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A piece of material is subjected to two perpendicular stresses 112 N/mm2 (tensile) and p (compressive). If the strain energy due to both the stresses is same as a stress of 128 N/mm2 acting alone, then the compressive stress, p (in N/mm2), is;
[Take Poisson’s ratio (μ)= 0.3]
- a)104.10
- b)70.56
- c)36.89
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A piece of material is subjected to two perpendicular stresses 112 N/...
As we know the relation,
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Strain energy per unit volume,
Here, P1 = 112 N/mm2 (tensile)
P2 = p N/mm2 (compressive)
μ = 0.3
∴ U = 1/2E(1122 + p2 + 2 × 0.3 × 112p
U = 1/2E (12544 + 67.2p + p2) ...(i)
For a single stress, p’, acting alone , U = p1/2/2E per unit volume (wehere, p’ = 128 N/mm2)
⇒ 1282/2E = 1/2E(12544 + 67.2p + p2)
⇒ p2 + 67.2p - 3840 = 0
∴ p = 36.89 N/ mm2
Most Upvoted Answer
A piece of material is subjected to two perpendicular stresses 112 N/...
To solve this question, we need to use the concept of strain energy and the relationship between stress and strain in a material.
Given:
- Tensile stress (σt) = 112 N/mm²
- Compressive stress (p) = ?
- Stress acting alone (σ) = 128 N/mm²
- Poisson's ratio (μ) = 0.3
Now, let's proceed with the solution.
1. Understand the concept of strain energy:
Strain energy is the energy stored in a material when it is deformed due to the application of stress. It is given by the formula:
Strain energy (U) = (1/2) × stress × strain × volume
2. Calculate the strain energy for the tensile stress:
Since the strain energy is the same for both stresses, we can equate the strain energies for the tensile and compressive stresses.
For the tensile stress:
- Stress (σ) = 112 N/mm²
- Strain (εt) = σ/E
(where E is the Young's modulus of the material)
- Volume (V) = 1 mm³ (considering a unit volume for simplicity)
Substituting these values into the strain energy formula, we get:
U = (1/2) × 112 N/mm² × (112 N/mm² / E) × 1 mm³
3. Calculate the strain energy for the compressive stress:
For the compressive stress:
- Stress (σc) = -p (since it is compressive)
- Strain (εc) = -μ × εt
(where μ is the Poisson's ratio and εt is the tensile strain)
- Volume (V) = 1 mm³
Substituting these values into the strain energy formula, we get:
U = (1/2) × -p × (-μ × εt) × 1 mm³
= (1/2) × p × μ × εt × 1 mm³
4. Equate the strain energies:
Since the strain energies are the same, we can equate the expressions obtained in steps 2 and 3:
(1/2) × 112 N/mm² × (112 N/mm² / E) × 1 mm³ = (1/2) × p × μ × εt × 1 mm³
5. Simplify and solve for p:
Canceling out the common terms and rearranging the equation, we get:
112 N/mm² / E = p × μ × εt
Since E = σ / εt (from Hooke's law), we can substitute this into the equation:
112 N/mm² / (σ / εt) = p × μ × εt
Simplifying further, we get:
p = (112 N/mm² / σ) × μ
Substituting the given values, we have:
p = (112 N/mm² / 128 N/mm²) × 0.3
p ≈ 0.875 × 0.3
p ≈ 0.2625 N/mm²
Converting to two decimal places, we get:
p ≈ 0.26 N/mm²
Therefore, the compressive stress (p) is approximately 0.26 N/mm², which is closest to option C (
Given:
- Tensile stress (σt) = 112 N/mm²
- Compressive stress (p) = ?
- Stress acting alone (σ) = 128 N/mm²
- Poisson's ratio (μ) = 0.3
Now, let's proceed with the solution.
1. Understand the concept of strain energy:
Strain energy is the energy stored in a material when it is deformed due to the application of stress. It is given by the formula:
Strain energy (U) = (1/2) × stress × strain × volume
2. Calculate the strain energy for the tensile stress:
Since the strain energy is the same for both stresses, we can equate the strain energies for the tensile and compressive stresses.
For the tensile stress:
- Stress (σ) = 112 N/mm²
- Strain (εt) = σ/E
(where E is the Young's modulus of the material)
- Volume (V) = 1 mm³ (considering a unit volume for simplicity)
Substituting these values into the strain energy formula, we get:
U = (1/2) × 112 N/mm² × (112 N/mm² / E) × 1 mm³
3. Calculate the strain energy for the compressive stress:
For the compressive stress:
- Stress (σc) = -p (since it is compressive)
- Strain (εc) = -μ × εt
(where μ is the Poisson's ratio and εt is the tensile strain)
- Volume (V) = 1 mm³
Substituting these values into the strain energy formula, we get:
U = (1/2) × -p × (-μ × εt) × 1 mm³
= (1/2) × p × μ × εt × 1 mm³
4. Equate the strain energies:
Since the strain energies are the same, we can equate the expressions obtained in steps 2 and 3:
(1/2) × 112 N/mm² × (112 N/mm² / E) × 1 mm³ = (1/2) × p × μ × εt × 1 mm³
5. Simplify and solve for p:
Canceling out the common terms and rearranging the equation, we get:
112 N/mm² / E = p × μ × εt
Since E = σ / εt (from Hooke's law), we can substitute this into the equation:
112 N/mm² / (σ / εt) = p × μ × εt
Simplifying further, we get:
p = (112 N/mm² / σ) × μ
Substituting the given values, we have:
p = (112 N/mm² / 128 N/mm²) × 0.3
p ≈ 0.875 × 0.3
p ≈ 0.2625 N/mm²
Converting to two decimal places, we get:
p ≈ 0.26 N/mm²
Therefore, the compressive stress (p) is approximately 0.26 N/mm², which is closest to option C (
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A piece of material is subjected to two perpendicular stresses 112 N/mm2 (tensile) and p (compressive). If the strain energy due to both the stresses is same as a stress of 128 N/mm2 acting alone, then the compressive stress, p (in N/mm2), is;[Take Poisson’s ratio (μ)= 0.3]a)104.10b)70.56c)36.89d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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