2. deformation is always a function of the distribution of
material in a cross section. For the axial bar the cross
sectional area is key; for torsion the polar moment of
inertia shows up; for flexure the moment of inertia about
the axis of bending is an important property. These
properties are essential to determining the deformation
under load.
Most of the classical “tricks of the trade” that can be used
in service of the computation of geometric properties
amount to making use of the properties of simple
geometric pieces (often rectangles) to build up the overall
cross sectional properties. What we show in this set of
notes is that it is possible to derive the integrals needed to
compute the cross sectional properties for a general
triangle and then use that basic foundation to create a
method to compute the cross sectional properties of any
closed polygonal region. This approach is based on two
key observations: (1) all integrals can be divided into
4. For beam theory we need to compute area and moment
of inertia (about the centroid). To compute the moment of
inertia we need the location of the centroid.
The area is simply defined as the sum of the infinitesimal
areas that make up the cross section:
y
z
x
h
c
ρ
dA
2
e
3
e
r
O
C
5. A
The centroid of an area is defined as follows:
1
A
dA
A
In other words, it is the constant vector c that is the
average over the area of the position r of the infinitesimal
areas. Finally, the moment of inertia (tensor) is defined as
Where the vector r is the distance from the centroid C to
the infinitesimal area of integration.
2
1 1 1 2
1 2 2
2 1 2 2
T
v v v v
v v
11. c c c
c c c
c c c c
Thus, we can compute
Therefore, we can compute the moment of inertia tensor
in the original coordinate system as
A
We need the centroidal distance c, and we need to figure
CEE 213—Deformable Solids
16. dA d d
dA d d
d d
n m n m
x h
h x
x
e n m
n m e
n m e
CP 2—Properties of Areas
Integration over the triangle Integration over the triangular
region. It will
be convenient to set up the area integration in the
17. coordinate system n and m (which happens not to
be rectangular).
dAdxn
dhm
( )
0 0
0 0
( ) sin ( )
sin ( )
h w
A
b
h
h
dA d d
d d
x
x
18. dA
x n
h m
( )
b
w
h
The variables of integration are x (measures
distance in n direction) and h (measures distance
in m direction), which are in the range
Note how the limits of integration are done in
the box at right.
21. CP 2—Properties of Areas
Area of the general triangle
For beam theory we need to compute
area, centroidal location, and moment
22. of inertia (about the centroid): 1
2
dA
x n
h m
( )
b
w
h
Area of the general triangle. We can compute area of
the triangle as follows:
Note how “one half base times height” still seems to be
things: (1) that the triangle is not a right triangle and (2)
if it turns out to be negative then the area is “negative.”
The area of the general triangle is
CEE 213—Deformable Solids
26. p r
n m
n m
n m
n m
n m
n m
A
CP 2—Properties of Areas
The centroid of the triangle
We will sum the contributions for
27. each triangle so we will not compute
c (the location of the centroid) for the
cross section until we add up all of
the p contributions.
dA
x n
h m
( )
b
w
h
The centroid of the triangle. We can compute the
integral of r over the triangle as follows:
Note that p is one third of the distance from the edges,
but adjusted by the general area A and measured in the
directions n and m. The location of the centroid of the
general triangle is
31. r r n m n m
A B C
A B C
A B C
A B
3
0
3 2 2 31 1 1
4 8 12
2 21 1 1
2 4 6
sin
h
b
h
32. b h b h b h
A h bh b
C
A B C
A B C
A n n
B n m m n
C m m
A
33. B C
It is convenient to define tensors
CP 2—Properties of Areas
outer product of r with itself for the triangular region as
follows:
Hence, we have the following result (which we will
eventually use to compute the moment of inertia)
These tensors are constant over the
triangle and therefore can be pulled out of
the integral.
Note that we will not compute the actual
moment of inertia about the centroid at
this point because we plan to add together
multiple pieces to construct the moment
of inertia for polygonal regions. So we
will simply compute (and accumulate) the
39. According to the definition of cross product, the angle is
clockwise from the vector n to the vector m. You can see
how this plays out in the two examples below (the
triangles for sides 1-2 and 2-3 for our example).
The sine function is positive in the first and
second quadrant and negative in the third and
fourth, and this gives the integrals their
algebraic sign. Also, as the vectors tend to point
along the same line the magnitude gets smaller
(to reflect that the region of the triangle gets
smaller).
n
m
n
m
n
42. 1
sin
c. Compute the triangle contributions
, , and
d. Add the contributions to the whole
, ,
4. Finish the computation of and
/ ,
5. Com
i i i
i i i
i i i
A A A
i i i
A dA dA dA
A A A
A A
43. n m e
p r R r r
p p p R R R
c J
c p J R c c
pute principal values of
6. Print and plot results
J
1
2
3
4
The Cross SectionO
45. inertia of the cross section.
The way we computed J had the first step of laying down
a coordinate system (y, z). We found the location of the
centroid C and that allowed us to nail down the
coordinate system (x,h), which is simply a translation of
the original coordinate system to pass through the
centroid. These two coordinate systems were probably
chosen because of some other aspect of the problem we
are trying to solve (e.g., bending about the y-axis). But,
as far as the properties of the cross section are concerned,
the choice is arbitrary.
An interesting question to ask is this: Could we pick a
coordinate system (x',h') in such a way that the values of
Jyy and Jzz are the biggest or smallest possible? The
answer is “yes” and the coordinate system we need
points in the direction of the eigenvectors of J. Similarly,
the actual max/min values are the eigenvalues of J.
MATLAB gives us a very easy way to compute the
eigenvalues of a matrix:
2
46. e
3
e
C
x
h
y
z
Jmax = eig(J)
The maximum and minimum values of the
inertia tensor J are its eigenvalues. These are
the values of J that we would compute in the
(x',h') coordinate system (if we knew what it
was in advance). For a symmetric cross section
J is diagonal and the the max/min values are
equal to the diagonal elements of J.
Guidelines for Doing Computing Projects
47. The Mechanics Project
Keith D. Hjelmstad
08.19.2015
Introduction
The Computing Projects are designed to promote a different
kind of thinking about me-
chanics and to provide a different mechanism for demonstrating
understanding of the sub-
jects. The each computing project culminates in a written
report.
The main purpose of the computing projects is to experience an
authentic and complete
cycle of engineering problem solving. Recitation and exam
problems (and virtually all
problems that you might find in the back of a textbook) are very
limited in scope, very
prescriptive in what is given and what is required, and generally
more about a snapshot of
the state of a system in time rather than the evolution of that
system (for dynamics). Com-
puting projects usually involve computations that are not
practical to do by hand. Once you
have a working code, you can execute it for different inputs
very easily. That opens up the
possibility of examining the effects of the variables and seeing
how a system responds over
a broad array of input values. That is a prerequisite to any
design problem. The way we
can get to this view is through numerical computations. That is
what the computing projects
are all about.
48. This brief guide is meant to give you an outline of what you
should do and how you should
report on what you have done. You should also, of course,
consult the document Evaluation
of Computing Projects to see exactly how the reports will be
viewed, by your peers during
the Peer Review process and by the Instructional Team at final
grading.
What should be in the reports?
The report should have the following sections:
1. Introduction. In this part of the report you present the
problem statement. You can
summarize from the assignment sheet, but it needs to be your
take on the assign-
ment.
2. Theory. You need to present the equations that your computer
program is based
on. If the equations were not given to you then you need to
include a derivation of
those equations. If you present an equation without derivation
give reference to
where you got it. Make sure that every symbol you use in your
equations is defined
somewhere in the text (usually immediately after it is
introduced). Make sure that
The Mechanics Project Guidelines for Doing Computing
Projects
49. 2
your equations meet the standards of report presentation (i.e.,
they must be typeset
using Microsoft Equation 3.0 or equivalent). Do not use
MATLAB code notation in
the presentation of the theory.
It should go without saying that the code has to work—and that
does not just mean
that MATLAB is no longer bugging you about syntax errors, but
that the program is
actually computing what you want it to—i.e., it is computing
numerically what the
theory sets out theoretically.
Not only does the code have to work, you have to prove that it
works. This process
is called verification of the code. Verification means that you
can demonstrate that
the code generates results that match the solution to the problem
done by some
independent approach (e.g., using the exact classical solution)
or matches the re-
sults in some independent source in the reputable literature.
Verification is probably the most important and transferrable
engineering skill that
you will learn through these projects. How do you know when
an answer is correct
if there is no result in the back of the book to check it with?
This is what engineers
do. There are many ways to go about verification that span from
the qualitative to
the quantitative. Pay attention to it; document it.
50. 3. Study and Results. Every report has to be about something.
The purpose of these
projects is not only to write a program, but also to use that
program to explore and
discover something about the problem at hand. It is not about
“getting an answer”
but developing and using an engineering tool. Describe what
your study is about
(there is a specific section in the grading rubric that asks if you
have done that).
In this part of the report you will run the program with different
inputs and you
will need to summarize what you find. Do not simply paste the
output of the pro-
gram or graphics produced by the program into the report. If
you have, for exam-
ple, studied how things change when you vary one of the
parameters, put your
results into a spreadsheet, generate a professional plot, and put
that plot into the
report. Synthesize information; look for trends.
In your report, label every table and figure. Table labels go on
top of a table. Label
them Table 1, Table 2, etc. Figure labels go below a figure.
Label them Fig. 1, Fig.
2, etc. You might want to include text in your table and figure
labels to describe
what they are about. It is always a joy to read a report that can
be largely under-
stood by looking at pictures are reading the captions to those
pictures. But, even if
you do use fairly complete captions, you still need to refer to
every table and figure,
51. by number, in the text of your report. If you use references then
provide a reference
list in an accepted bibliographic format.
Your study needs to meet standards of scientific and
engineering competence.
Don’t just make stuff up. Use good scientific and engineering
judgment and use
the process to develop your scientific and engineering
judgment.
The Mechanics Project Guidelines for Doing Computing
Projects
3
4. Conclusions. The study must have conclusions and they must
relate to the study
(there is a specific section in the grading rubric that asks if your
conclusions con-
nect to information provided in the report). The conclusions do
not have to be
lengthy, but they should say something coherent about the
problem at hand. This
is not a place to speculate, it is not that useful to invent future
applications. In
general, it is not a good idea to talk about how valuable the
learning experience
was—let’s just take that as a given. The conclusions should
emanate from the
study.
5. Appendix A—MATLAB code. Always include your complete
52. MATLAB code in an
appendix to the report. This will allow the reader to easily look
over your code.
The code should be nicely presented, with comments,
indentation, and other fea-
tures that help to make it readable. It is very easy to select your
entire .m file and
paste it into Word keeping the source format. You probably will
need to change
the font size to 8 point to get the code to fit. It is also a good
idea to never go
beyond the vertical line on the right side of the MATLAB
editor. If you stay on
the left of that line then when you paste your code you will not
have word wrap on
a line of code.
You should think about your program as something someone
else will read, not
just something that you understand because you suffered
through every line. Even
you will forget what you were thinking a few weeks later if the
code is not well
documented. And remember, plagiarism applies to computer
code just like it does
to writing. Do your own work.
6. Appendix B—Derivations. Include any detailed derivations in
an appendix, not in
the body of the report. For many computing projects, the
derivation of the theory
is part of a recitation problem. In such a case you can simply
put the derivation
from the homework in the appendix. If the derivations are
complete in the Course
Notes then it is sufficient to refer to those results in the text. No
53. need to include the
derivations, even as an appendix, if the results are available in a
reference docu-
ment.
If you have these six sections in your report then you are off to
a good start. You can have
more sections, but you should not have fewer. Write the report
so that someone who was
not a part of the process can make sense of it.
Is there a difference between complete and good?
The report has to be both complete and good and there are a few
things that will make it
good. “Complete” means you get everything done. It means the
theory is there, the code
works, you proved that the code works (the report should deal
specifically with verification
of the code—this is very, very important), you did at least the
computations called out in
the assignment document, and you wrote a report to document
what you did.
The Mechanics Project Guidelines for Doing Computing
Projects
4
“Good” is not the same as complete. Good means that your
derivations are very well done.
It means that your code is not a big mess. It means that you
don’t say nonsensical things
54. about how the world works (there is a specific part of the
grading rubric that concerns
scientific competence—that is what it means: you say things
that any self-respecting tech-
nical person can interpret correctly and those things are not at
odds with what is known
about the universe at this time).
The quality of the report is about how well you write, your
attention to important details,
and how good the report looks. The report needs to look like it
was done by a professional.
If you put a plot in your report make sure it meets professional
standards. If you include a
sketch, make sure it is neat, correct, and easily interpreted. The
purpose of graphics is to
communicate ideas that are more difficult to convey in words. If
the reader cannot make
heads or tails of your graphics then you have not communicated.
If there is missing infor-
mation then you have not communicated. If the information is
ambiguous then you have
not communicated.
Most of these things are common sense and the best rule is the
“golden rule”: do unto others
as you would have them do unto you. Would you want to read
your report? If you had to
reproduce the results in your report from the information in the
report, could you? One of
the main reasons for including peer review is to give the
opportunity to take on the role of
reader of reports. What you learn from reading the work of
others should help you to avoid
things that make it hard to read.
55. You can always seek feedback from the Instructional Team
about how to do the projects
and how to write the reports.
In-class Code Checks
As with any larger task, the computing projects will take time
and may require a bit more
management of your time than ordinary homework does.
Because computer programming
is involved you really have nothing until your program works
(and then you have pretty
much everything). In order to write a report that will get you the
credit you want and need,
you will have to get the programs working with enough time left
to do the exploration and
report writing required. So plan accordingly!
To encourage good time management, we will execute in-class
code checks in accord with
the dates listed in the syllabus and course schedule. The code
checks are scheduled during
recitation time. Each student will be asked to run their code for
someone on the instruc-
tional team. You will be granted a significant amount of credit
towards the overall grade
for the CP based upon this code check. To get full credit come
to that recitation with a
working code!
The code check is generally scheduled one week prior to the due
date for the report, fos-
tering an appropriate amount of time for exploration and
discovery.
56. The Mechanics Project Guidelines for Doing Computing
Projects
5
Peer Review in Critviz
Written reports as learning vehicles are substantially better if
the student has an opportunity
to get feedback and to improve the report based on that
feedback. Furthermore, students
can gain valuable perspective on the overall task of inquiry and
communication through
reviewing the work of others. You can see examples from others
of good things to do and
you can provide suggestions to others on how to improve the
product. It creates a more
democratic answer to the question “what is the instructor
looking for?”1 In this case you
can learn a great deal from each other.
We will use the mechanism of peer review as part of the
Computing Project process. Critviz
is a program for managing the peer review process. It allows the
submission of work, it
randomly distributes the work for peer review, and it displays
the outcome of the process.
The peer review process is anonymous (i.e., you should not
know who’s report you are
reviewing). To maintain anonymity you are responsible for
eliminating identifying infor-
mation on your report. Use your ASU posting ID as the
57. identifier on your work.
Each project gets five reviews. Having five reviews provides
redundancy in the process
and enhances the likelihood of generating valuable comments.
As a peer reviewer, you are
responsible for writing comments in response to the submitted
projects of your five as-
signed peers. You will also rank your five projects (1 is the
best, …, 5 is the “least best”).
Rank each project uniquely with no ties.
An average ranking is computed by Critviz and the top few
projects are selected on this
basis to be highlighted in by Critviz. The remaining projects are
also displayed, but in al-
phabetical order.
It is important to note that after the review process is complete
the projects and reviews of
all students are visible to all students in the class on the Critviz
website (but not to anyone
outside of our class). This public forum gives you the
opportunity to show your work and
it gives you the opportunity to see how others have interpreted
the assignment and the
requirements. This knowledge should help you to improve
future reports.
Submission and Peer Review Process
Here is how the submission and peer review process will work:
1. Each student completes a project report by the due date and
submits that report to
the Critviz website (www.critviz.com). Your report should be
58. anonymous for the
peer review process so use your ASU posting ID rather than
your name on your
documents and in your code. All projects are due in accord with
the course sched-
ule.
1 While an expert opinion might be more fully developed, your
opinions are also valuable. Further, taking on
the responsibility of providing a critique of the work of others
is a fantastic way to advance your own per-
spective of your own work and to develop expertise.
The Mechanics Project Guidelines for Doing Computing
Projects
6
Reports must be submitted in .pdf format!
It is a good idea to upload a version early so that you are
guaranteed to have some-
thing. You can always upload again. When you do that, Critviz
replaces the file
submitted earlier. Click on the file you have just uploaded to
make sure it is there
and to make sure it is the one you intended to submit.
2. Critviz automatically (and randomly) assigns the peer
reviews at a specific time.
Only students who submit projects by the deadline are included
in the peer review
59. process for that project. If you miss the submission deadline
then you missed the
boat for peer review.
Don’t risk being late. Critviz has a countdown clock to help you
out with that.
3. Each project report is distributed to five other students for
their review and com-
ment. The reviews are due by the date established in the course
schedule. Note:
Critviz closes the review phase at a specific, preset time. So
late reviews cannot be
accepted.
Note that if peer review is not part of the submission (it is not
done for every course in The
Mechanics Project), then it is not necessary to maintain
anonymity. In such a case, make
sure your name and posting ID are on your work.
What makes a good peer review?
The reviews should be informed by the rubric outlined in the
document Evaluation of Com-
puting Projects. That is the guide that the instructional team
will use. However, your task
as a peer reviewer is not to “grade” the project (i.e., there is no
need to provide a numerical
score for each rubric area).
The task of the peer reviewer is to write comments for each of
the three main areas. Your
main role as reviewer is to provide feedback on the report. The
best feedback will promote
actions on the part of the author to improve the report. As such,
60. subjective comments on
quality (like “this is great” or “this is not great”) are not that
helpful. Comments that give
specific suggestions or specific direction are better. As you
write your comments put your-
self in the author’s position of needing to read those comments
and actually do something
about it. Are you making implementable suggestions or are you
asking the impossible? Do
unto others as you would have them do unto you!
This peer review process is exactly the process used for articles
submitted to research jour-
nals. In the real world, this is how the community helps to
decide what is worthy of publi-
cation and what is not. Generally, the reviewer is an “expert” in
the field and is asked to
provide an “expert” opinion. In practice, every article you
review in that arena is a new
contribution to the state of the art, so no person is really expert
enough to have already
solved that problem. So if you think of yourself as somehow not
capable of providing peer
review of work because you are still learning, think again. We
are all learning, but we all
The Mechanics Project Guidelines for Doing Computing
Projects
7
know something. And through the process of providing your
best review you improve your
61. ability to review.
Peer review is valuable only if the quality of the reviews is
high. Therefore, we will gen-
erally associate a portion of the project grade with the quality
of your peer reviews (i.e.,
how you review others counts in your project grade). Quality
will generally be associated
with insightful and helpful comments. Assigning rubric points is
not quality. Cheerleading
is not quality. Implementable guidance is quality. The peer
reviews do not have to be long
to get full marks for peer review (but a few sentences is
probably not enough).
CEE 213—Deformable Solids The Mechanics Project
Arizona State University CP 2—Properties of Areas
1
Computing Project 2
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
62. 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 formu-
las 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
63. section with a sequence of
points numbered in a clockwise 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.
64. 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 2—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 eigen-
value solver in MATLAB (eig) on the moment of inertia matrix
65. J. See the CP
Notes for more information on this task.
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 loca-
tion of the centroid of the cross section in the graphic along
with lines defining the
principal axes.
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.
66. 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.
c. 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 there is only one
submission for this problem (the
67. final submission).
CEE 213—Deformable Solids The Mechanics Project
Arizona State University CP 2—Properties of Areas
3
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.
