Analyzing a String on Elastic Foundations Using Finite Element Methods

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Homework Statement

Consider a string on elastic foundations as shown in the following figure:

[URL=http://img81.imageshack.us/i/imageqv.png/][PLAIN]http://img81.imageshack.us/img81/5255/imageqv.png[/URL][/PLAIN]

The Strong form is given by

[URL=http://img35.imageshack.us/i/image2yxq.png/][PLAIN]http://img35.imageshack.us/img35/2302/image2yxq.png[/URL][/PLAIN]

Where U(x) is the transverse displacement, p(x) is the stiffness of the elastic foundation, T is the tension in the string and f(x) is the distributed load. The string is fixed at both ends.


Questions

1) a. Write the weak form obtained from the Strong form.
b. Write the minimum principle describing the problem.

2) a. Using Galerkin’s method, formulate the corresponding problem
( Kd=F) and describe the properties of K.

Hint: In your formulations, the term p(x)u(x) should be the part of stiffness matrix

b. Show that a similar system is obtained from the Rayleigh-Ritz method, employing the functional given in 1b.

3) The string is discredited into elements of length h^e. Assuming linear shape functions and piece wise constant f(X) and p(x) in the elements ( i.e f^e and p^e are constant within the element e, Find the element stiffness matrix K^e and the element force matrix F^e. Use the natural coordinate system for this derivation.

Next, the following parameters are chosen: T =1, p(x)=1, f(x) =10, L=6

4) find the exact solution u_exactto the problem.( solve the problem analytically)

5) Subdivide the string into 2, 3 and 5 equal length elements. Assemble the elements and write down the global K and F for each case. Solve the system of equations and find the approximate solution vector d (The assembly should be done manually, however you can use MATLAB to solve the algebraic system of equations).


6. Plot the exact solution versus the three approximated solutions in one figure (use different colors, legend, and axis labeling).

7. Plot the derivative of the exact solution versus the derivative of the three approximated solutions in one figure.

8. What are your conclusions from questions 6 and 7? explain.

9. Assuming p(x) = 0 (no elastic foundations) and the other parameters are unchanged. Find the exact solution u_exact and repeat question 5. Compare the approximate solutions at element nodes with the exact solution – what do you get? explain. Repeat question 6 for this problem.


10- Assuming p(X) =10⁵ ( Very stiff elastic foundation) and other parameters unchanged.Find the exact solution u_exact and repeat question 5. Compare the approximate solutions at element nodes with the exact solution – what do you get? explain. Repeat question 6 for this problem.

Homework Equations

The Attempt at a Solution

Don't know how to solve this problem

If someone can solve these problems then please help me i have to submit this home work next Friday.

 

[cvm2126]

Any help on this problem would be greatly appreciated

Thanks,
Troubled Finite Elements Student