How Does a Bar's Flexibility Affect Spring Load Distribution?

[Infernorage]

Homework Statement

A very stiff horizontal bar, supported by four identical springs, as shown in the figure below, is subjected to a center load of 100N. What load is applied to each spring?

Repeat the previous problem, except assume that the horizontal bar as configured is not rigid and also has a spring constant of k.

2. The attempt at a solution
So I figured out the first question and the resulting equations were F1(force in the lower springs)=kx1=2/5*F and F2(force in the upper springs)=1/2kx1=1/5F. Just subbing in 100N for F, the force in the lower springs came out to be 40N and 20N in the upper springs.

My problem is with the second question. I am not really sure how to do the problem when the bar has a spring constant. I thought about just using the same equations I came to above, but instead of F being 100N it would have to be some equation involving the spring constant k and deflection, but I am not sure if that is the correct direction. If it is the correct direction, then what would the equation be for the force in that bar?

Thanks in advance for the help.

 

[Infernorage]

Can anyone help me out with this? The assignment with this question is due tomorrow morning so I need help quick.

 

[PhanthomJay]

I don't know what is meant by a spring constant for a horizontal beam, unless it implies that the beam deflects at its midpoint with a value equal to the deflection of an equivalent spring of spring constant k. In which case, it would be modeled like 3 springs in series at the top, effectively increasing the lower spring reactions, and decreasing the upper spring reaction, proportionally. But i don't know for sure.

 

[Infernorage]

I don't know what is meant by a spring constant for a horizontal beam, unless it implies that the beam deflects at its midpoint with a value equal to the deflection of an equivalent spring of spring constant k. In which case, it would be modeled like 3 springs in series at the top, effectively increasing the lower spring reactions, and decreasing the upper spring reaction, proportionally. But i don't know for sure.

Yea I couldn't really understand what it meant either. I was hoping someone would be able to clarify that. I think you might be right and it changes the problem by adding an additional spring to the upper springs in series, but I'm not sure.

 

[hotvette]

The standard interpretion of spring constant k relates force and deflection by the equation F = kx. In the case of a simply supported beam in bending with a center force, the deflection at the center is x = FL³/(48EI). Therefore, the beam spring constant would be k = 48EI/L³ and is determined by geometry only, just like a coil spring. For is problem, the details aren't important (i.e. you don't know the details of the formula for the coil spring constant either), but what is important is that you can model the flexible beam as just another coil spring with some spring constant k_b. Thus, it seems to me the net effect is adding a 3rd spring in series just above the rigid bar.