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Chapter 5:

Three-Dimensional Equilibrium

Chapter 5:

Three-Dimensional Equilibrium

or

Equilibrium of Non-Coplanar Forces

or

Equilibrium of Space Forces

Space Forces and Componentsy

O

F

zz

yy

xx

cosFF

cosFF

cosFF

2222 zyx FFFF.

d

F

z

F

y

F

x

F zyx

P(x, y, z)

Fy Fx

Fz

x

z

y

O

d = diagonal

x, y & z are dimensions

d

z

xx

z

y

222 zyxd:where

1. 3. or by ratio & proportion:

Ex. Determine the x, y & z components.

y

O

F=200N

d

F

z

F

y

F

x

F zyx

Fy

FxFz

z

x

N.Fx 42111

3m

2m

4m

222 243

200

243 zyx FFF

29

200

243 zyx FFF

N.Fy 56148

N.Fz 2874

Exercise: Determine the x, y & z components.

P=350N1m

1m

5m

1.

Q=1000N

6m

1m

3m

3.

R=500N

4m

2m

3.5m

2.

S=250N

6m

1m

3m

4.

Py

Px

Pz

5m

Ry

RxRz

Qy

Qx

Qz Sy

Sz

Tx

Uz

T=200N

U=750N

Ty

Ux

Sample Problem 5.6

The non-homogeneous plate weighing 60KN has its center of gravity at G.

It is supported in the horizontal plane by 3 vertical cables. Compute the

tension on each cable using the given FBD.

Parallel Space Forces

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Sample Problem 5.6

y

:Maxisy

0y +

Solution:

y

x

1.2m

3.6m

A

CBO

0.8m 2.2m

1.0m W

Therefore only TA and W will

rotate about the y-axis

Notes:

1. The axis of rotation is a LINE and not a

point.

2. All forces that are applied directly on the

chosen axis can not rotate about the said

axis.

KNTA 20

).(TA 63 2160 . 0

Top View:Sample Problem 5.6

:Maxiscx

0xC axis

+

Solution:

y

x

1.2m

3.6m

A

CBO

0.8m 2.2m

1.0m

W

Only TA . TB and W will rotate about the xc-axis

KN.TB 2727

)(TA 3 ).(TB 22 0

Top View:

All forces that are applied directly on the

chosen axis can not rotate about the said

axis. (Tc)

xC axis3m

)(260

2m

:=FZ 0∑

KN.TC 7312

060 CBA TTT

xC axis

Another Way 1:

:MaxisBx

0xB axis

+

Solution:

y

x

1.2m

3.6m

A

CBO

0.8m 2.2m

1.0m

W

Only TA . TC and W will rotate about the xB axis

KNTbut A 20

022206080 ).(T).().(T CA

Top View:

xB axis

0.8m

0.2m

:FZ 0

KN.TB 2727

060 CBA TTT

xB axis

KN.TC 7312

Another Way 2:

:MaxisGy

0+

Solution:

y

x

1.2m

3.6-1.2=2.4m

A

CBO

0.8m 2.2m

1.0m

W

Top View:

yG axis

yG axis

yG axisOnly TA . TB and TC will rotate about the yG axis

KNTbut A 20

0212142 ).(T).(T).(T CBA :=FZ 0∑

KN.TB 2727

06020 CB TT

KN.TC 7312

Parallel Space Forces

Prob. 5.38 The total mass of the L-shaped beam of constant cross

section is 1470kg. The beam is hoisted by 3 vertical cables attached

at O, A and B. Determine the distances a and b for which the

tensions in the cables are equal.

Parallel Space Forces

Prob. 5.38

W1 1.5m

2m

W2

a

b

PA PO

PB

x

y

Total Length of L-shaped beam =7m

7

481914701 ).(W

7

381914702 ).(W

NW 82401

NW 61802

Total mass of L-shaped beam =1470kg

A O

B

Equal Tensions: PA = PB = PC = PN

).(P 4807

3

8191470

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Parallel Space Forces

Prob. 5.38

W1=8240N1.5m

2m

W2=6180N

a

b

P=4807N

x

y

A O

B

:Maxisx

0x +

m.b 931

05161804807 ).()b(

P=4807N

P=4807N

:Maxisy

0

y

+

m.a 433

0282404870 )()a(

Concurrent Space Forces

Concurrent Space Forces

Find the force in each leg of the tripod if it can support a load

P=4600N. The legs of the tripod are connected by ball-and

socket joints to the platform. (Same fig. 5.52)

=4600N

Concurrent Space Forces

FBD:

PD

PC

PB

=4600N

PD

PC

PB

PD3m

8m

6m

Dz

Dy

Dx

Member AD:

222 836836 Dzyx PDDD

Dz

Dy

Dx

O

D

A

=4600N

109836

Dzyx PDDD

z

y

x

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PD

PC

PB

PC

3m

8m

6m

Cz

Cy

Cx

Member AC:

109836

Czyx PCCC

Dz

Dy

Dx

O

C

A

Cz

CyCx

=4600N

222 836836 Czyx PCCC PD

PC

PB

PB

8m

6m

Bz

By

Member AB:

22 8686 Bzy PBB

Dz

Dy

Dx

O B

A

Cz

CyCx

Bz

By

=4600N

1086

Bzy PBB

Dz

Dy

Dx

Cz

CyCx

Bz

By

Bz

P=4600N

P=4600N

Dz

PA, PB and PC were resolved into their components. These

components have the same effect as their corresponding forces.

Therefore you may remove PA, PB and PC from the FBD .

PD

PC

PB

Dz

Dy

Dx

Cz

CyCx

Bz

By

xC axis

:MaxiscX

0xC axis

+Only Bz and P will rotate about the xC axis

N.Bz 331533

)(Bz 9 )(34600 0

Notes: 1. All forces that are applied

directly on the chosen axis

can not rotate about the said

axis. (Dx,Dy, Dz, Cx, Cy & Cz)

1086

Bzy PBB

2. All forces passing through

the chosen axis will have

zero moment arm. (Bx)Bz

P=4600N

P=4600N

9m

3m

Recall:

N.PB 671916NBy 1150

Dz

Dy

Dx

Cz

CyCx

Bz

By

yC axis

:Maxiscy

0yC axis

Only Dz ,Bz and P will rotate about the yC axis

N.Dz 331533

)(BZ 6 )(64600 0

Notes: 1. All forces that are applied directly

on the chosen axis can not rotate

about the said axis. (Cx,Cy & Cz)

109836

Dzyx PDDD

2. All forces passing through the

chosen axis will have zero

moment arm. (Dx)

Bz

P=4600N

P=4600N

12m 6m

Recall:

N.PD 052001

3. All forces that are parallel to the

chosen axis can not rotate about the

said axis. (By & Dy)

Dz

)(DZ 12

NDy 575

NDx 1150

+

Dz

Dy

Dx

Cz

CyCx

Bz

By

=4600N

N.PB 671916

N.PD 052001

Answers:

:Fy

0

:Fz

0

:Fx

0

0 yyy BCD

4600=B+C+D zzz

xx CD N=D=C xx 1150∴

NCy 575

109836

Czyx PCCC

N.Cz 331533

N.Pc 052001

N.Pc 052001

Recall:

← I used this.

Use any of these equations:

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FIN.