CEE 213—Deformable Solids The Mechanics Project Arizona State University CP 1—Axial Bar

Computing Project 1 Axial Bar

The computing project Axial Bar concerns the solution of the problem of a prismatic bar in a uniaxial state of stress with a variable 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 or free at either end).

The theory needed to execute this project is contained in the set of notes (entitled CP 1— Axial Bar) that accompany this problem statement. Those notes provide an introduction to each aspect of the computation required to solve the problem. The general steps are as follows:

1. CP 1.1. Develop a routine based upon Simpson’s Rule to numerically integrate the applied loads and moments of the applied loads. This code segment will produce the quantities I0 and I1 that are mentioned in the CP1 notes. To get this part working it would be a good idea to get your code to integrate some functions that you can do easily by hand (e.g., the constant or linear functions). This step will be referred to as CP1.1, which is the first benchmark with an intermediate due date for this project. The main deliverable is the working code for Simpson’s Rule, verified for several functions.

2. CP 1.2. Develop a routine to set up and solve the system of equations that allow for the determination of the state variables (u and N) 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., bar fixed at one end with a uniformly distributed load). This step will be referred to as CP 1.2, which is the second benchmark with an intermediate due date for this project. The main deliverable is a working code that does the Simpson integration for I0 and I1 and then forms and solves the appropriate matrix equation to find the end state.

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 p, the axial displacement u, and the net axial force N as functions of x. Note that the number of generalized trapezoidal rule segments does not have to be the same as the number of Simpson segments. This step completes the code for CP 1. Include all three parts in the final report.

4. Structure your code so that you can easily change the loading function. Include simple load forms (constant, linear ramp up from left to right, linear ramp down from right to left, trapezoidal distribution, sinusoidal distribution, and a patch load over an interior part of the rod—from x=a to x=b). Verify, by hand calculation the code with a simple load form (e.g., uniform load). Use your collection of load functions to explore aspects of the problem. For example:

a. Explore how the distribution of load affects the displacement and internal variation of axial force.

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. If you find that the numerical integration rules are not accurate, increase the number of segments..

c. Explore how the state variables work to meet the boundary conditions (e.g., the slope of the displacement 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. Note that you can view the intermediate benchmark codes and reports and a single code and a single report to which you just keep adding information. So you are just doing one report, but you are handing in progress reports twice before the final submission.

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 re- view process for reports in this course.