2016 uhpc iowa_bi-lienar_v1

1. Flexural Design Procedures for UHPC Beams and
Slabs
Yiming Yao, Xinmeng Wang, Barzin Mobasher
School of Sustainable Engineering and the Built Environment Ira A.
Fulton Schools of Engineering
Arizona State University
Tempe, AZ 85287-5306
First International Interactive Symposium on UHPC, July, 2016, Des
Moines, Iowa

2. Introduction
 Traditional Design procedures were based on ultimate strength of a structure
defined by Inherent brittleness and low tensile strength.
 Improved strength, ductility, stiffness, and shear strength by adding fiber
reinforcement, to a densified low porosity matrix.
 Ultra-high performance concrete (UHPC), 150 MPa (22 ksi), 2-3% steel fibers
– High strength, high ductility, Low permeability
 Thin sections, Complex structural forms, Cast by pouring, injection, extrusion
 Strength, ductility, impact resistance, durability, serviceability in aggressive
environments

3. Thin sections and double curvature shapes
Photo courtesy of Szolyd
Development
Lafarge

4. UHPC girders and decks
Jakway Park Bridge in
Buchanan County, Iowa

5. Filed-cast UHPC Connections
UHPC connection between precast deck panels as deployed by NYSDOT on
I-81 in Syracuse, NY
UHPC connection between precast deck panels as
deployed by NYSDOT on CR47 over Trout Brook.
Deck-level connection between precast
deck panels.
Graybeal, B. (2014). Design and construction of field-cast UHPC Connections (No. FHWA-HRT-14-084).

6. Research Objectives
 A homogenized stress-strain tensile and compression material property model.
 Derivation of simplified deflection hardening bilinear moment [1].
 The load deflection response of simply supported beams and panels.
 A bilinear and trilinear moment-curvature model.
 Closed form equations for the deflection, stress, and strain field calculation are
obtained in parametric form
Soranakom C., Mobasher B. , “Closed-Form Solutions for Flexural Response of Fiber-Reinforced Concrete Beams”(ASCE)0733-
9399, 2007.

7. Outline
 Introduction
 Simplified Moment-Curvature Relationship
 Equilibrium Based Moment and Curvature Distributions
 Closed-Form Solutions for Load-Deflection
 Algorithm
 Closed-form deflection equations
 Parametric Study
 Field equations for Deflection, Curvature, stresses, and strains
 2-D Deflection Contour
 Experimental Verifications
 Discussions
Ductility Durability
Economy
FRC

8. Modelling Approach
 Materials evaluation
– Parametric linearized materials model
– Cross sectional analysis based on stress and strain diagrams
– Closed-form solutions of moment and curvature responses
– Back calculation of tensile responses and extraction of tensile properties
 Structural response check
– Forward simulation of flexural load-deflection responses
– Check serviceability states: curvature, deflection, post-crack stiffness
– Characterization of shear stresses using 2-D analysis
– Design recommendations: moment capacity, minimum reinforcement
ratio, serviceability limits
– Advance to structural analysis using the stiffness matrix

9. Stress-Strain for Hardening UHPC
t
t
E
cr trn=cr tu=tucr
cr=Ecr
cst=crE, <1
 Material parameters are described as a multiple of the first cracking tensile strain
(cr) and tensile modulus (E)
Tension model
Compression model
c
c
cy=cr
cy=cr γE
cu=λcucr
*tucr
l*cucr

10. Moment-Curvature Diagram
M
f
f
c
0 < t < tu
k
d
C2
T1
T2
T3
C1
stressstrain Moment curvature
diagram
 Incrementally impose 0 < t < tu
 Strain Distribution
 Stress Distribution
 SF = 0, determine k (Neutral axis)
 M = SCiyci+ STiyti and f=c/kd
 Normalization M’=M/M0 and f’=f/fcr
 1 10
kd
c cF b f y dy 
 1 10
1
kd
c c
c
b
y f y ydy
F
 

11. Closed Form Solutions for Strain Hardening/ or
Softening material
21
6
cr
cr cr
M =M' M
M bd E
'
2
cr
cr
cr
d
f f f

f


Soranakom C, Mobasher B. “Correlation of tensile and flexural responses of strain softening and strain
hardening cement composites”, Cem Concr Comp 2008;30:465–477.

12. Back Calculation of UHPC Material Beams,
size effect
Kim D-J, Naaman AE, El-Tawil S. “Correlation between Tensile and Bending Behavior of FRC Composites with Scale Effect”,
Proc FraMCoS-7, 7th International Conference on Fracture Mechanics of Concrete and Concrete Structures, May 23-28, 2010.
 Small: 50x25x300 mm
 Medium: 100x100x300 mm
 Large: 150x150x450 mm

13. Non-linear hinge
 Use of stress-strain law to model the
smear crack
 Plane sections remain plane
 Obtain analytical moment-rotation
relationship of the non-linear hinge
Moment
Curvature
M0
Mmax
Mfail
fj,Mj)
fj-1,Mj-1)
Loading
Unloading
Non-Localized
Zone
Localized
Zone
S S/2
cS
P Localized
Zone
Non-Localized
Zone
Axis of
Symmetry

14. Simplified Moment-Curvature Relationship
 Bilinear moment-curvature relationship
 The first cracking curvature and moment
 
1
1 1
1
m'( q') q' 0<m' 1 0<q' 1
m
m'( q') q' 1 m' 1 q'
q
  

    

 
g cr cr
u cr
cr cr cr u u
u cr
M( ) EI 0<M M 0<
M M
M( ) M M M M 1
 f  
    
 
  

      

'
cr
M ( )
m'( q')
M

 ' i
cr
q


 u
cr
M
m
M
 u
cr
q



21
6
cr crM bd E
2 cr
cr
d

 
cr
g
EI
EI
 

15. Moment & Curvature Distributions Due to equilibrium
 Use Static Equilibrium to get moment distribution
 Curvature distributions along the beam are generated based on simplified bilinear
moment-curvature model

16. Solutions are available for Other loading types
 Same approach different equilibrium equations affecting Moment distribution
 Simple supported beam and cantilever beam.
 8 different loading types

17. Closed-Form Solutions for Load-Deflection
 21 21 10
 
x
x dx C f 0   x
2
  Lx   22 21 21 20
   
x
x dx x dx C


 f f
0   x
2
  Lx   22 21 22 40
   
x
x dx x dx C


  
 21 21 30
x
x dx C  
 Rotation
Region I :
Region II:
 Deflection
Region I :
Region II
The constants of integration defined in terms of Ci are numerical
values which are determined based on the boundary conditions.

18. Example- Load-deflection of four point bending beam
The two regions of four-point bending beam:
 Region I (0 ≤ 𝑥 ≤ ξ ), pre-cracked region
 Region IIa ( ξ ≤ 𝑥 ≤ 𝐿/3), the post cracked region
 Region IIb ( 𝐿/3 ≤ 𝑥 ≤ 𝐿/2), post cracked, constant moment
Curvature:
 1
cr
x x



 0 x  
 
 
2
3 ' '
3
a cr
q x q x L
L
x




  


3
L
x  
 2 'b crx q 
3 2
L L
x 3 '
L
m
 

19.  Region I :  21 10
x cr
x xdx C



  0 x  
 Region IIa :  
 
22 20
3 ' '
3
xcr
a cr
q x q x L
L
x xdx dx C


 

 
  

    3
L
x  
 
 3
22 30 3
3 '
'
'
3
L xcr
b cr crL
q x q x
x xdx dx q
L
dx C
L


  



  

     
3 2
L L
x 
Boundary conditions :
2
0Lx
 
    21 22a   
   22 223 3a b
L L 
0 0x      21 22a       22 223 3a b
L L 
Example- Load-deflection of four point bending beam
Rotation :
 Region IIb :

20. Deflection :
   21 21 40
x
x x dx C  
 Region IIa:
 Region IIb:
     22 21 21 22 5( ) 0
x
a x x dx C

       
           22 21 21 22 22 22 63
0 3
x
b a a bL
x L x dx C            
Solutions: * 2
 cr L   is deflection coefficient
  2
*
21
3 21
3 ' 2 '
6
x x q L Lq
L
  

   
 
 3 3 2 2* 2 2 2 3
222
1
3 ' 3 3 9 ' 2 ' 9 ' 3 '
6 3
a x q x x L x q xL q xL xLq L q
L L
   

      


2
22 2
* ' ' 1 3 3 '
1 1 '
2 542
b
q x q x q
q
L L LL


   
       
  

 
*

Example- Load-deflection of four point bending beam
 Region I :

21. Parametric Curvature Distribution along the length
Distributed
 Due to symmetry:
 consider 1/2
model
 span L=1
 Normalized Moment
 m’=1.2, 1.5, 1.8
 Normalized Curvature
 q’=2, 2.5, 5
 Increasing m’, or q’
 Increases
curvature

22. 2-D Deflection Contour
Three-point bending
Four-point bending
Loading
type
Span
(L),mm
Width
(b),mm
Height
(d),mm
εcr, µstr E, Mpa η
3PB 300 100 100 244 20400 0.01
4PB 750 100 100 244 20400 0.01
h
Table 1 - Parameters of 2-D deflection distribution

23. 2D Contour – Distribution loading
Deflection distribution
X-Strain distribution

24. Experimental Verification
 Generalized approach
 Generate moment-curvature
response using cross sectional
analysis tools (SAP2000, SE::MC,
OpenSees, Soranakom and
Mobasher (2007,2008), etc.)
 Calculate the deflection coefficient δ*
using the proposed equation
 Substitution of the geometries b, h, L
and basic material properties: E, εcr
 Calibrate the model parameters m
and q with experimental data
 Serviceability limits check

25. Experimental Verification- UHPC beam
 Full size UHPC beam
 2% of smooth/twisted steel fiber
 fc’=201-232 MPa
 Ρ=0.94% or 1.5%
Yoo, D. Y., & Yoon, Y. S. (2015). Structural performance of ultra-high-performance concrete beams with different
steel fibers. Engineering Structures, 102, 409-423.

26. Experimental Verification- RC with Steel fibers
Yoo, D. Y., & Yoon, Y. S. (2015). Structural performance of ultra-high-performance concrete beams with different
steel fibers. Engineering Structures, 102, 409-423.

27. UHPC Pi-girder: FHWA Study
Jakway Park Bridge in
Buchanan County, Iowa

28. Modelling Approach

29. Model Simulation
Chen, L., & Graybeal, B. A. (2011). Modeling structural performance of second-generation ultrahigh-performance
concrete pi-girders. Journal of Bridge Engineering, 17(4), 634-643.

30. Conclusions
 Closed-form deflection equations of strain hardening fiber
reinforcement concrete beam for eight different loading
patterns were developed based on simplified bilinear and
trilinear moment-curvature models.
 The effect of normalized moment and normalized curvature on
the deflection distribution were studied.
 Accuracy of the closed-form deflection equations was identified
by comparing the simulated data with experimental data.

31. Closed-form Deflection equations
* 2
 cr L   *
 is deflection coefficient

32. Closed-form Deflection equations

33. Stress and Strain Distribution
Tensile Regions 1.0, 2.1 and 3.1
0 < β < 1 and λ < ω
 lctop cr=
 tbot cr=
1
1
hc1
ht1
kd
d
1
1
yc1
yt1
Fc1
Ft1
ft1
fc1
(1)
1 < β < α and λ < ω
 lctop cr=
 tbot cr=
1
1
hc1
ht1
kd
d
1
1
yc1
yt1
Fc1
yt2
ft1
fc1
2 ht2
cr
Ft22
ft2
Ft1
(2.1)
α < β < βtu and λ < ω
 lctop cr=
 tbot cr=
1
1
hc1
ht1
kd
d 1
1
yc1
yt1
Fc1
yt2ft1
fc1
2 ht2
cr
Ft22ft2
Ft1
(3.1)
3 ht3 3 Ft3
yt3
ft3
trn

34. Back calculation of UHPC tensile
properties
Mobasher, B., Bakhshi, M., & Barsby, C. (2014). Backcalculation of residual tensile strength of regular and high performance fiber
reinforced concrete from flexural tests. Construction and Building Materials, 70, 243-253.
Kim, D. J., Naaman, A. E., & El-Tawil, S. (2010, May). Correlation between tensile and bending behavior of FRC composites with
scale effect. InProceedings of FraMCoS-7, 7th international conference on fracture mechanics of concrete and concrete structures.
Jeju Island, South Korea.

35. Structural Analysis using Discrete Hinges
 Equally distributed
 Lp = crack spacing
 Tensile tests, tension stiffening model
DIC
observation

36. 2D Strain and Stress Distributions

37. Cont’d
~48°
tmax=2.21MPa

38. Finite Element Analysis on Shear Behavior
 Limitation of analytical based flexural model
– Cross sectional analysis
– Inclined growth
– Damage
 Finite element analysis in LS-DYNA
– Damage concrete model: MAT159 (*MAT_CSCM)
– Piecewise model for rebar: MAT024
(*MAT_PIECEWISE_LINEAR_PLASTICITY)
– Interface: *Lagrange_IN_SOLID
Minelli, F., Conforti, A., Cuenca, E., & Plizzari, G. (2014). Are steel fibres able to mitigate or eliminate size effect in
shear?. Materials and structures,47(3), 459-473.

39. Stress Distribution

40. Distributed Damage