Beam with uniformly distributed springs

[TheFerruccio]

Homework Statement

Suppose you have a beam with a transverse load P pointing down in the center, directly over the middle "spring". This beam is supported by 9 uniformly distributed springs of constant k. Compute the loads of the springs.

Homework Equations

You know that the problem is symmetric, and that the ends of the beam go off to infinity. You, thus, do not know the displacement of the endpoints. You know the spacing of the springs. This beam is assumed to have no mass.

The Attempt at a Solution

Basically, I have been racking my brain over how to set up the system of equations.

Since the problem is symmetric, I know that there are only 5 unknowns (spring loads 1 2 3 4 = springs 9 8 7 6, respectively)

I know that, since the beam is symmetric, the slope in the middle is 0. If I deconstruct the beam into halves, I can combine the loads of 4 of the springs to an initial shear force where the load is being applied, as well as an initial moment.

So, equation 1: slope in the middle is 0.

I need 4 more equations, but I do not know where to retrieve them from. In this case, there are no points where the slope is 0, or where the displacement is 0. I cannot find any points where the shear is 0, or where the moment is 0.

How do I find further boundary conditions to solve this?

 

[SteamKing]

You should state the problem is a concise fashion, in accordance with the homework template. It is very hard to follow what is given in the way it is expressed above.

With this admonition out of the way, it appears that what you have is a special case of an infinite beam on an elastic foundation.

The following article discusses some approaches for solution:

http://www.me.ust.hk/~meqpsun/Notes/Chapter4(202).PDF

See the example on p. 12

The solution to beams on elastic foundations is modified somewhat from the usual beam analysis. Read over the first part of the article for an illustration of the general procedure.

You can always google 'beams on elastic foundations' for more articles.

 

[TheFerruccio]

You should state the problem is a concise fashion, in accordance with the homework template. It is very hard to follow what is given in the way it is expressed above.

With this admonition out of the way, it appears that what you have is a special case of an infinite beam on an elastic foundation.

The following article discusses some approaches for solution:

http://www.me.ust.hk/~meqpsun/Notes/Chapter4(202).PDF

See the example on p. 12

The solution to beams on elastic foundations is modified somewhat from the usual beam analysis. Read over the first part of the article for an illustration of the general procedure.

You can always google 'beams on elastic foundations' for more articles.

This is not a case of smearing springs. I already solved that problem. This is specifically a case of solving for the forces and displacements of the individual springs using classical methods that do not involve a continuous elastic foundation.

Since I am unable to make custom pictures, there is only so much I can do to describe this. I will try again.

Horizontal beam of length 8c, properties EI. Springs of constant k under the beam distributed at 0, c, 2c, 3c, 4c, 5c, 6c, 7c, 8c. Downwards point load located at 4c. The ends are free. I need to solve this classically without superposition or without an elastic foundation.

 

[SteamKing]

Your original post said, "You know that the problem is symmetric, and that the ends of the beam go off to infinity. You, thus, do not know the displacement of the endpoints. You know the spacing of the springs. This beam is assumed to have no mass." That's why I pointed you to the article I did. Beams of infinite length are treated differently from beams with finite length. Discrete springs are a limiting case for continuously distributed support.

Now, if you have a beam of finite length with a finite number of supports > 2, you've got a continuous beam. There are several different approaches you can take to solve for the reactions. This article illustrates the Three-Moment Theorem with a beam on spring suppports:

http://www.facweb.iitkgp.ernet.in/~baidurya/CE21004/online_lecture_notes/m2l13.pdf

In regular continuous beam analysis, the deflection at the supports is taken to be zero. With spring supports, the reaction force is proportional to the deflection of the spring.

 

[rezeew]

I would approach this problem by first defining the variables and assumptions. The variables in this problem are the transverse load P, the spring constant k, and the loads of the individual springs. The assumptions are that the beam is symmetric, the ends of the beam go off to infinity, and the beam has no mass.

Next, I would draw a free body diagram of the beam to visualize the forces acting on it. This would help me identify the points where the shear and moment are zero, as well as the points where the slope and displacement are zero. From the problem statement, we know that the load P is applied at the center of the beam, directly over the middle spring. We also know that the beam is supported by 9 uniformly distributed springs, so the loads of the springs can be assumed to be evenly distributed.

Using the free body diagram, I would then apply the equations of static equilibrium (sum of forces and sum of moments equal to zero) to determine the remaining equations needed to solve for the spring loads. Since the beam is symmetric, we can split it into two halves and only consider one half for our calculations. This would give us a total of 6 equations (3 for force equilibrium and 3 for moment equilibrium) and 6 unknowns (loads of 4 springs and the reaction force at the support).

Finally, I would solve the system of equations using algebra or numerical methods to find the loads of the individual springs. It may also be helpful to check the solution by substituting the calculated loads into the equations of equilibrium to ensure that they satisfy the conditions of the problem.

In summary, to solve this problem, we need to use the equations of static equilibrium and apply them to a free body diagram of the beam. This will give us the necessary equations to solve for the spring loads, taking into account the symmetry and assumptions of the problem.