Reliability of sheet pile walls and the influence of corrosion – structuralreliability analysis with finite elements
T. SchweckendiekTU Delft/TNO Built Environment and Geosciences, Delft, The Netherlands
W.M.G. CourageTNO Built Environment and Geosciences, Delft, The Netherlands
P.H.A.J.M. van GelderTU Delft, Hydraulic Engineering Section, Delft, The Netherlands
ABSTRACT: The Finite Element Method is well accepted in design practice nowadays. It can be used formodeling complex structures and systems. The constitutive models are improving, which enables us to makemore accurate predictions of the real world behavior. On the other hand, especially in the field of geotechnicalengineering, the uncertainties in the input variables, namely the soil properties, are relatively high. Therefore,the use of probabilistic techniques is attractive. In this paper we present an approach that combines efficientreliability methods with FEM. A sheet pile wall serves as example. The structural reliability of this structure willbe determined accounting for the uncertainties in the soil properties and the groundwater levels as well as thestrength reduction of the structural elements due to corrosion. It is demonstrated that reliability techniques canbe combined in combination with FEM and that reliability analysis can be carried out with reasonable effort.
1 INTRODUCTION
There is a trend in the development of safety con-cepts as well as in economical approaches to structuraldesign to imply more probabilistic concepts. Recentdevelopments were the introduction of partial safetyconcepts such as Load and Resistance Factor Design(semi-probabilistic) or risk based approaches like theDutch regulations for dike safety.Also Life Cycle CostAssessment (LCCA) or maintenance strategies arebased on structural reliability considerations respec-tively the development of the structural reliability overtime.
Probability and reliability theory form the founda-tion for these concepts. Furthermore, the determina-tion of the reliability of a structural design respectivelya structure is an essential subtask within these ideas.In this paper an attempt is made to contribute to thisdevelopment by describing how structural reliabilityanalysis can be carried out for sheet pile structures,respectively deep excavations.
The Finite Element Method is used to an increasingamount in design practice. We can use it for model-ing complex structures and systems. The constitutivemodels are improving, which enables us to make moreaccurate predictions of the real world behavior. Onthe other hand, especially in the field of geotechnical
engineering, the uncertainties in the input variables,namely the soil properties, are relatively high. There-fore, the use of probabilistic techniques is attractive. Inthis paper we present an approach that combines effi-cient reliability methods with FEM. It is demonstratedthat reliability techniques can be combined in combi-nation with FEM and that reliability analysis can becarried out with reasonable effort.
Current design codes are based on partial safetyconcepts. The load and material factors are ideallycalibrated by means of probabilistic analysis. Thesefactors might be suitable for a wide range of typicalapplications, but they are certainly not defined for spe-cific, e.g. extreme cases like very deep excavations.Reliability analysis allows us in principle to deter-mine the reliability of any structure directly and thesuitability of the prescribed partial safety factors canbe assessed. This way the target reliability levels ofthe design codes can be compared with the reliabilityobtained by the analyses.
2 RELIABILITY METHODS
TNO Built Environment and Geosciences have devel-oped ProBox, a generic tool for reliability analysis.
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The limit states to be analyzed may contain models inform of analytical expressions that can be defined, andalternatively external models, such as FEM codes canbe used for the limit state evaluation (coupled analy-sis). In this case the FEM code Plaxis 8.2 was used formodeling the geotechnical structures respectively thesheet pile structure.
The following reliability techniques were availablefor the analyses:
– First Order Reliability Method (FORM)– Second Order Reliability Method (SORM)– Numerical Integration (NI)– Directional Sampling (DS)– Directional Adaptive Response Surface Sampling
(DARS)– Crude Monte Carlo Sampling (MC)– Increased Variance Sampling
In principle all input and model variables can beassigned statistical distributions to account for theuncertainties. ProBox comprises 14 distribution typesand allows the definition of distributions by means oftables. The correlations between the variables can beintroduced in form of a (product moment) correlationmatrix.
The program is under constant development andhas generic couplings for some specific programslike FEM-codes as well as standard programs likeExcel, Matlab or Mathcad already established. Alsoself-defined dll-routines can be used.
In this paper we will primarily make use of FORM(level II) and level III methods like Directional Sam-pling and DARS. The level III methods are either usedto confirm the applicability of FORM, i.e. as valida-tion tool, or for limit states that are non-linear andthat include system effects. They have furthermore theadvantage that their performance in terms of preci-sion does not depend on subjective choices, like e.g.the choice of a response distribution as necessary for
Figure 1. ProBox Screenshot.
e.g. the Point Estimate Method or Latin HypercubeSampling.
The use of Crude Monte Carlo Sampling is notalways attractive for structural reliability problems. Itscalculation effort depends on the probability of failure,which is for this kind of problem ideally in the orderof magnitude of Pf = 10−4.
The expected number of calculations can be calcu-lated with (see [Waarts 2000]):
where Pf is the probability of failure (The expressionholds for an accepted error of 10% of Pf .).
This means that even for a ‘light’ FEM-model thatrequires e.g. 10 seconds of calculation time, a reliabil-ity analysis with Crude Monte Carlo Sampling wouldrequire approximately 15 months!
3 COUPLED CALCULATIONS
In the proposed framework the calculations are con-trolled by the reliability algorithm (see fig 2).
After defining the models, assigning the statisticalproperties to the input variables and specifying thereliability method, the program determines the inputfor each model evaluation. The corresponding Plaxisdata files are amended accordingly and the calcula-tion is triggered. After each calculation the relevantoutcomes are read from the corresponding Plaxis datafiles and the limit state function is evaluated. This
Figure 2. Coupling Scheme ProBox-Plaxis.
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procedure is repeated until the pre-defined stop respec-tively convergence criteria are satisfied and the resultsare presented. These consist of:
– Reliability index β– Probability of failure Pf– Influence coefficients αi– Design point values
(For methods other than FORM/SORM approxi-mations for the design point values and the influencefactors are applied.)
4 LIMIT STATES
The definition of failure respectively the limit statefunction is crucial for a reliability analysis. It canhave also considerable effects on the efficiency of theapplied reliability analysis algorithm. In the follow-ing sections the most important failure mechanismsfor sheet pile structures and according limit stateformulations are discussed.
4.1 Retaining system
Fault trees can be used for system reliability analysisand provide an overview over the critical failure mech-anisms. In figure 3 such a fault tree is presented for asheet pile wall. From a design point of view there areUltimate Limit State (ULS) and Serviceability Limit
Figure 3. Fault Tree of a Sheet Pile Wall.
State (SLS) criteria that have to be fulfilled. We willconcentrate on the ULS. There are basically three fail-ure modes, for which the reliability respectively thefailure probability is to be determined:
– Failure of the sheet pile (Z1).– Failure of the support (Z2).– Failure of the soil (Z3).
Each of these failure modes consist of several failuremechanisms and the whole system can be considereda serial system:
In this paper we will focus on the failure modes sepa-rately. the green color in figure 3 indicates that a failuremechanisms or mode can be determined directly bymeans of reliability analysis, the yellow objects canbe determined based on these results and white oneseither cannot be determined separately or are of minorinterest.
For more information on system reliability consid-erations refer to [Schweckendiek 2006].
4.2 Sheet pile
In common design codes the principal load on the sheetpiles is considered to be the bending moment due tohorizontal loads generated by soil and groundwater.
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The design moment (in [kNm/m]) is usually deter-mined by:
where Wel [m3/m] is the elastic section modulus andfy [kPa] is the steel yield strength.
In fact, using this expression, we consider theexceedance of the yield stress in the outer fibre as fail-ure. In anchored sheet pile walls there is also an axialforce contribution to the stresses in the wall, mainlydue to the vertical component of the anchor force. Con-sidering these two components, bending moments andaxial forces, the stresses in the outer fibre of the sheetpile are determined by:
where M [kNm/m] is the bending moment, FN [kN/m]the axial force and ASP [m2/m] the cross sectional areaof the sheet pile wall.
For the reliability analysis it is convenient to deter-mine the limit state function as:
That means that using this limit state function Z1 wedetermine the probability that the yield strength isexceeded anywhere in the sheet pile. Note that allthe relevant variables are depth (z-direction) depen-dent. The load variables M and FN have a distributionover depth and the strength variables, the geometri-cal properties of the sheet piles Wel and ASP , can alsobe variable over depth. Especially as we will considercorrosion models that determine the thickness loss�e(z)[mm] as a function of depth.
The section modulus decreases almost linearly withthe decreasing thickness of the sheet pile wall and cantherefore be approximated by:
where Wel,0 [m3/m] is the initial section modulus, i.e.before corrosion (see [Schweckendiek 2006]).
The cross sectional area including corrosion can beapproximated by:
where ASP,0 [m2/m] is the initial cross sectional are ande the thickness of the flanges for sake of simplicity.
Using the above expressions we can determine allthe load variables in the limit state function (equa-tion 5) by means of the FEM-analysis and the strengthvariables can be updated using the corrosion modeloutcomes �e(z)[mm].
Also plastic moments could be used, if one wants togo beyond the limits of elasticity. This is just a matterof the failure respectively the limit state definition.
4.3 Anchors
For anchors we can follow a similar approach, usingthe exceedance of the yield stress as failure criterion.The stresses in an anchor are determined by:
where FA [kN/m] is the anchor force that is assumed tobe constant over the free anchor length and AA [m2]isthe cross sectional area of the anchor.
Similarly to eq. 5 the limit state function can bedefined as:
Note that the limit state function is again a spatiallyvariable function due to the fact that the strength reduc-tion by corrosion is not necessarily uniform over thewhole anchor.
The determination of the waling failure limit stateis trivial in most cases, because the waling design isusually based on the design anchor force, which itselfhas already a certain exceedance probability. There-fore the waling reliability is usually higher than theanchor reliability.
4.4 Soil shear failure
In any retaining structure the soil forms part of thestructural system. Figure 4 gives an overview over themost relevant failure mechanisms involving the soilnot only as load element, but where also the strengthis determined by the soil properties.
The determination of the failure probabilities forthese mechanisms is difficult and these probabili-ties can usually not be determined separately. Thereare failure mechanisms that are more dominant thanothers. The consequence is that using FEM we cannotassess the less dominant ones, because the dominantones will occur first and the calculation does notconverge anymore.
What we can do is determine the common failureprobability of all these mechanisms, i.e. the probabilitythat any of these mechanisms occurs or its converse,the reliability against soil failure.
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Figure 4. Soil Shear Failure Mechanisms in RetainingWalls.
Figure 5. Directional Sampling with Limit Equilibrium.
Some approaches to determine Pf ,soil are:
– Deformation-based limit state functions.– Safety-factor-based limit state functions.
(e.g. phi-c-reduction, see [Brinkgreve 1991])– Mobilized shear strength in potential slip planes.– Limit equilibrium criteria (convergence of the
FEM-calculation).
In this paper we will restrict ourselves to the lastapproach using limit equilibrium criteria. For informa-tion on the other possibilities refer to [Schweckendiek2006].
The basic idea of the approach is that we considerfailure to occur, if an FEM-calculation does not reachequilibrium, i.e. does not converge. Consequentlyall FEM-calculations with parameter combinationsrespectively realizations that reach equilibrium in allconstruction stages are considered to be in the safedomain. In fact similar definitions are used for thephi-c-reduction technique and even in the Dutch Tech-nical Recommendation for sheet pile walls CUR166.The problem, as in all reliability analysis problems, isto integrate the probability density either over the safeor over the failure domain. To this end the DirectionalSampling (see [Deak 1980]) method is adopted to theneeds of this binary criterion (see fig. 5).
For each directional sample the ‘distance’λbetweenthe origin of parameter space and the limit state hasto be determined. Since the only information we haveabout the value of the limit state function after an FEM-analysis is, whether it is positive (safe) or negative
(failed), we assign the value Z = 1 to the evaluations inthe safe domain and Z = −1 to the failures.A bisectionalgorithm is used as iterative procedure to determine λ.
In Directional Sampling the failure probability isdetermined by:
where χ2(X , n) is the chi-squared distribution with ndegrees of freedom and N is the number of (direc-tional) samples.
As convergence criterion for the reliability analy-sis we can use the variance of the failure probability(estimate):
When the variance respectively the variation coeffi-cient of the estimate of the failure probability dropsbelow a pre-defined acceptable value the analysis isstopped.
This criterion does not require any information fromthe FEM-analysis apart from the fact, whether equi-librium has been reached in all phases or not. Thismakes it a robust method. For more information on itsapplication refer to [Schweckendiek 2006].
4.5 Summary
The limit state definitions in the previous sections arebased on quantities that are either input or output vari-ables of an FEM-analysis or they can be determinedby analytical or other simple models. That means thatwe have gathered all the ingredients for a reliabilityanalysis of a sheet pile structure using FEM-analysis.
5 CALCULATION EXAMPLE
The application of the presented methodology isdemonstrated by a sheet pile wall in layered soil withone anchor layer. Two variants are discussed. Variant1 treats only the soil properties, i.e. the load on thestructure, as stochastic quantities. The rest of the vari-ables assume either nominal or mean values. In variant2 a stochastic thickness loss on the sheet pile due tonatural corrosion is considered.
The structural dimensions are the result of a struc-tural design calculation. The results of the reliabilityanalysis are compared with the target reliability of thedesign guideline.
5.1 Description
The calculation example is a sheet pile wall in lay-ered soil and one anchor layer. The top layers are soft
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(peat and clay) down to a depth of −11.0 m. The baselayer is a stiff dense sand layer. Groundwater is present.Figure 6 shows the geometry in the final excavationstage.
The pit will is excavated stepwise and the ground-water level inside the pit is lowered before the finalexcavation.
5.2 Parameters
For the soil parameters typical values for Dutchsoil conditions were assumed. Their distributions aresummarized in table 1.
Figure 6. Geometry Calculation Example.
Table 1. Soil Parameter Distributions.
Peat, medium COV Mean STD Lower Bound Upper Bound Distribution Unit
It is not accounted for any correlations between thevariables in this example. The structural parameterswere determined in a deterministic design based on theCUR166 (see [CUR166]) with safety class two, whichis based on a target reliability of β=3.4. The structuraldimensions used in the FEM – model are indicatedin figure 6. The water levels were set to the designwater levels used in the CUR166 design at −5.84 mrespectively −1.5 m.
5.3 FEM-model
The Finite Element Model was made with the codePlaxis 8.2. The model and the mesh are illustrated infigure 7.
The construction stages are simulated as follows:
– Gravity loading (generation of initial stresses).– Excavation to −1.0 m.– Placing of the anchor and prestress with 80 kN/m.– Groundwater lowering inside the excavation and
final excavation to −5.0 m.
The Mohr-Coulomb Model was used for the cal-culations with the respective soil parameters fromtable 1.The structural elements were modeled as linearelastic.
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5.4 Results – Sheet pile failure
5.4.1 Variant 1 – Stochastic soil propertiesIn variant 1 we consider only the relevant soil proper-ties as stochastic variables, whilst the geometry, thepore pressures and the properties of the structuralmembers are treated deterministically. The calcula-tions were carried out with FORM.
Based on the results in table 2, the calculations wererepeated with a reduced number of random variables.The 5 most influential parameters were kept and therest of the variables was set to their expectations andtreated deterministically. Due to the decrease in inputuncertainty, the reliability index increased to β = 4.38(Pf = 5.97 E-6). This confirms that the impact of theneglected variables on the reliability was small and it
shows that previous sensitivity analysis can decreasethe calculation effort considerably, in this case from99 to 37 limit state function evaluations with FORM.
These level II results were also assessed with aDARS-analysis (level III). It gave basically the sameresult using about 1,500 limit state function evalua-tions. Hereby the applicability of FORM (linearizationof the limit state) could be confirmed and the calcula-tion effort was still acceptable considering that withCrude Monte Carlo several millions of evaluationswould have been necessary for the same accuracy.
The influence factors (also called ‘importance fac-tors’) in table 3 also reveal that for this limit state theclearly dominant parameters are the stiffness param-eters of the clay layer in this case. A design point iscalculated in which the he sheet pile fails before thestrength parameters of the soil become important. Theproblem remains in the elastic domain for this limitstate. That is especially important considering thatthe current design codes mostly focus on the strengthproperties of the soil for the calibration of partial safetyfactors. In fact, for using FEM in design the sets of loadand material factors should be calibrated separately.
5.4.2 Variant 2 – Stochastic corrosionVariant 2 is an extension of variant 1, in which wealso account for the uncertainty in the strength reduc-tion due to natural corrosion of the sheet pile. To thisend the following distributions for the thickness lossin 100 years were assumed, based on 95%-quantiles(characteristic values) from EC3:
Taking this thickness loss into account we obtain theresults as presented in table 4, in an similar analysis
Table 3. Thickness Loss (�e) Distributions per Soil Type.
Figure 8. Reliability Index vs. Initial Section Modulus.
as carried out in 5.4.1 with FORM with the reducednumber of variables.
The reliability decreases considerably (fromβ = 4.38 to β = 1.98) and the corrosion in the peatlayer becomes the most influential parameter. In fact,the maximum stresses were calculated at the lowerboundary of the peat layer, where relatively high bend-ing moments are combined with the high thickness lossin the aggressive peat environment.
Figure 8 shows the results of a parametric study onthe sheet pile type. According to this study an AZ25is required instead of the AZ18 without corrosionallowance.
5.5 Results – Anchor failure
The anchor reliability was determined by FORMand checked with DARS, similarly to the proceduredescribed in 5.4.1. The results are presented in table 5.
The anchors were over-designed with respect to thetarget reliability of β = 3.4, which is basically a resultof the design process and the limited choices of anchorsizes. The influence factors show a similar picture asfor the sheet pile.
5.6 Results – Soil shear failure
The probability of soil shear failure was determinedwith the limit equilibrium approach that was describedin 4.4. The Directional Sampling analysis required1,653 model evaluations. The results are presented intable 6.
For this limit state there is clearly a shift in the influ-ence coefficients from the stiffness parameters that
dominate the horizontal load on the wall to a combi-nation of this horizontal load and the shear strength ofthe soil. It should be noted that the shear strength notsolely depends on the strength parameters but also onthe effective stress field, and therefore implicitly alsoon the weight of the soft top layers.
6 CONCLUSIONS
We can draw the following conclusions:
– A fully probabilistic reliability analysis of the rel-evant limit states of a sheet pile structure has been
carried out, taking uncertainties in the soil prop-erties and the strength reduction by corrosion intoaccount.
– The reliability analysis provides valuable informa-tion in form of influence coefficients, which can beused in optimization and to better understand thephysical problem itself. E.g. the sheet pile failurewas clearly dominated by the stiffness parametersin this example.
– The presented methodology proved to work well forlimit states where the soil represents the load on thestructure, whilst for soil failure further research isnecessary.
– The presented approach can be used in probabilis-tic and risk-based design concepts. Furthermore,it allows us to compare the target reliability ofdesign codes with the ‘actual‘ (calculated) relia-bility. Therefore it can be used for calibration ofload and resistance factors, when FEM is used fordesign.
– This probabilistic approach is not restricted to struc-tural reliability problems. It can be applied to allkinds of problems, where input uncertainty has tobe propagated through a model and especially forthe computation of small failure probabilities.
We would like to thank Plaxis bv and TNO Built Envi-ronment and Geosciences as well as the research groupof Prof. Vrijling at TU Delft who have supported thisresearch.
REFERENCES
Baecher, G.B. and Christian, J.T., Reliability and Statistics inGeotechnical Engineering, Wiley, West Sussex, England,2003.
Brinkgreve, R.B.J. and Bakker, H.L., Non-linear finite ele-ment analysis of safety factors, Computer Methods andAdvances in Geotechnics, 1991.
Brinkgreve, R.B.J. and Broere, W. Plaxis Finite ElementCode,Version 8.2, Plaxis bv, Delft,The Netherlands, 2004.
CUR 166. Damwandconstructies, Technical Recommenda-tion, 4e herziene uitgave 2005, The Netherlands, 2005.
Deak, I., Three digit accurate multiple normal probabilities,Num. Math., 35, p. 369–380, 1980.
ProBox, A Generic Probabilistic Toolbox, information anddemo-version available on www.tno.nl/probox.
Schweckendiek, T., Structural Reliability Applied to DeepExcavations –Coupling Reliability Methods With FiniteElements-, MSc-thesis, Delft University of Technology,Delft, The Netherlands, 2006.
Waarts, P.H., Structural ReliabilityAnalysis Using Finite Ele-ments, PhD-thesis, Delft University of Technology, Delft,The Netherlands, 2000.
