MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar...

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.


TENSION SOFTENING

Tensile Failure of Axial Force Member:


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.


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


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: ∆` = ∆è + ∆`s

∆` =∆σe

Eeè +

∆σs

Es`s

and

∆σaxial =EeEs

Ee`s + Esè∆`

Controllable Softening Range as long as:

Ee`s + Esè > 0


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


TENSION SOFTENING

Cohesive Interface Approach: Strong Discontinuity

∆σ =KsEe

Ee + Ksè∆` snap-back when è

crit = 2EeG

critf

f 2t

Note: ècrit 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


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


TENSION SOFTENING

3-D Cross Effects: of Damage-Plasticity Model in Abaqus

Displacement Continuity introduces lateral confinement


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.


TENSION SOFTENING

Cohesive Interface Elements Eliminate 3-D Cross Effects:

No lateral confinement due to loss of bond.


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]


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.


COMPRESSION SOFTENING

Issue of Material vs Structural Response:

Axial Force Member: compression response of weak element



Full 3-D Cross Effect:

Lateral confinement introduces uniform triaxial state of stress (elastic if no cap)



3-D Cross Effect: Confinement Introduces Elastic Triaxial Compression

No softening of Damage-Plasticity Model in Abaqus because of missing cap.



Reduction of 3-D Cross Effect:

Cohesive Interface Elements: eliminate lateral confinement



Cohesive Interface Elements Eliminate 3-D Cross Effects

Localization of compression failure in weak element.



Snap-Back due Localization of Compression Failure in Weak Element

Cohesive Interface elements eliminate lateral confinement.


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


TENSION STIFFENING


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


TENSION STIFFENING


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 [-]


TENSION STIFFENING


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


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


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


TENSION STIFFENING


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


TENSION STIFFENING


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


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


MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar...

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