engineering mechanics: statics in si units, 12e [compatibility … · internal forces • external...

3/15/2015

1

Equilibrium of a Rigid Body55

Engineering Mechanics:

Statics in SI Units, 12e

Copyright © 2010 Pearson Education South Asia Pte Ltd


Chapter Objectives

• Develop the equations of equilibrium for a rigid body

• Concept of the free-body diagram for a rigid body

• Solve rigid-body equilibrium problems using the

equations of equilibrium

3/15/2015

2


Chapter Outline

1. Conditions for Rigid Equilibrium

2. Free-Body Diagrams

3. Equations of Equilibrium

4. Two and Three-Force Members

5. Free Body Diagrams

6. Equations of Equilibrium

7. Constraints and Statical Determinacy


5.1 Conditions for Rigid-Body Equilibrium

• The equilibrium of a body is expressed as

• Consider summing moments about some other point,

such as point A, we require

( ) ∑∑

==

==

0

0

OOR

R

MM

FF

( )∑ =+×= 0ORRA MFrM

3/15/2015

3


5.2 Free Body Diagrams

Support Reactions

• If a support prevents the translation of a body in a

given direction, then a force is developed on the body

in that direction.

• If rotation is prevented, a couple moment is exerted on

the body.



3/15/2015

4





Internal Forces

• External and internal forces can act on a rigid body

• For FBD, internal forces act between particles which

are contained within the boundary of the FBD, are not

represented

• Particles outside this boundary exert external forces

on the system

3/15/2015

5



Weight and Center of Gravity

• Each particle has a specified weight

• System can be represented by a single resultant force,

known as weight W of the body

• Location of the force application is known as the

center of gravity



Procedure for Drawing a FBD

1. Draw Outlined Shape

• Imagine body to be isolated or cut free from its

constraints

• Draw outline shape

2. Show All Forces and Couple Moments

• Identify all external forces and couple moments that

act on the body

3/15/2015

6



3. Identify Each Loading and Give Dimensions

• Indicate dimensions for calculation of forces

• Known forces and couple moments should be properly

labeled with their magnitudes and directions


Example 5.1

Draw the free-body diagram of the uniform beam. The

beam has a mass of 100kg.

3/15/2015

7


Solution

Free-Body Diagram


Solution

Free-Body Diagram

• Support at A is a fixed wall

• Three forces acting on the beam at A denoted as Ax, Ay,

MA, drawn in an arbitrary direction

• Unknown magnitudes of these vectors

• Assume sense of these vectors

• For uniform beam,

Weight, W = 100(9.81) = 981N

acting through beam’s center of gravity, 3m from A

3/15/2015

8



3/15/2015

9


Examples and Problems

• Resolve the following problems for Homework: 5.1, 5.4

and 5.10.


3/15/2015

10


5.3 Equations of Equilibrium

• For equilibrium of a rigid body in 2D,

∑Fx = 0; ∑Fy = 0; ∑MO = 0

• ∑Fx and ∑Fy represent sums of x and y components of

all the forces

• ∑MO represents the sum of the couple moments and

moments of the force components

Please refer to the Companion CD for the animation: Equilibrium of a Free Body



Alternative Sets of Equilibrium Equations

• For coplanar equilibrium problems,

∑Fx = 0; ∑Fy = 0; ∑MO = 0

• 2 alternative sets of 3 independent equilibrium

equations,

∑Fa = 0; ∑MA = 0; ∑MB = 0


3/15/2015

11



Procedure for Analysis

Free-Body Diagram

• Force or couple moment having an unknown

magnitude but known line of action can be assumed

• Indicate the dimensions of the body necessary for

computing the moments of forces





Equations of Equilibrium

• Apply ∑MO = 0 about a point O

• Unknowns moments of are zero about O and a direct

solution the third unknown can be obtained

• Orient the x and y axes along the lines that will provide

the simplest resolution of the forces into their x and y

components

• Negative result scalar is opposite to that was assumed

on the FBD


3/15/2015

12


Example 5.5

Determine the horizontal and vertical components of

reaction for the beam loaded. Neglect the weight of the

beam in the calculations.


Solution

Free Body Diagrams

• 600N represented by x and y components

• 200N force acts on the beam at B

3/15/2015

13


Solution


NBBNF xxx 424045cos600 ;0 =⇒=−=∑→+ o

NB

BNNNN

F

NA

mAmNmNmN

M

y

y

y

y

y

B

405

020010045sin600319

;0

319

0)7()2.0)(45cos600()5)(45sin600()2(100

;0

=

=+−−−

=∑↑+

=

=−−+

=∑

o

oo


3/15/2015

14



3/15/2015

15



3/15/2015

16

Examples and Problems

• Resolve the following problems for Homework: 5.21,

5.41, 5.42 and 5.91.



3/15/2015

17



3/15/2015

18



5.4 Two- and Three-Force Members

Two-Force Members

• When forces are applied at only two points on a

member, the member is called a two-force member

• Only force magnitude must be determined

3/15/2015

19


5.4 Two- and Three-Force Members

Three-Force Members

• When subjected to three forces, the forces are

concurrent or parallel


Example 5.13

The lever ABC is pin-supported at A and connected to a

short link BD. If the weight of the members are negligible,

determine the force of the pin on the lever at A.

3/15/2015

20


Solution

Free Body Diagrams

• BD is a two-force member

• Lever ABC is a three-force member


Solving,

kNF

kNFA

32.1

07.1

=

=

045sin3.60sin ;0

040045cos3.60cos ;0

3.604.0

7.0tan

1

=−=∑↑+

=+−=∑→+

=

= −

oo

oo

o

FFF

NFFF

Ay

Ax

θ


5.5 Free-Body Diagrams

Support Reactions

As in the two-dimensional case:

• A force is developed by a support

• A couple moment is developed when rotation of the

attached member is prevented

• The force’s orientation is defined by the coordinate

angles α, β and γ

3/15/2015

21





3/15/2015

22


Example 5.14

Several examples of objects along with their associated

free-body diagrams are shown. In all cases, the x, y and z

axes are established and the unknown reaction

components are indicated in the positive sense. The

weight of the objects is neglected.


Solution

3/15/2015

23



Vector Equations of Equilibrium

• For two conditions for equilibrium of a rigid body in

vector form,

∑F = 0 ∑MO = 0

Scalar Equations of Equilibrium

• If all external forces and couple moments are

expressed in Cartesian vector form

∑F = ∑Fxi + ∑Fyj + ∑Fzk = 0

∑MO = ∑Mxi + ∑Myj + ∑Mzk = 0


5.7 Constraints for a Rigid Body

Redundant Constraints

• More support than needed for equilibrium

• Statically indeterminate: more unknown

loadings than equations of equilibrium

3/15/2015

24



Improper Constraints

• Instability caused by the improper constraining by the

supports

• When all reactive forces are concurrent at this point,

the body is improperly constrained




Free Body Diagram

• Draw an outlined shape of the body

• Show all the forces and couple moments acting on the

body

• Show all the unknown components having a positive

sense

• Indicate the dimensions of the body necessary for

computing the moments of forces

3/15/2015

25





• Apply the six scalar equations of equilibrium or vector

equations

• Any set of non-orthogonal axes may be chosen for this

purpose


• Choose the direction of an axis for moment summation

such that it insects the lines of action of as many

unknown forces as possible


Example 5.15

The homogenous plate has a mass of 100kg and is

subjected to a force and couple moment along its edges.

If it is supported in the horizontal plane by means of a

roller at A, a ball and socket joint at B, and a cord at C,

determine the components of reactions at the supports.

3/15/2015

26


Solution

Free Body Diagrams

• Five unknown reactions acting on the plate

• Each reaction assumed to act in a positive coordinate

direction


0981300;0

0;0

0;0

=−−++=∑

==∑

==∑

NNTBAF

BF

BF

Czzz

yy

xx


Solution


• Components of force at B can be eliminated if x’, y’ and

z’ axes are used

0)3(.200)5.1(981)5.1(300

;0

0)2()2(300)1(981;0

0.200)3()3()5.1(981)5.1(300

;0

0)2()1(981)2(;0

'

'

=+−−

=∑

=−+=∑

=−−−+

=∑

=+−=∑

mTmNmNmN

M

mAmNmNM

mNmAmBmNmN

M

mBmNmTM

C

y

zx

zz

y

ZCx

3/15/2015

27


Solution

Solving,

Az = 790N Bz = -217N TC = 707N

• The negative sign indicates Bz acts downward

• The plate is partially constrained as the supports

cannot prevent it from turning about the z axis if a force

is applied in the x-y plane


QUIZ

1. If a support prevents translation of a body, then the support exerts a ___________ on the body.

A) Couple moment

B) Force

C) Both A and B.

D) None of the above

2. Internal forces are _________ shown on the free body diagram of a whole body.

A) Always

B) Often

C) Rarely

D) Never

3/15/2015

28


QUIZ

3. 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


QUIZ

4. Internal forces are not shown on a free-body diagram

because the internal forces are_____.

A) Equal to zero

B) Equal and opposite and they do not affect the

calculations

C) Negligibly small

D) Not important

3/15/2015

29


QUIZ

5. The three scalar equations ∑ FX = ∑ FY = ∑ MO = 0,

are ____ equations of equilibrium in two dimensions.

A) Incorrect B) The only correct

C) The most commonly used D) Not sufficient

6. A rigid body is subjected to forces.

This body can be considered

as a ______ member.

A) Single-force B) Two-force

C) Three-force D) Six-force


QUIZ

7. For this beam, how many support reactions are there

and is the problem statically determinate?

A) (2, Yes) B) (2, No) C) (3, Yes) D) (3, No)

8. The beam AB is loaded as shown: a) how many

support reactions are there on the beam, b) is this

problem statically determinate, and c) is the structure

stable?

A) (4, Yes, No) B) (4, No, Yes)

C) (5, Yes, No) D) (5, No, Yes)

F F F F

FFixed

support

A B

3/15/2015

30


QUIZ

9. Which equation of equilibrium allows you to determine

FB right away?

A) ∑ FX = 0 B) ∑ FY = 0

C) ∑ MA = 0

D) Any one of the above.

10. A beam is supported by a pin joint and a roller. How

many support reactions are there and is the structure

stable for all types of loadings?

A) (3, Yes) B) (3, No)

C) (4, Yes) D) (4, No)

AX

A B

FBA

Y

100 lb


QUIZ

11. If a support prevents rotation of a body about an

axis, then the support exerts a ________ on the

body about that axis.

A) Couple moment B) Force

C) Both A and B. D) None of the above.

12. When doing a 3-D problem analysis, you have

________ scalar equations of equilibrium.

A) 3 B) 4

C) 5 D) 6

3/15/2015

31


QUIZ

13. 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


QUIZ

14. 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.

3/15/2015

32


QUIZ

15. 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


QUIZ

16. 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

engineering mechanics: statics in si units, 12e [compatibility … · internal forces • external...

Documents