Equilibrium of Rigid Bodies

1
Lecture 15

• Equilibrium of Rigid Bodies
• Equations of Statics
• Free Body Diagrams
• Idealization of 2D Supports and Connections
• 7 Example Problems

2
Equations of Statics
y

• SFx 0 SMx 0
• SFy 0 SMy 0
• SFz 0 SMz 0

x
o
z
Use the right hand rule !
Lecture 15
3
Free Body Diagrams

• Isolate the rigid body (or part) from all other
  interacting bodies and draw it to scale.
• Draw all forces, known and unknown acting on that
  body as vectors (magnitude, direction, sense)
• Show all dimensions and the coordinate system.

Lecture 15
4
Idealization of 2D Supports and Connections

• gravitational attraction

Line of action of the force W passes through the
center of gravity of the body.
W
W
Lecture 15
5
Idealization of 2D Supports and Connections

• flexible chord, rope, chain of cable

(tension only)
q
reference line
R
line of action
Lecture 15
6
Idealization of 2D Supports and Connectors

• rigid link

two force member (tension or compression)
line of action
Lecture 15
7

Idealization of 2D Supports andConnections

• ball, roller or rocker

Lecture 15
R

• Line of action is perpendicular to the
  surface supporting
• the roller.

• If smooth surface normal force (R)
  perpendicular to the
• surface (contact force - no friction).

• If rough surface normal force (R) and
  friction force (mN),
• where m -
  coefficient of sliding friction .

8
Idealization of 2D Supports andConnections

• pin or hinge

Rx
Ry

• Free to rotate.

• Fixed against translation into x and y
  directions.

Lecture 15
9
Idealization of 2D Supports andConnections
Idealization of 2D Supports andConnections

• pin in smooth guide

Ry
line of action

• Free to rotate.

• Free to translate into x direction.

• Fixed against translation into y direction.

Lecture 15
10
Idealization of 2D Supports andConnections

• pin in smooth guide

Ry
line of action

• Transmits force (Ry), only perpendicular to
  the surface
• of the guide.

(similar to a collar on a smooth shaft pinned
connected)
Lecture 15
11
Idealization of 2D Supports andConnections

• fixed support

Mz
Rx
Ry
Lecture 15
12
Idealization of 2D Supports andConnections

• linear elastic spring

F ks
k spring constant (N/mm)
s
s deformation of the spring (mm)
k
line of action
Lecture 15
13
Idealization of 2D Supports andConnections

• pulley

T
Rx
Ry
T
Lecture 15
14
Example 1

• Determine the equations of statics.

h
o
30
Lecture 15
15
Example 1
All lines of action intersect in the
weight center of the disk.
FBD
W
o
30
F1
h
q
F2
o
30
Lecture 15
16
Example 1
W
FBD
SFx 0
o
30
F1
F1 cos(q 30) - F2 cos(60) 0
h
q
F2
o
30
SFy 0
-W F1 sin(q 30) F2 sin(60) 0
Lecture 15
17
Example 2

• Determine the equations of statics.

P
A
B
o
45
l/2
l/2
Lecture 15
18
Example 2
P
y
A
Ax
Bx
B
x
B
o
Ay
By
45
l/2
l/2
FBD
Lecture 15
19
Example 2
FBD
P
y
A
Ax
Bx
B
x
B
o
Ay
By
45
l/2
l/2
SFx 0 ?
Ax - B cos(45) 0 (1)
Lecture 15
20
Example 2
FBD
P
y
A
Ax
Bx
B
x
B
o
Ay
By
45
l/2
l/2
SFy 0 ?
Ay - P B sin(45) 0 (2)
Lecture 15
21
Example 2
FBD
P
y
A
Bx
B
x
B
o
Ay
By
45
l/2
l/2
SMz(A) 0
-P l/2 B sin(45) l 0 (3)

Lecture 15
22
Example 3

• Determine the equations of statics.

A
D
l/3
B
o
30
2l/3
Lecture 15
23
Example 3
W
FBD - 1
Dx
A
C
Ax
C
o
30
l/3
B
Ay
o
30
2l/3
FBD - 2
By
Lecture 15
24
Example 3
W
Dx
FBD-1
C
o
30

• equilibrium equations

SFx 0
-Dx Csin(30) 0
SFy 0
-W Ccos(30) 0
SMz 0
Lecture 15
25
Example 3
A
Ax
C
FBD-2
Ay
l/3
B
o
30
2l/3

• equilibrium equations

By
Ax - C cos(60) 0 (1)
SFx 0
Ay - C sin(60) By 0 (2)
SFy 0
SMz(A) 0
-C l/3 By l cos(30) 0 (3)

Lecture 15
26
Example 4

• Determine the equations of statics.

M
A
B
4
P2
P1
C
5/2
5
Lecture 15
27
Example 4 FBDs
Lecture 15
A
M
B
By
M
A
B
Ax
Bx
P2
Ay
5
P2
P1
FBD -1
4
By
C
5/2
B
Bx
5
4
5
Cx
C
P1
FBD -2
Cy
28
Example 4 FDB - 1
By
M
Ax
A
B
FBD -1
Bx
Ay
5
P2
SFx 0
Ax Bx 0
SFy 0
Ay By - P2 0
SMA 0
M By(5) - P2(5) 0

Lecture 15
29
Example 4 FDB - 2

SMC 0
-P1(5/2) - By(5) Bx(4) 0
SFx 0
Cx - Bx 0
By
SFy 0
Cy - By - P1 0
B
Bx
4
5
Cx
C
P1
FBD-2
Cy
Lecture 15
30
Example 4 Summary
Lecture 15
FBD-1
By
M
Ax
A
B
SFx 0
Ax Bx 0
Bx
Ay
Ay By - P2 0
SFy 0
5
P2

SMA 0
M By(5) - P2(5) 0
By
NOTE 6 equations, 6 unknowns
B
Bx

SMC 0
-P1(5/2) - By(5) Bx(4) 0
4
5
SFx 0
Cx - Bx 0
Cx
C
P1
FBD-2
SFy 0
Cy
Cy - By - P1 0
31
Example 5

• Determine the displacement of the point F.

C
1
D
B
2
E
P1
F
P2
k
1
2
A
Lecture 15
32
Example 5 FBDs
Lecture 15
C
D
B
E
Cy
F
P1
s
Cx
F
P2
C
T
k
1
A
D
2
T ks
T
E
B
P1
Bx
P2
2
1
By
FBD -1
FBD -2
T
33
Example 5 FBD - 1
T
B
Bx
By
T
SFx 0
Bx - T 0
FBD -1
SFy 0
By - T 0
Lecture 15
34
Example 5 FBD - 2
SFx 0
P1 - T Cx 0
SFy 0
Cy - P2 0
SMC 0
P1(3) - T(1) P2(3) 0

Cy
Cx
C
1
T
D
2
E
P1
P2
2
1
FBD -2
Lecture 15
35
Example 6

• Determine the reactions at A and B.

20 Nm
A
B
C
800 mm
300 mm
frictionless surface at B
Lecture 15
36
Example 6 Solution
20 Nm
A
B
Ax
C
Ay
By
300 mm
800 mm

• equations of statics

SFx 0 Ax 0
SFy 0 Ay By 0
SMA 0 20 By(1.1m) 0

Ay 18.18 N
By 18.18 N
Lecture 15
37
Example 7

• Determine the reactions at A.

2 KN
3 KNm
A
B
C
2m
2m
Lecture 15
38
Example 7 Solution

• step 1 draw a FBD

2 KN
3 KNm
A
Ax
C
B
MA
FBD
Ay

• step 2 equations of statics

SFx 0 Ax 0
SFy 0 Ay - 2 0
SMA 0 MA - 3 - 2(4m) 0

Ax 0, Ay 2 kN, MA 11 kNm
Lecture 15
39
Free Body Diagrams and Equationsof Statics

• Suggested Problems
• 5-2, 4, 9, 10, 25, 26, 37, 46, 59