University of Sulaimani
College of Engineering
Civil Engineering Department
Engineering Mechanics I
First Year
Chapter 3: Resultant of Force Systems
Prepared by:
Assist Professor Dr. Serwan Kh. Rafiq
Shuaaib A. Mohammed
2023-2024
3.1 MOMENTS OF FORCES
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3.1 MOMENTS OF FORCES
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3.1 MOMENTS OF FORCES
Example 3.1
Calculate the magnitude of the moment of the 600=N force about Point O.
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3.1 MOMENTS OF FORCES
Example 3.2
Determine the moment of the 5=kN force about Point O.
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3.1 MOMENTS OF FORCES
Homework 3.1
Determine the moment of the force about Point O.
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3.1 MOMENTS OF FORCES
Example 3.3
Determine the resultant moment produced by the forces about Point O.
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3.1 MOMENTS OF FORCES
Example 3.4
Determine the resultant moment produced by the forces about Point O.
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3.1 MOMENTS OF FORCES
Homework 3.2
Determine the resultant moment produced by the forces about Point A.
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3.1 MOMENTS OF FORCES
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3.1 MOMENTS OF FORCES
Example 3.5
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3.1 MOMENTS OF FORCES
Example 3.6
Determine the moment of force F about z axis.
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3.1 MOMENTS OF FORCES
Homework 3.3
The chain AB exerts a force of 20 lb on the door at B. Determine the magnitude of the
moment of this force along the hinged axis x of the door.
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3.1 MOMENTS OF FORCES
Couples
A couple consists of two forces which have equal magnitude and parallel
non-collinear lines of action, but which are opposite in sense.
The forces of a couple cannot be combined into a single force because
their sum in every direction is zero. Therefore, a couple has no tendency
to translate a body in any direction but tends only to rotate the body on
which it acts.
The characteristics of a couple, which indicate its external effect on a rigid body, are:
a) The magnitude of the moment of the couple.
b) The slope of the plane of the couple.
c) The sense of rotation of the couple.
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3.1 MOMENTS OF FORCES
Couples
The moment of a couple is the algebraic sum of the moments of its forces about
any axis perpendicular to the plane of the couple. Consider the action of two
equal and opposite forces F and –F a distance d apart, the combined moment of
the two forces about an axis normal to their plane and passing through any point
such as O in their plane is the couple M.
This couple has a magnitude:
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3.1 MOMENTS OF FORCES
Couples
The moment of a couple has the same dimensions and common units as the
moment of a force. Since a couple is a vector quantity, it is sometimes
convenient to represent a couple by a vector. The vector is drawn perpendicular
to the plane of the couple to indicate the aspect of the plane.
The magnitude of the moment of the couple is represented by the length of the
vector. The sense of rotation is indicated by the arrowhead on the vector which
points in the direction a right-hand screw would advance if turned in the
direction of the sense of rotation of the couple.
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3.1 MOMENTS OF FORCES
Transformation of a couple
Transformations of a couple can be defined as the number of operations on the couple that do not change
any of its characteristics. The characteristics of a couple are not changed if:
a) the couple is rotated in its plane,
b) the couple is moved to a parallel position in its plane,
c) the couple is moved to a parallel plane.
d) if the distance between the forces of the couple and the magnitude of the forces are changed, provided
the moment remains the same.
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3.1 MOMENTS OF FORCES
Transformation of a couple
None of these transformations involves any change in either the magnitude of the moment of the couple or the
sense of rotation of the couple. The use of the transformations of a couple should help to emphasize the
invariant characteristics of couples and the external effect of a couple on a body.
The figure shows four different configurations of the same couple M. In each of the four cases, the couples are
equivalent and are described by the same free vector which represents the identical tendencies to rotate the
bodies.
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3.1 MOMENTS OF FORCES
Transformation of a couple
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3.1 MOMENTS OF FORCES
Example 3.7
Replace the 10-kN force acting on the steel column by an equivalent force–couple system at
point O. This replacement is frequently done in the design of structures.
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3.1 MOMENTS OF FORCES
Example 3.8
Replace the loading system acting on the beam by an equivalent force
and a couple at point O.
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3.1 MOMENTS OF FORCES
Example 3.9
By means of the transformation of a couple, replace the three forces by a
single force and specify where the resultant’s line of action intersects the
beam measured from O.
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3.1 MOMENTS OF FORCES
Example 3.10
By means of the transformation of a couple, replace the
loading acting on the beam by a single force. Specify
where the force acts, measured from B.
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3.1 MOMENTS OF FORCES
Homework 3.4
Replace the loading system by an equivalent force and a couple acting at
point A.
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3.1 MOMENTS OF FORCES
Example 3.11
By using the transformation of a couple, replace the three couples by one
couple acting on the beam with the forces acting horizontally at A and B.
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3.1 MOMENTS OF FORCES
Example 3.12
Determine the magnitude of F so that the resultant couple moment
acting on the beam is 450 lb.ft counterclockwise.
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3.1 MOMENTS OF FORCES
Homework 3.5
Determine the magnitude of F so that the resultant couple moment acting on
the beam is 1.5 kN.m clockwise.
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3.1 MOMENTS OF FORCES
Homework 3.6
By means of the transformation of a couple, replace the loading on the frame by a
single force. Specify where its line of action intersects a horizontal line along member
CB, measured from end C.
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3.1 MOMENTS OF FORCES
Homework 3.7
Two couples act on the frame. If the resultant couple moment is to be zero, determine
the distance d between the 80-lb couple forces.
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