ME 211 Homework 8 - Centroids and Moments of Inertia (1)
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Date
Apr 3, 2024
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ME 211 Homework 8 - Centroids and Moments of Inertia
SMK Deliverables. Centroids. Moments of Inertia. Parallel-Axis Theorem.
50 points
To receive full credit for each (written, non-SMK) problem, include a short reflection at the end of your completed
work about what learning concepts you practiced and/or any questions you still have on it (e.g., if you have made a
reasonable effort but are unable to fully complete the solution).
Final SMK Checklist (25 points)
Mental Cutting Test
Retake the Mental Cutting Test on Canvas (
ME 211-03 Link
)(
ME 211-04 Link
). Instructions are given within the
assignment. The time limit is 25 minutes.
Statics Concept Inventory
Complete the Statics Concept Inventory on
Concept Warehouse
. There is no time limit, but it generally takes around
an hour to finish. This is also a good opportunity to review the course and identify topics to focus on for the final
exam.
Informed Consent and Demographics Survey
Complete the Informed Consent and Demographics Survey on Canvas (
ME 211-03 Link
)(
ME 211-04 Link
).
The
purpose of this survey is to collect some demographic information that may help the instructor better understand
how learning activities in this course work for different students.
This course is part of a larger research study
involving multiple colleges. The researchers are asking for permission to include some of your work from this class
in their data analysis.
Your name will not be shared with the researchers.
The choice to provide or not provide
permission will not affect your grade in this class in any way.
Problem 1 (10 points)
As discussed in class, cats are not a good substitute for structural beam elements. In any case, when studying general
loading scenarios of beams one of the first parameters to consider is the normal stress
σ
of a beam in bending.
For the coordinate system shown, the normal stress on the cross section for a beam bending along the
x
-axis is
σ
x
=
My
I
where
M
is the applied bending moment,
y
is the distance along the cross section, and
I
is the area moment of inertia
about the
z
-axis. A lower stress corresponds to lower bending deflection and failure rate in this bending direction.
For the purposes of this question, let us make a broad generalization that a higher area moment of inertia results
in a stronger beam (in reality, we would have to consider more factors such as the load distribution as well as the
specific point on the beam cross section we are analyzing).
Consider the following standard beam cross sections a) - e) and design your own for f), making sure to stay within
the bounds of an
a
×
a
square. Let
a
= 6 cm. Calculate
I
z
for each cross section and comment on what characteristics
might result in a stronger beam.
As an optional thought exercise, consider the stiffness-to-mass ratio of each cross section. This metric is important to
maximize the strength of a structure while minimizing weight, material usage, and cost. With certain assumptions,
the stiffness-to-mass ratio can be approximated as the ratio of area moment of inertia
I
z
to the cross sectional area
A
. Calculate the stiffness-to-mass ratio for each cross section and comment on the ways that certain cross sections
achieve a high stiffness-to-mass ratio.
Answer:
I
a
= 108 cm
4
,
I
b
= 63
.
6 cm
4
,
I
c
= 86
.
67 cm
4
,
I
d
= 51 cm
4
,
I
e
= 20 cm
4
Problem 2 (15 points)
For the composite area shown below, find the centroid (¯
x
c
and ¯
y
c
), the area moments of inertia (
I
x
and
I
y
), and the
centroidal moments of inertia (
¯
I
x
and
¯
I
y
).
Answer: ¯
x
c
= 0
.
391 in, ¯
y
c
= 0
.
191 in,
I
x
= 5
.
19e-3 in
4
,
I
y
= 1
.
78e-2 in
4
,
¯
I
x
= 1
.
22e-3 in
4
,
¯
I
y
= 1
.
22e-3 in
4
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