Fatigue and Fracture Behavior of Airfield

Concrete SlabsProf. S.P. Shah (Northwestern University)Prof. J.R. Roesler (UIUC)Dr. Bin MuDavid Ey (NWU)Amanda Bordelon (UIUC)

FAA Center Annual Review – Champaign, IL, October 7, 2004

Research Work Plan

1. Finite Element Simulation of Cracked Slab

2. Concrete slab compliance

3. Develop preliminary R-curve for concrete slab

4. Small-scale fracture parameters

5. Fatigue crack growth model

6. Model Validation

Large-Scale Concrete Slab Tests

Typical S-N Curves for Concrete Fatigue

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06

Repetitions to Failure (N)

Ap

plie

d S

tre

ss

/ F

lex

ura

l Str

en

gth

(S

)

Roesler(1998) Darter(1977)

FAA 150/5320-6D Rollings(1988)

Parker(1979) Foxworthy(1985)

PCA(1985) Darter(1989)

NCHRP 1-26 (1992) LEDFAA

Beam Curves

Slab Curves

The load – crack length (compliance) response obtained from static loading acts as an envelope curve for fatigue loading.

The condition KI = KIC can be used to predict fatigue failure.

Fatigue crack growth rate has two stages: deceleration stage and acceleration stage.

Summary of Approach

Static loading acts as an envelope curve for fatigue loading

(Subramaniam, K. V., Popovics, J.S., & Shah, S. P. (2002), Journal of Engineering Mechanics, ASCE 128(6): 668-676.)

Static Envelope

The crack growth in deceleration stage is governed by R-curve.

The crack growth in acceleration stage is governed by KI.

1)( 01naaC

N

a

2)(2

nIKC

N

a

Static and Fatigue Envelope

Fatigue

Static envelope governed by KI=KIC

Crack length

Load

# of cycles

Fatigue crack growth

A

B

C

acritical

0

0.2

0.4

0.6

0.8

1

0 0.3 0.6 0.9 1.2

Normalized cycles, N/Nf

Cra

ck

len

gth

(in

)

90%-5%

Inflection point

Crack growth during fatigue test

(a) crack length vs. cycles (b) rate of crack growth

-4

-3

-2

-1

0.2 0.4 0.6 0.8

Crack length (in)

Lo

g(

a/

N)

90%-5%

deceleration acceleration

a inflection

Step 1

Phase –1: Fatigue test

FEM C=C(a) and KI=KI(a)

FEM Simulation of Cracked Slab

a

Experimental setup and FEM mesh

Elastic support

2000 mm

1000 mm

aSymmetric line

200 mm

100 mm UIUC setup

FEM mesh with a=400 mm

Deformation

(a=400 mm)

Node force

(a=400mm)

FEM Contours

Calculation of KI: A modified crack closure integral

Rybicki, E. F., and Kanninen, M. F., Eng. Fracture Mech., 9, 931-938, 1977.

Young, M. J., Sun C. T., Int J Fracture 60, 227-247, 1993.

a

c

b

d

e

f

Element-1 Element-2

Element-4Element-3

Y, v

X, uO’

Fc

Finite element mesh near a crack tip

a a a

)(2

1)(

2

1

00dce

adcc

aI vvF

avvF

aG LimLim

)1(,

2

EGorEGK I

II

If < 20% crack length, then accuracies are within 6% of the reference solutions.

a

KI Determination

Vertical displacement at the mid point of edge

0

2

4

6

8

10

0 400 800 1200 1600 2000

Crack length (mm)

Dis

p. a

t lo

ad

ing

po

int

(mm

)

FEM

UIUC experiment

Westergaard solution

No crack

Transverse through crack

Deflection vs. Crack Length

y = 1E-07x2 - 1E-06x + 1.0228

0

0.3

0.6

0.9

1.2

1.5

1.8

0 400 800 1200 1600 2000

Crack length (mm)

No

rma

lize

d c

om

plia

nc

e

FEM

Poly. (FEM)

0

500

1000

1500

2000

2500

1 1.2 1.4 1.6 1.8 2

Normalized compliance

Cra

ck

len

gth

(m

m)

FEM

Poly. (FEM)

y=-7176.2*x2+22074.9*x-14898.7

Compliance and crack length

FEM Compliance Results

Stress intensity factor and crack length

y = 2E-06x + 0.0007

y = 5E-10x2 - 9E-07x + 0.0014

0

0.0005

0.001

0.0015

0.002

0 400 800 1200 1600 2000

Crack length (mm)

KI/P

(m

m-3

/2)

FEM-1

FEM-2

Linear (FEM-1)

Poly. (FEM-2)

KI vs Crack Length (a)

0

0.05

0.1

0.15

0.2

0.25

0 500 1000 1500 2000

Crack length (mm)

CM

OD

(m

m)

FEM

Poly. (FEM)

y=-1E-7x2+2.9E-4x

CMOD vs Crack Length

Processing Lab Fatigue Data

Single pulse loading

Tridem pulse loading

Single Pulse Fatigue Loading (1 Cycle)

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250 300

Time (sec)

Lo

ad

(k

N)

Loading Unloading

Pmax

Pmin

Tridem Pulse Fatigue Loading (1 Cycle)

0

10

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200

Time (sec)

Lo

ad

(k

N)

Loading L1

Loading L2

Loading L3

Unloading U1

Unloading U2

Unloading L3

Pmax

Pint

Pmin

Deflection vs. Number of Cycles

(Single Pulse Slab 4)

Deflection vs. Number of Cycles

(Tridem Pulse Slab 7)

Compliance Plots Loading vs. Unloading Compliance

Single vs. Tridem Pulses Need to measure CMOD in future!!!

Single Pulse Loading vs. Unloading Compliance

Load vs Rebound Deflection for S4 Cycle 85529

0

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2 2.5

Rebound Deflection (mm)

Lo

ad

(k

N)

Loading Compliance

Unloading Compliance

Single Pulse Compliance (Slab 9)S9 Compliance (Including Loaded and Unloaded)

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 50 100 150 200 250 300 350 400

Number of Cycles

Co

mp

lian

ce (

mm

/kN

)

Loading Unloading

S9 Cycle 176

52.3

52.4

52.5

52.6

52.7

52.8

52.9

53

0 50 100 150 200 250 300

Time (sec)

Pmax = 96.9 kNPmin = 67.7 kNNfail = 352

Tridem Pulse Loading vs. Unloading Compliance

Load vs Rebound Deflection for T4 Cycle 3968

0

10

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5 3 3.5

Rebound Deflection (mm)

Lo

ad

(k

N)

Loading L1 Compliance

Unloading U3 Compliance

Unloading U1, Loading L2, Unloading L2 and

Loading L3 Compliances

Tridem Pulse Compliance (Slab 2)T2 Compliance (Including Loaded and Unloaded)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 10,000 20,000 30,000 40,000 50,000 60,000 70,000

Number of Cycles

Co

mp

lian

ce (

mm

/kN

)

L1 U1 L2 U2 L3 U3

Pmax = 91.5 kNPmin = 7.0 kNNfail = 61,184

T2 Cycle 36000

0

10

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200

Time (sec)

Tridem Pulse Compliance (Slab 4)T4 Compliance (Including Loaded and Unloaded)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000

Number of Cycles

Co

mp

lian

ce (

mm

/kN

)

L1 U1 L2 U2 L3 U3

T4 Cycle 4352

0

10

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200

Time (sec)

Pmax = 90.7 kNPmin = 7.5 kNNfail = 4,384

Slab 4 Compliance

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 20000 40000 60000 80000 100000 120000 140000 160000

Number of Cycles

Co

mp

lian

ce (

mm

/kN

)

Pmax = 69.6 kNPmin = 6.5 kNNfail = 143,896k = 1373 psi/in

0

0.4

0.8

1.2

1.6

2

0 0.3 0.6 0.9 1.2

Normalized cycles (N/Nf)

No

rmai

lized

co

mp

lian

ce

slab-4

Slab-4

Normalized Compliance

0

500

1000

1500

2000

2500

0 0.3 0.6 0.9 1.2

Normalized cycles, N/Nf

Cra

ck

len

gth

(m

m)

70 ~ 6.5 kN

-4

-3

-2

-1

0

1

0 500 1000 1500 2000 2500

Crack length (mm)

Lo

g(d

a/d

N)

slab-4

Compliance, crack length and da/dN for Slab-4

Single Pulse Slab4

T2 Compliance

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 10000 20000 30000 40000 50000 60000 70000

Number of Cycles

Co

mp

lian

ce (

mm

/kN

)

Pmax = 91.5 kNPmin = 18.2 kNNfail = 61,184

T-2

0

0.4

0.8

1.2

1.6

2

0 0.3 0.6 0.9 1.2

Normalized cycles, N/Nf

No

rma

lize

d c

om

plia

nc

e T-2

Tridem Slab (T2)

0

500

1000

1500

2000

2500

0 0.3 0.6 0.9 1.2

Normalized cycles, N/Nf

Cra

ck

len

gth

(m

m) 92 ~ 18 kN

Compliance, crack length and da/dN for T-2

-3

-2

-1

0

1

0 500 1000 1500 2000 2500

Crack length (mm)

Lo

g(d

a/d

N)

T-2

Crack Growth for Slab T2

T4 Compliance

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 1000 2000 3000 4000 5000 6000

Number of Cycles

Co

mp

lian

ce (

mm

/kN

)

Pmax = 90.7 kNPmin = 8.6 kNNfail = 4,384

0

0.4

0.8

1.2

1.6

2

0 0.3 0.6 0.9 1.2

Normalized cycles (N/Nf)

No

rma

lize

d c

om

plia

nc

e T-4

T-4

Tridem Slab (T4)

0

500

1000

1500

2000

2500

0 0.3 0.6 0.9 1.2

Normalized cycles (N/Nf)

Cra

ck

len

gth

(m

m)

90.7 ~ 8.6 kN

Compliance, crack length and da/dN for T-4

-1.5

-1

-0.5

0

0.5

1

1.5

0 500 1000 1500 2000 2500

Crack length (mm)

Lo

g(d

a/d

N)

T-4

Crack Growth for Slab T2

-4

-3

-2

-1

0

1

0 500 1000 1500 2000 2500

Crack length (mm)

Lo

g(

a/

N)

70 ~ 6.5 kN

Decelaration

Acceleration

-2

-1.5

-1

-0.5

0

0.5

1

0 500 1000 1500 2000 2500

Crack length (mm)

Lo

g(

a/

N)

91 ~ 9 kN

Deceleration

Acceleration

-3

-2

-1

0

1

2

0 500 1000 1500 2000 2500

Crack length (mm)

Lo

g(

a/

N)

92 ~ 18 kN

Deceleration

Acceleration

Models for Slab-4, T2 & T4

1)( 01naaC

N

a

2)(2n

IKCN

a

Fatigue Crack Growth Model

Accel.

Decel.

Challenges Need to calibrate material constants C1,n1,

C2, n2 with slab monotonic data and small-scale results

Explore other crack configurations modes (partial depth and quarter-elliptical cracks)

Size Effect….

Concrete Property Testing Test Setup

Two Parameter Fracture Model (KI and CTODc)

Size Effect Law (KIf and cf)

Concrete Material Property Setup Three Beam Sizes

Small Medium Large

Size

Depth Width Length Span Notch Length Notch Width

(mm) (in) (mm) (in) (mm) (in) (mm) (in) (mm) (in) (mm) (in)

1 62.5 2.461 80 3.15 350 13.78 250 9.843 20.8 0.82 3 0.118

2 150 5.906 80 3.15 700 27.56 600 23.62 50 1.969 3 0.118

3 250 9.843 80 3.15 1100 43.31 1000 39.37 83.3 3.281 3 0.118

S = 1 m

D = 250 mm

50 mm

W = 80 mm

Large Beam

Initial crack length

= 83 mm

Clip gauge CMOD

50 mm

notch

LVDT

Top View

LVDT

10 mmCMO

D

Testing Apparatus

Before Loading After Loading

Load vs. CMOD (Small Beam)

Cast Date: 06-14-04 Test Date: 06-22-04

Small Beam

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

2,000

0 1 2 3 4 5 6

CMOD (10^-2 mm)

Lo

ad

(N

)

1st cycle

2nd cycle

3rd cycle

4th cycle

5th cycle

Load vs. CMOD (Large Beam)

Cast Date: 06-14-04 Test Date: 06-22-04

Large Beam

0

1,000

2,000

3,000

4,000

5,000

6,000

0 1 2 3 4 5 6 7 8 9 10

CMOD (10 -̂2 mm)

1st cycle

2nd cycle

3rd cycle

4th cycle

5th cycle

Two Parameter Fracture Model Results

Test #

Dimensions (mm) ft

c

w/c

da

(mm)

E

ao/b ac/b

KsIc

CTODc

(mm)

GsIc

(N/m)bS b t (MPa) (GPa)

(MPa m1/2)

1 250 62.5 80 35.7 0.45 19 27.3 0.333 0.417 1.177 0.0072 50.73

2 600 150 80 35.7 0.45 19 39.6 0.333 0.538 1.735 0.0402 76.08

3 1000 250 80 35.7 0.45 19 39.4 0.333 0.460 1.788 0.0321 81.06

4 250 62.5 80 37.9 0.45 19 28.0 0.333 0.524 1.314 0.0254 61.67

5 600 150 80 37.9 0.45 19 46.1 0.333 0.515 1.699 0.0292 62.63

6 1000 250 80 37.9 0.45 19 34.0 0.333 0.461 1.693 0.0352 84.18

Jenq and Shah

Size Effect Law Results

E

S b t (GPa)1 250 62.5 80 35.7 0.45 19 0.3332 600 150 80 35.7 0.45 19 0.3333 1000 250 80 35.7 0.45 19 0.3334 1367 341.7 80 37.9 0.45 19 0.3335 1742 435.4 80 37.9 0.45 19 0.3336 2117 529.2 80 37.9 0.45 19 0.333

cf

0.095 0.1558

αo

(ao/b)

CB

(MPa-2)

0.04538 4.111

m (<.2)

17.396

17.396

AB (mm-1

MPa-2)Series

Number

Dimensions (mm) f'c (MPa)

da

(mm)w/c

Gf

(N/m)

ωA

(<.1)

0.095 0.1558

35.0 27.68

36.0 26.910.04538 4.111

Bazant et al

Partial Depth Crack

Edge Notch Crack

Quarter-Elliptical Crack

a

b

cd

Slab Tests

Analysis of Slabs on Elastic Analysis of Slabs on Elastic

Foundation using FM- OverviewFoundation using FM- Overview

ba

L

Foundationp = k0 * w * y

Applied total load (P)r

ba0

L

Foundation

a0b

P

S t

L

Slab on Elastic Foundation

Beam on Elastic Foundation Beam

CMOD

Load

KIC

Ci Cu

Static Mode I SIF

1

1.2

1.4

1.6

0 0.5 1

Crack length (in)

No

rma

lize

d c

om

plia

nc

e

Experiment

Fitting curve

y=3.35x2-1.80x+1

Compliance vs. crack length

Crack Growth Validation from Monotonic Slab Tests

Future Direction Complete Monotonic Slab Testing**

develop failure envelope

Validate for fatigue edge notch slabs**

Validate for fully-supported beams** testing and FEM

Develop Partial-Depth Notch and Size Effect

Incorporate small-scale fracture parameters into fatigue crack growth model

Compliance vs. Crack Length for

Fully Supported Beam λ4 (1 - e-λw cos (λ w)) = 3(k2 b w C) / (d2 q) λ2 / (e-λw sin (λ w)) = 3(q √(π a0) F(α0)) / (KIC b d2)

λ = characteristic (dimension is length-1) w = ½ the length of load distribution k = modulus of subgrade reaction b = width of the beam C = Compliance d = depth of the beam q = distributed load a0 = crack length F(α0) = -3.035α0

4 + 2.540α03 + 1.137α0

2 – 0.690α0 + 1.334 α0 = a0 / b KIC = Critical Stress Intensity Factor for Mode I

q

w

a0