Review for Exam II

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Review for Exam II Moment of Inertia

• ENGR 221
• March 31, 2003

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Lecture Goals

• Review
• Parallel Axis Theorem
• Composite Bodies

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2-D Trusses
Understand method of sections and method of
joints.
Free-body diagram, find reactions, make the
section and find the stresses. Always draw the
force in tension and let sign decide if the
member is in tension or compression.
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3-D Trusses
Use method of joints and sections
Draw the free-body diagram, find reactions, make
the section and find the stresses. Always draw
the force in tension and let sign decide if the
member is in tension or compression.
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Frames and Mechanisms
Determine the forces by doing disassembly of the
frame or machine, draw the free-body diagram.
Determine if members have multiple forces (shear
and axial loads, moments). Solve for the forces.
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Shear and Bending Moment Diagrams
Draw free-body diagram Draw the shear diagram
using the force to determine the slope on the
shear diagram. (Point loads and distributed loads
(uniform, ramp)) Draw the moment diagram using
the magnitude of shear from the shear diagram to
determine the slope on the moment diagram and
concentrated moments are considered point loads.
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Friction
Draw the free-body diagram with the direction of
motion. (friction is in the opposite direction).
Consider motions in both directions. Determine
the magnitude of the force required to find
impending motion. Check to see if the member is
going to tip, by taking the moment about any
point on the body to find where the normal force,
N, is going to be located. If N is under the
body, it is stable and will slide. If N is not
under the body, it will tip over.
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Moment of Inertia - Parallel Axis Theorem
Parallel axis theorem Consider the moment of
inertia Ix of an area A with respect to an axis
AA. Denote by y the distance from an element of
area dA to AA.
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Moment of Inertia - Parallel Axis Theorem
Consider an axis BB parallel to AA through the
centroid C of the area, known as the centroidal
axis. The equation of the moment inertia becomes

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Moment of Inertia - Parallel Axis Theorem
The first integral is the moment of inertia about
the centroid.
The second component is the first moment area
about the centroid
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Moment of Inertia - Parallel Axis Theorem
Modify the equation obtain the parallel axis
theorem.
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Example Moment of Inertia
Compute the moment of inertia in the x about the
AA plane.
AA
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Example Moment of Inertia
Compute the moment of inertia in the x about the
AA plane.
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Example Moment of Inertia
From earlier lecture, the moment of inertia about
the centroid
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Example Moment of Inertia
Using the parallel axis theorem
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Parallel Axis - Why?
Recall that the method of finding centroids of
composite bodies? - Table technique How would
you be able to find the moment of inertia of the
body. Use a similar technique, table method, to
find the moment of inertia of the body.
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Parallel Axis - Why?
Use a similar technique, table method, to find
the moment of inertia of the body.
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Moment of Inertia
Use a set of tables
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Example - Moment of Inertia
Find the moment of inertia of the body, Ix and
the radius of gyration, kx (rx)
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Example - Moment of Inertia
Set up the reference axis at AB and find the
centroid
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Example - Moment of Inertia
From the table to find the moment of inertia
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Example - Moment of Inertia
Compute the radius of gyration, rx.
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Class Problem - Moment of Inertia
Find the moment of inertia of the body, Ix and
the radius of gyration, kx (rx)
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Example - Moment of Inertia
Find the moment of inertia of the body, Ix and
the radius of gyration, kx (rx)
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Example - Moment of Inertia
The components of the two bodies and subtract the
center area from the total area.
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Example - Moment of Inertia
Compute the radius of gyration, rx.
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Class Problem - Moment of Inertia
Find the moment of inertia of the body, Iy and
the radius of gyration, ky (ry)
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Moment of Inertia (Tables)
Using tables of steel items, I-beams, C-sections,
L-beams, the centroid and moment of inertia can
be computed.
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Example - Moment of Inertia
Find the moment of inertia of the body, Ix and
the radius of gyration, kx (rx). If the beam is
a W18 x76 and 12 in. x 1.00 in. plate.
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Example - Moment of Inertia
Set the reference axis at the bottom.
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Example - Moment of Inertia
Compute the radius of gyration, rx.
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Example - Moment of Inertia
Find the moment of inertia of the body, Ix and
the radius of gyration, kx (rx). If the beam is
composed of 4 x (L 5 x 3-1/2 x 3/4) and 32 in. x
7/16 in. plate.
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Homework (Due 4/7/03)
Problems
10-34,10-37,10-39,10-43