Engineering Mechanics: Statics
Force System Resultants
Chapter Objectives
To discuss the concept of the moment of a force and show how to calculate it in 2-D and 3-D systems.
Definition of the moment of a couple.
Chapter Objectives
• To present methods for determining the resultants of non-concurrent force systems in 2D and 3D systems.
• Reducing the given system of forces and couple moments into an equivalent force and couple moment at any point.
Chapter Outline
Moment of a Force
Force Couple
Principles of Moments
Moment of a Force
Carpenters often use a hammer in this way to pull a stubborn nail.
Through what sort of action does the force FH at the handle pull the
nail? How can you mathematically model the effect of force FH at
point O?
Moment of a force about a point (or an axis) is
a measure of the tendency of the force to
cause a body to rotate about the point or axis.
Moment of a Force 2D System
Case 1:
Fx horizontal and acts perpendicular to
the handle of the wrench and
is located dy from the point O
Fx tends to turn the pipe about the z axis
The larger the force or the distance dy the greater the turning effect
Moment of a Force 2D System
Moment of a Force 2D System
The larger the force ‘W’ or the distance ‘D’ the greater the turning effect at point ‘P’.
Note that:
Moment axis (z) is perpendicular to shaded plane (x-y). i.e. The remaining third axis: „z‟
Fx and dy lies on the shaded plane (x-y)
Moment axis (z) intersects
the plane at point O
Moment of a Force 2D System
Case 2:
Apply force Fz to the wrench Pipe does not rotate about z axis The pipe not actually rotates but Fz creates a tendency for rotation so causing (producing)moment along (Mo)x. Moment axis (x) is perpendicular to the shaded plane (y-z) Fz and dy lies on the shaded plane (y-z)
Moment of a Force 2D System
Case 3:
Apply force Fy to the wrench
No moment is produced about point O
Lack of tendency to rotate
as line of action passes
through O
Note that, Fy and dy both
lies on the same line (and not forming any
plane) hence No Moment is produced.
Moment of a Force 2D System
Moment of a force does not always cause rotation
Force F;
tends to rotate the beam clockwise about A with moment MA = dAF
tends to rotate the beam counterclockwise about B with moment MB = dBF
Moment of a Force 2D System
In General
Consider the force F and the point O which lies in the shaded plane
The moment MO about point O,
or about an axis passing
through O and which is
perpendicular to the plane, is a vector quantity.
Moment of a Force 2D System
Moment MO is a vector having specified magnitude and direction.
Moment of a Force 2D System
M = F x d M = Magnitude of the moment
about point or axis [N.m]
F= Magnitude of the force [N]
d = perpendicular distance [m]
Direction is determined by using the right hand rule
Moment of a Force 2D System
Positive direction of the moment :
Anti clockwise
Moment of a Force 2D System
Positive moment
Negative moment
Moment of a Force 2D System
Magnitude:
Use simple multiplication;
For magnitude of MO: MO = F . d d = moment arm or perpendicular distance
from the axis at point O to its line of action
of the force.
F = Magnitude of the force
Units for moment is N.m, kN.m
Moment of a Force 2D System
Scalar Formulation
Moment of a Force 2D System
Scalar Formulation Direction:
Direction of MO is specified by using “right hand rule”
Fingers of the right hand are curled to follow the sense of rotation when force rotates about point O.
Direction:
Thumb points along the moment axis to give the direction and sense of the moment vector
Moment vector is upwards and perpendicular to the shaded plane
Moment of a Force 2D System
Scalar Formulation
Direction MO is shown by a vector arrow with a curl
to distinguish it from force vector Fig b.
MO is represented by the counterclockwise curl, which indicates the action of F.
Arrowhead shows the sense of rotation caused by F.
Using the right hand rule, the direction and sense of the moment vector points out of the page.
Moment of a Force 2D System
Scalar Formulation
FrM
r Position vector which runs from the
moment reference point to any point
on the line of action of the force.
In some two dimensional problems and
most of the three dimensional problems,
it is convenient to use a vector approach
for moment calculations.
The MOMENT of a force about point A
may be represented by the cross product
expression.
Moment of a Force 2D System
Vector Formulation
Without using the right hand rule directly apply the equation through
cross product of vectors.
MO = d X F
The cross product directly gives the magnitude and the direction. Units for moment is N.m, kN.m
Moment of a Force 2D System
Vector Formulation
FrMo
Sarrus’ Rule
+k - k
)kFr)kFrMxyyxo
( (
Moment of a Force 2D System
Vector Formulation
Example:
For each case, determine the moment of the
force about point O
Moment of a Force 2D System
Solution
Line of action is extended as a dashed line to establish moment arm d
Tendency to rotate is indicated and the orbit is shown as a colored curl
o
o
(a)M (2m)(100N) 200.000 N.m (CW)
(b)M (0.75m)(50N) 37.500 N.m (CW)
Moment of a Force 2D System
Solution
o
o
o
(c) M (4m 2cos30 m)(40N) 229.282 N.m (CW)
(d) M (1sin 45 m)(60N) 42.426 N.m (CCW)
(e) M (4m 1m)(7kN) 21.000 kN.m (CCW)
Moment of a Force 2D System
Example:
Determine the moments of
the 800 N force acting on the
frame about points A, B, C
and D.
Moment of a Force 2D System
Solution
(Scalar Analysis)
Line of action of F passes through C
A
B
C
D
M = (2.5m)(800N) = 2000 N.m (CW)
M = (1.5m)(800N) = 1200 N.m (CW)
M = (0m)(800N)= 0 N.m
M = (0.5m)(800N) = 400 N.m (CCW)
Moment of a Force 2D System
Moment of a Force 2D System
Principles of Moments
Principles of Moments
Also known as Varignon‟s Theorem
This principle states that the moment of a force about a point is equal to the sum of moments of the force’s components about the point.
Principles of Moments
“Moment of a force about a point is equal to the sum of the moments of the forces‟ components about the point”
Principles of Moments
Principles of Moments
Solution Method 1: From trigonometry using triangle BCD, CB = d = 100cos45° = 70.7107mm
Thus, MA =dF= (0.07071m) 200N
= 14142.136 N.mm (CCW)
= 14.142 N.m (CCW)
As a Cartesian vector,
MA = {14.142 k} N.m
Principles of Moments
Solution
Method 2:
Resolve 200 N force into x and y components
Principle of Moments
MA = ∑dF
MA =(200)(200sin45°) – (100)(200cos45°)
= 14142.136 N.mm (CCW)
= 14.142 N.m (CCW)
Principles of Moments
“Moment of a force about a point is equal to the sum of the moments of the forces‟ components about the point
m 898.275sin3 d
mkN 489.14
898.25
FdMO
mkN 489.14
30cos345sin530sin345cos5
xyyxO dFdFM
Example:
Example:
The force F acts at the end of the angle
bracket. Determine the moment of the force
about point O.
Moment of a Force 2D System
Solution
Method 1:
Resolve the given force into components and than apply the moment equation.
MO = 400sin30°N(0.2m)-400cos30°N(0.4m)
= -98.5641 N.m
As a Cartesian vector,
MO = {-98. 5641k} N.m
Moment of a Force 2D System
Solution
Method 2:
Express as Cartesian vector
r = {0.4i – 0.2j} m
F = {400sin30°i – 400cos30°j} N
= {200.000i – 346.410j}N
For moment,
O
i j k
M rXF 0.4 0.2 0
200.000 346.410 0
-98.564k N.m
Moment of a Force 2D System
Example (T):
Determine the moment of the 600 N force with respect to
point O in both scalar and vector product approaches.
Moment of a Force 2D System
Resultant Moment of
System of Coplanar Forces
Resultant moment MRo = addition of the moments
produced by all the forces
algebraically since all
moment forces are
collinear (for 2D case).
MRo = ∑F.d
taking counterclockwise (CCW), to be positive.
Resultant moment, MRo = addition of the moments produced by all the forces algebraically, since all moment forces are collinear (for 2D case).
MRo =M1 – M2 + M3
=∑dF= d1 F1 – d2 F2 + d3 F3
taking counterclockwise (CCW)
to be positive.
Resultant Moment of
System of Coplanar Forces
Counterclockwise is positive
ORM Fd
Resultant Moment of
System of Coplanar Forces
+
Example:
Determine the resultant moment of the four
forces acting on the rod about point O.
Resultant Moment of
System of Coplanar Forces
Solution: (by scalar analysis)
Note that always positive moments acts in the +k
direction, CCW
)CW(m.N923.333
m.N923.333
)m30cos3m4)(N40(
)m30sin3)(N20()m0)(N60()m2)(N50(M
d.FM
Ro
Ro
Resultant Moment of
System of Coplanar Forces
Solution: (by vector analysis)
(CW) or (-k) N.m333.923
(-k)] [263.923 (k)] [30 [0] k)]( [100
(-j)] 40 X )(i )3cos30(4 [
(i )] 20 X (-j) [3s in30(i )] 60 X [0(-j)] 50 X (i ) [2MRo
dXFMRo
Resultant Moment of
System of Coplanar Forces
Moment (Revision)
Moment
Moment force F about point O can be expressed using cross product
MO = r X F where r represents position vector
from O to any point lying
on the line of action of F.
• Remember M = r (vector) X F (vector)
• Find the length ‘r’ vectorially for each
force ‘F’
• if not given, find the vectorial
representation of ‘F’ also.
Moment
In some two dimensional problems and
many three dimensional problems, it is
convenient to use a vector approach for
moment calculations. The MOMENT of
a force about point A may be
represented by the cross product
expression
FrM
Moment
FrM
r
Position vector which runs from the
moment reference point to any point
on the line of action of the force
A
F
1r
2r
3r
MA
Due to the principle of
transmissibility, can act at any point
along its line of action and still create
the same moment about point A.
F
FrFrFrM A
321
Moment
The Moment Vector
The result obtained from r X F doesn‟t depend on where the vector r intersects the line of action of F:
r = r’ + u
r F = (r’ + u) F = r’ F
because the cross product of the parallel vectors u and F is zero.
Moment of a Couple
Couple - two parallel forces
- same magnitude but opposite direction
- separated by perpendicular distance d
Resultant force = 0
Tendency to rotate in specified direction
Couple moment = sum of
moments of both couple
forces about any arbitrary point
Moment of a Couple
A couple is defined as two
parallel forces with the same
magnitude but opposite in
direction separated by a
perpendicular distance d.
The moment of a couple is defined as:
MO = F . d (using a scalar analysis; right hand rule for direction),
MO = d X F (using vector analysis).
Moment of a Couple
The net external effect of a couple is zero since
the net force equals zero and the magnitude of
the net moment equals F.d
Moments due to couples can be added using
the same rules as adding any vectors.
The moment of a couple is a free vector.
It can be moved anywhere on the body
and have the same external effect on the
body.
O
a
d
F
F A
B
C
MO= F (a+d) – F a = F d
MO=MA=MB=MC
Moment of a couple has the same value for all
moment centers.
Moment of a Couple „2D‟
M M
M M
2D CCW couple
2D CW couple
Moment of a Couple „2D‟
The moment of a couple is a free vector. It can be moved
anywhere on the body and have the same external effect on
the body.
=
Moment of a Couple „2D‟
Moment of a Couple „2D‟
APPLICATIONS
Moment of a Couple „2D‟
APPLICATIONS
(continued)
Moment of a Couple „2D‟
Scalar Formulation Magnitude of couple moment
M = F.d
Direction and sense are determined by right hand rule
In all cases, M acts perpendicular to plane containing the forces.
Moment of a Couple „2D‟
Vectorial Formulation
M = d X F
In all cases, M acts perpendicular
to plane containing the forces.
Moment of a Couple „2D‟
Example:
A couple acts on the gear teeth. Replace it
by an equivalent couple having a pair of
forces that act through points A and B.
=
Moment of a Couple „2D‟
Solution
Magnitude of couple
M = 24 N.m
Direction out of the page since
forces tend to rotate CCW
M is a free vector and can
be placed anywhere.
Moment of a Couple „2D‟
Solution
To preserve CCW motion,
vertical forces acting through points A and B must be directed as shown
For magnitude of each force,
M = F.d
24 N.m = F (0.2m)
F = 120.000 N
Moment of a Couple „2D‟
Example (T):
Two different couples are equivalent if they produce the same moment, (magnitude as well as direction).
Equivalent Couples
Moment of a Couple „2D‟
Two couples are equivalent if they produce the same moment
with magnitude and direction.
= = = =
Moment of a Couple „2D‟
Example (T):
Moment of a Couple „2D‟