Rigid body equilibrium

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Equilibrium of a Rigid Body5

Engineering Mechanics: Statics in SI Units, 12e


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


Chapter Outline

1. Conditions for Rigid Equilibrium2. Free-Body Diagrams3. Equations of Equilibrium4. Two and Three-Force Members5. Free Body Diagrams6. Equations of Equilibrium7. 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


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.







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



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. 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.


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, Az, 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


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




Procedure for AnalysisFree-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




Procedure for AnalysisEquations 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



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


Solution

Equations of EquilibriumNBBNM xxB 424045cos600 ;0

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


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


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.


Solution

Free Body Diagrams• BD is a two-force member• Lever ABC is a three-force memberEquations of Equilibrium

Solving,

kNFkNFA

32.107.1

045sin3.60sin ;0

040045cos3.60cos ;0

3.604.07.0tan 1

FFF

NFFF

Ay

Ax


5.5 Free-Body Diagrams

Support ReactionsAs 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






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



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



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



Procedure for AnalysisFree 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



Procedure for AnalysisEquations of Equilibrium• Apply the six scalar equations of equilibrium or vector

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

purpose

Equations of Equilibrium• 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 N, and a cord at C, determine the components of reactions at the supports.


Solution

Free Body Diagrams• Five unknown reactions acting on the plate• Each reaction assumed to act in a positive coordinate

directionEquations of Equilibrium

0981300;0

0;00;0

NNTBAF

BFBF

Czzz

yy

xx


Solution

Equations of Equilibrium

• 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

;00)2()2(300)1(981;0

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

;00)2()1(981)2(;0

'

'

mTmNmNmN

MmAmNmNM

mNmAmBmNmN

MmBmNmTM

C

y

zx

zz

y

ZCx


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


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 momentB) 3 forces and 1 couple momentC) 3 forcesD) 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


QUIZ

5. The three scalar equations FX = FY = MO = 0, are ____ equations of equilibrium in two dimensions.A) Incorrect B) The only correctC) 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-forceC) 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


QUIZ

9. Which equation of equilibrium allows you to determine FB right away?

A) FX = 0 B) FY = 0C) 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

FBAY

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) ForceC) 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) 4C) 5 D) 6


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 reactionsC) 3 force and 3 moment reactionsD) 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.


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 momentsB) 6 forcesC) 5 forcesD) 4 forces and 1 moment


QUIZ

16. What will be the easiest way to determine the force reaction BZ ?

A) Scalar equation FZ = 0B) Vector equation MA = 0C) Scalar equation MZ = 0D) Scalar equation MY = 0