Other Lindenthal Projects
Smithfield Bridge, Pittsburgh 1882
Smithfield
Royal Albert Bridge at Saltash 1859
Balance of Forces in Equilibrium
Isambard Kingdom Brunel, Engineer
Brunel
Saltash
Balance of Forces in Equilibrium
Statics:Hibbeler
PULLEYS, FRAMES, AND MACHINES
Compute the forces in pulleys, ropes, beam, and frame members that
require multiple free-body diagrams.
Compute the reactions for compound beams, frames, or similar
devices.
Show why it is advantageous to: Separate elements at pins and draw
multiple FBDs & identify 2-force members
1
Pulley Systems
Pulley Systems – Mechanical Advantage
T
W
Free Body Diagrams of Pulley System
Pulley Systems – Mechanical Advantage
Reaction
FBD Top
T
T
T T T
FBD A
W
W
Pulley Systems – Other Cuts and FBDs
Pulley Systems – Other Cuts and FBDs
FBD
Fbar
FBD
Fbar
b
T
T
T
T T
W
Statics:Hibbeler
W
W
2
Pulleys – Method of Joints Approach
Pulleys – Method of Joints Approach
Pulleys – Method of Joints Approach
Cy
FBD 2
Ay
Pulleys – Method of Sections Approach
10 lb
P
R
R
R
P
R
P
FBD 3
10 lb
10 lb
P
FBD 1
100 lb
Pulleys – Method of Sections Approach
R
R
P
10 lb
10 lb
100 lb
Statics:Hibbeler
3
QUICK PROBLEM SOLVING
Given: A frame and loads
as shown.
Find: The reactions that
the pins exert on the
frame at A, B, and C.
Plan:
a) Draw a FBD of members AB and BC.
b) Apply the equations of equilibrium to each FBD to solve for
the six unknowns.
QUICK PROBLEM SOLVING (continued)
FBDs of members AB and BC:
QUICK PROBLEM SOLVING (continued)
FBDs of members AB and BC:
BY
B
1000N
AX
BX
BX
BY
BY
B
B
1000N
0.4m
C
0.2m 0.2m
0.2m
0.4m
CY
Summing moments about A and C on each member, we get:
+ MA = BX (0.4) + BY (0.4) – 1000 (0.2) = 0
+ MC = -BX (0.4) + BY (0.6) + 500 (0.4) = 0
BY = 0
and
BX = 500 N
BY
BX B
0.4m
500N
C
A X A 45º
500N
A 45º
AY
BX
0.2m 0.2m
For FBD AB:
0.2m
AY
 + FX = AX – 500 = 0 ;
AX =
 + FY = AY – 1000 = 0 ;
AY = 1,000 N
0.4m
CY
500 N
For FBD of BC:
 + FX = 500 – CX = 0 ;
CX = 500 N
 + FY = CY – 500 = 0 ;
CY = 500 N
WORKING EXAMPLE
Given: The wall crane supports an
external load of 700 lb.
Find: The force in the cable at the
winch motor W and the
horizontal and vertical
components of the pin
reactions at A, B, C, and D.
Plan:
a) Draw FBDs of the frame’s members and pulleys.
b) Apply the equations of equilibrium and solve for the
unknowns.
Statics:Hibbeler
4
EXAMPLE (continued)
350 lb
FBD of the Pulley E
CX
T
T
350 lb
FBD of pulley C
E
700 lb
350 lb
Equations of Equilibrium:
BY
+↑ Fy = 2 T – 700 lb = 0
30°
350 lb
T = 350 lb
TBD
A
AY
45°
4 ft
BX
+Fx = Cx – 350 = 0
lb
Cx = 350
+↑ Fy = Cy – 350 = 0
Cy = 350
lb
+FX = – Bx + 350 – 350 sin 30° = 0
Bx = 175 lb
B
+↑ Fy = By – 350 cos 30° = 0
FBD of pulley B
By = 303.1 lb
EXAMPLE (continued)
Note that member BD is a two-force
member
AX
C
CY
A FBD of member BD
350 lb
B
175 lb
303.11 lb
4 ft
2409 lb
D
700 lb
FBD of member ABC
45
°
+MA = TBD sin 45° (4 ft) – 303.1 (4 ft) – 700 (8 ft) = 0
TBD = 2409 lb
+FY = AY + 2409 sin 45° – 303.1 – 700 = 0
At D, the x and y component are
AY = – 700 lb
 +FX = AX – 2409 cos 45° + 175 – 350 = 0
 + DX = –2409 cos 45° = –1700 lb
B
2409 lb
 + DY = 2409 sin 45° = 1700 lb
AX = 1880 lb
Why are trusses like donuts?
Analysis of Machines
Donuts are not good in compression either
After the 5th one you feel kinda sick
Both = fun!
You can’t drive to school without them
If they fall to the ground they are not any
good anymore
Statics:Hibbeler
5
Pliers, Cutters, and Grips
APPLICATIONS
Frames are commonly used
to support external loads.
Find vertical clamping force at E
Find all reaction forces and forces at
pinned connections in press
How is a frame different than a
t
truss?
?
How can you determine the
forces at the joints and supports
of a frame?
Statics:Hibbeler
6
Elevation View of Cairo Bridge
(Source: S. Nam, University of Illinois)
View of Finite Element Model
Acc. (g)
2D Schematic of Cairo Bridge
1
0.5
0
-0.5
-1
0
10
20
30
40
Time (sec)
(Source: S. Nam, University of Illinois)
Statics:Hibbeler
7
Animation of Bridge Deflection
Graph of Maximum Deflection
Vertical Displacem ent
10
8
6
4
2
0
0
500
1000
1500
2000
2500
3000
3500
4000
bridge cood. (ft)
(Source: S. Nam, University of Illinois)
BACK
TEDA Stadium
Tianjin, China
Statics:Hibbeler
8