How to Calculate and Solve for Mohr – Coulomb Criterion | Rock Mechanics

September 26, 2019March 9, 2024 Loveth Idoko

Last Updated on March 9, 2024

The image above represents mohr – coulomb criterion. To calculate mohr – coulomb criterion, three essential parameters are needed and these parameters are Cohesion (|τ_o), Co-efficient of Friction (μ) and Normal Stress (σ_n).

The formula for calculating the mohr – coulomb criterion:

|τ| = τ_o + μσ_n

Where;

|τ = Mohr – Coulomb Criterion
|τ_o = Cohesion
μ = Co-efficient of Friction
σ_n = Normal Stress

Let’s solve an example;
Find the mohr – coulomb criterion when the cohesion is 7, the co-efficient of friction is 30 and the normal stress is 28.

This implies that;

|τ_o = Cohesion = 7
μ = Co-efficient of Friction = 30
σ_n = Normal Stress = 28

|τ| = τ_o + μσ_n
|τ| = 7 + 30(28)
So, |τ| = 7 + 840
|τ| = 847

Therefore, the mohr – coulomb criterion is 847.

Calculating the Cohesion when the Mohr – Coulomb Criterion, the Co-efficient of Friction and the Normal Stress are Given

|τ_o = |τ| – μσ_n

Where;

|τ_o = Cohesion
|τ = Mohr – Coulomb Criterion
μ = Co-efficient of Friction
σ_n = Normal Stress

Let’s solve an example;
Find the cohesion when the mohr – coulomb criterion is 44, the co-efficient of friction is 10 and the normal stress is 4.

This implies that;

|τ = Mohr – Coulomb Criterion = 44
μ = Co-efficient of Friction = 10
σ_n = Normal Stress = 4

|τ_o = |τ| – μσ_n
So, |τ_o = 44 – 10(4)
|τ_o = 44 – 40
|τ_o = 4

Therefore, the cohesion is 4.

Calculating the Co-efficient of Friction when the Mohr – Coulomb Criterion, the Cohesion and the Normal Stress are Given

μ = ^{|τ| – |τ_o} / _{σ_n}

Where;

μ = Co-efficient of Friction
|τ = Mohr – Coulomb Criterion
|τ_o = Cohesion
σ_n = Normal Stress

Let’s solve an example;
Find the co-efficient of friction when the cohesion is 10, the mohr – coulomb criterion is 24 and the normal stress is 12.

This implies that;

|τ = Mohr – Coulomb Criterion = 24
|τ_o = Cohesion = 10
σ_n = Normal Stress = 12

μ = ^{|τ| – |τ_o} / _{σ_n}
Then, μ = ^{24 – 10} / ₁₂
μ = ¹⁴ / ₁₂
μ = 1.667

Therefore, the co-efficient of friction is 1.667.

Calculating the Normal Stress when the Mohr – Coulomb Criterion, the Co-efficient of Friction and the Cohesion are Given

σ_n = ^{|τ| – |τ_o} / _μ

Where;

σ_n = Normal Stress
|τ = Mohr – Coulomb Criterion
|τ_o = Cohesion
μ = Co-efficient of Friction

Let’s solve an example;
Find the normal stress when the mohr – coulomb criterion is 30, the cohesion is 14, and the co-efficient of friction is 5.

This implies that;

|τ = Mohr – Coulomb Criterion = 30
|τ_o = Cohesion = 14
μ = Co-efficient of Friction = 5

σ_n = ^{|τ| – |τ_o} / _μ
So, σ_n = ^{30 – 14} / ₅
σ_n = ¹⁶ / ₅
σ_n = 3.2

Therefore, the normal stress is 3.2.

How to Calculate Mohr – Coulomb Criterion Using Nickzom Calculator

Nickzom Calculator – The Calculator Encyclopedia is capable of calculating the mohr – coulomb criterion.

To get the answer and workings of the mohr – coulomb criterion using the Nickzom Calculator – The Calculator Encyclopedia. First, you need to obtain the app.

You can get this app via any of these means:

Web – https://www.nickzom.org/calculator-plus

To get access to the professional version via web, you need to register and subscribe for NGN 1,500 per annum to have utter access to all functionalities.
You can also try the demo version via https://www.nickzom.org/calculator

Android (Paid) – https://play.google.com/store/apps/details?id=org.nickzom.nickzomcalculator
Android (Free) – https://play.google.com/store/apps/details?id=com.nickzom.nickzomcalculator
Apple (Paid) – https://itunes.apple.com/us/app/nickzom-calculator/id1331162702?mt=8
Once, you have obtained the calculator encyclopedia app, proceed to the Calculator Map, then click on Geology under Add-on.

Now, Click on Rock Mechanics under Geology

Now, Click on Mohr – Coulomb Criterion under Rock Mechanics

The screenshot below displays the page or activity to enter your values, to get the answer for the mohr – coulomb criterion according to the respective parameters which are the Cohesion (|τ_o), Co-efficient of Friction (μ) and Normal Stress (σ_n).

Now, enter the values appropriately and accordingly for the parameters as required by the Cohesion (|τ_o) is 7, Co-efficient of Friction (μ) is 30 and Normal Stress (σ_n) is 28.

Finally, Click on Calculate

As you can see from the screenshot above, Nickzom Calculator– The Calculator Encyclopedia solves for the mohr – coulomb criterion and presents the formula, workings and steps too.