Chapter2-Resultants of Force Systems

Forces are analyzed in a number of ways;

it is common approach to establish a coordinate system to quantify the forces and their effects in a system or body. Since it is customary to assign the axes, the analysis may be coplanar (two-dimensional) or non-coplanar (three-dimensional).

A system of forces may be represented by a resultant force which has the same effect as the system.

The resultant force, much like any other force, has magnitude and direction. The geometric sum of the forces will yield the resultant.

Forces on an object

Equivalent Resultant Force

2N

1N

3N

2N 4N

6N

Resultant =0

3N

6N

3N

Coplanar Force Systems analyze forces acting

on a body by taking their components along two designated axes.

A force system can be identified into two main types:

concurrent

non-concurrent.

Concurrent Forces are forces whose lines of action intersect at a common point. The resultant of concurrent forces originates from the intersection.

Point of Intersection

The resultant of concurrent forces must be defined by magnitude and direction. Magnitude represents the length of the vector while the direction is referred from the defined axis. Resultant

Force

To determine the magnitude, first take the algebraic sum of the force components for each axis. The geometric sum (or SRSS: square root of the sum of squares) is then computed from these. The direction, for convention, is reckoned from the positive x-axis.

Example 1: Find the resultant of the force system.

Example 2:

If the tension in rope AB is 100N, what is the tension in rope BC? (Hint: The resultant of forces AB and BC is in the direction of the boat.)

Example 3:

Two horses on opposite banks of canal pull a barge moving parallel to the banks by means of two ropes. The tension in these ropes are 200lb and 240lb while the angle between them is 60 degrees. Find the

resultant pull on the barge and

the angle between each of the

ropes and the sides of the canal.

α

θ

Example 4:

If the resultant force

is required to act along

the positive u-axis and

have a magnitude of

5 KN, determine the

required magnitude

of FB and its

direction θ.

Example 5:

Determine the magnitude

and direction of the resultant

force of the three forces acting

on ring A. If F1 = 500N and

the angle F1 makes with

the y-axis = 20 degrees.

Non-concurrent Forces are forces whose lines of action are parallel. The resultant of parallel force has a magnitude equal to the algebraic sum of the forces and is located somewhere between them.

5KN

10KN/m

25KN/m

10KN A B

3m 3m 1m

Resultant Force

The equivalent load of load diagrams – rectangular or uniform, triangular or uniformly varying – is equal to its area which is located at its geometric centroid.

To determine the location of the resultant, apply

Varignon’s Theorem – which states that the effect of the whole is equal to the sum of the effects of it components.

Using Varignon’s Theorem on taking moment about point A, the location can be found. Recall that the moment of a force about a point or axis is simply the magnitude of the force multiplied by its level arm or perpendicular distance to the point.

Also, remember that,

Moment = O if:

1. The Force intersects

the axis

2. The force is parallel to the axis

Example 6: Assuming clockwise moments to be positive, compute for the moment of force F=450 lb and of force P=361 lb about points A,B,C, and D.

A

C

D B

F

P

Example 7: Find the resultant of the force system

and its location from point A.

Example 8: Find the resultant of the non-concurrent force system, its direction and location from point A.

5KN

10KN 10KN/m

10KN

A B

2m 1m 1m 3m

θ=60

Example 9:

The 16-ft wing of an airplane is subjected to a lift which varies from zero at the tip to 360lb/ft at the fuselage according to y=90x1/2, where x is measured from the tip. Compute the resultant and its location from the wing tip.

Example 10: Determine the resultant of three forces acting on the dam shown and

locate its intersection

with the base AB. For

good design, this inter-

section should occur

within the middle

third of the base.

Does it?

24000 lbs

Example 11: Serious neck injuries can occur when a football player is struck in the face guard of his helmet

in the manner shown, giving rise to a guillotine mechanism. Determine the

moment of the knee force

P=50lb about a point A.

What would be the magnitude

of the neck force F so that it

gives the counterbalancing

moment about point A.

Example 12: Find the resultant of the non-concurrent force system, its direction and location from point A.

A B

50KN 80KN 1m 2m 1.5m 1.5m 1m

10KN/m 10KN/m 10KN/m

20KN/m

A couple is comprised of two parallel, non-collinear forces that are equal in magnitude but oppositely directed.

Moment of a Couple = F*d

The moment of a couple is constant and independent of the moment center

F

F

d

X

A

A force can be resolved into another force and a couple.

10KN 10KN

10KN

10KN 2m 2m

20KN-m

Example 13:

Replace the System of forces

acting on the frame by a

resultant, R, acting at

Point A and a couple

acting Horizontally

through B and C.

Example 14:

A couple consists of two vertical forces of 60 lb each.

One force acts up through A and the other acts down

through D. Transform the couple into an equivalent

couple having horizontal

forces acting through

E and F.

Example 15: The three-step pulley shown in the figure is subjected to the given couples. Compute the value

of the resultant couple. Also

determine the forces acting

at the rim of the middle

pulley that are required to

balance the given system.