REVIEW
Final Exam
1
Final Exam Information
DATE: Monday, Dec 14
TIME: 10:00 AM - 12:00 noon
ROOM ASSIGNMENT:
Butler-Carlton Hall
Room 125
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Final Exam Information
• Comprehensive exam covers all topics
on syllabus.
• Equivalent to taking two regular exams.
• Format is similar to what you have seen
on hourly exams throughout the
semester.
3
Strategy for Studying
• Review topics:
– From the first half of the semester.
– Topics that you are least comfortable with
• WORK homework problems and
review problems and examples from the
text.
4
Test taking strategy
• Get to bed by midnight the night before.
• Eat a good breakfast.
• Work the problems you CAN and go
back to the rest later.
• If you have time, take a two minute
mental break and check over the whole
exam once before turning in.
5
Sign Convention Rules
• Always use a right handed coordinate
system
• Designate the equation you are writing
and indicate positive direction.
• Use right hand rule when evaluating
moments.
6
Vectors
Types of vectors:
• Position vectors ( r )
• Unit vectors ( u )
• Vectors expressing a force ( F ) / moment ( M )
Vector operations:
• Addition / subtraction
• Dot product
– i*i + j*j + k*k
– Scalar answer (NO < i, j, k > components; just magnitude)
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Particle Equilibrium - 2D & 3D
Static Equilibrium Equations:
FX FY (for 2D)
FZ
(for 3D)
Solving these problems:
• Draw Free Body Diagram of point!
• Springs add additional equation FSP = ks
F = 0 (for equilibrium problems)
OR
F = FR (for resultant force)
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Moment Created by Forces
Static Equilibrium Equation:
MPOINT
Solving these problems:
• Draw A PICTURE of the system!
• Scalar notation (M = F*d):
– Make sure moment arm is  to
force
– Watch sign convention and
direction of moment created by
each component of each force.
9
3D Moment vectors
10
Moment Created by Forces
Solving these problems (…continued) :
• Vector notation ( M = r x F ):
– Watch direction of position vector ( r )
– r is always first in cross product.
• Moment about a line:
– Find moment about point ON line
– Then M = M  uline
11
Couple moments
Can sum moments for two forces creating a
couple at any point and the resulting moment
will be the same value.
12
Equivalent Systems
• Locate point of interest. If you have to find it, then use the problem to help
you find a preliminary “stand in” point.
• Sum forces (F = FR)
• Sum moments at point of interest (M = MR)
• If problem asks for resultant force ONLY, then find location (d) using
equation MR = FR*d
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Distributed Loads
• Reduce to a single force by
integrating the area under the load
diagram.
• Find location of resultant force
using the same method as for
equivalent systems
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Equilibrium of a 2D Rigid Body
Equilibrium Equations for 2D:
FX
FY
Mpoint
Solving these problems:
•
•
•
•
Identify all forces and support reactions
Draw Free Body Diagram of object!
Sum forces in x and y direction.
Sum moments at a point
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Trusses: 2D Rigid Bodies
• Consist entirely of 2-force members.
• Direction of forces in members act along
the direction of member (axially)
• Note axial direction of forces using (T) for
tension and (C) for compression.
• Analyze forces by cutting through internal
members
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Analyzing Trusses
• When cutting through members, this results in Free
Body Diagrams of:
– Single joint / pin connection
• Max of 2 unknowns
FX FY
– Section of structure (two or more joints)
• Max of 3 unknowns
FX FY Mpoint
• Always assume unknown forces to be in tension
(positive)  negative answer indicates compression
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Frames and Machines
• DO NOT consist entirely of 2-force members.
– Know how to identify 2-force members to
help eliminate unknowns
– DO NOT cut through members, but take the
structure apart at the connections.
• Direction of forces acting at connections is
unknown, so make an assumption.
• Make sure forces on contacting members act
in opposite directions.
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Analyzing Frames & Machines
• Can establish equilibrium for each Free Body
Diagram.
• Look for the most efficient path. Try to begin with a
FBD you can solve… or get as close as you can…
• You may have more than 3 unknowns on a particular
diagram, so just solve what you can and move on …
• …Using the process of elimination to solve unknowns.
• The TOTAL number of unknowns cannot exceed
the TOTAL number of equilibrium equations you
can write.
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Internal Structural Loads
• Multi-force members carry
– Moment load: bending (M) or torsional (T)
– Shear force (V)
– Normal force (N)
• To evaluate, cut through members
• “Expose” internal loads at the
cut…
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Types of Problems:
Evaluating Internal Loads
1. Cut through structure at specific point.
2. Divide structure into regions bounded by
discontinuities. Cut through structure within a
region at a random point ‘x’.
3.
Integrate load on beam (2x) and solve for
constants of integration.
4. Construct diagrams using “area method.”
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(does not give equations, just a graph)
Arbitrary Cut
•
Whenever there is a change in load, a discontinuity is created.
•
Divide beam into regions determined by discontinuity in loading.
•
The beam shown below has three distinct regions bounded by
(0 ≤ x < 3)
(3 ≤ x < 5)
(5 ≤ x < 8)
•
Set origin (x = 0) at left end of beam.
•
Cannot evaluate internal shear EXACTLY at the point of an applied
force. Must be just to the right or left of load.
Cut through
structure at
arbitrary
point.
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Drawing V & M Diagrams
• Plot directly under beam
• x = 0 at left end
• Plot shear first
• Then plot moment
• Shear and moment are
measured on y-axis
• Label each axis and
graph including
units
• Label all minimums and
maximums, including
points where shear
crosses x-axis.
• SHOW calculations used
to get values and draw
graphs.
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Six Rules for V & M Diagrams
1. Concentrated force creates a jump in the shear diagram
Creates downward jump
Creates upward jump
2. Change in shear equals area under load diagram
3. Slope of shear diagram equals value of distributed load
4. Change in moment equals area under shear diagram
5. Slope of moment diagram equals value of shear diagram.
6. Concentrated moment creates a jump in moment diagram
Creates downward jump
Creates upward jump
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Dry Friction (Fs = s N)
• We always evaluate the case for
impending motion
• # of unknowns = # of equil. eqns.
+ friction eqn.
• Dimensions must be given to
check for tipping.
Mo to check for tipping
•If (x < l / 2) then block will slide.
•If (x > l / 2) then block will tip.
x = Ph
W
• Position of applied force
influences how object responds
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Friction Wedges
impending
motion
• Uses the same friction
equation as for dry
friction – there are just
more pieces.
• Be very careful when
evaluating that you look
at the direction of
impending motion for
each piece.
• Free Body Diagrams
are absolutely
essential.
impending
motion
remains
stationary
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Belt Friction
T2 = T1e
• T2 is in the direction
of impending motion
• T1 opposes direction
of impending motion
•  = coefficient of
friction between belt
and surface of
contact
•  = angle of contact
between belt and
surface
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Finding Centroids
Calculate as a weighted average:
1. Calculate the first “moment” of each differential
element [weight, mass, volume, area, length] about an axis
2. Divide by total [weight, mass, volume, area, length]
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Using Single Integration
1)
2)
3)
4)
5)
6)
7)
DRAW an ‘element’ on the graph.
~ ~
Label the centroid (x,
y)
Label the point where the element intersects the curve (x, y)
Write down the general equation
Define each term according the problem statement
Determine limits of integration
Integrate
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Finding Centroids of Composite Shapes
1)
2)
3)
4)
5)
Divide the object into simple shapes.
Establish a coordinate axis system on the sketch
~ ~
Label the centroid (x, y) of each simple shape
Set up a table as shown below to calculate values
Subtract empty areas instead of adding them.
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First Theorem of Pappus & Guldinus
• The distance the centroid
travels for a surface area of
revolution will be in the
shape of a circle.
• Arc length for a circle is
defined as S = r 
y
x
As = LD
_
As = r  L
or for composite shapes:
_
TORUS: http://upload.wikimedia.org/wikipedia/commons/c/c6/Simple_Torus.svg
By GYassineMrabetTalk; This vector image was created with Inkscape. (Own work)
As =  (ri Li)
Second Theorem of Pappus & Guldinus
y
x
The volume of a body of
revolution (a solid)
equals the
generating area
times the
distance traveled by the
centroid of the area
while the body is being
generated.
V = AD
TORUS: http://upload.wikimedia.org/wikipedia/commons/c/c6/Simple_Torus.svg
By GYassineMrabetTalk; This vector image was created with Inkscape. (Own work)
Resultant Force for Fluid on Surface
Flat surfaces
Solve for perpendicular
resultant force
_
FR =  z A
or
FR = (w1+w2)(1/2)(L)
Curved surfaces
Solve for vertical and
horizontal components of
resultant force and THEN
find resultant.
FV =  (volume)
FH = FR
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