How to Calculate and Solve for Vertical Strain | Rock Mechanics

September 25, 2019March 8, 2024 Loveth Idoko

Last Updated on March 8, 2024

To calculate vertical strain, five essential parameters are needed, and these parameters are Vertical Stress (σ_v), Poisson’s Ratio (v), Principal Horizontal Stress Component 1 (σ_H1), Principal Horizontal Stress Component 2 (σ_H2), and Young’s Modulus (E).

The formula for calculating the vertical strain:

ε_v = ^{(σ_v – vσ_H1 – vσ_H2)} / _E

Where:

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

So, let’s solve an example;
Find the vertical strain with vertical stress of 44, Poisson’s ratio of 5, principal horizontal stress component 1 of 6, principal horizontal stress component 2 of 4, and then young’s modulus of 12.

Hence, this implies that;

σ_v = Vertical Stress = 44
v = Poisson’s Ratio = 5
Then, σ_H1 = Principal Horizontal Stress Component 1 = 6
σ_H2 = Principal Horizontal Stress Component 2 = 4
E = Young’s Modulus = 12

ε_v = ^{(σ_v – vσ_H1 – vσ_H2)} / _E
ε_v = ^{(44 – 5(6) – 5(4))} / ₁₂
So, ε_v = ^{(44 – 30 – 20)} / ₁₂
ε_v = ^-6 / ₁₂
ε_v = -0.5

Therefore, the vertical strain is -0.5.

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

σ_v = (ε_v x E) + vσ_H1 + vσ_H2

Where:

σ_v = Vertical Stress
ε_v = Vertical Strain
v = Poisson’s Ratio
Also, σ_H1 = Principal Horizontal Stress Component 1
σ_H2 = Principal Horizontal Stress Component 2
E = Young’s Modulus

So, let’s solve an example;
Find the vertical stress with a vertical strain of 52, Poisson’s ratio of 12, principal horizontal stress component 1 of 15, principal horizontal stress component 2 of 10 and then young’s modulus of 13.

Hence, this implies that;

ε_v = Vertical Strain = 52
v = Poisson’s Ratio = 12
σ_H1 = Principal Horizontal Stress Component 1 = 15
σ_H2 = Principal Horizontal Stress Component 2 = 10
And then, E = Young’s Modulus = 13

σ_v = (ε_v x E) + vσ_H1 + vσ_H2
Then, σ_v = (52 x 13) + 12(15) + 15(10)
σ_v = 676 + 180 + 150
σ_v = 1006

Therefore, the vertical stress is 1006.

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

v = ^{σ_v – (ε_v x E)} / _{σ_H1 + σ_H2}

Where:

v = Poisson’s Ratio
ε_v = Vertical Strain
σ_v = Vertical Stress
σ_H1 = Principal Horizontal Stress Component 1
σ_H2 = Principal Horizontal Stress Component 2
E = Young’s Modulus

So, let’s solve an example;
Find the Poisson’s ratio with a vertical strain of 16, vertical stress of 110, principal horizontal stress component 1 of 7, principal horizontal stress component 2 of 13 and then young’s modulus of 6.

Hence, this implies that;

ε_v = Vertical Strain = 16
σ_v = Vertical Stress = 110
σ_H1 = Principal Horizontal Stress Component 1 = 7
σ_H2 = Principal Horizontal Stress Component 2 = 13
E = Young’s Modulus = 6

v = ^{σ_v – (ε_v x E)} / _{σ_H1 + σ_H2}
v = ^{110 – (16 x 6)} / _{7 + 13}
Hence, v = ^{110 – 96} / ₂₀
v = ¹⁴ / ₂₀
v = 0.7

Therefore, the Poisson’s ratio is 0.7.

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

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

Where;

σ_H1 = Principal Horizontal Stress Component 1
ε_v = Vertical Strain
σ_v = Vertical Stress
v = Poisson’s Ratio
σ_H2 = Principal Horizontal Stress Component 2
E = Young’s Modulus

Then, let’s solve an example;
Find the principal horizontal stress component 1 with a vertical strain of 15, Poisson’s ratio of 10, vertical stress of 11, principal horizontal stress component 2 of 8 and young’s modulus of 14.

Hence, this implies that;

ε_v = Vertical Strain = 15
σ_v = Vertical Stress = 11
v = Poisson’s Ratio = 10
σ_H2 = Principal Horizontal Stress Component 2 = 8
E = Young’s Modulus = 14

σ_H1 = -(^{(E_v x E) – σ_v + vσ_H2} / _v)
σ_H1 = -(^{(15 x 14) – 11 + 10(8)} / ₁₀)
So, σ_H1 = -(^{210 – 11 + 80} / ₁₀)
σ_H1 = -(^{199 + 80} / ₁₀)
σ_H1 = -(²⁷⁹ / ₁₀)
Then, σ_H1 = -(27.9)
σ_H1 = -27.9

Therefore, the principal horizontal stress component 1 is -27.9.

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

σ_H2 = -(^{(E_v x E) – σ_v + vσ_H1} / _v)

Where;

σ_H2 = Principal Horizontal Stress Component 2
ε_v = Vertical Strain
σ_v = Vertical Stress
v = Poisson’s Ratio
σ_H1 = Principal Horizontal Stress Component 1
E = Young’s Modulus

So, let’s solve an example;
Find the principal horizontal stress component 2 with a vertical strain of 30, Poisson’s ratio of 18, vertical stress of 17, principal horizontal stress component 1 of 11 and then young’s modulus of 19.

Hence, this implies that;

ε_v = Vertical Strain = 30
σ_v = Vertical Stress = 17
v = Poisson’s Ratio = 18
σ_H1 = Principal Horizontal Stress Component 1 = 11
E = Young’s Modulus = 19

σ_H2 = -(^{(E_v x E) – σ_v + vσ_H1} / _v)
σ_H2 = -(^{(30 x 19) – 17 + 18(11)} / ₁₈)
Then, σ_H2 = -(^{570 – 17 + 198} / ₁₈)
σ_H2 = -(^{533 + 198} / ₁₈)
σ_H2 = -(⁷⁵¹ / ₁₈)
So, σ_H2 = -(41.72)
σ_H2 = -41.72

Therefore, the principal horizontal stress component 2 is -41.72.

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

E = ^{(σ_v – vσ_H1 – vσ_H2)} / _{ε_v}

Where;

E = Young’s Modulus
ε_v = Vertical Strain
σ_v = Vertical Stress
v = Poisson’s Ratio
σ_H1 = Principal Horizontal Stress Component 1
And also, σ_H2 = Principal Horizontal Stress Component 2

Then, let’s solve an example;
Find the young’s modulus with a vertical strain of 20, vertical stress of 60, Poisson’s ratio of 2, principal horizontal stress component 1 of 7, and then the principal horizontal stress component 2 of 9.

Hence, this implies that;

ε_v = Vertical Strain = 20
σ_v = Vertical Stress = 60
And then, v = Poisson’s Ratio = 2
σ_H1 = Principal Horizontal Stress Component 1 = 7
σ_H2 = Principal Horizontal Stress Component 2 = 9

E = ^{(σ_v – v(σ_H1) – v(σ_H2))} / _{ε_v}
E = ^{(60 – 2(7) – 2(9))} / ₂₀
Then, E = ^{(60 – 14 – 18)} / ₂₀
E = ⁽²⁸⁾ / ₂₀
E = 1.4

Therefore, the young’s modulus is 1.4.

How to Calculate Vertical Strain Using the Nickzom Calculator

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

Hence, to get the answer and workings of the vertical strain, use the Nickzom Calculator – The Calculator Encyclopedia. First, you need to obtain the app.

Moreover, you can get this app via multiple means, such as

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.

So, Click on Rock Mechanics under Geology

Then, click on Vertical Strain under Rock Mechanics

So, the screenshot below displays the page or activity to enter your values, to get the answer for the vertical strain according to the respective parameters which are the Vertical Stress (σ_v), Poisson’s Ratio (v), Principal Horizontal Stress Component 1 (σ_H1), Principal Horizontal Stress Component 2 (σ_H2) and Young’s Modulus (E).

Then, enter the values appropriately and accordingly for the parameters as required by the Vertical Stress (σ_v) is 44, Poisson’s Ratio (v) is 5, Principal Horizontal Stress Component 1 (σ_H1) is 6, Principal Horizontal Stress Component 2 (σ_H2) is 4 and Young’s Modulus (E) is 12.

Finally, Click on Calculate

As you can see from the screenshot above, Nickzom Calculator– The Calculator Encyclopedia solves the vertical strain and also presents the formula, workings and steps too.