Analysis and Design Procedures for Strain Hardening Beams and Panels

Analysis and Design Procedures for Strain
Hardening Flexural Beam and Panels
B. Mobasher, Y. Yao, N. Neithalath, K. Aswani, X. Wang
School of Sustainable Engineering and the Built Environment
Arizona State University
International RILEM Conference
Strain-hardening Cement-based Composites SHCC4
September 2017, Dresden, Germany

Presentation Outline
 Introduction
 Parametric Material Models for SHCC
 Derivation of Moment-Curvature Relationship
 Analytical Deflection Solutions for 1-D and 2-D Members
 Experimental Verification
 Conclusions

Motivation
Sustainability in construction
 Challenges for civil infrastructure systems
o Economic growth
o Efficient resource utilization
o Global warming
 Sustainability – A need of the hour
o Energy efficient
o Cost effective
o Durable and Safe
 Composite Systems – A solution
o Fiber reinforcement and textile composites
o Ductility and Crack control
o Light weight
o Low cement concrete
Geospatial world, “Sustainable Infrastructure: Geospatial tech in the forefront

SHCC Engineering Applications
 Applications
o Pavements/slabs, Pre-cast components, Shotcrete
o Canal Lining, Tunnel Lining, Elevated slabs
 Benefits
o Energy absorption, Fatigue life
o Impact and seismic resistance
o Freeze-thaw resistance
o Construction costs
o Steel reinforcement requirements
o Crack width
Improve
Reduce
Ductility Durability
Economy
FRC

Classification –
Fiber reinforced cement based composites
 Two categories of tensile response
o Strain hardening
o Strain softening
 Strain softening behavior
o Discrete fiber systems
o SFRC, GFRC, PP-FRC
 Strain hardening behavior
o Discrete & continuous fiber systems
o TRC, SHCC, UHPFRC
GFRC, SHCC
ECC
TRC
e
Tensile Behavior
s
Paste
SFRC
Deflection softening
Deflection hardening
UHPFRC

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 of parameters at serviceability states: curvature, deflection, post-crack
stiffness
– Design recommendations: moment capacity, minimum reinforcement ratio,
serviceability limits
– Characterization of shear stresses using 2-D analysis
– Advance to structural analysis using the stiffness matrix

SHCC Multiple cracking in tension and flexure
Tensile test
Flexural test
Moment-curvature

 Material Properties
– Strain hardening material models
Design Procedure for Modelling SHCC
Panels for Serviceability
 Moment-Load
– 1-D: beam statics
– 2-D: Limit State Analysis (Yield Line Theory)
1
2
3
4
b
Y
X X
Y
45
a
 Curvature-Deflection
– 1-D: Double integration
– 2-D: Kinematically admissible deflections
δmax
θ
 Moment–Curvature Relationship
– Cross-sectional analysis

Material Models for Strain Hardening and
Softening
Compression Model Tension Model
 Material parameters
E – Young's modulus
ecr – First creaking strain
 Normalized parameters
Compression: w, g, lcu,
Tension: a, btu, h, µ
Soranakom, Chote, and Barzin Mobasher. "Correlation of tensile and flexural responses of strain softening and strain hardening cement
composites."Cement and concrete Composites 30.6 (2008): 465-477.

Design using Serviceability based Strain Limits
 Flexural members can be designed given serviceability limits
 Allowable stress, strain, crack width, curvature, deflection
 Different modes of failure can be identified
 Interactions between tension and compression behavior
Compression Model Tension Model

Derivation of Moment-Curvature Relationship
Strain Stress
Incrementally impose
0 < et < etu
Strain Distribution
Stress Distribution
SF = 0, determine k
Moment: M = SFciyci+ SFtiyti
Curvature: φ=ec/kh
Simplified
bilinear moment-curvature
Stage : l>w, b>a

Moment-Curvature Diagram
M
f
f
ec
0 < et < etu
k
d
stressstrain Moment curvature
diagram
 Incrementally impose 0 < et < etu
 Strain Distribution
 Stress Distribution
 SF = 0, determine k (Neutral axis)
 M = SCiyci+ STiyti and f=ec/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
 
C2
T1
T2
T3
C1
FS

Bilinear Moment-Curvature Model
 
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
   
    
 
  

      

 
1
1 1
1
m'( q') q' 0<m' 1 0<q' 1
m
m'( q') q' 1 m' 1 q'
q
  

    

'
cr
M ( )
m'( q')
M


'
'
cr
q 


u
cr
M
m
M
 u
cr
q


 cr
g
EI
EI
h 
First cracking moment 21
6
cr crM bd Ee
First cracking curvature
2 cr
cr
d
e
 
Normalized moment-curvature

1-D Member: Moment-Curvature
Distributions
 Use static equilibrium to get
moment distributions
 Moment distributions
3PB 4PB
Bilinear moment-curvature
Curvature distributions

Solutions are available for other loading types
 Different equilibrium equations affecting Moment distribution
 Simply supported beam and cantilever beam.
 8 different loading types

 21 21 10
x
x dx C  
0 x 
2
Lx    22 21 21 20
x
x dx x dx C


     
0 x 
2
Lx    22 21 22 40
   
x
x dx x dx C


  
 21 21 30
x
x dx C  
 Rotation
Region ① :
Region ②:
 Deflection
The constants of integration defined in terms of Ci are
numerical values which are determined based on the
boundary conditions.
Analytical Deflection Solutions
Region ① :
Region ②:

Example – Simply Support Beam
Deflection solutions for four point bending beam
Stage I: M ’ (x=L/2)≤ Mcr
 Region ①
 1
3 '
a x x
L

 
 1 'b x 
0
3
Lx 
0
3
Lx 
3 2
L Lx 
Rotation:
 1 10
3 'x
a x xdx C
L

  
  3
1 20
3
3 '
'
L x
b L
x xdx dx C
L

    
0
3
Lx 
3 2
L Lx 
Boundary Conditions :  1 0
2b
L 
   1 13 3a b
L L 
2C1C
Curvature:
a: (0 ≤ 𝑥 ≤ 𝐿/3), Pre-cracked
b: ( 𝐿/3 ≤ 𝑥 ≤ 𝐿/2), Pre-cracked, constant moment
x

Deflection:
 Region ① -a:
   1 1 30
x
a ax x dx C  
     3 2
1 1 1 40
3
L L
b a bL
x x dx x dx C     
 Region ①-b:
0
3
Lx 
3 2
L Lx 
Boundary Conditions :
 1 0 0a x  
   1 13 3a b
L L 
3C 4C
Solution:
3
*
1 3
'
32
a
x x
q
LL

 
   
 
2
*
1 2
1
'
2 542
b
x x
q
LL

 
    
 
* 2
 cr L  *
 deflection coefficient
Maximum Deflection
   
3
1
23
2 1296
b
PLLx
EI
   
'
'
6g g
M PL
EI EI
   

 Region ①: (0 ≤ 𝑥 ≤ 𝜉 ), Pre-cracked region
Stage II: M ’ (x=L/2)> Mcr
Transition point:
From pre-cracked region to post-cracked region: x=ξ
 ' 2 3
crM
M x L L



 ' 2
'
cr
M x L
m
M


3 '
L
m
 
 Region ②
a: (𝜉 ≤ 𝑥 ≤ 𝐿/3), Post-cracked
b: ( 𝐿/3 ≤ 𝑥 ≤ 𝐿/2), Post-cracked, constant moment

Rotation
Boundary conditions
 2 0x L      21 22a       22 223 3a b
L L 
Curvature
 1
cr
x x



 0 x  
 
 
2
3 ' '
3
a cr
q x q x L
L
x




  


 2 'b crx q 
3
L
x  
3 2
L L
x 
 21 10
x cr
x xdx C



 
 
 
22 20
3 ' '
3
xcr
a cr
q x q x L
L
x xdx dx C


 

 
  

   
 
 3
22 30 3
3 '
'
'
3
L xcr
b cr crL
q x q x
x x dx dx q
L
dx C
L


  



  

     
0 x  
3
L
x  
3 2
L L
x 
 0 0x      21 22a       22 223 3a b
L L 

Deflection    21 21 40
x
x x dx C  
     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            
0 x  
3
L
x  
3 2
L L
x 
Solutions
 Region ① :
 Region ②-a:
 Region ②-b:
  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


   
       
  

 
* 2
 cr L  

Analytical Deflection Solutions for Stage I
SS – Simply supported beam, C– Cantilever beam
*S is the distance from start point to loading point

*
 is deflection coefficient
Analytical Deflection Solutions for Stage II

Analytical Deflection Solutions for Stage II
Notations

2-D Member
Model Approach Type of Members
Round Panel Rectangle Panel Square Panel
Boundary conditions
Simply supported
Clamped
 Stage I: Elastic Solution
 Stage II : Yield Line Approach

Elastic Solution for Panels
Assumptions
 Material is elastic, homogeneous and isotropic
 Plate is initially flat
 Deflection of the midplane is small compared with thickness of the plate
 Straight lines normal to the mid-surface remain straight and remain normal to the
mid-surface after deformation
 gxz, gyz, ez sz neglected
4 4 4
4 2 2 4
2
w w w p
x x y y D
  
  
  Governing Differential Equation
3
2
12(1 )
Et
D 

w(x,y) – Deflection function, p – Applied load, t– thickness
D – Flexural rigidity, E – Young’s Modulus , v – Poisson’s ratio
Ventsel, Eduard, and Theodor Krauthammer. Thin plates and shells: theory: analysis, and applications. CRC press, 2001.

Elastic Solution:
Maximum Deflection Equations
Case 1.1 Case 1.2
Case 2.1
Case 2.2
2
3
16 1
PR v
D v
 
 
 
2
16
PR
D
4
0.00406
qa
D  
4
2
3
0.032
1
2
qa
v
Et
 
  
 
4
qa
D
a
 
 
2 4
4 3
0.032 1
1
v qa
Eta
b
  
 
 
Case 3.1 Case 3.2
Clamped support
Simply supported
Free support
Moment Rotation
Round Panel – Point load (P) at center
Rectangular/Square panel— Uniform load (q)
Case 2.2 Westergaard approximate solution
Boresi, Arthur Peter, Richard Joseph Schmidt, and Omar M. Sidebottom.Advanced mechanics of materials. Vol. 6. New York: Wiley, 1993.

Plastic Solution: Yield Line Moment–
Applied Load
ext intW W
( ) ( )N ml   
Assumptions:
 Failure takes place according to the assumed pattern
 Yield lines are straight and end at slab boundary
 Yield lines at vertexes are at 45° to the edges in case of square and rectangular
slabs
 Hogging moment about the yield lines and sagging moment about the supports
are equal
Work done in moving loads = Work done in rotating yield lines
N – Load, δ – Defl., m – moment abt. YL, l – length of YL, θ - Rotation
Kennedy G., Goodchild C., “Practical yield line design”

Yield Line Moment – Applied Load
Case study
Square panel with simply supported edges Uniform load: q
2
1
( ) 4
4 3
ext
a
W N q
 
     
 

ext intW W
( ) ( )N m l   
Equivalent point load
δmax is unit
Deflection of the centroid
int
1
( ) 4
0.5
W m l m a
a
 
   
 
 
Aswani, Karan. Design procedures for Strain Hardening Cement Composites (SHCC) and measurement of their shear properties by
mechanical and 2-D Digital Image Correlation (DIC) method. Diss. Arizona State University, 2014.
2
24
qa
m 

Yield Line Moment – Applied Load
Case 1.1 Case 1.2
Case 2.1
Case 2.2
Case 3.1 Case 3.2
Clamped support
Simply supported
Free support
Moment Rotation
Round panel – Point load (P) at center
Rectangular/Square panel— Uniform load(q)
2
P
m 
 4
P
m 

 2
3
12(2 2 )
qb b a
m
b a



 2
3
12(4 4 )
qb b a
m
b a



2
24
qa
m 
2
48
qa
m 

Curvature-Deflection Relationship for
Square panels
Square panel
2
( ) 0.5 0a x a z  2 2
( ) 0.5 0a x a z a   
Equation of plane # 1 Equation of plane # 3
Angle between plane # 1 and #3
   
2 2 4 2 2
2 22 2 4 2 2 4
0.25 4
cos2 cos( 2 *)
40.25 0.25
a a a
L
aa a a a
 
   
 
 
  
 
1 cos2 *
2 1 cos2 *
a L
L






Simplifying
1
2
4
3
L* is the hinge length

θ
2R
δ
Section A-A
dα
R
P
n n( cos( ), sin( ), )OA R Ra a  
uuur
( ,0, )OB R 
uuur
n n( cos( ), sin( ), )OC R Ra a 
uuur
n
2
n

a 
 2
1 n n nsin( ), (1 cos( )), sin( )n OA OB R R R a  a a    
uur uuur uuur
 2
2 n n nsin( ), (1 cos( )), sin( )n OB OC R R R a  a a    
uur uuur uuur
2 2 2
1 11 2 n n
2 2 2
n1 2
cos( ) 2cos( )1 1
cos cos
* * cos( ) 2
n n R R
L L R Rn n
a a 

a 
 
       
    
      
uur uur
g
uur uur
Round panel
Curvature-Deflection Relationship for
Round panels
Yao, Y., Wang, X., Aswani, K., & Mobasher, B. Analytical procedures for design of strain softening and hardening cement
composites. International Journal of Advances in Engineering Sciences and Applied Mathematics 2017.

Curvature-deflection relationship
φL* - Rotation, L* - Hinge length, L – Length, φ – Curvature, δ - Deflection
Rectangular panel
δ
a
Section X-X
δ
b
Section Y-Y
φL* φL*
φL* φL*
Round panel
δmax
2R
δ2θ
φL* φL*
2 2cos( *)
1 2cos( *)
L
R
L






14
14
1 cos2( *)
2 cos2( *)
La
L





1
2
3
4
b
K L
P
Y
MN
O
X X
Y
45
ab
b a
a

Experimental Verification
Input
Dimension: b, d, L
Material prosperities: E, εcr
Calculate
Mcr=bd2Eεcr /6
φcr=2εcr/d
Select position x
Assume maximum q’and η
Loop from q’=0 to q’max
If q’>1
If q’<1
Stage II
Calculate
M(i), P(i), m’, ξ(i)
Stage I
Calculate
M(i), P(i), δ (i)= δ1
If Region ①
0<x<ξ(i)
δ (i)= δ21
If Region ②
ξ(i)<x<L/2
δ (i)= δ22
Moment-curvature Load - deflection

Data Set 1 – Textile Reinforced Concrete
 Three-point bending test
 30 (b) × 9(d) × 220 (L) mm
 Textile fabrics:
 Polypropylene
 Aramid
ID εcr,µstr E, Gpa η m q EI, 107
EIcr, 107
100P 130 22 0.01 4.28 298 4.0 0.044
100A 130 22 0.1 17 157 4.0 0.41
25A75P 130 22 0.05 10.48 198 4.0 0.202
100P
Normalized curvature vs. Normalized moment Deflection at mid-span vs. Applied load
Simulated Parameters (Avg.)
Mobasher, Barzin, et al. "Correlation of constitutive response of hybrid textile reinforced concrete from tensile and flexural tests."
Cement and Concrete Composites 53 (2014): 148-161.

100A
25A75P
Data Set 1 – Textile Reinforced Concrete
Mobasher, Barzin, et al. "Correlation of constitutive response of hybrid textile reinforced concrete from tensile and flexural tests."
Cement and Concrete Composites 53 (2014): 148-161.

Data Set 2 – Square Slab
Square slab with point load at center
680 mm x 680 mm, thickness is 80 mm
Steel fiber vf = 1.0% and 1.5%
Khaloo, A.R., Afshari, M.: Flexural behaviour of small steel fibre reinforced concrete slabs. Cem. Concr. Compos. 27,
141–149 (2005).

Simplified Design Approach
 Step 1: Determine the full range moment-curvature relationship using the closed-form equations
 Step 2: Obtain simplified relations using polynomial curve fit
For clear span of 680 mm, thickness 80 mm, cracking stress as
1.5 MPa, moment–curvature relationship of Stage 2.1 can be shown as
Similar expression can also be calculated for Stage 3.1:
-4 2 -3
1:
1.882
2.1: 2.805 ( )
(10 ) (10 ) 2.45 ( )
3
7.
.1
9
:
66
Stage m
Stage m ascending
m descending
Stage


 

 
  
2
3949
0.592m

 
2
6
7 2 3
1.882 3.570
2.806 4489.6 ( )
/ (1.5 )
7.111( ) 8497.1
1.5 1 80
610
10 13 0.920 ( ) ( )
M ascending
M desce
N mm
N mm nding

 

 
   
 
 
   
  
     
   
5
2
1.422(10 )
946.72M N mm


 
  



Curve fit of the moment-
curvature relationship

Cont’d
 Step 3: The boundary conditions are simply
supported on all sides. The moment-load
relationship for a point load for square slab is as
follows:
 Step 4: The curvature-deflection relationship for
a square slab in cracked stages (2.1 and 3.1) are
derived as follows:
 Step 5: Calculate the load-deflection result using
the results from Steps 3 and 4.
08P m
 
2 3
2
0.00406
: ,
12(1
cos * 1 cos *
:
2 co
)
s *
L LL
Cracked
PL Et
stage
L
Elastic stage D
D




 
 



Data Set 3 – Round Panel
Round panel with point load at center
Diameter is 750 mm, thickness is 80 mm
Steel fiber vf = 1.0% and 1.25%
Montaignac, R. de, Massicotte, B., Charron, J.-P., Nour, A.: Design of SFRC structural elements: post-cracking tensile
strength measurement. Mater. Struct. 45, 609–622 (2012).

Conclusions
 Parametric material models were used to characterize the compressive and
tensile behavior of SHCC
 Linearized moment-curvature model is generated from stress-strain models
 Analytical deflection equations of 1-D and 2-D flexural members were
derived
 Equations of maximum deflection for various types of beams and panel are
derived
 Accuracy of the analytical deflection equations is identified by comparing
the simulated data with experimental data
 Simplified moment-curvature model is applied for analysis of panels

Analysis and Design Procedures for Strain Hardening Beams and Panels

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