This document provides an overview of engineering statics concepts related to force systems. It defines key terms like force, vector, moment, and couple. It also describes methods for analyzing both 2D and 3D force systems, including resolving forces into rectangular components, calculating moments and couples, and determining resultant forces and wrench resultants. The examples show how to use these methods to solve static equilibrium problems involving various force combinations and configurations.

1. Engineering Mechanics:
Statics
Chapter 2: Force Systems

2. Force Systems
Part A: Two Dimensional Force Systems

3. Force
 An action of one body on another
 Vector quantity
 External and Internal forces
 Mechanics of Rigid bodies: Principle of Transmissibility
• Specify magnitude, direction, line of action
• No need to specify point of application
 Concurrent forces
• Lines of action intersect at a point

4. Vector Components
 A vector can be resolved into several vector components
 Vector sum of the components must equal the original vector
 Do not confused vector components with perpendicular projections

5.  2D force systems
• Most common 2D resolution of a force vector
• Express in terms of unit vectors ,
Rectangular Components
F
x
y
i
x
F
y
F
j î ˆj
ˆ ˆ
cos , sin
x y x y
x y
F F F Fi F j
F F F F
 
   
 
2 2
x y
F F F F
  

1
tan y
x
F
F
 

Scalar components – can
be positive and negative

6. 2D Force Systems
 Rectangular components are convenient for finding the sum
or resultant of two (or more) forces which are concurrent
R
1 2 1 1 2 2
1 2 1 2
ˆ ˆ ˆ ˆ
( ) ( )
ˆ ˆ
= ( ) ( )
x y x y
x x y y
R F F F i F j F i F j
F F i F F j
     
  
Actual problems do not come with reference axes. Choose the most convenient one!

7. Example 2.1
 The link is subjected to two forces F1
and F2. Determine the magnitude and
direction of the resultant force.
   
2 2
236.8 582.8
629 N
R
F N N
 

1 582.8
tan
236.8
67.9
N
N
   
  
 

Solution

8. Example 2/1 (p. 29)
Determine the x and y scalar components of each of the three forces

9.  Unit vectors
• = Unit vector in direction of
cos direction cosine
x
x
V
V

 
Rectangular components
V
n
x
y
i
x
V
y
V
j
ˆ ˆ
ˆ ˆ
ˆ ˆ
cos cos
x y y
x
x y
V i V j V
V
V
n i j
V V V V
i j
 

   
 
n V
x
y
2 2
cos cos 1
x y
 
 

10. The line of action of the 34-kN force runs through the points A and B as
shown in the figure.
(a) Determine the x and y scalar component of F.
(b) Write F in vector form.
Problem 2/4

11. Moment
 In addition to tendency to move a body in the
direction of its application, a force tends to
rotate a body about an axis.
 The axis is any line which neither intersects
nor is parallel to the line of action
 This rotational tendency is known as the
moment M of the force
 Proportional to force F and the
perpendicular distance from the axis to
the line of action of the force d
 The magnitude of M is
M = Fd

12. Moment
 The moment is a vector M perpendicular to
the plane of the body.
 Sense of M is determined by the right-hand
rule
 Direction of the thumb = arrowhead
 Fingers curled in the direction of the
rotational tendency
 In a given plane (2D),we may speak of
moment about a point which means moment
with respect to an axis normal to the plane
and passing through the point.
 +, - signs are used for moment directions –
must be consistent throughout the problem!

13. Moment
 A vector approach for moment calculations is
proper for 3D problems.
 Moment of F about point A maybe
represented by the cross-product
where r = a position vector from point A to
any point on the line of action of F
M = r x F
M = Fr sin a = Fd

14. Example 2/5 (p. 40)
Calculate the magnitude of the moment about
the base point O of the 600-N force by using
both scalar and vector approaches.

15. Problem 2/50
(a) Calculate the moment of the 90-N force about
point O for the condition  = 15º.
(b) Determine the value of  for which the moment
about O is (b.1) zero (b.2) a maximum

16. Couple
 Moment produced by two equal, opposite, and
noncollinear forces = couple
 Moment of a couple has the same value for
all moment center
 Vector approach
 Couple M is a free vector
M = F(a+d) – Fa = Fd
M = rA x F + rB x (-F) = (rA - rB) x F = r x F

17. Couple
 Equivalent couples
 Change of values F and d
 Force in different directions but parallel plane
 Product Fd remains the same

18. Force-Couple Systems
 Replacement of a force by a force and a couple
 Force F is replaced by a parallel force F and a counterclockwise
couple Fd
Example Replace the force by an equivalent system at point O
Also, reverse the problem by the replacement of
a force and a couple by a single force

19. Problem 2/76
The device shown is a part of an automobile seat-
back-release mechanism.
The part is subjected to the 4-N force exerted at A
and a 300-N-mm restoring moment exerted by a
hidden torsional spring.
Determine the y-intercept of the line of action of the
single equivalent force.

20. Resultants
 The simplest force combination which can
replace the original forces without changing the
external effect on the rigid body
 Resultant = a force-couple system
1 2 3
2 2
-1
, , ( ) ( )
= tan
x x y y x y
y
x
R F F F F
R F R F R F F
R
R

     
       

21. Resultants
 Choose a reference point (point O) and move
all forces to that point
 Add all forces at O to form the resultant force R
and add all moment to form the resultant couple
MO
 Find the line of action of R by requiring R to
have a moment of MO
( )
=
O
O
R F
M M Fd
Rd M
 
   

22. Problem 2/87
Replace the three forces acting on the bent pipe by a
single equivalent force R. Specify the distance x from
point O to the point on the x-axis through which the
line of action of R passes.

23. Force Systems
Part B: Three Dimensional Force Systems

24.  Rectangular components in 3D
• Express in terms of unit vectors , ,
• cosx, cosy , cosz are the direction cosines
• cosx = l, cosy = m, cos z= n
Three-Dimensional Force System
ˆ ˆ ˆ
x y z
F Fi F j Fk
  
2 2 2
x y z
F F F F
  
î ˆj k̂
cos , cos , cos
x x y y z z
F F F F F F
  
  
ˆ ˆ ˆ
( )
F Fli mj nk
  

25.  Rectangular components in 3D
• If the coordinates of points A and B on the line of
action are known,
• If two angles  and f which orient the line of action
of the force are known,
Three-Dimensional Force System
2 1 2 1 2 1
2 2 2
2 1 2 1 2 1
ˆ ˆ ˆ
( ) ( ) ( )
( ) ( ) ( )
F
x x i y y j z z k
AB
F Fn F F
AB x x y y z z
    
  
    
cos , sin
cos cos , cos sin
xy z
x y
F F F F
F F F F
f f
f  f 
 
 

26. Problem 2/98
 The cable exerts a tension of 2 kN on the fixed bracket at A.
Write the vector expression for the tension T.

27.  Dot product
 Orthogonal projection of Fcosa of F in the direction of Q
 Orthogonal projection of Qcosa of Q in the direction of F
 We can express Fx = Fcosx of the force F as Fx =
 If the projection of F in the n-direction is
Three-Dimensional Force System
cos
P Q PQ a
 
F i

F n


28. Example
 Find the projection of T along the line OA

29.  Moment of force F about the axis through point O is
 r runs from O to any point on the line of action of F
 Point O and force F establish a plane A
 The vector Mo is normal to the plane in the direction
established by the right-hand rule
 Evaluating the cross product
Moment and Couple
MO = r x F
ˆ ˆ ˆ
O x y z
x y z
i j k
M r r r
F F F


30.  Moment about an arbitrary axis
known as triple scalar product (see appendix C/7)
 The triple scalar product may be represented by the
determinant
where l, m, n are the direction cosines of the unit vector n
Moment and Couple
( )
M r F n n
   
x y z
x y z
r r r
M M F F F
l m n
 
 

31. A tension T of magniture 10 kN is applied to the cable
attached to the top A of the rigid mast and secured to
the ground at B. Determine the moment Mz of T
about the z-axis passing through the base O.
Sample Problem 2/10

32.  A force system can be reduced to a resultant force and a resultant
couple
Resultants
1 2 3
1 2 3 ( )
R F F F F
M M M M r F
    
      

33.  The motor mounted on the bracket is acted on by its 160-N weight,
and its shaft resists the 120-N thrust and 25-N.m couple applied to it.
Determine the resultant of the force system shown in terms of a
force R at A and a couple M.
Problem 2/154

34.  Any general force systems can be represented by a wrench
Wrench Resultants

35.  Replace the two forces and single couple by an equivalent force-
couple system at point A
 Determine the wrench resultant and the coordinate in the xy plane
through which the resultant force of the wrench acts
Problem 2/143

36.  Special cases
• Concurrent forces – no moments about point of concurrency
• Coplanar forces – 2D
• Parallel forces (not in the same plane) – magnitude of resultant =
algebraic sum of the forces
• Wrench resultant – resultant couple M is parallel to the resultant
force R
• Example of positive wrench = screw driver
Resultants

37.  Replace the resultant of the force system acting on the pipe
assembly by a single force R at A and a couple M
 Determine the wrench resultant and the coordinate in the xy plane
through which the resultant force of the wrench acts
Problem 2/151