- #1

Tygra

- 19

- 4

- Homework Statement
- Solving a Continuous Beam

- Relevant Equations
- Fourth Order Differential Equation (FDM)

Hi all,

I am learning how to solve differential equtions using the finite diference method. In particular, for beams under a uniformly distributed load. For a simply supported beam this is quite easy. The boundary conditions are that at each end the displacement equal zero, and using the fourth order finite difference equation you construct the matrix. The FDM for the fourth order differential equation is:

For a 10m beam using 12 points, and applying the boundary conditions, the matrix looks like:

6 -4 1 0 0 0 0 0 0 0

-4 6 -4 1 0 0 0 0 0 0

1 -4 6 -4 1 0 0 0 0 0

0 1 -4 6 -4 1 0 0 0 0

0 0 1 -4 6 -4 1 0 0 0

0 0 0 1 -4 6 -4 1 0 0

0 0 0 0 1 -4 6 -4 1 0

0 0 0 0 0 1 -4 6 -4 1

0 0 0 0 0 0 1 -4 6 -4

0 0 0 0 0 0 0 1 -4 6

The fourth order differential is equal to the uniformly distributed load q(x). So the right-hand side is:

So, I have solved the displacements for a simply supported beam quite easily, but now I want to go further and use the FDM to solve for a continuous beam that say has 3 supports - pin support at one end with a roller support in the middle of the beam and at the other end.

There are 3 boundary conditions: The displacement is zero at each support. My question is: how do I construct the matrix using the FDM for this problem? I believe it would be called a multi-point boundary value problem. Everything I have searched only considers a two-point boundary value problem, like for the simply supported beam. I am wondering if anyone on here has knowledge on how to use the FDM for a boundary value problem for more than two points?

Many thanks in advance!

Tygra

I am learning how to solve differential equtions using the finite diference method. In particular, for beams under a uniformly distributed load. For a simply supported beam this is quite easy. The boundary conditions are that at each end the displacement equal zero, and using the fourth order finite difference equation you construct the matrix. The FDM for the fourth order differential equation is:

For a 10m beam using 12 points, and applying the boundary conditions, the matrix looks like:

6 -4 1 0 0 0 0 0 0 0

-4 6 -4 1 0 0 0 0 0 0

1 -4 6 -4 1 0 0 0 0 0

0 1 -4 6 -4 1 0 0 0 0

0 0 1 -4 6 -4 1 0 0 0

0 0 0 1 -4 6 -4 1 0 0

0 0 0 0 1 -4 6 -4 1 0

0 0 0 0 0 1 -4 6 -4 1

0 0 0 0 0 0 1 -4 6 -4

0 0 0 0 0 0 0 1 -4 6

The fourth order differential is equal to the uniformly distributed load q(x). So the right-hand side is:

So, I have solved the displacements for a simply supported beam quite easily, but now I want to go further and use the FDM to solve for a continuous beam that say has 3 supports - pin support at one end with a roller support in the middle of the beam and at the other end.

There are 3 boundary conditions: The displacement is zero at each support. My question is: how do I construct the matrix using the FDM for this problem? I believe it would be called a multi-point boundary value problem. Everything I have searched only considers a two-point boundary value problem, like for the simply supported beam. I am wondering if anyone on here has knowledge on how to use the FDM for a boundary value problem for more than two points?

Many thanks in advance!

Tygra