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1http://www.block.arch.ethz.ch/eq/
Structural Design IPhilippe Block ∙ Joseph Schwartz
2
3
Graphic statics
>> Case study - suspension bridge
Relation between form and forces in cables
4Case study - suspension bridge
5Case study - suspension bridge
6 m 6 m
12 m
N 2
N 3
B
F=12 kN
F=12 kN
I
I
N1
N1
A B
N3
N 2
A
32
32
2 3
1
1
1
Subsystems
6Case study - suspension bridge
6 m 6 m
12 m
N 2
N 3
B
F=12 kN
F=12 kN
I
I
N1
N1
A B
N3
N 2
A
32
32
2 3
1
1
1
Subsystems
F=12 kNN1
7Case study - suspension bridge
8Case study - suspension bridge
3
1
2
9Case study - suspension bridge
3
1
2
10Case study - suspension bridge
N2
11Case study - suspension bridge
N3
N2
12Case study - suspension bridge
N2
N3
13Case study - suspension bridge
12 kN
8.5 kN
3
2
8.5 kN
I
1
F
FBA I
32
Force diagram
Scale 1 cm = 1 kN
Form diagram
N2
N3
14Case study - suspension bridge
F 12 kN
8.5 kN
B
8.5 kN
2
A
1
F
BA
2 3
3
Force diagram
Scale 1 cm = 1 kN
Form diagram
N2
N3
N1
15Case study - suspension bridge
F 12 kN
2
A 2
12 kN
A11
A 2B2
F
BOption 1
Option 2
12 kN
B2
3
1
2 3
Force diagram
Scale 1 cm = 1 kN
Form diagram
N2
N3
N1
16Case study - suspension bridge
8.8 kN
1
F 12 kN
8.8 kN
BA1
A 3
B 3
I
A 3
B 3
F
1Option 1
Option 3
3
2
2
3
Force diagram
Scale 1 cm = 1 kN
Form diagram
N2
N3
N1
17Case study - suspension bridge
A 4
11 kN
B4
F
Fractured rock
9 kN
B4
3
F 12 kNA 4
B1
1
2
3
2
Force diagram
Scale 1 cm = 1 kN
Form diagram
N2
N3
N1
18Case study - suspension bridge
12 kN20 kN F
A 5
B5
F
B1
14 kN
B5
3
A 5
2
Fractured rock
3
2
1
Scale 1 : 100
Force diagram
Scale 1 cm = 1 kN
Form diagram
N2
N3
N1
19Case study - suspension bridge
14 kN
B5
3
20 kN
2
Fractured rock
3
12 kNFA1B1
A1
F
B5
1
2
Scale 1 : 100
Force diagram
Scale 1 cm = 2 kN
Form diagram
N2
N3
N1
20Case study - suspension bridge
eQ: Pedestrian Bridge 1
http://www.block.arch.ethz.ch/eq/drawing/view/2
21Case study - suspension bridge
12 m
4 m 4 m 4 m
F2 F1
BA
5 3
24 1
22Case study - suspension bridge
F1
8 kN
N5
II
N 4 N1
N5 N3
N3
N1
I
N 4
N2N2
F1 = 8 kNF2 = 8 kN
BA
5
5
2 2
4
3
3
4 1
1
Subsystems
Force diagram
N1
23Case study - suspension bridge
8 kN
I
IN2
1
F2 F1
BA
N1
N3
2
3
2
3
1
SubsystemForm diagram
Force diagram
N1
24Case study - suspension bridge
8 kN
I
I
I
B
N2
1
F2 F1
N3
N1
A
3
2
3
2
1
SubsystemForm diagram
Force diagram
N1
25Case study - suspension bridge
I
I
N3
N1
A B
I
N2
F2 F1
2
3
2
3
1
1
SubsystemForm diagram
Force diagram
N1N3
N2
26Case study - suspension bridge
9 kN
II
N5
II
N 4
N 2
A B
4
F2 F1
5
5
4
2
2
SubsystemForm diagram
Force diagram
N2
27Case study - suspension bridge
8 kN
9 kN
II
II
4
F2 F1
N5
II
B
N 2
N 4
A
5
4
2
5
2
SubsystemForm diagram
Force diagram
N4
N2
28Case study - suspension bridge
12 kN
9 kN
8 kN
II
II
N5
II
N 4
N 2
A B
F2 F1
42
5
2
4
5
SubsystemForm diagram
Force diagram
N4
N2
N5
29Case study - suspension bridge
A B
F2 F14 F2
A
F1
BB
A4 F2
F1
3
3
5 3
24
5 5
22
2
1
1
1
Force diagramForm diagram
N4
N2
N5
N1N3
N2
N4
N2
N5
N1N3
30Case study - suspension bridge
F1
F2
A B
F2 F1
5 3
24 1
Force diagramForm diagram
31Case study - suspension bridge
F2F2 F1
BA F1
5 3
24 1
Force diagramForm diagram
N2
N3
32Case study - suspension bridge
F2
A
F2 F1
F1B 3
5 3
24
5
2
1
Force diagramForm diagram
N2
N3
N5
33Case study - suspension bridge
A
2 F1
F1
4 F2F5
A
BB 3
3
4
5
2
2
1
1
Force diagramForm diagram
5
N2
N3
N5 N4
N1
34Case study - suspension bridge
F1
4 F2
A Fractured rock
AF2 F1
42
35
5
2
3
1
1
Force diagramForm diagram
N2
N3
N5 N4
N1
35Case study - suspension bridge
B
A Fractured rock
A4 F2
F1
F2 F15
2
3
3
5
24 1
1
Force diagramForm diagram
N2
N3
N5 N4
N1
36Case study - suspension bridge
F1
B
F2 F1
Fractured rockA
4 F2
A
32
5
5
24
1
1
Force diagramForm diagram
N2
N5 N4
N1
37Case study - suspension bridge
F2 F1
B3
B
A 4 F2
F1Fractured rock
3
1
5
2
1
Force diagramForm diagram
N2
N3
N5
N4
N1
38Case study - suspension bridge
F2 F1
B3
B
A 4 F2
F1Fractured rock
3
2
1
5
2
1
Force diagramForm diagram
N2
N3
N5
N4
N1
39Case study - suspension bridge
F2 F1
Fractured rock F1
4 F2A
B
B3
4
2
5
3
2
1
1
Force diagramForm diagram
N2
N3
N5
N4
N1
40Case study - suspension bridge
F2 F1
B
B
A 4 F2
F1Fractured rockA
3
2
3
5
2
4
5
1
1
Force diagramForm diagram
N2
N3
N5
N4
N1
41Case study - suspension bridge
B
B
A 4 F2
F1
A
R
Fractured rock
F2 F1
2
3
5
2
4
3
5
1
1
Force diagramForm diagram
N2
N3
N5
N4
N1
42Case study - suspension bridge
A
B
B
F2 F1
R
AFractured rock F1
4 F2
2
3
5
2
4
3
5
1
1
Force diagramForm diagram
N2
N3
N5
N4
N1
R
43Case study - suspension bridge
A
B
B
F2 F1
R
AFractured rock F1
F2
24
3
5
1
Force diagramForm diagram
R
44Case study - suspension bridge
eQ: Pedestrian Bridge 2
http://www.block.arch.ethz.ch/eq/drawing/view/3
45Relation between form and forces in cables
H 1
A
B
F
F
F
F
F
V
IV
III
II
I
A B
IIIIIIIV
V
F F F F F
Force diagramScale 1 cm = 1 kN
Form diagramScale 1 : 100
„Pole“
46Case study - suspension bridge
47Case study - suspension bridge
F
A B
B
B
F
B A
A
B
F
A
F
B
B
F
F
B
F
F
A A
A
A
Case study - suspension bridge 43
48
3.0 m
8.0 m
4.0 m 4.0 m
Q = 25 kN
B
F tot
F tot= 25.0 kN + 5 kN = 30kN
N1 =A =25 kN
N2 =B =25 kN
BA
A
G = 5 kN
21
Repetition Dimensioning
Today: What is the dimension of a cable‘s profile so that the structure does not collapse?
So far: What is the magnitude of a force?
N2
N1
49
4.0 m 4.0 m
8.0 m
3.0 m
1
A
G d= 6.7 kN
Q d = 1.35* 25.0 kN + 1.5* 5 kN = 44.2 kN
A B
B2
Fd
N1,d =B =36.9 kN
N 2,d=A =36.9 kN
Nd 36900 N2
Aerf ≥ += 2 = 165 mm
f s1,d235 N/mm / 1.05
Q d= 37.5 kN
21
F = 1.35* G +1.5* Qd
Fd = 1.35* 25 kN +1.5* 5 kN = 44.2 kN
N1 = A = 36.9 kN (aus Kräfteplan)
N2 = B = 36.9 kN (aus Kräfteplan)
Nd 36‘900 N
fs1,d
+ 235 N/mm2 /1.05
Aerf
= = 165 mm2
Repetition Dimensioning: ULS (Ultimate limit state)
=
N2
N1
50
4.0 m
3.0 m
8.0 m
4.0 m
Q k = 1.0* 25.0 kN + 1.0* 5 kN = 30kN
N1 =A =25 kN
N 2 =B =25 kN
N i * l i 25000 N * 5000 mm∆l 1 = ∆l 2 = = 2 2 = 3.4 mm
A i * E 177 mm * 210000 N/mm
B
F k
B
G = 5 kN
w=
5.5
mm
A
A
Q = 25 kN
2
2
1
1
Fk = 1.0* G +1.0* Q
Fk = 1.0* 25 kN +1.0* 5 kN = 30 kN
N1 = A = 25 kN (aus Kräfteplan)
N2 = B = 25 kN (aus Kräfteplan)
Ni * li
Ai * E
25‘000 N *5000 mm
177 mm2 * 210‘000 N/mm2
∆ l i=
∆ l 1=∆ l 2= = 3.36 mm
Repetition Dimensioning: SLS (serviceability limit state)
N2
N1
51
Graphic statics
Case study - suspension bridge
>> Relation between form and forces in cables
52Relation between form and forces in cables
L/2L/2
L/2 L/2
F
A BB
2F
BA
2F
A
B
2
2
1
1
Force diagramScale 1 cm = 1 kN
Form diagramScale 1 : 100
A
F
N2
N1
N2
N1
53Relation between form and forces in cables
L/2
L/2
L/2
L/2
A
B
F
A
F
B
F
A B
A B
F
2
2
1
1
Force diagramScale 1 cm = 1 kN
Form diagramScale 1 : 100
N2
N1
N2
N1
54Relation between form and forces in cables
1/4 L
L/2L/2
3/4 L
F
A
F
B
A
F
B
F
A B
AB
2
2 1
1
Force diagramScale 1 cm = 1 kN
Form diagramScale 1 : 100
N2
N1
N2
N1
55Relation between form and forces in cables
L/2L/2
A
F
B
F
BA
A
F
2
B
F
A B
2
1
1
Force diagramScale 1 cm = 1 kN
Form diagramScale 1 : 100
N2
N1
N2
N1
56Relation between form and forces in cables
H 1
A
B
F
F
F
F
F
V
IV
III
II
I
A B
IIIIIIIV
V
F F F F F
Force diagramScale 1 cm = 1 kN
Form diagramScale 1 : 100
57Relation between form and forces in cables
Schlaich Bergermann: Factory building, Dachwig, 1993
58Relation between form and forces in cables
H 2
H 1
I
IIIII
IV
V
A B
V
IV
III
II
I
A
B
F
F
F
F
F
F F F F F
Force diagramScale 1 cm = 1 kN
Form diagramScale 1 : 100
59Relation between form and forces in cables
Schlaich Bergermann: Pedestrian bridge, Bad Windsheim, 1988
60Relation between form and forces in cables
H 3
H 1
V
IV
III
II
I
F
F
F
F
A
B
F
I
IIIII
IV
V
A B
F
FF
F
F
Force diagramScale 1 cm = 1 kN
Form diagramScale 1 : 100
61Relation between form and forces in cables
Julia Bohn: Pedestrian bridge, München, 1992
62Relation between form and forces in cables
Lord Norman Foster: Renault Center, Swindon GB, 1983
63Relation between form and forces in cables
Lord Norman Foster: Renault Center, Swindon GB, 1983
64
