PROBABILISTIC MODELLING FOR RELIABILITY ANALYSIS OF JACKETS. Det Norske Veritas, N-1322 Høvik, Norway.

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1 Proceedings o OMAE4 3rd International onerence on Oshore Mechanics and Arctic Engineering June -5, 4, Vancouver, British olumbia, anada OM AE4-57 POBABIII MODEING FO EIABIIY ANAYI OF JAE Jan Mathisen Det Norske Veritas, N-3 Høvik, Norway Jan.Mathisen@dnv.com Gudinnur igurdsson Det Norske Veritas, N-3 Høvik, Norway nut O. onold Det Norske Veritas, N-3 Høvik, Norway nut.onold@dnv.com ABA Experience rom recent reliability analyses o jacket platorms is used to discuss selected aspects o probabilistic modelling in more detail. hese modelling details can have a signiicant eect on the computed reliabilities. An overview o basic considerations and ailure modes in jacket reliability analysis is included to set the various details into context. Ultimate limit states or jackets in relatively shallow water are emphasised; i.e. quasi-static structural response is applicable. he ollowing topics are considered: (a) Failure modes and some requirements to load and resistance analysis. (b) Directionality in loading and resistance. (c) (d) andom periods o individual extreme waves. Foundations axial and lateral capacity modelling or multiple piles and model uncertainty or pile capacity. eywords: jackets, structural reliability, directional eects, pile capacity. INODUION wo o the present authors published a paper on the reliability analysis o a jacket at OMAE last year [3]. hat paper and subsequent jacket analysis work gave us the idea o exploring a ew aspects o the reliability analysis o jackets in somewhat more detail, and led to the present paper. Although this paper is less extensive than we planned, we hope it may generate some discussion and contribute a little to the development o reliability analysis procedures or jackets. he exploration o random directionality presented here arises naturally in relation to jackets, but should have implications or reliability analysis o other types o structures, too. NOMENAUE g limit state unction l applied base shear orce l characteristic base shear capacity l M mean load in an environmental state l load standard deviation in an environmental state l,l load parameters r resistance r characteristic resistance r, r resistance parameters u model uncertainty actor on capacity u model uncertainty actor on loading capacity actor, normalised wrt. individual load cases λ A λ B θ θ θ ς OVEVIEW capacity actor, normalised wrt. characteristic base shear capacity direction direction with largest load coeicient direction with largest resistance height o line o action o base shear orce Basic Analysis onsiderations In relatively shallow water, the natural periods o the dominant modes o global lateral vibration o jackets tend to lie well below the periods o the incoming regular waves. his permits quasi-static structural analysis. Dynamic structural opyright 4 by AME

2 analysis is required in deeper water, when the natural periods are longer and can be excited by the incoming waves. Nonlinear load-eects due to drag orces and to wave elevation can produce excitation orces at other requencies, typically at two or three times the wave requency. hese eects may also have to be taken into account when choosing the method o structural analysis. he sub-division between response analysis methods is also relected in the methods applied to load analysis. he relative contributions o the various wave periods in a sea state are important or accurate assessment o dynamic response, and tend to require application o a wave spectrum, through requency or time domain methods. For quasi-static structural response, the peak values o the applied loads are o most importance, permitting use o regular waves; i.e. a single wave period in the load model. O course, the regular wave height and period have to be careully selected to replace the more detailed spectral representation o the sea state. Dynamic ampliication actors can be used to extend the quasi-static analysis procedure in regular waves to include jackets with some dynamic response. However, detailed comparison o dynamic and quasi-static analyses results may be required to derive accurate dynamic ampliication actors. he structural elements o jackets tend to be relatively slender, such that drag orces give rise to a signiicant portion o the hydrodynamic loads due to waves and current. he nonlinear nature o the drag orces is a little awkward in a requency domain analysis and requires some orm o linearization. he nonlinear drag orces do not present any diiculty to load analysis in regular waves and hence, tend to avour this approach. Furthermore, it is straightorward to represent simultaneous wave kinematics at all points within the jacket in regular wave analysis, but more arduous to do so in irregular waves. he present paper is primarily concerned with jackets in relatively shallow water, implying quasi-static structural analysis and load analysis in regular waves. Failure Modes he ultimate limit state (U) tends to be critical or the global design o jackets in relatively shallow water, whereas the atigue limit state (F) can be more critical in deeper water, with dynamic response. he F can sometimes be critical or local design in shallow water. Pushover analysis is commonly applied to the U; e.g. as implemented in re.[] & []. he passage o a regular wave past the jacket is discretised into a number o time steps with associated hydrodynamic (and wind) loads. he instant giving the highest load is selected. his environmental load is gradually applied in the pushover analysis and urther incremented by a load actor, while the displacement is computed to describe a series o equilibrium states. Plastic behaviour o the structural elements o the jacket is taken into account. Nonlinear models or the behaviour o the oundation piles are included. he maximum capacity is ound at the peak o the load and displacement curve, as indicated in Figure. hus, the capacity may be expressed in terms o a capacity actor λ A, which is equal to the value o the load actor at this point. I the capacity actor equals., then the speciied load lies exactly on the limit state surace. he details o the ailure mode may be ound rom the underlying pushover analysis, and include plastic collapse o struts or legs, lateral soil ailure, axial pile ailure, and combinations o these modes, etc. oad actor Pushover analysis 3 Displacement Figure oad displacement curve rom pushover analysis in a regular wave. he maximum is indicated as a capacity actor. Overturning ae zone Failed zone Base shear orce Figure Interaction sketch or jacket capacity, where cross-hatched zone indicates range o probable limit states in shallow water In shallow water the load and capacity is airly well characterised in terms o the base shear orce acting on the jacket. I only base shear is considered, then a limit state unction might be deined as: opyright 4 by AME

3 (capacity or base shear orce) - (applied base shear load) O course, the environmental load is really distributed over the jacket structure and is more ully characterised in terms o the base shear orce and an overturning moment, as indicated in Figure. he limit between sae and ailed zones may be thought o as the locus o all combinations o base shear and overturning moment with a capacity actor o λ A.. Hence, a more precise limit state unction may be ormulated as: g λ () A here are iner details in the eects o the load and capacity distributions over the jacket on the U which are not brought out in Figure, but they are included in Eq.(), when the capacity actor is based on a detailed pushover analysis. he capacity actor in eq.() is a unction o all the physical random variables deining the structural capacity and applied loads in the pushover analysis. I the details o the load distribution over the structure are o less importance, then the load and capacity can be separated to some extent. In this case, the load capacity actor λ B may be normalised relative to a characteristic base shear orce l and expressed as a unction o height o the line o action o the orce ς and the compass direction o the orce θ. hen the limit state unction may be written in terms o the base shear orce as g λ ( ς, θ ) l l () B where l is the applied base shear orce. he base shear orce l, the orce height ς and the orce direction direction θ are all dependent on the environmental conditions, including the directions o wind, waves and current. he characteristic orce l is not varied with direction. Dependency on the height o the orce can be omitted in some cases; e.g. or purely lateral ailure o the oundation. Model uncertainty actors or capacity u and load can conveniently be included in the limit state unction as g u λ ( ς, θ ) l u l (3) B he loads and capacity are dependent on a large number o physical parameters, many o which may need to be modelled as random variables. Details o one case study are given in [3], where response surace methods are applied to handle these dependencies in the reliability analysis. ome aspects o the dependencies on load direction and on wave period are discussed in the ollowing sections DIEIONA EFFE It tends to be computationally arduous to take ull account o directional eects in a reliability analysis. A simpliied model is developed in the ollowing, to permit investigation o u some aspects. Fourier polynomials with only a ew terms are used to model some typical eatures o jacket load and resistance here. his type o unction can be extended with additional terms, or increased accuracy, and used to interpolate between numerical results rom detailed jacket analyses at discrete headings. Directional model g Directional limit state he limit state unction is written as r( θ ) l( θ ) (4) where both resistance r and load l are dependent on direction θ, and the same direction is assumed or both environmental eects and the induced loads. ymmetry about a vertical plane in the longitudinal axis through the centre o the jacket is assumed. Directional resistance he resistance is expressed by ( r + r cos[ ( θ θ )]) r( θ ) r (5) where r is the characteristic resistance, r, r are directional resistance parameters and θ indicates the direction with the greatest resistance; i.e. along the longitudinal axis through the centre o the jacket. Qr esistance coeicient Angle (deg) Figure 3 Variation in resistance coeicient with direction. his Fourier polynomial provides symmetry o resistance about longitudinal and transverse axes, a lower resistance in the transverse directions than in the longitudinal directions, and smooth variation through intermediate directions. hese properties are typical o jacket resistance when length and breadth are unequal. An eight-legged jacket in 8 m water 3 opyright 4 by AME

4 r depth might typically have r.5.85, as illustrated in Figure 3, based on calculations o lateral oundation ailure similar to those reported in re.[3]. ubsequent parameter variation is carried out such that r + r.. Directional load he load coeicient is expressed by ( l + l cos[ ( θ θ )]) Q ( θ ) (6) where l, l are directional load parameters and θ π / indicates the direction with the highest load coeicient; i.e. along the transverse axis through the centre o the jacket. imilar reasoning is applied in the choice o the Fourier polynomial or the load coeicient, as is used or the resistance, but the peak load coeicient is ound in the transverse, rather than the longitudinal direction. When the breadth is smaller than the length o the jacket, then the loads on the various elements tend to be more nearly in phase when waves are propagating in the transverse direction. O course, this is also dependent on the ratio o the wavelength to the platorm dimensions. A typical jacket, as mentioned above, l might have Q_ l..9, as illustrated in Figure 4. oad coeicient Angle (deg) Figure 4 Variation in load coeicient with direction. he mean and standard deviation o the load in a short term environmental state are expressed by lm k Q (θ ) (7) l k Q (θ ) (8) where k is an environmental intensity actor, that is also dependent on direction. he load maxima are assumed to be distributed according to the ayleigh distribution in a short term environmental state, with a mean period o 8s between maxima. he maxima are assumed independent and the distribution o short term extreme largest in a 3-hour state * ( θ ;3hr) is obtained rom a Gumbel distribution, via a probability transormation rom ayleigh to an auxillary exponential distribution. his random variable is applied as the load in the limit state equation (4). his type o distribution model is commonly applied to wave loads. Directional environment he long term probability density o environmental directions is written as ( + a cos( θ θ ))/( ) ( θ ) π (9) E where the st order coeicient is set to a. 333 in the present example and the dominant direction is speciied by θ E.. his distribution is illustrated in Figure 5. he present example is chosen such that the dominant direction leads to the highest environmental intensity, and this direction is aligned with the platorm direction that tends to maximise the resistance and minimise the load coeicient. uch situations are not uncommon, and may be ound in the outh hina ea, or example, where the north-east monsoon is dominant or part o the year and produces the highest waves. However, the present model is not speciically itted to this location. Note that the density is deined on a range ( π, π ) and is zero elsewhere. he length o this range is ixed, but the location is arbitrary. Hence, the mean value o the direction is also arbitrary. his is somewhat unusual, as compared to other types o random variables, and needs some extra consideration when applying asymptotic reliability methods. (Ed) (/rad) Prob.density o envn. direction Angle (deg) Figure 5 Probability density unction or environmental direction (scaled w.r.t. radians, but with angle shown in degrees). In the reliability analysis, the direction distribution can be obtained by a probability transormation rom an auxillary variable with a uniorm distribution. his was originally set up with the uniorm variable deined on the range (-,+). he 4 opyright 4 by AME

5 present symmetrical case is then conveniently handled by redeining the uniorm variable on the range (,), without modiying the original probability transormation. he short term environmental intensity k takes a long term Weibull distribution, written as k F ( k; θ ) exp α ( θ ) where the distribution parameters are β. and ( k + k cos( θ θ )) β () α ( θ ) () with coeicients k.7, k. 3, and most severe direction θ.. he parameter o the environmental intensity distribution is illustrated in Figure 6. alpha_k Alpha param. or envn. intensity Angle (deg) Figure 6 Variation in distribution parameter or environmental intensity with direction. he simpliied environmental loads are intended to represent the eects o wind, waves and current on a typical 8- legged jacket, with only one plane o symmetry. he distribution o environmental direction and intensity resembles the directional distributions ound or these eects, and the distributions commonly applied or signiicant wave height or wind speed. Probabilistic model No time-independent random variables are included in this analysis, in order to allow more scope or computational investigation o directional eects; i.e. there is no model uncertainty on load or resistance and although the resistance is dependent on direction, it is otherwise deterministic. he marginal probability o ailure in a single, random short term state is obtained rom the limit state equation (4) and integration with respect to the distributions o environmental direction, environmental intensity and short term extreme load. P (3hr) Θ g< *, Θ ( θ ) Θ ( k θ ) ( l k, θ;3hr) dθ dk dl () he annual probability o ailure is subsequently obtained by taking account o the number o short term states in a year N, as yr N [ P (3hr ] yr P ( yr) ) (3) he POBAN program is used or the reliability calculations [4]. Directional results he model is scaled to a typical probability level by inserting a characteristic resistance o r 69. to be multiplied by the resistance coeicient in (5). esults rom a parameter study on the mean resistance parameter are shown in Figure 7. Four types o reliability methods are applied in the calculation. he second order reliability method (OM) and directional simulation agree closely, while the irst order reliability method overestimates the probability o ailure by a actor o rom to 5. hese directional simulation results are unbiased and have been computed with a coeicient o variation o 6% to 7%. One point has been checked by Monte arlo simulation o 8 sample points, yielding a coeicient o variation o %, and showing good agreement with directional simulation and OM. he results seem airly typical o directional problems, which tend to show suicient curvature in the limit state unction (in the transormed standard normal space) such that the FOM results are inaccurate. P.E-.E-.E-3.E-4.E Dir.sim. FOM OM M.. Figure 7 Annual probability o ailure as a unction o mean resistance parameter or 4 types o calculation. imilar problems with FOM have previously been observed by the authors in another reliability problem with strong directional content; viz. the ultimate limit state or a single mooring line in a spread mooring. A simple explanation o the 5 opyright 4 by AME

6 dierent perormance o FOM with respect to directional random variables as opposed to other types o random variables may perhaps be provided by remembering that FOM provides a linear approximation to the ailure surace, such that: more eective approximation is provided to eects that behave monotonically; e.g. loads tend to increase with random wave height and lateral resistance tends to increase with soil strength, whereas less eective approximation is provided to eects that do not behave monotonically; e.g. the load coeicient tends to decrease on both sides o the peak angles in Figure 4 and the resistance tends to increase on both sides o the troughs in Figure 3. OM may be expected to be more eective, since it is based on a second order approximation to the ailure surace. WAVE PEIOD onguet-higgins [5] provides an expression or the short term distribution o the period o an individual wave, conditioned on the wave height. his distribution may also be ound in Massel s textbook [6]. It is more amenable or use in reliability analysis than the distributions given by avanie et al. [7] and by indgren and ychlik [8]. etaining most o the notation rom the paper [5], this conditional probability density may be written as where ( t r) r exp ν t r πνf( r ν ) t (4) t τ τ is a normalised wave period, with τ as the r ρ m wave period and τ as the mean wave period. is a normalised wave amplitude, with ρ as the wave amplitude and m as the zero order moment o the wave spectrum. he wave amplitude is taken to be hal the wave height. mm ν is a spectral width parameter, with m m and m as the irst and second moments o the wave spectrum. Now the unction F ( r ν ) is a well-known error unction, that may be neglected or our purposes, in relatively large waves, when r >. his distribution is applicable to the period o the individual wave applied to load a jacket platorm, when the wave height is drawn rom the annual extreme wave period. When implementing this distribution in a reliability analysis, it is convenient to employ a transormation to an auxiliary variable u, with a standard normal distribution unction, as deined by φ ( u ) du ( t r) dt (5) u r ν t where φ (.) is the normal density unction. he auxiliary variable is modelled as an input variable in the reliability analysis, and a realisation o the individual wave period can be obtained as ollows, in terms o more convenient parameters Z ν + τ uνh h (6) where h is the individual wave height, H is the associated signiicant wave height, and Z is the zero-up-crossing period. here is a possibility that the transormation may lead to negative wave periods or large values o u. his possibility is related to the correction made by the error unction in the original distribution. he transormation is not allowed at the transition between positive and negative values, at u h. ome bound needs to be set to avoid these νh values in the reliability computation, because they may give rise to numerical diiculties. Another bound should be set to avoid wave height and period combinations that imply breaking waves. Both bounds can easily be implemented using a truncated normal distribution or the auxiliary variable u. An example o the conditional wave period distribution is shown in Figure 8. he distribution tends to become more narrow and centered on the mean period as the individual wave height increases. he upper bound on u has no real inluence in this example, but the bound or breaking waves obviously has appreciable inluence. he example applies to a severe sea state, and to an extreme individual wave height in that state ( h H.). he bound or breaking waves would have less eect in milder conditions. his wave period distribution is primarily based on mathematical theory or narrow-banded processes, without making direct use o the physical properties o ocean waves. he bound or breaking waves takes some o these properties into account, but it also seems that there may be urther scope or improvement o the distribution model or individual wave periods under extreme conditions. 6 opyright 4 by AME

7 Probability density Hs 6 H z Wave period (s) Figure 8 onditional distribution o periods o individual waves or h m, H s 6m, Z 8s, ν. 3. A deepwater bound or breaking waves is indicated at 7.3 s. Prior to introducing this conditional wave period distribution it might typically be assumed that the period corresponding to the extreme individual waves would be close to the peak period o the wave spectrum. he conditional distribution tends to lead to a shorter wave period, closer to the average period o the spectrum. his change in wave period was ound to lead to an appreciable increase in wave loads in the reliability analysis o a jacket platorm. GEOEHNIA IUE he axial capacities o the piles in a jacket oundation provide the resistance against the overturning moment on the jacket, while the lateral capacities provide the resistance against the base shear. For evaluation o the overall stability o a jacket oundation under extreme loading conditions, the spatially averaged soil strength properties over the extent o the oundation are o interest. oil strength properties exhibit spatial connectivity vertically as well as laterally, i.e. there is correlation between the soil strengths rom one point to another within the soil volume. he horizontal correlation length o the soil strength ield is usually much larger than the vertical correlation length. Axial pile capacity he axial capacities o the piles in the oundation come about as the skin riction integrated over the respective airly long pile lengths. For the axial capacities, it can thereore be assumed that the eects o the local luctuations o the skin riction rom point to point along each pile will average out over the length o the pile, and the axial capacities o all piles can thus be represented by capacities calculated rom average skin riction properties only, without considering any local variability ateral pile capacity he lateral capacities o the piles come about rom a much more localized soil strength, i.e. it arises rom the soil strength in a limited zone near the soil surace. he vertical extent o this zone is so limited that or practical purposes it will not be reasonable to count on any eect o spatial averaging vertically. his leaves to consider spatial averaging horizontally or its inluence on the lateral capacities o the piles in the oundation. he lateral pile capacity is proportional to the undrained shear strength o the soil. he spatial average o the lateral pile capacities over the lateral extent o the oundation will thereore come about in the same manner as the spatial average o the undrained shear strength over this extent. In the structural reliability analysis, it will thereore suice to represent the lateral capacity o each pile as the lateral capacity that comes about rom a calculation on the basis o a spatially averaged undrained shear strength. In a reliability analysis, the spatially averaged undrained shear strength can be expressed as [ ] + U λ σ u, spatial E u (7) in which E[ u ] and σ denote the mean value and the standard deviation, respectively, o the local undrained shear strength. U denotes a standard normally distributed variable, and λ is a variance reduction actor associated with the spatial averaging in the horizontal plane and whose value is less than.. his representation is based on an assumption o a Gaussian strength ield. For a jacket oundation o N piles, the value o λ can be established rom Monte-arlo simulations o joint outcomes o an N-dimensional standard normal variable X whose correlation matrix is an NxN matrix with entries calculated according to an expression or the correlation coeicient ρ( r) between two piles located a horizontal distance r apart. he ith element X i in X represents the local variability in the strength at the ith pile among the N piles in the oundation. For the entry in the ith row and jth column o the correlation matrix, r comes about as the distance between the ith pile and the jth pile in the oundation. When a joint outcome o X(X,...X N ) is simulated, this gives one outcome o the derived variable (8) Y N N X i i he distribution o Y can be established rom an adequate number o Monte-arlo simulations o X (and thus o Y). he variance reduction actor λ can be interpreted as the variance o Y that results rom the simulations and will be a unction o the model or the horizontal correlation structure or the soil strength and o the horizontal correlation length in this model. he commonly assumed quadratic exponential decay model, 7 opyright 4 by AME

8 r ρ ( r) exp( ( ) ) (9) in which the correlation length represents the horizontal scale o luctuation, will produce a variance reduction actor λ as indicated in Figure 9 or an example oundation. λ,8,6,4, (m ) Figure 9 Example o standard deviation reduction actor vs. horizontal correlation length or an 8-legged jacket. Note that the emphasis in the above considerations has been placed on the total lateral capacity o the entire pile oundation, as there is no reason to expect that a variation in lateral capacity between the individual piles will lead to ailure o one pile and thereby initiate a progressive ailure scenario. his is so, because i one (weak) pile would tend to ail, then part o the load on this pile would immediately become redistributed through the jacket structure to the other (stronger) piles. he ailure mode or the jacket oundation in lateral loading is thus a global ailure mode, governed by the soilstructure interaction and the total lateral capacity o the piles. Model uncertainty Another geotechnical issue o great importance or a reliability analysis o a jacket oundation is the model uncertainty associated with prediction o the axial pile capacities. In a reliability analysis, the model uncertainty can be represented by a random model uncertainty actor F. he actor is deined as the ratio between the true axial pile capacity and the predicted axial pile capacity and is applied as a actor on the capacity as predicted by the chosen capacity model in the limit state unction. he distribution o the stochastic model uncertainty actor F can be assessed based on data rom ull-scale tests on piles, rom which the observed pile capacity, measured as the axial pile load at ailure, can be interpreted as a measure o the true pile capacity. For establishing the distribution o F, data bases o results rom ull-scale tests on piles need to be consulted. everal such data bases exist, however, they cover a wide range o soil types and in many cases a very limited number o tests are available rom each location. his puts a limit on which data can be o use or the piles o a particular jacket oundation. A mean value o F dierent rom. indicates a biased prediction model. A data base, which is commonly reerred to, is the so-called API data base presented in re.[9]. ONUION ome eatures o reliability analysis o jackets have been discussed and may be summarised as ollows: Detailed reliability analysis with respect to random directions o environmental eects is demanding, and can useully be explored with simpliied models. FOM analysis tends to provide inaccurate results or the eects o random directions, while OM appears to be reasonably accurate, when the conditional probability o ailure with respect to direction has a single, dominant peak. he mean direction is arbitrary and should be chosen such that the dominant peak is kept well away rom the ends o the direction range, when OM (or FOM) is applied. he individual wave period associated with an extreme wave height may be modelled using a distribution unction due to onguet-higgins [5]. his distribution appears to need a bound against breaking waves. horter wave periods than the spectral peak period tend to be associated with the highest waves when using this distribution. Average skin riction properties may be applied to establish the axial capacity o piles, without considering local variability. patial averaging o the undrained shear strength o the soil is required to establish the lateral capacity o a pile oundation. ANOWEDGMEN he authors grateully acknowledge permission to publish this paper rom Det Norske Veritas. Any opinions stated here are the responsibility o the authors and do not necessarily relect the views o this company. EFEENE [] kallerud, B., Amdahl, J., (), Nonlinear analysis o Oshore tructures, esearch tudies Press td., Baldock, England. [] UFO (996); UFO - A omputer Program or Progressive ollapse Analysis o teel Oshore tructures, INEF eport no. F7 F8839, Dated [3] igurdsson, G., Mathisen, J., trøm, P., Goh,.., (3), eliability eassessment o a Jacket Platorm with Gas eepage in the outh hina ea, nd Int. on. Oshore Mechanics and Arctic Engng., paper no.omae3-3747, ancun, Mexico. 8 opyright 4 by AME

9 [4] esam, (996), User s manual, POBAN, General Purpose Probabilistic Analysis Program, DNV otware report no.9-749, rev.no., Høvik. [5] onguet-higgins, M.., (983), On the joint distribution o wave periods and amplitudes in a random wave ield, Proc.. oc. ond., A 389, pp [6] Massel,.., (996), Ocean urace Waves: heir Physics and Prediction, Advanced series on Ocean Engineering,World cientiic, ingapore. [7] avanie, A., Arhan, M., Ezraty,., (976), A statistical relationship between individual heights and periods o storm waves, Proc. ymp. Behaviour o Oshore tructures, pp [8] indgren, G., ychlik, I., (98), Wave characteristic distribution or Gaussian waves wavelength, amplitude and steepness, Ocean Engineering 9, pp [9] Olson,.E., Winter, D.G., Final eport. Project: eview and ompilation o Pile est esults. Axial Pile apacity. ponsored by he American Petroleum Institute, Austin, exas, March opyright 4 by AME