# Solid mechanics, castiglianos second theorem

• kantti
In summary, the conversation discusses using Castigliano's second theorem to determine the deflection of a beam under an applied force. The formula to calculate the deflection is delta = dU / dP, where U is the strain energy. The conversation also touches on using this theorem to determine support reactions for a given structure.
kantti
http://img237.imageshack.us/i/lahaka.jpg/
http://img237.imageshack.us/i/lahaka.jpg/
So... I need to determine the deflection of the beam under applied force P due to flexural deformation using castiglianos second theorem.

Simply, delta = dU / dP

How do I determine the strain energy U for the given system?

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first calculate M and V ...from there calculate U using formulae...
as in one using moment is...
U=integral(0 to L) (M^2/2*E*L) dl...
similarly go wid V..
den differentiate...to get final displacement...

Now I'm stuck on a different problem. I want to determine the support reactions for given structure using the same theorem (see attachment). Axial deformations are considered negligible. This is what i got so far, but now I'm a bit clueless on how to continue. Be gentle, I'm lacking some necessary literature =)

#### Attachments

• _DSC0002.JPG
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well i can solve it...
but there is some kind of policy that for new problm u have to make a new thread...

## 1. What is Castigliano's second theorem in solid mechanics?

Castigliano's second theorem, also known as the principle of virtual work, is a method used to determine the deflection and deformation of a solid body under external loads. It states that the partial derivative of the strain energy with respect to the applied load is equal to the displacement of the point of application of the load in the direction of the load.

## 2. How is Castigliano's second theorem used in engineering?

In engineering, Castigliano's second theorem is often used in structural analysis to determine the deflection and stress of a structure. It is particularly useful for analyzing complex structures and non-linear materials, as it allows for the calculation of deflections and stresses without having to solve the entire system of equations.

## 3. What is the difference between Castigliano's first and second theorem?

Castigliano's first theorem, also known as the principle of complementary energy, is used to determine the deflection of a structure by considering the strain energy stored in the structure. On the other hand, Castigliano's second theorem is used to determine the deflection by considering the external work done on the structure by the applied loads. Both theorems are based on the principle of virtual work, but they use different approaches and assumptions.

## 4. Can Castigliano's second theorem be applied to all types of materials?

Castigliano's second theorem can be applied to most types of materials, including linear and non-linear materials. However, it is important to note that the theorem assumes that the material is elastic and follows Hooke's law, which may not be true for all materials. Therefore, it is necessary to verify the applicability of the theorem for the specific material being analyzed.

## 5. What are the limitations of Castigliano's second theorem?

One of the main limitations of Castigliano's second theorem is that it can only be used for small deformations. This means that it may not accurately predict the behavior of structures under large deformations or failure. Additionally, the theorem assumes that the material is isotropic, homogeneous, and follows Hooke's law, which may not be true for all materials. Therefore, it is important to carefully consider the assumptions and limitations when using this theorem.

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