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Ishik University / Sulaimani

Civil Engineering Department

Chapter -4-

Force System Resultant 1

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4.3 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.

Example;

Position vectors rA and rB are directed from O to A and B, lying on the line of

action of F and –F.

Couple moment about O

M = rA X (-F) + rB X (F)

Couple moment about A

M = r X F

since moment of –F about A = 0

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A couple moment is a free vector

- It can act at any point since M depends only on the position vector

r directed between forces and not position vectors rA and rB,

directed from O to the forces.

- Unlike moment of force, it do not require a definite point or axis.

Scalar Formulation Magnitude of couple moment

M = Fd

Direction and sense are determined by

right hand rule.

In all cases, M acts perpendicular to

plane containing the forces.

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4.3 MOMENT OF A COUPLE

Vector Formulation

For couple moment,

M = r X F

If moments are taken about point A, moment of –F is zero about this

point.

r is crossed with the force to which it is directed.

Resultant Couple Moment

Couple moments are free vectors and

may be applied to any point P and added

vectorially.

For resultant moment of two couples at

point P,

MR = M1 + M2

For more than 2 moments,

MR = ∑(r X F)

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4.3 MOMENT OF A COUPLE

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Equivalent Couples

• Two couples are equivalent if they produce the same moment.

• Since moment produced by the couple is always perpendicular to the plane

containing the forces, forces of equal couples either lie on the same plane or

plane parallel to one another.

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Example 4.6 : Force Couple

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Example 4.7 : Force Couple

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Example 4.8

Determine tile resultant couple moment acting on the beam.

Ans:

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Determine the resultant couple moment acting on the triangular

plate.

Example 4.9

Ans:

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Example 4.10

If F=200 Ib, determine the resultant couple moment. Ish

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Solution;

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Example 4.11

Determine the required magnitude of force F if the resultant

couple moment on the frame is 200 Ib.ft, clockwise.

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Solution;

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4.4 Equivalent System

A force has the effect of both translating and rotating a body.

The extent of the effect depends on how and where the force

is applied.

We can simplify a system of forces and moments into a

single resultant and moment acting at a specified point O.

A system of forces and moments is then equivalent to the

single resultant force and moment acting at a specified point

O.

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Point O is on the Line of Action

Consider body subjected to force F applied to point A.

Apply force to point O without altering external effects on

body.

- Apply equal but opposite forces F and –F at O.

4.4 Equivalent System

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Point O is on the Line of Action

- Two forces indicated by the slash across them can be

cancelled, leaving force at point O.

- An equivalent system has be maintained between each of

the diagrams, shown by the equal signs.

4.4 Equivalent System

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Point O is on the Line of Action

- Force has been simply transmitted along its line of action from point A to point O.

- External effects remain unchanged after force is moved.

- Internal effects depend on location of F.

4.4 Equivalent System

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4.4 Simplification of a Force and Couple System

Simplifying any force and couple system,

FR = ∑F

MR = ∑MC + ∑MO

If the force system lies on the x-y plane and any couple moments are perpendicular to this plane,

FRx = ∑Fx

FRy = ∑Fy

MRo = ∑MC + ∑MO

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4.4 Simplification of a Force and Couple System

MOVING A FORCE OFF OF ITS LINE OF ACTION

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Example 4.12

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Example 4.13

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Example 4.14

Replace the force system acting on the beam by an equivalent

force and couple moment at point A.

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solution:

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Example 4.15

Replace the force system acting on the beam by an equivalent

force and couple moment at point A.

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solution: