Calculate the rotation "theta" at O

[Cranky]

Homework Statement

a frame is composed as it is shown in the figure, each of length L. bending stiffness of AO is EI, of OB is 3EI. Force P is acting in the middle of AO, i.e at L/2. the question is to calculate the rotation "theta" at O.

Homework Equations

The Attempt at a Solution

the problem is statically undetermined.
to calculate the rotation we have to use castigliano's teorem and take the partial derivative with the respect to the ficticious moment M0 at point O, putting the M0=0:
theta=d/dMo(complementary elastic energy) at M0=0.

in order to get W we have to have 2 beams A0 and OB: (by superposition principle)
W= W(ao)+W(ob)

W(ao)=L/2EI*I (M1^2 dx)
W(ob)=L/6EI *I (M2^2 dx)

where I= integral from 0 to L
M1 and M2 internal moments in beams AO and BO respectively.

I am stuck on equillibrium equations:
in X dir: Rax-Rbx=0
in Y dir: Ray+Rby-P=0
and then there should be moment equillibrium also:
p.A: Mo+PL/2+M1 (??)=0 (by connecting points A and B)?
p.B Mo-M2=0 (?)
is there an internal moment near points A and B?

I'm mixed up about the equillibrium equation.Will be greatful for any help and suggestions

 

[anathema]

Thank you for your question. I understand your confusion regarding the equilibrium equations and internal moments near points A and B. Let me try to clarify it for you.

Firstly, it is important to note that the frame shown in the figure is a statically indeterminate structure. This means that the number of unknown forces and moments is greater than the number of equilibrium equations (3 in this case). Therefore, we need to use additional methods, such as Castigliano's theorem, to solve for the unknowns.

Now, let's look at the equilibrium equations. The first two equations you have written are correct, representing the equilibrium of forces in the x and y directions. However, the third equation you have written is not necessary. In fact, we do not need to consider the equilibrium of moments in this problem because the frame is simply supported at points A and B, meaning there are no external moments acting at these points.

Moving on to the internal moments near points A and B, these are known as point moments and are caused by the applied force P at the midpoint of beam AO. In order to solve for these moments, we can use the method of sections. By taking a section through the frame at point A, we can find the internal moment at that point, which can then be used to solve for the rotation at O using Castigliano's theorem.

I hope this helps clarify the problem for you. If you have any further questions, please do not hesitate to ask.
Scientist

 

[tomcue]

I would like to first clarify that I am not able to provide a solution to this specific problem without further information or context. However, I can provide some general guidance and suggestions for approaching this type of problem.

Firstly, it is important to clearly define the problem and identify any assumptions or simplifications that are being made. This will help in understanding the problem and determining the appropriate equations and methods to use.

Based on the given information, it seems that this problem involves a frame composed of two beams (AO and OB) of equal length, with different bending stiffnesses (EI and 3EI respectively). A force (P) is acting in the middle of beam AO, and the goal is to calculate the rotation (theta) at point O.

One approach to solving this problem is to use the principle of virtual work, which states that the work done by external forces on a system is equal to the change in complementary elastic energy of the system. This can be expressed mathematically as: W = delta Wc, where W is the external work and Wc is the complementary elastic energy.

In this case, the external work (W) can be calculated by considering the work done by the applied force P on beam AO, and the work done by the internal moments (M1 and M2) on beams AO and OB respectively. The complementary elastic energy (Wc) can be calculated using the bending stiffnesses and the internal moments.

As you mentioned, Castigliano's theorem can also be used to calculate the rotation at point O. This theorem states that the partial derivative of the complementary elastic energy with respect to a fictitious moment at a specific point is equal to the rotation at that point. In this case, the fictitious moment (M0) can be placed at point O and the partial derivative can be taken to calculate the rotation theta.

In terms of equilibrium equations, it is important to consider both force and moment equilibrium. The force equilibrium equations can be used to determine the reaction forces (Rax, Ray, Rbx, Rby) at the supports, while the moment equilibrium equations can be used to determine the internal moments (M1 and M2).

To determine the internal moments, you can use the moment equilibrium equations at specific points along the beams. For example, at point A, the moment equilibrium equation can be written as: M1 + PL/2 = 0, where M1 is the internal moment in