3.4 THREE-DIMENSIONAL FORCE SYSTEMS Today’s Objectives:

Students will be able to solve 3-D particle equilibrium problems by a) Drawing a 3-D free body diagram, and, b) Applying the three scalar equations (based on one vector equation) of equilibrium.

R. Michael PE 8/14/2012

READING QUIZ

1. Particle P is in equilibrium with five (5) forces acting on it in 3-D space. How many scalar equations of equilibrium can be written for point P?

A) 2 D) 5 B) 3 E) 6 C) 4 2. In 3-D, when a particle is in equilibrium, which of the following equations apply?

A) (  F x )

i

B) 

F =

0 + (  F y ) C)  F x =  F y = 

j

+ (  F z = 0 F z )

k

= 0 D) All of the above.

E) None of the above.

APPLICATIONS

You know the weights of the electromagnet and its load. But, you need to know the forces in the chains to see if it is a safe assembly. How would you do this?

APPLICATIONS

(continued) Offset distance This shear leg derrick is to be designed to lift a maximum of 200 kg of fish. How would you find the effect of different offset distances on the forces in the cable and derrick legs?

THE EQUATIONS OF 3-D EQUILIBRIUM

When a particle is in equilibrium, the vector sum of all the forces acting on it must be zero ( 

F

= 0 ) .

This equation can be written in terms of its x, y and z components. This form is written as follows. (  F x )

i

+ (  F y )

j

+ (  F z )

k

= 0 This vector equation will be satisfied only when  F x = 0  F y = 0  F z = 0 These equations are the three scalar equations of equilibrium . They are valid for any point in equilibrium and allow you to solve for up to three unknowns.

EXAMPLE #1 Given:

The four forces and geometry shown.

Find:

The force

F 5

required to keep particle O in equilibrium.

Plan:

1) Draw a FBD of particle O.

2) Write the unknown force as

F 5

= {F x

i

+ F y

j

+ F z

k

} N 3) Write

F 1

,

F 2

,

F 3

,

F 4

and

F 5

in Cartesian vector form.

4) Apply the three equilibrium equations to solve for the three unknowns F x , F y , and F z .

EXAMPLE #1

(continued)

F 1

= {300(4/5)

j

+ 300 (3/5)

k

} N

F 1

= {240

j

+ 180

k

} N

F 2

= {– 600

i

} N

F 3

= {– 900

k

} N

F 4

= F 4 (

r B

/ r B ) = 200 N [(3

i

– 4

j

+ 6

k

)/(3 2 + 4 2 + 6 2 ) ½ ] = {76.8

i

– 102.4

j

+ 153.6

k

} N

F 5

= { F x

i

– F y

j

+ F z

k

} N

EXAMPLE #1

(continued) Equating the respective

i

,

j

,

k

components to zero, we have  F x = 76.8 – 600 + F x  F y = 240 – 102.4 + F = 0 ; solving gives y F = 0 ; solving gives x F y = 523.2 N = – 137.6 N  F z = 180 – 900 + 153.6 + F z = 0 ; solving gives F z = 566.4 N Thus,

F 5

= { 523

i

– 138

j

+ 566

k

} N Using this force vector, you can determine the force’s magnitude and coordinate direction angles as needed.

EXAMPLE #2 Given:

A 600 N load is supported by three cords with the geometry as shown.

Find:

The tension in cords AB, AC and AD.

Plan:

1) Draw a free body diagram of Point A. Let the unknown force magnitudes be F B , F C , F D .

2) Represent each force in the Cartesian vector form.

3) Apply equilibrium equations to solve for the three unknowns.

EXAMPLE #2

(continued) FBD at A

F D

z

2 m

F C

1 m 2 m

A

x

600 N

F B F C

= F B (sin 30 

i

+ cos 30 

j

) N = {0

.

5 F B

i

+ 0

.

866 F B

j

} N = – F C

i

N

F D

= F D (

r AD

/r AD ) = F D { (1

i

– 2

j

= { 0

.

333 F D + 2

k

)

/

(1

i

– 0

.

667 F D

j

2 + 2 2 + 2 2 ) ½ } N + 0

.

667 F D

k

} N 30˚

F B

y

EXAMPLE #2

(continued) Now equate the respective

i

components to zero.

,

j

,

k

 F x = 0.5 F B – F C + 0

.

333 F D = 0  F y = 0.866 F B – 0

.

667 F D = 0  F z = 0.667 F D – 600 = 0 FBD at A

F D

2 m 1 m

x

2 m

A

z

600 N F C

30˚

F B

y

Solving the three simultaneous equations yields F C = 646 N F D = 900 N F B = 693 N

CONCEPT QUIZ

1. In 3-D, when you know the direction of a force but not its magnitude, how many unknowns corresponding to that force remain?

A) One B) Two C) Three D) Four 2. If a particle has 3-D forces acting on it and is in static equilibrium , the components of the resultant force (  and  F z ) ___ . F x ,  F y , A) have to sum to zero, e.g., -5

i

+ 3

j

+ 2

k

B) have to equal zero, e.g., 0

i

+ 0

j

+ 0

k

C) have to be positive, e.g., 5

i

+ 5

j

+ 5

k

D) have to be negative, e.g., -5

i

- 5

j

- 5

k

GROUP PROBLEM SOLVING Given:

A 3500 lb motor and plate, as shown, are in equilibrium and supported by three cables and d = 4 ft.

Find:

Magnitude of the tension in each of the cables.

Plan:

1) Draw a free body diagram of Point A. Let the unknown force magnitudes be F B , F C , F D .

2) Represent each force in the Cartesian vector form.

3) Apply equilibrium equations to solve for the three unknowns.

GROUP PROBLEM SOLVING

(continued) FBD of Point A

z

W

x

F B F C F D W

= load or weight of unit = 3500

k

lb

F F B C F D

= F B (

r AB

/r AB ) = F B {(4

i

– 3

j

– 10

k

) / (11.2)} lb = F C (

r AC

/r AC ) = F C { (3

j

– 10

k

) / (10.4 ) }lb = F D (

r AD

/r AD ) = F D { (– 4

i

+ 1

j

–10

k

)

/ (

10.8) }lb

y

GROUP PROBLEM SOLVING

(continued) The particle A is in equilibrium, hence

F B

+

F C

+

F D

+

W

= 0 Now equate the respective

i

,

j

,

k

components to zero (i.e., apply the three scalar equations of equilibrium).

 F x = (4/ 11.2)F B – (4/ 10.8)F D = 0  F y = (– 3/ 11.2)F B + (3/ 10.4)F C + (1/ 10.8)F D  F z = (– 10/ 11.2)F B – (10/ 10.4)F C – (10/ 10.8)F = 0 D + 3500 = 0 Solving the three simultaneous equations gives F B F C F D = 1467 lb = 914 lb = 1420 lb

ATTENTION QUIZ

1. Four forces act at point A and point A is in equilibrium . Select the correct force vector

P .

P

A) {-20

i

+ 10

j

– 10

k

}lb B) {-10

i

– 20

j

– 10

k

} lb C) {+ 20

i

– 10

j

– 10

k

}lb F 1 = 20 lb

x

A

z

F 3 = 10 lb F 2 = 10 lb

y

D) None of the above.

2. In 3-D, when you don’t know the direction or the magnitude of a force, how many unknowns do you have corresponding to that force? A) One B) Two C) Three D) Four

In-class Example – 3D Equilibruim Example 3-7: Determine the force in each cable used to support the 40 lb crate. Note, will look at two ways: with scalars only (Fx = dx/d, etc.), and position vectors (books approach). They are essentially the same!! Break forces into components, apply static equilibrium equations and solve for unknowns.

3.47 The shear leg derrick is used to haul the 200-kg net of fish onto the dock. Determine the compressive force along each of the legs

AB

and

CB

and the tension in the winch cable

DB

. Assume the force in each leg acts along its axis.

This shear leg derrick is to be designed to lift a maximum of 200 kg of fish. How would you find the effect of different offset distances on the forces in the cable and derrick legs?