Talk:Mohr's circle

Clarification of shear stresses

In the explanation of "Drawing Mohr's circle", and the accompanying Figure 1, the fist step is drawing points A and A at position $(\sigma _{x},\tau _{xy})$ and $(\sigma _{x},-\tau _{xy})$. However, should point B not have the coordinates $(\sigma _{x},\tau _{yx})$. If $\tau _{yx}=-\tau _{xy}$, then this should be made clear. — Preceding unsigned comment added by Citizenthom (talk • contribs) 10:44, 27 April 2012 (UTC)

Hopefully, the latest update on the sign convention for drawing the Mohr circle addresses your question or suggestion. sanpaz (talk) 05:33, 17 July 2013 (UTC)

Hi, I think, this topic is not resolved yet. I perfectly agree with the lhs (physical space) of figure 5. There, it is clearly shown, which coordinate-systems (x/y and x'/y') are used. Also, the angle theta, in which you "cut", is clearly shown.

However, no "sign convention" should be necessary at all, because: sigma_n and tau_n are (from lhs of figure 5) defined as the **components** of the traction vector in x'- and y'-direction respectively. And no sign convention is needed for components of vectors.

Also, I think, the (correct) formulae in the grey box of fig. 6 are *not* in agreement with the construction algorithm "Drawing Mohr's Circle" in combination with the green circle in fig. 6... Kassbohm (talk) 08:13, 23 March 2014 (UTC)

When you say that no sign convention should be necessary, you are implicitly assuming the convention that a positive traction component should correspond to a positive coordinate direction. While that may be true for materials that behave the same in tension and compression, it becomes problematic if the material cannot support any (or little) tension (soils/foams/rocks etc.). For such materials you will be forced to work with negative numbers all the time. Some people don't like that and use the convention that compression is positive. I haven't checked figure 6 but don't think User:Sanpaz will make an elementary mistake. Bbanerje (talk) 20:33, 24 March 2014 (UTC)

Hi Kasbohm. I checked Figure 6 again (it is always good to be questioned) and I see that the Figure is correct: The angle $2\theta$ should start counterclockwise from point B (not from point A as you suggest). Please see in Figure 6 the first blue square where the components of the stress tensor act on plane B (vertical plan B) which corresponds according to the sign convention to point B in the Mohr circle. According to the Mechanical Engineering convention, the normal stress component $\sigma _{x}$ is positive (outward from the plane) and the shear component $\tau _{xy}$ is negative (counterclockwise around the center of the blue square or point P). The angle $\theta$ in the square is between the normal vector to the plane B (vertical plane B) and the horizontal axis x, then it rotates counterclockwise from plane B (in the blue square), or point B (in the Mohr circle) as can be seen in the blue squares below and the Mohr circle.

A sign convention is needed, as explained in the article, to address the needs of the user (either a mechanical, structural, or geotechnical engineer), as also pointed out above by User:Bbanerje. If the explanation in the article is not clear, please let me know where more emphasis is needed so there is no confusion.

The information provided in the article is referenced. I suggest also you check the references to get more clarity. sanpaz (talk) 22:13, 24 March 2014 (UTC)

Hi Sanpaz, thanks for response. Assuming the following:

* The stress tensor components are (sigma_x, tau_xy, sigma_y) = (1,2,4).

* the x-direction is pointing to the right and the y-direction is pointing to the top, as indicated in fig. 4.

With these two assumptions, the stress state is fully defined. Now assume:

* a cutting-line at theta=0 degrees, as indicated in fig.4 (but for theta=0), so that this cutting line is vertical.

Then: From the formulae, you get (sigma_n_0, tau_n_0) = (1, 2) -- regardless of if you call this point A or B. Do you agree until here? Kassbohm (talk) 06:16, 25 March 2014 (UTC)

Before we go into your example please review the example presented in the article (Yellow circle). Follow the steps from that example using your values. Remember that in the calculations using the formulas you have to take into account the sign convection, which means that $\sigma _{x}=+1$ and $\tau _{xy}=+2$ in the physical space and $\sigma _{x}=+1$ and $\tau _{xy}=-2$ when plotted in the Mohr circle. I know the article is correct. If you are not getting the same results there are some signs missing in your calculations. One thing you can do is present all your calculations in this discussion and we can see where the issue is.

$\sigma _{\mathrm {n} }={\frac {1}{2}}(1+(+2))+{\frac {1}{2}}(1-(+2))\cos 2\theta +\tau _{xy}\sin 2\theta$

$\tau _{\mathrm {n} }=-{\frac {1}{2}}(1-(+2))\sin 2\theta +(+2)\cos 2\theta$

One last thing. Make sure you are using the same sign convention as the article. In the sign convention section there is reference to 3 different sign conventions that different authors use. No single sign convention is correct, they just fulfill different needs for different users. The one chosen for the article is the most common (at least in mechanical and structural engineering) and I think the easiest to deal with for most people. I'm a geotechnical engineer and I prefer the geotechnical convention, for example.

Another thing for clarity: The values you obtain with these two formulas above will plot the circle following the sign convention # 1 from Figure 5. But the article follows convention #3 which requires a different interpretation for the sign of the shear stresses. sanpaz (talk) 16:13, 25 March 2014 (UTC)

---

Hello Sanpaz, thanks for your response. My point was: You have to agree on only 3 things:

1) the definition of the system, the stress tensor components are related to, i.e. the x- and y-axis, to keep things simple.

2) the definition of the cutting angle theta, i.e. where is theta=0 and counting direction. And this is in the free-body diagram and in the Mohr-Circle.

3) the definition of sigma_n and tau_n, i.e. counting direction.

Fig. 4 is properly defining all these things. All the rest is basic mechanics and a tiny little bit of math. The stress tensor has been defined a long time ago. And the definition of how to compute cutting forces / traction vectors from the stress tensor and the cutting direction people have also agreed on. So everything should be clear without any more definitons... If you do *not* agree with this argument: Please tell me, what information you think would be missing to compute sigma_n and tau_n from the information mentioned above.

I think, the article should be simplified and clarified in regards to the questions:

1) Do we need a sign convention? What is missing once the 3 points mentioned above are properly defined?

2) In the example, the stress tensor components are given wrt x' and y'. Why? Above in Fig. 4 the stress state components are given wrt x and y. Why use 2 different points of departure?

3) In Fig. 6 you cut "through" the (red) particle to compute points C, D and E. Why do you not cut "through" the particle for A and B as well?

4) If a sign convention is used, fig. 4 should be replaced, because fig. 4 indicates, that *just one* sign convention is used (i.e. the one in fig. 4). However, later on, this sign convention is not used anymore...

Kassbohm (talk) 22:01, 25 March 2014 (UTC)

Part of the confusion arises from there being two Figure 6s. I agree with User:Kassbohm that the entire convention issue needlessly confuses matters. However, if User:Sanpaz chooses to go with any single convention someone with come and change the article claiming it's wrong because it doesn't use the convention they are familiar with. I think a better way of talking about these issues is to define the two major conventions, the right-hand rule and the left-hand rule, clearly and with separate diagrams. Bbanerje (talk) 04:54, 26 March 2014 (UTC)

I just fixed the issue with having two Figure 6.

I will try to follow your argument step-by-step. But before that, I want to make the point that we are not discussing the veracity of the article nor redefining the topic. Everything that is in the article is based on the literature and not on opinion. What we are discussing is the presentation of the information and how to make it more clear.

You are presenting 3 points. I agree with each individually:

1. The stress tensor is defined in a coordinate system, for example (x, y ,z)

2. There is a variable $\theta$ which represents the orientation of the plane where the stress component is evaluated.

3. There are formulas for the stress component normal ($\sigma _{n}$) and tangential ($\tau _{n}$) to a plane of interest.

Later you say, and I agree, that Figure 4 properly defines all those things. And later using statics you can find the relation between $\sigma _{n}$ and $\tau _{n}$ with respect to the components of the stress tensor ($\sigma _{x}$, $\sigma _{y}$, $\tau _{xy}$).

All that is fine.

Now you have 4 questions:

1) Do we need a sign convention? What is missing once the 3 points mentioned above are properly defined?'.

Yes, you need a sign convention. Force has a magnitude and a direction. You have to specify what direction is positive and what is negative. What is compression and what is tension. Is the force acting on an area or away from the area. At the same time, depending on your field of work, you want to make things easier to work with. A structural engineer or mechanical engineer prefers (according to literature and years of usage) to consider the orientation of stresses as positive according to Figure 4, or to Figure 5. On the other hand, geomechanics people prefer a different sign convention because they deal mostly with compressive forces or compressive stresses. I do not know how else you are proposing to address the topic of the Mohr circle. That is how things are. There is no other way (unless there has been advances in coordinate free mechanics -I think Walter Noll has something like that for continuum mechanics but that is way advance for what we are discussing here). Again, the article is only showing what the literature presents on the subject.

2)In the example, the stress tensor components are given wrt x' and y'. Why? Above in Fig. 4 the stress state components are given wrt x and y. Why use 2 different points of departure?

I don't know what 'wrt' is, but I understand the question. In Figure 4 the element has an orientation that matches the x and y axis. In the example, the element does not align with those axis (horizontal and vertical), so it is better to name the local coordinate system for the element as x' and y'. I do not know what you mean by "two different points of departure". I think what you are trying to say is "two different states of stress", which is the case: Figure 4 is an element oriented differently that the one in the example. The state of stress is at a point, and the orientation of the element you are considering is up to you to decide (it depends on your needs).

3)In Fig. 6 you cut "through" the (red) particle to compute points C, D and E. Why do you not cut "through" the particle for A and B as well?

That is a good point. That might be confusing. I did it that way because we are showing an infinitesimal element around a point P, so it does not mater if it is not shown passing exactly through point P. That is why you can have a Figure like Figure 3 or Figure 4: you do not have to show the stress components coming out of the centre of the square element (point P). But I get your point and I will modify that. I'll think about how to change it, if possible.

4)If a sign convention is used, fig. 4 should be replaced, because fig. 4 indicates, that *just one* sign convention is used (i.e. the one in fig. 4). However, later on, this sign convention is not used anymore..

Figure 4 is an just aid to illustrate how the equation of the Mohr circle is derived. One could have used a different Figure 4; one with a different sign convention. But I did not see the need for repeating the derivation for the circle with another sign conventions. It can be done, though.

Later in the article, we mention the different sign conventions to show that there are other sign conventions. Then the article uses one convention in particular (I chose that one) for illustrating an example. Again, we can show other examples with other sign conventions. That can be done. And to make it clear, you need to chose a sign convention.

I hope this answer your questions. sanpaz (talk) 18:29, 26 March 2014 (UTC)

Hi Sanpaz, Thanks a lot for your answer. I meant wrt="with respect to". Sorry about this.

Yes. We are trying to improve the article together. And we are using information from books.

It's good, that we do agree on the three basic facts regarding the article:

1. The stress tensor is defined in a coordinate system, for example (x, y ,z)

2. There is a variable theta indicating the cutting plane direction, at which the stress/traction vector and its components is evaluated.

3. There are derived formulas for stress components normal () and tangential () to a plane of interest.

I also really appreciate, that you want to improve the tiny issue "cut through red particle".

As to sign-conventions: Maybe you or the cited authors are right. And it does make sense to use different sign conventions for sigma_n and tau_n or even for stress tensor components. However, for *explaining* or *presenting* the topic, I think, it is always best to stick to one convention first, explain what needs to be explained, and *afterwards* or later in the article, describe additional "sign conventions". I agree an what you say about that for forces you do need sign conventions. But remember: sigma_n and tau_n are not *forces*, but these are the *components* of forces stresses. Thus, we only need to define a "counting direction". This is done perfectly by your figure 4 with the red arrows for sigma_n and tau_n. Also this is the most "natural" sign convention, I think, because it is "like the counting direction for x and y" - rotated positively about z by angle theta.

Instead of criticizing and asking questions, I'd like to just mention how I would try to improve the article, so that it becomes really good:

Everything is just perfect at the beginning. The sections "Sign convention for Mohr's circle" and "Engineering mechanics sign conventions" I would cut and paste below. Instead, I would rightaway show how to "Draw the Circle" - using the "natural" sign convention. Afterwards: I'd keep everything as is. Finally, the "Example" I would modify in such a way, that the stress state is given with respect to x and y - as it was given for the construction of the circle. This is maybe the most powerful improvement, because firstly: The reader can use the same "point of departure" again, i.e. stress state given with respect to (x,y). And secondly, this emphasizes the relation between computing (sigma_n, tau_n) on the one hand - and applying the tensor (coordinate) transformation on the other hand. Because, as you know: Computing points (sigma_n, tau_n) on the circle from (sigma_x, tau_xy, sigma_y) and theta is exactly the same as transforming the tensor components from c.s. (x,y) to c.s. (x',y'). Except the fact, that only (sigma_x', tau_xy')=(sigma_n, tau_n) is computed in the circle-approach - but not sigma_y'.

Well, what's your idea on this? Kassbohm (talk) 21:10, 26 March 2014 (UTC)

I'll explain why you must explain first the sign convention before you even start plotting a Mohr circle. By the same logic you say we can draw the circle "using the 'natural' sign convention you in effect are agreeing that there is a sign convention that needs to be followed to begin with. There is no natural sign convention, there is just sign conventions.

First of all, we are talking about two different things: One is the sign convention for the state of stresses in the physical world (Figure 4) and another completely different thing is the convention for the Mohr circle. Each sign convention is arbitrary. What you call normal space sign convention (Figure 4) is the Mechanical engineering convention. You are suggesting following from this convention the plotting of the Mohr Circle. In fact, the convention of Figure 4 is the convention that it is used for the rest of the article (see Figure 5 left bluish side) (I could have used the geomechanics space sign convention, but I did not because I did not see the need, even though I mentioned it). Therefore, the article actually agrees with what you are saying.

Now, the Mohr-sign convention is the other sign convention that has 3 different options, shown in Figure 5 on the Yellow side (Please see that the left-Bluish side of Figure 5 is in fact the 'normal' space convention). The article explains what happens if you plot the Mohr circle according to the derivation from Figure 4, which is Option 1. Then it explains the limitations of using that option (the clockwise rotation of $\theta$ which is in the opposite direction from $\theta$ in Figure 4. Then, to make the rotation of $\theta$ the same in both space and mohr-space people use Option 2 (inverting the $\tau _{n}$ axis). Then to avoid this inversion and keep the rotation $\theta$ in both spaces (mohr and physical) in the same direction, people use Option 3.

I think your issue is that the explanation of the sign conventions in the article is confusing. I thought I wrote it very clearly but it does not seem to be the case.

The last thing is the Example. I used a 'harder' example to illustrate the fact that the state of stress in a point is not necessarily in the direction of the x-y orientation, that it can be in any other direction (e.g. 10 degrees). And that you can have any "point of departure" to use the Mohr circle. A more simple example can be added (with the state of stress similar to Figure 4). Actually, the section on Drawing the Mohr circle, Figure 6, and the following sections on finding the principle stresses, are in fact a 'simple' example but with variables.

In conclusion:

- I don't agree on changing the order of presentation of ideas.

- The explanation on sign conventions can be improved (I have to figure out how).

- A more 'simple' example can be added, without removing the current example.

sanpaz (talk) 02:47, 27 March 2014 (UTC)

Hello Sanpaz. Too bad. --- I have no more arguments, which I haven't mentioned already. All the best. Kassbohm (talk) 06:10, 27 March 2014 (UTC)

Kassbohm, I hope the arguments I presented make sense to you. The discussion brought up some issues with the article which I hope were captured. The sign convention adopted in the article is the most common, and that is why I used it. sanpaz (talk) 16:10, 27 March 2014 (UTC)

Kassbohm, You were right. I agree with your assertion that Also, I think, the (correct) formulae in the grey box of fig. 6 are *not* in agreement with the construction algorithm "Drawing Mohr's Circle" in combination with the green circle in fig. 6.... The formula for the shear stress $\tau _{n}$ only works for sign conventions #1 and #2, but not for sign convention #3 (Figure 5). If one wants to plot the circle using the parametric equations for $\sigma _{n}$ and $\tau _{n}$ and using the sign convention #3, then the formula for $\tau _{n}$ needs to change signs:

$\tau _{\mathrm {n} }=+{\frac {1}{2}}(\sigma _{x}-\sigma _{y})\sin 2\theta -\tau _{xy}\cos 2\theta$

Figure 6 needs to be fixed to reflect that change, which makes me think about re-evaluating the presentation of ideas and consider closely what you suggested previously.sanpaz (talk) 15:58, 31 March 2014 (UTC)

Info missing

Sorry, but it seems that some information is missing. I find it difficult to understand what the documented procedure is supposed to be achieving. There is one mention of point "P" in the text, but what is that about? I cannot see it in any pictures.

Point "P" refers to the center of stressed square. Corresponding figures 1 and 2 were replaced by Ecass-NJITWILL 01:34, 24 November 2010. I cannot find the reason for that replasement. Finally, I suggest to undo the figure replacement. --Temaotheos (talk) 11:48, 31 January 2011 (UTC)

Have tried to add in the text and figures that were deleted by Ecass-NJITWILL. The article still needs work for consistency to be achieved. Bbanerje (talk) 04:03, 1 February 2011 (UTC)

Figure is wrong i think. Step one shows that the midpoint of mohr's circle should (sigmax+sigmay)/2, when it should be (sigmax-sigmay)/2. —Preceding unsigned comment added by 72.50.164.10 (talk) 02:52, 20 October 2008 (UTC)

Figure 1 is not wrong Sanpaz (talk) 03:32, 20 October 2008 (UTC)

Figure numbers don't match the text in the latest version. Bbanerje (talk) 00:10, 29 April 2010 (UTC)

Figure 2 (I presume) is wrong. There are three τyx and only one τxy on the finite element. Text also crosses arrows, it's not a terribly clear diagram. 86.9.78.134 (talk) 10:11, 8 December 2010 (UTC)

Wikiversity link

Sorry for saying that the last edit on Jun 24, 2010 by user Taltastic was vandalism. I realized it was not a spam link when it was to late. That link should be placed at the end of the article. Please do so. Sorry again. sanpaz (talk) 15:05, 24 June 2010 (UTC)

Mohr Diagram

I hope that's the same thing? since I've created a redirect (was requested article). A presentation I found on Mohr diagrams had basically the same content as this article. DS Belgium (talk) 14:08, 27 October 2011 (UTC)

Mohr's Circle when the shear stresses are not equal.

I think the descriptions on how to draw a Mohr circle assumes that the shear stresses are equal in magnitude on opposite faces. Sometimes the states of stress are not given on perfectly square soil elements, so the shear stresses are not equal. (Basically the generalization of how to draw a Mohr circle isn't general enough to meet all cases.)

I'm just learning this topic myself, so I don't have enough expertise at the moment to fix this page. Hopefully someone else can fix it up.

70.31.105.204 (talk) 17:56, 31 March 2012 (UTC)

Why is Mohr's Circle "true"?

As a naive reader, I think this page is marvelous. However, I have trouble seeing why the relationships revealed by Mohr's Circle apply to the stresses in a physical situation. (I come to Mohr's Circle as a naive reader of "Soil Mechanics", another marvelous page.) What I do not understand is why you can use Mohr's Circle to calculate the stresses that impinge on a particular segment of soil. I remember enough trigonometry to follow the algebraic proof of the Mohr's Circle formulas given at http://www.roymech.co.uk/Useful_Tables/Mechanics/Mohrs_circle.html . So, it might be nice to include some version of that proof in the Wikipedia article. However, it would still be nice to understand how the geometry of Mohr's Circle is related to the geometry of the stresses impinging on my soil segment. Otherwise, I am left with the feeling that Mohr's Circle is "true" because of coincidence or magic. Nkennington (talk) 22:57, 23 May 2013 (UTC)

Good point. I'll try to address that. sanpaz (talk) 15:46, 24 May 2013 (UTC)

To be able to understand what the Mohr circle is and its relation to a physical situation, it is necessary to understand the need to do a stress tensor transformation. This was originally done in the stress (mechanics) article, which this article use to belong to. However, when the article was moved, that connection was lost. I am trying to incorporate this explanation (connection) back to this article. I will add more either today or this weekend. sanpaz (talk) 22:05, 24 May 2013 (UTC)

Any feedback on the new additions to better explain Mohr's circle is appreciated. sanpaz (talk) 05:22, 26 May 2013 (UTC)

Clarification of the term "Reactive Forces"

The first line under the heading 'Motivation for the Mohr Circle' talks of "Internal reactive forces" being produced "as a reaction" to applied forces. This statement is perfectly right, and in fact, this is the explanation for the development of stress in a material when subjected to forces. However, the terms "reactive forces" and "reaction" are likely to be confused with the "reaction force" defined in Newton's Third Law of Motion. It is important to realize that Newton's "Reaction Force" is a force which is applied by a body which is subjected to a force, on the body which applies the force, which has the same magnitude as the force applied on it, but acts in a direction opposite to the applied force. The "reactive forces" mentioned here, however, are developed within the material which is subjected to a force.

Could someone please try to clarify this point in the article? I find it difficult to put this briefly. And, yes, if there's something wrong in what I've said, do correct me. Yetanotherwriter (talk) 06:22, 11 April 2014 (UTC)

Actually, it is correct to call them reaction forces. These forces actually arise from Newton's Third law. In the same paragraph there is mention about this. In continuum mechanics Euler's law of motion is equivalent to Newton's third law.

It is important to point out (as you say) that they are "internal" reactive forces as oppose to external reactive forces. Is there a need to specify this in the article in more detail? sanpaz (talk) 18:08, 9 April 2014 (UTC)

Thanks for your reply. I agree that it can be stated that these forces arise from Newton's 3rd Law. If we cut down our structural member into a number of sections, the force exerted on one section by another may be stated to arise because of Newton's 3rd Law. However, it is incorrect to state that these internal forces themselves arise as a "reaction" (i.e, "reaction" as defined by Newton's 3rd Law) to the external load applied on the body, since both the "action" and "reaction" would be acting on the same body, i.e, the structural member. Kindly have a look at the topic "Equal and Opposite" under Reaction (physics). Further, there is no mention of Euler's Third Law in Euler's laws of motion. Kindly correct me if I am still mistaken. Yetanotherwriter (talk) 05:30, 11 April 2014 (UTC)

Please read Cauchy stress tensor#Cauchy’s fundamental lemma, and see the reference mentioned in that section.

I read the article Reaction (physics). That article is correct but the explanation they give is not very straight forward. What they are trying to say is that just because two forces are equal and opposite does not mean that they are a result of Newton's third law. What one has to realize when looking at the interaction of objects is the system of forces involved in the objects under consideration (the resultant of forces in one object and the resultant of forces in the second object), and how those resultants interact with each other. In the case of a continuum body that is divided into two imaginary objects by an imaginary plane, the resultant of forces on the imaginary surface of the first imaginary object is equal and opposite to the resultant of forces on the imaginary surface of the second object(that is the Cauchy fundamental lemma).

In conclusion, the internal forces in the continuum body are a reaction to the action of the total external forces on the object. sanpaz (talk) 23:39, 13 April 2014 (UTC)

Value of $\sigma _{x}$, $\sigma _{y}$, and $\tau _{xy}$

What is the possible value of $\sigma _{x}$, $\sigma _{y}$, and $\tau _{xy}$ in the article? Are they supposed to be always positive, such that the direction is given by the diagram? --IngenieroLoco (talk) 20:55, 12 April 2016 (UTC)