How to Calculate and Solve for Horizontal Strain | Rock Mechanics

September 26, 2019March 9, 2024 Loveth Idoko

Last Updated on March 9, 2024

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 (σ_H1), Principal Horizontal Stress Component 2 (σ_H2), Poisson’s Ratio (v), Vertical Stress (σ_v) and Young’s Modulus (E).

The formula for calculating the horizontal strain:

ε_H = ^{(σ_H1 – vσ_H2 – vσ_v)} / _E

Where:

ε_H = Horizontal Strain
σ_H1 = Principal Horizontal Stress Component 1
σ_H2 = Principal Horizontal Stress Component 2
v = Poisson’s Ratio
σ_v = 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;

σ_H1 = Principal Horizontal Stress Component 1 = 7
σ_H2 = Principal Horizontal Stress Component 2 = 9
v = Poisson’s Ratio = 11
σ_v = Vertical Stress = 21
E = Young’s Modulus = 27

ε_H = ^{(σ_H1 – vσ_H2 – vσ_v)} / _E
ε_H = ^{(7 – 11(9) – 11(21))} / ₂₇
So, ε_H = ^{(7 – 99 – 231)} / ₂₇
ε_H = ^-323 / ₂₇
ε_H = -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

σ_H1 = (ε_H x E) + vσ_H2 + vσ_v

Where;

σ_H1 = Principal Horizontal Stress Component 1
ε_H = Horizontal Strain
σ_H2 = Principal Horizontal Stress Component 2
v = Poisson’s Ratio
σ_v = 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.

Hence, this implies that;

ε_H = Horizontal Strain = 22
σ_H2 = Principal Horizontal Stress Component 2 = 10
v = Poisson’s Ratio = 7
σ_v = Vertical Stress = 12
E = Young’s Modulus = 3

σ_H1 = (ε_H x E) + vσ_H2 + vσ_v
σ_H1 = (22 x 3) + 7(10) + 7(12)
That is, σ_H1 = 66 + 70 + 84
σ_H1 = 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

σ_H2 = -(^{(ε_H x E) – σ_H1 + vσ_v} / _v)

Where;

σ_H2 = Principal Horizontal Stress Component 2
ε_H = Horizontal Strain
σ_H1 = Principal Horizontal Stress Component 1
v = Poisson’s Ratio
σ_v = 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;

ε_H = Horizontal Strain =28
σ_H1 = Principal Horizontal Stress Component 1 = 14
v = Poisson’s Ratio = 5
σ_v = Vertical Stress = 17
And then, E = Young’s Modulus = 7

σ_H2 = -(^{(ε_H x E) – σ_H1 + vσ_v} / _v)
σ_H2 = -(^{(28 x 7) – 14 + 5(17)} / ₅)
Then, σ_H2 = -(^{196 – 14 + 85} / ₅)
σ_H2 = -(^{182 + 85} / ₅)
σ_H2 = -(²⁶⁷ / ₅)
So, σ_H2 = -(53.4)
σ_H2 = -53.4

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

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 = ^{σ_H1 – (ε_H x E)} / _{σ_H1 + σ_H2}

Where:

v = Poisson’s Ratio
ε_H = Horizontal Strain
σ_H1 = Principal Horizontal Stress Component 1
σ_H2 = Principal Horizontal Stress Component 2
σ_v = 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;

ε_H = Horizontal Strain = 19
σ_H1 = Principal Horizontal Stress Component 1 = 2
σ_H2 = Principal Horizontal Stress Component 2 = 4
σ_v = Vertical Stress = 26
E = Young’s Modulus = 12

So, v = ^{σ_H1 – (ε_H x E)} / _{σ_H1 + σ_H2}
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

σ_v = -(^{(ε_H x E) – σ_H1 + vσ_H2} / _v)

Where:

σ_v = Vertical Stress
ε_H = Horizontal Strain
And then, σ_H1 = Principal Horizontal Stress Component 1
σ_H2 = 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;

ε_H = Horizontal Strain = 27
σ_H1 = Principal Horizontal Stress Component 1 = 12
σ_H2 = Principal Horizontal Stress Component 2 = 9
v = Poisson’s Ratio = 24
E = Young’s Modulus = 8

σ_v = -(^{(ε_H x E) – σ_H1 + vσ_H2} / _v)
σ_v = -(^{(27 x 8) – 12 + 8(9)} / ₂₄)
Then, σ_v = -(^{216 – 12 + 72} / ₂₄)
σ_v = -(^{204 + 72} / ₂₄)
σ_v = -(²⁷⁶ / ₂₄)
So, σ_v = -(11.5)
σ_v = -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 = ^{(σ_H1 – vσ_H2 – vσ_v)} / _{ε_H}

Where:

E = Young’s Modulus
ε_H = Horizontal Strain
And then, σ_H1 = Principal Horizontal Stress Component 1
σ_H2 = Principal Horizontal Stress Component 2
v = Poisson’s Ratio
σ_v = 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;

ε_H = Horizontal Strain = 44
σ_H1 = Principal Horizontal Stress Component 1 = 28
And then, σ_H2 = Principal Horizontal Stress Component 2 = 5
v = Poisson’s Ratio = 4
σ_v = Vertical Stress = 3

E = ^{(σ_H1 – vσ_H2 – vσ_v)} / _{ε_H}
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.

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 (σ_H1), Principal Horizontal Stress Component 2 (σ_H2), Poisson’s Ratio (v), Vertical Stress (σ_v) and Young’s Modulus (E).

Then, enter the values appropriately and accordingly for the parameters as required by the Principal Horizontal Stress Component 1 (σ_H1) is 7, Principal Horizontal Stress Component 2 (σ_H2) is 9, Poisson’s Ratio (v) is 11, Vertical Stress (σ_v) is 21 and Young’s Modulus (E) is 27.

Finally, Click on Calculate

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.