2. Force Systems
2.1 Introduction
2.2 Force
- A force is an action of one body on another.
- A force is an action which tends to cause acceleration of a body (in
dynamics).
- A force is a vector quantity.
- The complete specification of the action of a force must include its
magnitude, direction, and point of application.
External and internal effects
- External forces can be either applied forces or reaction forces.
- Internal forces are resultants of stresses caused by external forces.
A
A
A
N
N
F
F
F
A, F : external
N
: internal
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Principle of Transmissibility
- Assume: rigid body
- The principle of transmissibility states that a force may be applied at
any point on its given line of action without altering the resultant
effects of the force external to the rigid body
→ The external effects at O and C (support reactions) remain the same
regardless on which point of the line of action of the force, the force
acts.
- In this case, the force may be treated as sliding vector.
- The principle of transmissibility is not valid for elastic bodes if we
study the internal effects of a force.
→ In this case, the force is a fixed vector
Force Classification
- Forces are classified as either contact or body forces
- Contact forces are caused by the direct contact of one body with the
surface of another
- A body force occurs when one body exerts a force on another body
without direct physical contact between the bodies. Examples include
the effects caused by the earth’s gravitation or its electromagnetic
field.
- Force may be further classified as either concentrated or distributed
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Action and reaction
- The action of a force is always accompanied by an equal and opposite
reaction.
- It is important to distinguish between the action and the reaction in a
pairs of forces.
F
A
B
F
A
R
F : Action force
R : Reaction force, exerted on body A by body B.
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Concurrent Forces
- Two or more forces are said to be concurrent at a point if their lines
of action intersect at that point.
- Use parallelogram law to determine the resultant
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- We can also use the triangle law to obtain R, but we need to move the
line of action of one of the forces. If we add the same two forces, we
correctly preserve the magnitude and direction of R, but we lose the
correct line of action, because R obtained in this way does not pass
through A. Therefore this type of combination should be avoided.
- The relationship between a force and its vector components along
given axes must not be confused with the relationship between a force
and its perpendicular projections onto the same axes.
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Vector Components
- We often need to replace a force by its vector components in direction
which are convenient for a given application
Note:
Fa ≠ Fcosθ
But Fx = Fcosθ (rectangular components)
A special Case of Vector Addition
- Used when the resultant of two parallel forces is to be determined.
- Applicable on rigid bodies only
- R is the resultant of the forces F1 and F2 and
is correct in magnitude, direction, and line of
action.
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2.3 Rectangular Components of a Force
cos sin || tan F is the magnitude of the force vector F and is always positive, where Fx
and Fy can be positive or negative.
Fx, Fy are the vector components of F in the x- and y-directions.
Fx, Fy are the scalar components of F.
When both a force and its vector components appear in a diagram, it is
desirable to show the vector components of the force with dashed lines and
show the force with a solid line, or vice versa. (Only one force F, or two
components Fx and Fy, but not 3 separate forces).
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The choice of a coordinate system is arbitrary. A logical choice simplifies
the solution of the problem.
y
y
y
F
F
F
x
γ
α
β
x
x
cos β sin cos sin β cos sin 0
Note: In all three cases the magnitude F is the same and is defined as
|| 8
Determining the Components of a Force
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Determining the Resultant of two Forces
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2.4 Moment
- A force can tend to rotate a body about an axis which neither
intersects nor is parallel to the line of action of the force. This
rotational tendency is known as the moment M of a force.
- The moment M of a force F about a point A is defined using cross
product as
where r is a position vector which runs from the moment reference
point A to any point on the line of action of F.
- Note .
- Moment about a point A means here:
Moment with respect to an axis normal to the plane and passing
through the point A.
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- The magnitude M of the moment is
defined as:
! " sin #
where d is a moment arm and is
defined as the perpendicular
distance between the line of action
of the force and the moment center.
- The moment M is a vector quantity. Its direction is perpendicular to
the r-F-plane.
- The sense of M depends on the direction in which F tends to rotate the
body → right-hand rule
+ : counterclockwise rotation.
- : clockwise rotation.
Sign consistency within a given problem is very important.
- The moment M may be considered as a sliding vector with a line of
action coinciding with the moment axis.
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Varignon’s Theorem
- Valid for any number of concurrent forces
- States that the moment of a force (R) about any point is equal to the
sum of the moments of the components (P, Q) of the force about the
same point.
- The theorem can be stated in another way as:
The resultant moment M of the moments Mi of the concurrent forces
Fi is equal to the moment of their resultant R.
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2.5 Couple
- The moment produced by two equal, opposite, parallel, and
noncollinear forces is called a couple.
- The force resultant of a couple is zero. Its only effect is to produce a
tendency of rotation.
- The moment M of a couple is defined as
$ % $ % ! where rA and rB are position vectors which run from point O to
arbitrary points A and B on the lines of action of F and –F.
- The moment expression contains no reference
to the moment center O and, therefore, is the
same for all moment centers
→ the moment of a couple is a free vector.
- The sense of the moment M is established by the right-hand rule.
Counterclockwise couple (+)
Clockwise couple (-)
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- The magnitude M of the moment M of the couple is given as
& # ! $ &
#
where d is the perpendicular distance
between the couple forces (F, -F).
- The magnitude of the couple is
independent of the distance a.
Equivalent Couples
- Changing the values of F and d does not change a given couple as
long as the product Fd remains the same.
- A couple is not affected if the forces act in a different but parallel
plane.
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Force-Couple Systems
- The effect of a force acting on a body is:
a) the tendency to push or pull the body in the direction of the
force, and
b) to rotate the body about any fixed axis which does not intersect
the line of action of the force ( force does not go through the
mass center of the body).
→ We can represent this dual effect more easily by replacing the
given force by an equal parallel force and a couple to compensate
for the change in the moment of the force.
- Also we can combine a given couple and a force which lies in the
plane of the couple to produce a single, equivalent force.
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2/6 Resultants
Reduce the system of forces and moments to a resultant force R only or to a
resultant force R and a resultant moment MO.
The resultant of a system of forces is the simplest force combination which
can replace the original forces without altering the external effect on the
rigid body to which the forces are applied.
Assume: all forces act in one plane (x-y plane) and the moments act about an
axis perpendicular to this plane .
The resultant of a system of forces is the simplest force combination which
can replace the original forces without altering the external effect on the
rigid body to which the forces are applied.
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a. Force Polygon Method
Yields magnitude and direction of the resultant only
Line of action of the resultant remains undetermined.
b. Parallelogram Law
Yields: Magnitude, direction, and the correct line of action
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c. Algebraic Method
Example: Three force-system
1. Choose a convenient reference point O and move all forces to that
point with considering the couples M1, M2, and M3 resulting from
the transfer of forces F1, F2, and F3 from their respective original
lines of action to parallel lines of action through point O.
2. Add all forces at O to form the resultant force R, and add all
couples to form the resultant couple MO.
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3. Find the line of action of R by requiring R to have a moment of
MO about point O.
-
-
For concurrent system of forces where the lines of action of all forces
pass through a common point O, the moment sum ∑Mo about that point
is zero.
For a parallel force system, select a coordinate axis in the direction of the
forces.
If the resultant force R for a given force system is zero, the resultant of
the system need not be zero because the resultant may be a couple.
Principle of moments
States that the moment of the resultant force about any point O equals the
sum of the moments of the original forces of the system about the same
point. The may be concurrent or nonconcurrent.
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