1. No category

By
Ravi Mullapudi, PhD
University of Houston
Houston, Texas, USA
1
OUTLINE
I.
INTRODUCTION
II. THEORITICAL MODEL
III. EXPERIMENTAL PROGRAM
IV. ANALYTICAL STUDY
V. TEST RESULTS
VI. SUMMARY AND CONCLUDING REMARKS
2
I. INTRODUCTION : General Behavior of RC Columns
Investigate the complex behavior of bridge columns under combined loading and its impact
on the response of bridge system through experimental and analytical studies
Kobe
Earthquake
Shear Failure
Shinkansen bridge
Japan
Kobe
Earthquake
Hanshin
Express Way
Failure by
Combined Loadings
(Axial + Shear + Bending + Torsion)
Northridge
Earthquake in
California
Bending Failure
3
I. INTRODUCTION : NEES Project Overview
Investigation of Combined Actions on Bridge Earthquake
Response
UNR
Dynamic
Shake
Table
Tests
UH
Finite Element
Models &
Development of
Models
Missouri S&T
Slow cyclic tests
&
Development of
Models
UIUC
Pseudodynamic
tests
UCLA
Soil Structure
Interaction
WU
Educational,
Outreach,
and Training
Modules
 Final Deliverables:
1. Further Improved Understanding of Bridge Columns under Combined Loadings
2. Develop Finite Element Simulating Models
3. Interaction Diagram of Moment, Shear and Torsion
4. Development of Design Guidelines
5. Educational Modules
4
I. INTRODUCTION
Investigate the structural response of
individual column under following
loading conditions and develop FE
Models
•
•
•
Axial + Shear +Bending
Axial + Torsion
Axial + Shear + Bending + Torsion
Analytical
Study
Experimental
Study
 Develop Useful
Design Tools
 Upgrade
Design Details
for Columns
Investigate the impact of column
behavior on whole bridge system
Estimate possible critical loading
conditions due to
•
•
Geometrical conditions
External load (Earthquake etc..)
5
I. INTRODUCTION : Research Significance
 No analytical models to include the effect of torsion and
combined loadings
 No simplified design guidelines and interaction diagrams
 No information on quantification of degree of damage and
correlation with engineering limit states under combined
loading
To identify the changes in existing performance levels in
the presence of torsional loadings
6
52" (4.33')
175.75" (14.65')
16"
7"
Main frame
left column
Adding 10 more out
of plane hydraulic
20 Yokes
cylinders
55"
175.75"
55"
Main frame
bottom beam
40 in plane hydraulic cylinders 8“ bore size, @ 5000 psi,
Four out
plane
beams 200 kip tension
Apply
250of
kip
compression,
Supporting
beams
Main frame
right column
Out of plane
frame
7
II. THEORETICAL MODEL : Universal Panel Tester
Main frame
top beam
II. THEORETICAL MODEL : Fiber Element
 Micro Models
 Macro Models
– Based on 2D
– Based on Frame-Type
Membrane/3D Brick
Elements
Elements
– Suitable for System
– Suitable for Component
Evaluation (e.g.
Evaluation (e.g. Joints)
Buildings/Bridges)
8
II. THEORETICAL MODEL : Fiber Element
 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 Concrete Biaxial Constitutive Models (Softened
Membrane Model)
9
II. THEORETICAL MODEL : Macro-Modeling of RC Elements
y Y
j
z
j
Z
Y
Z
X
10
II. THEORETICAL MODEL : Softening (ς)
P
Stress/Strain Curve in Principal
1-2 Coordinate of Applied Stress
Non Softened
Curve
cracks
Softened Curve
y
 lt
 xy
 1  12  2
1  2
ς
x
2
 1c

c
12
 12  1
Concrete Element in
Concrete Element in Principal
Cartesian Coordinate System 1-2 Coordinate
of Applied Stress System
 2c

c
12
 2c
 1c
12c  G 12
11
II. THEORETICAL MODEL : Material - Concrete
Hsu (1993)
fc
Non Softened
f c'
ε20   ε0
f
εc   ε0
'
c

A
Softened
0.2ςfc'
O
0.8ςfc'
 o
o
2εo
fc
B
c
ε20
C
c
A (  ε0 , fC )
'
Mullapudi and Ayoub (2010)
D (1 , f 1 )
m m
Stress, fc
G
EC
O
EC
EC
1F
H
B(ε20 ,0.2 fC )
'
E
0.5Er
C
0.2 fC
'
(1t ,0)
1
Er
(-εr ,-fr )
R
 r
E20
 fr
 20
Strain, εc
12
II. THEORETICAL MODEL : Material - Steel
1
2
C
2
(ε0 ,σ0 )
1
(εr ,σ r )
E1
B
R = R0-
With Isotropic Strain Hardening
Approach (Filippou 1983)
a1 ε y (εm - ε0 )
a2 ε y + εm - ε0
Stress σ
E0
εy
σ = bε +
*
*
ε* =
2
2
(ε r , σ r )
A
1
1
(ε0 ,σ0 )
Strain
*
(1 - b)ε*
(1+ ε*R )1/R
ε - εr
ε0 - εr
σ =
σ - σr
σ0 - σ r
ε
Bare steel bar
Steel bar in concrete
Stress
fy
fn
With Smeared Steel Approach
(Belarbi and Hsu 1994, 1995)
 n y
Strain
Bauschinger Effect
fy
13
II. THEORETICAL MODEL : Softening Models
The in-plane element considered in this model
(x-y) Coordinate system.......Longitudinal &Transverse steel direction
(1-2) Coordinate system……Applied principle stresses direction…reinforced
concrete
(r-d) Coordinate system……Concrete principle stresses direction…
unreinforced concrete
αr1β
τxy
 12c
τxy
1
σx x
σ2
σ1
 12c
r
1
σ1
stresses of RC element
Rotating Angle αr …Concrete
principle stresses
Deviation Angle  ….source of
Contribution of the concrete shear
Stress
RA-STM………α
r
SMM/FA-STM………….. 1
r
σy
σ2
 Fixed Angle1 …. Applied principle
d
y
2
14
II. THEORETICAL MODEL : Mohr Circle - Angles
Rotating Matrix
2
 cos 
 R(  ) =  sin2
-cos sin

sin 
cos 2
cos sin
2
2cos sin 

-2cos sin 
cos 2 - sin 2 
2

( xc ,  xy )
21
2 r
d
( 2c , 12c )
2αr1

r
2 12c
r
σy
αr1
τxy
c
 12
γ
2
σ 2c

d
( 2 , 0.5 12 )
τxy
σ
c
12
( x , 0.5 xy )
σ 1c
21
2 r
2αr1
(1c , 12c )
2αr1
y
d
r
1
1
σx
c
1
σ 2c
(1 , 0.5 12 )

r
2αr1
 12
2α*r
( y , 0.5 xy )
( yc ,  xy )
1   2
 1c   2c
tan 2αr 
2 xy
 
c
x
c
y
tan 2α1 
2 xy
 x  y
 xy
tan 2αr 
x y
15
x
II. THEORETICAL MODEL : Constitutive Model
Compression/Tension region (Hsu and
Zhu 2002)
 
 2 p   f c
f1 1  f 2  f c f3   
1
5.8

f1  1  
f 2  f c  
 0.9 f     1 
3
1  4001
fc
24
Biaxial Tension region
Biaxial Compression region (Kupfer et al.)
 ip   i fc
  
  
 i  1  0.92  ip   0.76  ip 
 f c 
 f c 
2
Tensile strength is constant
16
II. THEORETICAL MODEL : Fiber Element – 2D Modeling
Local Stiffness
Uniaxial Strains
 x 
 1 




  2    R(α1 )   y 
0.5 
0.5 
12 
xy 


 Dlo 
c
Equivalent Uniaxial Strains
 1 
 1 




 2      2 
0.5 
0.5 
12 
12 


12
1  12 21
1
1  12 21
0

0


0


1

E1c 12
1  12  21
E2c
1  12  21
0

0 


0 

c 
G12


c
12
1c   2c


0.5 12
1   2
c
G12
Hsu/Zhu Ratios (Modified Poisson Ratios)

1

1  12 21
 21
   
1  12 21

0


 E1c

1  12  21
 E2c  21

1  12  21

0


    D   
c
12
c
lo
12
Global Stiffness
 Dgl    R(1 ) Dlo   R(1 )
c
 x 
 x 
 
 


Dgl
*
 y
 y 
 
 
xy
 
 xy 
c
17
II. THEORETICAL MODEL : Fiber Element – Equilibrium
 At each fiber equilibrium equations that relate the applied stresses to the
internal stress of reinforcements and of concrete are as follows:
 σ x   cos 2 α1
  
2
 σ y  =  sin α1
 τ  cosα sinα
1
1
 xy  
sin 2 α1
cos 2 θ
-cosα1 sinα1
-2cosα1 sinα1   σ 1c   f stx 
  

2cosα1 sinα1   σ 2c  +  ρsty f sty 
c  

cos 2 α1 - sin 2 α1  τ12
0
 

 Transverse strain can be evaluated with help of equilibrium equations
18
II. THEORETICAL MODEL : Force Based Formulation
N
Force-based frame models assume “exact”
force Interpolation functions
P
 Element result is exact for the beam theory
Very few structural DOF’s
Accurate in representation of curvature
localization at plastic zone
L
x

1

Force Interpolation N ( x)  0

function

0

Section Force =

0
0 

1
1


L
L
x
x 

1
L
L 
N 
N 
 V   N ( x)  M 
 
 1
 M 
 M 2 
Moment Mu
Curvature κ
 0 ( x) 
N 


 
Strains  ( x)  f ( x) V


 
  ( x) 
 M 
f ( x) = Flexibility Matrix
19
III. EXPERIMENT : MS&T Column
Axial Force using Prestressing
(approximately 7 % of predicted axial capacity)
Belarbi et al.(2007)
Strong Wall Load Cell
Hydralic Jack
Load Stub
Two Hydraulic
Actuators
Steel Strands
(Inside Column)
12 ft
Test Unit
Support Blocks
Strong Floor
f c' = 4.85 ksi
Longitudinal yield stress = 66.4 ksi
Transverse yield stress = 65.2 ksi
2 ft
20
III. EXPERIMENT : Test Matrix and Column Details
Reinforcement Detail
Reinforcement Ratios
12 #8 (25 mm)
Longitudinal bars
l
For Flexure
Design
Purpose
#3 (9.5 mm) or #4 (12.7 mm)
Spiral at 2.75 in. (70 mm) C/C
t
100 Al
 2.10%
Ag
100
 dc Asp
sAc
 0.73% or
= 1.32%
1 in. (25 mm)
Clear Cover
COLUMNS
PROPERTY
H/D(6)
T/M(0)
H/D(6)
T/M(∞)
H/D(3)
T/M(∞ )
H/D(6)
T/M(0.4)
H/D(6)
T/M(0.2)
H/D(6)
T/M(0.4)
37.9
27.2
25.7
41.2
Average
Compressive
Strength26.5
was 40 MPa
H/D(6)
T/M(0.2)
Compressive Strength (f’c,MPa)
33.4
Modulus of Rupture (fcr, MPa)
3.52
3.86
3.25
3.52
3.38
3.93
3.86
Spiral Reinforcement Ratio (%)
0.73
0.73
1.32
0.73
0.73
1.32
1.32
Longitudinal Reinforcement Ratio (%)
2.10
Spiral Yield Strength (fty, MPa)
450
Longitudinal Yield Strength (fly, MPa)
457
41.2
21
1V. ANALYTICAL STUDY : Cyclic Curve
12 ft Column
MS&T Column
Axial Load = 113 kip
60
Experiment
Load (k)
30
Fiber Element
Analysis
0
-30
-60
-10
0
10
20
Displacement (in)
Strength and energy dissipation are well
represented by the developed element.
22
V. TEST RESULTS
Progression of Damage under Flexure
Buckling of Longitudinal
Rebars
Ductility-1
At yielding of the
longitudinal reinforcement
in the first layer
Ductility-3
At ultimate
Ductility-12
The test column could not resist
any further load due to the buckling
of the longitudinal reinforcement
23
1V. ANALYTICAL STUDY : Load Displacement
MS&T Column
6 ft Column
Axial Load = 113 kip
Shear Force (kip)
200
160
120
Experiment
Force Model - 1 Element
Displ. Model - 1 Element
Displ. Model - 5 Element
Displ. Model - 15 Element
80
40
0
0
2
4
6
Displacement (in)
8
10
Force-based model was able to capture the behavior with only one element.
15 elements of the displacement-based model are required to reach convergence.
24
1V. ANALYTICAL STUDY : Earthquake Analysis
Axial Load = 355 kN
UNR Column- 9S1
Laplace et al. (2001)
Cover Region
0.4
Core Region
Acceleration (g)
Transverse
Reinforcement
0.92%
Longitudinal
Reinforcement
3.5%
406.4 mm (16 in.)
Dia.
1.0 x EL Centro
0.3
0.2
0.1
0
-0.1
-0.2
16”
-0.3
-0.4
0
10
20
30
40
50
60
Time (sec)
2.5 x EL Centro NS
25
1V. ANALYTICAL STUDY : Load Displacement
UNR Column- 9S1
500
400
300
Experiment
Shear Element
Base shear (kN)
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)
26
1V. ANALYTICAL STUDY : Longitudinal Reinforcement
UNR Column- 9S1
1400
.
Shear Element - 2.5 x El Centro
Experiment - 2.5 x El Centro
Shear Element - 1.0 x El Centro
Experiment - 1.0 x El Centro
Column Height (mm)
1200
1000
800
Loading
600
Gauge 1
400
200
0
-0.002
-0.001
0
0.001 0.002
Bar Strain (mm/mm)
0.003
0.004
Shear element predicted fairly accurate comparing to experimental values
27
1V. ANALYTICAL STUDY : Spiral Reinforcement
1400
.
UNR Column- 9S1
1200
Column Height (mm)
1000
800
Shear Element - 2.5 El Centro
Experiment - 2.5 El Centro
Shear Element - 1.0 El Centro
Experiment - 1.0 El Centro
600
400
Loading
Gauge 2
200
0
0
0.001
0.002
0.003
Bar Strain (mm/mm)
0.004
The shear model accurately predicted the transverse strains along the
plastic hinge length, but slightly overestimated the values near the column
ends. This is in part due to the presence of the loading blocks which were not
accounted for in the finite element model.
28
II. THEORETICAL MODEL : 3D Modeling
To Develop a Beam Element for Analysis of combined loadings including the
torsion.
y
 yy
 yz
 zy
 zz
z
 yx
 zx
 xy
 xz
 xx
x
29
II. THEORETICAL MODEL : 3D – Beam Element
Y

u1
1
x
 z2
 z1
v1
v2
X
u2
L, EI
 x2
w2
w1
 1y
 y2
Y
V1
1
x
T
N1
M 1z
M z2
V2
X
N2
L, EI
Tx2
W2
W1
M 1y
M y2
Timoshenko Beam formulation
Displacement and Force Formulations
STM/SMM Constitutive Laws
30
II. THEORETICAL MODEL : 3D – Constitutive Model
Vecchio & Selby Approach
  xx 
 
 xx 
 yy 
 
  zz 
yy
 
 
 zz 
  xy 
 D 
2
  xy 
 
 
  yz 

 yz 
 2 
 
 
 xz 
  xz 
 2 
  f1 1  f 2  fc f3   

 



1
 0.9  
  1  4001
f c' ( MPa)

5.8

 
 1 
  24 

No Complete 3D- Model available
Assuming the same relation ship for
intermediate stress calculation
Implemented the 3D procedure to SMM
 3  f ( 3 , 1 )
1  f (1 ,  3 )  2  f ( 2 , 1 )
31
1V. ANALYTICAL STUDY : Wall
Peng and Wong (2010)
Longitudinal Steel:
8, 12 mm dia.
Transverse Steel:
10 mm dia, 200 C/C
f c' = 40.2 Mpa
Longitudinal yield stress = 535 MPa
Transverse yield stress = 564 MPai
Eccentricities
0mm
100 mm
400 mm
32
1V. ANALYTICAL STUDY : Wall
Peng and Wong (2010)
400 mm Eccentricity
SW10-400 wall
350
Flexural Shear (kN)
300
250
SW10-0 Analysis
SW10-400 Experiment
SW10-400 Analysis
200
150
100
50
0
0
5
10
15
20
25
30
Lateral Displacement (mm)
Failed in shear mode
Flexural strength was reduced to 60% and the flexural ductility reduced to 90%
comparing to the SW10-0 wall response.
33
1V. ANALYTICAL STUDY : Wall
Peng and Wong (2010)
SW10-400 wall
400 mm Eccentricity
70
Torque (kNm)
60
50
40
SW10-400 Experiment
SW10-400 Analysis
30
20
10
0
0
1
2
3
Twist Angle (deg./m)
4
5
At first yield, transverse reinforcement was yielded first but at
peak load both the longitudinal and transverse steel has been
yielded.
34
III. EXPERIMENT : MS&T Column
Axial Force using Prestressing
(approximately 7 % of predicted axial capacity)
Belarbi et al.
Strong Wall Load Cell
Hydralic Jack
Load Stub
Two Hydraulic
Actuators
Steel Strands
(Inside Column)
12 ft
Test Unit
Support Blocks
Strong Floor
f c' = 4.85 ksi
Longitudinal yield stress = 66.4 ksi
Transverse yield stress = 65.2 ksi
2 ft
Spiral reinforcement ratio 0.73%
Longitudinal reinforcement ratio 2.1%
35
1V. ANALYTICAL STUDY : Load Displacement
MS&T Column (Combined Bending/Torsion)
Column H/D(6)-T/M(0.2)
60
50
Load (kip)
40
Experiment T/M 0.2
Analysis T/M 0
Analysis T/M 0.2
30
20
10
0
0
2
4
6
8
10
Displacement (in)
Load - Displacement response matched well with the experiment
Increase of the torque reduced the load resisting capacity
36
1V. ANALYTICAL STUDY : Load - Steel Strain
MS&T Column (Combined Bending/Torsion)
Column H/D(6)-T/M(0.2)
50
40
Experiment T/M 0.2
Analysis T/M 0.2
Load (kip)
30
20
Gauge 1
17 in
10
0
0
500
1000
1500
2000
2500
3000
3500
4000
Micro Strain
Longitudinal steel response matched well with the experiment
Model captured the yielding and strains values
37
1V. ANALYTICAL STUDY : Load Displacement
MS&T Column (Combined Bending/Shear/Torsion) Column H/D(3)-T/M(0.2)
Spiral reinforcement ratio 1.32%
Longitudinal reinforcement ratio 2.1%
120
Shear Force (kip)
100
80
Analysis
Experiment
60
40
20
0
0
1
2
3
4
5
Displacement (in)
Column failed in flexure shear mode
Column showed stable flexural capacity because of the higher spiral ratio
38
1V. ANALYTICAL STUDY : Torque -Twist
MS&T Column (Combined Bending/Torsion)
Column H/D(3)-T/M(0.2)
Spiral reinforcement ratio 1.32%
Longitudinal reinforcement ratio 2.1%
1600
Analysis
Experiment
1400
Torque (k-in)
1200
1000
800
600
400
200
0
0
1
2
3
4
Twist (deg.)
Peak torsional moment was reached at a twisting angle of 0.85 deg.
At peak longitudinal steel at bottom of the column was yielded first and after
spiral reinforcement was yielded at bottom of the column.
39
V. TEST RESULTS
H/D(6)-T/M(0)
0.84 m
0.67 m
0.54 m
2.43 m
3.70 m
Effect of Torsional Moment (Long Column)
H/D(6)-T/M(0.1)
H/D(6)-T/M(0.2)
Torsional Moment Increase
H/D(6)-T/M(0.4)
H/D(6)-T/M(∞)
40
V. TEST RESULTS
H/D(3)-T/M(0)
1.83 m
0.73 m
0.45 m
1.3 m
Effect of Torsional Moment (Short Column)
H/D(3)-T/M(0.2)
H/D(3)-T/M(0.4)
Torsional Moment Increase
H/D(3)-T/M(∞)
41
III. EXPERIMENT : Earthquake Analysis
UNR Column- C1
Arias-Acosta and Sanders (2010)
f
'
= 32 MPa
c
X
Y
0.6x Petrolia at Mendocino (1992)
42
Earthquake
1V. ANALYTICAL STUDY : Load - Displacement
UNR Column- C1
X C om ponent
Y C om ponent
60
150
E x p e r im e n t
E x p e r im e n t
100
40
A n a ly s is
B a se S h e a r (k N )
B a se S h e a r (k N )
A n a ly s is
50
0
-5 0
-1 0 0
20
0
-2 0
-4 0
-1 5 0
-6 0
-4 0
-2 0
0
20
D ix p la c e m e n t ( m m )
40
60
-3 0
-2 0
-1 0
0
10
20
30
D ix p la c e m e n t ( m m )
Shear element predicted fairly accurate comparing to experimental values
43
V. TEST RESULTS: Effect of Spiral Ratio on T-M Interaction
T-M Interaction at Peak Torque
450
400
Spiral Ratio-0.73%
350
Spiral Ratio-1.32%
Analytical Predictions
T (kN-m)
300
250
Prediction from
Analytical
Model
200
150
100
50
First longitudinal
bar yielding
Flexural
cracking
0
0
100
200
300
400
500
600
700
800
900
1000
M (kN-m)
Increase in torsional and flexural strength under combined bending and torsion and pure torsion is
clearly evident in the interaction diagram with increase in spiral ratio
44
VI. SUMMARY AND CONCLUDING REMARKS
The numerical method of analysis developed in the study, which is based on
SMM, is able to predict the response of beams, walls, columns subjected to
static, reverse cyclic, and dynamic loading. The stiffness, cracking, yield point,
ultimate strength, energy dissipation and failure modes have been predicted
satisfactorily in most cases.
Displacement and force based fiber beam element for analysis of combined
loadings including torsion was developed.
For bending-shear load, columns with aspect ratio more than three and
walls with aspect ratio more than two are failed under flexure-shear mode
with the formation of plastic hinge at the base of the column or wall followed
by core degradation.
Failure of columns under combined loadings including torsion, the location
of the plastic hinge zone shift upwards from the base of the column or wall
based on the increase of T/M ratio.
45
VI. SUMMARY AND CONCLUDING REMARKS
Increase of bending and torsional moments reduces the torsional moment required
to cause yielding of the transverse reinforcement and yielding of the longitudinal
reinforcement.
An increase of spiral reinforcement ratio increases the confinement to the concrete
core, decreases the spiral yielding occurrence, increases the peak torsional strength
and improves the torsional strength and twist ductility with a better redistribution of
the shear stresses.
With a reduction in aspect ratio, reduces the displacement and twist at ultimate
resisting load and prone to fail predominantly in shear.
Fiber element accounts for the torsional sensitive structures with 1D frame
elements.
Model is accurately predicting the combined loadings including torsion.
46
ACKNOWLEDGEMENTS
Network for Earthquake
Engineering Simulation
National Science Foundation
47