Fardis EC8-3 member models

MODELLING THE BEHAVIOR OF CONCRETE MEMBERS:
DEVELOPMENTS SINCE THE COMPLETION OF EN 19981998-3:2005
Michael N. Fardis
University of Patras
Idealized skeleton curve - envelope to hysteresis loops
M
Effective elastic
Yield
stiffness:
secant-to-yielding moment
Ultimate moment, Mu=My
My
Mres < 0.8M
0 8Mu
member deformation:
chord-rotation
chord
rotation
δy
section deformation:
Yield deformation
curvature
δu
Ultimate deformation
δ
Practical
act ca e
expressions
p ess o s
for the yield & failure properties
developed for EN 19981998-3:2005
- their advancement
ad ancement since 2005
D. BISKINIS, G. ROUPAKIAS & M.N. FARDIS, Cyclic Deformation Capacity of Shear-Critical RC Elements, Proceeding of fib Symposium, Athens, May 2003.
M.N.FARDIS & D.BISKINIS, Deformation Capacity of RC Members, as Controlled by Flexure or Shear. Proceedings of International Symposium on
Performance-based Engineering for Earthquake Resistant Structures honoring Prof. Shunsuke Otani University of Tokyo, Sept. 2003, pp. 511-530.
D.BISKINIS, G.K. ROUPAKIAS & M.N.FARDIS, Resistance and Design of RC Members for Cyclic Shear, 14th Greek National Concrete Conference, Kos,
Oct 2003,
Oct.
2003 Vol.
Vol B,
B pp.
pp 363
363-374
374.
D.BISKINIS & M.N.FARDIS, Cyclic Strength and Deformation Capacity of RC Members, including Members Retrofitted for Earthquake Resistance, Proc. 5th
International Ph.D Symposium in Civil Engineering, Delft, June 2004, Balkema, Rotterdam, p. 1125-1133.
D.BISKINIS, G.K. ROUPAKIAS & M.N.FARDIS, Degradation of Shear Strength of RC Members with Inelastic Cyclic Displacements, ACI Structural Journal,
Vol. 101, No. 6, Nov.-Dec. 2004, pp.773-783.
M.N.FARDIS,
S D.BISKINIS,
S
S A. KOSMOPOULOS,
OS O O OS S.N.
S
BOUSIAS
O S S & A.-S.
S SPATHIS,
S
S Seismic
S
Retrofitting
f
Techniques for
f Concrete
C
Buildings, Proc. SPEAR
S
Workshop – An event to honour the memory of Jean Donea, Ispra, April 2005 (M.N.Fardis & P. Negro, eds.)
S.N. BOUSIAS, M.N.FARDIS & D.BISKINIS, Retrofitting of RC Columns with Deficient Lap-Splices, fib Symposium, Budapest, May 2005.
S.N. BOUSIAS, M.N.FARDIS, A.-S. SPATHIS & D.BISKINIS, Shotcrete or FRP Jacketing of Concrete Columns for Seismic Retrofitting, Proc. International
Workshop:
p Advances in Earthquake
q
Engineering
g
g for Urban Risk Reduction,, Istanbul,, May–June
y
2005.
D.BISKINIS & M.N.FARDIS, Assessment and Upgrading of Resistance and Deformation Capacity of RC Piers, Paper no.315, 1st European Conference on
Earthquake Engineering & Seismology (a joint event of the 13th ECEE & the 30th General Assembly of the ESC), Geneva, September 2006.
D.BISKINIS & M.N.FARDIS, Cyclic Resistance and Deformation Capacity of RC Members, with or without Retrofitting, 15th Greek National Concrete
Conference, Alexandroupolis, Oct. 2006, Vol. B, pp. 495-506.
D BISKINIS M.N.FARDIS,
D.BISKINIS,
M N FARDIS Effect of Lap Splices on Flexural Resistance and Cyclic Deformation Capacity of RC Members
Members, Beton & Stahlbetonbau,
Stahlbetonbau 102,
102 2007
th
D.BISKINIS & M.N.FARDIS, Cyclic Deformation Capacity of FRP-Wrapped RC Columns or Piers, with Continuous or Lap-Spliced Bars, 8 International
Symposium on Fiber Reinforced Polymer Reinforcement for Concrete Structures (FRPRCS-8), Patras, July 2007.
D.BISKINIS & M.N.FARDIS, Upgrading of Resistance and Cyclic Deformation Capacity of Deficient Concrete Columns, in “Seismic Risk Assessment and
Retrofitting with special emphasis on existing low rise structures” (A.Ilki, ed.), Proceedings, Workshop, Istanbul, Nov. 2007, Springer, Dordrecht.
D.BISKINIS & M.N.FARDIS, Cyclic Deformation Capacity, Resistance and Effective Stiffness of RC Members with or without Retrofitting, 14th World
Conference on Earthquake Engineering, Beijing, paper 05-03-0153, Oct. 2008.
D.BISKINIS & M.N.FARDIS, Chapter 15: “Upgrading of Resistance and Cyclic Deformation Capacity of Deficient Concrete Columns” , in “Seismic Risk
Assessment and Retrofitting with special emphasis on existing low rise structures” (Ilki A et al, eds.), Springer, Dordrecht, 2009
D BISKINIS & M.N.FARDIS,
D.BISKINIS
M N FARDIS Ultimate Deformation of FRP
FRP-Wrapped
Wrapped RC Members
Members, 16th Greek Nat.
Nat Concrete Conference
Conference, Paphos,
Paphos Oct.
Oct 2009,
2009 paper 171106
D.BISKINIS & M.N.FARDIS, Flexure-Controlled Ultimate Deformations of Members with Continuous or Lap-Spliced Bars, Structural Concrete, Vol. 11, No. 2,
June 2010, 93-108.
D.BISKINIS & M.N.FARDIS, Deformations at Flexural Yielding of Members with Continuous or Lap-Spliced Bars, Structural Concrete, Vol. 11, No. 3,
September 2010, 127-138.
D.BISKINIS & M.N.FARDIS, Effective stiffness and cyclic ultimate deformation of circular RC columns including effects of lap-splicing and FRP wrapping.
Paper 1128, 15th World Conference on Earthquake Engineering, Lisbon, Sept. 2012.
D.BISKINIS & M.N.FARDIS, Stiffness and Cyclic Deformation Capacity of Circular RC Columns with or without Lap-Splices and FRP Wrapping, Bulletin of
Earthquake Engineering, Vol. 11, 2013, DOI 10.1007/s10518-013-9442-7.
D.BISKINIS & M.N.FARDIS, Models for FRP-wrapped rectangular RC columns with continuous or lap-spliced bars, Engineering Structures, 2013 (submitted).
Experimental Database
1 Range/mean of parameters in tests for calibration of expressions
1.
for the member chord rotation and secant stiffness at yielding
1653 rectangular
Parameter
beams/columns
214 rectangular 229 members of nonnon
walls
rectangular section
307 circular
columns
min/max mean min/max mean
min/max
mean
min/max mean
section depth or diameter, h (m)
0.1 / 2.4
0.31
0.2 / 3.96
1.4
0.15 /1.83 0.43
shear-span-to-depth ratio, Ls/h
1 / 13.3
3.78 0.45 / 5.53 1.92 0.45 / 8.33
2.03
1.1 / 10
3.77
section aspect ratio, h/bw
0.2 / 4
1.3
16.7
1.0
1.0
40.4
16.7 / 90
36.7
0.06
-0.1 / 0.7 0.137
0.7
0 / 8.83
fc (MPa)
9.6 / 175
axial-load-ratio, N/Acfc
-0.05 / 0.9 0.126
transverse steel ratio, w (%)
total longitudinal steel ratio tot (%)
diagonal steel ratio, d (%)
0 / 3.54
0.4 / 3.0
4 / 30
1.19
10.9
37.7 13.5 / 109 35.8 13.5 / 101.8
0 / 0.86
0.10
0 / 0.50
0.62 0.05 / 2.18 0.54 0.04 / 2.44
0 2 / 8.55
0.2
8 55 1.97
1 97 0.07
0 07 / 4.27
4 27 1.5
15
0 / 1.68
2.5 / 57
0.027
0 / 0.25
0.005
0 205 / 6.2
0.205
62
1 23
1.23
-
-
0.86
0 53 / 55.88 2.34
0.53
2 34
-
-
transverse steel yyield stress fyyw, MPa 118 / 2050 468 220 / 1375 443 160 / 1375
504
200 / 1728 454.2
longitudinal steel yield stress fy, MPa 247 / 1200 440.2 276 / 1273 470
453
240 / 648 414.3
209 / 900
Experimental Database
2. Range/mean of parameters in tests for calibration of
2
expressions for ultimate curvature in monotonic or cyclic loading
Parameter
415 rectangular beams/columns
254 monotonic
160 cyclic
min/max mean min/max mean
section depth or diameter, h (m)
0.12 / 0.8
0.31 0.22 / 2.4
05/2
0.5
59 rectangular walls
13 monotonic
min/max
mean
0.41
2.39 / 2.41
2.4
46 cyclic
min/max mean
1.7 / 2.0
1.99
section aspect ratio,
ratio h/bw
0 225 / 3.73
0.225
3 73 1.54
1 54
1 21
1.21
21 2 / 23.4
21.2
23 4
22 4
22.4
13 3 / 28.3
13.3
28 3 13.7
13 7
fc (MPa)
19.7 / 99.4 34.8 17.7 / 102.2 38.6
34.5 / 40.8
35.3
26.2 / 45.6 40.6
0.236 0.063 / 0.077 0.07
0.05 / 0.11 0.07
axial-load-ratio, N/Acfc
0 / 0.78
0.08
transverse steel ratio, w (%)
0 / 2.38
0.345 0.04 / 2.96 0.656 0.41 / 0.82
total longitudinal steel ratio tot (%)
0 / 3.68
1.4 0.37 / 4.19 1.82
t
transverse
steel
t l yield
i ld stress
t
fyw, MPa
MP
0 / 596
longitudinal steel yield stress fy, MPa 277 / 596
0 / 0.77
0.66
0.11 / 0.25 0.24
0.88 / 1.76
1.35
0.07 / 0.77 0.63
419 255 / 1402 477.2
477 2
440 / 483
443 3
443.3
465 / 562
502
490
444 / 510
466.6
523 / 580
552
341 / 573 493.3
Experimental Database
3. Range/mean of parameters in tests for calibration of
3
expressions for the member ultimate cyclic chord rotation
Parameter
1159 rectangular
95 rectangular
53 members of nonnon
143 circular
beams/columns
walls
rectangular section
columns
min/max mean min/max mean
min/max
mean
min/max mean
section depth or diameter, h (m)
0.1 / 2.4
0.33 0.4 / 2.75 1.15
0.2 / 3.4
1.21
0.2 / 1.83 0.45
shear-span-to-depth ratio, Ls/h
1 / 13.3
3.7
2.85
1.77 / 10 4.22
section aspect ratio, h/bw
0.2 / 6
1.18 2.5 / 28.3 9.75
0.5 / 5.53 2.15 0.65 / 8.33
2.5 / 36
9.95
1.0
1.0
fc (MPa)
12.2 / 175 43.7 13.5 / 109 35.9 20.8 / 83.6
38.5
23.1 / 90 38.0
axial-load-ratio, N/Acfc
-0.1 / 0.9 0.165
0.07
-0.09 / 0.7 0.15
0.59
0.1 / 8.83 0.94
0.2 / 6.19
1.32
0.75 / 5.5 2.05
-
-
transverse steel ratio, w (%)
0 / 0.86
0.116
0 / 0.30
0.015 / 3.37 0.82 0.05 / 2.18 0.63 0.04 / 2.09
total longitudinal steel ratio tot (%)
0 / 6.29
2.08 0.07 / 4.27 1.37
diagonal steel ratio, d (%)
0 / 1.68
0.028
0 / 0.25
0.004
-
-
t
transverse
steel
t l yield
i ld stress
t
fyw, MPa
MP 118 / 1497 497 220 / 1375 435.3
435 3 178 / 1375
513
200 / 1569 482.5
482 5
longitudinal steel yield stress fy, MPa 281 / 1275 467.5 276 / 1273 471.2 331 / 596
438.7
240 / 648 428
Yield
e d & failure
a u ep
properties
ope t es o
of RC
C
sections
M-φ at yielding of section w/ rectangular
p
zone (width
(
b,, effective depth
p d)) –
compression
section analysis
• Yield moment (from moment-equilibrium & elastic σ-ε laws):
2


 y  Es 

V


y 
 0.51  1    

1  1 1  1 
1   y 1   y  1  2 
  y  Ec
3

2 
3 2 
6
bd




• 1, 2 : tension & compression reinforcement ratios, v: “web” reinforcement ratio,
~uniformly distributed between 1, 2 : (all normalized to bd); 1 = d1/d.
My
• Curvature
C
t
att yielding
i ldi off ttension
i steel:
t l
•
from axial force-equilibrium & elastic σ-ε laws ( = Es/Ec):
y
 
 2B 
A
2
B  1   2 1  0.5 v 1   1  
• Curvature at ~onset of nonlinearity of concrete:
A  1   2   v 
E s 1   y d
1/ 2
 A
N
,
A  1   2   v 
bdff y
2
y 
fy
N
bdff y
c
1.8 f c
y 

 y d E c y d
N
N
 1   2   v 
, B  1   21  0.5 v 1  1 
 c Es bd
1.8 bdfc
Moment at corner of bilinear envelope to experimental momentdeformation curve vs yield moment from section analysis
Left: 2085 beams/columns, CoV:16.3%; Right: 224 rect. walls, CoV:16.9%
10500
median: M y,exp= 0.99M y,pred
4000
9000
median:
di
M y,exp=1.025M
1 025M y,pred
7500
My,exp (kN
Nm)
My,exp (k
kNm)
3000
2000
1000
6000
4500
3000
1500
0
0
1000
2000
My,pred (kNm)
3000
4000
0
0
1500
3000
4500
6000
7500
9000 10500
My,pred (kNm)
Bias by +2.5%
+2 5% or -1%,
1% because corner of bilinear envelope of the
experimental moment-deformation curve ≠ 1st yielding in section. Same bias
considered to apply to predicted yield curvature.
Empirical formulas for yield curvature - section w/
rectangular compression zone
for beams or columns:
for walls:
y 
y 
1.54 f y
Es d
1.37 f y
Es d
y 
y 
1.75 f y
Es h
1.47 f y
Es h
Empirical expressions don’t have a bias w.r.to experimental
yield moment;; but scatter is larger:
y
g
In ~2100 test beams, columns or walls: CoV: ~18%
Flexural failures - columns
Flexural failures - beams
Conventional definition of ultimate deformation
250
250
200
200
150
150
100
100
50
0
-50
-100
-150
-200
-250
250-125 -100
-75
-50
-100
-150ultimate
conventional
a -200
deformation-250coincides
-125 -100 -75 -50 -25
0
25
-50 -25 w/
0
25
50 failure
75 100 250
125
real
Deflection (mm)
Deflection (mm)
150
Applied force (kN)
150
100
50
0
-50
a
50
75
50
0
-50
-100
100
conventional ultimate deformation
before “real” failure
-200
125
-150
-120
-150
b
-200
-75
-50
-25
0
25
Deflection (mm)
50
75
50
0
-50
-150
b
-200
-250
-125 -100
100 125
-90
-60
-30
0
30
Displacement (mm)
100
-100
-100
Q RC
Q-RC
100
0
200
Ap
pplied force (kN)
150
50
200
-250
-125 -100
200
Force (kN
N)
Applied forrce (kN)
Applied forrce (kN)
The value beyond which, any increase in deformation cannot
increase the resistance above 80% of the maximum previous
p
(ultimate) resistance.
-75
-50
-25
0
25
Deflection (mm)
50
75
100
125
60
90
Ultimate curvature of section with rectangular
compression zone,
zone from section analysis
•
C
Concrete
t σ-ε law:
l
– Parabolic up to fc, εco,
– constant stress (rectangular) for εco< ε < εcu
•
Steel σ-ε
σ ε law:
– Elastic-perfectly plastic, if steel strain rather low and
concrete
t fails
f il first;
fi t
– Elastic-linearly strain-hardening, if steel strains are
relatively high and steel breaks at stress and strain ft,
εsu.
Possibilities for ultimate curvature:
1.
Section fails by rupture of tension steel, εs1= εsu, before extreme compression
fibers reach their ultimate strain (spalling), εc < εcu →
 su
Ultimate curvature occurs in unspalled section,
section due to steel rupture: su 
1   su d
2. Compression fibres reach their ultimate strain (spalling): εc = εcu →
the confined concrete core becomes now the member section.
Two possibilities:
i. The moment capacity of the spalled section, ΜRo, never increases above
80% of the moment at spalling,
spalling ΜRc: ΜRo< 0.8Μ
0 8ΜRc →
 cu


Ultimate curvature occurs in unspalled section, due to the concrete: cu
 cu d
ii. Moment capacity
p
y of spalled
p
section increases above
80% of the moment at spalling: ΜRo> 0.8ΜRc →
The confined concrete core is now the member section and Cases 1 and
2(i) - applied
li d for
f the
th confined
fi d core - are the
th ttwo possibilities
ibiliti ffor attainment
tt i
t
of the ultimate curvature → φsu, φcu calculated as above but for the
confined core; the minimum of the two is the ultimate curvature.
M Rc
 (1   )(   )


2
1
1
2
 bd f c 
 
2
1  1



1  f y
(   1 )(1   )  
3   s  cu




2

1  

 
 co
3 cu

 2
1
 co  
   
  
4 cu  
2

Ultimate curvature of section w/ rectangular compression zone
 su
for steel rupture:
 su 

1   su d
Steel ruptures before concrete crushes, after compression steel yields, if ν:
 1 su   y  1   1 
 su   y
•
 co
 co


cu
3     f t   1  f t   su   y   
3     f t   1  f t   su 1   1    cu 1   1 
2
1
v
2
1
v




fy
f
2






f
y
y 
su
y
cu
su

 f y  1   1  su   cu 
ξsu calculated from axial force equilibrium for:

 su 
1  1   1




   1   1 1  f t v
  2 
f y 


 
f 
  1  t v

f y 
 
ft

 2  co
3 su
fy

1  1 1   co

3 su
ν=N/bdfc, ω1, ω 2 : tension & compression mech. reinforcement ratios, ωv: “web” mech.
y ); 1=d1/d.
reinforcement ratio ~uniform distribution between ω1, ω 2 ((ω= fy/fc);


•
Steel ruptures before concrete crushes or compression steel yields, if ν:
 1 su   y  1   1 
 co
3     f t   1  f t   su   y
v
2
1
 su   y
fy
f y  2  su   y

 
v  f t
 su  2
ft
co



1


1

 
ξsu from:


3

2
(
1


)
f
E



su
1 
y
s su
y 




ft
 su
  f t
ft
 su    co
ft
 su
  f t
ft
2 co
2  su 


























1
1
1

 
0
1
2
1
1
2 1
1







f


f
E



f


f
E


3
(
1
)
3
2
(
1
)


1 
1 
su
y
y
y
s su
y 
su
y
y
y
s su
y 
Ultimate curvature of section w/ rectangular compression zone
 cu
for concrete crushing:
 cu 
 cu d

Concrete crushes after tension steel yields, w/o compression steel yielding, if ν :
  2  1 
•
 cu 

v   cu   y
3
1
 1   1

1   1   cu   y
 cu   y

ξcu from axial force equilibrium, for:

 cu   y 2  2 
 co
v



    1   2 cu 
1 
 y 1 1
 3 cu 2(1   1 )  cu  y 


 co

 
1  cu 1

y



 v  1   cu  1
   2 
0


21   1    y


Concrete crushes w/ tension & compression
steel yielding
yielding, if ν :

 cu 
co
 cu 
 co


v   cu   y
v   cu   y
3
3

 1  1
   2  1 
 1  
2  1 
1

  cu   y
1   1   cu   y
1   1   cu   y
 cu   y


 cu 
1  1   1  2   1  1 v
•
ξcu from:

cu

Concrete crushes after compression
steel
yields
yields, w/o tension steel yielding
yielding, if ν :
•

1  1 1   co
3
ξcu from:
2  1 
 cu 

  2v

 co

v   cu   y
3 
 1  


1   1   cu   y
  cu   y


 cu   y 2  2 
 co
v



1 
   2  1 cu   
3 cu 2(1   1 )  cu  y 
1  1
y


  cu


v   cu

  1   1 
  0

 y


2
1


1 
y



Unconfined full section – Steel rupture
yes
no
δ1 satisfies
Eq.(3.38)?
yes
ν<νs,y2 - LHS
Eq.(3.39)?
su from
yes
ν<νs,c - RHS
Eq.(3.39)?
Eq.(3.41)
no
φsu from
Eq.(3.36)
q(
)
no
yes
ν<νs,c - RHS
Eq.(3.39)?
Confined core after spalling of concrete cover.
Parameters are denoted by an asterisk and computed with:
 b, d, d1 replaced by geometric parameters of the core: bc, dc, dc1;
 N,
N 1, 2, v normalized
li d to
t bcdc, instead
i t d off bd;
bd
 σ-ε parameters of confined concrete, fcc, cc, used in lieu of fc, cu
su from
Eq.(3.40)
ν*<ν*s,y2 - LHS
Eq.(3.39)?
no
Unconfined full section – Spalling of concrete cover
*su from
Eq.(3.41)
no
δ1 satisfies
Eq.(3.42a)?
yes
cu from
yes
ν*<ν*s,c - RHS
Eq.(3.39)?
νν<  c,y1
c y1 - LHS
Eq.(3.48)?
Eq.(3.47)
no
no
cu from
*su from
Eq.(3.40)
Failure of compression zone (concrete)
no
ν<  c,y2 - RHS
E (3 48)?
Eq.(3.48)?
Eq (3 46)
Eq.(3.46)
yes
yes
cu from
ν*<ν*c,y2 - LHS
Eq.(3.44)?
cu from
Eq.(3.45)
Eq.(3.49)
Compute moment resistances:
 M Rc (of full, unspalled section) and
 M Ro (of confined core, after spalling of cover).
M Ro
0 8 M Rc
R < 0.8
R ?
yes
φsu from
Eq.(3.36)
no
no
ν<νc,y1 - RHS
Eq (3 44)?
Eq.(3.44)?
yes
no
yes
ν<νc,y2
c y2 - LHS
Eq.(3.44)?
Rupture of tension steel
yes
no
Ultimate curvature of confined core after spalling of concrete cover
yes
*cu from
Eq.(3.47)
no
φcu from
Eq.(3.37)
ν*<ν*c,y1 - RHS
Eq (3 44)?
Eq.(3.44)?
yes
*cu from
Eq.(3.45)
no
*cu from
Eq (3 46)
Eq.(3.46)
φcu from
Eq.(3.37)
Test results vs ultimate curvature w/ failure strains for cyclic flexure
per EN 1998-3:2005 :
Steel: εsu: 2.5%, 5%, 6% for steel class A, B, C per Eurocode 2
εcu,c  0.004  0.5αρs f yw / f cc 
08
0,8
0,7
median: φu,exp =0.96φu,pred
0,6
cyclic test results
φu,eexp (1/m)
0,5
,
Cyclic
all
Concrete
crushing
Steel
rupture
Median
0 96
0.96
0 90
0.90
0 98
0.98
C.o.V.
46.7%
55.1%
38.0%
205
97
108
0,4
0,3
0,2
No
No.
cyclic, slip
0,1
cyclic, no-slip
0
0
0,1
0,2
0,3
0,4
φu,pred (1/m)
0,5
0,6
0,7
0,8
Test results vs ultimate curvature w/ failure strains for cyclic flexure
per Biskinis & Fardis 2010 (adopted in fib MC2010):
• Monotonic flexure:
•max. available steel strain: (7/12)εsu
• Cyclic flexure:
•max. available steel strain: (3/8)εsu
1.2
2
 10 
w
*
  0.285
 cu  0.0035  
1 K
 hc (mm) 2
 10 
w
*
  0.2
 cu  0.0035  
1 K
 hc (mm) 
monotonic & cyclic
data, no. 474,
median=1 00
median=1.00,
CoV=49.7%
median: φu,exp =φu,pred
1
φu,exp (1
1/m)
0.8
06
0.6
cyclic test data:
0.4
cyclic, slip
cyclic, no-slip
monotonic, slip
monotonic,
t i no-slip
li
0.2
0
0
0.2
0.4
0.6
φu,pred (1/m)
0.8
1
1.2
Cyclic
all
Concrete
crushing
Steel
rapture
Median
0.99
0.99
1.01
C.o.V.
44.2%
52.6%
34.2%
205
97
108
No.
Yield
e d & failure
a u ep
properties
ope t es o
of RC
C
members
Flexural behavior at member level ((Moment
Moment-chord rotation)
Definition of chord rotations, θ, at member ends
Elastic moments at ends A, B from chord rotations at A, B:
• ΜΑ = (2ΕΙ/L)(2θΑ+θΒ),
• ΜΒ = (2ΕΙ/L)(2θΒ+θΑ)
Fixed-end rotation of member end due to bar slippage
from their anchorage zone beyond member end
• Sli
Slippage off ttension
i b
bars ffrom region
i b
beyond
d end
d section
ti ((e.g. ffrom jjoin
i
or footing) →
rigid body rotation of entire shear span = fixed-end
rigid-body
fixed end rotation
rotation, θslip
(included in measured chord-rotations of test specimen w.r. to base or joint; doesn’t
affect measured relative rotations between any two member sections).
• If s = slippage of tension bars from anchorage → θslip= s/(1-ξ)d
g length
g lb of bar beyond
y
section of maximum
• If bond stress uniform over straight
moment → bar stress decreases along lb from σs (=fyL at yielding) at section o
maximum M to zero at end of lb → s=σslb/(2Es)
• lb= bond
b d fforce d
demand
d per unit
it llength
th ((=A
Asσs/(d
/( dbL)=d
) dbLσs/4),
/4) divided
di id d b
by ~bon
b
strength (assume =√fc)
σs/Es)/(1
)/(1-)d = φ
• εs((=σ
• At yielding of member end section
 y ,slip 
 y d bL f y
8 fc
(fyL, fc in MPa )
Fixed-end rotation of member end due to rebar pull-out from
anchorage zone beyond member end, at member yielding
φy,measured/(φy,predicted+y,slip,/lgauge) no.160 measurements w/ slip:
median = 1.0, C.o.V = 34%
2.5
2.25
2
φy,exxp / (φy,1st-priinciples+φ y,duee to slip)
Ratio:
experimental-topredicted yield
curvature
t
(w/
( /
correction for
fixed-endfixed
end
rotation) in terms
of gauge length
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
lgauge / h
1
1.25
1.5
Chord rotation of shear span at yielding of end section
per EN 1998-3:2005
• Rect. beams or columns:

Ls  aV z
h

y y
 0.0014 1  1.5
3
Ls

• Rect.
R t or non-rect.
t walls:
ll
Ls  aV z
y y
 0.0013  asl y , slip
3
“Shift
Shift rule”:
rule :

  asl y , slip

Diagonal cracking shifts value of force in tension reinforcement to a section
at a distance from member end equal to z (internal lever arm)
• z = d-d
d d1 in beams
beams, columns
columns, or walls of barbelled or T
T-section,
section
• z = 0.8h in rectangular walls.
– av = 0, if VRc > My/Ls;
– av = 1,
1 if VRc ≤ My/Ls .
VRc = force at diagonal cracking per Eurocode 2
(in kN, dimensions in m, fc in MPa):


0.2 1 / 6 
0.2  1 / 3
N

1/ 3
 f c  0.15 
VR ,c  max 180 100 1  , 35 1 
f c 1 
bwd

d
d 
Ac 





– asl = 0, if no slip from anchorage zone beyond end section;
– asl = 1, if there is slip from anchorage zone beyond end section.
Test-model comparison – θy
Beams/rect. columns, no. tests: 1653
3.5
3
median=1.01,
CoV=32.1%
θy,exp (%
%)
2.5
median:
θy,exp=1.01θy,pred
2
1.5
1
0.5
rect. beams / columns
0
0
0.5
1
1.5
2
θy,pred (%)
2.5
3
3.5
1,2
1,2
1
1
0,8
0,8
θy,exp (%)
θy,exp (%)
Test-model comparison – θy
Walls,
a s, no.
o tests:
tests 386
median:
θy,exp=0.97θy,pred
0,6
04
0,4
0,2
median:
θy,exp =1.01θy,pred
0,6
04
0,4
0,2
Rectangular walls
Rectangular walls
Non-rectangular Walls
0
Non-rectangular Walls
0
0
0,2
0,4
0,6
θy,pred (%)
0,8
1
1,2
0
0,2
0,4
0,6
0,8
1
1,2
θy,pred (%)
Expression in EN 1998-3:2005 Modified expression adopted in MC2010
median=0.97, CoV=31.1%

Ls  aV z
5h 
y  y
 0.00045
0 00045 1 
  asl y , slip
3
 3Ls 
median=1.01, CoV=30.9%
Test-model comparisons - θy
Circular columns - not in EN 1998-3:2005: no. tests: 291
Ls  aV z

 2 Ls  
y  y
 0.0027 1  min 1;
   asl y , slipp
3
 15D  

2,5
median=1.00,, CoV=31.7%
median:
θy,exp=θy,pred
θy,exp (%)
2
1,5
1
0,5
circular
0
0
0,5
1
1,5
θy,pred (%)
2
2,5
Effective elastic stiffness, EIeff
(for linear or nonlinear analysis)
 Part 1of EC8 (for design of new buildings):
 EIeff : secant stiffness at yielding =50% of
uncracked gross-section stiffness.
- OK in force-based design of new buildings (safesided for forces);
- Unsafe in displacement-based assessment for
displacement demands).
 More realistic:
EI eff 
M y Ls
3 y
secant stiffness at yielding of
end of shear span Ls=M/V
Test-model comparison – EIeff
Beams/rect. columns, no. tests: 1616
600
500
EI eff 
M y Ls
3 y
(MyLs/3
3θy)exp (M
MNm2)
median=1.00,
CoV=32.1%
median:
(MyLs/3θy)exp =MyLs/3θy)pred
400
300
200
100
rect. beams / columns
0
0
100
200
300
400
(MyLs/3θy)pred (MNm2)
500
600
Test-model comparison – EIeff Walls, no. tests: 386
detail
M y Ls
EI eff 
3 y
1500
median:
(My Ls/3θy )exp =1.035(My Ls/3θy )pred
median:
(My Ls/3θy )exp =1.035(My Ls/3θy )pred
1250
3000
2000
1000
non-rectangular walls
(MyLs/3θy)exp (M
MNm2)
With expression
i ffor θy
in EN 1998-3:2005
median=1 035
median=1.035
CoV=42.8%
(MyLs/3θy)exp (M
MNm2)
4000
1000
750
500
250
non-rectangular walls
rectangular walls
rectangular walls
0
0
0
1000
2000
3000
0
4000
2)
( yLs/3θ
(M
/ θy)pred (MNm
(
250
1500
median:
(My Ls/3θy )exp =0
0.98(M
98(My Ls/3θy )predd
750
1000
1250
1500
detail
median:
(My Ls/3θy )exp =0
0.98(M
98(My Ls/3θy )predd
1250
3000
2000
1000
non-rectangular walls
(M
MyLs/3θy)expp (MNm2)
With new expression
for θy of walls
(adopted in MC2010)
median=0.98
CoV=41.1%
(M
MyLs/3θy)expp (MNm2)
4000
500
2)
( yLs/3θ
(M
/ θy)pred (MNm
(
1000
750
500
250
non-rectangular walls
rectangular walls
0
rectangular walls
0
0
1000
2000
(MyLs/3θy)pred
3000
(MNm2)
4000
0
250
500
750
(MyLs/3θy)pred
1000
1250
(MNm2)
1500
Test-model comparison – EIeff
Circular columns - not in EN 1998-3:2005,, no. tests: 273
EI eff 
M y Ls
3 y
median=0.99,
ed a 0 99, Co
CoV=31.2%
3 %
4000
175
median: (My Ls/3θy )exp =0.99(M
0.99(My Ls/3θy )pred
3500
150
(M
MyLs/3θy)eexp (MNm
m2 )
(M
MyLs/3θy)eexp (MNm
m2 )
3000
2500
2000
1500
1000
500
median: (My Ls/3θy )exp =0.99(M
0.99(My Ls/3θy )pred
125
100
75
50
25
detail
circular
0
circular
0
0
500
1000 1500 2000 2500 3000 3500 4000
(MyLs/3θy)pred (MNm2)
0
25
50
75
100
125
150
(MyLs/3θy)pred (MNm2)
175
Empirical secant stiffness to yielding, EIeff, independent of
amount of reinforcement - not in EN 1998-3:2005
EI eff



 
Ls
N


 a 0.8  ln  max ;0.6   1  0.048 min[ 50;
( MPa)] 
Ec I c
Ac
 h
  



(all variables known before dimensioning the longitudinal reinforcement)
– If there is slippage of longitudinal bars from their anchorage zone
beyond the member end:
• a = 0.081 for columns;;
• a = 0.10 for beams or non-rectangular walls (barbelled, T-, H-section);
• a = 0.115 for rectangular walls;
• a = 0.12 for members with circular section.
– If there is no slippage of longitudinal bars: effective stiffness x 4/3
Test-model comparison – Empirical EIeff, independent
of amount of reinforcement - not in EN 1998
1998-3:2005
3:2005
600
Beams/columns
no. tests: 1616
median=1.00
CoV=36 1%
CoV=36.1%
500
(M
MyLs/3θ
θy)exp (M
MNm2)
median:
((M yLs/3θy)exp =EIpred
400
300
200
100
0
0
100
200
300
400
EIpred (MNm2)
500
600
4000
median:
(My Ls/3θy )exp =EIpred
3000
2000
1000
non-rectangular
non
rectangular walls
1000
500
non-rectangular
non
rectangular walls
rectangular walls
rectangular walls
0
0
0
1000
EIpred
2000
3000
4000
0
EIpred
175
median: (My Ls/3θy ) exp =EIpred
3500
2500
2000
1500
1000
500
detail
150
((MyLs/3θy)exxp (MNm2)
3000
((MyLs/3θy)exxp (MNm2)
500
(MNm2)
4000
Circular columns
no tests: 273
no.
median=0.995
CoV=31.4%
detail
detail
median:
(MyLs/3θy ) exp =EIpred
(M
MyLs/3θy)exxp (MNm2)
Walls
no tests: 386
no.
median=1.00
CoV=44.6%
1500
(M
MyLs/3θy)exxp (MNm2)
Test-model
comparison –
Empirical EIeffff
independent of
reinforcement , not
in EN1998-3: 2005
1000
1500
(MNm2)
median: (My Ls/3θy ) exp =EIpred
125
100
75
50
25
circular
circular
0
0
0
500
1000 1500 2000 2500 3000 3500 4000
EIpred (MNm2)
0
25
50
75
100
EIpred (MNm2)
125
150
175
Flexure-controlled ultimate chord rotation from curvature
& plastic hinge length per EN 1998-3:2005
 0.5L pl 
pl

θu  θ y  θu  θ y  (φu  φ y ) L pl 1 

L
s    cu,c



su

• y: yield curvature (section analysis);
,
u  min 

 cu,c d c (1   su )d 
Option 1: Confinement per Eurocode 2
O ti
Option
2:
2 New,
N
per EN 1998
1998-3:2005
3 2005
αρsx f yw  0.05 f c : f cc  f c  5αρsx f yw
0.86
αρsx f yyw  0.05 f c : f cc  1.125 f c  2.5αρsx f yyw f cc  f c 1  3.7 sx f yyw / f c

 cu ,c   co  0.2sx f yw / f c 
Lpl  0.1Ls  0.17 h  0.24 d b f y

 cu ,c  0.004  0.5sx f yw
fc
 
/f 
cc
Lpl  Ls / 30  0.2h  0.11d b f y / f c
2
: confinement effectiveness:

sh 
sh   bi 
index c: confined;
1 
1 
α  1 
– rectangular section:

ρs: stirrup ratio;
  2bc s  2 2hc  6bc hc 
h
–
circular section & hoops:

α

1

Ls=M/V: shear span at member end;
 2 D 
c 

sh: centerline spacing of stirrups,
h: section depth;
Dc, bc, hc: confined core dimensions to centerline of hoop;
db: bar diameter;
bi: centerline spacing along section perimeter of longitudinal
fy, fc: MPa
bars (index: i) engaged by a stirrup corner or cross-tie.
Test-model comparison – ultimate chord rotation from curvatures
& plastic hinge length per EN 1998-3:2005, no. tests: 1100
Option 1: confinement per EC2
median=0.88, CoV=52.3% Option 2: new confinement per EC8-3
median=0.91,
di
0 91 CoV=52.2%
C V 52 2%
17,5
Cyclic loading
15
median: θu,exp =0.88θ
=0 88θu,pred
12 5
12,5
12,5
median: θu,exp =0.91θ
=0 91θu,pred
10
θu,eexp (%)
θu,eexp (%)
Cyclic
y
loading
g
10
7,5
7,5
5
5
2,5
beams & columns
rect. walls
non-rect. sections
0
0
2,5
5
7,5
10
θu,pred (%)
12,5
15
17,5
2,5
beams & columns
rect. walls
non-rect. sections
0
0
2,5
5
7,5
θu,pred (%)
10
12,5
Flexure-controlled ultimate chord rotation of rect. beams, columns,
walls, non-rect. walls & circular columns from curvatures & plastic
hi
hinge
length
l
th by
b Biskinis
Bi ki i & Fardis
F di 2010,
2010 2013 - adopted
d t d in
i fibMC2010
Flexure-controlled ultimate chord rotation, accounting separately for
slippage in yield-penetration
yield penetration length, from yielding till ultimate deformation:
 u   y  a sl  u ,slip
 L pl
 ( u   y ) L pl 1 
 2 Ls




Confinement per fib MC2010:
34

 w f yw  
 
f cc  f c 1  3.5
fc  



– Monotonic loading
g - Rect.
beams,, columns,, walls,, non-rectangular
g
walls:
2
αρw f yw
 10 
7


εcu,c  0.0035  
 0.57
,  su , mon 
 su ,no min al

f cc
12
 ho ( mm ) 

 Ls  
L pl,m on  h1.1  0.04 min
i  9;

h 


2
αρ f

3
  0.4 w yw ,  su ,cy   su ,no min al
– Cyclic loading:
 10
εcu,c  0.0035  
f cc
8
 ho ( mm ) 
• Rect. beams/columns/walls, non-rect. walls: L
• Circular columns:
L pl ,cy,cir

1
 Ls  
pl,cy  0.2h
1  3 min  9; h 




1
 Ls  
 0.6 D1  min  9;  
6
 D 

Post-yield fixed-end rotation of member end due to bar slippage
from yield penetration length beyond member end, from yielding
till ultimate
lti t flexural
fl
l deformation
d f
ti per Biskinis
Bi ki i & F
Fardis
di 2010,
2010 2013 adopted in fibMC2010 (not in EN 1998-3:2005)
• Monotonic loading: u , sli p  9 .5d bL u
or
 u , slip
li  5.5d bL  u
or
• Cyclic loading:
Complete
C
l t pull-out
ll t off
beam bars, due to
short anchorage in
corner joint
←
→


 10d bL  y   u  / 2
 16d bL  y   u / 2
Test-model comparison – Cyclic ultimate chord rotation from
curvatures & plastic hinge length per Biskinis & Fardis 2010, 2012
Rect. beams/columns/walls,
non-rect. walls - no. tests: 1100
median=1.00, CoV=43.2%
Circular columns – no. tests:
143, median=1.00, CoV=30.3%
16
C li loading
Cyclic
l di
12,5
median: θu,exp =θu,pred
14
12
θu,exp (%)
10
θu,expp (%)
median:
θu,exp=θu,pred
7,5
5
10
8
6
4
2,5
beams & columns
rect. walls
non-rect.
o ect sections
sect o s
0
2
circular
0
0
2,5
5
7,5
θu,pred (%)
10
12,5
0
2
4
6
8
10
θu,pred (%)
12
14
16
Empirical cyclic flexure-controlled ultimate chord rotation of rect.
beams/columns/walls, non-rect. walls in EN1998-3:2005 & fibMC2010
θum
or:
 max (0.01; ω' ) 
fc 
 0.016  (0.3 ) 
 max (0.01; ω) 
ν
0.225
 Ls 
 
h
0.35
0, 3

f yw
 αρ
sx

fc

25




(1.25100 ρd )
0.35

f 
 αρ yw 
sx

f c 

 max(0.01; ω' ) 
0.2  Ls 
pl
θum
f
 θum  θy  0.0145 (0.25ν )
 c   25
h
 max(0.01; ω) 
(1.275100 ρd )
, ':
:
mechanical ratio of tension (including web) & compression steel;
p
zone; N>0 for compression);
p
)
N/bhfc ((b: width of compression
Ls/h :
M/Vh: shear span ratio;
α:
sx:
d:

sh 
s
1  h
confinement effectiveness factor :
α  1 
 2bc  2hc
Ash/bwsh: transverse steel ratio // direction of loading;
ratio of diagonal reinforcement.
• Walls:
W ll
• Cold-worked brittle steel:
  bi2 

1 
 6b h 
c c 

1st expression
i di
divided
id d b
by 1
1.6;
6 2nd multiplied
lti li d b
by 0
0.6
6
1st expression divided by 1.6; 2nd by 2.0
Non-seismically detailed members with continuous bars
• Plastic part, plum=θum-y, of ultimate chord rotation: divided by 1.2.
Test-model comparison – Empirical cyclic ultimate chord rotation of
members with seismic detailing per EN 1998-3:2005 – no. tests 1100
Model for total um
Model with um=y+pl
median=1.00, CoV=37.8%
median=1.00, CoV=37.6%
Cyclic loading
Cyclic loading
median: θu,exp
, p =θu,pred
,p
12,5
,
12,5
10
θu,,exp (%)
10
θu,exp (%)
median:
median:θθu,exp
=θu,pred
, p =θ
,p
u,exp
u,pred
7,5
5
5% f ractile:
θ u,exp=0.5θ u,pred
2,5
beams & columns
0
7,5
5
5% f ractile:
θ u,exp=0.52θ u,pred
2,5
beams & columns
rect. walls
rect. walls
non-rect. sections
non-rect. sections
0
0
2,5
5
7,5
θu,pred (%)
10
12,5
0
2,5
5
7,5
θu,pred (%)
10
12,5
Test-model comparison – Empirical cyclic ultimate chord rotation of
members without seismic detailing per EN 1998-3:2005 – no. tests 48
Model for total um
Model with um=y+pl
median=1.00, CoV=30.6%
median=0.99, CoV=31.8%
9
9
median: θu,exp =θu,predd
8
7
7
6
6
θu,exxp (%)
θu,exxp (%)
8
5
4
5
4
3
3
2
2
1
1
0
0
0
1
2
3
4
5
θu,pred (%)
6
7
8
9
median: θu,exp =0.99θu,pred
0
1
2
3
4
5
θu,pred (%)
6
7
8
9
General empirical ultimate chord rotation of rect. beams/columns/
walls & non-rect. walls per Biskinis & Fardis 2010, adopted in fib
MC2010 (not in EN1998-3:2005)
EN1998 3:2005)
1
3
 w f yw 

f c 
1.225100 d




 h   
 max(0.01;2 )
L

0
.
2

pl
hbw
s

min 9;   fc 25
u  st (1  0.525acy )1  0.6asl  1  0.052max1.5; min10;   0.2 



 h 
 max(0.01;1)
 bw   


cyclic loading median: θ
median: θu,exp
=θu,pred
u,exp=θ
u,pred
12,5
10
θu,exp (%)
αsthbw:
0.022 for hot-rolled
hot rolled or Tempcore bars;
0.0095 for brittle cold-worked bars;
αcy:
= 1 ffor cyclic
li lloading,
di
= 0 for monotonic;
αsl:
= 1 if slippage of long. bars from
anchorage zone is possible,
= 0 otherwise;
bw: width of (one) web
7,5
5
5% f ractile:
θ u,exp=0.51θ u,pred
25
2,5
b
beams
& columns
l
rect. walls
non-rect. sections
0
0
25
2,5
5
75
7,5
10
12 5
12,5
Non-seismically detailed members:
θu,pred (%)
pl
•  um=θum-y divided by 1.2.
no. tests: 1100, median=1.00, CoV=37.6%
Effect
ect o
of laplap
ap-sp
splicing
c g the
t e column
co u
bars in the plastic hinge region
Members with ribbed bars lap-spliced over length lo inside
the plastic hinge region per EN 1998-3:2005 or other options
• EN 1998-3:2005:
1. Both bars in pair of lapped compression bars count as compression steel.
2. For the yield properties (My, y , θy), the stress fs of tension bars is:
fs = fy(lo/loy,min), if lo< loy,min=(0.3fy/√fc)db (fy, fc in MPa)
3. Ultimate chord rotation
θu=y+plu(lo/lou,min), if lo < lou,min = dbfy/[(1.05+14.5αrsωsx)√fc],
– fy, fc in MPa, ωsx=ρsxfyw/fc: mech. transverse steel ratio // loading,
– αrs=(1-sh/2bo)(1-sh/2bo)nrestr/ntot
(nrestr/ntot: restrained-to-total lap-spliced bars).
• Or,
O Eligehausen
Eli h
& Lettow
L tt
2007 ffor fib MC2010:
0.55
0.25
 lb   f c  

20




f s  51.2    
 d b   20   max(d b ; 20 mm ) 
• cd = min[a/2;
[ ; c1; c]] ≥ db , cd ≤ 3db
• cmax = max[α/2; c1; c] ≤ 5db
• fy, fc in MPa
0.2
 c 1/ 3  c  0.1

1  k s nl Ash 
max
d




  
 kKtrt   f y , kKtr 



 d b   cd 

nb d b  sh 
Test-model comparison: Yield moment & chord rotation, effective stiffness,
ultimate cyclic chord rotation - lapped ribbed bars per EN 1998-3:2005
2
2000
median: θy,exp =1.05θy,pred
median: My,exp =My,pred
1750
median=1.00
CoV=11.8%
1,5
1250
no.tests: 92
1000
My
750
500
median=1 05
median=1.05
θy,exp (%)
no.tests: 114
My,eexp (KNm)
1500
θy
0,5
CoV=20%
250
1
0
0
0
250
500
750
1000 1250 1500 1750 2000
0
0,5
1
My,pred (KNm)
1,5
2
θy,pred (%)
250
12
median: θu,exp =1.035θu,pred
median:
(M yLs/3θy)exp=0.955(M yLs/3θy)pred
no tests: 92
no.tests:
median=0.955
CoV=24.8%
10
8
150
θu,exp (%)
(M
MyLs/3θy)exp (MNm2)
200
100
4
EIeff
50
0
0
50
100
150
200
(MyLs/3θy)pred (MNm )
2
6
250
no tests: 81
no.tests:
2
median=1.035
0
CoV=39.3%
θu
0
2
4
6
8
θu,pred (%)
10
12
Test-model comparison: Yield moment & chord
rotation, effective stiffness, - lapped bars,
steel stress per Eligehausen & Letow 2007
2000
median:
di
My,exp =0.98M
0 98My,pred
1750
no.tests: 114
Median=0.98
CoV=12%
no tests: 92
no.tests:
My,exp (KNm
m)
1500
1250
1000
750
500
median=1.03
250
CoV=20.5%
CoV
20.5%
0
0
250
500
750
1000 1250 1500 1750 2000
My,pred (KNm)
2
250
median: θy,exp =1.03θy,pred
median:
(M yLs/3θy)exp=0.97(M
=0 97(M yLs/3θy)pred
200
1
0,5
θy
no tests: 92
no.tests:
median=0.97
0
0
0,5
1
θy,pred (%)
1,5
2
CoV=24.6%
(M
MyLs/3θy)exp (MNm2)
1,5
θy,exp (%)
My
150
100
EIeff
50
0
0
50
100
(M L /3θ )
150
d
200
(MNm2)
250
Members with smooth hooked bars, with or without lap
splice (of length lo) in the plastic hinge per EN 1998-3:2005
Provided that lapping lo ≥ 15db:
1 Yield properties My, y, θy:
1.
– As in members with continuous ribbed bars.
2 Ultimate cyclic chord rotation:
2.
– For continuous bars:
• 0.8θum (: ~0.95 for smooth bars /1.2 for no seismic detailing)
or
• θum=y+0.75plum (: ~0.9 for smooth bars /1.2 for no seismic detailing)
– For lapping lo ≥ 15db :
• um x 0.019(10+min(40, lo/db)) or
•
θum=y+plum
x 0.019 min(40, lo/db)
– Ls for
o plum not
ot reduced
educed by lo
T t
Test-model-ratio
d l ti for
f θum
no laps
laps
M di
Median
1 015
1.015
1 03
1.03
C.o.V.
33.3%
33.4%
34
11
no tests
no.
(ultimate condition not necessarily controlled by region right above the lap)
Cyclic
Cyc
cs
shear
ea resistance
es sta ce o
of RC
C
members
Cyclic shear resistance per EN 1998-3:2005
• Shear resistance in p
pl. hinge
g after flex. yyielding,
g, as controlled byy stirrups
p
(linear decay of Vc & Vw with cyclic plastic rotation ductility ratio θpl=(-y)/y>0


hx
 L
pl
VR 
min N ; 0.55 Ac f c   1  0.05 min 5; μθ 0.16 max( 0.5;100 ρtot )1  0.16 min  5; s
2 Ls
 h





  f c Ac  Vw 


Vw=ρwbwzfyw, due to stirrups (bw: web width, z: internal lever arm; ρw: shear reinf. ratio)
ρtot: total
t t l longitudinal
l
it di l reinforcement
i f
t ratio
ti
h: section depth
x : depth
p of compression
p
zone at yyielding
g
Ac= bwd
• Shear resistance as controlled by web crushing (diagonal compression)
 Walls, before flexural yielding (θpl = 0) or after (cyclic θpl > 0):


VR  0.85 1  0.06 min 5,
pl




1  1.8 min  0.15, N  1  0.25 max((1.75, 100  tot ) 1  0.2 min  2, Ls   f c bw z



A
f
 h 

c
c



Squat columns (Ls/h ≤ 2) after flexural yielding (cyclic θpl > 0):
VR 


4
1  0.02 min 5, pll
7
1  1.35 ANf

c

1  0.45  100  tot  min  f c , 40 bw z sin 2
c 
δ: angle between axis and diagonal of column (tanδ=0.5h/Ls)
Test-model comparison: Cyclic shear resistance in plastic hinge
(after flexural yielding) as controlled by stirrups, per EN 1998-3:2005
1000
no. tests: 306
median=0.995
CoV=14.7%
800
Vexp (kN
N)
600
400
Rectangular
Circular
walls & piers
200
0
0
200
400
600
Vpred (kN)
800
1000
Test-model comparison: Cyclic shear resistance as controlled
by web crushing (diagonal compression), per EN 1998-3:2005
Walls
Squat columns
2500
800
2000
600
Vexp (kN)
Vexp (kN)
1500
400
1000
200
500
0
0
0
500
1000
1500
2000
Vpred (kN)
no. tests: 62, median=1.00,
CoV=19.3%
2500
0
200
400
600
Vpred (kN)
no. tests: 40, median=1.00,
CoV=9.6%
800
FRP Jackets per EN1998EN1998-3:2005
Experimental Database
4 Range and mean values of parameters in the tests of
4.
FRP-jacketed rectangular columns in the database
Parameter
219 columns with continuous bars (145
with CFRP, 24 with GFRP, 27 with
AFRP and 23 with other composite)
p
)
min-max
mean
effective depth, d (mm)
170-720
296
shear-span-to-depth
shear
span to depth ratio,
ratio Ls/h
11-77.44
3 55
3.55
concrete strength, fc (MPa)
10.6-90
31.8
vertical bar yield stress, fy (MPa)
295-816
431
stirrup yield stress
stress, fyw (MPa)
200 750
200-750
388
axial-load-ratio, N/Acfc
0-0.85
0.255
0-1.18
0.24
transverse steel ratio, w (%)
0 81
0.815-7.6
6
2 08
2.08
total vertical steel ratio, tot (%)
geometric ratio of FRP, ρf (%)
0.01-5.31
0.605
nominal FRP strength (MPa)
113-4830
2755
elastic modulus of FRP, Ef (GPa)
5.8-390
166
lapping-to-bar-diameter ratio, lo/db
-
45 columns with lapspliced bars (42 with
CFRP,, 3 with GFRP))
min-max
mean
180-720
393
22-66.66
45
4.5
11.7-55
31
331-617
492
280 535
280-535
442
0-0.4
0.143
0-0.445
0.212
0 81 3 9
0.815-3.9
1 88
1.88
0.13-7.5
1.04
532-4430
2621
17.8-390
184
15-45
30.4
FRP-wrapping of plastic hinges in rectangular
per EN 1998-3:2005
members with continuous bars p
• MR, My: Enhanced by FRP jacket (by 9% w.r.to calculated w/o confinement)
– EN 1998-3:2005: increase neglected.
• Effective (elastic) stiffness EIeff: unaffected by FRP; pre
pre-damage:
damage: 35% drop
• EN1998-3:2005: Flexure-controlled ultimate chord rotation, θu:
Confinement byy FRP increases that due to the stirrups
p by
y αfρfff,e
f e/fc, where:
– ρf=2tf/bw : FRP ratio // direction of loading;
– ff,e: FRP effective strength:
min  f u,f , εu,f Ef ρf

f f,e  min  f u,f , εu,f Ef min  0.5; 1  0.7
fc




fu,f, Ef : FRP tensile strength & Modulus;
εu,f: FRP limit strain. CFRP/AFRP: εu,f=1.5%; GFRP: εu,f=2%
– FRP-confinement
FRP confinement effectiveness:
2
2

h  2 R   b  2 R 
f  1 
3bh
b, h: sides of section;
R: radius at section corner
Test-model comparison - yield properties for FRP-wrapping of
rectangular columns with continuous bars per EN 1998-3:2005
no.tests: 203 (pre-damaged or not)
median=1.09, CoV=19.4%
no.tests: 159 (no pre-damage)
median=1.03, CoV=37.3%
3
2500
median: My,exp =1.09M
=1 09My,predd
2000
non-predamaged
predamaged
median (non-predamaged)
median (predamaged)
2,5
1500
θyy,exp (%)
My,exp (KNm)
2
1000
1,5
1
500
My
0
0
500
1000
1500
My,pred (KNm)
2000
θy
05
0,5
2500
0
0
05
0,5
1
15
1,5
θy,pred (%)
2
25
2,5
EIeff (no pre-damage), no.tests: 159, median=1.02, CoV=30%
3
Yield properties for FRP-wrapping of the plastic hinge
and continuous bars – Biskinis & Fardis 2007, 2013
• Yield moment, My:
– Strength of FRP-confined concrete fcc, instead of fc, in section-analysis
for φy, My, including the calculation of the concrete Elastic Modulus
Modulus, Ec;
– Ec may be estimated per fibMC2010, using fcc instead of fc:
2
Ec=10000(f
0000( cc((MPa))
a))1/3


f cc
bx  a f  f f u , f

 1  3 .3
– fcc from (widely used) Lam & Teng 2003:
b 
f
f
c

fu,f: effective strength of FRP: fu,f=Ef(keffεu,f)
Ef: Elastic Modulus of FRP,
εu,f: FRP failure strain,
keffff: FRP effectiveness factor
factor, equal to 0
0.6
6 per Lam & Teng
y

c
(~same results using other FRP-confinement models: Teng et al 2009,
Samaan et al 1998, Bisby et al 2005, Ilki et al 2008, Wang et al 2012)
• Chord rotation at yielding, θy, effective stiffness, EIeff :
– Yield curvature φy calculated with fcc (as above) & multiplied times
correction factor 1.06
– EIeff = MyLs/3y
Test-model comparison - yield properties for FRP-wrapping of rect.
columns with continuous bars per Biskinis & Fardis 2007, 2013
no.tests: 203 (pre-damaged or not)
median=1.06, CoV=19.3%
no.tests: 159 (no pre-damage)
median=1.01, CoV=37.3%
2500
3
median: My,exp =1.06My,pred
non-predamaged
predamaged
median (non-predamaged)
median (predamaged)
2.5
2000
1500
θy,exp (%
%)
My,exxp (KNm
m)
2
1000
1.5
1
500
My
0.5
0
0
500
1000
1500
My,pred (KNm)
(
)
2000
2500
θy
0
0
0.5
1
1.5
θy,pred (%)
2
2.5
EIeff (no pre-damage): no.tests: 159, median=0.99, CoV=30.4%
3
Extension of empirical ultimate plastic chord rotation of
rect. columns with continuous bars & FRP-wrapping Biskinis & Fardis 2007, 2013
• Pre-damaged or not:
L 
 a 
ν  max0.01, ω' 
 f c0.2  s 
θupl  0.0185  1  0.52acy 1  sl 0.25 
h
 1.6 
 max0.01, ω 
0.3
0.35
 αρ f

 w yw  αρfu
 f
 fc
 c

25
with:
ρ f fu , f
 aρfu 



 a f c f min
i 0.4;
 f 
fc
 c  f , eff

ρ f fu , f



i 0.4;
1  0.5 min
fc



 

cf=1
1.8
8 for CFRP or polyacetal fiber (PAF) sheets
sheets,
cf=0.8 for GFRP or AFRP.
fu,f=Ef(keffεu,f): effective FRP strength


 f ,eff




1.275100 ρd
Test-model comparison – ultimate cyclic chord rotation for FRP-wrapping
of rect. columns with continuous bars: no. tests 128 (pre-damaged or not)
EN 1998
1998-3:2005
3:2005 v Biskinis & Fardis 2007
2007, 2013
EN 1998-3:2005
median=1 09 CoV=30.6%
median=1.09,
CoV=30 6%
Biskinis & Fardis 2013
median=1 025 CoV=30.4%
median=1.025,
CoV=30 4%
25
25
median (all): θu,exp =1.09θu,pred
median (all): θu,exp =1.025θu,pred
20
θu,eexp (%)
θu,eexp (%)
20
15
10
15
10
CFRP jacket
CFRP jacket
AFRP jacket
5
0
0
5
10
15
θu,pred (%)
AFRP jacket
5
GFRP jacket
GFRP jacket
PAF jacket
PAF jacket
20
0
25
0
5
10
15
θu,pred (%)
20
25
Alternative ultimate cyclic chord rotation for continuous bars &
FRP-wrapping – per GCSI (KANEPE)
f cc
 1 .125  1 .25 wd
fc
Test-to-predicted ultimate cyclic
chord rotation, θu – no.tests: 128,
median=1.75,
di
1 75 CoV=54.5%
C V 54 5%
ωwd: FRP volumetric
l
t i ratio
ti
α: FRP-confinement effectiveness factor
25
CFRP (adopted here also for
AFRP and PAF):  cu  0 .0035  f cc f c 2
20
 cu  0 .007  f cc f c 2
FRP effective strength:
f f ,e  f u , f 
ψ: reduction factor for number of
layers (k);
ψ 1 for k<4
ψ=1
  k 1 4 for k≥4
θu
μθ 
 μ δ  μ φ  2  3
θy
θu,eexp (%)
GFRP:
median (all):
θu,exp
1.75θ
75θu,pred
e p =1
pred
15
10
CFRP jjacket
AFRP jacket
5
GFRP jacket
PAF jacket
0
0
5
10
15
θu pred (%)
20
25
FRP-wrapped rectangular members with ribbed bars lap-spliced
over length lo in the plastic hinge: EN 1998-3:2005 & other options
• EN 1998-3:2005:
1. Both bars in pair of lapped compression bars count as compression steel.
2. For the yield properties (My, y , θy), the stress fs of tension bars is:
fs=fy(lo/loy,min), if lo< loy,min=(0.2fy/√fc)db (fy, fc in MPa), loy,min: one-third shorter
th without
than
ith t FRP
FRP.
3. Ultimate chord rotation
θu=y+plu(lo/lou,min),
) if lo < lou,min = dbfy/[(1.05+14.5
/[(1 05+14 5 αlρf ff,e)√fc],
– fc: MPa, ρf=2tf/bw: FRP ratio // loading,
– αl = αf(4/ntot)
ff,e: effect. FRP strength (MPa),
(ntot: total lap-spliced bars; only the 4 corner bars restrained)
restrained).
• Or, Eligehausen & Lettow 2007 for fibMC2010:
0.55
0.25
 lb   f c  

20




f s  51.2    
 d b   20   max(d b ; 20 mm ) 
• cd = min[a/2;
[ ; c1; c]] ≥ db , cd ≤ 3db
• cmax = max[α/2; c1; c] ≤ 5db
• fy, fc in MPa
0.2
 c 1/ 3  c  0.1

d
max
  kKtr   f y , kK  1
  
tr
 d b   cd 

nb d b
 k s nl Ashh k f n f t f E f


 s
Es
h





Test-model comparison - Yield properties for FRP-wrapping of rect.
columns with bars lap-spliced over length lo per EN1998-3:2005, no.tests 45
My: median=1.075, CoV=10.5%
θy: median=1.075, CoV=17.7%
2
2000
median: My,exp =1.075y,pred
1.75
1500
1.5
1250
1 25
1.25
θy,exp (%)
My,exp (KNm)
1750
median: θy,exp =1.075θy,pred
1000
750
1
0.75
0.5
500
My
250
θy
0.25
0
0
0
250
500
750
1000 1250 1500 1750 2000
My,pred (KNm)
0
0 25
0.25
05
0.5
0 75
0.75
1
1 25
1.25
θy,pred (%)
15
1.5
1 75
1.75
2
no.tests: 42
median=1.00
no tests: 42
no.tests:
5000
median: My,exp =My,pred
4000
My,exp ((KNm)
Test-model comparison - yield
properties of FRP-wrapped circular
columns
l
with
ith lap-spliced
l
li d bars
b
per
EN1998-3:2005 (Biskinis & Fardis 2007,
2013)
3000
2000
My
1000
C V 15 2%
CoV=15.2%
median=0.98
0
0
CoV=22%
CoV
22%
1000
2000
4000
5000
My,pred
(KNm))
y pred (
1000
2
3000
θy,exxp (%)
median θy,exp =0.98θ
median:
0 98θy,pred
1
no.tests: 42
θy
0,5
median=0.96
(MyLs/3θy)exxp (MNm2)
800
15
1,5
EIeff
600
400
median:
(M yLs/3θy)exp=0
0.96(M
96(M yLs/3θy)pred
200
CoV=22.2%
0
0
0
0,5
1
θy,pred (%)
1,5
2
0
200
400
600
800
(MyLs/3θy)pred (MNm2)
1000
Extension of empirical ultimate plastic chord rotation for
rectangular members with lap-spliced bars & FRP-wrapping –
Biskinis & Fardis 2007, 2013
• Required lapping for no adverse effect of lap-splice on ultimate deformation
lou , min 
db f y

1.05  14.5 2
n

 tension




2
 aρ f u

 fc


 f , eff

 f
 c

ρ f fu , f



i 0.4;
 a f c f min
 f 
fc
 c  f , eff

with:  aρfu 
ρ f fu , f



i 0.4;
1  0.5 min
fc



 

fu,f=Ef(keffεu,f)
ntension: number of lapped bars on tension side of section
• If
lo<lou,min, θupl
is reduced as:

lo
 min 1,
 lou ,min
i

≤0.5
upl,lap
• (αρfu/fc )f,eff neglected in θupl if 2/ntension
 pl
 u


Test-model comparison – ultimate cyclic chord rotation of FRP-wrapped of
rectangular columns w/ lap-spliced bars, no.tests 44 (pre-damaged or not)
EN1998-3:2005 v Biskinis & Fardis 2007
2007, 2013
Biskinis & Fardis 2013
median=1 005 CoV=23.2%
median=1.005,
CoV=23 2%
EN 1998-3:2005
median=0 89 CoV=36.3%
median=0.89,
CoV=36 3%
12
12
median: θu,exp =1.005θu,pred
10
10
8
8
θu,exxp (%)
θu,exxp (%)
median: θu,exp
, p =0.885θu,pred
,p
6
4
6
4
2
2
0
0
0
2
4
6
θu,pred (%)
8
10
12
0
2
4
6
θu,pred (%)
8
10
12
Extension of ultimate plastic chord rotation model with
curvatures and plastic hinge length to circular columns with
l
lap-spliced
li d bars
b
& FRP-wrapping
FRP
i
– Biskinis
Bi ki i & F
Fardis
di 2013
u   y  (u   y )Lpl 1  0.5Lpl Ls   asl u, slip
  u , slip  5 .5 d bL  u
Lpl, fcc, εcu as in members (FRP-wrapped or not) with continuous bars
6
Steel strain at ultimate deformation of
member depends on lap length, if lo<
lou,min
i :


f
l
l

with: lou,min 
o
lou, min
5
y
 0.2 su  o
loy, min Es

d bL f y
5 6  6a sh f yh / f c 
fc
4
θu,exp (%
%)
 su,l  1.2
median: θu,exp =0.97θu,pred
3
2
1
Test-to-prediction ratio, no.tests: 37
median=0.97, CoV=38.6%
0
0
1
2
3
θu pred (%)
4
5
6
Cyclic shear resistance of FRP-wrapped rectangular members
as controlled by diagonal tension, per EN1998-3:2005
VR 



hx
 L 
min  N , 0.55 Ac f c   1  0.05 min 5, pl 0.16 max( 0.5, 100 tot )1  0.16 min  5, s   f c Ac  Vw   V f
2 Ls
 h 






• Vf=min(ε
( u,f
y
shear resistance:
u fEu,f
u f, fu,f
u f)ρf bwz/2 : FRP-contribution to cyclic
• ρf :FRP ratio, ρf = 2tf/bw;
• fu,f:FRP tensile strength.
Vw=ρwbwzffyw: contribution
t ib ti off web
b steel
t l (bw: web
b width,
idth z: iint.
t llever arm; ρw: steel
t l ratio)
ti )
ρtot: total longitudinal steel ratio
1,4
p
h: section depth
12
1,2
x : depth of compression zone
1
Ac= bwd
Test-to-prediction ratio vs μ, no. tests 12
median=0.99,, CoV=14.8%:
Vu,exp /V
Vu,pred
•
•
•
•
•
0,8
0,6
0,4
0,2
0
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
d tilit
ductility
• In a FRP-retrofitted member the shear resistance as controlled by diagonal tension
cannot exceed the shear resistance of old member as controlled by web crushing
5
RC jackets per EN1998EN1998-3:2005
Concrete Jackets (continued/anchored in joint; w/ or w/o lap splices in old member)
Calculation assumptions:
•
•
•
•
•
•
•
Fullll composite
F
it action
ti off jacket
j k t & old
ld concrete
t assumed
d (jacketed
(j k t d member:
b
monolithic”), even for minimal shear connection at interface (roughened interface,
steel dowels epoxied into old concrete: useful but not essential);
fc of “monolithic
monolithic member”=
member = that of the jacket (avoid large differences in old & new fc)
Axial load considered to act on full, composite section;
Longitudinal reinforcement of jacketed column: mainly that of the jacket. Vertical
bars of old column considered at actual location between tension & compression
bars of composite member (~ “web” longitudinal reinforcement), with its own fy;
Only the transverse reinforcement of the jacket is considered for confinement;
For shear resistance,
resistance the old transverse reinforcement taken into account only in
walls, if anchored in the (new) boundary elements
The detailing & any lap-splicing of jacket reinforcement are taken into account.
Then:




MR & My of jacketed member:
θy of jacketed member for pre-yield (elastic) stiffness:
Sh
Shear
resistance
i t
off jacketed
j k t d member:
b
Flexure-controlled ultimate deformation θu:
~100%
~105%
~100%
100%
~100%
those of “monolithic member” calculated w/ assumptions
p
above.
If jacket bars are not continued/anchored in the joint:
The jacket is considered only to confine fully the old column section.
of
of
off
of
54 jacketed members w/ or w/o lap splices:
test-to-calculated as monolithic
1.8
1.2
1.6
1.4
M y,exp/ My,calc
θ y,exp/ θ yy,Eq.(1)&(2)
1
0.8
0.6
0.4
My
0.2
continuous bars
plain 15db laps
deformed 15db laps
plain 25db laps
deformed 30db laps
deformed 45db laps
non-anchored jacket bars
group average
st. dev. of group mean
1.2
1
0.8
0.6
04
0.4
0.2
0
2.5
0
1.6
2.25
1.4
θy
2
pl
Eq.((3))
1.2
1.5
y + θu
θ u,exp / ( θ*
EIexp / EI*eff
1.75
1.25
1
0.75
EIeff
1
0.8
06
0.6
0.4
0.5
0.2
0.25
0
θu
a
b
c
d
e
f
g
h
i
j
0
k
a
b
c
d
e
f
g
h
i
j
a: no treatment, b: no treatment, predamage, c: welded U-bars, d: dowels, e: roughened, f: roughened / predamage,
g: U-bars+ roughened, h: U-bars+roughened / predamage, i: roughened+dowels, j: roughened+dowels / predamage
Steel jackets per EN1998EN1998-3:2005
Steel Jackets (not continued/anchored in joint)
J k t stops
Jacket
t
before
b f
the
th joint
j i t (several
(
l mm gap to
t joint
j i t face)
f
)
• Flexural resistance, pre-yield (elastic) stiffness & flexurecontrolled ultimate deformation of RC member is not
enhanced by jacket (flexural deformation capacity ~same as
i old
in
ld member
b iinside
id jjacket,
k t no b
benefit
fit ffrom confinement);
fi
t)
• 50% of shear resistance of steel jacket, Vj=Ajfyjh, can be
relied upon for shear resistance of retrofitted member
(suppression of shear failure before or after flexural
yielding);
)
• Lap-splice clamping via friction mechanism at jacketmember interface, if jacket extends to ~1.5 times splice
length and is bolt-anchored to member at end of splice
region & ~1/3 its height from the joint face (anchor bolts at
third-point of side)