ta3111 mohr circle

7/29/2019 TA3111 Mohr Circle

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Mohrs Circle for Plane Stress:

Real Space and Mohr Space

Compiled by

Ridho K. Wattimena

2004

The Mohr Circle is a tremendously usefulway to plot stress and strain.Unfortunately, to derive it, we have to plowthrough a bit of trigonometry first.


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Assumptions Compression is positive, tension is negative.

The system is not rotating or moving (it might be,but we imagine ourselves moving along with it.We're only interested in the forces that are

acting to deform the rock on a local scale).

The stresses can be defined in terms of twoperpendicular longitudinal stresses. We call this

a principal axis system. For any stress system,we can always find a principal axis coordinatesystem.


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Choose a suitable scale and plot S1 and S2 on the horizontal axis.

Draw a circle centered on the midpoint between S1 and S2 and passing

through both.

If the pole to the plane makes an angle A with the S1 direction,

measure off 2A on the Mohr Circle.

Point X represents the stress on the plane, with normal stress measured

horizontally and shear measured vertically (green lines).

Three Fundamental Principlesin Mohr Circles

Directions of planes are alwaysrepresented by their poles

Angles on the Mohr Circle are double thecorresponding angles in real life.

Always measure angles in the same sensein both real life and on Mohr Circles


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What Does the Bottom Half of

the Mohr Circle Represent If we define positive normal stress as

compression, then negative normal stressis obviously tension.

Similarly, above the horizontal axisrepresents positive shear, and belowrepresents negative.

Only what exactly do posit ive andnegative shear stress mean?

In the example we've been using, S1 is the larger stressand, given the tilt of the plane, slippage would tend to be

left lateral if it were a fault. On this Mohr Circle, then, positive shear means left-

lateral slip. If we were to flip the diagram vertically, we'dhave right-lateral slip and negative shear.


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But suppose we're looking at a dip-slip fault. Positiveshear would correspond to thrust faulting, negativeshear to normal faulting.

Whatever the orientation of the fault, when slipoccurs on the plane shown, it will result incounterclockwise rotation. For this diagram, then,positive is counterclockwise and negative isclockwise.

However, we cannot give a universal definition

because different workers define the signs ofstresses differently.

All we can say is that positive and negative shearcorrespond to the different senses of shear motion(right lateral-left lateral, normal-thrust, clockwise-counterclockwise).


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The only sure way to ascertain which is which is towork from the real-world stress situation and seewhere each type of shear plots on the Mohr Circle.

Often we don't worry about the sense of shear andsimply portray only the top half of the Mohr Circle tosave space.

Recall that Shear Stress = ((S1 - S2)/2) sin 2A. Nowwe have defined terms so that S1 is always greaterthan S2, and sin 2A is always positive for A < 90degrees.

So in the real-world diagrams above, shear is always

positive. Now if S2 = 0, then only S1 can exert ashear force on the plane.

Therefore the shear sense is always determinedby the larger stress. So ignore the smaller stressand observe which way the larger stress is pushingrocks on either side of the plane.

A Useful Tip


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Example 1

Given the stressesshown at left, whatare the normal andshear stresses onthe plane shown?

Example 1

Recall that everythingin Mohr space is donewith reference to thepole to the plane, so thefirst thing to do is

construct the pole andmeasure the anglesbetween the pole andthe stresses as shown.


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Example 1Next (below) we construct a graph and plot the Mohr Circle.

Its center is at (Smax+ Smin)/2 and its radius is (Smax -Smin)/2. It passes through both Smax and Smin.

Example 1

Next (below) we plot the stress. Recall that allangles on the Mohr Circle are double the values inthe real world, and all angles are measured in thesame sense. When we plot the angles we find weare in the lower half of the diagram. We can seethat in the real world the stresses would result inright-lateral slip, so on the Mohr Circle, negativeshear means right-lateral shear. Remember thatyou have to determine the relationship betweenshear sense and sign f rom the real-worldsituation. Negative may not always be right-lateral!


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Example 1

Example 1

Finally, we simplymeasure off thestresses as shown.


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Example 2

Here we have thesame geometry asbefore, but now oneof the stresses istensional.

Example 2


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Example 2 In the case above, where the two stresses were pushing in

opposite directions along the plane, we had to think a bit

to see that the shear on the plane was right lateral.

Here, both stresses are pushing in the same direction

along the plane, and the sense of shear is obvious.

Also, since both principal stresses are acting in the same

direction along the plane, the magnitude of the shear

stress will be greater.

Sign Conventions

Since most stresses in geology are compressionstresses, many workers prefer to definecompression as positive and tension asnegative.

In situations where mathematical consistency isimportant, especially in more theoretical studies,

it's best to define positive vectors as pointing inthe positive axis direction. Thus positive stresspoints in the positive x-direction, away from theorigin. In this system, tension is positive andcompression is negative.


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Sign ConventionsTo quote the Hitchhiker's Guide to the

Galaxy, don't panic.

Mohr Circles work equally well in eitherconvention.

J ust be sure you know which convention isin use, follow it consistently, and always

measure angles on the Mohr Circle in thesame sense as in real life.

ta3111 mohr circle

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