Axial Deformation on Statically indeterminate beam

[SteliosVas]

Homework Statement

I am trying to find a solution to the below attached image.
The area for AB=CD = 0.8m^2
Area of BC = 0.3m^2

Force at B = 140 kN (left)
Force at C = 60 kN (right)

AB=CD = Concrete with a E of 25,000N/mm^2
CD = Steel with an E = 200,000N/mm^2

Homework Equations

PL/AE

The Attempt at a Solution

This is where I get confused because there is a force acting at B and a force acting at C I am not sure if I need to make a cut at say B or make a cut at C.

I understand the the sum of the forces in X direction have to equal Ax + Dx = 80 kN (As a result of 140 - 60 kN)

Now taking BC as the redundant section

PL/AE for section AB should that of section CD?

I am not sure if I am going about it the right way?

If it was one continuous section with forces only in one direction or even two I understand but because of that section BC it has thrown me out .

 

[SteamKing]

There's no image attached to your post.

 

[SteliosVas]

Oops... Here we go

 

[SteliosVas]

Second Attempt at solution

Taking Section AB

δA/D = Fab*L(ab)/AE = Fcd*L(cd)/AE

Now F in AB = 140 kN
and F in CD = -60 kN

*** Edit ***

My compatibility equation of Fab + Fcd = 80 kN?

 

[rezeew]

I would advise approaching this problem by considering the static equilibrium of the beam. This means that the sum of all forces acting on the beam must equal zero, and the sum of all moments (torques) acting on the beam must also equal zero.

First, let's consider the forces acting on the beam. We have a force of 140 kN acting to the left at point B, and a force of 60 kN acting to the right at point C. These forces create a resultant force of 80 kN to the left, as you mentioned. Since the beam is statically indeterminate, we cannot determine the internal forces (such as Ax and Dx) simply by looking at the external forces. Therefore, we need to consider the beam's deformation to find a solution.

To do this, we can make a cut at point B or C and consider the forces and moments acting on each section. Since we have more information about the section AB, let's make a cut at point B. This means that we are considering the forces and moments acting on section AB and the forces acting on section BC.

For section AB, we can use the equation PL/AE to determine the deformation caused by the 140 kN force. However, we also need to consider the effect of the force at point C on the deformation of section AB. This can be done by considering the equilibrium of forces and moments at point B. This will give us an equation that relates the forces at B and C to the deformation of section AB.

For section BC, we can use the same equation PL/AE to determine the deformation caused by the 60 kN force. Keep in mind that this section is made of steel with a different Young's modulus, so you will need to use the appropriate value in the equation.

Once you have equations for the deformations of both sections, you can use them to solve for the unknown internal forces (Ax and Dx) by setting the total deformation of the beam equal to zero. This means that the deformations of section AB and BC must cancel each other out, resulting in a net deformation of zero for the entire beam.

In summary, the key to solving this problem is to consider the equilibrium of forces and moments at various points along the beam, and to use the equation PL/AE to relate these forces to the deformations of each section. I hope this helps guide you in the right direction.