1. FE/Graduate Seminar Review Notes
Department of Civil and Environment Engineering
CGN 4980/CGN 6939
FE/Graduate Seminar Review Notes
Fall 2005

2. TOPICS Forces Moments Equilibrium Trusses and Frames Plane Areas
Friction

3. A FORCE is caused by the action of one body on the another body
A FORCE is caused by the action of one body on the another body. Forces are described by:
Its point of application.
Its magnitude.
Its direction or two points along its line of action

4. Types of Forces Concurrent coplanar Non-Concurrent coplanar
Concurrent non-coplanar
Non-Concurrent non-coplanar
External forces
Internal forces

5. Representation of Forces
A vector is represented graphically by an arrow which defines the magnitude direction and sense

6. Vector Operations

7. Resultant of Coplanar forces
Cartesian Vector Notation
Component vectors
F1 = Fxi + Fyj
F1 = F1xi + F1yj
F2 = -F2xi + F2yj
F3 = -F3xi - F3yj

8. Resultant - Vector notation
Vector Resultant
FR = F1 + F2+ F3
= (FRx)i + (FRy)j
FR = F1 + F2+ F3
= (FRx)i + (FRy)j
Resultant - Scalar notation
FRx =  Fx
FRy =  Fy

9. Rectangular Components of a Vector
A = Ax + Ay+ Az
Unit vector
uA = A
=> A = A uA A
Cartesian vector representation
A = Axi + Ayj+ Azk
Magnitude of a cartesian vector

10. Direction of a Cartesian Vector
Unit vector
Important relations
A = AuA
= Axi + Ayj+ Azk

11. Position Vector Direction Vector
Position vector r is defined by
r = xi + yj+ zk
rA + r = rB
Position vector r is a fixed vector which locates a point in space relative to another point.
Direction Vector
Direction vector rAB is defined by
rAB = (xB – xA)i + (yB – yA)j + (zB – zA)k

12. 2 Multiplication by a scalar
Dot Product
A.B = AB cos  0
1 Commutative law
A.B = B.A
i.i = (1) (1) cos 0 o = 0
i.j = (1) (1) cos 90 o = 0
2 Multiplication by a scalar
a(A.B) = (aA).B =A.(aB) = a(B.A)
i.i = j.j = k.k = 1
i.j = j.k = k.j = 0
3 Distributive law
(to determine component of forces parallel and perpendicular to a line )
A.(B+D) = A.B + (A.D)
A.B = AxBx + AyBy + AzBz

13. Dot Product Applications
1) Angle formed between two vectors or lines
A// = |A| cos  u
A// = (A.u)u
A.B
A = A + A
A= A - A
2) Component of a vector // or perpendicular to a line
Projection of A along a line is the dot product of A and the unit vector u which defines the direction of the line

14. Dot Product Applications
F= (300j) N A B z 6m 2m 3m x y
Determine the magnitude of components of the force F, parallel and perpendicular to member AB
The magnitude of the component of F along AB is equal to the dot product of F and the unit vector uB, which defines the direction of AB.

15. 2 Multiplication by scalar A x B B x A
Cross Product
C = A x B
= AB sin 
Magnitude of C
C = AB sin 
Direction of C
C = A x B
=( AB sin  )uc
1 Commutative law
2 Multiplication by scalar
A x B B x A
a(A x B) = (aA) x B =A x (aB) = (B x A) a
A x B = - B x A
3 Distributive law
A x (B + D) = (A x B) + (A x D)

16. Cartesian Vector Formation
i x j = k i x k = -j i x i = 0
j x k = i j x i = -k j x j = 0
k x i = j k x j = -i k x k = 0
AxB = (Ax i + Ay j + Az k) x (Bx i + By j + Bz k)
A x B
NB : j element has a –ve sign

17. Equilibrium of Forces
1 Two dimensional Forces
 F = 0
 Fxi +  Fyj = 0
 Fx = 0
 Fy = 0
=>
2 Three dimensional Forces
 Fx = 0
Fy = 0
Fz = 0
 F = 0
 Fxi +  Fyj +  Fzk = 0
=>

18. Moment of a Forces -Vector
M0 = r x F
1) Magnitude
M0 = rF sin =
F( r sin )
= Fd
2) Direction determined by right hand rule
Cartesian Vector Formulation
M0 = rB x F
M0 = rC x F
Transmissibility
M0 = (ryFz – rzFy)i – (rxFz – rzFx)j + (rxFy – ryFx)k

19. Moment of a Forces -Vector
Resultant Moment of Forces -Vector
MRo = ( r x F)
Resultant Moment of Forces -Scalar
M0 = Fd
MR0 =  Fd

20. Moment of a Force About a Specific Axis
1 Scalar analysis
Ma = F da
da is the perpendicular or shortest distance from the force line of action to the axis
2 Vector analysis
Mb = ub . (r x F) Mb
Mb = Mb ub

21. Moment of a Couple
- A Couple is a pair of equal and opposite parallel forces
- Two Couples producing the same moment are equivalent
1 Scalar analysis
2 Vector analysis
M = F d
M = r x F

22. Equilibrium of a Rigid Body
Equilibrium in 2D

23. Equilibrium in 3D
F (x,y,z) = 0
M0(x,y,z) = 0

24. Structural Analysis
Assumptions
All loads are applied at the joints
Members are joined together by smooth pins

25. Structural Analysis
Method of Joints
Draw free body diagram
Establish sense of known forces
Orient x an y axes
Apply equilibrium equation
Start from one simple joint and proceed to others

26. Structural Analysis
Zero Force Members
If two members form a truss joint and no external load or support reaction is applied to the joint, the member must be zero-force member

27. Structural Analysis
Zero Force Members
If three members form a truss joint for which two of the members are co-linear, the third member is a zero-force member provided no external force or support reaction is applied to that joint,

28. Structural Analysis
Zero Force Members
(a) a
(b)
(d)
(c)
(e)

29. Structural Analysis
Method of Sections
Determine external reaction
Cut members through sections
where force is to be determined
Draw free body diagram
Apply equilibrium equations

30. Friction Impending Motion (static) F s = sN  s = sN
Motion (P > Fk kinetics)
F k = kN

31. Friction

32. Friction

33. Friction

34. Friction

35. Sample Problem
The rod has a weight W and rests against the floor and wal for which the coefficients of static friction are mA and mB, respectively. Determine the smallest value of q for which the rod will not move.

36. slipping must occur at A & B
Impending Motion at All Points FB W NB
slipping must occur at A & B
L sin q FA NA
Equilibrium Eqs.

37. Similar equations for line and volumes
Center of Gravity
Similar equations for line and volumes

38. Find the x and y coordinates of the centroid
Example
Find the x and y coordinates of the centroid
Find the Centroid

39. 1 2 3

40. Find the x and y coordinates of the centroid
Example
Find the x and y coordinates of the centroid 1 2 3

41. Polar Moment of Inertia

42. Moment of Inertia
Parallel axis theory
Polar Moment of Inertia
Radius of Gyration