How to Calculate and Solve for Horizontal Strain | Rock Mechanics

The image above represents horizontal strain. To calculate the horizontal strain, five essential parameters are needed, and these parameters are Principal Horizontal Stress Component 1 (σH1), Principal Horizontal Stress Component 2 (σH2), Poisson’s Ratio (v), Vertical Stress (σv) and Young’s Modulus (E).

The formula for calculating the horizontal strain:

εH = (σH1 – vσH2 – vσv) / E

Where:

εH = Horizontal Strain
σH1 = Principal Horizontal Stress Component 1
σH2 = Principal Horizontal Stress Component 2
v = Poisson’s Ratio
σv = Vertical Stress
E = Young’s Modulus

So, let’s solve an example;
Find the horizontal strain when the principal horizontal stress component 1 is 7, the principal horizontal component 2 is 9, the Poisson’s ratio is 11, the vertical stress is 21 and then young’s modulus is 27.

Hence, this implies that;

σH1 = Principal Horizontal Stress Component 1 = 7
σH2 = Principal Horizontal Stress Component 2 = 9
v = Poisson’s Ratio = 11
σv = Vertical Stress = 21
E = Young’s Modulus = 27

εH = (σH1 – vσH2 – vσv) / E
εH = (7 – 11(9) – 11(21)) / 27
So, εH = (7 – 99 – 231) / 27
εH = -323 / 27
εH = -11.96

Therefore, the horizontal strain is -11.96.

Calculating the Principal Horizontal Stress Component 1 when the Horizontal Stress, Principal Horizontal Stress Component 2, Poisson’s Ratio, Vertical Stress and Young’s Modulus are Given

σH1 = (εH x E) + vσH2 + vσv

Where;

σH1 = Principal Horizontal Stress Component 1
εH = Horizontal Strain
σH2 = Principal Horizontal Stress Component 2
v = Poisson’s Ratio
σv = Vertical Stress
E = Young’s Modulus

So, let’s solve an example;
Find the principal horizontal stress component 1 when the horizontal strain is 22, the principal horizontal component 2 is 10, the Poisson’s ratio is 7, the vertical stress is 12 and then young’s modulus is 3.

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Hence, this implies that;

εH = Horizontal Strain = 22
σH2 = Principal Horizontal Stress Component 2 = 10
v = Poisson’s Ratio = 7
σv = Vertical Stress = 12
E = Young’s Modulus = 3

σH1 = (εH x E) + vσH2 + vσv
σH1 = (22 x 3) + 7(10) + 7(12)
That is, σH1 = 66 + 70 + 84
σH1 = 220

Therefore, the principal horizontal stress component 1 is 220.

Calculating the Principal Horizontal Stress Component 2 when the Horizontal Stress, Principal Horizontal Stress Component 1, Poisson’s Ratio, Vertical Stress and Young’s Modulus are Given

σH2 = -((εH x E) – σH1 + vσv / v)

Where;

σH2 = Principal Horizontal Stress Component 2
εH = Horizontal Strain
σH1 = Principal Horizontal Stress Component 1
v = Poisson’s Ratio
σv = Vertical Stress
And then, E = Young’s Modulus

So, let’s solve an example;
Find the principal horizontal stress component 2 when the horizontal strain is 28, the principal horizontal component 1 is 14, the Poisson’s ratio is 5, the vertical stress is 17 and then young’s modulus is 7.

Hence, this implies that;

εH = Horizontal Strain =28
σH1 = Principal Horizontal Stress Component 1 = 14
v = Poisson’s Ratio = 5
σv = Vertical Stress = 17
And then, E = Young’s Modulus = 7

σH2 = -((εH x E) – σH1 + vσv / v)
σH2 = -((28 x 7) – 14 + 5(17) / 5)
Then, σH2 = -(196 – 14 + 85 / 5)
σH2 = -(182 + 85 / 5)
σH2 = -(267 / 5)
So, σH2 = -(53.4)
σH2 = -53.4

Therefore, the principal horizontal stress component 2 is -53.4.

Read more: How to Calculate and Solve for Lateral Strain | Rock Mechanics

Calculating the Poisson’s Ratio when the Horizontal Stress, Principal Horizontal Stress Component 1, Principal Horizontal Stress Component 2, Vertical Stress and Young’s Modulus are Given

v = σH1 – (εH x E) / σH1 + σH2

Where:

v = Poisson’s Ratio
εH = Horizontal Strain
σH1 = Principal Horizontal Stress Component 1
σH2 = Principal Horizontal Stress Component 2
σv = Vertical Stress
And then, E = Young’s Modulus

So, let’s solve an example;
Find the Poisson’s ratio when the principal horizontal stress component 1 is 2, the principal horizontal component 2 is 4, the horizontal strain is 19, the vertical stress is 26 and then young’s modulus is 12.

Hence, this implies that;

εH = Horizontal Strain = 19
σH1 = Principal Horizontal Stress Component 1 = 2
σH2 = Principal Horizontal Stress Component 2 = 4
σv = Vertical Stress = 26
E = Young’s Modulus = 12

So, v = σH1 – (εH x E) / σH1 + σH2
v = 2 – (19 x 12) / 2 + 4
That is, v = 2 – 228 / 6
v = -226 / 6
v = -37.6

Therefore, the Poisson’s ratio is -37.6.

Calculating the Vertical Stress when the Horizontal Stress, Principal Horizontal Stress Component 1, Principal Horizontal Stress Component 2, Poisson’s Ratio and Young’s Modulus are Given

σv = -((εH x E) – σH1 + vσH2 / v)

Where:

σv = Vertical Stress
εH = Horizontal Strain
And then, σH1 = Principal Horizontal Stress Component 1
σH2 = Principal Horizontal Stress Component 2
v = Poisson’s Ratio
E = Young’s Modulus

So, let’s solve an example;
Find the vertical stress when the horizontal strain is 27, the principal horizontal stress component 1 is 12, the principal horizontal component 2 is 9, the Poisson’s ratio is 24 and then young’s modulus is 8.

Hence, this implies that;

εH = Horizontal Strain = 27
σH1 = Principal Horizontal Stress Component 1 = 12
σH2 = Principal Horizontal Stress Component 2 = 9
v = Poisson’s Ratio = 24
E = Young’s Modulus = 8

σv = -((εH x E) – σH1 + vσH2 / v)
σv = -((27 x 8) – 12 + 8(9) / 24)
Then, σv = -(216 – 12 + 72 / 24)
σv = -(204 + 72 / 24)
σv = -(276 / 24)
So, σv = -(11.5)
σv = -11.5

Therefore, the vertical stress is -11.5.

Calculating the Young’s Modulus when the Horizontal Stress, Principal Horizontal Stress Component 1, Principal Horizontal Stress Component 2, Poisson’s Ratio and Vertical Stress are Given

E = (σH1 – vσH2 – vσv) / εH

Where:

E = Young’s Modulus
εH = Horizontal Strain
And then, σH1 = Principal Horizontal Stress Component 1
σH2 = Principal Horizontal Stress Component 2
v = Poisson’s Ratio
σv = Vertical Stress

So, let’s solve an example;
Find the young’s modulus when the horizontal strain is 44, principal horizontal stress component 1 is 28, the principal horizontal component 2 is 5, the Poisson’s ratio is 4 and then the vertical stress is 3.

Hence, this implies that;

εH = Horizontal Strain = 44
σH1 = Principal Horizontal Stress Component 1 = 28
And then, σH2 = Principal Horizontal Stress Component 2 = 5
v = Poisson’s Ratio = 4
σv = Vertical Stress = 3

E = (σH1 – vσH2 – vσv) / εH
E = (28 – 4(5) – 4(3)) / 44
So, E = (28 – 20 – 12) / 44
E = (8 – 12) / 44
Then, E = (-4) / 44
E = -0.09

Therefore, the young’s modulus is -0.09.

How to Calculate and Solve for Horizontal Strain Using the Nickzom Calculator

Nickzom Calculator – The Calculator Encyclopedia is capable of calculating the horizontal strain.

To get the answer and workings of the horizontal strain 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.

How to Calculate and Solve for Horizontal Strain | Rock Mechanics

Then, Click on Rock Mechanics under Geology

Then, Click on Horizontal Strain under Rock Mechanics

So, the screenshot below displays the page or activity to enter your values, to get the answer for the horizontal strain according to the respective parameters which are the Principal Horizontal Stress Component 1 (σH1), Principal Horizontal Stress Component 2 (σH2), Poisson’s Ratio (v), Vertical Stress (σv) and Young’s Modulus (E).

How to Calculate and Solve for Horizontal Strain | Rock Mechanics

Then, enter the values appropriately and accordingly for the parameters as required by the Principal Horizontal Stress Component 1 (σH1) is 7, Principal Horizontal Stress Component 2 (σH2) is 9, Poisson’s Ratio (v) is 11, Vertical Stress (σv) is 21 and Young’s Modulus (E) is 27.

Finally, Click on Calculate

How to Calculate and Solve for Horizontal Strain | Rock Mechanics

As you can see from the screenshot above, Nickzom Calculator– The Calculator Encyclopedia can calculate the horizontal strain and present the formula, workings, and steps.

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