Advanced Structural Concrete · 2020. 11. 11. · Colloquium 2 11/11/2020 ETH Zurich | Chair of...

Advanced Structural Concrete

11/11/2020 ETH Zurich | Chair of Concrete Structures and Bridge Design | Advanced Structural Concrete 1

Colloquium 2

Colloquium 2


Exercise 2

Dimension the slab using the strip method.

Dimension the slab using an elastic FEM-

calculation (e.g. with CEDRUS-7, [4]).

Determine an upper limit value of the ultimate load

using the yield line method.

Educational content

• FEM-Calculation:

Application in practice

• Strip method:

Pre-dimensioning / Control of FEM-calculations

• Transformation of skew reinforcement

Derivation of yield conditions

→ Separate document on website

z

y

x

10 m

60 °

7.5 m

Free edge

Legend

Simple support

Colloquium 2


Dimensioning workflow with FEM

Documentation / check of:

• Geometry (Interaction CAD)

• Loads

• Combination of results and specifications of the limit states

• Deflections (Attention: elastic → Ultimate state: Factor 3-4)

• Reactions

• Internal forces

• Required reinforcement reinforcement layout

Colloquium 2


Reinforcement layout

• Representative frames

→ 2 to 3 gradations for typical slabs

• Look for repetitions

• Peaks do not necessary need to be

“covered”, averaging possible, (however, no

real lower limit value)

• Caution when averaging over high shear

forces (punching)

• Slight over-dimensioning of the field

reinforcement is preferred to slight over-

dimensioning of the reinforcement above the

column

G1

G2

G3

G4

C3 C1

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

-1.332

-2.313

-1.055

-2.279

-0.714

-2.288

-0.203

-0.479

-1.085

-0.520

-2.778

-0.771

-3.051

-4.272

-0.683

-1.763

-2.316

-1.365

-3.631

-5.432

-2.552

-1.926

-6.112

-1.296

-2.128

-7.633

-1.394

-13.385

-6.397

-1.241

-0.991

-0.628-4.228

-3.790

-34.242-9.681

-1.534

-0.723

-2.565

-9.417

-24.967

-9.716

-20.571

-47.447

-2.645

-4.207

-3.004

-3.299

-3.495

-43.871

-38.351

-28.497

-2.699

-5.329

-2.889-1.327

-39.919

-7.335

-4.974

-2.817 -3.219

-49.511

-22.798

-15.126

-44.546

-1.939

-11.276

-3.289

-16.810

-27.525

-3.420

-0.856

-3.271

-5.069

-19.135

-23.693

-27.919

-4.368

-0.122

-7.154

-10.843

-2.220

-21.704

-19.759-20.852

-2.624

-0.860

-3.174

-2.430

-3.037

-0.153

-1.924

-2.197

-0.739

-0.546

-0.924

-0.420

-9.020-24.210

-1.924-1.544

-1.524

-0.730

-1.083

-0.690

-43.906

-9.987

-3.064

-1.860

-1.648

-38.653

-2.126

-4.436

-5.508

-0.074

-52.104

-1.007

-3.233

-2.161

-1.897

-6.557

-1.950

-7.982

-3.796

-8.499

-5.651

-4.402

-2.637

-1.015

-6.189

-2.237

-2.329

-1.014

-1.042

-3.657

-1.096

-6.617

-7.145

-1.820

-2.023

-15.000

-15.000

-15.000

-15.000

-1.000

-1.000-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000-1.000 -1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-1.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-3.000

-7.000-7.000

-7.000

-7.000

-7.000

-7.000

-7.000

-7.000

-7.000

-7.000

-11.000

Cedrus-7: axt [cm2/m] an [cm2/m]

Reference lines 5 cm2/m,

Equidistance 2 cm2/m

G1

G2

G3

G4

C3 C1

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

-21.704

Section

Dimensioning with FEM

Colloquium 2


Edited section of the

reinforcement layout

Dimensioning with FEM

Reinforcement layout

• Representative frames

→ 2 to 3 gradations for typical slabs

• Look for repetitions

• Peaks do not necessary need to be

“covered”, averaging possible, (however, no

real lower limit value)

• Caution when averaging over high shear

forces (punching)

• Slight over-dimensioning of the field

reinforcement is preferred to slight over-

dimensioning of the reinforcement above the

column

Colloquium 2


Colloquium 2

Slab with

• h = 0.45 m

• Concrete C30/37

→ fcd = 20 MPa

• Steel B500B

→ fsd = 435 MPa

• Concrete cover

cnom = 55 mm

Task:

Dimension the slab by using the

results from an elastic FEM-

calculation

→ here only bending moments

60°

3 2

2 2

1.35 0.45 m 25 kN m 3 kN m

1.5 15 kN m 42 kN m

dq

z

y

x

10 m

60 °

7.5 m

Free edge

Legend

Simple support

Colloquium 2


FEM-calculations with reference load

F1

p=-100 kN/m2I1:

h=0.45m

Isotropic

10.00

yx

3.75

420.42

100

dq

p Factor

6 210 kN mSupport : Vertical stiffness

Colloquium 2


w 1

-1

w 2

-1

7.50

3.75 10.00

X-Richtung

Y-R

ichtu

ng

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e-6

kN

/m2, R

ota

tionen fre

i

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e6 k

N/m

2, R

ota

tionen fre

i

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e -6 k

N/m

2, R

ota

tionen fre

i

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e -6 k

N/m

2, R

ota

tionen fre

i

250.0 500.0

-28.4

750.0

-62.9

953.4

-80.0

750.0

-99.2

750.0

-99.2

-80.0

953.4

-62.9

-31.6

250.0 0.0

x


1

2 3

4

5

6

7

8

9

10

11

12

y

[kNm/m]xm390

365

335

32

34

38kNm m

0

0

280

235

175

110

xm

Contour lines at: 50 kNm/m

w 1

-1

w 2

-1

7.50

3.75 10.00

X-Richtung

Y-R

ichtu

ng

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e6 k

N/m

2, R

ota

tionen fre

i

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e -6 k

N/m

2, R

ota

tionen fre

i

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e -6 k

N/m

2, R

ota

tionen fre

i

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e -6 k

N/m

2, R

ota

tionen fre

i

0.00.0

99.5

-100.0

-291.3

85.4

85.4

-291.3

99.5

-100.0

Colloquium 2


x


1

2 3

4

5

6

7

8

9

10

11

12

y

35

0

0

65

75

17kNm m

0

0

8

27

46

17

ym


[kNm/m]ym

w 1

-1

w 2

-1

7.50

3.75 10.00

X-Richtung

Y-R

ichtu

ng

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e6 k

N/m

2, R

ota

tionen fre

i

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e -6 k

N/m

2, R

ota

tionen fre

i

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e -6 k

N/m

2, R

ota

tionen fre

i

Lageru

ng: E

inse

nku

ngen S

teifi

gke

it 1e -6 k

N/m

2, R

ota

tionen fre

i

20.2

-50.0

18.4 -150.0

-200.0

-50.0

-250.0

-250.0

-100.0

-300.0

-307.4

-150.0

-250.0

-324.5

-300.0

-200.0

-307.4

-250.0 18.4

20.2

Colloquium 2


x


1

2 3

4

5

6

7

8

9

10

11

12

y

135

92

130

21

53

13kNm m

8

4

125

110

107

88

xym


[kNm/m]xym

w 1

-1-1

w 2

-1-2

Colloquium 2



Principal moments

Colloquium 2


Reinforcement 1./2. layer

26 ( 100 mm)

26 ( 100 mm)

26 ( 100 mm)

14 ( 200 mm)

14 ( 200 mm)

14 ( 200 mm)

14 ( 200 mm)

14 ( 200 mm)

26 ( 100 mm)

26 ( 100 mm)

26 ( 100 mm)

26 ( 100 mm)

inf

s

s

s

s

s

s

s

s

s

s

s

s

20

14

14

14

14

14mm mm

14

14

14

14

14

14

inf

1. Layer: -Direction:


1 1450 mm 2 382 mmnomd c

2 1 2450 mm 2 359 mmnomd c

z

y

x, ξ

10 m

60 °

7.5 m

η

1

2 3

5

6

7

910

11

12

8

4

s =

20

0 m

m

z

y

x, ξ

10 m

60 °

7.5 m

η

Colloquium 2


Reinforcement 3./4. layer

14 14

14 14

14 14

14 14

14 14

14 14mm mm

14 14

14 14

14 14

14 14

14 14

14 14

sup sup



4 4450 mm 2 388 mmnomd c

3 4 3450 mm 2 374 mmnomd c

1

2 3

5

6

7

910

11

12

8

4

s =

20

0 m

m

s =

20

0 m

m

Colloquium 2


Bending resistance in skewed reinforcement directions

, 1

, 2

, ,

2

2

749

749

749

127

127

127kNm/m

127

127

749

749

749

749

s ,inf sd

u inf s ,inf sd

cd

s ,inf sd

u inf s ,inf sd

cd

u inf u i

a fm a f d

f

a fm a f d

f

m m

Lower reinforcement :

234

118

118

118

118

118kNm/m

118

118

118

118

118

118

nf

,

, , 4

,

, , 3

, ,

2

2

127

127

127

127

127

127kNm/m

127

127

127

127

127

127

s sup sd

u sup s sup sd

cd

s sup sd

u sup s sup sd

cd

u sup u s

a fm a f d

f

a fm a f d

f

m m

Upper reinforcement :

122

122

122

122

122

122kNm/m

122

122

122

122

122

122

up

Colloquium 2


Transformation of the bending resistance in the global x,y coordinates

0 , 60 2 2

, , ,

2 2

, , ,

, , ,

cos cos

sin sin

sin cos sin cos

xu inf u inf u inf

yu inf u inf u inf

xyu inf u inf u inf

m m m

m m m

m m m

Lower reinforcement :

, , ,

807 175 101

778 89 51

778 89 51

157 89 51

157 89 51

157 89 51kNm/m kNm/m

157 89 51

157 89 51

778 89 51

778 89 51

778 89 51

778 89 51

xu inf yu inf xyu infm m m

kNm/m

,x

y

1

sinym sinxym

cosyxm

cosxm , ,infum

, , ,

158 92 53

158 92 53

158 92 53

158 92 53

158 92 53

158 92 53kNm/m kNm/m

158 92 53

158 92 53

158 92 53

158 92 53

158 92 53

158 92 53

xu sup yu sup xyu supm m m

kNm/m

2 2

, , ,

2 2

, , ,

, , ,

cos cos

sin sin

sin cos sin cos

xu sup us up u sup

yu inf u sup u sup

xyu inf u sup u sup

m m m

m m m

m m m

Upper reinforcement :

Colloquium 2


Transformation of the bending resistance in the global x,y coordinates

0 , 60

,x

y

1

sinym sinxym

cosyxm

cosxm , ,infum

Colloquium 2


Control of the yield condition

2 2

, , ,

2742

16193

6524

13957

20366

16469kNm/m 0

12043

11681

9208

7581

56302

51331

inf xyu inf xy xu inf x yu inf yY m m m m m m

for positive bending resistance :

for negative b

2

, , , ...sup xyu sup xy xu sup x yu sup yY m m m m m m

ending resistance :

z

y

x, ξ60 °

η

1

2 3

5

6

7

910

11

12

8

4

Advanced Structural Concrete · 2020. 11. 11. · Colloquium 2 11/11/2020 ETH Zurich | Chair of...

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