Reliability assessment of concrete structures

Software package under development:- Project GA 1O3/02/1030 (Czech Grand Agency)

- International project SARA

BUT, Faculty of Civil Engineering, Brno

Cervenka Consulting, Prague

BOKU and TU Vienna

SARA studio

ATENA

FREET

Modular concept

ATENA Interface

NNononlinelineaar r fracture mechanics fracture mechanics – – FEM FEM software ATENA (Červenka Consulting)software ATENA (Červenka Consulting)

NNononlinelineaar r fracture mechanics fracture mechanics – – FEM FEM software ATENA (Červenka Consulting)software ATENA (Červenka Consulting)

• SBETA material model for concrete damage: 2D failure criterion• Tension:

– nonlinear fracture mechanics, exponential softening – smeared cracks: fictitious crack approach, crack band theory– two crack models: fixed and rotated cracks

• Compression: – non-linear behaviour including softening (crushing)– compressive strength reduction after crack propagation

• Shear: – shear strength reduction of cracked concrete

• Solution methods (equations of equilibrium): – Newton Raphson, Arc-length.

Failure criterion for concrete Failure criterion for concrete in in 2D 2D plane stress stateplane stress state

Failure criterion for concrete Failure criterion for concrete in in 2D 2D plane stress stateplane stress state

Crack Band ModelCrack Band ModelCrack Band ModelCrack Band Model

Independence of FEM mesh (localization limiter) (Bažant and Oh, 1983).

.constGLG Ff )(. inputglobalconstG

LtorelatedenergyfractureG

F

f

• Uncertainties modelling - random variables by PDF, described by mean value, variance and other statistical parameters.

• Statistical simulation - random input parameters are generated according to their PDF using LHS.

• Generated realizations of random parameters are then used as inputs for ATENA computational model. The complex nonlinear solution is performed and results (response) are saved.

• Previous two steps are repeated N-times (N is the number of simulations used).

• Statistical assessment - the resulting set of structural responses is statistically evaluated: Mean value, variance, coefficient of skewness, histogram, empirical cumulative probability density function of structural response, etc.

Probabilistic module -Probabilistic module - FREET FREET implementation into ATENAimplementation into ATENA

Probabilistic module -Probabilistic module - FREET FREET implementation into ATENAimplementation into ATENA

Practical example - PORR bridge in Vienna

SARA studio

Longitudinal section

Ground plan

Cross section

PORR bridge

Variable Ec ft fc Gfc

Ec 1 0.7 0.9 0.5 0.9ft 0.77 1 0.8 0.9 0.6fc 0.83 0.82 1 0.6 0.9Gf 0.69 0.83 0.75 1 0.5c 0.84 0.77 0.83 0.69 1

Table 2: Correlation matrix for concrete properties

Random variable description Symbol Units Mean value COV Distribution type Reference Concrete grade B500Modulus of elasticity Ec GPa 36.95 0.15 Lognormal 6

Poisson's ratio - 0.2 0.05 Lognormal EstimationTensile strength ft MPa 3.257 0.18 Weibull 6

Compressive strength fc MPa 42.5 0.10 Lognormal 6,7

Specific fracture energy Gf N/m 81.43 0.20 Weibull 8

Uniaxial compressive strain c - 0.0023 0.15 Lognormal 6

Reduction of strength cRed - 0.8 0.06 Rectangular EstimationCritical comp displacement wd m 0.0005 0.10 Lognormal EstimationSpecific material weight MN/m3 0.023 0.10 Normal 9

Prestressing strandsModulus of elasticity Es GPa 200.0 0.03 Lognormal 10

Yield stress fy MPa 1600.0 0.07 Lognormal 10

Prestressing force F MN 21.85 0.04 Normal 9

Area of strands As m2 0.0237 0.001 Normal 9

Table 1: Statistical properties of random variablesTable 1: Statistical properties of random variables

Correlation between variablesusing simulated annealing

Variable Ec ft fc Gf εc

Ec 1 0.7 0.9 0.5 0.9ft 0.77 1 0.8 0.9 0.6fc 0.83 0.82 1 0.6 0.9Gf 0.69 0.83 0.75 1 0.5εc 0.84 0.77 0.83 0.69 1

upper triangle - presribed correlation matrixlower triangle - generated correlation matrix

(8 samples)

Left -Load-deflection curves (8 samples), right - Histogram of ultimate loads (32 samples)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

deflection [m]

load

[M

N]

D1

D2

D3

D4

D5

D6

D7

D8

D mean

Mean value Variance Standard deviation Coeff. of variation Number of samples[MN] [MN] [MN] -

8 2.52 0.0707 0.266 0.10532 2.51 0.0649 0.255 0.102

Table 3: Estimation of basic statistical parameters of the ultimate load

012345678

0.5 1 1.5 2 2.5

Load [MN]

Rel

iab

ilit

y in

dex

COV of load = 0.1COV of load = 0.2

Eurocode

Reliability assessment (Prof. Novak and Mrs. Eichinger)