Structural Design II - ETH Z · Structural design I Structural design II Structural design I+II 1. ... Reaction forces ... Distortion of a bent beam segment 11. Bending

Structural Design II

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

Philippe Block ∙ Joseph Schwartz

Course overview

Structural design I Structural design II

Structural design I+II

1. Introduction

2. Equilibrium and graphic static

3.+4. Cables

5. Cable-net and membrane struc-tures

7. Arches

8.+9. Vaults, domes and shells

11. Spatial arch-cable-structures

10. Arch-cable-structures

6. Materials and dimensioning

12.+13. Trusses

12. Spacial trusses

14.+15. Beams and frames

16. Shear walls and plates

20. Columns

17.+18.+19. Bending

2

A

B

F

F

F

R

B

A

A

R

R

F F

FF F FR

RA

B

B

R

F

F

FR

R

Repetition

Comparison of structures consideration as an ideal line system, without and with girder height

1. Resultants

2. Reaction forces

without height with height

3

y2

T2

T1

z2

z1

T1

y2

T2

y1

y1

T2

T1

c2

A B

F F

F

F

F

F

F

A

NF

A

F

1

F

A

M = T1*y1

F

N

F

B

B

A

A

N

F

A

N

N

F

N

FNc1

t1

t1

c1

N2c2

M = T2*y2

t2

t2

F

F

Repetition

Comparison structures consideration as an ideal line system without and with girder height

Analogy thrust line - funicular line internal forces with arch-cable

4

M1 = M2 M1 = M2

F F

A B

B

FN

t

F

FN

Nv

F

F

Nc2

N

t

B

N

c

NB

Av c1

F

F

A B

BA

FF

A

t

cNN

Nv

Repetition


Shear diagram(V-line)

Subsystems

Moment diagram (M-line)

5

Repetition


Shear diagram(V-line)

Moment diagram (M-line)

Truss

Arch-cable structure

Stress field

6

Repetition


7

M

V

N

Bending

Bending and flectionBending stiffnessDeformation of bending beamsResistance of bending beamsStructures efficiency

8

Nc cN

tNNt

A B

F F

Bending and flection

Bending beam with flection

9

2

∆l

∆l

∆l

∆l∆l

1

5 5

2

∆l

∆l

44

∆l

1

Nc5

Nc4

c5N

c4N

Nt1 Nt1

Nt2 Nt2


Distribution of the deformation along the girders heigth

10

rc

φ

r

th r z

cN

Nt

Nc

Nt

x = 1/r

𝑥 = 1 r


Distortion of a bent beam segment

11

Bending


12

φ

r

M

2

2

φ

* z = * z = * z = * z =

1 z

21

z r

1

c1

t2

t1

c1

t1

t2

c2

c2 c2

c1

t1

t1 NN

N

N

N

N

N

N

NN

N

N

φ

r

M

2

2

φ

* z = * z = * z = * z =

1 z

21

z r

1

c1

t2

t1

c1

t1

t2

c2

c2 c2

c1

t1

t1 NN

N

N

N

N

N

N

NN

N

N

Bending stiffness

Bending at different beams heights

13

∆l

∆l [mm]120

10

20

30

240 360

1∆l [mm]N [kN]S =

N

∆l

N [kN]

N = S *

Bending stiffness

Force-deformation diagram

14

φ

zr

* z = * z = M [Nmm]NN c

χ = 1/r [1/mm]

tN

cN

tN

Nc

t

ΧZ * zB =

1 φ

zr

* z = * z = M [Nmm]NN c

χ = 1/r [1/mm]

tN

cN

tN

Nc

t

ΧZ * zB =

1

χ = 1 r

χ

Bending stiffness

Bending moment-flection diagram

15

σ

z

σ

h

φ

r

t

Nc

tN

cN

Nt

z = 2/3 * h

Bending stiffness

Internal forces in a bent beam with rectangualr section at linear-elastic material behaviour

16

φ/2

2*h

h

t

φ/8

φ

h2*h

t

φ/2

2*t

φ φ/8

2*t

NcN

Nc

tN Nt Nt Nt

Nt Nt

cN cNcN

c

Bending stiffness

Influence of width and height of the cross-section to the bending strenght

17

Flansch

Flansch

Steg

A1 = A2

Steg

A2A1

Bending stiffness

Rectangular section and I-section with same height and same amount of material

Web

Flange

Flange

18

φ

r

b

z

t

h

Nt

Nc

cN

tN

Bending stiffness

Internal forces in a bent beam with I-section at linear-elastic material behaviour

19

z b

t

t

φ

r

cN /2

tN /2N /2c

N /2c

N /2c

tN /2

tN /2tN /2

z = 2/3 * b

Bending stiffness

Internal forces in a 90o bent beam with I-section at linear-elastic material behaviour

20

Bending


21

w

h

l/4

t

l/4

l

A B

A B

F F

F F

Deformation of bending beams

Deflection of a beam with two single loads and the influence of the span l on the bending

22

2*l/4 2*l/4

8*w

2*l

BA

A B

F

FF

F


Deflection of a beam with two single loads and the influence of the span l on the bending

23

wz

l

cN

Nt

F

B

AF

F

Bv


Deflection of a cantilever with point load

24

w`

l

w

0.5*l

tN

cN

q

BA

q

Bv

w > w`


Deflection of a beam and a cantilever with distributed load

25

f

l

l`/l = 0.5

l`

l`/l = 0.4

0.84f

w = 38%

f

l`/l = 0.35

w = 23%

0.16fl

l`

0.5f

w

l`

l

w = 80%

l`

0.33f

l`

l`/l = 0.2

l` = 0

w = 100%

w = -20%

l`

0.5f

0.67f

l

l` l l`


Arch-cable structures and the deflection of a overhanging beam

26

Zug: Verlängerung

Zug: Verlängerung

Zug: Verlängerung

Druck: Verkürzung

Druck: Verkürzung

Druck: Verkürzung


Deflection of continuos beam

Tension: extension

Tension: extension

Tension: extension

Compression: shortening



27

Bending


28

Bruch

Materialverhalten

Χ = 1/r [1/mm]

M

linear elastischesMaterialverhalten

1

z h

* z = * z = M [Nmm]

BB

1

plastisches

Nt Nc

Nt

Nc

χ = 1 r

Resistance of bending beams

Bending-flection relation at plastic material behaviour

linear elastic material behaviour

plasticmaterial behaviour

29

l

h/2

h/2

h

t

h/2

a

f * (t*h)/2

a

h/2

R,dQQR,d

QR,dQR,d

R,dQ

R,dQ

R,dQ R,dQ

R,dQ QR,d

QR,d

Nt

cN

d


Mechanical resistance of a beam at plastic material behaviour

30

a

h

t

h

a

h

a

2h

f * (t*h)/2

f * (t*h)

h

a

t

l

l

h

R,dQQR,d

R,dQR,d

QR,dQR,d

R,dQ QR,d

QR,d

QR,d

Q

d

d


Collapse load comparison between a beam with double height and two beams on top of the other

31

h

b

f * t *b

z = h

- t

t

a a

t

l

R,dQ

QR,d QR,d

QR,d QR,d

d f

f

f


Collapse load comparison of a beam with I-section

32

Bending


33

Structures efficiency

Comparison of cross-sections (under vertical loads) with increasing efficiency under bending stress from left to right

efficiency under bending stress

34

F

F

F B

B

B

A

A

A

A

F

A

F

B

B


Possible cross-sections for a beam loaded in the middle

35

12

8

f

2

205

14[m3]

Volumen (Q *l)

0

10

4

15 25

variable Querschnitte

konstante Querschnitte

6

l/h10

d

d


Required amount of material as a function of the slenderness l/h

constant section

variable section

36

l/h2010

0.004

0.012

0

variable Querschnitte

0.010

0.014

konstante Querschnitte

15 255

0.002

0.006

w/l

hw

l0.008

F

BA


Deflections at the midspan as a result of a centered load as a function of the slenderness l/h (εmax = 0.001)

constant section

variable section

37

prof. schwartz Tragwerksentwurf III 2Allgemein

Links: Erforderliche Materialmengen in Funktion der Schlankheit l/h Rechts: Richtwerte für die Schlankheit von Tragwerken

* Fachwerk mit konstanten Diagonalen * Bei Konsolen entspricht l der zweifachen Spannweite der Konsole, bei Balken

mit Konsolen bzw. Durchlaufträgern entspricht l dem Abstand der beiden

zwischen den Auflagern liegenden Gelenken.

l

l

l

l/h


Useful slenderness rate of structures

Material Structure

Vierendeel girderTruss girderSpace trussBeamBeamPlate

BeamPlate

Beam

economic slenderness

Steel

Wood

Prestressedconcrete

Reinforcedconcrete

38

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Structural Design II - ETH Z · Structural design I Structural design II Structural design I+II 1. ... Reaction forces ... Distortion of a bent beam segment 11. Bending