Deformable solids matlab code and report

CEE 213—Deformable Solids The Mechanics Project Arizona State University CP 3—Plane Beam

Computing Project 3 Plane Beam

The computing project Plane Beam concerns the solution of the problem of a prismatic beam subjected to a variable transverse load. The goal is to write a MATLAB program that will allow the solution for a variety of load distributions and for all possible boundary conditions (i.e., fixed, simply supported, or free at either end).

The theory needed to execute this project is contained in the set of notes (entitled CP 3— Plane Beam) that accompany this problem statement. Those notes provide an introduction to each aspect of the computation required to solve the problem. This project is very similar to CP 1—Axial Bar, and it would be a very good idea to use the code for that project as the starting point for this one.

One of the things you might notice in the CP3 notes is that the sign convention on shear and transverse displacement are slightly different. It would be a good idea to stick with the sign convention in the CP3 notes for this project to assure that you do not mix sign con- ventions (both are acceptable, but mixing features of each is not). The sign convention manifests is the governing differential equations and in the interpretation of the plots at the end.

The general steps are as follows: 1. Develop a routine based upon Simpson’s Rule to numerically integrate the applied

loading terms that produce the quantities I0, I1, I2, and I3 that are mentioned in the CP3 notes. To get this part working dust off the Simpson code segment from CP1 and add the new pieces (i.e., I2, and I3). Verify the numerical integration by com- paring with integrals you can do by hand.

2. Develop a routine to set up and solve the system of equations that allow for the determination of the state variables (w, θ, M, and V) at both ends of the bar. This step will require some logic to make it work easily for different boundary condition cases (it should cover all of them). Debug your code with a problem that you can solve by hand (e.g., beam fixed at one end with a uniformly distributed load).

3. Develop a routine to integrate the governing equations from the left end to the right end using generalized trapezoidal rule to do the integration numerically. Store the results at each step along the axis and provide a plot of the applied load q, the transverse displacement w, the rotation of the cross section θ, the bending moment M, and the net shear force V as functions of x. A good way to verify this step is to just plot the functions and verify that they go to the right values at x=L.

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CEE 213—Deformable Solids The Mechanics Project Arizona State University CP 3—Plane Beam

4. Generate a library of load forms, including some simple ones (e.g., uniform load) to verify the code. Use the library to explore aspects of the problem. For example:

a. Explore how the distribution of load affects the displacement, rotation, moment and shear along the length of the beam.

b. Investigate a distributed load over a small segment of the bar (with no load over the rest of it) to see how a distributed load in the limit does the same thing as a concentrated load. Make sure to increase the amplitude of the distributed load as you decrease the length of its action so that the total force is equal to a fixed value (i.e., the value of the concentrated force you are comparing it to). What happens to the shear diagram? What does the moment diagram look like?

c. Explore how the state variables work to meet the boundary conditions (e.g., the slope of the rotation function should go to zero at a free end).

d. Explore any other feature of the problem that you find interesting.

Write a report documenting your work and the results (in accord with the specification given in the document Guidelines for Doing Computing Projects). Post it to the Critviz website prior to the deadline.

Please consult the document Evaluation of Computing Projects to see how your project will be evaluated to make sure that you can get full marks. Note that there is no peer review process for reports in this course.

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