About Mohr's circle

Page 1

Mohr's circle

Created summer 1996 by James M. StroutV1.0 JMS released August 1996V2.0 JMS released May 1998 - Excel 5.0 and Excel 97 formats

Department of Geotechnical Engineering The Norwegian University of Science and Technology N-7034 Trondheim, Norway http:// www.geotek.unit.no

All rights reserved. This program is intended for educational use only. It is provided for use as is, without any warranty (either expressed or implied). No guarantee is made regarding calculated values.

About Mohr's circle

Page 2

released May 1998 - Excel 5.0 and Excel 97 formats

Department of Geotechnical Engineering The Norwegian University of Science and Technology N-7034 Trondheim, Norway http:// www.geotek.unit.no

All rights reserved. This program is intended for educational use only. It is provided for use as is, without any warranty (either expressed or implied). No guarantee is made regarding calculated values.

Input data

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Initial stress stateVertical stress 0

Horizontal stress 8Shear stress -3

Rotations200

20 20

Show original cube? yes

Current rotation qIncrement Dq

0 2 4 6 8 10 12 14

-8

-6

-4

-2

0

2

4

6

8 Stresses on vertical plane

Stresses on horizontal plane

Pole

Pole (0,-3)Sigma 1 = 9 kPa

Sigma 3 = -1 kPa

s

t

0.0 kPa8.0 kPa

-3.0 kPa3.0 kPa

s t

Input data

Page 4

Adjusted scale:

See Help! for instructions about adjusting the scale

y axis maximum value (t)x axis maximum value (s)

0 2 4 6 8 10 12 14

-8

-6

-4

-2

0

2

4

6

8 Stresses on vertical plane

Stresses on horizontal plane

Pole

Pole (0,-3)Sigma 1 = 9 kPa

Sigma 3 = -1 kPa

s

t

0.0 kPa8.0 kPa

-3.0 kPa3.0 kPa

s t

Help!

Page 5

The mohr circle and stress element plots are wrong...

Scaling the plot

Stress definitions

Pole definitions

The solution for the mohr circle has a bug in it which I can't find. The result is that under some conditions (certain stress combinations and angles of rotation) the calculated stresses and plots are wrong. It is obvious when this happens, because the calculated stresses and the plots seem to 'hang up', or in other words the mohr circle plot does not change although you vary the rotation angle.

You may see for yourself: Enter 200 kpa for the vertical stress, and 20 kpa for the horizontal and shear stresses. Set the rotation angle to 89 degrees, and the increment to 1 degree. Click on the Rotate + Ds button a number of times, and watch what happens to the plots as the rotation angle goes from 89 degrees through 103 degrees.

Just pay attention to what is going on with the stresses and the graphical solution, and this bug will not be a problem

The Mohr circle plot has an automatic scaling function to hopefully produce a true circle. If for some reason the circle is not round, you may try to adjust the x and y scales. Note that the scaling of the plot are a function of Excel, and the numbers you enter may not necessarily change the scale, these numbers will only influence the automatic scaling done by Excel. It may be necessary to play around with both of the scales until something reasonable with the plot happens. Good Luck.

Naming convention for stresses follows geotechnical practice: Compressive normal stresses are positive, and shear stresses causing counterclockwise rotation of the body are positive. See sketch.

A line drawn from the pole and intersecting the circle defines the stresses on a plane perpendicular to that line. See sketch.

Pole

Help!

Page 6

The solution for the mohr circle has a bug in it which I can't find. The result is that under some conditions (certain stress combinations and angles of rotation) the calculated stresses and plots are wrong. It is obvious when this happens, because the calculated stresses and the plots seem to 'hang up', or in other words the mohr circle plot does not change although you vary the rotation angle.

You may see for yourself: Enter 200 kpa for the vertical stress, and 20 kpa for the horizontal and shear stresses. Set the rotation angle to 89 degrees, and the increment to 1 degree. Click on the Rotate + Ds button a number of times, and watch what happens to the plots as the rotation angle goes from 89 degrees through 103 degrees.

Just pay attention to what is going on with the stresses and the graphical solution, and this bug will not be a problem

The Mohr circle plot has an automatic scaling function to hopefully produce a true circle. If for some reason the circle is not round, you may try to adjust the x and y scales. Note that the scaling of the plot are a function of Excel, and the numbers you enter may not necessarily change the scale, these numbers will only influence the automatic scaling done by Excel. It may be necessary to play around with both of the scales until something reasonable with the plot happens. Good Luck.

Positive normal and shear stresses

Plot data

Page 7

rotation initial alpha sin alpha cos alphaAngles 20 0 1.13446401 0.90630779 0.42261826names alpha sin cos

Plot of the square elements.The original square Rotated by alpha

0.7071 0.7071 0.4226 0.9063-0.7071 0.7071 -0.9063 0.4226-0.7071 -0.7071 -0.4226 -0.90630.7071 -0.7071 0.9063 -0.42260.7071 0.7071 0.4226 0.9063

Show original square? 1

Sigma Taux 12 0y 0 6-y 0 -6

Stress Vectors

Position 0.85Scale 1

Scale sigma tauBlue face 0.000 -0.150

Green face 0.889 0.150

Angle q

The input angle is converted to radians and the sin and cosine are evaluated. The initial angle is intended for cases where the input data is given on aribitrary planes. This is not used yet. The names are defined for these data, and these data may be refered to in this way in later formulas. Note that these angles are adjusted by 45 degrees. Why? Well, the user is presented with an upright cube, but the program works with vectors describing the corners of a cube. For an upright cube, these vectors need to point at a 45 degree angle from the faces. See the left drawing below.

These are the square element plots. The original is plotted as a dotted line on Input data, the rotate is plotted by a solid line. The squares are defined by the corners of the square, which is basically a vector field of 4 vectors, each of unit length, which are put through a vector rotation using the angle given by the user.

Plot scaling for the mohr circle plot. The sigma is defaulted as 10% larger than sigma 1, and the tau defaults to half of the sigma scale. This may be controlled by the user by setting the plot scale on the input data sheet. Minimum sigma scale is 100. The shear scale needs to be extended to the -y side to allow the full circle to be plotted.

This is the reference position of the stress vectors above the face of the cube. Since the cube is defined by unit vectors at 45 degrees from vertical and horizontal, the face of the unrotated cube is at 0.707 in a unit plot. A position of 0.85 is slightly above the face. The scale is just a multiplier for adjusting the relative size of the vectors.

The stresses on the faces are scaled: The sigmas are scaled to sigma 1, and the taus are scaled to (sigma 1 - sigma 3)/2. The name sigmaplane refers to the normal stress on the blue face, and sigmaperp that on the green face. The tau scale is corrected by a factor of 2. See the note at the end of this page.

The rotation given by the user. This is not corrected by 45 degrees, since the stress vectors are shown above the face, not at the corners. These angles are given the names sinb and cosb for the vector definitions below.

Many of the formulas below use names, that is variable references instead of normal cell references (like alpha instead of E10). These variable definitions may be viewed: Choose Insert, then Names and finally Define.

Plot data

Page 8

radians sine cosine0.349 0.342 0.940

sigmasx y

0.799 0.2910.799 0.291

-0.799 -0.291-0.799 -0.291

-0.291 0.799-0.595 1.634

0.291 -0.7990.595 -1.634

Tausx y

0.850 0.1500.747 0.432

-0.850 -0.150-0.747 -0.432

-0.150 0.850-0.432 0.747

0.150 -0.8500.432 -0.747

The rotation given by the user. This is not corrected by 45 degrees, since the stress vectors are shown above the face, not at the corners. These angles are given the names sinb and cosb for the vector definitions below.

The positioning of these vectors is really tricky. Consider the first set of four numbers shown in color. The first number, which is blue, is the starting point of the vector. This is given by the position defined above, which itself is rotated through the user defined rotation of the cube. The equation: l*cosb where l is a name refering to the position given in cell C39. The second number, colored green, is the position plus the scaled length of the vector (defined in cell c43, named hs). This number is then rotated through the user given angle, as (l+hs)*cosb. These two numbers (blue and green) together with the other half of the vector (light blue and light green) define a series, which make one of the stress arrows on the stress cube plot on the input data page. The yellow and orange numbers are the negative of the blue and green, and define the related vector on the opposite side of the cube. The same sort of approach is followed for all of the normal stress vectors.

The same sort of idea is used for the tau vectors. However, the endpoints need to be adjusted so that the center of the vector acts through the location point. This allows the shear stress vector to be centered on the face of the element. The normal stress vectors are aligned so that the end of the vector is at the location point.

The endpoints are adjusted as a function of the length of the shear stress vector and the user given angle. In order to avoid doubling the length of the shear stress vector through this process, the actual scale of this vector is reduced by half. See cell D50

Plot data

Page 9

The input angle is converted to radians and the sin and cosine are evaluated. The initial angle is intended for cases where the input data is given on aribitrary planes. This is not used yet. The names are defined for these data, and these data may be refered to in this way in later formulas. Note that these angles are adjusted by 45 degrees. Why? Well, the user is presented with an upright cube, but the program works with vectors describing the corners of a cube. For an upright cube, these vectors need to point at a 45 degree angle from the faces. See the left drawing below.

These are the square element plots. The original is plotted as a dotted line on Input data, the rotate is plotted by a solid line. The squares are defined by the corners of the square, which is basically a vector field of 4 vectors, each of unit length, which are put through a vector rotation using the angle given by the user.

Rotated by q

Plot scaling for the mohr circle plot. The sigma is defaulted as 10% larger than sigma 1, and the tau defaults to half of the sigma scale. This may be controlled by the user by setting the plot scale on the input data sheet. Minimum sigma scale is 100. The shear scale needs to be extended to the -y side to allow the full circle to be plotted.

This is the reference position of the stress vectors above the face of the cube. Since the cube is defined by unit vectors at 45 degrees from vertical and horizontal, the face of the unrotated cube is at 0.707 in a unit plot. A position of 0.85 is slightly above the face. The scale is just a multiplier for adjusting the relative size of the vectors.

The stresses on the faces are scaled: The sigmas are scaled to sigma 1, and the taus are scaled to (sigma 1 - sigma 3)/2. The name sigmaplane refers to the normal stress on the blue face, and sigmaperp that on the green face. The tau scale is corrected by a factor of 2. See the note at the end of this page.

The rotation given by the user. This is not corrected by 45 degrees, since the stress vectors are shown above the face, not at the corners. These angles are given the names sinb and cosb for the vector definitions below.

Many of the formulas below use names, that is variable references instead of normal cell references (like alpha instead of E10). These variable definitions may be viewed: Choose Insert, then Names and finally Define.

Plot data

Page 10

The rotation given by the user. This is not corrected by 45 degrees, since the stress vectors are shown above the face, not at the corners. These angles are given the names sinb and cosb for the vector definitions below.

The positioning of these vectors is really tricky. Consider the first set of four numbers shown in color. The first number, which is blue, is the starting point of the vector. This is given by the position defined above, which itself is rotated through the user defined rotation of the cube. The equation: l*cosb where l is a name refering to the position given in cell C39. The second number, colored green, is the position plus the scaled length of the vector (defined in cell c43, named hs). This number is then rotated through the user given angle, as (l+hs)*cosb. These two numbers (blue and green) together with the other half of the vector (light blue and light green) define a series, which make one of the stress arrows on the stress cube plot on the input data page. The yellow and orange numbers are the negative of the blue and green, and define the related vector on the opposite side of the cube. The same sort of approach is followed for all of the normal stress vectors.

The same sort of idea is used for the tau vectors. However, the endpoints need to be adjusted so that the center of the vector acts through the location point. This allows the shear stress vector to be centered on the face of the element. The normal stress vectors are aligned so that the end of the vector is at the location point.

The endpoints are adjusted as a function of the length of the shear stress vector and the user given angle. In order to avoid doubling the length of the shear stress vector through this process, the actual scale of this vector is reduced by half. See cell D50

Calculations

Page 11

Input data

Initial stress state Rotationssigma green 0 User value 20 20 20Sigma blue 8tau -3 theta 20 0Initial angle (not used yet) 0 Show origina yes yes

Solving for the principle stresses and the pole

Values for the stress elementsHelper values Input (fixed)

tan(q) 0 0.364a 1 1.132b -8 -10.184c 0 0.000m 0 0.364

intercept -3 -3.000

Plot valuespole 0 -3 0.000 -3.000

Sigma blue 8 -3 0.000 -3.000sigma green 0 3 8.000 3.000

pole 0 -3 0.000 -3.000

phi 1 -71.5651116phi2 18.4349644

Sigma 1 9Sigma 3 -1

r 5xo 4

step 0.1

Plotting the circle itselfSigma Tau

Variable with q

Info necessary for plotting the circle. r is the circle radius, and step is the size of the sigma increase for each data point in the plot data below.

The plot is just a circle function using the calculated center (xo) and the radius. xo and r are obtained from sigma 1 and sigma 3

These calculations are not well described at all. Sorry. This is actually more complicated than it needs to be for the current workbook. The original model allowed any original orientation to be used for defining the initial stress state (an aribtrary set of mutually perpendicular planes, rather than require vertical and horizontal planes). This forced a solution which was much more general and much more complicated.

The basic idea is to find the intersection between a circle defined by the given stresses, and a line drawn from the pole with the slope defined by the angle given by the user. Good luck on sorting this out.

Calculations

Page 12

0.000 Plot labels-1.000 0.000 Sigma 1 = 9 kPa-0.900 0.995 Sigma 3 = -1 kPa-0.800 1.400 Pole (0,-3)-0.700 1.706-0.600 1.960-0.500 2.179-0.400 2.375-0.300 2.551-0.200 2.713-0.100 2.8620.000 3.0000.100 3.1290.200 3.2500.300 3.3630.400 3.4700.500 3.5710.600 3.6660.700 3.7560.800 3.8420.900 3.9231.000 4.0001.100 4.0731.200 4.1421.300 4.2081.400 4.2711.500 4.3301.600 4.3861.700 4.4401.800 4.4901.900 4.5382.000 4.5832.100 4.6252.200 4.6652.300 4.7022.400 4.7372.500 4.7702.600 4.8002.700 4.8282.800 4.8542.900 4.8773.000 4.8993.100 4.9183.200 4.9363.300 4.9513.400 4.9643.500 4.9753.600 4.9843.700 4.9913.800 4.9963.900 4.9994.000 5.0004.100 4.9994.200 4.996

Calculations

Page 13

4.300 4.9914.400 4.9844.500 4.9754.600 4.9644.700 4.9514.800 4.9364.900 4.9185.000 4.8995.100 4.8775.200 4.8545.300 4.8285.400 4.8005.500 4.7705.600 4.7375.700 4.7025.800 4.6655.900 4.6256.000 4.5836.100 4.5386.200 4.4906.300 4.4406.400 4.3866.500 4.3306.600 4.2716.700 4.2086.800 4.1426.900 4.0737.000 4.0007.100 3.9237.200 3.8427.300 3.7567.400 3.6667.500 3.5717.600 3.4707.700 3.3637.800 3.2507.900 3.1298.000 3.0008.100 2.8628.200 2.7138.300 2.5518.400 2.3758.500 2.1798.600 1.9608.700 1.7068.800 1.4008.900 0.9959.000 0.000

-1 0 This is the bottom half of the circle-0.9 -0.99498744-0.8 -1.4-0.7 -1.70587221-0.6 -1.95959179

Calculations

Page 14

-0.5 -2.17944947-0.4 -2.37486842-0.3 -2.55147016-0.2 -2.71293199-0.1 -2.8618176

0 -30.1 -3.128897570.2 -3.249615360.3 -3.363034340.4 -3.469870310.5 -3.570714210.6 -3.666060560.7 -3.756327990.8 -3.841874540.9 -3.92300905

1 -41.1 -4.073082371.2 -4.142463041.3 -4.208325081.4 -4.27083131.5 -4.330127021.6 -4.386342441.7 -4.439594581.8 -4.489988861.9 -4.53762052

2 -4.582575692.1 -4.624932432.2 -4.664761522.3 -4.702127182.4 -4.737087712.5 -4.769696012.6 -4.82.7 -4.828043082.8 -4.853864442.9 -4.87749936

3 -4.898979493.1 -4.918333053.2 -4.935585073.3 -4.950757523.4 -4.963869463.5 -4.974937193.6 -4.983974323.7 -4.990991893.8 -4.99599843.9 -4.9989999

4 -54.1 -4.99899994.2 -4.99599844.3 -4.990991894.4 -4.983974324.5 -4.974937194.6 -4.963869464.7 -4.950757524.8 -4.93558507

Calculations

Page 15

4.9 -4.918333055 -4.89897949

5.1 -4.877499365.2 -4.853864445.3 -4.828043085.4 -4.85.5 -4.769696015.6 -4.737087715.7 -4.702127185.8 -4.664761525.9 -4.62493243

6 -4.582575696.09999999999999 -4.537620526.19999999999999 -4.489988866.29999999999999 -4.439594586.39999999999999 -4.386342446.49999999999999 -4.330127026.59999999999999 -4.27083136.69999999999999 -4.208325086.79999999999999 -4.142463046.89999999999999 -4.073082376.99999999999999 -47.09999999999999 -3.923009057.19999999999999 -3.841874547.29999999999999 -3.756327997.39999999999999 -3.666060567.49999999999999 -3.570714217.59999999999999 -3.469870317.69999999999999 -3.363034347.79999999999999 -3.249615367.89999999999999 -3.128897577.99999999999999 -38.09999999999999 -2.86181768.19999999999999 -2.712931998.29999999999999 -2.551470168.39999999999999 -2.374868428.49999999999999 -2.179449478.59999999999999 -1.959591798.69999999999999 -1.705872218.79999999999999 -1.48.89999999999998 -0.994987448.99999999999998 0

Calculations

Page 16

181 2

Labels for the plot:0.0 kPa -3.0 kPa8.0 kPa 3.0 kPa

Info necessary for plotting the circle. r is the circle radius, and step is the size of the sigma increase for each data point in the plot data below.

The plot is just a circle function using the calculated center (xo) and the radius. xo and r are obtained from sigma 1 and sigma 3

These calculations are not well described at all. Sorry. This is actually more complicated than it needs to be for the current workbook. The original model allowed any original orientation to be used for defining the initial stress state (an aribtrary set of mutually perpendicular planes, rather than require vertical and horizontal planes). This forced a solution which was much more general and much more complicated.

The basic idea is to find the intersection between a circle defined by the given stresses, and a line drawn from the pole with the slope defined by the angle given by the user. Good luck on sorting this out.