[elsevier] reliability of reinforced concrete structures under fatigue

8/13/2019 [Elsevier] Reliability of Reinforced Concrete Structures Under Fatigue

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Reliability of reinforced concrete structures under fatigue

Y.S. Petryna*, D. Pfanner, F. Stangenberg, W.B. Kratzig

Department of Civil Engineering, Ruhr-University Bochum, Universitatsstr. 150, D-44780 Bochum, Germany

Abstract

This paper focuses on time-variant reliability assessment of deteriorating reinforced concrete structures under fatigue conditions. Astrategy combining two time scales, namely the micro-scale of instantaneous structural dynamics (or statics) and the macro-scale of structural

lifetime, is proposed. Non-linear response of reinforced concrete structures is simulated by means of the finite element method with adequate

material model. A phenomenological fatigue damage model of reinforced concrete is developed and calibrated against experimental results

available in the literature. Reliability estimates are obtained within the response surface method using the importance/adaptive sampling

techniques and the time-integrated approach. The proposed assessment strategy is illustrated by an example of a concrete arch under fatigue

loading. The obtained results show a general inapplicability of local and linear fatigue models to system level of structures.q 2002 Elsevier

Science Ltd. All rights reserved.

Keywords:Structural reliability; Reinforced concrete; Fatigue; Damage

1. Introduction

The necessity of time-variant reliability assessment of

deteriorating structures is becoming increasingly recog-

nized [l4]. However, technical service life limited by

structural failure is mainly addressed by defining either

critical load or material parameters [4,5]. Degradation

analysis of structural capacity is often wrongly con-

sidered as durability assessment of constructional

materials or single components overlooking the impact

of local damage on global failure conditions.

A similar situation may also be observed in fatigue of

reinforced concrete structures focused mainly on the

local material level [68]. Fatigue failure itself plays in

concrete engineering a secondary role, but it caninfluence usual failure mechanisms and accelerate

structural degradation. Such effects can be studied solely

on the system level taking interaction of damage

mechanisms and stress redistribution within entire

structure into account.

Response of reinforced concrete as composite construc-

tional material exhibits strongly non-linear character of

response and corresponding damage processes [9]. Struc-

tural reliability assessment under such conditions implies a

system approach, which combines damage mechanics, non-

linear structural analysis and probabilistic methods in a

rational way. In our opinion, such a general and unified

approach is still missing.This work proposes an assessment strategy for structural

reliability problems under fatigue conditions and illustrates

this assessment by a test numerical example. The results

show a fundamental inapplicability of local and linear

fatigue damage models to structural level.

2. Material model of reinforced concrete

Realistic material models of reinforced concrete suitable

to be used in structural analysis must correctly simulate at

least four different components: a ductile concrete responseto compression and a brittle response to tension, a ductile

behavior of steel reinforcement and highly non-linear bond

effects[9]. The material model applied in the present study

has recently been developed by Polling[10]and bases on

the uniform elasto-plastic continuum damage theory [11,

12]. It describes the time-independent static material

response to monotonic or cyclic loads.

For the model formulation, the hardening and softening

functions of inelastic concrete response are of importance

(Fig. 1). The model fits the ascending branches of the

uniaxial stressstrain relationships stated in the Model

Code 90 [13], whereas the applied softening laws in the

post-peak ranges depend on the dissipated energy in the

localized process zones. The descending branch of

0951-8320/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 9 5 1 - 8 3 2 0 ( 0 2 ) 0 0 0 5 8 - 3

Reliability Engineering and System Safety 77 (2002) 253261www.elsevier.com/locate/ress

* Corresponding author. Fax: 49-234-32-14370.

E-mail address:[email protected] (Y.S. Petryna).
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the stress strain curve for uniaxial compression reads

sc1 21

2gcfc1c2fc

gc1 gc

21c12

1

with

gc p

2fc1c

2 gpcl 21

2fc 1c1 2 b b

fc

Ec

2 ; gpcl Gclleq :

Herein,1crepresents the peak strain and fcthe compressive

strength,gccontrols the area under the curve and therewith

the dissipated energy depending on the initial Youngs

modulusEc, the crushing energy Gcl, the equivalent length

leq of the finite element and on the ratio b of plastic and

damage components of inelastic strains.

For tensile stresses, the softening law reads

sct1 fctexp 1

gt

fct

Ec2 1

; gt

Gf

leqfct2

1

2

fct

Ec;

2

where fct denotes the tensile strength, gt controls the area

under the stressstrain curve and Gf indicates the fracture

energy.

According to the envelope concept confirmed byexperiments, the monotonic stressstrain curves provide a

bound for all hysteretic loops. Cyclic un- and reloading

paths of the material are modeled linearly with a gradient

corresponding to the damaged stiffness.

The uniaxial material model for reinforcing steelcorresponds generally to the classical elasto-plasticity.

The yield surface has been modified according to Ref. [14]

in order to correctly describe the Bauschingers effect. The

stressstrain relationship in the plastic hardening domain

reads

s1 fy Bs 1 2 1h 21 2 1h

gs

gs1f 2 1hgs 21

3

with

gs Bs1f 2 1h

fs 2fy 2Bs1f 2 1h

;

where fy denotes the yield stress, fs the tensile strength, 1hthe strain at the beginning of plastic hardening, Bs, the

plastic hardening modulus and 1f the constriction strain

[10].

3. Fatigue damage model of reinforced concrete

General strategy. For complex structures under high-

cycle fatigue loading (N . 103 load repetitions), an explicit

tracing of each single load cycle becomes unacceptable due

to enormous computational efforts. An alternative way

consists in the application of prognostic models predicting

the increments of fatigue damage, similar to the increments

of creep strains according toRef. [15]. Starting with a stress

state after a single cycle of fatigue loading, the model

computes the degradation of material properties for n1repetitions. Structural simulation with damaged parameters,

which are evidently different in each material point,

provides the estimate of residual carrying capacity. Foreach of the following increments ni, an updating iteration

must be carried out taking redistribution of stresses and

damage into account.

Fatigue failure criterion. The following S N curve

according to Ref. [6] has been applied to determine thenumber of load cycles to failure Nf:

Smax 1 2 0:06621 2 0:556RlogNf 2 0:0294 logT; 4

where Smax denotes the ratio of maximum stress to static

strength,Rthe stress amplitude and Tthe period of one load

cycle. Since the envelope stressstrain curve corresponding

to monotonic loading can be assumed to cover the strain

values of arbitrary load histories [7], fatigue failure isassociated with the moment, when a regular loading path

intersects the envelope (point B,Fig. 1). The stiffness andstrain values at the moment of fatigue failure read for

compression

Ec;fail 1fail1 2 b

Smaxfc

b

Ec

21

;

1fail 1c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi21cgcfc

11

Smax

;

s 5

and tension

Ect;fail Smaxfct1t;fail

; 1t;fail fct

Ec2 gt ln Smax: 6

Fig. 1. Material law and cyclic loading for uniaxial compression and tension.

Y.S. Petryna et al. / Reliability Engineering and System Safety 77 (2002) 253261254


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Fatigue damage evolution. Fatigue damage accumu-

lation in concrete observed in experiments cannot be

described by classical linear theories since it is a non-linear

process. This accumulation is found to cohere with the

evolution of total strains 1tot (Fig. 5), which proceeds

parabolically in the initial and in the final stage of fatigue

life, whereas in the intermediate phase a constant strain rate

can be observed. A regression analysis of experimental data

of several authors leads to the following evolution functions:

1tot

n

1

350d80 2 4d20n 2 5d80 35d20 1RS

10max

1fail 2 10 for n# 0:2; 7

1tot

n

5

3d80 2 d20 1RS

10max 1fail 2 10

for 0:2 , n # 0:8; 8

1tot

n

50

3 d20 2 4d80n 2 55d80 2 15d20

1RS10max

50n 2 40

1fail 2 10 for n . 0:8; 9

where ndenotesn=Nfwithnapplied load cycles, 10indicatesthe initial strain at the beginning of the fatigue process (point

A,Fig. l),d20,d80represent the measured relative changes of

total strains after 20 and 80% of the fatigue life, respectively.The strain evolution according to Eqs. (7)(9) is depicted in

Fig. 5.

Due to the fact that total strains in the first stage of life are

governed by creep processes, which are not attributed to

mechanical damage, a modified cyclic creep compliance

function Jt; t0 according to Ref. [16] is introduced tocalculate the fatigue strains 1fat associated with fatigue

damage:

1fat 10 1tot 2 10nkfat

Jn; t0 2 EcJ1; t0 2 Ec

: 10

Herein,kfat

controls the fatigue strain rate in dependence of

the period Tof cyclic loads acting since time t0. Since the

calculated fatigue strains are imbedded into the constitutive

equations by the fatigue failure criteria (5) and (6), the scalar

parameter of fatigue damage for the uniaxial ease can be

directly derived:

Dc;fat 1fat 2 10

1fail 2 101 2

Ec;fail

Ec

: 11

Its evolution is depicted inFig. 2.

Damaged material parameters. The reduced elasticity

modulus defining the slope of the reloading path for the

current fatigue state reads:

Ec;fat Ec1 2Dc;fat: 12

The values of residual strengths fc,fat, fct,fat(Fig. 2) and the

new peak strain 1c,fat of a damaged material can easily be

determined by means of the original stress strain relations(1) and (2) presuming the knowledge of the intersection

point of the current loading path with the initial envelope (at

strain 1int). Due to difficulties in determining this point in

closed-form, iteration algorithms have been applied. Then,

one obtains for the modified localized crushing energy (in

compression):

Gcl;fat 1

2fc;fat 1c;fat1 2 b b

fc;fat

Ec;fat

!lim

1c;fat1

2 sc1d1: 13

The first term corresponds to the area under the ascendingbranch of the damaged stress strain curve, whereas the

integral represents the area under the descending branch.

For tension domain, the new localized fracture energy

reads:

Gf;fat leq gtfctexp 1

gt

fct

Ec2 1int

1

2

f2ct;fat

Ec;fat

" #: 14

These modified parameters define a new, damaged material

law for concrete (Fig. 3) used in the further structural

simulations.

Reinforcement. In contrast to concrete, the known

Palmgren-Miner hypothesis provides acceptable estimates

of damage accumulation in steel under elastic deformations,

Fig. 2. Evolution of damage, degradation of compressive and tensile strength.

Y.S. Petryna et al. / Reliability Engineering and System Safety 77 (2002) 253261 255


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presuming the absence of inherent stresses or initial

damage. The following modified rule has been applied

below in order to allow for physical interpretation of

damage variable:

Ds;fat 21

a 1 2 1 2 e2a

1

1 h

Xmk1

nkDsk

NfDsk

" #

withXmk1

nkDsk

NfDsk# 1;

15

where Dsk; represents the stress amplitude, Nf the numberof load cycles to failure according to Ref. [13] and nk the

number of load cycles within the corresponding stress range.

The parameter Ds,fat allows for damage-oriented formu-

lation of the material law for steel depicted in Fig. 3.

4. Probabilistic model calibration

Probabilistic aspects of fatigue. Experiments show that

fatigue processes are generally subject of great variability

both in load cycles to failure and in deformation history. The

scatter ofNfhas been confirmed to fit the known Weibulldistribution [8]. Several procedures to estimate fatigue

reliability have been discussed in Ref. [17]. However, they

are mainly based on stochastic fracture mechanics and

hardly applicable to real concrete structures exhibiting

usually multiple cracks.

Definition of random variables. The predictions offatigue damage presented above depend on two parameters

of major influence: the fatigue life Nf and the relativechanges of total strains d20and d80. The evolution of these

values and their scattering over time has been determined

based on experimental data available in Refs. [8,18]. It

seems to be rational to introduce these values as principal

random variables into the model. Such an approach

overcomes difficulties in finding adequate time-dependent

distributions of conventional random variables such as

strength and stiffness, since fatigue life Nf and strain

evolution naturally include these values.

Parameterization of random variables. Because of its

physical assumptions, the Weibull distribution of fatigue lifeis proved to fit best the results of fatigue experiments in

general [19] and for concrete in particular [4]. However,

such parameters as the characteristic life Vs, the minimum

life n0,s and the shape parameter ss of the distribution

function

FNn ; P{Nf # n} 1 2 exp 2n 2 n0;s

Vs 2 n0;s

!as16

have seldom been specified with regard to concrete as in

Ref. [8].A plenty of experimental data have been processed

in this work in order to obtain reliable values of parameters

of the Weibull distribution. The differences of these data

due to unknown experimental conditions do not allow for an

overall statistical analysis. Consequently, data sets of

several authors have been separately studied by the methods

described inRef. [20]. The results validated by statistical fit

tests correspond to basic assumptions of the Weibulldistributions[21] and exhibit a strong dependence on the

maximum stress level Smax(Fig. 4).

Regarding the strain evolution, experimental data

available in the literature [18] have been evaluated as

before with respect to the distribution functions for the total

strains 1tot (Fig. 5) and found to follow a two parameter

lognorma1 distribution. The mean values can be calculatedaccording to Eqs. (7) (9), whereas the coefficient of

variation exhibits dependence on the upper stress level

Smax. The implementation of the distribution functions into

the fatigue model causes the corresponding scatter of the

material state predictions depicted for compression inFig. 5.

5. Structural simulation under fatigue

Fatigue load modeling. In contrast to linear approaches,

damage in general, and fatigue damage in particular,

immediately make the problem strongly non-linear and

path-dependent, so that the useful superposition rules for

load combinations[22]become inapplicable. The aspect of

representative load histories applicable to non-linear

systems remains still generally unsolved[23].

Although both static and dynamic structural responses

may be considered, this contribution concentrates on slow

(quasi-static) cyclic loading processes. The fatigue load isassumed to have a constant period and amplitude within a

Fig. 3. Change of stressstrain relationships for concrete (compression and tension) and steel reinforcement due to fatigue damage.



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given lifetime interval Ti. These two parameters can,

however, change between two successive time intervals.

Then, the duration of time intervals DTi; the amplitudePmax;iand the frequencyaiof cyclic loading constitute three

potential random variables of the fatigue load history

consisting ofn intervals:

Pfat [ni1

PfatDTi; Pmax;i;ai: 17

Non-linear finite element analysis. Structural damage

processes in a rigor treatment are always attached to non-

linear responses. Non-linear response of an arbitrarystructure discretized by finite elements under static loading

can be incrementally traced on the basis of the known

tangent stiffness equation[24]:

KTV; dDV DlPFIV; d; 18

whereV, DVdenote the vector of nodal displacements and

its increment, Dl the increment of the load factor, P the

vector of nodal forces, KTV; d the non-linear tangentstiffness matrix depending on the current damage extent d.

FIV; ddenotes the vector-functional of internal forces.For numerical simulation of quasi-static structural

response processes described by Eq. (18), a large variety

of NewtonRaphson, BFGS or path-following methods[25,26] offers approved techniques. The structural simulations

of the present study have been carried out within the

FEMAS software[27]already used in numerous structural

damage simulations[28].

Lifetime simulation. Life assessment of structures either

implicitly or explicitly deal with two principal time scales.

The micro-time scale tdescribes usually the processes of

structural dynamics measured in seconds to hours or simply

the time-independent structural response. Just in this time

scale, the limit states may be properly detected. On the other

hand, fatigue processes last over the macro-scale T

(lifetime) usually measured in years. Since both scales are

relevant, but not directly compatible, structural simulations

currently proceed in turns. The entire life history is

subdivided by discrete time instants Ti; i 1; ; n intointervals of generally non-equal duration DTi Ti 2 Ti21;

i 1; ; n 2 1:The fatigue damage Dc,fat is predicted for lifetime

intervals DTi according to the model presented in Section

4. After a correction of material properties due to damage at

the end of each time interval, structural simulations are

carried out anew:

KTV; dTiDV DlPFIV; dTi: 19

Such a simulation strategy provides complete informationnecessary for reliability examinations including stability,

Fig. 4. Estimation of Weibull parameters and S Ncurve with Weibull distribution.

Fig. 5. Scatter of strain evolution and of stressstrain relationship.



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deflections, strains, stresses, crack widths, etc. during the

entire service life of the structure under consideration.

6. Time-variant reliability assessment

Reliability assessment of structures exposed to fatigue

loading needs an approach combining methods of non-

linear structural analysis, damage mechanics and probabil-

istic reliability theory. A majority of publications on

reliability degradation due to damage still concerns simple

eases, when local damage in a single cross-section may be

directly associated with a global limit state. A typical

example is fatigue crack growth[17].However, the fatigue

limit state in reinforced concrete structures is difficult to find

as a single cause of global failure. Fatigue interacts withother failure mechanisms and can rather influence typical

limit states of carrying capacity or serviceability. Due to

enormous computational difficulties of such complex

problems, a general and unified approach is still missing.

The present work proposes an assessment strategy appli-

cable to such complex reliability problems, depicted in

Fig. 6and described below.

Since the number of random variables may be crucial for

efficiency of stochastic reliability assessment, they are

defined on the basis of preliminary sensitivity analysis and

describe generally statistical uncertainties of load, material

and structural origin. Additionally, statistical distributions

and their eventual changes during the lifetime shall bedefined for each of the selected random variable.

Among various approaches to reliability of reinforced

concrete structures, the response surface method (RSM)

[29] is of advantage, if closed-form models either are not

available or may be based on too rough structural

idealizations. The method combines the full-scale non-

linear structural analysis with statistical simulations by

means of a uniform interface in the form of limit state points

calculated deterministically. The latter are simply critical

combinations of governing parameters X1; ;Xm; whichsatisfy the limit state condition

gX RX2

SX 0; 20where R and S denote the conjugate resistance and load

variables. For example, the critical load Pmax at collapse

(Fig. 6) and concrete strength fc form a limit state pointX Pmax;fcof two components.

According to the RSM[29],an analytical approximation

of the implicit limit state function (20) is searched for,

currently in the form of the second order polynomial:

gX a0 Xki1

biXiXki1

Xkj1

cijXiXj: 21

Having the analytical function (21) at disposal, one can

further calculate the instantaneous failure probabilities,practically the failure rates hTi; by means of the most

efficient statistical simulation tools available. The adaptive/

importance sampling procedures within the COSSAN

software[30]are currently used.

The failure rates hTi obtained for instantaneous values

of the resistanceRTiand the load effect STican then be

numerically integrated over the macro-timeT[3]providing

finally the cumulative time-dependent failure probability:

pfT 1 2 2expT

0hTdT

1 2 2expXni0

hTiDTi

!" #: 22

7. Example: concrete arch

In order to illustrate the entire assessment procedure on a

possibly simple example, we consider solely the concrete

response to cyclic compression observed in the arch under

uniformly distributed normal pressure (Fig. 7). The arch is

made of concrete with compressive strength fc 35 MPa

and elasticity modulusEc 33; 200 MPa:It was discretizedfor structural analysis by 20 finite elements.

The non-linear load displacement diagram computed

for the mid-span point are shown inFig. 8. Structural failure

occurs each time due to the overloading of the upper

concrete layer at mid-span following by an immediate lossof global equilibrium and collapse.

For efficiency purposes, only two random variables are

considered in the present example: compression strength of

concrete fc and external pressure qls. Both variables are

assumed to be normally distributed with the parameters in

Table 1.

The concrete strength is assumed to be uniformlydistributed along the arch in the initial state, it changes

then according to the fatigue damage model in each point

individually. The reference pressure q lsis taken stationary,

i.e. its amplitude does not depend on time.

The high-cycle fatigue loading is modeled as cyclic

pressure of constant frequency and the magnitude varyingfrom zero to qfat 2:0 MPa:It amounts to about 0.53qmax,where qmax denotes the ultimate load capacity of the archwithout damage. The stresses in the arch vary at that from

zero to Smax 0:70fc: The load history is assumed to beregular high-cycle loading over the entire life-span. The

number of load cycles is taken according to a typical

European traffic situation measured at Auxerre [31] and

corresponding to 150 heavy trucks per day and approxi-

matelyN 32,000 cycles per year.

In order to estimate residual capacity and reliability, the

arch has been statically loaded at each of time instants Tiup

to collapse (Fig. 8). The corresponding reduction of qmaxdue to accumulation of fatigue damage is depicted also inFig. 9.



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Fig. 6. Concept of structural reliability assessment.



8/9

Finally, the instantaneous probability of collapse (failure

rate)

hTi pHCFf Ti pqmaxfc; qfat; dTi # qlsTi 23

is calculated by means of the RSM and depicted inFig. 9.

The results show a strongly non-linear character of fatigue

damage evolution in concrete even under regular cyclic

loading. This fact makes the application of linear accumu-

lation rules [32] to concrete structures on system level

problematic.

8. Conclusions

Reliability assessment of structures under fatigue

conditions is a highly complicated problem, which implies

interaction of different scientific fields such as damage and

continuum mechanics, non-linear structural analysis andprobabilistic reliability theory. Due to the evident complexity

of such a general formulation, a majority of publications

still concerns simple eases of local damage in a single cross-

section and associate them with a global limit state, such as

in typical fatigue crack growth models.

Fatigue evolution in reinforced concrete structures

exhibit rather global character with multiple cracks and

large material volumes affected by damage, such that local

assessment becomes unacceptable. Only a global approach

taking stress redistribution within the structure and

interaction of various degradation mechanisms into accountcan provide correct predictions of structural response and

reliability. The present contribution proposes main elements

of such an approach.

A fatigue damage model of reinforced concrete capable

to simulate arbitrary damage states under cyclic loading has

been developed. The model is embedded into the elasto-

plastic material law for concrete with damage component.

The damage evolution function depending on fatigue strains

has been fitted to a plenty of fatigue test data available in the

literature.

This model has been further incorporated into a new

structural simulation strategy based on interval assessmentof damage accumulation processes in the lifetime scale. The

Fig. 7. Geometry and loading of concrete arch.

Fig. 8. Loaddisplacement diagrams of arch under fatigue.

Table 1

Variable Mean value Standard deviation Coefficient of variation

fc(MPa) 35.0 5.25 0.15

qls(MPa) 2.1 0.315 0.15

Fig. 9. Degradation of carrying capacity and increase of failure rate over lifetime.



9/9

interaction of instantaneous and long-time damage mech-

anisms can then be traced over the entire lifetime

contributing to a prognosis of structural response.

The time-variant reliability assessment is proposed to

carry out by means of the RSM combined with time

integration of the calculated instantaneous failure rates.

Such an approach advantageously combines deterministic

damage-oriented structural analysis with probabilistic

assessment procedures.

The results obtained in a test problem show that linear

damage accumulation rules widely used on local level

deliver incorrect predictions of global structural behavior.

Acknowledgements

This work has been carried out within the Projects Cl,C2

of the Collaborative Research Center 398 (SFB 398) at the

Ruhr-University Bochum. Financial support of the German

Science Foundation (DFG) is gratefully acknowledged.

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