Statically Indeterminate Bar with alternating area and length

[hboyd3]

http://img197.imageshack.us/img197/5981/probie.png

So first I did a force balance (dF_x=0)

P=P_B-P_C=8.5kN

R_D= R_A - P
R_A = R_D + P

I know the total change in length is equal to 0, so

d_AB + d_BC + d_CD = 0

==>

(R_A*L₁)/(E*A₁) + (P*L₂)/(E*A₂) + ((R_A-P)*L₁)/(E*A₁) = 0

but when I do the algebra I can never get it to work. my numbers never match the answer no matter which way I approach it. I can do problems with only 2 different sections but this 3 sectioned one has me baffled.

the answers are
R_A= 10.5kN to the left
R_D= 2kN to the right

anyone have any insight?

 

[PhanthomJay]

You are assuming that the force in the middle segment is P. It is not. Cut a section (Free Body Diagram) thru the middle bar and determine the force in that bar as a function of the known and unknown variables.

 

[Mene]

I can offer some insight into your problem. First, it's important to understand the concept of a statically indeterminate bar. This means that the bar cannot be fully analyzed using only the equations of static equilibrium. In this case, the bar has three different sections with different areas and lengths, making it difficult to solve using traditional methods.

One approach to solving this problem is to use the method of virtual work. This method involves considering the bar as a whole and determining the total virtual work done on it. This virtual work should be equal to zero, as the bar is in equilibrium. From there, you can set up equations for the virtual work done by each section of the bar and solve for the unknown reactions.

Another approach is to use compatibility equations, which relate the changes in length of each section to the overall change in length of the bar. These equations can be used in conjunction with the equations of static equilibrium to solve for the reactions.

Regardless of the method you choose, it's important to carefully consider all the forces acting on each section of the bar and how they contribute to the overall equilibrium of the system. It may also be helpful to draw a free body diagram for each section to better visualize the forces at play.

In addition, it's important to double check your calculations and make sure all units are consistent. Small errors in calculation or unit conversions can lead to significant discrepancies in the final answer.

Overall, solving problems involving statically indeterminate bars can be challenging, but with careful consideration and application of principles, it is possible to arrive at the correct solution.