Rigid body equilibrium

Equilibrium of a Rigid Body5
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

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

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

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

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

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
NBBNM 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 member
Solving,
kNF
kNFA
32.1
07.1


045sin3.60sin;0
040045cos3.60cos;0
3.60
4.0
7.0
tan 1








 



FFF
NFFF
Ay
Ax


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

5.5 Free-Body Diagrams

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

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

• 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 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
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

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 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

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

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
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) 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

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.

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

Rigid body equilibrium

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