1. REVIEW
Final Exam
Review_Final Exam

2. Final Exam Information
DATE: THURSDAY, MAY 14
TIME: 10:00 AM - 12:00 noon
ROOM ASSIGNMENT:
Butler-Carlton Hall
Room 315

3. 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.
Review_Final Exam

4. 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.

5. 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.

6. 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.

7. 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)

8. 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
Be able to do 3x3 matrix on calculator
F = 0 (for equilibrium problems) OR
F = FR (for resultant force)
3D: Often better to use vector notation

9. 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.

10. 3D Moment vectors

11. 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

12. Couple moments
Can sum moments for two forces creating a couple at any point and the resulting moment will be the same value.

13. 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

14. 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

15. Equilibrium of a 2D Rigid Body
Equilibrium Equations for 2D:
SFX SFY SMpoint
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
Don’t forget couple moments!

16. 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. DROP NEG. SIGN!!
Analyze forces by cutting through internal members

17. Analyzing Trusses SFX SFY SFX SFY SMpoint
When cutting through members, this results in Free Body Diagrams of:
Single joint / pin connection
Max of 2 unknowns
SFX   SFY
Section of structure (two or more joints)
Max of 3 unknowns
SFX      SFY  SMpoint
Always assume unknown forces to be in tension (positive)  negative answer indicates compression

18. 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.

19. Analyzing Frames & Machines
Write 3 equilibrium equations for each Free Body Diagram.
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.

20. Comparison Table Characteristics Trusses Frames & Machines
Consist of _______ members
2-force
Multiforce
FBDs
Cut through members
Take apart at joints
Assume direction of unknown is
always tension
Doesn’t matter (except for cables)

21. 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 so they become external
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22. Sign Convention for Internal Loads
Review_Final Exam 22

23. Types of Problems: Evaluating Internal Loads
Cut through structure at specific point.
Cut through structure at arbitrary point.
Integrate load on beam (2x) and solve for constants of integration (p ).
Construct diagrams using “area method.” (does not give equations, just a graph)
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24. Arbitrary Cut Cut through structure at arbitrary point.
Whenever there is a change in load, a discontinuity is created.
Divide beam into regions determined by discontinuity.
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.
Review_Final Exam 24

25. 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.
Review_Final Exam 25

26. Six Rules for V & M Diagrams
Concentrated force creates a jump in the shear diagram
Creates downward jump Creates upward jump
Change in shear equals area under load diagram
Slope of shear diagram equals value of distributed load
Change in moment equals area under shear diagram
Slope of moment diagram equals value of shear diagram.
Concentrated moment creates a jump in moment diagram
Creates downward jump Creates upward jump
Review_Final Exam 26

27. Dry Friction (Fs = s N) x = Ph W
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.
Position of applied force influences how object responds
x = Ph W

28. 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

29. 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
Review_Final Exam 29

30. Finding Centroids Calculate as a weighted average:
Compute the “moment” of each integral
element [weight, mass, volume, area, length] about an axis
Divide by total [weight, mass, volume, area, length]

31. Using Single Integration
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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32. Finding Centroids of Composite Shapes
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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33. Resultant Force for Fluid on Surface
Curved surfaces
Solve for vertical and horizontal components of resultant force and THEN find resultant.
Fv = (volume)
Fh = Fperp
Flat surfaces
Solve for perpendicular resultant force
Fperp = z A or
Fperp = (w1+w2)(1/2)(L) _