Structural Design I · Cantilever truss Transfer truss. 8 Terminology and nomenclature >> Statical determinacy of trusses ... The system is statically indeterminate. Statical determinacy

1http://www.block.arch.ethz.ch/eq/

Structural Design IIPhilippe Block ∙ Joseph Schwartz

2

6. Trusses 7. Beams 8. Frames 9. Slabs

1. IntroductionArchitecture and Structure

3. Cables 4. Arches 5. Arch-Cable Structures2. Graphic Statics

10. Columns 11. Spatial Structures

Structural Design II

Structural Design I

3

>> Terminology and nomenclature

Statical determinacy of trusses

Analysis of trusses

Trusses as a combination of arch-cable structures

Determination of internal forces

Influence of the position of external forces

Determination of the maximal member force

Geometrical variations of trusses

Different types of trusses

Additional types of trusses

Construction of trusses

Planar trusses

4Terminology and nomenclature

Renzo Piano, Richard Rogers, Peter Rice: Centre Pompidou, Paris, 1977


8m 2m

2m

2m

Depth or Rise

(statische Höhe)Diagonal (Strebe)

Diagonal member

Members:

Pile (Pfosten)

Vertical member

Column

(Stütze)

Node / Joint

(Knoten)

Top chord (Obergurt)

Horizontal member

Bottom chord (Untergurt)

Horizontal member


- All connections are pin joints

- All elements are straight

- Loads only apply on nodes

- Only axial forces occur

- Trusses consist out of stiff triangles

Assumptions


Trussed beam Trussed frame

Cantilever truss Transfer truss

8

Terminology and nomenclature

>> Statical determinacy of trusses

Analysis of trusses









Planar trusses

9Statical determinacy of trusses

S = Number of members (Stäbe)

K = Number of nodes (Knoten)

A = Number of support restraints (Auflager)

S + A < 2 * K

S + A = 2 * K

S + A > 2 * K

The system is unstable/ a mechanism

The system is statically determinate

The system is statically indeterminate


IV

III

I

II

IVI

II III

A

B

BV

BV

BH

A

BV

BH

A

H

II

I

III

IV

2

2

2

4

4

4

5

5

6 3

3

3

1

1

1

S + A = 7 < 2 * K = 8

The system is unstable/ a mechanism.

S + A = 8 = 2 * K = 8

The system is statically determinate.

S + A = 9 > 2 * K = 8

The system is statically indeterminate.

S = 4

A = 3

K = 4

S = 5

A = 3

K = 4

S = 6

A = 3

K = 4


IV

III

I

II

IVI

II III

A

B

BV

BV

BH

A

BV

BH

A

H

II

I

III

IV

2

2

2

4

4

4

5

5

6 3

3

3

1

1

1I

II

III

IV

BV

BH

AV

AH

2

31

II

I

III

IV

BV

BH

AV

AH

2

31

S + A = 7 < 2 * K = 8

The system is unstable/ a mechanism.

S = 3

A = 4

K = 4

12



>> Analysis of trusses









Planar trusses

13Analysis of trusses

7 kN 7 kN 7 kN7 kN7 kN7 kN 7 kN


R


24.5 kN24.5 kN

R

A

B

R

BA

Form diagramScale 1 : 100

Force diagram

Scale 1 cm ≙ 1 kN


AForm diagram

Scale 1 : 100

Force diagram

Scale 1 cm ≙ 1 kN



Force diagram

Scale 1 cm ≙ 1 kN



Force diagram

Scale 1 cm ≙ 1 kN



Force diagram

Scale 1 cm ≙ 1 kN



Force diagram

Scale 1 cm ≙ 1 kN



Force diagram

Scale 1 cm ≙ 1 kN



Force diagram

Scale 1 cm ≙ 1 kN



Force diagram

Scale 1 cm ≙ 1 kN



Force diagram

Scale 1 cm ≙ 1 kN


R

R




eQ: Pratt / Howe truss

http://www.block.arch.ethz.ch/eq/drawing/view/37



29

Introduction



Analysis of trusses


>> Trusses as a combination of arch-cable structures







Case study: Lufthansa maintenance hall

Planar trusses

30Trusses as a combination of arch-cable structures

Q/2

Q

Q

Q/2

Q/2 Q/2

Combination


Q

Q

Q/2Q/2

Q/2Q/2

Combination


Q

Q

Q/2Q/2

Q/2Q/2

Combination


R. Piano, R. Rogers, P. Rice: Centre Georges Pompidou, Paris, 1977


Ludwig Mies van der Rohe: Project, National theater, Mannheim, 1953


Q

Q

Q/2 Q/2

Q/2 Q/2

Force distribution


Q

Q

Q/2Q/2

Q/2Q/2

Force distribution


Q

Q

Q/2 Q/2

Q/2 Q/2

Force distribution

38



Analysis of trusses


>> Determination of internal forces







Planar trusses

39Determination of internal forces

500 kN

1000 kN

500 kN

1000 kN

500 kN

500 kN


Force diagram

Scale 1 cm ≙ 1 kN

40Determination of internal forces


Force diagram

Scale 1 cm ≙ 1 kN

41



Analysis of trusses



>> Influence of the position of external forces






Planar trusses

42Influence of the position of external forces

200 kN 800 kN

700 kN

R

800 kN

R

200 kN

700 kN

300 kN

300 kN

R


Force diagram

Scale 1 cm ≙ 1 kN


800 kN 200 kN200 kN

700 kN

800 kN

300 kN

700 kN 300 kN


Force diagram

Scale 1 cm ≙ 1 kN



Force diagram

Scale 1 cm ≙ 1 kN

45



Analysis of trusses




>> Determination of the maximal member force





Planar trusses

46Determination of the maximal member force


800 kN

700 kN

700 kN

800 kN

200 kN200 kN

300 kN

N3

N 2 N

N1

200 kN 800 kN

700 kN

N3

N

N1

N2


N3

700 kN

200 kN

N 2

300 kN

800 kN

300 kN

N N3

N 2

N1

N1N 300 kN


N5 N6

5 6

N7N N

300 kN

N4

700 kN

300 kN

N4

700 kN

N7

800 kN200 kN


Force diagram

Scale 1 cm ≙ 1 kN

50



Analysis of trusses





>> Geometrical variations of trusses




Planar trusses

51Geometrical variations of trusses

Q

Q/2

Q

Q/2

Q/2

Q

Q/2

Q/2 Q/2

Q/2Q/2

Q


Force diagram

Scale 1 cm ≙ 1 kN


Q1

Q1

A

A

A

B

Q 2

Q 2 B

A

Q 2Q1

Q1 Q 2

B

B

h

2h


Force diagram

Scale 1 cm ≙ 1 kN


A

Q 2

Q1

Q1 Q 2

A

Q 2

A

A

B

Q1

Q 2

B

Q1

B

B

h

h/2


Force diagram

Scale 1 cm ≙ 1 kN

54



Analysis of trusses






>> Different types of trusses



Trusses

55Different types of trusses

Truss with V-shaped diagonals (System Warren)



R. Piano, R. Rogers, P. Rice: Centre Georges Pompidou, Paris, 1977


A

A

A

=

+


Joseph Paxton, Fox Henderson: Crystal palace, London, 1851


Truss with N-shaped diagonals (System Howe), patent 1840


Structure proposed by A. Palladio in 1570 („The four books of architecture“, 3. book, chapter VIII)


Truss with N-shaped diagonals (System Pratt), patent 1844


Ludwig Mies van der Rohe: Project, National theater, Mannheim, 1953


A

A

A +

=


Truss with X-shaped diagonals and posts (System Long), patents 1830 and 1839


Double decker Fairey Swordfish I, Great Britain, 1934


Truss with X-shaped diagonals without posts





A B

A B


Friedrich August von Pauli, Heinrich Gerber, Ludwig Werder: Isar bridge, Grosshesselohe, 1857


A B


BA


BA


B

BA

A

Truss system Polonceau


Jakob Ignatz Hittorf: Gare du Nord, Paris, 1865


Jakob Ignatz Hittorf: Gare du Nord, Paris, 1865

79



Analysis of trusses







>> Additional types of trusses


Trusses

80Additional types of trusses

100 kN100 kN100 kN

100 kN

100 kN

100 kN

100 kN

100 kN

RA

A

BB

R

R


Force diagram

Scale 1 cm ≙ 1 kN


RA

A

B

R

B


Force diagram

Scale 1 cm ≙ 1 kN


RA

A

BB


Force diagram

Scale 1 cm ≙ 1 kN


A

B

Q 2

Q1

Q 3 Q 4 Q 5 Q 6Q1 Q 2

A

Q 3

Q 5

Q 4

Q 6

B


Force diagram

Scale 1 cm ≙ 1 kN


Skidmore, Owings & Merrill, F. Kahn: John Hancock Center, Chicago, 1969


180 kN180 kN

200 kN

180 kN

180 kN

200 kN

180 kN

180 kN

200 kN

R

BA

R

B

A


Force diagram

Scale 1 cm ≙ 1 kN


180 kN

200 kN

180 kN

200 kN

180 kN

180 kN

V

IV

III

II

I A

I V

II

IV

A

BIII B

180 kN

171 kN

113 kN

200 kN

447 kN

180 kN

171 kN

308 kN 247 kN 308 kN

308 kN 308 kN

300 kN

171 kN

247 kN

171 kN

300 kN


Force diagram

Scale 1 cm ≙ 1 kN


0.5*l

l

l

l

f 1

f 3 f 4f 2

88



Analysis of trusses








>> Construction of trusses

Planar trusses

89Construction of trusses

Gustav Eiffel, Theophile Seyring: Maria Pia bridge, Porto, 1877






Squire Whipple, Cairo Rail Bridge, Illinois, USA, 1899


Ceiling tile

Lighting fixture

Air-conditioning slit

Minoru Yamasaki, John Skilling, Leslie Robertson: World Trade Center, New York, 1973, destroyed 2001

Concrete Floor tiles

Telephone and power outlets

Corrugated steel truss covering

Heating , ventilation, air-conditioning


Squire Whipple, Cairo Rail Bridge, Illinois, USA, 1899


Sir John Fowler, Benjamin Baker, Firth of Forth Bridge, Edinburgh, 1890


Sir Horace Jones, Sir John Wolfe Barry: Tower Bridge, London, 1894


Office Eiffel, Maurice Koechlin: Eiffel tower, Paris, 1889


Frederic-Auguste Bartholdi, Gustave Eiffel: Statue of Liberty, New York, 1886



Lifschutz, Techniker: Royal Victoria Dock Bridge, London, 1996


Schulhaus Leutschenbach Zürich, 2009, Architekt: C. Kerez, Bauingenieur: J. Schwartz





http://www.bgs-architekten.ch/projektseiten/NSL.html

Documents

Structural Design I · Cantilever truss Transfer truss. 8 Terminology and nomenclature >> Statical determinacy of trusses ... The system is statically indeterminate. Statical determinacy