to a B term is going to cancel out from both that while adding 3 & 4 okay so plus 0 now just by looking at these equations do you are you able to recall
like which type of graphic figure this equation is representative so if you look closely this one is exactly equation of circle with Sigma and tau TS horizontal and vertical axis so that will be like if you compare this thing with X minus a square plus y square is equal to R square so this is an equation of circle with Center at a comma 0 and radius R so what we can say is that we can get just by transforming this thing Sigma and tau T these things can be represented as a circle with Center Sigma X plus Sigma Y by 2 and radius equal to Sigma X minus Sigma by 2 square plus tau square XY so here what we have done is that we are mathematically proved that what will be the center of this Mohr circle and the radius of the Mohr circle now let's see that without going into this mathematics how we can plot just by using the graphical chart papers how you can derive the Mis get the Mohr circle
Derivation Of Mohr's Circle
Ok so let's first take a simple element that will be subjected to the planar stress system so the stress system our Sigma X Sigma Y and tau XY so the first thing that we have to do while creating
the Mohr's circle is to decide a horizontal and vertical axis so for the Mohr circle the horizontal axis is your Sigma that corresponds to the normal stress at any data tau will be your shear stress that will be the vertical axis okay so it's going to look like
this now the next thing is that we have to decide the sign conventions that we are going to use so the sign conventions that we are going to use is that if the forces are tensile in nature they will be lying on the positive horizontal axis if they are compressive they will lie on the negative part of the horizontal axis similarly for shear stress like if the stresses on the faces are producing anti clockwise rotation in the element they will be they will lie on the negative shear stress axis negative vertical axis like in this reason and if those forces are producing clockwise rotation about the centroid of any element they will lie in this part of the axis like positive part of this vertical axis at this point like if we want to represent this element and the forces that are acting or the stresses that are acting on this element in this chart of the Mohr circle so let's do that so if we first consider this phase so on this phase what we can do is that we can represent this x and y forces that will be the normal and shear stress you as a point because now the horizontal and the vertical axis are Sigma and tau so first thing is that if we look at the tensile stresses what Sigma X what we can say is that it is tensile in nature it means that it will lie in this region second thing is that if we look at the shear stress and the centroid so what if this force is doing is that it is producing a anti-clockwise rotation it means that it will lie in the negative part of this vertical axis so this X point is located somewhere now for this case what we have done is that we have assumed that Sigma Y is smaller than Sigma X it can be any value but for this case we are assuming same magnitude of Sigma Y is a smaller than magnitude of Sigma X okay we can give it this magnitude now and here if we again look at this Sigma Y point it is tensile in nature it means it's going to lie in this spot of this vertical x horizontal axis and now the tau X Y this tau X Y is creating clockwise moment about this centroid so this point will lie in somewhere in this first quadrant so if we plot this line plot this point what we are going to get is that for this two orthogonal faces we are going to get this two points x and y now third exist a third step for creating the Mohr circle is that we are going to join these two lines find the midpoint of the line that is joining this x and y so if we are going to join this line here you can see that the midpoint automatically lies on the horizontal axis so this will be the center of your Mohr circle and the radius of the Mohr circle will be CX is equal to c1 now here we are not going to going into much detail but what we can do is that we can take the center of the circle and radius at CX and CY and draw circle so now this circle is known as Mohr circle now at this point if suppose if we want to derive the values of the center like at what location of the horizontal axis it is going to lie because for its center is simply equal equal to something a comma 0 so without using the formula we are going to do that just by using the geometrical relationship now if we look at this point this point is Sigma while this point is Sigma X so the distance between these two lines these two points will be Sigma X minus Sigma Y now if we further divide because these two points if we are going to take a mid section of this line what we are going to do is that the midpoint will be the C so this side and this side will be equal for these cases it means that this side will be from here to here the value of this will be Sigma X minus Sigma Y by 2 now if we want to compute the location from this origin then we have to add the value to this Sigma Y Sigma Y plus Sigma X minus Sigma Y by 2 so this will be equal to Sigma X plus Sigma Y by 2 now if you compare with what we had discussed in our last slide it is
Mathematically equal to the same value Sigma X plus Sigma Y by 2 so we had mathematically today but we can also graphically derive those things now if we want to also compute the radius that what is going to radius now here we already know this value that is equal to Sigma X minus Sigma Y by 2 now this vertical axis will be your tau X Y now simply if we are going to use the Pythagoras theorem we are what we are going to get is that R is equal to a square root of Sigma X minus Sigma Y by 2 square plus tau square XY so no mathematics will be required if we are directly playing with the Mohr circle so if we are starting with the strength of material or mechanics of materials it's good very good practice to start with the Mohr circle because in those cases you don't have to remember any formula you can directly using the Mohr circle derive any values and we are going to look at those things in our upcoming slides ok so don't worry about those things
Steps to Draw Mohr's Circle
So let's discuss about the second thing like using the mole circle how we can derive the principle and the maximum shear planes so in this slide we are going to discuss about how we can find the values of the principal planes and the principal stresses so suppose we have already created one more so by using the values of Sigma X Sigma Y in the Tau XY now if you want to find the principal stresses so what we can do is that we can recall the definition of the principal stress so the definition of the principal stress was that any plane on which the shear stress is going to be zero and the normal stresses are going to be maximum so for this circle there are only two points on which the shear stresses are 0 so these two points represent your principal stresses so that will be Sigma 1 and Sigma 2 now the angle that this corresponds with this horizontal axis will be your first principal stress or that will be your second principal stress so here maybe at first look you may get confused like why we are measuring the angle with respect to this line so the reason is that like if we take a simple uniaxial case ok or miss a simple element so the forces that will be acting at this phase will be represent are represented by this point X and that's the reason we are measuring any angle with respect to 2 theta P 1 and these are in the anti-clockwise direction so what it means is that suppose initially the normal is inclined in this of this face is included in this direction so when we are measuring this plane we have to measure in this anti-clockwise direction so it means the plane corresponding to this 2 theta P 1 will be something like this and if we are cutting this thing if the plane will cut at this angle similarly if we take this 90 degree that will be something like this.
Relationship B/W Principle & Max Shear Plane
Ok so next thing is that how to find the value of normal stresses so the normal stress what it says is that first
normal stresses are the planes on which the shear stresses are going to be maximum so there are two points on which your seri stress are maximum so this points directly light above and bottom of the center of this Mohr circle because if we are going to take any other point the value of the Tau XY corresponding to this will always be smaller than the radius of this Mohr circle so tau x max is equal to radius of your Mohr circles okay similarly if we have to find the plane corresponding to this maximum shear plane we can again measure this planes like now in this case what we are doing is that if suppose initially the plane and kept the normal of this plane is oriented in this direction so in this case we are measuring in clockwise direction so it means the plane will be inclined at this angle another thing to notice from the Mohr circle in this case is that the maximum shear planes and the principal stress planes are making and 90 degree it means that in actually miss in Mohr circle we always talk in terms of the two theta not in terms of the theta so in actual this mode circuit though the difference between maximum plane maximum sorry maximum shear plane and the principal strength will be two theta is equal to 90 degree and theta will be 45 degree so always the maximum shear planes and this shear stress principle shear stress plane banks a 45-degree between each other so it's directly clear from the Mohr circle without going into all the derivations and everything so that's why Mohr circle is such a beautiful concept you can duh just by looking at this Mohr circle and using some geometrical and trigonometric relation you can come up with any of the equations that we had derived using some mathematical formulations okay
Value Of Normal Stress At Max Shear Plane
So next thing is that let's revise the properties of the Mohr circle that we have discussed in our previous slide so first thing is that the center of the Mohr circle is Sigma X plus Sigma Y back to the radius of the Mohr circle is a square root of Sigma X minus Sigma Y squared plus tau square XY and the third thing is your sign convention so sign convention when the face creates an anti-clockwise moment it lies in the negative part of this vertical similarly clockwise positive part of the vertical and also like in many books you will you may find different sign conventions but for this video and just to explain once you get familiar it's all about then then it's all about the sign conventions you can use any sign conventions on your side it's not an issue okay so I'm going to write down those formulas for you okay one more thing that I maybe forget forgot to highlight
Properties Of Mohr's Circle
like if we look at this more circle we can also define
Value Of Normal Stress At Max Shear Plane
this values of R and C in terms of the principal stresses like for this case if you look this will be your Sigma 2 and this distance is simply Sigma 1 minu Sigma 2 now if we are going to add it to find the center of this Mohr circle that will be Sigma 1 Sigma 2 plus Sigma 1 minus Sigma 2 divided by 2 and it will get Sigma 1 plus Sigma 2 by 2 similarly if you want to find the radius so the radius is simply equal to Sigma 1 minus Sigma 2 by 2 so that that's what I wanted to highlight that in terms of this principal stresses we can also write the center and radius of this Mohr circle now just to and get familiar with the Mohr circle in a much deeper way let's take two different cases so you can try solving these things on your site with me so that you will have a get familiar with these things so the first thing is that like suppose if I am going to take a uniaxial case so for uniaxial cases how the plane stress is going how the Mohr circle is going to look so form or circle first thing is that if we look at this face it's going to give us some Sigma value and tau is 0 so it means that if this point is going to lie somewhere on this horizontal axis now for this face if we are going to look this Sigma is equal to 0 and tau is equal to 0 it means that it's going to lie on the origin now if you are going to connect this two points and take the midpoint of this line that's getting generated after connecting those two points we are going to get the center of the Mohr circle and the circle is going to look something like this so for this case the circle will pass through the origin and ok the circle will pass through the origin and it completely lies in the first and fourth quadrant ok so just if this thing's injured in in mind because if you are appearing for any
competitive nations like gait ISRO PS use any examination so these things will definitely help you next thing is that if you are going to simply reverse the Daleks and what is going to happen so if you are going to reverse the direction simply this more circular is going to shift to your second and third quadrant next case is that if we take in case of PI XEL case in which your Sigma X is equal to Sigma Y so if we look at this face this represents a point on the horizontal axis because it is a tensile in nature similarly this Sigma Y also represents a same point on the horizontal axis so what's the physical significance like because this is in a special case it's simple it's represent it doesn't represent a Mohr circle complete you can call it a circle but it's simply a point on the horizontal axis so the physical significance of this is that like if
there is any stress system like this is present in your system and if you are going to cut at any plane ami plane the
shear stress will be on that plane will be zero means all that brilliance theta is equal to zero or 94 45 60 every plane is itself a principal plane so this is a special case.
Different Cases
next if we are going to reverse the directions simply this point is going to shift in this negative on this negative horizontal axis okay now let us discuss another case in which your Sigma Y is greater than Sigma X so if Sigma Y is greater than Sigma s and shear stress is zero on this element this circle is going to sift away from the origin in this first and the fourth quadrant okay next case is a pure shear case so purport for your shear case if we look at this face so this is something important so if you look at this face what you get you will get a point corresponding to minus tau and Sigma will be 0 so 0 comma minus tau so this point will be somewhere similarly for the orthogonal face you will get zero comma so that point will be here now if we connect this point and find the midpoint of this line you will get the origin origin of this so this circle is again passing through origin means the center of this Mohr circle will be the origin and it will the radius so the radius of this small circle will be equal to tau only now for this case you can see for a pure shear case the principal stresses values are also equal to tau now let's take another special case like if this for this case only the magnitudes are equal but the directions can be reversed so first thing is that on this face the tensile forces are acting and on this face compressive forces are acting now for this case it again represents the same Mohr circle so what's the unique thing about this case so the unique thing
about this is that for all the case that we had discussed that for all the shear stresses like for this case the point at which shear stress is maximum there are some normal stress present in the system now for this case if you look this is a principal plane and this will be your maximum shear plane so at maximum shear plane your normal stress is zero for this case what it means is that this represents a pure stress configuration so it means that if I am going to rotate this thing by 45 degree it means if I am going to this is a 45 if I am going to cut it like this the stress that was going to be generated at this surface will not carry any normal stress this only CL stresses are going to be presented that surface okay so these are the different cases that I wanted to discuss with you
Different Cases
now what we had discussed mainly about the Mohr circle is that Mohr circle this deals about the transformation from one set to another set and it's simply a graphical representation of those transformation equations so it doesn't mean it is only so this means that it is not only applicable to the stresses or a strain it can be applied to any other applications in which the transformation from one set to another set is happening so let's take few other application areas let's talk about few other application areas we will talk about these things in detail in our applique do so for now let's see first thing is that we can compute the area moment of inertia at any treati means if we are going to rotate like if we have some L section like this okay sorry for bad drawing.
now if I am going to rotate this L section in this configuration now what will be my I xx test so we can also compute these things using the Mohr circle so we will look into these things in our upcoming videos similarly we can compute the mass moment of inertia we can compute the strain transformation and also 3d stress transformations and the strain transpose
Other Application
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