# Finding deflection and twisting using Castiglianos and Energy Methods

• Payam30
In summary, the conversation discusses a problem related to structural engineering and the use of Castigliano's theorem to find solutions. The person asking for help has provided a figure and their approach to solving the problem, but is unsure if it is correct. The expert suggests using Castigliano's theorem and provides six steps to follow. There is also a discussion about the given information and how to solve for different forces. Ultimately, the expert suggests seeking help from someone familiar with the class and offers to review the solution provided by the person asking for help.
Payam30
Hi,
This is not an homework. I'm doing a old exam question and have diffuclities. Since there is not a solution I want to get confirmed before I goes further with other similar question.
I have the figure belove.
The problem is here

I do like this. Since I should have some bending moment I insert a fictive bending moment which is made by the force N. Then I do draw the forces in different part and solve the system. When I have N then I use the relation N = k.d where K is the stiffness coefficient of the spring. and then I solve for d. Forces R1 and R2 är equal since N is in the middle of the sctructure. But if I insert the fictive force They take other values right? or am I wrong?

I wonder if this is the right wat to do these kind of problem. Please guide me! thanks :)

Your solution is most likely wrong and too simplistic, I think.

Did you apply that prescribed theta and the applied moment or was the part of the initial problem?

I'm assuming that those were NOT part of the initial problem.

I also assume that you were given more information than you are showing: e.x. E, I, L

Pretty sure you can't solve this problem without knowing such things because the structure is statically indeterminate (there could certainly be horizontal reactions at those pins)

Most likely, you are expected to use Castigliano's theorem in the following way:
1) Recognize that your structure is statically indeterminate to degree 1.
2) Change one of the pins to a roller, and replace the missing force with your own force, call it "H."
3) Find all of the moments in the frame as functions of x and y due to "H" and the applied load N (this is possible now since you have made your structure statically determinate).
4) Apply Castigliano's Theorem, which will involve taking the derivative with respect to "H," integrating your moments, etc. (you should know how to do this if you have learned Castigliano's theorem)
5) The motion in direction "H" is actually zero, so you use that constraint to obtain one equation, with one unknown, "H."
6) Since you now know "H," you can use static equilibrium to solve for any other forces of interest..

E.x. 1 at THIS LINK is a statically indeterminate problem of degree 2, if you want to study the procedure.

Last edited:
afreiden said:
Your solution is most likely wrong and too simplistic, I think.

Did you apply that prescribed theta and the applied moment or was the part of the initial problem?

I'm assuming that those were NOT part of the initial problem.

I also assume that you were given more information than you are showing: e.x. E, I, L

Pretty sure you can't solve this problem without knowing such things because the structure is statically indeterminate (there could certainly be horizontal reactions at those pins)

Most likely, you are expected to use Castigliano's theorem in the following way:
1) Recognize that your structure is statically indeterminate to degree 1.
2) Change one of the pins to a roller, and replace the missing force with your own force, call it "H."
3) Find all of the moments in the frame as functions of x and y due to "H" and the applied load N (this is possible now since you have made your structure statically determinate).
4) Apply Castigliano's Theorem, which will involve taking the derivative with respect to "H," integrating your moments, etc. (you should know how to do this if you have learned Castigliano's theorem)
5) The motion in direction "H" is actually zero, so you use that constraint to obtain one equation, with one unknown, "H."
6) Since you now know "H," you can use static equilibrium to solve for any other forces of interest..

E.x. 1 at THIS LINK is a statically indeterminate problem of degree 2, if you want to study the procedure.

Hi Thank you for your help. Theta is given at the place I showed you. E and I and L are given.and mass of the thin = m and g =9.81 I didnt actully understand what you ment by the steps you represented. :(

Anyone?

When I do as you say. I get a vertical force too. and I will have N, V and H. is N =mg?

Payam30 said:
When I do as you say. I get a vertical force too. and I will have N, V and H. is N =mg?

Not sure what N,V,H you are referring to, but the "N" I was talking about was consistent with your "N" from your first post - N=k*d.

If you are expected to give a solution in terms of the given angle, then you should do so, in which case you still do my six steps, but instead of "H," you'll have "M." This is sort've what you drew in your original post. However, you ought to split up your structure into two parts: the beam, and the column. Apply the M equal and opposite onto each piece. Go through my steps. Your final solution would be in terms of the angle change.

If you think that my method is too complicated, can you ask a real life person who is familiar with your particular class?

I will post my solution here so you can see if I do something wrong. becouse the total energy is too complicated now! and the derivation will not give a good solution.
I paste my solutions here so you can tell me where do I do wrong. Yes but I already send an email but not response yet.

Im expected to determine deflektion theta in the spring and the rotation in theta..where I showed it from beginning

Doesn't look like you're using Castigliano's method like you said in the title..
or did you just skip the step where you write down the integrals?
Regardless of what method you are using, however (Castigliano's method, moment distribution method..), you should be able to solve for everything in terms of theta and the spring deflection without splitting the structure into so many different segments.. why don't you just split it into two segments -- the beam and the column?

I don't know how to do that Could you please write down what you mean? in mathematical terms

Look at the link in my first post. If you take the time to understand that problem, then you will be able to do your problem.

afreiden said:
Look at the link in my first post. If you take the time to understand that problem, then you will be able to do your problem.

Im running ubuntu and it has some problem with the link. Can we talk in private message?

I just can't get it work sorry.!

Could you please just draw the figure you are thinking about?

H vill not give a moment in the x axis.. I don't know how you mean. and I saw the file... I will get problems

## 1. How do Castigliano's and Energy Methods differ in their approach to finding deflection and twisting?

Castigliano's method involves finding the first derivative of the strain energy with respect to the external load and setting it equal to zero. Energy methods, on the other hand, use the principle of virtual work and the equations of equilibrium to determine deflection and twisting.

## 2. Can Castigliano's and Energy Methods be used for all types of structures?

Yes, both methods can be applied to any type of structure as long as the equations of equilibrium and compatibility are satisfied. However, the complexity of the structure may affect the applicability and accuracy of the methods.

## 3. Is it necessary to consider both bending and shear effects when using these methods?

Yes, both methods take into account both bending and shear effects. Castigliano's method considers bending moments, while energy methods consider the total strain energy of the structure.

## 4. What are the assumptions made when using Castigliano's and Energy Methods?

The main assumptions for both methods include linear elastic behavior, small deformations, and a statically determinate structure. Additionally, Castigliano's method assumes that the deflection is a continuous function, while energy methods assume that the structure is homogeneous and isotropic.

## 5. Can these methods be used to find deflection and twisting in non-uniform or variable cross-sections?

Yes, both Castigliano's and Energy Methods can be applied to structures with non-uniform or variable cross-sections. However, the calculations may become more complex and may require the use of numerical methods to obtain accurate results.

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