Castiglianos second theorem for indeterminate structures

In summary, the question is asking to find the reaction at C and internal force Q of the rod BD, which only has axial internal forces (Q) and AL2=10*I. The answers are supposed to be Cy=0.346P and Q=0.042P. The structure is twice statically indeterminate, so 2 equations are needed to find the 2 unknowns. The calculations have been attempted 4 times without getting the correct answer. L/4 has been defined as a to avoid confusion. The assumption is made that the energy of normal forces is negligible compared to moments.
  • #1
Dell
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in the following question it is known that the rod BD has only axial internal forces, (which i called Q) if AL2=10*I, find the reaction at C and the internal force Q.
Assume that the energy of normal forces is negligible compared to moment where there are both.

question.PNG

instead of using L/4 i defined L/4=a

the correct answers are supposed to be Cy=0.346P and Q=0.042P

these are my calculations

i have tried solving it 4 times now and can't get the correct answer.
001.jpg

002.jpg
 
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  • #2
the structure is twice statically indeterminate therefore i need to use 2 equations to find the 2 unknowns
 

1. What is Castigliano's second theorem for indeterminate structures?

Castigliano's second theorem for indeterminate structures is a method used to determine the partial derivative of the total strain energy of a structure with respect to a particular force or displacement. It is commonly used in structural engineering to analyze the behavior of indeterminate structures.

2. How is Castigliano's second theorem applied in real-world scenarios?

Castigliano's second theorem can be applied in real-world scenarios by using it to calculate the deflection of a structure under a specific load or to determine the internal forces and moments in a structure. It is commonly used in the design and analysis of bridges, buildings, and other structures.

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

Castigliano's first theorem is used to calculate the deflection of a structure under a particular load, while the second theorem is used to determine the partial derivative of the total strain energy with respect to a specific force or displacement. The first theorem is applicable to both determinate and indeterminate structures, while the second theorem is mainly used for indeterminate structures.

4. What are the assumptions made in Castigliano's second theorem?

The main assumptions made in Castigliano's second theorem are that the material of the structure is linearly elastic, the structure is statically determinate, and the deflections are small. These assumptions help simplify the calculations and provide accurate results.

5. Are there any limitations to using Castigliano's second theorem?

Yes, there are some limitations to using Castigliano's second theorem. It is only applicable to linearly elastic materials and small deflections. It also does not account for the effects of material nonlinearity, geometric nonlinearity, and large deformations. Additionally, it may not provide accurate results for structures with highly nonlinear behavior.

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