chapter vector mechanics for engineers: 4 …ybu.edu.tr/fozturk/contents/files/354chapter-4.pdf ·...

1

VECTOR MECHANICS FOR ENGINEERS:

STATICS

Ninth Edition

Ferdinand P. Beer

E. Russell Johnston, Jr.

Lecture Notes:

J. Walt Oler

Texas Tech University

CHAPTER

© 2010 The McGraw-Hill Companies, Inc. All rights reserved.

4Equilibrium of Rigid

Bodies

© 2010The McGraw-Hill Companies, Inc. All rights reserved.

Vector Mechanics for Engineers: Statics

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Design of a support

2 - 2

1. Identify the main structure.

2. Where is the load on the

structure?

3. How is the structure fixed to

the ground?

4. How do we decide on the

dimensions of the ground

support?

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Contents and Objectives

4 - 3

Draw Free-Body Diagram

Identify Reactions at Supports for a Two-Dimensional Structure

Solve Problems of Equilibrium of a Rigid Body in Two Dimensions

Identify Statically Indeterminate Reactions

Recognize a Two-Force Body

Recognize a Three-Force Body

Solve Problems of Equilibrium of a Rigid Body in Three Dimensions

Recognize Reactions at Supports and Connections for a Three-Dimensional

Structure



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Introduction

4 - 4

• The necessary and sufficient condition for the static equilibrium of a

body are that the resultant force and couple from all external forces

form a system equivalent to zero.

00 FrMF O

000

000

zyx

zyx

MMM

FFF

• Resolving each force and moment into its rectangular components

leads to 6 scalar equations which also express the conditions for static

equilibrium,

• For a rigid body in static equilibrium, the external forces and

moments are balanced and will impart no translational or rotational

motion to the body.

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2 - 5

Free-Body Diagram

The truck ramp has a weight of 1800 N.

The ramp is pinned to the body of the truck and held in the position by the

cable. How can we determine the cable tension and support reactions ?

How are the idealized model and the free body diagram used to do this?

Which diagram above is the idealized model?

Geometry Geometry + Loads



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Free-Body Diagram

4 - 6

First step in static equilibrium analysis of a rigid

body is identification of all forces acting on the

body with a free-body diagram.

• Draw an outlined shape. Cut the body free

form its constraints.

• Label loads and dimensions on FBD necessary

to compute the moments of the forces.

• Indicate point of application and assumed

direction of unknown applied forces. These

usually consist of reactions through which the

ground and other bodies oppose the possible

motion of the rigid body.

• Indicate point of application, magnitude, and

direction of external forces, including the rigid

body weight and couple moments.

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Reactions at Supports and Connections for a Two-Dimensional Structure

4 - 7

• Reactions equivalent to a

force with known line of

action.



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Reactions at Supports and Connections for a Two-Dimensional Structure

4 - 8


force of unknown direction

and magnitude.


force of unknown

direction and magnitude

and a couple of unknown

magnitude.

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Quiz 1

2 - 9

1. The beam and the cable (with a frictionless pulley at D) support

an 80 kg load at C. In a FBD of only the beam, there are how

many unknowns?

A) 2 forces and 1 couple moment

B) 3 forces and 1 couple moment

C) 3 forces

D) 4 forces



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Quiz 2

2 - 10

2. If the directions of the force and the couple moments are

both reversed, what will happen to the beam?

A) The beam will lift from A.

B) The beam will lift at B.

C) The beam will be restrained.

D) The beam will break.

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Quiz 3

2 - 11

Draw a FBD of member ABC, which is supported

by a smooth collar at A, roller at B, and link CD.



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Quiz 4

2 - 12

How many unknown support reactions

are there in this problem?

A) 2 forces and 2 couple moments

B) 1 force and 2 couple moments

C) 3 forces

D) 3 forces and 1 couple moment

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Practical Example

2 - 13

A 4000 N of engine is supported by three chains, which are

attached to the spreader bar of a hoist.

You need to check to see if the breaking strength of any of the

chains is going to be exceeded. How can you determine the

force acting in each of the chains?



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Equilibrium of a Rigid Body in Two Dimensions

4 - 14

• For all forces and moments acting on a two-

dimensional structure,

Ozyxz MMMMF 00

• Equations of equilibrium become

000 Ayx MFF

where A is any point in the plane of the

structure.

• The 3 equations can be solved for no more

than 3 unknowns.

• The 3 equations can not be augmented with

additional equations, but they can be replaced.

000 BAx MMF

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STEPS FOR SOLVING 2-D EQUILIBRIUM PROBLEMS

2 - 15

1. If not given, establish a suitable x - y coordinate system.

2. Draw a free body diagram (FBD) of the object under analysis.

3. Apply the three equations of equilibrium to solve for the

unknowns.



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Sample Problem 4.1

4 - 16

A fixed crane has a mass of 1000 kg

and is used to lift a 2400 kg crate. It

is held in place by a pin at A and a

rocker at B. The center of gravity of

the crane is located at G.

Determine the components of the

reactions at A and B.

SOLUTION:

• Create a free-body diagram for the crane.

• Determine B by solving the equation for

the sum of the moments of all forces

about A. Note there will be no

contribution from the unknown

reactions at A.

• Determine the reactions at A by

solving the equations for the sum of

all horizontal force components and

all vertical force components.

• Check the values obtained for the

reactions by verifying that the sum of

the moments about B of all forces is

zero.

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Sample Problem 4.1

4 - 17

• Create the free-body diagram.

• Check the values obtained.

• Determine B by solving the equation for the

sum of the moments of all forces about A.

0m6kN5.23

m2kN81.9m5.1:0

BM A

107.1kNB

• Determine the reactions at A by solving the

equations for the sum of all horizontal forces

and all vertical forces.

0:0 BAF xx

107.1kNxA

0kN5.23kN81.9:0 yy AF

33.3 kNyA



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Sample Problem 4.3

4 - 18

A loading car is at rest on an inclined

track. The gross weight of the car and

its load is 25 kN, and it is applied at at

G. The cart is held in position by the

cable.

Determine the tension in the cable and

the reaction at each pair of wheels.

SOLUTION:

• Create a free-body diagram for the car

with the coordinate system aligned

with the track.

• Determine the reactions at the wheels

by solving equations for the sum of

moments about points above each axle.

• Determine the cable tension by

solving the equation for the sum of

force components parallel to the track.

• Check the values obtained by verifying

that the sum of force components

perpendicular to the track are zero.

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Sample Problem 4.3

4 - 19

• Create a free-body diagram

kN 10.5

25sinkN 25

kN 22.65

25coskN 25

y

x

W

W

• Determine the reactions at the wheels.

0mm 1250

mm 150kN 22.65mm 625kN 10.5:0

2

R

M A

2 8 kNR

0mm 1250

mm 150kN 22.65mm 625kN 10.5:0

1

R

M B

1 2.5 kNR

• Determine the cable tension.

0TkN 22.65:0 xF

22.7 kNT



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Sample Problem 4.4

4 - 20

A 6-m telephone pole of 1600 N is

used to support the wires. Wires T1 =

600 N and T2 = 375 N.

Determine the reaction at the fixed

end A.

SOLUTION:

• Create a free-body diagram for the

telephone cable.

• Solve 3 equilibrium equations for the

reaction force components and

couple at A.

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Sample Problem 4.4

4 - 21

• Create a free-body diagram for

the frame and cable.

• Solve 3 equilibrium equations for the

reaction force components and couple.

010cos)N600(20cos)N375(:0 xx AF

N 238.50xA

020sinN 37510sin600NN1600:0 yy AF

N 1832.45yA

:0AM

0 (6m)

20cos375N)6(10cos600N

mM A

N.m00.1431AM



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Notes

2 - 22

1. If there are more unknowns than the number of independent equations,

then we have a statically indeterminate situation. We cannot solve these

problems using just statics.

2. The order in which we apply equations may affect the simplicity of the

solution. For example, if we have two unknown vertical forces and one

unknown horizontal force,

then solving FX = 0 first allows us to find the horizontal unknown

quickly.

3. If the answer for an unknown comes out as negative number, then the

sense (direction) of the unknown force is opposite to that assumed when

starting the problem.

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Equilibrium of a Two-Force Body

4 - 23

• Consider a plate subjected to two forces F1 and F2 .

• For static equilibrium, the sum of moments about A

must be zero. The moment of F2 must be zero. It

follows that the line of action of F2 must pass

through A.

• Similarly, the line of action of F1 must pass through

B for the sum of moments about B to be zero.

• Requiring that the sum of forces in any direction be

zero leads to the conclusion that F1 and F2 must have

equal magnitude but opposite sense.



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Applications of 2-Force Bodies

2 - 24

In the cases above, members AB can be considered as two-force members,

provided that their weight is neglected.

This fact simplifies the equilibrium analysis of

some rigid bodies since the directions of the

resultant forces at A and B are thus known (along

the line joining points A and B).

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Equilibrium of a Three-Force Body

4 - 25

• Consider a rigid body subjected to forces acting at

only 3 points.

• Assuming that their lines of action intersect, the

moment of F1 and F2 about the point of intersection

represented by D is zero.

• Since the rigid body is in equilibrium, the sum of the

moments of F1, F2, and F3 about any axis must be

zero. It follows that the moment of F3 about D must

be zero as well and that the line of action of F3 must

pass through D.

• The lines of action of the three forces must be

concurrent or parallel.



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Sample Problem 4.6

4 - 26

A man raises a 10 kg joist, of

length 4 m, by pulling on a rope.

Find the tension in the rope and

the reaction at A.

SOLUTION:

• Create a free-body diagram of the joist.

Note that the joist is a 3 force body acted

upon by the rope, its weight, and the

reaction at A.

• The three forces must be concurrent for

static equilibrium. Therefore, the reaction

R must pass through the intersection of the

lines of action of the weight and rope

forces. Determine the direction of the

reaction force R.

• Utilize a force triangle to determine the

magnitude of the reaction force R.

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Sample Problem 4.6

4 - 27

• Create a free-body diagram of the joist.

• Determine the direction of the reaction

force R.

636.1414.1

313.2tan

m 2.313m 515.0828.2

m 515.020tanm 414.1)2045cot(

m 414.1

m828.245cosm445cos

21

AE

CE

BDBFCE

CDBD

AFAECD

ABAF

6.58



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Sample Problem 4.6

4 - 28

• Determine the magnitude of the reaction

force R.

38.6sin

N 1.98

110sin4.31sin

RT

N 8.147

N9.81

R

T

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Statically Indeterminate Reactions

4 - 29

• More unknowns than

equations

Redundant Constraints: When a body has more

supports than necessary to hold it in equilibrium, it

becomes statically indeterminate.

A problem that is statically indeterminate has

more unknowns than equations of equilibrium.

Are statically indeterminate structures used in

practice? Why or why not?



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Improper Constraints

2 - 30

• Equal number unknowns

and equations but

improperly constrained.

Here, we have 6 unknowns but there is nothing

restricting rotation about the AB axis.

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Improper Constraints

2 - 31

In some cases, there may be as many unknown reactions as

there are equations of equilibrium.

However, if the supports are not properly constrained, the body

may become unstable for some loading cases.

0AM



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Equilibrium of a Rigid Body in Three Dimensions

4 - 32

• Six scalar equations are required to express the conditions

for the equilibrium of a rigid body in the general three

dimensional case.

000

000

zyx

zyx

MMM

FFF

• These equations can be solved for no more than 6 unknowns

which generally represent reactions at supports or connections.

• The scalar equations are conveniently obtained by applying the

vector forms of the conditions for equilibrium,

00 FrMF O

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Applications

2 - 33

Ball-and-socket joints and journal bearings are often used in

mechanical systems. To design the joints or bearings, the support

reactions at these joints and the loads must be determined.



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Applications

2 - 34

If A is moved to a lower position D, will

the force in the rod change or remain the

same? By making such a change without

understanding if there is a change in

forces, failure might occur.

The tie rod from point A is used to support

the overhang at the entrance of a building.

It is pin connected to the wall at A and to

the center of the overhang B.

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Application

2 - 35

The crane, which weighs 1550 N, is supporting a oil drum.

How do you determine the largest oil drum weight that the

crane can support without overturning ?



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Reactions at Supports and Connections for a Three-Dimensional Structure

4 - 36

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Reactions at Supports and Connections for a Three-

Dimensional Structure

4 - 37



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2 - 38

Exchange arm for a suspension

Universal joint in transmission shaft of

a truck

//upload.wikimedia.org/wikipedia/commons/2/22/Universal_joints_shaft.jpg

//upload.wikimedia.org/wikipedia/commons/2/22/Universal_joints_shaft.jpg

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2 - 39

Bearing for trolley wheels



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Sample Problem 4.8

4 - 40

A sign of uniform density weighs

1200 N and is supported by a ball-

and-socket joint at A and by two

cables.

Determine the tension in each cable

and the reaction at A.

SOLUTION:

• Create a free-body diagram for the sign.

• Apply the conditions for static

equilibrium to develop equations for

the unknown reactions.

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Sample Problem 4.8

4 - 41

• Create a free-body diagram for the

sign.

Since there are only 5 unknowns,

the sign is partially constrain. It is

free to rotate about the x axis. It is,

however, in equilibrium for the

given loading.

kjiT

kjiT

rr

rrTT

kjiT

kjiT

rr

rrTT

EC

EC

EC

ECECEC

BD

BD

BD

BDBDBD

72

73

76

32

31

32

7

236

6.3

4.22.14.2



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Sample Problem 4.8

4 - 42

• Apply the conditions for

static equilibrium to

develop equations for the

unknown reactions.

0m . N 1440771.08.0:

0514.06.1:

0N 1200m 1.2

0:

0N 1200:

0:

0N 1200

72

32

73

31

76

32

ECBD

ECBD

ECEBDBA

ECBDz

ECBDy

ECBDx

ECBD

TTk

TTj

jiTrTrM

TTAk

TTAj

TTAi

jTTAF

kjiA

TT ECBD

N 100.1N 419N 1502

N 1402N 451

Solve the 5 equations for the 5 unknowns,

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Quiz 5

2 - 43

1. The rod AB is supported using

two cables at B and a ball-and-

socket joint at A. How many

unknown support reactions exist

in this problem?

A) 5 force and 1 moment reaction

B) 5 force reactions

C) 3 force and 3 moment

reactions

D) 4 force and 2 moment

reactions



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2 - 44

2. If an additional couple moment in the

vertical direction is applied to rod AB at

point C, then what will happen to the

rod?

A) The rod remains in equilibrium as the

cables provide the necessary support

reactions.

B) The rod remains in equilibrium as the

ball-and-socket joint will provide the

necessary resistive reactions.

C) The rod becomes unstable as the cables

cannot support compressive forces.

D) The rod becomes unstable since a

moment about AB cannot be restricted.

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Quiz 6

2 - 45

1. A plate is supported by a ball-and-

socket joint at A, a roller joint at B,

and a cable at C. How many

unknown support reactions are

there in this problem?

A) 4 forces and 2 moments

B) 6 forces

C) 5 forces

D) 4 forces and 1 moment



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2 - 46

2. What will be the easiest way to determine the force

reaction BZ ?

A) Scalar equation FZ = 0

B) Vector equation MA = 0

C) Scalar equation MZ = 0

D) Scalar equation MY = 0

chapter vector mechanics for engineers: 4 …ybu.edu.tr/fozturk/contents/files/354chapter-4.pdf ·...

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