Statically indederminate bar undergoing axial load

[member 392791]

To be honest, this problem was gone over symbolically in my discussion session, but I didn't fully understand it. I am having difficulties using the superposition method (or even force method) to solve these problems where a load is applied on the bar and it is constrained at both ends. I don't know where to make the cuts.

I sort of get the idea that the purpose is to remove one constraint then solve for the elongation if 1 constraint was missing, then from there I am to use that elongation to say that the elongation of the reaction force of the removed constraint is the same, which should solve the reaction force there.

For example what I wrote in my solution, I can't tell how to determine which f's to add up for the elongations, meaning why was the elongation of the first one (f1 + f2 + 2f3), elongation 2 -(f2 + 2f3) etc. I don't know how these are determined.

 

[SteamKing]

It's not clear from your work how you came up with these solutions either.

I suggest that you analyze the problem starting with the left bar (i.e. make your cut at the interface between the two materials). Make a FBD of this bar and include all loads and reactions. You can then write an equation for the deflection at the interface. Do likewise for the bar on the right. You will have one equation of equilibrium from statics and a second equation which will equate the deflections at the interface of the two bars. That should give you enough information to solve for the reactions.

You have to solve these types of problems in a logical fashion. Don't skip steps, and don't ignore the equations of statics.

 

[Chestermiller]

To be honest, this problem was gone over symbolically in my discussion session, but I didn't fully understand it. I am having difficulties using the superposition method (or even force method) to solve these problems where a load is applied on the bar and it is constrained at both ends. I don't know where to make the cuts.

I sort of get the idea that the purpose is to remove one constraint then solve for the elongation if 1 constraint was missing, then from there I am to use that elongation to say that the elongation of the reaction force of the removed constraint is the same, which should solve the reaction force there.

For example what I wrote in my solution, I can't tell how to determine which f's to add up for the elongations, meaning why was the elongation of the first one (f1 + f2 + 2f3), elongation 2 -(f2 + 2f3) etc. I don't know how these are determined.

I don't know how to do the problem the way you described, but I do know how to do the problem. If R_A is the tensile force exerted on the left end of the bar by the wall at A and you make a cut on the bar at a location between A and B (such as the dotted cut you have shown in the figure), then you can do a free body diagram on the section of the bar between the cut and the left end. If T represents the force that the part of the bar to the right of the cut exerts on the free body to the left of the cut, how is T related to R_A? No matter where you made the cut between points A and B, T would have the same value.

Next, make a cut between points B and C. Now let your free body be the section of the bar between the left end of the bar and this second cut. (Forget about the first cut). Let F represent the force that the part of the bar to the right of this new cut exerts on the new free body to the left of the new cut. From your force balance on this free body, how is F related to the combination of R_A and the 60 kip force applied at location B?

I think by now you are beginning to get the idea. If you make a cut between points C and D, what is the tensile force to the right of this free body? If you make a cut between points D and E, what is the tensile force to the right of this free body?

You now know the tension in all for sections of the body in terms of R_A. Now, algebraically, in terms of R_A, what is the tensile stress in each of the four sections? From this, what is the tensile strain in each of the four sections? What is the displacement of the right end of each section relative to the left end of each section? In terms of R_A, what is the cumulative displacement of the right end of the entire body relative to the left end? But, under the constraints of the problem, this displacement must be zero. This gives you an equation for calculating R_A.

 

[member 392791]

I found the reaction forces, but now I am stuck for finding the deflection at a point. Is that the same as finding the elongation of bar CE or bar AC? Or do you just subtract those two values? Do I find the deflection of segments AB, BC, CD, and DE? Is it based on internal force?

 

[Chestermiller]

I found the reaction forces, but now I am stuck for finding the deflection at a point. Is that the same as finding the elongation of bar CE or bar AC? Or do you just subtract those two values?

No and no.

Do I find the deflection of segments AB, BC, CD, and DE? Is it based on internal force?

Yes. You find out how much each of these segments stretches, which is the same thing as the displacement of the right end of the segment relative to the left end of the segment. The internal force represents the tension in each segment (or compression if the internal force turns out to be negative). In this problem, the internal force is uniform in each of the four segments, since it doesn't matter where you made the cut in each of the segments.

You then add up the displacements of the individual segments to get the overall displacement of the right side of the composite bar relative to the left side of the composite bar. This must be zero, since the right side of the bar is constrained not to move relative to the left side. This gives you an equation for calculating the reaction force at A.

Chet