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'In a linearly elastic structure, partial derivativeof the strain enrgy with respect to a load is equalto the deflection of the point where the load isacting, the deflection being measured in the direction of the load'. This is known as
- a)Castigliano's theorem
- b)Bernauli's theorem
- c)Work energy principal
- d)unit load method
Correct answer is option 'A'. Can you explain this answer?
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'In a linearly elastic structure, partial derivativeof the strain ...
The correct answer is option 'A', which refers to Castigliano's theorem. Castigliano's theorem is a fundamental principle in structural analysis that relates the partial derivative of strain energy with respect to a load to the deflection at the point where the load is applied.
Castigliano's theorem is derived from the principle of virtual work, which states that the work done by external loads in deforming a structure is equal to the strain energy stored in the structure. By differentiating the strain energy with respect to a particular load, we can determine the corresponding deflection.
Here's a detailed explanation of Castigliano's theorem:
1. Principle of Virtual Work:
The principle of virtual work states that the total virtual work done by external loads and internal forces acting on a structure in a state of equilibrium is zero. This principle is based on the concept of virtual displacements, which are hypothetical small displacements used to analyze the behavior of a structure.
2. Strain Energy:
Strain energy is the energy stored in a structure due to the deformation caused by external loads. It is a measure of the potential energy stored in the elastic material. The strain energy can be calculated using the formula:
U = (1/2) ∫(σ * ε) dV
where U is the strain energy, σ is the stress, ε is the strain, and dV is the differential volume element.
3. Castigliano's Theorem:
Castigliano's theorem states that the partial derivative of strain energy with respect to a load is equal to the deflection at the point where the load is applied. Mathematically, it can be expressed as:
∂U/∂P = δ
where ∂U/∂P is the partial derivative of strain energy with respect to the load P, and δ is the deflection at the point where the load is applied.
This theorem allows us to determine the deflection caused by a specific load by taking the partial derivative of the strain energy with respect to that load. It provides a powerful tool for analyzing and designing structures by relating loads to deflections.
In summary, Castigliano's theorem is a fundamental principle in structural analysis that relates the partial derivative of strain energy with respect to a load to the deflection at the point where the load is applied. It is derived from the principle of virtual work and provides a valuable tool for analyzing and designing linearly elastic structures.
Castigliano's theorem is derived from the principle of virtual work, which states that the work done by external loads in deforming a structure is equal to the strain energy stored in the structure. By differentiating the strain energy with respect to a particular load, we can determine the corresponding deflection.
Here's a detailed explanation of Castigliano's theorem:
1. Principle of Virtual Work:
The principle of virtual work states that the total virtual work done by external loads and internal forces acting on a structure in a state of equilibrium is zero. This principle is based on the concept of virtual displacements, which are hypothetical small displacements used to analyze the behavior of a structure.
2. Strain Energy:
Strain energy is the energy stored in a structure due to the deformation caused by external loads. It is a measure of the potential energy stored in the elastic material. The strain energy can be calculated using the formula:
U = (1/2) ∫(σ * ε) dV
where U is the strain energy, σ is the stress, ε is the strain, and dV is the differential volume element.
3. Castigliano's Theorem:
Castigliano's theorem states that the partial derivative of strain energy with respect to a load is equal to the deflection at the point where the load is applied. Mathematically, it can be expressed as:
∂U/∂P = δ
where ∂U/∂P is the partial derivative of strain energy with respect to the load P, and δ is the deflection at the point where the load is applied.
This theorem allows us to determine the deflection caused by a specific load by taking the partial derivative of the strain energy with respect to that load. It provides a powerful tool for analyzing and designing structures by relating loads to deflections.
In summary, Castigliano's theorem is a fundamental principle in structural analysis that relates the partial derivative of strain energy with respect to a load to the deflection at the point where the load is applied. It is derived from the principle of virtual work and provides a valuable tool for analyzing and designing linearly elastic structures.
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'In a linearly elastic structure, partial derivativeof the strain enrgy with respect to a load is equalto the deflection of the point where the load isacting, the deflection being measured in the direction of the load'. This is known asa)Castigliano's theoremb)Bernauli's theoremc)Work energy principald)unit load methodCorrect answer is option 'A'. Can you explain this answer?
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