Structural Design II - ETH Z · 2017-04-27 · Structural Design II ... Bending-flection relation at plastic material behaviour linear elastic material behaviour plastic material

Structural Design II

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

Philippe Block ∙ Joseph Schwartz

Structural design I+II

1. Introduction

2. Equilibrium and graphic static

3.+4. Cables

13. Cable-net and membrane structures

5.+7. Arches

14.+15. Vaults, domes and shells

16. Spatial arch-cable-structures

6.+8. Arch-cable-structures

12. Materials and dimensioning

9. Trusses

17. Spacial trusses

10.+11. Beams and frames

16. Shear walls and plates

20. Columns

19. Bending

Structural design I

Structural design II

Course overview

Bending

Bending and flectionBending stiffnessDeformation of bending beamsResistance of bending beamsResistance of framesStructures efficiency

3

Nc cN

tNNt

A B

F F

Bending and flection

Bending beam with flection

4

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

5

rc

φ

r

th r z

cN

Nt

Nc

Nt

x = 1/r

𝑥 = 1 r


Distortion of a bent beam segment

6

Bending


7

r

M

r2

* z = * z =

z

1

1

φ φ

2 * z = * z = 2

z

1 t2

c1

t1

c1

t1

c2

t2

c1

c2

t1 NN N

N

N

N

N

N

NN

N

N

c2

t2

r

M

r2

* z = * z =

z

1

1

φ φ

2 * z = * z = 2

z

1 t2

c1

t1

c1

t1

c2

t2

c1

c2

t1 NN N

N

N

N

N

N

NN

N

N

c2

t2

Bending stiffness

Bending at different beams heights

8

∆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

9

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

10

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

11

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

12

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

13

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

14

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

15

Bending


16

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

17

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

18

wz

l

cN

Nt

F

B

AF

F

Bv


Deflection of a cantilever with point load

19

w`

l

w

0.5*l

tN

cN

q

BA

q

Bv

w > w`


Deflection of a beam and a cantilever with distributed load

20

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

21

Zug: Verlängerung

Zug: Verlängerung

Zug: Verlängerung

Druck: Verkürzung

Druck: Verkürzung

Druck: Verkürzung


Deflection of continuos beam

Tension: extension

Tension: extension

Tension: extension

Compression: shortening



22

Bending


23

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

24

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

25

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

26

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

27

Bending


28

ξ = 1/3

ξ = 0

ξ = 1

ξ = 3

ξ = inf

1

5QQ

1.143=

1

3

Q

1

2

Q

2Q =Q

Q =Q

1Q

Q3

Q

1 31/3

Q4

ξ Q

1.455=

3Q

QQ

QQ

4

4

5

52

3

Q

Q

3QQ

2

2QQ

1QQ

Q

1

5

1

1

4

4

Widerstandder Flansche

ξ * h

h

Maximum capacity load (collapse load) analysis

Resistance of frames 29

ξ = 3

ξ = inf

ξ = 1

ξ = 1/3

ξ = 0

Q =Q

Q

1Q

3

1

Q

Q =QQQ

1

2

Q

2

=Q

A

Q

1Q

=

4

5

Q

3Q

A

Q3

1.143

5

4

1

5

4

1

4

4

QQ

QQQ

1

Q

3

Q

2

4

1Q

2

Q1

Q

4

Q

1

3

A

1.455

1

Q

2

5

5

A

A

A

3

3

3

Q

A

2

5

A

4

2

2

A

5

A


Resistance of frames


30

ξ = 1/3

ξ = 0

ξ = 1

ξ = 3

ξ = inf

1

5QQ

1.143=

1

3

Q

1

2

Q

2Q =Q

Q =Q

1Q

Q3

Q

1 31/3

Q4

ξ Q

1.455=

3Q

QQ

QQ

4

4

5

52

3

Q

Q

3QQ

2

2QQ

1QQ

Q

1

5

1

1

4

4


ξ * h

h



eQ



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

BeamPlateBeam

economic slenderness

Steel

Wood

Prestressedconcrete

Reinforcedconcrete

38

Documents

Structural Design II - ETH Z · 2017-04-27 · Structural Design II ... Bending-flection relation at plastic material behaviour linear elastic material behaviour plastic material