MODELING OF CONCRETE MATERIALS AND STRUCTURES
Kaspar Willam, University of Colorado at BoulderIn cooperation with Bernd Koberl, Technical University of Vienna, Austria.
Class Meeting #6: Tension Softening vs Tension Stiffening
Smeared Crack Approach: Plastic Softening (isotropic case)
Axial Force Member in Tension and Compression: Snap-Back Effect
Cross-Effect: Lateral Confinement due Mismatch in 3-D
Tension Stiffening: Debonding in Reinforced Concrete
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION SOFTENING AND APPARENT DUCTILITY
Tensile Cracking:Smeared Crack Approach vs Plastic Softening.
Axial Force Problem:Serial Structure-Localization in Weakest Link.
Localization of Axial Deformation:Snap-Back and 3-D Cross-Effect when elastic energy release exceedsdissipation in softening domain.
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION SOFTENING
Tensile Failure of Axial Force Member:
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
1-D PLASTIC HARDENING/SOFTENING
Elastic-Plastic Decomposition:
ε = εe + εp where εe =σ
E eand εp =
σ
Ep
Consequently,
ε =σ
E e+
σ
Ep=
σ
Etan
Elastoplastic Tangent Stiffness Relationship:
σ = Etanε where Etan =EeEp
Ee + Ep
Note: Etan = −∞when Ecrit
p = −Ee.
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
3-D PLASTIC SOFTENING
Rankine Criterion for Tension Cut-Off: FR(σσσ, κ) = σ1 − ft(κ) = 0
Associated Plastic Flow Rule: εεεp = λmmm where εp1 = λ sign(σ1)
Isotropic Strain Softening Rule: ft(κ) = ft + Epκ where −E < Ep < 0.
Plastic Consistency: FR(σσσ, κ) = σ1 − Epκ = 0.
Strain-driven Format: from κ = εp1 = λ we find λ = E
E+Epε1
Tangent Stiffness Format: σ1 = E[ε1 − λ] = Etanε1 where Etan =EEp
E+Ep
Fracture Energy Based Softening:
Ep = dσ1dε
p1
= dσ1
dufN
dufN
dεp1
= Kp s
where s = crack separation
GIf =
∫ f
u σ1dufN = 1
2ft ufcr
Critical Softening:Ecrit
p = Kcritp s = −Ee
or Kcritp = −Ee
s
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
WEAK ELEMENT IN AXIAL FORCE MEMBER
Snap-Back Analysis of Serial Structure:
Static Equilibrium: ∆σaxial = ∆σe = ∆σs
Total Change of Length of Axial Force Member: ∆` = ∆`e + ∆`s
∆` =∆σe
Ee`e +
∆σs
Es`s
and
∆σaxial =EeEs
Ee`s + Es`e∆`
Controllable Softening Range as long as:
Ee`s + Es`e > 0
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION SOFTENING
Critical Size of Softening Zone for Snap-Back:
`crits = −Es
Ee`e
Note Snap-Back in spite of Constant Fracture Energy: Gf = const
0
1000
2000
3000
4000
5000
6000
0.0E+00 2.0E-05 4.0E-05 6.0E-05 8.0E-05 1.0E-04 1.2E-04
displacement [m]
forc
e [N
]
Gf = 100 NmLs=0,0095m=Ls,critGf = 100 NmLs=0.0448m>Ls,critGf = 100 NmLs=0.0077m<Ls,crit
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION SOFTENING
Cohesive Interface Approach: Strong Discontinuity
∆σ =KsEe
Ee + Ks`e∆` snap-back when `e
crit = 2EeG
critf
f 2t
Note: `ecrit compares with characteristic length of Hillerborg et al.
Effect of different Gf values on Structural Softening:
0
200
400
600
800
1000
1200
1400
0.00E+00 2.00E-05 4.00E-05 6.00E-05
displacement [m]
forc
e [N
]
Gf = Gf,crit = 34Nm
Gf = 32Nm < Gf,crit
Gf = 50Nm > Gf,crit
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION SOFTENING
Fracture energy-based softening: Linear vs Exponential Format
Mesh-size dependent softening modulus: Es = dσduf
duf
dεf= Kshel
0
1000
2000
3000
4000
5000
6000
0.00E+00 3.00E-05 6.00E-05 9.00E-05 1.20E-04
displacement [m]
forc
e [N
]
Gf=120Nm - linear softening DP Gf=120Nm - exp. softening DPGf=120Nm - linear softening - SC Gf=120Nm - exp. softening SC
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION SOFTENING
3-D Cross Effects: of Damage-Plasticity Model in Abaqus
Displacement Continuity introduces lateral confinement
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION SOFTENING
3-D Cross Effects Eliminated by Shear Slip and Loss of Bond
Insert zero shear interface elements between weak softening element and elasticunloading elements.
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION SOFTENING
Cohesive Interface Elements Eliminate 3-D Cross Effects:
No lateral confinement due to loss of bond.
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
MISMATCH AT PLASTIC SOFTENING-ELASTIC UNLOADING INTERFACE
• Perfect Bond: No Separation-Delamination:us
axial = ueaxial and us
lat = uelat with εs
lat = εelat
• Statics: σsaxial = σe
axial = σaxial
• Plastic Softening-Elastic Unloading in Axial Tension:
- Strain Rate in Plastic Softening Domain: εεεs = EEE−1s σσσ + εεεp
- Strain Rate in Elastic Unloading Domain: εεεe = EEE−1e σσσ
- Parabolic Drucker-Prager Yield Condition: F = J2 + αI1 − β = 0where α = 1
3[fc − ft] and β = 13fcft
- Associated Plastic Flow Rule: εεεp = λmmm = λ[sss + α111]
- Lateral Plastic Strain Rate: εlat = λmlat = λ[13(σslat − σaxial) + α]
• Elastic-Plastic Mismatch due Axial Tension:Introduces lateral contraction in softening domain:
σslat =
νsEe − νeEs
Ee(1− νs) + Ls
LeEs(1− νe)σaxial − λEe[
1
3(σs
lat − σaxial) + α]
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
BIMATERIAL INTERFACE CONDITIONS
Perfect Bond:
[|uN |] = ueN − us
N = 0 and [|tN |] = teN − ts
N = 0
Weak Discontinuities: all strain components exhibit jumps across interfaceexcept for εe
TT = εsTT restraint.
Note: Jump of tangential normal stress, σeTT 6= σs
TT .
Imperfect Contact:
[|uN |] = ueN − us
N 6= 0 whereas [|tN |] = teN − ts
N = 0
Strong Discontinuities: all displacement components exhibit jumps acrossinterface.
Note: FE Displacement method enforces traction continuity in ‘weak’ senseonly, hence [|tN |] 6= 0.
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
COMPRESSION SOFTENING
Issue of Material vs Structural Response:
Axial Force Member: compression response of weak element
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
COMPRESSION SOFTENING
Full 3-D Cross Effect:
Lateral confinement introduces uniform triaxial state of stress (elastic if no cap)
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
COMPRESSION SOFTENING
3-D Cross Effect: Confinement Introduces Elastic Triaxial Compression
No softening of Damage-Plasticity Model in Abaqus because of missing cap.
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
COMPRESSION SOFTENING
Reduction of 3-D Cross Effect:
Cohesive Interface Elements: eliminate lateral confinement
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
COMPRESSION SOFTENING
Cohesive Interface Elements Eliminate 3-D Cross Effects
Localization of compression failure in weak element.
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
COMPRESSION SOFTENING
Snap-Back due Localization of Compression Failure in Weak Element
Cohesive Interface elements eliminate lateral confinement.
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION STIFFENING
Stress Transfer of Parallel System: Full Bond
Kinking iff embedded rebar has no shear and bending stiffness
(Avg: 75%)PE, PE33
+6.352e−06+5.290e−03+1.057e−02+1.586e−02+2.114e−02+2.643e−02+3.171e−02+3.699e−02+4.228e−02+4.756e−02+5.285e−02+5.813e−02+6.341e−02
1
2
3
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION STIFFENING
Stress Transfer of Parallel System: Full Bond
Mesh Effect for Constant Fracture Energy: Gf = const.
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 0.0004 0.0008 0.0012 0.0016 0.002
displacement [m]
forc
e [N
]
single - rebar
1x1 Gf=32Nm
4x4 sym Gf=32Nm eq.ef=0.0026 [-]
8x8 sym Gf=32Nm
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION STIFFENING
Stress Transfer of Parallel System: Full Bond
Mesh Effect for Constant Cracking Strain: εf = const.
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 0.0004 0.0008 0.0012 0.0016 0.002
displacement [m]
forc
e [N
]
single rebar
1x1sym ef=0.0026 [-]
4x4 sym ef=0.0026 [-] eq. Gf=32Nm8x8 sym ef=0.0026 [-]
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION STIFFENING
Stress Transfer of Parallel System: Full Bond
Regular crack spacing ‘independent’ of mesh size.
(Avg: 75%)PE, PE33
+6.352e−06+5.290e−03+1.057e−02+1.586e−02+2.114e−02+2.643e−02+3.171e−02+3.699e−02+4.228e−02+4.756e−02+5.285e−02+5.813e−02+6.341e−02
1
2
3
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION STIFFENING
Local Study of Stress Transfer in Segment between Adjacent Cracks:
Effect of fracture energy mode II for modeling shear debonding.
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 0.00025 0.0005 0.00075
displacement [m]
forc
e [N
]
GfII = 10xGfI
GfII = GfI
naked steel
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION STIFFENING
Stress Transfer of Parallel System near Center Crack
Shear transfer in steel rebar (von Mises)
(Avg: 75%)S, Mises
+6.000e+04+6.133e+04+6.267e+04+6.400e+04+6.533e+04+6.667e+04+6.800e+04+6.933e+04+7.067e+04+7.200e+04+7.333e+04+7.467e+04+7.600e+04
+0.000e+00
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION STIFFENING
Stress Transfer of Parallel System near Center Crack
Shear transfer in concrete (von Mises stress)
(Avg: 75%)S, Mises
+0.000e+00+5.833e+01+1.167e+02+1.750e+02+2.333e+02+2.917e+02+3.500e+02+4.083e+02+4.667e+02+5.250e+02+5.833e+02+6.417e+02+7.000e+02+7.600e+04
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
TENSION STIFFENING
Stress Transfer of Parallel System near Center Crack
Axial stress transfer at steel-concrete interface
4.30E+084.40E+084.50E+084.60E+084.70E+084.80E+084.90E+085.00E+085.10E+085.20E+085.30E+08
0 0.03 0.06 0.09 0.12 0.15
length of the specimen [m]
stre
ss in
reba
r[N/m
²]
0.00E+00
5.00E+05
1.00E+06
1.50E+06
2.00E+06
2.50E+06
3.00E+06
3.50E+06
4.00E+06
4.50E+06
stre
ss in
con
cret
e[N
/m²]
SM of rebar S33 of concrete
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
CONCLUDING REMARKS
Main Lessons from Class # 6:
Tension Softening vs Tension Stiffening:Both Serial and Parallel Systems Exhibit Snap-Back Conditions.
Loss of Bond at Weak Element Interface:Loss of Triaxial Confinement-No Cross Effects
Loss of Bond at Steel-Concrete Interface:Tensile Cracking Followed by Shear Debonding
Class #6 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007