Seismic analysis of tall anchored sheet-pile walls

Soil Dynamics and Earthquake Engineering xx (xxxx) xxxx–xxxx

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Seismic analysis of tall anchored sheet-pile walls ⁎

G. Gazetas , E. Garini, A. Zafeirakos National Technical University, Athens, Greece

A R T I C L E I N F O

A BS T RAC T

Keywords: Retaining wall Seismic analysis Anchor Sheetpile Hydrodynamics Mononobe-Okabe

The state of practice in designing of anchored steel sheet-pile (SSP) walls for strong earthquake shaking in nonliquefiable ground is reviewed, prior to investigating the performance of a typical quay-wall with a free height of 18 m, embedded into relatively dense sandy “foundation” soil. Supporting moderately dense silty sand, the SSP wall is subjected to seismic motions of various intensities with respect to the PGA at the rock-outcrop level, namely 0.15 g, 0.30 g, and 0.50 g. The long-established simplified design methods of (i) pseudo-static limit equilibrium (pLEM) and (ii) beam-on-Winkler-foundation (BWF), in conjunction with the Mononobe-Okabe (MO) method, are shown to lead to results for bending moments that in some cases are larger than, but in other cases quite similar to, those computed with two commercially available finite element (FE) codes, ABAQUS and PLAXIS. The pLEM neglects the effect of the anchor on the distribution of earth pressures on the wall, while at the same time the predetermined MO actions and reactions do not take advantage of the arching in the backfill due to the wall flexure. In contrast, the numerical FE analyses can capture well the physical phenomena of this complex interaction problem leading to reliable results, although the computed deflection of the wall is quite sensitive to the soil constitutive model. The hydrodynamic effects on the seaward side of the wall are shown to have a small effect on the wall performance, contrary to the pseudo-statically applied Westergaard pressures which exaggerate the effect.

1. Introduction: past seismic performance of anchored sheetpile walls

performance of anchored bulkheads (SSP walls), subjected to strong earthquake shaking:

Harbor facilities have often suffered damage in strong earthquakes, causing among other problems disruptions of post-earthquake emergency operations with serious economic consequences for the affected regions. Concrete caissons, currently the most frequent type of quay walls, have had a rather poor performance not only in the devastating Kobe 1995 earthquake, but also in milder seismic events (such as the Lefkada 2003 and Cephalonia 2014 earthquakes in Greece — see [15,18]. This, despite the fact that they were conservatively designed and constructed. Anchored steel sheet-pile walls, schematically depicted in Fig. 1, are also used as retaining structures in wharves and quays, owing mainly to their easy installation and because the soft or loose soils that usually underline such waterfront structures may not support the additional weight of gravity concrete walls without massive soil replacement or improvement [32,33]. Thus, in many cases SSP walls are cheaper than gravity walls on piles. In addition, they do not require dry-docks for their construction, as are caisson type walls. Consequently, numerous quay-walls have been constructed with this method and expectedly, many of the reported seismic failures, refer to these types of quay-walls (e.g. [21,12,2]). The following conclusions emerge from a study of the

1. Most of the observed earthquake failures resulted from large-scale liquefaction of loose cohesionless soils mainly in the backfill/ retained soil, but sometimes in the supporting foundation soil as well. Such soils are common in port and harbor facility sites. 2. Another frequent but not as dramatic type of SSP wall damage, takes the form of excessive permanent seaward “bulging” and tilting of the sheet-pile wall, accompanied by excessive movement of the anchor wall relatively to the surrounding soil; such an anchor-wall movement manifests itself in the form of ground settlement and cracking of the concrete apron directly behind it. Apparently, such failures are due either to inadequate passive anchor-wall resistance, or to insufficient strength of the main sheet-pile wall, or to both. 3. Development of detrimental excess pore-water pressures in the backfill next to the wall, once thought to be a contributor to large deformations and failure, is now recognized as rather unlikely to occur when substantial seaward bulging takes place [10,31,9].

⁎

These observations raise questions as to whether properly designed SSP walls are capable to withstand moderate and strong levels of seismic shaking; and whether their design can be based on conven-

Corresponding author.

http://dx.doi.org/10.1016/j.soildyn.2016.09.031 Received 2 March 2016; Received in revised form 13 September 2016; Accepted 23 September 2016 Available online xxxx 0267-7261/ © 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: Gazetas, G., Soil Dynamics and Earthquake Engineering (2016), http://dx.doi.org/10.1016/j.soildyn.2016.09.031

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TieRod

SSP Wall

Anchor Wall FILL

4m

h H = 18 m

SOIL 1

12 m

(a) D (variable)

SOIL 2

L (variable)

(b)

Fig. 1. (a) Definition of anchored Steel Sheet-Pile (SSP) wall geometry, and (b) detail of the sheetpile cross-section utilised in this study.

tional limit – equilibrium analyses. This paper attempts to answer or at least shed some light on these questions.

(θΑΕ = θΡΕ = 0), when the effective acceleration becomes equal to:

α = AC / g = tan φ ≈ 0.58

(1)

2. State-of-practice seismic design procedures Beyond this value of acceleration MO solution does not exist. Yet, walls have survived larger acceleration levels (e.g., in the 1995 Kobe earthquake). More interestingly, for a more realistic strong excitation, e.g. acceleration coefficient α = 0.40, θΑΕ ≈32° and θΡΕ ≈20°, and an anchor-wall of a length h=H/2, where H=the free height of the wall (Fig. 4), the required length of the anchor (or tie-rod) would be

The difficulty of providing a comprehensive rigorous analytical method arises from several factors: the complicated wave diffraction pattern due to “ground-step” geometry; the presence of two different but interconnected structural elements in contact with the soil; the inevitably nonlinear hysteretic behavior of soil in strong shaking, often including pore-pressure buildup and degradation, both in front and behind the sheet-pile; the no-tension behavior of the soil-SSP interface; the presence of radiation damping effects due to stress waves propagating away from the wall in the backfill and in the foundation; and the hydrodynamic action on the SSP wall on its seaward side. Before the advent of reliable and relatively user-friendly FE and FD codes, which could properly handle all these phenomena, pseudo-static simplified procedures were and are still used in practice. Attempts to provide refinement and sophistication to such methods continue to this day. Specifically:

Lanch = Hd / tan θΑΕ + h / tan θPE ≫ 3.2H

(2)

For an Hd =H +Df ≈28 m, where Df is defined in Fig. 4, this leads to a substantial length of Lanch =65 m. It must be noted that Df increases with the level of excitation; the Df ≈10 m assumed here is appropriate for the 0.30g peak acceleration (as will be defined in the next sections), but with larger excitation Df tends to 18 m! For smaller levels of acceleration, a semi-empirical variant of the MO method had been developed by [14]. Utilizing numerous case history data, referring to successful, moderately damaged, severely damaged, and collapsed SSP walls, interpreted using the MO method, a Seismic Design Chart was constructed, as shown in Fig. 5. The Chart’s axes are: •The effective anchor index EAI=d/H, where d is the horizontal distance of the anchor–tie point from the MO active failure surface, assumed to originate at the effective point of rotation of the wall; H is the free height of the wall. •The embedment participation index EPI =(FPE/FAE) {1+Df /(Df +H)} , where Df is the depth from the mud-line to the point of rotation, often estimated as a fraction of the depth of embedment D (e.g. D/2 in the above example). These two variables have been related to the degree of recorded damage and the chart was calibrated using case histories and analyses reported in [26]. In this approach, three types of expected damage are

(1) Pseudo-static Limit Equilibrium methods [pLEM] determine dynamic lateral earth pressures using the Mononobe-Okabe (MO) analysis [26,28] (Fig. 2). Differences among the several variants of the method arise primarily from the assumed point of application of the resultant active and passive forces FAE and FPE (on the two sides of the SSP wall), the way water is taken into account, and the partial factors of safety introduced in the design. Among other problems, MO method produces seismic earth pressure active and passive coefficients, KAE and KPE, that are sensitive to (large values of) the effective (“driving”) acceleration ── not verified with rigorous FE analyses and with field observations in many earthquakes. As an example, for a φ=30° sand Fig. 3 shows that active and passive coefficients equalize, achieving a common value KAE=KPE ≈1.35, while the critical angles vanish 2

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γeff z (KPE – KAE) γ eff KAE 1

Mononobe -Okabe Passive Pressures Mononobe -Okabe

γeff KPE 1

Active Pressures

Fig. 2. Elements of the state-of-practice pseudo-Static Limit Equilibrium method (pLEM).

have been a favorite method in engineering practice. Because of the reliance of these methods on the MO active and passive earth pressures they may lead to conservative results.

distinguished: small or no deformation (zone I) with seaward top displacement of 0–30 cm; moderate deformation involving seaward top displacement of 30–80 cm (Zone II); and, finally, excessive displacement ( > 80 cm up to collapse) (Zone III). This chart, however, is good only for a crude preliminary evaluation of SSP walls of small to medium height (H≤10 m), in relatively dense non-liquefiable soil. The chart does not discriminate with reference to the nature of the ground shaking (frequency content, number of significant cycles, etc.), does not reflect the layering of the retained and the foundation soil, and, perhaps more significantly, does not account for the absolute value of the height H. (2) Beam-on-Winkler-Foundation [BWF] models treat the SSP wall just as a pile foundation, with suitable one-sided plane-strain linear Winkler springs (or non-linear p–y reaction “springs”) on both the active and passive sides of the wall, with an elasto-plastic support for the anchor (Fig. 6). Two 1-D shear beams are attached to the ends of the springs and transmit the seismic motion to the system. The ensuing kinematic response could reproduce the flexural response of the wall with reasonable accuracy, but only for very small levels of excitation. When the acceleration level is high enough to induce wedge-type failure mechanisms and the anchor is activated passively, the results are not necessarily reliable. To alleviate this shortcoming the Winkler springs become elastoplastic, the ultimate resistance of which is calibrated with the active and passive failure mechanisms. (3) Hybrid procedures, combining the BWF with the pLEM methods

Overall, the simplified methods may produce unrealistically exaggerated bending moments for the wall and required distance of the anchoring point, as it will be demonstrated below.

3. The Mononobe-Okabe method, soil amplification, the effect of submergence One of the key problems with all limit equilibrium methods is the meaning of the (constant throughout the backfill) “effective driving acceleration”, αh , or as usually symbolized: kh. For a short rigid gravity wall on stiff soil, the code specified ground acceleration, A, for a particular region and site may be a reasonable approximation for the single value of acceleration. However, for tall walls and, especially if underlain by deep and soft soil, amplification of the incoming waves as they propagate in the soil will make the distribution of acceleration non-uniform (in the vertical and horizontal direction) and the peak acceleration values at the surface quite different (probably larger) than the acceleration at the base. [25] have shown than even an 8 m–tall gravity wall would experience an amplified ground motion at (lowperiod) resonance. With taller, flexible, and founded on deformable soil retaining systems, the phenomenon may become significant. This issue is further addressed in this paper.

Fig. 3. Effect of acceleration coefficient on the angle of the active, θΑΕ, and passive, θΡΕ , sliding surfaces (left), and on the active and passive seismic earth-pressure coefficients (right). [adopted from [11]].

3

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Fig. 4. Conventionally-prudent location of the anchor wall, explaining implicitly the increase of LANCHOR with increasing driving acceleration. Df depends on the stiffness of the soils and the intensity of seismic shaking.

Fig. 6. Sketch of the Beam-on-Winkler-Foundation model for the kinematic response of the “Anchored Sheet-Pile Wall”.

ψ = arctan (αh′)

(4)

The value of α΄h includes also the effect of submergence in water [13,22]. For dry retained soil αh′ = αh , and the unit weight γ in Eq. (3a) is the dry unit weight γdry. For a fully submerged backfill

ah′ = ah Fig. 5. The modified empirical seismic Design Chart, originally proposed by [14] and modified utilizing observations by Iai (PIANC), for wall heights H < 10 m. Data of Iai shown with asterisk.

KAE =

1 2 γH KAE 2

(5)

and the submerged unit weight γb=γsat –γw must replace γ in Eq. (3a) to compute FAE. The underlying assumptions for the above treatment of a submerged backfill are that: (a) pore water pressures do not change as a result of horizontal motion; and (b) that backfill permeability is low enough for the water to move as a unit with the mineral skeleton. According to the [26] manual the threshold permeability for the latter condition is of the order of k≈10 −4 m/s. For a partially submerged backfill, as in this case (the water table is 4 m below the top of the retained soil), weighted thrusts based on the volume of soil in the failure wedge below and above the water surface result in the following expressions for the apparent seismic coefficient of an equivalent “homogeneous” soil [26].

The M-O expression for the total (static plus dynamic) active earthpressure force is:

FAE =

γsat γb

(3a)

cos2 (φ − ψ ) 2 ⎧ 1 ⎫ cos ψ ⋅ cos(ψ + δ ) ⎨1 + [sin(φ + δ )⋅ sin(φ − ψ )/ cos(δ + ψ )] 2 ⎬ ⎩ ⎭ (3b)

In which φ is the angle of shearing resistance of the retained soil, and δ is the angle of friction along the vertical SSP wall-soil interface. The angle ψ is a function of the apparent seismic coefficient α′h :

ah′ = ah

4

2 γsat Hw2 + γHsur + 2γHw Hsur + qH /2 2 γb Hw2 + γHsur + 2γHw Hsur + qH /2

(6)

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Fig. 7. Finite element mesh of the examined sheet-pile wall system and details of several key features: (a) with ABAQUS, and (b) PLAXIS finite element codes.

4. Comparative study

Table 1 Soil parameters.

Fill Soil 1 Soil 2

c (kPa)

φ (degrees)

E (MPa)

1 8 10

32.5 35 37.5

100 200 300

4.1. Problem definition The 32 m long SSP wall (18 m free-height and 14 m embedded), shown in Fig. 7, is analyzed dynamically with two different commercially available FE codes. The main SSP wall is embedded in a dense sandy layer, while the backfill soil is comprised of a medium-dense (but not liquefiable) silty sand, overlain by a cohesionless fill. The strength and stiffness parameters of the three layers are given in Table 1 and are typical of those encountered in harbors. The wall cross-section has the form of Fig. 1(b) with parametrically varying dimensions and thickness. The rigidity ranges from EI ≈1×106 to 2.3x106 kNm2/m the ultimate moment capacity from Mult≈4000 kNm/m to 9000 kNm/m, and the distance of the anchor wall (length of the tie-rod), Lanch=45– 55 m from the main SSP wall.

And the effective unit weight of soil,

⎡ ⎛ H ⎞2 ⎤ ⎛ H ⎞2 γ′ = γb ⎜ w ⎟ + γ ⎢1 − ⎜ w ⎟ ⎥ ⎢⎣ ⎝ H ⎠ ⎥⎦ ⎝H⎠

(7)

in which H=Hw+Hsur is the total height of the wall; Hw is the height below the water surface, where the buoyant unit weight γb controls; and Hsur is the height above the water surface, where the soil is not in buoyancy and has a unit weight γ. Eqs. (6) and (7) are used in Eqs. (3) and (4) to determine the effective static-plus-dynamic earth thrust. Being an extension of the Coulomb method, the MO method does not compute distribution of earth pressures. Only by invoking the Rankine solution is a triangular shape established. We follow the idea by distributing the computed as above total FAE linearly with depth.

4.2. The seismic excitation To draw conclusions on the capability of SSP walls to withstand ground shaking in regions of moderate, high, and very high seismicity, three sets of acceleration time histories were developed. The effective ground accelerations were 0.15g, 0.30g and 0.50g, respectively. The 5

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Fig. 8. (a) Design “rock outcrop” target response spectra; (b) Comparison of the “rock outcrop” target and fitted spectra, with the computed “base” motion spectra; and (c) examples of “base” motion acceleration time histories.

Fig. 7b compares the three design target spectra, the corresponding spectra of three fitted “rock-outcrop” motions, and the spectra of the resulting “base” motions (accounting for a “rock” shear wave velocity of 800 m/s). Three acceleration records were used in Fig. 7c, namely Cogswell, Aegion and Jensen. Note that the match with target spectra was not perfect—this was done deliberately, so as to preserve as much as possible the natural features of the records. 4.3. Methods of analysis Two FE computer codes were used in the analyses, ABAQUS [1] and PLAXIS. Only 2-D analyses are performed. It is worth mentioning the work of Bielak and Cristiano [8] as one of the first finite-element analysis of this type. In all our FE analyses the models were symmetric, as already shown in Fig. 8, in order to (a) ameliorate the lateral boundary effects, and (b) to examine the effect of the inherent asymmetry of the accelerogram (“polarity” effect) in a single dynamic analysis (e.g. [17]). Thus, effectively, each analysis produces two results. More specifically:

Fig. 9. Definition of the “effective” apparent coefficient of friction, μ, as a function of the steel-soil interface friction coefficient, and the geometric details of the sheet-pile cross section.

design response spectra of EC8 for Soil Class B (“rock” of Vs > 800 m/s) are adopted as the free-field rock-outcrop spectra (Fig. 7a). Actual recorded accelerograms were modulated so that their response spectra matched the appropriate target spectrum, and then were applied as “rock-outcrop” motions—not as motions of the base.

– PLAXIS: The FE mesh consisted of triangular 6-node elements Fig. 8b. The maximum finite element size was at least 10 times smaller than the minimum wavelength of significance, thus avoiding

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Fig. 10. Typical result of FE analysis for motion type I: Bending moments and lateral deflection of the SSP wall with and without consideration of hydrodynamic effects.

Fig. 11. Free-field spectral amplification for: (a) the backfill and (b) the foundation soil, for Type I excitation. Comparison of results from two state-of-practice analyses.

spurious wave filtering effects. The adhesion between the soil and the SSP wall was taken into account by adding positive and negative interface elements between the wall and the soil. Interface strength value of Rinter =0.67 is considered. To avoid spurious oscillations at very small deformations (nearly elastic response), mass and stiffness proportional Rayleigh damping was also introduced into the model, accounting for an equivalent hysteretic damping between the eigenfrequency of the soil deposit and the dominant frequency of the earthquake ground motion, equal to 2%. The seismic loading was applied at the base of the soil profile. For the lateral boundaries of the model, kinematic constraints are imposed to the edges of the model (tied degrees of freedom), allowing it to move as the free field in the direction of the seismic motion (This condition is representative of the boundaries imposed in the laminar boxes, which are used in shaking table and dynamic centrifuge experiments.) For the undrained effective stress analysis, the Hardening-Soil Small [HSS] model is used [27]. This model is able to treat smallstrain stiffness nonlinearity of the soil, which is essential in the accurate simulation of seismic problems [6,7]. Obviously, any threat

of liquefaction or substantial excess pore-water pressure has to be excluded. – ABAQUS: The FE domain shown in Fig. 8a is discretized using quadrilateral solid plane-strain fine elements 0.5×0.5 m2, capable of transmitting without bias the wave frequencies of significance. Interface between wall and soil was modeled as tension-less but frictional, using special elements that allow both separation and sliding, the latter controlled by coefficients of friction μ. For the soil materials, a model developed by Gerolymos and Gazetas (2005) [19] and Anastasopoulos et al., (2011) [3] was used. This is a nonlinear soil model with kinematic hardening, Von Mises failure criterion, and an associative flow rule. The evolution law consists of two components: a nonlinear kinematic hardening component describing the translation of the yield surface in stress space, and an isotropic hardening component which defines the size of the yield surface sο as a function of plastic deformation. The normal-stress dependence of the failure surface is controlled by sο. Details and validation of the model can be found in the afore-cited references. To capture radiation damping, normal and shear viscous dashpots

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Fig. 12. Comparison of acceleration time-histories at the top of the SP wall and of the base, along with their response spectral ratio.

Fig. 13. Deformed shape with superimposed contours of plastic deformations (PEMAG) for moderate intensity of excitation (Type I ). Deformation Scale Factor=5.

ρVS and ρVP (per unit area) were placed at the vertical boundaries between the soil domain and the vertical free-field columns, to model a transmitting boundary.

contact surfaces between the two different materials: soil and wall. In reality the interfaces are nor just two planes (one in front, one in the back) but rather a multitude of contact surfaces around the perimeter of the sheet-pile. Fig. 9 schematically illustrates the meaning of the effective coefficient of friction μ. Hence the “real” friction between wall and soil is much larger than what two single-planes offer. Therefore the correct friction coefficient of the assumed plane interface elements, μ, is larger than the steel–soil interface coefficient μsteel–soil. Especially

4.4. The “effective” coefficient of interface friction, μ In our 2–D modeling, the two vertical lines representing the wall– soil interface (in the back and front of the wall) are only fictitious 8

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Fig. 14. Comparison of the distributions of maximum bending moments and lateral deflections of the main SSP wall for Type I excitation from the two FE codes and the pseudostatic limit equilibrium method (pLEM).

Fig. 15. Distribution of earth pressures against the SSP wall computed with FE at the instance of maximum thrust in frond and back of the wall. Comparison is given with the active MO pressures.

for the particular SSP walls studied here, even if μsteel–soil≈0.30– 0.40, an “effective” coefficient of friction μ≈0.70 is appropriate. This value was adopted in all our analyses.

consideration of hydrodynamic action. The two methods gave almost identical results: the hydrodynamic action has a rather negligible effect, on the order of 5% for bending moments (for acceleration levels between 0.15–0.45g). The deformations, however, are usually slightly more sensitive. All the subsequent ABAQUS results include the hydrodynamic effect (not possible with PLAXIS at this time), as well as a rise in the water table within the backfill of 1 m above the sea water level (in prudent anticipation of future flooding).

4.5. Hydrodynamic effects Proper modeling of the dynamic action of water in front of the wall was explored in ABAQUS [1]. Two different approaches were explored: (a) attaching to the wall the “added masses” of the Westergaard theory,

m (z ) =

7 ρw Hz 8

5. Results

(8)

Among the many questions that our analyses try to answer, the most significant are summarized as follows:

where H=4 m is the water depth and z=the depth from the free water surface, and (b) modeling the water with acoustic elements. These special-purpose elements, with fluid pressure as the only stress component, are commonly used in the simulation of pressure and sound wave propagation. However, these can also be successfully used in problems where a volume of water is subjected to excitation (e.g., Mutto et al., 2012 [24] and Goldgruber et al., 2015 [20]). Fig. 10 compares distress (in terms of bending moments) and horizontal deflection of the SSP wall as obtained with and without

(a) Can properly-designed SSP walls withstand the action of strong seismic shaking? If not, what level of seismic acceleration can be undertaken safely and economically? (b) Is soil-amplification an important factor in the response of SSP walls? (c) What are the qualitative and quantitative differences between SSP walls and the gravity caisson quay walls, with respect to the overall 9

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Fig. 16. Deformed shape with superimposed contours of plastic deformations for the very strong Type III excitation. The coalescence of the active wedge of the main wall with the origins of the passive wedge of the anchor wall is evident.

two regions: in the passive side (in front) of the wall near the mudline, and on active (back) side of the anchor wall. The active wedge on the back side of the wall itself has just begun to form and it has almost reached the surface at a clear distance from the anchor. There is no indication that passive strains developed in front of the anchor wall; therefore the design of the particular anchoring system for this level of excitation is more than just adequate, which can lead to a more economic design. Fig. 14 shows the bending moments and displacements of the SSP wall obtained by all methods (ABAQUS, PLAXIS, pLEM) and Fig. 15 shows the distribution of horizontal earth pressures on the wall. These results show that:

response and the earth pressures? (d) Are the MO earth pressures reasonable or conservative for this particular type of wall? 5.1. Response to moderate shaking For a first answer to these questions, a complete set of results elucidating the response of a SSP wall subjected to moderate seismic excitation type I (α=0.15g) is shown in Figs. 11–15 for the Cogswell record. Fig. 11(a) shows the spectral amplification ratio, SAtop/SAbase , for three particular shaking base motions Type I and the two different FE codes (ABAQUS and PLAXIS). The free-field soil amplification is evident, with the fundamental period being to T≈0.85 s. Some differences between the two methods are noticeable, especially the higher resonant amplification computed with PLAXIS. Similar is the trend in the spectral amplification ratio at the free-field at the front of the SSP wall (seabed), as shown in Fig. 11(b). The fundamental period is about 0.45 s, almost ½ of TS. Fig. 12 plots the acceleration time histories at the top of the wall and at the base of the model, and their response spectral ratio. Note the amplification at about the same period, T≈Ts≈0.8 s, as the fundamental period of the free-field soil, implying that the SSP wall vibration is driven by the free-field ground motion, at least for the imposed moderate level of shaking. In this respect the response of the SSP wall matches that of gravity caisson quay-walls. Fig. 13 shows the deformed shape of the system with superimposed the contours of plastic strains. The snapshot is taken at the time of the maximum thrust on the wall. Note the intense plastic deformations in

1. The peak bending moments occur at just below the middle of the free height (H≈18 m) of the wall, which is about 12 m below the top. The peak values, of the order of 1650 ± 100 kN/m, are insensitive to the details of the excitation as well as to the exact elastoplastic constitutive relation of the soil layers. 2. The simplified pLEM method predicts larger flexural distress for the SSP wall than the one obtained from the FE analyses. 3. In contrast to the bending moments, the computed displacements are sensitive to the soil constitutive relations, ranging from about 6– 10 cm at the top of the wall. 4. 4. The distribution of horizontal earth pressures ph on the wall compared with the KAE and KPE diagrams of MO reveals the main causes for the possible (but slight) over-prediction of SSP bending moments by the pseudo-static methods. On the active side the (unavoidable) seaward “bulging” of the wall creates “arching” conditions with a respective decrease of ph (Gazetas et al 10

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Fig. 17. Comparison of the distributions of the maximum bending moments and lateral deflection of the main SSP wall for: (a) Type II, and (b) Type III excitation, obtained with the two FE codes and the pseudostatic limit equilibrium method (pLEM). Note: for the Type III motion the ultimate state was reached at A=0.40g.

tive accelerations of A =0.30g and 0.50g, respectively: Fig. 16 depicts the deformed shape with the contours of plastic deformations at the instance of maximum active thrust on the SSP wall, for the strongest excitation. The conclusions are similar to those drawn on the basis of Fig. 13 for the moderate excitation, except that the soil reaches plastic state over a larger zone and the deformation of the system is significantly larger. Moreover, the “active” failure surface seems to be forming at a much reduced angle with respect to the horizontal and has interfered with the passive wedge of the anchor, as the latter has started developing. This implies that the length L of the tie-rod is becoming less effective. Yet, although the displacement of the main SSP wall is appreciable, the anchor wall has not reached passive failure. Fig. 17 plots the distributions of the maximum bending moments and lateral deflections for the two types of excitation, II and III, calculated with all methods. It should be noted that for Type III motion the limit state using the pLEM was reached at A =0.40g, contrary to the A =0.50g used (as the effective rock-outcrop acceleration in the FE analyses). Therefore, the respective results are not directly comparable. It was nevertheless seemed useful to present them alongside, in order to highlight the limitations and applicability of each method. From the results we observe the disproportionate increase of moments and deflections as the peak acceleration increases from 0.30g to 0.50g.

2005[b]). Similar conclusions were drawn from centrifuge tests on different types of cantilever walls supporting dry soil, by Sitar and his co-workers [23,30,4,5]. 5. The distance (L=45 m) of the anchor from the main wall was computed with the EC8-5 expression:

L = Lstatic (1 + 1.5a S ) ≈ Lstatic (1 + 1.5x 0.5x 1.2) = 1.9 Lstatic

(9)

derived utilizing the original version of the Chart of Fig. 5 [14]. It appears that this is quite adequate: only active state of stress develops behind the anchor wall ── the crucially important passive failure is far from developing. So there is a sufficiently ample margin of anchor resistance. Consequently, according to ABAQUS and PLAXIS analyses, the response of the selected SSP wall against the Type I motion is quite satisfactory. 5.2. Response to very strong shaking Only a limited set of the obtained results are presented here for the strong and very strong excitations, with reference rock-outcrop effec-

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Z :m

Z :m 4

3

0

0

at max P

-3

at max P

M o no no be-Okabe

-6

M o no no be-Okabe

-4 -8

-9

-1 2

-14 m -1 2

-14 m

-1 6 -1 5 -20 -1 8 -24

-21

-28

-24

-32

-27 -400

-300

-200

-100

-1200

0

Pressures Behind Wall : kPa

-1000

-800

-600

-400

-200

0

Pressures Behind Wall : kPa

(b)

(a)

Fig. 18. Distribution of earth pressures against the SP wall for the two types of excitation: (a) Type II; (b) Type III. The circles are the FE results. Note that the MO solution for type III excitation is based on A=0.40g.

and strong (A ≤0.30 g) shaking may slightly exceed the pressures computed using FE analysis. The main difference stems from the arching, taking place about the midspan of the above seabed SSP wall, which can be captured only using FE analysis. (b) For the strongest Type III shaking (A =0.50 g), the MO expression does not “converge”, since

For the latter, the largest moment varies between 6800 and 7500 kNm/ m, thus approaching─ but not exceeding ─ the ultimate capacity of the particular SSP section. For the very strong Type III motion the computed displacements can be found within the range of 0.5─1.0 m (undrained effective stress analysis), highlighting the strong dependence of deformations on soil constitutive modeling and method of analysis. However, the final evaluation of the performance of this type of quay-wall will always be made according to specific performancebased criteria set for the particular project. In any case, our methodology showed that with levels of acceleration (at the rock-outcrop level) exceeding 0.45g the SSP wall reaches a critical point of its economic application. Fig. 18 compares the distribution of the lateral earth pressures obtained from the FE analysis and from MO. Regarding the Type II strong motion, the conclusion drawn for the moderate Type I excitation (Fig. 15) remains valid: the MO based method exaggerates the applied ph , since it ignores the arching effect as well as soil amplification. With the Type III motion (A=0.50g), MO did not converge. The pLEM results are for A=0.40g. Yet, even with the reduced acting acceleration the MO method predicts larger pressures than those of the FE analyses. (Note however that MO just as Coulomb’s analysis does not provide the distribution of active/passive pressures. The linear distribution assumed here is just a conventional one, adequate for comparing the resultant force on the wall. The [29] recommendation would require an inverse triangular distribution to have the resultant force at 2/3H from the base). However, such a distribution ceases to be valid when the wall structural or rotational flexibility are accounted for [16].

tanψ = (γsat / γb ) tanψ ≈ 2α from which ψ΄=45ο and Eq. (3b) breaks down. By contrast, FE analysis gives finite earth pressures and acceptable SSP distress, as long as development of excess pore water pressure is insignificant. (c) Soil amplification is of great importance in the seismic response of such tall walls. In practical situations and significant projects, it is advisable that soil amplification be first studied independently of the design of the wall. (d) The dynamic interaction between soil, main SSP wall, and anchorwall can be captured adequately with FE methods. Beam-onWinkler-Foundation models could possibly approximate such interaction effects, but the development of regions of concentrated plastic deformation (surrogates of Coulomb sliding surfaces) cannot always be adequately represented in such models even when nonlinear p–y type of “springs” are used in the model. (e) The hydrodynamic pressures (positive and negative) have a minor effect on the wall performance (of the order of 5% or less). (f) It was shown that tall SSP walls can be designed to safely withstand moderate and strong seismic ground motions (Type I and II), in terms of both displacements and structural distress. In the extreme case of Type III motions, the feasibility of the SSP design depends on the particular nature and strength of the retained as well as the foundation soils. If the retained soil is cohesionless and likely to develop substantial excess pore water pressures (let alone to liquefy), or if the foundation soil is very soft clay, suitable soil improvement will be needed. After all, such soil improvements are also necessary with caisson-type walls, as the Kobe harbor case (among others) has demonstrated.

6. Conclusions The paper presents a case study of the seismic response of a deep anchored steel sheet-pile wall supporting 18 m of soil. The results suggest the following: (a) The earth pressures, ph, computed using the Mononobe-Okabe (MO) method for partially-submerged backfill and for moderate 12

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