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```ENGI 1313 Mechanics I
Lecture 25:
Equilibrium of a Rigid Body
Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]
Lecture Objective
to illustrate application of 2D equations of
equilibrium for a rigid body
 to examine concepts for analyzing
equilibrium of a rigid body in 3D

2
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
Example 25-01

3
Determine the force
P needed to pull the
50-kg roller over the
smooth step. Take θ
= 60°.
© 2007 S. Kenny, Ph.D., P.Eng.
=
ENGI 1313 Statics I – Lecture 25
Example 25-01 (cont.)

What XY-coordinate
System be Established?
=
4
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
Example 25-01 (cont.)

Establish FBD

=
w = mg = (50 kg)(9.807 m/s2) = 490 N
5
© 2007 S. Kenny, Ph.D., P.Eng.
NB
NA
ENGI 1313 Statics I – Lecture 25
Example 25-01 (cont.)

Determine Force Angles

Roller self-weight


70
 = 20
 = 20
=
w = 490 N
6
© 2007 S. Kenny, Ph.D., P.Eng.
NB
NA
ENGI 1313 Statics I – Lecture 25
Example 25-01 (cont.)

Determine Force Angles

Normal reaction force at A

90
=
NA
7
© 2007 S. Kenny, Ph.D., P.Eng.
w = 490 N
NB
NA
ENGI 1313 Statics I – Lecture 25
Example 25-01 (cont.)

Determine Force Angles

Normal reaction force at B

yB = (0.6 m – 0.1 m)
= 0.5 m
  a cos
1
r = 0.6 m
NB
 0 .5 m 

  33 . 56
 0 .6 m 
=
x B  r sin   0 . 6 m sin 33 . 56
8
© 2007 S. Kenny, Ph.D., P.Eng.
w = 490 N


NB
NA
 0 . 3317 m
ENGI 1313 Statics I – Lecture 25
Example 25-01 (cont.)

Draw FBD
w = 490 N
 = 20
P
 = 60

NB
NA= 0 N
9
© 2007 S. Kenny, Ph.D., P.Eng.
=
w = 490 N
NB
NA
ENGI 1313 Statics I – Lecture 25
Example 25-01 (cont.)

What Equilibrium Equation
should be Used to Find P?

w = 490 N
 = 20
MB = 0
P
 = 60
w sin   y B   w cos   x B   
P cos   y B   P sin   x B   0
yB = 0.5 m xB = 0.3317 m
490 N sin 20  0 . 5 m   490 N cos 20  0 . 3317
P cos 60  0 . 5 m   P sin 60  0 . 3317   0



m

P  6 . 35 kN
10
© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 25
NA
NB
Comprehension Quiz 25-01

If a support prevents rotation
of a body about an axis,
then the support exerts a
________ on the body





11
A) Couple moment
B) Force
C) Both A and B
D) None of the above.
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
3-D Equilibrium

Basic Equations






F
x
0
M
x
0
F
y
0
M
y
0
M
z
0
F
z
0
Moment equations can also be
the rigid body. Typically the
point selected is where the
most unknown forces are
applied. This procedure helps
to simplify the solution.
12
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
Application to 3D Structures (cont.)

Engineering Design
Basic analysis
 Check more rigorous methods

13
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
Application to 3D Structures (cont.)
Axial Forces

Design of Experimental
Test Frame
Lateral
14
© 2007 S. Kenny, Ph.D., P.Eng.
Couple Forces
For Bending
ENGI 1313 Statics I – Lecture 25
3-D Structural Connections

Ball and Socket

15
Three orthogonal forces
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
3-D Structural Connections (cont.)

Single Journal Bearing

Two forces and two couple moments
• Frictionless
• Circular shaft

16
Orthogonal to longitudinal bearing axis
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
3-D Structural Connections (cont.)

Journal Bearing (cont.)

17
Two or more (properly aligned) journal
bearings will generate only support reaction
forces
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
3-D Structural Connections (cont.)

Single Hinge
Three orthogonal forces
 Two couple moments
orthogonal to hinge axis

18
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
3-D Structural Connections (cont.)

Hinge Design

19
Two or more
(properly aligned)
hinges will
generate only
support
reaction
forces
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
Rigid Body Constraints

What is the
Common
Characteristic?

20
Statically
determinate
system
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
Redundant Constraints

Statically Indeterminate System

21
Support reactions > equilibrium equations
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
Improper Constraints

Rigid Body Instability

2-D problem
• Concurrent reaction forces

22
Intersects an out-of-plane axis
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
Improper Constraints (cont.)

Rigid Body Instability

3-D problem
• Support reactions intersect a common axis
23
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
Improper Constraints (cont.)

Rigid
Body
Instability

24
Parallel
reaction
forces
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1

25
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 25
```
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