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A cube having each side of length a, is constrained in all directions and is heated uniformly so that the temperature is raised by T℃ . If α is the thermal coefficient of expansion of the cube material and E the modulus of elasticity, the stress developed in the cube is
- a)αTE/μ
- b)αTE/(1 − 2μ)
- c)αTE/2μ
- d)αTE/(1 + 2μ)
Correct answer is option 'B'. Can you explain this answer?
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A cube having each side of length a, is constrained in all directions...
Since the cube is restrained in all directions,
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⇒ εv = εx + εy + εz ⋯ ①
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A cube having each side of length a, is constrained in all directions...
Stress developed in the cube:
To determine the stress developed in the cube when it is heated uniformly, we can use the formula for thermal stress:
Formula for thermal stress:
σ = αTE / (1 - 2μ)
where:
σ = stress developed in the cube
α = thermal coefficient of expansion
T = temperature change
E = modulus of elasticity
μ = Poisson's ratio
Explanation:
- Thermal stress is caused by a change in temperature, leading to a change in dimensions of the material.
- The thermal coefficient of expansion (α) represents how much a material expands or contracts per degree change in temperature.
- The modulus of elasticity (E) measures the material's stiffness or ability to deform under stress.
- Poisson's ratio (μ) relates to the material's lateral and axial strains when subjected to stress.
Calculation:
Given the formula for thermal stress:
σ = αTE / (1 - 2μ)
Therefore, the correct answer is option 'B': αTE / (1 - 2μ), which represents the stress developed in the cube when heated uniformly.
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A cube having each side of length a, is constrained in all directions...
Since the cube is restrained in all directions,
⇒ εv = εx + εy + εz ⋯ ①
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A cube having each side of length a, is constrained in all directions and is heated uniformly so that the temperature is raised by T℃ . If α is the thermal coefficient of expansion of the cube material and E the modulus of elasticity, the stress developed in the cube isa) αTE/μb) αTE/(1 − 2μ)c) αTE/2μd) αTE/(1 + 2μ)Correct answer is option 'B'. Can you explain this answer?
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