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CEE 213—Deformable Solids The Mechanics Project
Arizona State University CP 3—Properties of Areas
1
Computing Project 3
Properties of Areas
The computing project Properties of Areas concerns the computation of various properties
of cross sectional areas. In each of our theories (i.e., axial bar, torsion, and beams) we arrive
at a point where we need certain properties of the cross section to advance the analysis. For
the axial bar we needed only the area; for torsion we needed the polar moment of inertia;
for the beam we will need moment of inertia of the cross section about the centroid of the
cross section.
We can develop an algorithm that allows the computation of all of the properties of a cross
section if the cross section can be described as a polygon. The algorithm is built on formulas
for the properties of a triangle. What that program will do is create a triangle from the
origin and the two vertex nodes associated with a side of the polygon. Whether this polygon
adds or subtracts from the accumulating properties will be determined from the vectors
defining the sides of the polygon (see the CP Notes for further clarification). If you loop
over all of the sides, the end result will be the properties of the entire cross section.
The general steps are as follows:
1. Develop a routine that allows you to describe the cross section with a sequence of
points numbered in a counterclockwise fashion starting with 1. The last point
should be a repeat of the first one in order to close the polygon. Some suggestions:
a. Store the (x,y) coordinates of each point in a single array x(N,2), where N
is the number of points required to describe the cross section (including
the repeat of the first point as the last point) and the first column contains
the x values of the coordinate and the second column contains the values
of the coordinate and the second column contains the y value.
b. It will eventually be a good idea to put the input into a MATLAB function
and call the function from your main program. That way you can build up
a library of cross sectional shapes without changing your main program.
c. If you need a negative area region (for a cutout section like in an open
tube) then number the points in that region in a counter-clockwise fashion.
Just keep numbering the vertices in order (no need to start over for the
negative areas).
2. Develop a routine to loop over all of the edges of the polygon and compute (and
accumulate) the contributions of the triangle defined by the vectors from the origin
to the two vertices of the current side of the triangle (that gives two sides) and the
CEE 213—Deformable Solids The Mechanics Project
Arizona State University CP 3—Properties of Areas
2
vector that points from the first to the second vertex (in numerical order). Calculate
the area, centroid, and outer-product contributions to the properties (see the CP
Notes for clarification of this issue).
3. Compute the orientation of the principal axes of the cross section using the eigenvalue
solver in MATLAB (eig) on the moment of inertia matrix J. See the CP
4. Create an output table (print into the Command Window) giving the relevant cross
sectional properties. Develop a routine to plot the cross section. Include the location
of the centroid of the cross section in the graphic (along with lines defining
the principal axes if you can figure out how to do that).
5. Generate a library of cross sections, including some simple ones (e.g., a rectangular
cross sections) to verify the code. Include in your library as many of the following
cross sections as you can get done:
a. Solid rectangle with width b and height h.
b. Solid circle of radius R.
c. Rectangular tube with different wall thickness on top and bottom.
d. I-beam with flange width b, web depth d, flange thickness tf, and web
thickness tw.
e. Angle section with different leg lengths and leg thicknesses.
f. Circular tube with outside radius R and wall thickness t.
g. T-beam.
6. Use the program to explore aspects of the problem. For example,
a. Why is it more efficient to use an open circular tube for torsion rather than
a solid cylinder?
b. For beam bending we can control deflections and reduce stresses with a
large moment of inertia about the axis of bending. Show the trade-offs
available in an I-beam when you can select different web and flange depths
and thicknesses. What is the ideal allocation of material? Why would we
never actually do that in practice?
c. Demonstrate that the principal axes of a symmetric cross section lie along
the lines of symmetry. You can do this by showing that the off-diagonal
elements of J are zero for symmetric sections with axes so chosen.
d. Explore any other feature of the problem that you find interesting.
CEE 213—Deformable Solids The Mechanics Project
Arizona State University CP 3—Properties of Areas
3
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 there is only one submission for this problem (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 review process for reports in this course.

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