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Behaviour of Complex Reinforced

Concrete Structures under Three-

Dimensional Loads

Ashraf Ayoub, PhD, PE, FACI

RAE Chair of Nuclear Infrastructure

Dept. of Civil Eng.

City University London

Research Centre for Civil Engineering Structures

January 13, 2016

OUTLINE

• Experimental Program and Results of Panels

subjected to 3D Loads

• Conclusion and On-Going Work

• Background and Objective

• FE Model of RC Elements subjected to 3D

Loads and Correlation Studies

x

y

z

Fig. 1 Three-Dimensional Panel Elements under Various Types of Stresses

Structures subjected to tri-directional state of stresses

1-Columns subjected to multi-directional loads

2-Beams subjected to bi-directional loads

3-Shells and plates subjected to combination of

membrane and transverse shear

BEHAVIOR OF RC COLUMNS

Shear Failure Bending Failure

Failure by

Combined Loadings

(Axial + Shear + Bending )

Kobe

Earthquake

Northridge

Earthquake in

California

Shinkansen bridge

Japan

P

R

T (http://www.wsdot.wa.gov/projects/viaduct/deis/chapter1_1.htm)

Seattle, WA, US

TORSION IN BRIDGE MEMBERS

Shear Stresses in Containment Wall

• Tangential Shear (in-plane)

• Peripheral Shear (punching)

• Radial Shear (out-of –plane)

Available Experimental Data for Concrete

Elements subjected to 3D Loads

• Tests conducted on cubes or cylinders using

hydraulic triaxial cells (e.g. K. Gurstle, and

Van Mier)

• Specimens are small-scale and unreinforced

• Specimens subjected to compressive loads

only

Experimental Panel Tests

(Adebar, University of Toronto)

0

1

2

3

4

5

6

7

8

9

10

-30 -20 -10 0 10 20 30

Toronto

SP7: (-10.48,2.62)

SP9: (-16.72,2.09)

SP4: (14.32,3.58) SP3: (0,2.93)

SP8: (-27.82,0) SP8: (27.82,0)

τxy / √fc

τ 23

/√f c

Normalized

No

rmal

ized

3D Universal Panel Tester

(University of Houston)

Experimental Program

Reinforcement elevation Reinforcement Sideview

8.0

4.5

30.0

4.5

8.0

16

Group B panels

2#4

2#4

2#4

2#4

A

B

C

D

7

16.0

12.5

11.5 0.5 10.0

0.5

55

55

0

200

400

600

800

1000

1200

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020

In-Plane Shear Capacity of Panels

OP6 (72%)

OP5 (68%)

OP4 (46%)

OP2 (22%)

OP1 (13%)

OP3 (28%)

OP0

In-p

lan

e sh

ear

stre

ss (

psi

)

In-plane shear strain (in./in.)

• Concrete crushing at the south

bottom side

• Severe cracks and spalling of concrete

at the north side

Group B: OP4

South Side North Side

46% of out-of-plan shear capacity

• In-plane shear load/Jack=26 kip

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30

Interaction Diagram between Out-of-Plane and In-Plane

Shear Strength

Normalized in-plane shear strength

No

rmal

ized

ou

t-o

f-p

lane

shea

r st

reng

th

τxy / √fc

τ 23

/√f c

OP0 (14.15,0)

OP2 (13.91,1.23)

OP1 (12.18,0.73)

OP3 (11.32,1.59)

OP4 (8.18,2.91)

OP5 (8.56,4.06)

OP6 (4.17,4.62)

OPR (5.65,0)

OPR

OP3

OP1

OP0

1.0

0.68

0.48

0.29

0.13

0.58 0.60

0.80 0.86

OP5

OP4

1.0 0.68 0.48 0.29

1.0

τxy

τxz

τyz

Interaction Diagram between Tri-directional Shear Stresses

SL

NB

South side

NT

North side

NR

SR

ST

SB

NL

1

2

3

45◦

x y

z

τxy

τxy

τyz

τxz

τxz τyz =

Concrete Behavior in Compression

Concrete in Compression

Softening Coefficient z

'ζ

p

cf

p

'

cf

is compressive strength in test

is compressive strength for cylinder test

15

9.08.5

'

'

c

cf

ff

1

14001

1

f

241f

z ffff c 1

Constitutive Relationships of Concrete

Concrete in Tension

1

1

1

1

    when  

c

crc cr crf

1 1   when c c crE

crf

'47,000 ( )psic cE f'3.7 (psi)5 ccr ff

0.00008cr

Concrete in Compression

2

2

'

cfz

0z

2

' 2 22 2 c

c

o o

fz

z z

2

2'

2

/ 11

4 / 1

c ocfz

z

z

c

c

16

12υ 0.2 850ε

sf       ε ε 

sf y

12υ 1.9                      ε ε 

sf y

Poisson Ratio for RC (Zhu and Hsu, 2002)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Pang's series

Belarbi's series

OP series

Soft

enin

g co

effi

cien

t

Lateral horz. strain (in./in.)

Softening coefficient

100000178.01

241

4001

19.0

8.5

1

'

xx

fc

z

100ou

ox

OP2

OP3

OP5

OP4

OP1

OP6

Out-of-plane shear

Reduction factor

Macro Models

– Based on Frame-Type

Elements

– Suitable for System

Evaluation (e.g.

Buildings/Bridges)

Micro Models

– Based on 2D

Membranne/3D Brick

Elements

– Suitable for Component

Evaluation (e.g. Joints)

3D FEM Analysis of RC Structures

Nonlinear Frame Elements

Plastic Hinges

– Lumped Plasticity

– Computationally

Efficient

– Suitable for Evaluation

of Global Parameters

(e.g. Drifts)

– Not suitable for

Evaluation of Local

Details (e.g.

Reinforcement and

Concrete Strains)

– Input Parameters not

Easily Defined

Fiber Beam Elements

– Distributed Plasticity

– Suitable for Evaluation of

both Global and Local

Parameters

– Input Parameters based

on Basic Material

Properties

– Computationally more

Expensive

X

Y

Z

Y

Z

yj

zj

FEM Analysis - Fiber Element for RC Columns

22

Moment Mu Curvature κ

N

P

Mixed frame models assume “exact” force

Interpolation functions

Element result is exact for the beam theory

Same structural DOF’s as plastic hinge

Accurate in representation of curvature

localization at plastic zone

1 0 0

1 1( ) 0

0 1

N xL L

x x

L L

Force Interpolation

function

Section Force = 1

2

( )

N N

V N x M

M M

0 ( )

( ) ( )

( )

x N

x f x V

x M

( )f x = Flexibility Matrix

Strains

x

L

THEORETICAL MODEL : Mixed Formulation

FIBER ELEMENT - Accounting for Shear Effects

• Adopt a Timoshenko Beam Formulation with Shear

Deformations

• Impose Equilibrium in Transverse Direction to Determine

Concrete Lateral Strain

• Rotate Stresses to Principal (Crack) Directions

• Use 3D Concrete Constitutive Models

FIBER ELEMENT -Timoshenko Formulation

ys yc

Concrete Beam Stirrups

x

b

y

x

y

STIRRUP - Equilibrium in Transverse Direction

Equilibrium in Vertical Direction:

y

, ,, , 0y c y s

i i i i

y c y sA A , ,, , 0

z c z s

i i i i

z c z sA A

z

Ay,s Stirrup Area

Longitudinal steel Area

i

Ax,s

t

Az,s Stirrup Area

y

z

y

z

y’

z’ '

Stirrup

Fiberi

y

z

y’

z’ '

Stirrup

Fiberi

RC Circular Section (No Warping)

2 ' ' ' 2 '

y' yy yz zzε = cos (θ )ε - 2cos(θ )sin(θ ) sin (θ )ε

Fz xy Fy

xz

y' y'

2 '

y'

2 '

y'

= f(ε )

= cos (θ )

= sin (θ )

yy

zz

Fiber Element – Softened Membrane Model

Concrete Element in

Cartesian Coordinate System Concrete Element in Principal

1-2 Coordinate

of Applied Stress System

cracks

P Stress/Strain Curve in Principal

1-2 Coordinate of Applied Stress

Non Softened

Curve

Softened Curve

1

212

1

2

l

t

12

lt

lt

1 2

1

c

12

c 1

c

2

c

2

c

12

c

12 12

c G

ς

2D BIAXIAL STRENGTH ENVELOPE - Constitutive Model

Biaxial Compression region Compression/Tension region

Biaxial Tension region

Tensile strength is constant

ip i cf z

2

1 0.92 0.76ip ip

i

c cf f

z

2 p cf z

1 1 2 3cf f f fz

2

5.80.9c

c

f ff

3 1

24f

1 1

1

1

1 400f

FE Model I: In-Plane

Analysis under the

Effect of Out-of-Plane

Loads

Select

Assume

Assume

Calculate , and , Eqs. and

Calculate , , , , and Eqs. , , , and

Calculate and Eq. to

Calculate and from and

Calculate and Eqs. and

Calculate and Eqs. and

End

No (assume )

No (assume )

No Yes

Yes

Yes

2

12

1

t 11 12

zc

2 c

1 12c 15 14 13 16 17

f tf 18a 18e

1)( tt ff 1)( tt ff f tf

2)( tt ff 2)( tt ff 19 20

?0)()( 12 tttt ffff

?0)()( 12 tttt ffff

t t 3 6a

? Is 02

modified

Verified with the

experimental data

Prediction of Model for In-Plane Shear Strength of OP Series:

0

200

400

600

800

1000

1200

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020

Experimental

Analytical

Shea

r st

ress

(p

si)

Shear strain (in./in.)

OP4 (46%)

OP2 (22%)

OP1 (13%)

OP3 (28%)

FE Analysis and Prediction Yoshimura 1996

Tested short RC columns under bi-directional shear loads

1- Lateral shear load equal to 0, 33, 55, 77 kip was applied and kept constant in

direction-2.

2- Shear load in direction-1 was applied and increased till failure.

FE Analysis and Prediction Kobayashi et. al. 1986

0

25

50

75

100

125

0 0.05 0.1 0.15 0.2 0.25 0.3

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8

Lo

ad (

kip

)

Deflection (in.)

Lo

ad (

kN

)

Deflection (mm)

D00 experimental

D00 analytical

0

25

50

75

100

125

0 0.05 0.1 0.15 0.2 0.25 0.3

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8

Lo

ad (

kip

)

Deflection (in.)

Lo

ad (

kN

)

Deflection (mm)

S35 experimental

S35 analytical

Fig. (7) 3D Concrete failure Surface

3D Concrete Constitutive Model

Menetrey-Willam (1995) 3-Parameter

Model

2

( , , ) ( , ) 0f f ff A B r e C cz z

2 2 2

2 1/ 2

4(1 )cos (2 1)( , )

2(1 )cos (2 1)

e er e

e e D

1.5 1 1; ;

6 3f f f

c c c

A B Cf f f

FE Model II: Stress De/Re-Composition Approach

Maekawa model (2003)

Vecchio and Selby Model (1991)

3D strain field is decomposed into

three in-plane sub-spaces

In each sub-space, the component stresses are

computed using the 2D crack model and

constitutive laws

3 3 1( , )f

1 1( )f

2 2 1( , )f

100000178.01

241

4001

19.0

8.5

1

'

xx

fc

z

xx

zz yy

xy

xz

36

Support Blocks

Strong Floor

Hydralic JackLoad Cell

Steel Strands

(Inside Column)

Strong Wall

Test Unit

Two Hydraulic

Actuators

Load Stub

Axial Force using Prestressing

(approximately 7 % of predicted axial capacity)

12 ft

2 ft

Belarbi et al.(2007)

'

cf = 4.85 ksi

Longitudinal yield stress = 66.4 ksi

Transverse yield stress = 65.2 ksi

MS&T Column

MS&T Column (Combined Bending/Torsion) – T/M =0.2

NUMERICAL RESULTS – Static Load Displacement

0

10

20

30

40

50

60

0 2 4 6 8 10

Displacement (in)

Lo

ad (

kip

)

Experiment T/M 0.2Analysis T/M 0Analysis T/M 0.2

Load - Displacement response matched well with the experiment

Increase of the torque reduced the load resisting capacity

-50

-40

-30

-20

-10

0

10

20

30

40

50

-10 -8 -6 -4 -2 0 2 4 6 8 10

Displacement (in)

Lo

ad

(k

ip)

Analysis T/M 0.4

Experiment T/M 0.4

MS&T Column (Combined Bending/Torsion) – T/M =0.4

NUMERICAL RESULTS – Cyclic Load Displacement

3D Element predicted well the cyclic response

NUMERICAL RESULTS – Earthquake Analysis

Axial Load = 356 kN

Concrete Strength = 36.7 MPa ksi

Spiral Strength = 397.4 MPa

Long. Bar Strength =458.2 MPa

Long. Steel Ratio = 3.5% (16 #6)

Spiral Ratio = 0.92% ( 0.25”, 1.5” C/C)

UNR Column - Laplace et al.

EL Centro (1940) record

Squat Column

Longitudinal

Reinforcement 3.5%

Shell Region

Core Region

406.4 mm Dia.

Transverse Reinforcement

0.92%

2.5 x EL Centro NS

NUMERICAL RESULTS – Time History

UNR Column- 9S1

-40

-30

-20

-10

0

10

20

30

40

0 5 10 15 20 25 30

Time (s)

Dis

pla

cem

ent

(mm

)Experiment

Flexural Element

Flexure Element is not able to predict the displacement

NUMERICAL RESULTS – Time History

UNR Column 9S1

-50

-40

-30

-20

-10

0

10

20

30

40

50

0 5 10 15 20 25 30

Time (s)

Dis

pla

cem

ent

(mm

)

Experiment

Shear Element

Shear Element is able to predict the displacements rather well

NUMERICAL RESULTS- Shear Element Load Displacement

UNR Column- 9S1

-500

-400

-300

-200

-100

0

100

200

300

400

500

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50

Displacement (mm)

Bas

e sh

ear

(kN

)

ExperimentShear Element

Shear Element response is rather well

Arias-Acosta and Sanders (2010)

UNR Column- C1

'

cf = 32 MPa

0.6x Petrolia at Mendocino (1992)

Earthquake

X

Y

III. EXPERIMENT : Earthquake Analysis

UNR Column- C1

X C o m p o n e n t

-1 5 0

-1 0 0

-5 0

0

5 0

1 0 0

1 5 0

-4 0 -2 0 0 2 0 4 0 6 0

D ix p la c e m e n t (m m )

Ba

se

Sh

ea

r (

kN

)

E x p e r im e n t

A n a ly s is

Y C o m p o n e n t

-6 0

-4 0

-2 0

0

2 0

4 0

6 0

-3 0 -2 0 -1 0 0 1 0 2 0 3 0

D ix p la c e m e n t (m m )

Ba

se

Sh

ea

r (

kN

)

E x p e r im e n t

A n a ly s is

3D element prediction is fairly accurate compared to experimental values

ANALYTICAL STUDY : Load - Displacement

Ongoing Research

Evaluation of the behavior of concrete nuclear structures

under multi-hazards:

Long term effects: Temperature, Radiation,

Shrinkage, Creep, Corrosion

Natural and man-made hazards: Earthquake,

Impact/Blast, Loss of Coolant (LOCA)

Multi-Scale Analysis of Concrete Structures

Meso-Modeling for long term deterioration effects

Macro-Modeling for hazard loads

Improving the design and behavior of concrete structures

Experimental evaluation of new design systems

Enhancement of Concrete Properties

• Fibre Reinforced Concrete

• Steel Fibre Reinforced Concrete (SFRC)

• Carbon Nano-Fibre Reinforced Concrete

(CNFRC)

• Reactive Powder Concrete

• Engineering Cementitious Composites (ECC)

New Construction Techniques for Nuclear Structures:

Steel-Concrete (SC) Walls

FRCM/TRM Composites

Resin is made of inorganic materials

Suitable for use in high temperatures

Summary and Conclusion

• New test data for full scale reinforced concrete panels subjected to

tri-directional loads were obtained

• The softening coefficient for concrete subjected to tri-directional

load was established

• Based on the test results, a new finite element model for concrete

structures subjected to 3D loads was developed

• The model was implemented in FE fiber beam elements, and the

behavior of structures subjected to multi-directional loads was

accurately correlated with experimental results

Future Work

• Experimental and analytical evaluation of the constitutive law to

model FRC (using steel & CNF fibers)

• Evaluation and Improvement of the behavior of SC panels

• Simulation of Impact/Blast loads (SHPB tests, large deformation,

explicit algorithm)

• Modeling of bond-slip (RC & SC panels) and simulation of Steel

Buckling

• Behavior under elevated temperature – Multi-hazard analysis

• Evaluation of strengthening techniques using FRCM/TRM

composites