The image above represents vertical strain.

To compute for 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} = ^{(σv – vσH1 – vσH2)} / _{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

E = Young’s Modulus

Let’s solve an example;

Find the vertical strain with a vertical stress of 44, Poisson’s ratio of 5, principal horizontal stress component 1 of 6, principal horizontal stress component 2 of 4 and young’s modulus of 12.

This implies that;

σ_{v} = Vertical Stress = 44

v = Poisson’s Ratio = 5

σ_{H1} = Principal Horizontal Stress Component 1 = 6

σ_{H2} = Principal Horizontal Stress Component 2 = 4

E = Young’s Modulus = 12

ε_{v} = ^{(σv – vσH1 – vσH2)} / _{E}

ε_{v} = ^{(44 – 5(6) – 5(4))} / _{12}

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 is Given.**

σ_{v} = (ε_{v} x E) + vσ_{H1} + vσ_{H2}

Where:

σ_{v} = Vertical Stress

ε_{v} = Vertical Strain

v = Poisson’s Ratio

σ_{H1} = Principal Horizontal Stress Component 1

σ_{H2} = Principal Horizontal Stress Component 2

E = Young’s Modulus

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 young’s modulus of 13.

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

E = Young’s Modulus = 13

σ_{v} = (ε_{v} x E) + vσ_{H1} + vσ_{H2}

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 is Given.**

v = ^{σv – (εv 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

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 young’s modulus of 6.

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 = ^{σv – (εv x E)} / _{σ}_{H1} + σ_{H2}

v = ^{110 – (16 x 6)} / _{7 + 13}

v = ^{110 – 96} / _{20}

v = ^{14} / _{20}

v = 0.7

Therefore, the **P****oisson’s ratio **is **0.7.**

Continue reading How to Calculate and Solve for Vertical Strain | Rock Mechanics