Analyzing a String on Elastic Foundations Using Finite Element Methods

In summary, the conversation discusses a problem involving a string on elastic foundations with a distributed load and fixed ends. The strong form and weak form of the problem are given, along with the corresponding minimum principle. Galerkin's method and Rayleigh-Ritz method are used to formulate the problem, and the element stiffness and force matrices are derived. The exact solution and approximate solutions for different number of equal length elements are found and compared. The effect of changing the stiffness of the elastic foundation is also explored.
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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 he. Assuming linear shape functions and piece wise constant f(X) and p(x) in the elements ( i.e fe and pe are constant within the element e, Find the element stiffness matrix Ke and the element force matrix Fe. 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 uexactto 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 uexact 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 uexact 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.
 
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  • #2
Any help on this problem would be greatly appreciated

Thanks,
Troubled Finite Elements Student
 

1. What is a string on elastic foundation?

A string on elastic foundation is a physical model used to describe the behavior of a string or cable that is under tension and has a small amount of flexibility or elasticity. This model assumes that the string is resting on a surface that provides resistance to movement, such as a table or a pulley, and that the string is subject to an external force or load.

2. How does the string on elastic foundation model work?

The string on elastic foundation model works by using mathematical equations to describe the relationship between the tension in the string, the stiffness of the foundation, and the displacement or deflection of the string. This model takes into account the effects of both the external load and the elasticity of the foundation on the behavior of the string.

3. What are the applications of the string on elastic foundation model?

The string on elastic foundation model has many practical applications in engineering and physics. It can be used to analyze the behavior of suspension bridges, cables, and ropes under different loads and conditions. It is also commonly used in the design and analysis of musical instruments, such as guitars and violins, which rely on the tension and elasticity of strings for their sound.

4. What are the limitations of the string on elastic foundation model?

Although the string on elastic foundation model is a useful tool for understanding the behavior of strings and cables, it does have some limitations. For example, this model assumes that the string is perfectly flexible and does not account for any nonlinearities in the material. It also does not take into account any friction or damping effects, which can have a significant impact on the behavior of the string in real-world situations.

5. How is the string on elastic foundation model different from other string models?

The string on elastic foundation model is different from other string models, such as the simple string model or the wave equation, in that it considers the effects of the foundation or surface on which the string is resting. This allows for a more accurate representation of the behavior of real strings and cables, which are often subject to external forces and have some level of flexibility or elasticity. Additionally, this model can be used to analyze strings that are not in a straight line or have varying properties along their length.

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