Resilience of Buildings with Deep Reinforced Concrete ... presentation - orbi.pdfDiagonal strut...

Kinematics-Based Modelling of Deep Transfer

Girders in Reinforced Concrete Frame Structures

Liège, Belgium

14-06-2019

Jian LIU

Doctoral defense

Jian Liu Kinematics-Based Modelling of Deep Transfer Girders in RC Frame Structures

1. Background and objectives

2. Comparative study on models for deep beams

3. Macroelement for complete shear behaviour of deep beams

4. Mixed-type modelling with slender and deep beam elements

5. Shear strength of deep beams with openings

6. Conclusions and future work

Jian Liu Outline

Outline

Background and Objectives

Jian Liu Background and Objectives

Characteristics of deep transfer girders

• Transfer heavy loads from

discontinuous columns/ walls

• Small aspect ratio: a/d ≤ 2.5

• Crucial to structural safety

Jian Liu Background and Objectives 01/61

(Photo by J. G. MacGregor.)

d

Deep transfer girders in structures

(By Evan Bentz, Toronto, 2008)


(Train station of Leuven)

(Grand Chancellor Hotel, New Zealand. By Kam et al., 2011)

(Brunswick building, Chicago: J. G. MacGregor)

(https://iarjset.com/upload/2017/march-17/IARJSET%2024.pdf)

Floor diaphragm

Other application of deep beams

(https://civildigital.com/the-five-major-parts-of-bridges-concrete-span-bridge/)

(https://www.kore-system.com/blog_list/insulation-series-what-type-of-

foundation-is-right-for-me/)

(https://photo.xuite.net/hspsj60440/4103822/1.jpg)

Strip footings

Cap beams

Raft footings


0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7

V/(

bd

f c')

a/d

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7

V/(

bd

f c')

a/d

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7

V/(

bd

f c')

a/d

fc’ = 27.2 MPa

ag,max = 19 mm

d = 538 mm

b = 155 mm

As = 2277 mm2

fy = 372 MPa

Plate size:

152×152×25mm3

152×229×51mm3

152×76×9.5mm3

Difference between slender and deep beams

Strut and tie model

Sectional model

a

d

V V

V V

~2.5

Beam action Strut action

(tests by Kani in 1979, adapted from Collins and Mitchell, 1997)


0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7

V/(

bd

f c')

a/d

V

M

V

Shear behaviour of deep transfer girders

fc = 40 MPa ag = 14 mm

d = 3840 mm b = 250 m

ρl = 0.656% fy = 573 MPa

P=2162kN

w (self weight) = 24.4 kN/m

4 m

7 m 12 mΔ

EastWest

P = 2162 kN

w (self weight) = 24.4 kN/m

West East

Concrete crushing


0

500

1000

1500

2000

2500

0 10 20 30 40

P,

KN

Δ, mm

P, k

N

0 500 1000 1500 2000 2500 3000 3500 4000

P, kN

American

Canadian

European

Other

Un

ive

rsity

28

Pre

dic

tio

ns

Ind

ustr

y1

6 P

red

iction

s

CSA Sect.1485

Flex.2730 kN

Exp.2162

CSA S&T 1660

ACI Sect. 2090

Difficulty in predicting shear strength of deep transfer girders

Objective 1): To evaluate the accuracy of existing models for shear resistance of deep

beams by using a large database of laboratory tests.

P, kN


0 500 1000 1500 2000 2500 3000 3500 4000

P, kN

American

Canadian

European

Other

Un

ive

rsity

28

Pre

dic

tio

ns

Ind

ustr

y1

6 P

red

iction

s

CSA Sect.1485

Flex.2730 kN

Exp.2162

CSA S&T 1660

ACI Sect. 2090

0 500 1000 1500 2000 2500 3000 3500 4000

P, kN

American

Canadian

European

Other

Un

ive

rsity

28

Pre

dic

tio

ns

Ind

ustr

y1

6 P

red

iction

s

CSA Sect.1485

Flex.2730 kN

Exp.2162

CSA S&T 1660

ACI Sect. 2090

0 500 1000 1500 2000 2500 3000 3500 4000

P, kN

American

Canadian

European

Other

Un

ive

rsity

28

Pre

dic

tio

ns

Ind

ustr

y1

6 P

red

ictio

ns

CSA Sect.1485

Flex.2730 kN

Exp.2162

CSA S&T 1660

ACI Sect. 2090

0 500 1000 1500 2000 2500 3000 3500 4000

P, kN

American

Canadian

European

Other

Un

ive

rsity

28

Pre

dic

tio

ns

Ind

ustr

y1

6 P

red

iction

s

CSA Sect.1485

Flex.2730 kN

Exp.2162

CSA S&T 1660

ACI Sect. 2090

P

Δ

shear capacity

Complete shear response of deep transfer girders


• Serviceability

• Ductility

• Resilience

• Structure-soil interaction

• …

• Deep transfer girder • Complete shear response

transfergirder

P

Δ

transfergirder

Objective 2):

To develop 1D element for deep

beams combining accuracy and

efficiency.

Large frame structure with deep transfer girders


• Model with 1D frame elements • Model with 2D elements

(Program VecTor5)

• computationally efficient

• inaccurate for deep beams

• complex for large structures

• suitable for deep beams

(Program VecTor2)

Modelling of frame structures with deep transfer girders


Modelling of large structures with deep beams

Jian Liu

Deep element Slender elements

Objective 3): To integrate the new model into a framework of frame structures with both

slender and deep elements.

• Large frame structure with

deep transfer girder

• Model with 1D slender and

deep elements

Background and Objectives 10/61

Deep transfer girder with web openings

Transfer girder

openings

Objective 4): To propose a model to predict the shear capacity of RC deep beams

with web openings.


Comparative Study on Models for Shear

Strength of RC Deep Beams

Jian Liu Comparative Study on Models for Shear Strength of RC Deep Beams

Classification of models

73 existing models published between 1987 and 2014:

Jian Liu Comparative Study on Models for Shear Strength of RC Deep Beams 12/61

Collect a large database of tested deep beams

Implement ten representative models

Predicted vs. experimental shear strength

Comparative study procedure


…

Database of 574 RC deep beams


Ten implemented models


at,avg

V

Pb1(V/P)L

b1L

c

r

(1+ )

CLZ

=0

b1eL =

h

v

t,min

t,max t,avg

x

z

① ② ③

④ ⑤ ⑥ ⑦

⑧ ⑨ ⑩

# of test = 289

Avg = 0.91

COV = 26.0%

429

0.81

21.3%

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

401

0.97

25.2%

465

1.35

34.8%

334

0.91

24.3%

350

1.00

19.8%

465

1.19

32.5%

350

1.08

31.2%

392

1.08

15.4%

411

1.14

26.3%

Shear strength predictions

Russo et al. 2005

350

1.00

19.8%

Mihaylov et al. 2013

392

1.08

15.4%

⑦

⑩


v = c1(kχfccosθ ) + c2ρhfyhcotθ + c3 ρvfyv

nodal

zone

d

aV

P

T

C

1

32

1

2

3

Diagonal strut

Vertical web reinforcement

Horizontal web reinforcement

strut

tie

θ

Strut-and-tie model by Russo et al., 2005

v = = vc + vw

V

bd

a

d

0.76 0.25 0.35


c

h

DOF c

c

a t,avg

V

Pb1(V/P)L

b1L

c

r(1+ )

CLZ

=0

b1eL =

d

kL

v

t,min

t,max

t,avg

x

z

t

DOF t,avg

kl =l0

t,min

at,avg

(1+ )

x

z c

+ =

at,avg

V

Pb1(V/P)L

b1L

c

r

(1+ )

CLZ

=0

b1eL =

h

v

t,min

t,max t,avg

x

z

Two-parameter kinematic theory (2PKT) by Mihaylov et al., 2013


c

h

DOF c

ckL

t

DOF t,avg

t,min

at,avg

(1+ )

x

z c

kL

Rigid

block

Shear components and solution procedure in 2PKT

TminVd

V

Avfv

vci

a

Vs 1

d=

1095 m

m

=

w≈3.5 mm

~

160 m

m

S

tirr

up

slip

2.

4 m

m

~~

w≈3.5 mm

~

160 m

m

S

tirr

up

slip

2.

4 m

m

~~

w≈3.5 mm

~

160 m

m

S

tirr

up

slip

2.

4 m

m

~~

Aggregate interlock

Critical Loading Zone

VCLZ

V c

d

V = VCLZ + Vs + Vci + Vd = T(0.9d)/a


Vertical

equilibrium

Moment

equilibrium

Solution procedure for 2PKT

V = VCLZ + Vs + Vci + Vd


= T(0.9d)/a

0.00

0

V

εt

Vd

VCLZ

Vs

Vci

Equilibrium

Size effect in shear


~ 3

000 m

m ~3000 m

m

(tested by Zhang & Tan, 2007)

3000

0.20

Predicting size effect in shear

STM by Russo et al., 2005

3000

2PKT by Mihaylov et al. (2013) provides adequate predictions.

2PKT by Mihaylov et al., 2013

3000


Below the crack:

• Displacement at (x, z)

• Crack width and slip:

Deformation prediction of 2PKT


δx x, z = εt,avg h − z cotα

δz x, z = εt,avgxcotα + Δc

δx x, z = εt,avgx

δz x, z =εt,avgx

2

h − z

w = εt,avglk

2sinα1+ Δccosα1

s = Δcsinα1

Δ = Δc + εt,avgacotα

• Deflection:

Above the crack:

at,avg

V

Pb1(V/P)L

b1L

c

r

(1+ )

CLZ

=0

b1eL =

h

v

t,min

t,max t,avg

x

z


Predicted displacement capacity, 53 tests

Δ = Δc + εt,avgacotα 0 2 4 6 8 10 12 14 16 18

0

2

4

6

8

10

12

14

16

18

pred

, mm

e

xp,

mm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

pred

, in

e

xp,

in

at,avg

V

Pc

r

(1+ )

CLZ

=0

d

v

t,min

t,max

t,avg

x

z

Predicted deformed shapes

• a/d = 1.55

Pred.

Exp.

P/Pu= 92%

P/Pu= 78%

P/Pu= 91%

P/Pu= 100%

P

P/Pu= 100%

P/Pu= 97%

P/Pu= 92%

P/Pu= 93%

• a/d = 2.29

P


=

Three-parameter kinematic theory (3PKT) by Mihaylov et al., 2015


d

1

x

z c

x

z

x

z

kl

c

CLZ

xz

a+ d cott,avg 2

x

z

x

z

d2

1

a+ d cotb,avg 1

b,min

+

d1

x

z c

x

z

x

z

kl

c

CLZ

xz

a+ d cott,avg 2

x

z

x

z

d2

1

a+ d cotb,avg 1

b,min

Pb1L

b1eL

1

b2LP2

c

d h1d

2

gaw

v

b,min

b,max

t,avg

CLZ

x

z1

a+ d cotb,avg 1

a+ d cott,avg 2

b,min

d1

d2

x

z c

kl 0l =

c

t,min

t,max

b,avg

Macroelement for Complete Shear

Behaviour of Deep Beams

Jian Liu Macroelement for Complete Shear Behaviour of Deep Beams

Critical loading zone (CLZ)

Vext

Shear behaviour of deep beams

Jian Liu Macroelement for Complete Shear Behaviour of Deep Beams 27/61

(tested by Mihaylov et al., 2015)

P

Vext


Vext

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

1600

1800

, mmP

, kN





Vext

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

1600

1800

, mmP

, kN




concrete

crushed

DOFs εt1 and εt2 (or θ1 and θ2)

Complete deformation pattern

= + DOF Δc

P2

CLZ εt2

εt1

T2

M2

θ2

P1

Δc

T1

M1

θ1

εt2

εt1θ1

θ2

Δc

Three-parameter kinematic model for deep beams


Δc

θ2

θ1

k1 k3k2

Rigid bar Rigid bar

Section 1 Section 2

a

CLZΔc

εt2

εt1

M2

M1

θ2

θ1

a

d Macroelement for deep beams

θ1 = εt1 a / d

θ2 = εt2 a / d

M1 (θ1) + M2 (θ2) = V (Δc, θ1 ,θ2) a

k1 θ1 + k2 θ2 = k3 Δc a

k1 = M1 / θ1

k2 = M2 / θ2

k3 = V / Δc


0 9cr crM N . d

i 2

6kAGEI 4EI Lk

kAGL 312EI kAGL

Timoshenko beam theory

Constitutive relationship of rotational spring

M2

M1

θ2

θ1

a



M2

M1

θ2

θ1

a

Jian Liu

0 2 crT N

0 0 0 9M T . d

Parabola

Ncr - cracking force of zone

influenced by bottom

reinforcement.

'

cr c,eff s c s cN = A + E E -1 A 0.63 f

Macroelement for Complete Shear Behaviour of Deep Beams 30/61


M2

M1

θ2

θ1

a

Jian Liu

T is the tension in long. reinf.:

M ≈ T (0.9d)

εt = θ d / a

0 33

1 200

cs s t

t

s y

. fT = E A +

A f



M2

M1

θ2

θ1

a

Jian Liu

T is the tension in long. reinf.:

M ≈ T (0.9d)

εt = θ d / a

0 33

1 200

cs s t

t

s y

. fT = E A +

A f



M2

M1

θ2

θ1

a


Shear components from:

VCLZ — critical loading zone

Vci — aggregate interlock

Vs — stirrups

Vd — dowel action

VCLZ

Vs

Vci

Vd Va

0.9dd

-V

VΔc

θ2

θ1

Δc

a

Transverse springs (shear behaviour)


Springs of the four shear mechanisms

Jian Liu

0

50

100

0.0 5.0 10.0

Δc, mm

Vd, kN

εt1,avg=0

εt1,avg=1x10-3

0

125

250

0.0 5.0 10.0

Δc, mm

εt1,avg=0

εt1,avg=1x10-3

Vs, kN

0

250

500

0.0 5.0 10.0

Δc, mm

εt1,avg=0

εt1,avg=1x10-3

Vci,

kN

Vci,

kN

• Critical loading zone • Aggregate interlock

• Stirrups • Dowel action

Contact density model

(Li et al., 1989)

0

300

600

0.0 5.0 10.0

Δc, mm

εt2,avg=0

εt2,avg=1x10-3V

CL

Z,

kN

(beam S1M tested by Mihaylov et al. 2013)


Δc

V0

Δcp

V

1

ki

k1 k3k2

Rigid bar Rigid bar

Section 1 Section 2

a

M2

-VM1

V

c ci 2

c c

c2

c c

( )

6kG AE Ik

12E I kG Aa

3kG Aa2

a 6E I kG Aa

Timoshenko beam theory

Jian Liu

Constitutive relationship of shear spring


Δc

V0

Δcp

V

1

ki

Δc

V0

Δcp

V

1

ki

Diagonal crack

forms

k1 k3k2

Rigid bar Rigid bar

Section 1 Section 2

a

M2

-VM1

V

Jian Liu



Δc

V0

Δcp

V

1

ki

Δc

V0

Δcp

V

1

ki

Δc

V0

Δcp

V

1

ki

k1 k3k2

Rigid bar Rigid bar

Section 1 Section 2

a

M2

-VM1

V

Jian Liu


V = VCLZ + Vs + Vci + Vd


Δc

V0

Δcp

V

1

ki

Δc

V0

Δcp

V

1

ki

Δc

V0

Δcp

V

1

ki

Δc

1

k0

V0

Δcp

V

1

ki

Δc

1

k0

k0

1V0

Δcp

V

1

ki

k1 k3k2

Rigid bar Rigid bar

Section 1 Section 2

a

M2

-VM1

V

Jian Liu



k1 k3k2

Rigid bar Rigid bar

Section 1 Section 2

φ2

v2

v1 Δc

θ2

θ1

Δc

x

y

a

φ1

k1 k3k2

Rigid bar Rigid bar

Section 1 Section 2

φ2

v2

v1 Δc

θ2

θ1

Δc

x

y

a

φ1

Solution procedure of macroelement

from previous

converged step

P P

… …

Global stiffness matrix [K]

21 2 3 1 3

1 3 3 1 2 2 3 3 1 2

2

3

2 3 2 1 3

3 1 2

1 2 1

21 2 3 1 2 2

1

3

1 3 2 23 3

1

-k k k k a

kk k k a -

k k k a -k k a

-k k a k k k -k k a -k k k

-k k a

k

k k k k a

k k a k a

k k k a

-k k k k k k

Structure

{Δ} = {P} \ [K] Global linear analysis


0

0.0

M1

θ1

k11

0

0.0

M2

θ2

k21

0

0.0

V

Δc

1

k3

Element

11

12

2

2c

vT

v

22 3

32

3 1 3

1

2 3

2

1 321 2 3

1 12

2

1

-k k a

T

-

k k a -k a

-k a k k-k k ak k k a

a

k k kk a kk a

r2

y2

r1

y1 Δc

θ2

θ1

Update k1 = M1 / θ1, k2 = M2 / θ2, k3 = V / Δc

Calculate M1, M2 and V from θ1, θ2 and Δc Until convergence

reached at element level

from {Δ}

Jian Liu



k1 k3k2

Rigid bar Rigid bar

Section 1 Section 2

φ2

v2

v1 Δc

θ2

θ1

Δc

x

y

a

φ1

k1 k3k2

Rigid bar Rigid bar

Section 1 Section 2

φ2

v2

v1 Δc

θ2

θ1

Δc

x

y

a

φ1

P P

… …

Global stiffness matrix [K]

21 2 3 1 3

1 3 3 1 2 2 3 3 1 2

2

3

2 3 2 1 3

3 1 2

1 2 1

21 2 3 1 2 2

1

3

1 3 2 23 3

1

-k k k k a

kk k k a -

k k k a -k k a

-k k a k k k -k k a -k k k

-k k a

k

k k k k a

k k a k a

k k k a

-k k k k k k

{Δ} = {P} \ [K] Global linear analysis

Until convergence

reached at structural level

Jian Liu


Structure


P

Δc1 Δc2

ΔV V

(S1M tested by Mihaylov et al. 2013)

Complete shear response predicted with macroelement

P

Δc1 Δc2

ΔV V

a/d=1.55 d=1095mm

ρl=0.70% ρv=0.70%

fc=33.0MPa

0

500

1000

0 1 2 3

Δc1, mm

VkN

VdVci

Vs

VCLZ

0 10 20

Δc2, mm

Vs

VCLZ

Vci

Vd

V

0

200

400

600

800

1000

0 4 8 12 16

V, k

N

Δ, mm

pred.

exp.


Jian Liu

0

1600

0 18

P, k

N

Δ, mm

S0M

a/d=1.55

ρv=00

2000

0 20

P, kN

Δ, mm

S1M/C

a/d=1.55

ρv=0.1%

pred.

exp.

0

1400

0 20

P, k

N

Δ, mm

L1M/C

a/d=2.28

ρv=0.1%0

1000

0 16

P, k

N

Δ, mm

L0M/C

a/d=2.28

ρv=0

Simply-supported deep beam



Jian Liu


0

500

1000

1500

0 3 6 9

Δ, mm

kN pred.

Vint

P

Vext

kN

exp.

force redistribution

ductile

.35 .35 .05 .3 .55 .4

.45.55.75

.2.35

0

.25

LS11

.05

.1.05

.05

1.9

2.5

1.8

.9

.05

.15.05

3

1.8

2.65

.85

.95

.6

.4

0.55

.3.2

.35

.55

.25

.1.2.4.15

.05 .3.3.15.3.4.35

.85.2 .3 .35

.7.55

.45

.2.5

.2 .1 .45

.2

P

2Vint

Vext Vext

P


• Simply-supported deep beam • Continuous deep beam

0

500

1000

1500

2000

0 2 4

P =

2V

, kN

w, mm

S1M/C

L1M/C

exp. - S1M / L1M

exp. - S1C / L1C

pred.

0

400

800

1200

1600

0 2 4 6 8 10

P,k

N

w, mm

exp.

pred.

P

Jian Liu

Crack widths predicted with macroelement

measured w

estimated P safe?

Pmax


Mixed-type Modelling of Structures with

Slender and Deep Beam Elements

Jian Liu Mixed-type Modelling of Structures with Slender and Deep Beam Elements

Model with 1D slender and

deep elements

Modelling of large structures with deep beams

Deep element Slender elements

Jian Liu Mixed-type Modelling of Structures with Slender and Deep Beam Elements 40/61

• 1D fiber-based element for slender beams

• Distributed plasticity approach for shear behavior

• Excellent predictions for plane frames reported

Existing FE program: VecTor5

(http://vectoranalysisgroup.com/vector5.html)

Solution procedure of VecTor5

Global frame

analysis

Nonlinear

sectional

analysis

Member deformation

Unbalanced forces

(Classical stiffness-based

finite element formulation)

(Distributed-nonlinearity

fiber section approach)

Unbalanced forces = Global forces - Sectional forces

( If unbalanced forces =0, iteration stops.)

φ2v2φ1 v1

u1 u2

φ2

v2

φ1

v1


Deep beam

analysis

Global frame

analysis

Nonlinear

sectional

analysis

Solution procedure of modified VecTor5

Member deformation

Unbalanced forces

Deep

beam

Slender

beam

Unbalanced forces = Global forces - Sectional forces

(If unbalanced forces =0, iteration stops.)

φ2v2φ1 v1

u1 u2

φ2

v2

φ1

v1


Deep ele.

Slender ele.

• 1D mixed-type model

Application to simply-supported deep beams

P

400 200

100

900 mm

40

0

45

0 m

m

300 mm

4Φ29

2Φ16

Φ13

@100

100

Δ

P

a Lf/2

Lb1

L

d hb

As1

As2

φv@ss

(Tested by Tanimura and Sato in 2005)

• 2D finite element model

2 2 2

1

11

3

nui ui ui

i i i i

N V MCF

n N V M

Convergence factor (CF):

0

50

100

0.999

1.000

1.001

0 10 20 30 40 50

No

. o

f it

er.

CF

Load step No.

max No. of iter. = 100

peak load


Prediction of entire shear response


0 0.3

0

1000

0

1000

0 8

Δ, mm

2

in.

a/d = 0.5

a = 200 mm

ρv = 0.21 %

fc' =23.2 MPa

0 0.3

0

1000

0

1000

0 8

Δ, mm

3

in.

a/d = 0.5

a = 200 mm

ρv = 0.48 %

fc' = 23.2 MPa

0 0.3

0

200

0

1000

0 8

Δ, mm

4

in.

kip

a/d = 0.5

a = 200 mm

ρv = 0.84 %

fc' = 23.2 MPa

0 0.2

0

800

0

1000

0 5

V,

kN

Δ, mm

5

in.

a/d =1.0

a = 400 mm

ρv = 0.00 %

fc' = 29.0 MPa

0 0.2

0

1000

0

1000

0 5

Δ, mm

6

in.

a/d = 1.0

a = 400 mm

ρv = 0.21 %

fc' = 29.1 MPa

0 0.2

0

1000

0

1000

0 5

Δ, mm

7

in.

a/d = 1.0

a = 400 mm

ρv = 0.48 %

fc' = 29.2 MPa

0 0.2

0

800

0

800

0 6

V,

kN

Δ, mm

9

in.

a/d = 1.5 a = 600 mm

ρv = 0.00 % fc' = 22.9 MPa

0 0.4

0

800

0

800

0 10

Δ, mm

10

in.

a/d = 1.5 a = 600 mm

ρv = 0.21 % fc' = 22.5 MPa

0 0.4

0

800

0

800

0 10

Δ, mm

11

in.

a/d = 1.5 a = 600 mm

ρv = 0.48 % fc' = 23.0 MPa

0 0.2

0

200

0

1000

0 5

Δ, mm

8

in.

kip

a/d = 1.0

a = 400 mm

ρv = 0.84 %

fc' = 29.3 MPa

exp.

1D

mixed-type

modelling

2D

FEM

0 0.3

0

1000

0

1000

0 8

V,

kN

Δ, mm

Exp.

2D high-hidelity FEM

1D mixed-typemodelling

1

in.

a/d = 0.5

a = 200 mm

ρv = 0.00 %

fc' = 23.2 MPa

0 0.4

0

160

0

800

0 10

Δ, mm

Exp.

1D mixed-type

modelling

2D high-fidelity FEM

12

in.

kipa/d = 1.5 a = 600 mm

ρv = 0.84 % fc' = 23.5 MPa

0 0.2

0

800

0

800

0 6

V,

kN

Δ, mm

9

in.

a/d = 1.5 a = 600 mm

ρv = 0.00 % fc' = 22.9 MPa

0 0.4

0

800

0

800

0 10

Δ, mm

11

in.

a/d = 1.5 a = 600 mm

ρv = 0.48 % fc' = 23.0 MPa

0 0.8

0

1000

0

5000

0 25

V,

kN

Δ, mm

B-18 kip

in.

a/d = 1.5 a = 1400 mm

d = 2100 mm fc' = 23.5 MPa

ρv = 0.4 % ρl = 2.05 %


0 0.4

0

800

0

800

0 10

Δ, mm

10

in.

a/d = 1.5 a = 600 mm

ρv = 0.21 % fc' = 22.5 MPa

0 0.4

0

160

0

800

0 10

Δ, mm

Exp.

1D mixed-type

modelling


12

in.

kipa/d = 1.5 a = 600 mm

ρv = 0.84 % fc' = 23.5 MPa

0.0 1.0

0

600

0

3000

0 30

V,

kN

Δ, mm

B-17 kip

in.

a/d = 1.5 d = 1000 mm

ρv = 0.4 % ρl = 2.04 %

fc' = 28.7 MPa

0 0.8

0

600

0

3000

0 25

V,

kN

Δ, mm

B-15

in.

kip

a/d = 1.5

d = 1200 mm

ρv = 0.0 %

ρl = 1.99 %

fc' = 27.0 MPa


Exp.

1D mixed-type modelling

VecTor2

DIANA

exp.

1D

mixed-type

modelling

2D

FEM

0 0.5

0

250

0

1200

0 15

V,

kN

Δ, mm

B-13-2

in.kip

a/d = 1.5 d = 800 mm

ρv = 0.0 % ρl = 2.07 %

fc' = 24.0 MPa

Prediction of entire shear response

0 0.5

0

80

0

400

0 15

V,

kN

Δ, mm

B-10-2 Exp.



kip

in.

a/d = 1.5 d = 400 mm

ρv = 0.0 % ρl = 2.02 %

fc' = 23.0 MPa

Modelling a 20-storey frame

2700

5400 mm

3500

70

00 m

m

2700

3500

3

4

1 2

3

4

1 2

P/2

P/4 P/4

60

0 m

m

800 mm

16#11

600 mm

18

00

mm

4#10

#4

@150

#4

@250

40

0 m

m

69

600 mm

67

60

0 m

m

300mm

3#9

3#9

1-1

3-3

4-4

#4

@250

#4

@250

69

71 155 164 71155

2x7

2=

14

46

92

x7

2=

14

4

3x154=46269

67

4010#10

2-2

12#10

60

0 m

m

800 mm

16#11

600 mm

18

00

mm

4#10

#4

@150

#4

@250

40

0 m

m

69

600 mm

67

60

0 m

m

300mm

3#9

3#9

1-1

3-3

4-4

#4

@250

#4

@250

69

71 155 164 71155

2x7

2=

14

46

92

x7

2=

14

4

3x154=46269

67

4010#10

2-2

12#10

60

0 m

m

800 mm

16#11

600 mm

18

00

mm

4#10

#4

@150

#4

@250

40

0 m

m

69

600 mm

67

60

0 m

m

300mm

3#9

3#9

1-1

3-3

4-4

#4

@250

#4

@250

69

71 155 164 71155

2x7

2=

14

46

92

x7

2=

14

4

3x154=46269

67

4010#10

2-2

12#10

60

0 m

m

800 mm

16#11

600 mm

18

00

mm

4#10

#4

@150

#4

@250

40

0 m

m

69

600 mm

67

60

0 m

m

300mm

3#9

3#9

1-1

3-3

4-4

#4

@250

#4

@250

69

71 155 164 71155

2x7

2=

14

46

92

x7

2=

14

4

3x154=46269

67

4010#10

2-2

12#10

(according to ACI 318)


Three modelling strategies


2-storey

2D FEM

2-storey


20-storey

a) 2D FEM

c) 1D mixed-type model of

entire 20-story frame

b) 1D mixed-type model

1

P/4 P/4P/2

Slender elements

Deep elements

P/38P/76 P/76

P/38P/76 P/76

20

Sto

rie

s

Slender elements

Deep elements

P/2P/4 P/4

Slender elements

Deep elements

P/38P/76 P/76

P/38P/76 P/76

20 S

tories


P/2P/4 P/4

Slender elements

Deep elements

40000

20000

P,

kN

Restrained

lateral displ.

• Load-disp. relationship • Response in each storey

of 20-storey frame


Prediction of loading response

0

10000

20000

30000

40000

0 2 4 6 8 10

P, k

N

Δ, mm

1D mixed-typemodel

2D FEM

2-storey

1D mixed-typemodel, 20-storey

2-storey

P/2P/4 P/4

Δ

Δ

P

Δ

P

1

10

100

1,000

10,000

100,000

1 10 100 1000 10000

An

aly

sis

tim

e, s

No. of unrestrained DOFs

Simply-supported beam

Continuous deep beam

2-storey frame

20-storey frame

Co

mp

uta

tio

n t

ime,

s

Efficiency of studied modelling strategies

• 40 load steps

• Office desktop

3.4 GHz quad-core processor

16 GB of RAM


6 hours

Shear Strength of RC Deep Beams with

Web Openings

Jian Liu Shear Strength of RC Deep Beams with Web Openings

Tests of deep beams with web openings (El-Maaddawy and Sherif, 2005) 5

00

mm

100 500600 mm

400 100P

25

0

200a

a0a

0

P50

a

a0

a0

80 mm

2×Φ8

4×Φ14

Φ6 @

150mm

75

P

a0

a0

75

50

50

0 m

m

100 500600 mm

400 100P

25

0

200a

a0

a0

P50

a

a0

a0

80 mm

2×Φ8

4×Φ14

Φ6 @

150mm

75

P

a0

a0

75

50

• Opening at centre • Opening at top near support

• Opening at bottom near load

50

0 m

m

100 500600 mm

400 100P

25

0

200a

a0

a0

P50

a

a0

a0

80 mm

2×Φ8

4×Φ14

Φ6 @

150mm

75

P

a0

a0

75

50

50

0 m

m

100 500600 mm

400 100P

25

0

200a

a0

a0

P50

a

a0

a0

80 mm

2×Φ8

4×Φ14

Φ6 @

150mm

75

P

a0

a0

75

50

fc = 21.0 MPa

fy = 420 MPa

fyv = 300 MPa

Jian Liu Shear Strength of RC Deep Beams with Web Openings 50/61

Two typical failure modes

P=103kN

V

P=53kN

V

NS-150-C

1 2

NS-250-C1

23

P=103kN

V

P=53kN

V

NS-150-C

1 2

NS-250-C1

23

• Small opening • Large opening

critical cracks

other cracks

horizontal

crack


• FEM model of beam NS-150-C

σ2,min

σ2,max

σ2,min

σ2,max

P

V

P

V

P

V

P

V

P

V

P

V

P

V

P

V

NS-150-C NS-250-C

σ2,min

σ2,max

σ2,min

σ2,max

P

V

P

V

P

V

P

V

P

V

P

V

P

V

P

V

NS-150-C NS-250-C

σ2,min

σ2,max

σ2,min

σ2,max

P

V

P

V

P

V

P

V

P

V

P

V

P

V

P

V

NS-150-C NS-250-C

Deep beams with web openings studied by FEM

(b) Deformation (×10) and crack pattern

(c) Principle compressive stress

Fig. 4 Deep beams with web openings studied by FEM

σ2,min

σ2,max

NS-150-C NS-250-C

P

V

P

V

P

V

P

V

P

V

P

V

P

V

P

V

NS-150-C NS-250-C

P

• Deformation (×10) and crack pattern

• Principle compressive stress

(With programme VecTor2)


σ2,min

σ2,max

σ2,min

σ2,max

P

V

P

V

P

V

P

V

P

V

P

V

P

V

P

V

NS-150-C NS-250-C

σ2,min

σ2,max

σ2,min

σ2,max

P

V

P

V

P

V

P

V

P

V

P

V

P

V

P

V

NS-150-C NS-250-C

σ2,min

σ2,max

σ2,min

σ2,max

P

V

P

V

P

V

P

V

P

V

P

V

P

V

P

V

NS-150-C NS-250-C

• Crack pattern of solid deep beams

Deep beams with web openings studied by FEM

(b) Deformation (×10) and crack pattern

(c) Principle compressive stress

Fig. 4 Deep beams with web openings studied by FEM

σ2,min

σ2,max

NS-150-C NS-250-C

P

V

P

V

P

V

P

V

P

V

P

V

P

V

P

V

NS-150-C NS-250-C

• Deformation (×10) and crack pattern

• Principle compressive stress

(With programme VecTor2)

Similar to solid deep beams


Kinematics of deep beams with openings

Δct

Δcb

εtt

εtb

+ CLZ-b

CLZ-t

z

x

DOFs εtb and εtt DOF Δcb and Δct

Δct

Δcb

εtt

εtb

+ CLZ-b

CLZ-t

z

x

DOFs εtb and εtt DOF Δcb and Δct

+

top sub shear

span

bottom sub shear

span

• DOF εtb and εtt • DOF Δcb and Δct

• 2PKT for solid deep beam


cracks

ab

hbdb

lb1e,b

at

ht

dt

lb1e,t

αt

αb

a

d h

Vt

Vt

Nt

Nt

Vb

Vb

Nt

β

eVt

Vt

Vt

Nt

Section A

Nt

e

lb1e,t

β

Equilibrium of forces in deep beams with openings

cracks

ab

hbdb

lb1e,b

at

ht

dt

lb1e,t

αt

αb

a

d h

Vt

Vt

Nt

Nt

Vb

Vb

Nt

β

eVt

Vt

Vt

Nt

Section A

Nt

e

lb1e,t

β

P

N

TV

C

β

P

T

C

aV

N

z

• Top L-shaped region • Bottom region

external

compression

external

tension


P

T

C

P

T

C

aV

N

z

P

T

C

aV

N

z

a

z

V

N

Vβ

0.00

0

V

εt

Vd

VCLZ

Vs

Vci

Equilibrium

0.00

0

V

εt

Vd

VCLZ

Vs

Vci

Equilibrium

0.00

0

V

εt

Vd

VCLZ

Vs

Vci

Equilibrium

Solution procedure for 2PKT in other load cases

Solution procedure under given Δc

(1)

(2)

(3)


(1)

(2)

(3)

P

T

C

P

T

C

aV

N

z

P

T

C

aV

N

z

a

z

V

P

T

C

P

T

C

aV

N

z

P

T

C

aV

N

z

a

z

V

P

T

C

P

T

C

aV

N

z

P

T

C

aV

N

z

a

z

V

N

Vβ

Failure along a crack

P=103kN

V

P=53kN

V

NS-150-C

1 2

NS-250-C1

23

eVt

Vt

Vt

Nt

Section A

Nt

e

β

Need to consider the

horizontal crack

• Section A under M-N interaction

• Vt.e ≤ Mu of section A


Calculation procedure

Top

BottomBottom

Top

Vb

VtP

V

A

Nt

Nt

Top

BottomBottom

Top

Vb

VtP

V

A

Nt

Nt

e

Vt

Sect. A

Top

BottomBottom

Top

Vb

VtP

V

A

Nt

Nt

Step 1

Isolate two shear spans

from the deep beam

Step 2

Calculate the shear strength

of top shear span Vt

Step 3

Check section “A” and

limit Vt if needed

Step 4

Calculate the shear strength

of bottom shear span Vb

Step 5

Obtain shear strength of the

deep beam V=Vt+Vb


Parametric study: opening size

a

m1a

m2h

k1a k2h

h

(tested by El-Maaddawy and Sherif, 2005)


0

120

0.2 0.3 0.4 0.5 0.6

V, k

N

m2

m1≈m2+0.1, k1=0.5, k2=0.5

exp.

pred.

V=Vt+Vb

Vt

Vb

0

400

0.0 0.5 1.0 1.5 2.0 2.5

V, k

N

a/d

pred.fc=80.4MPa

m1=0.5, m2=0.3, k1=0.5, k2=0.5

exp.

23.5MPa

sectional pred.

Parametric study: a/d ratio

≈ 1.5 ≈ 2.0

(tested by Yang et al., 2006)


a

d

Vexp/Vpred of 27 deep beams with openings:

Avg=1.03, COV=9.3%.

Conclusions and Future work

Jian Liu Conclusions and Future work

• Adequate models for shear strength of deep beams identified

• Efficient 1D macroelement formulated

• Complete shear response well predicted

• Mixed-type modelling framework proposed

• Complex structures under extreme loading analysed

• Kinematic model proposed for deep beams with openings

Summary and Conclusions

Jian Liu Conclusions and Future work 60/61

• Shear failures after flexural yielding ductility

• Effect of axial force in macroelement columns and shear walls

• Entire behaviour of RC deep beams with web openings

• Extend applicability of 3PKT, e.g. under cyclic loading

• Simplified 3PKT for design codes

Future work

Jian Liu Conclusions and Future work 61/61

Jian Liu Kinematics-Based Modelling of Deep Transfer Girders in RC Frame Structures

Thank you for your attention.

Date post:	07-Oct-2020
Category:	Documents
Upload:	others
View:	8 times
Download:	1 times

Resilience of Buildings with Deep Reinforced Concrete ... presentation - orbi.pdfDiagonal strut...

Documents