CE 201 - Statics Lecture 11. MOMENT OF A FORCE-VECTOR FORMULATION In scalar formulation, we have...

CE 201 - Statics

Lecture 11

MOMENT OF A FORCE-VECTOR FORMULATION

In scalar formulation, we have seen that the moment of a force F about point O or a bout a moment axis passing through point O is equal to:

+ MO = F dwhere d is the perpendicular distance between the force and point O.In vector form, the moment can be expressed as:

+ MO = r Fwhere r is the position vector drawn from O to any point on the line of action of F. The moment has both magnitude and direction.

Magnitude+ MO = r F = r F sin = F (r sin ) = F d

A

F

rO

MO

Moment Axis

A

F

rO

MO

Moment Axis

d

Transmissibility of a Force

F has a moment about point O equals to:

MO = r F

We have seen that r can be extended from point O to any point on the line of action of force F. Then:

MO = rA F = rB F = rC F

Then F has the property of a sliding vector and, therefore can act at any point along its line of action. F is therefore a transmissible force.

A

F

rA

O

MOrB

B

C

rC

x

y

z

Cartesian Vector Formulation

If F and r are expressed as Cartesian vectors, then

i j k

+ MO = r F = rx ry rz

Fx Fy Fz

If the determinant was determined, then,

+MO = (ryFz - rzFy) i – (rxFz – rzFx) j + (rxFy – ryFx) k

The vector formulation method has the benefit of solving three-dimensional problem easier. Position vectors can be easily formed rather than perpendicular distance ( d ).

F

r

O

MO

MomentAxis

x

y

z

z

y

x

r

rx

ry

rz

F

Fx

Fy

Fz

Resultant Moment of a System of Forces

+ MRO = ( r F )

F1

r1

F2

F3x

y

z

r2

r3

Principle of Moments

If F = F1 + F2+ MO = ( r F1 ) + ( r F2 ) = r (F1 + F2 ) = r F

The principle of moments states that “the moment of a force about a point is equal to the sum of the moments of the force’s components about the point”.

F

F1

F2r

O