A metal specimen has E = 10.4×10⁶ psi and Poisson's ratio ν = 0.26. Compute the bulk (volume) modulus of elasticity, the shear modulus, and the volumetric strain produced by a hydrostatic pressure of 14,000 psi.

Start with the thinking

• The bulk modulus links hydrostatic pressure to volume change; it blows up as ν approaches 0.5 (incompressible).
• E, G, K and ν are not independent — any two fix the others.

Step-by-step solution

1. Formula

   $$K = E / [3(1 - 2\nu)]$$

2. Substituting

   $$K = 10.4\times10^{6} / [3(1 - 2\times0.26)] = 7.222\times10^{6} psi$$

3. Formula

   $$G = E / [2(1 + \nu)]$$

4. Substituting

   $$G = 10.4\times10^{6} / [2(1 + 0.26)] = 4.127\times10^{6} psi$$

5. Formula — ΔV/V = −p/K

6. Substituting — ΔV/V = −14000/7,222,222 = -1.9385 × 10⁻³

Why the other options are there

• K = 2.28×10⁶ psi (sign in the bracket flipped)
• K = G (moduli confused)

Reference: FE Reference Handbook — Mechanics of Materials → Bulk (Volume) Modulus of Elasticity

A metal specimen has E = 29.0×10⁶ psi and Poisson's ratio ν = 0.32. Compute the bulk (volume) modulus of elasticity, the shear modulus, and the volumetric strain produced by a hydrostatic pressure of 27,000 psi.

Start with the thinking

• The bulk modulus links hydrostatic pressure to volume change; it blows up as ν approaches 0.5 (incompressible).
• E, G, K and ν are not independent — any two fix the others.

Step-by-step solution

1. Formula

   $$K = E / [3(1 - 2\nu)]$$

2. Substituting

   $$K = 29.0\times10^{6} / [3(1 - 2\times0.32)] = 26.852\times10^{6} psi$$

3. Formula

   $$G = E / [2(1 + \nu)]$$

4. Substituting

   $$G = 29.0\times10^{6} / [2(1 + 0.32)] = 10.985\times10^{6} psi$$

5. Formula — ΔV/V = −p/K

6. Substituting — ΔV/V = −27000/26,851,852 = -1.0055 × 10⁻³

Why the other options are there

• K = 5.89×10⁶ psi (sign in the bracket flipped)
• K = G (moduli confused)

Reference: FE Reference Handbook — Mechanics of Materials → Bulk (Volume) Modulus of Elasticity

A metal specimen has E = 16.0×10⁶ psi and Poisson's ratio ν = 0.32. Compute the bulk (volume) modulus of elasticity, the shear modulus, and the volumetric strain produced by a hydrostatic pressure of 8,000 psi.

Start with the thinking

• The bulk modulus links hydrostatic pressure to volume change; it blows up as ν approaches 0.5 (incompressible).
• E, G, K and ν are not independent — any two fix the others.

Step-by-step solution

1. Formula

   $$K = E / [3(1 - 2\nu)]$$

2. Substituting

   $$K = 16.0\times10^{6} / [3(1 - 2\times0.32)] = 14.815\times10^{6} psi$$

3. Formula

   $$G = E / [2(1 + \nu)]$$

4. Substituting

   $$G = 16.0\times10^{6} / [2(1 + 0.32)] = 6.061\times10^{6} psi$$

5. Formula — ΔV/V = −p/K

6. Substituting — ΔV/V = −8000/14,814,815 = -0.5400 × 10⁻³

Why the other options are there

• K = 3.25×10⁶ psi (sign in the bracket flipped)
• K = G (moduli confused)

Reference: FE Reference Handbook — Mechanics of Materials → Bulk (Volume) Modulus of Elasticity

A metal specimen has E = 30.0×10⁶ psi and Poisson's ratio ν = 0.31. Compute the bulk (volume) modulus of elasticity, the shear modulus, and the volumetric strain produced by a hydrostatic pressure of 23,000 psi.

Start with the thinking

• The bulk modulus links hydrostatic pressure to volume change; it blows up as ν approaches 0.5 (incompressible).
• E, G, K and ν are not independent — any two fix the others.

Step-by-step solution

1. Formula

   $$K = E / [3(1 - 2\nu)]$$

2. Substituting

   $$K = 30.0\times10^{6} / [3(1 - 2\times0.31)] = 26.316\times10^{6} psi$$

3. Formula

   $$G = E / [2(1 + \nu)]$$

4. Substituting

   $$G = 30.0\times10^{6} / [2(1 + 0.31)] = 11.450\times10^{6} psi$$

5. Formula — ΔV/V = −p/K

6. Substituting — ΔV/V = −23000/26,315,789 = -0.8740 × 10⁻³

Why the other options are there

• K = 6.17×10⁶ psi (sign in the bracket flipped)
• K = G (moduli confused)

Reference: FE Reference Handbook — Mechanics of Materials → Bulk (Volume) Modulus of Elasticity

A metal specimen has E = 29.0×10⁶ psi and Poisson's ratio ν = 0.30. Compute the bulk (volume) modulus of elasticity, the shear modulus, and the volumetric strain produced by a hydrostatic pressure of 19,000 psi.

Start with the thinking

• The bulk modulus links hydrostatic pressure to volume change; it blows up as ν approaches 0.5 (incompressible).
• E, G, K and ν are not independent — any two fix the others.

Step-by-step solution

1. Formula

   $$K = E / [3(1 - 2\nu)]$$

2. Substituting

   $$K = 29.0\times10^{6} / [3(1 - 2\times0.30)] = 24.167\times10^{6} psi$$

3. Formula

   $$G = E / [2(1 + \nu)]$$

4. Substituting

   $$G = 29.0\times10^{6} / [2(1 + 0.30)] = 11.154\times10^{6} psi$$

5. Formula — ΔV/V = −p/K

6. Substituting — ΔV/V = −19000/24,166,667 = -0.7862 × 10⁻³

Why the other options are there

• K = 6.04×10⁶ psi (sign in the bracket flipped)
• K = G (moduli confused)

Reference: FE Reference Handbook — Mechanics of Materials → Bulk (Volume) Modulus of Elasticity

A metal specimen has E = 10.4×10⁶ psi and Poisson's ratio ν = 0.33. Compute the bulk (volume) modulus of elasticity, the shear modulus, and the volumetric strain produced by a hydrostatic pressure of 34,000 psi.

Start with the thinking

• The bulk modulus links hydrostatic pressure to volume change; it blows up as ν approaches 0.5 (incompressible).
• E, G, K and ν are not independent — any two fix the others.

Step-by-step solution

1. Formula

   $$K = E / [3(1 - 2\nu)]$$

2. Substituting

   $$K = 10.4\times10^{6} / [3(1 - 2\times0.33)] = 10.196\times10^{6} psi$$

3. Formula

   $$G = E / [2(1 + \nu)]$$

4. Substituting

   $$G = 10.4\times10^{6} / [2(1 + 0.33)] = 3.910\times10^{6} psi$$

5. Formula — ΔV/V = −p/K

6. Substituting — ΔV/V = −34000/10,196,078 = -3.3346 × 10⁻³

Why the other options are there

• K = 2.09×10⁶ psi (sign in the bracket flipped)
• K = G (moduli confused)

Reference: FE Reference Handbook — Mechanics of Materials → Bulk (Volume) Modulus of Elasticity

A metal specimen has E = 29.0×10⁶ psi and Poisson's ratio ν = 0.31. Compute the bulk (volume) modulus of elasticity, the shear modulus, and the volumetric strain produced by a hydrostatic pressure of 9,000 psi.

Start with the thinking

• The bulk modulus links hydrostatic pressure to volume change; it blows up as ν approaches 0.5 (incompressible).
• E, G, K and ν are not independent — any two fix the others.

Step-by-step solution

1. Formula

   $$K = E / [3(1 - 2\nu)]$$

2. Substituting

   $$K = 29.0\times10^{6} / [3(1 - 2\times0.31)] = 25.439\times10^{6} psi$$

3. Formula

   $$G = E / [2(1 + \nu)]$$

4. Substituting

   $$G = 29.0\times10^{6} / [2(1 + 0.31)] = 11.069\times10^{6} psi$$

5. Formula — ΔV/V = −p/K

6. Substituting — ΔV/V = −9000/25,438,596 = -0.3538 × 10⁻³

Why the other options are there

• K = 5.97×10⁶ psi (sign in the bracket flipped)
• K = G (moduli confused)

Reference: FE Reference Handbook — Mechanics of Materials → Bulk (Volume) Modulus of Elasticity

A metal specimen has E = 30.0×10⁶ psi and Poisson's ratio ν = 0.27. Compute the bulk (volume) modulus of elasticity, the shear modulus, and the volumetric strain produced by a hydrostatic pressure of 40,000 psi.

Start with the thinking

• The bulk modulus links hydrostatic pressure to volume change; it blows up as ν approaches 0.5 (incompressible).
• E, G, K and ν are not independent — any two fix the others.

Step-by-step solution

1. Formula

   $$K = E / [3(1 - 2\nu)]$$

2. Substituting

   $$K = 30.0\times10^{6} / [3(1 - 2\times0.27)] = 21.739\times10^{6} psi$$

3. Formula

   $$G = E / [2(1 + \nu)]$$

4. Substituting

   $$G = 30.0\times10^{6} / [2(1 + 0.27)] = 11.811\times10^{6} psi$$

5. Formula — ΔV/V = −p/K

6. Substituting — ΔV/V = −40000/21,739,130 = -1.8400 × 10⁻³

Why the other options are there

• K = 6.49×10⁶ psi (sign in the bracket flipped)
• K = G (moduli confused)

Reference: FE Reference Handbook — Mechanics of Materials → Bulk (Volume) Modulus of Elasticity

A metal specimen has E = 10.4×10⁶ psi and Poisson's ratio ν = 0.31. Compute the bulk (volume) modulus of elasticity, the shear modulus, and the volumetric strain produced by a hydrostatic pressure of 40,000 psi.

Start with the thinking

• The bulk modulus links hydrostatic pressure to volume change; it blows up as ν approaches 0.5 (incompressible).
• E, G, K and ν are not independent — any two fix the others.

Step-by-step solution

1. Formula

   $$K = E / [3(1 - 2\nu)]$$

2. Substituting

   $$K = 10.4\times10^{6} / [3(1 - 2\times0.31)] = 9.123\times10^{6} psi$$

3. Formula

   $$G = E / [2(1 + \nu)]$$

4. Substituting

   $$G = 10.4\times10^{6} / [2(1 + 0.31)] = 3.969\times10^{6} psi$$

5. Formula — ΔV/V = −p/K

6. Substituting — ΔV/V = −40000/9,122,807 = -4.3846 × 10⁻³

Why the other options are there

• K = 2.14×10⁶ psi (sign in the bracket flipped)
• K = G (moduli confused)

Reference: FE Reference Handbook — Mechanics of Materials → Bulk (Volume) Modulus of Elasticity

A metal specimen has E = 16.0×10⁶ psi and Poisson's ratio ν = 0.31. Compute the bulk (volume) modulus of elasticity, the shear modulus, and the volumetric strain produced by a hydrostatic pressure of 19,000 psi.

Start with the thinking

• The bulk modulus links hydrostatic pressure to volume change; it blows up as ν approaches 0.5 (incompressible).
• E, G, K and ν are not independent — any two fix the others.

Step-by-step solution

1. Formula

   $$K = E / [3(1 - 2\nu)]$$

2. Substituting

   $$K = 16.0\times10^{6} / [3(1 - 2\times0.31)] = 14.035\times10^{6} psi$$

3. Formula

   $$G = E / [2(1 + \nu)]$$

4. Substituting

   $$G = 16.0\times10^{6} / [2(1 + 0.31)] = 6.107\times10^{6} psi$$

5. Formula — ΔV/V = −p/K

6. Substituting — ΔV/V = −19000/14,035,088 = -1.3538 × 10⁻³

Why the other options are there

• K = 3.29×10⁶ psi (sign in the bracket flipped)
• K = G (moduli confused)

Reference: FE Reference Handbook — Mechanics of Materials → Bulk (Volume) Modulus of Elasticity