Bearing capacity of shallow foundation under cyclic load on cohesive soil

Computers and Geotechnics 123 (2020) 103556

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Review

Bearing capacity of shallow foundation under cyclic load on cohesive soil a,⁎

a

Daniel R. Panique Lazcano , Rubén Galindo Aires , Hernán Patiño Nieto a b

T

b

ETSICCP, Universidad Politécnica de Madrid – Spain, c/Profesor Aranguren s/n, Madrid 28040, Spain ICIG, Instituto Colombiano de Investigaciones Geotecnicas, Colombia

A R T I C LE I N FO

A B S T R A C T

Keywords: Bearing capacity Pore pressure generation Cyclic softening Cyclic loading Finite difference method

The bearing capacity of shallow foundations under static loads has been widely studied by various authors. In contrast, the bearing capacity under dynamic loads has been solved indirectly by adopting an equivalent pseudostatic approach or by means of reduction coefficients. The study of the problems related to cyclic loads has focused mainly on the behavior of granular soils. However, in the case of cohesive soils, these began to be studied in detail as a result of the Northridge, Kocaeli and Chi-Chi earthquakes that took place in the 90s. The object of the present work is to evaluate the influence of pore pressure in the calculation of bearing capacity of shallow foundation on cohesive soil and dynamically requested. This research is based on the results of cyclic simple shear tests carried out with samples from the Prat port (Barcelona). A pore-water pressure generation equation is proposed that depends on the effective vertical stress “in situ” (σ’ov), static shear stress (τo), cyclic shear stress (τc), and void ratio (e0). The formulation was implemented to the finite difference software FLAC2D, by which the bearing capacity of foundations can be determined. The proposed formulation can calculate the maximum cyclic load (CL) that a cohesive soil can resist before failure by cyclic softening, as a consequence of cyclic loads transferred by a shallow foundation. The practical application of this research is to provide a chart to calculate the bearing capacity of cohesive soils requested by cyclic loading (CL). This application takes into account the properties of the ground, the static load capacity (Phe) and the effective load outside the foundation (q ). In addition, the formulation is validated in the application of a real case.

1. Introduction The bearing capacity under static loads has been extensively studied in soil mechanics over the years. The first works were developed by Prandtl [1] and Terzaghi [2]; subsequently, various authors have completed the theories Meyerhof [3,4], Brinch Hansen [5,6] and Vesic [7] for the estimation of static bearing capacity (Phe). These authors considered various factors such as geometry, depth, eccentricity, an inclination of the load, slope, and influence of the water table. Otherwise, the bearing capacity under dynamic loads (Phd) has often been addressed indirectly, by means of an equivalent pseudostatic approach or by means of reduction coefficients [8]. The equivalent pseudostatic approach corresponds to determining the bearing capacity factors using a dynamic internal friction angle [9]. Moreover, Meyerhof [3] and Shinohara et al. [10] worked with a pseudostatic approach in which horizontal and vertical accelerations are applied to the center of gravity of the structure and the problem is reduced to a static case with eccentric loads inclined [11]. The bearing capacity under cyclic loads in cohesive soil was not studied in detail until the Northridge earthquakes in 1994, Kocaeli in ⁎

1999 and Chi-Chi in 1999 [12,13]. In these earthquakes failures occurred in structures on cohesive soil, something not taken into account and little studied until that moment. Before, only the bearing capacity was studied due to dynamic loads in granular soils, associated with the phenomenon of liquefaction. Marcuson [14] defined this phenomenon as the transformation of granular soil from solid state to a liquefied state due to excess pore water pressure which reduces effective stress. The cyclic softening is a similar phenomenon that occurs in cohesive soils, but unlike granular soils, the effective tension does not reach a null value until the soil fails. The generation of pore water pressure has been studied by various authors from the 70s to the present, and this has served to generate various empirical formulations. The cyclic softening leads to engineering problems in the design of foundations of structures built along the coast, design of offshore structures, structures such as harbours, breakwaters and storm-surge barriers, the design of machinery foundations, the design of road and railway embankments, and other types of foundations such as footings, floating foundation or foundation slab, which are found on saturated cohesive soils [15]. These structures are subject to heavy wave loads, machinery vibrations or traffic-

Corresponding author. E-mail address: [email protected] (D.R. Panique Lazcano).

https://doi.org/10.1016/j.compgeo.2020.103556 Received 30 October 2019; Received in revised form 9 March 2020; Accepted 17 March 2020 0266-352X/ © 2020 Elsevier Ltd. All rights reserved.

Computers and Geotechnics 123 (2020) 103556

D.R. Panique Lazcano, et al.

Fig. 1. Variation of pore water pressure with cyclic shear stress and shear strains under undrained cyclic loading. Adapted from Andersen [30].

softening. Bray and Sancio [13] took into account factors such as confining pressure, initial static shear stress, stress-path, soil mineralogy, voids ratio, and the overconsolidation ratios. Patiño et al. [24] showed that the initial shear stiffness of a soft clay reduces after several load cycles, in the same way, the cyclic shear stress ratio necessary to reach the failure decreases as the number of cycles increases. Moreover, Leboeuf et al. [25] showed in their study that the cyclic softening in clay and silt was a phenomenon in which factors such as pore water pressure accumulation, fatigue loading, and rate effects are very important. Martinez et al. [26] proposed a methodology to study the risk of sudden failure, by testing different combinations of static and cyclic stresses to obtain formulas and graphs of the risk of sudden failure. This methodology allows for an adequate analysis of the degradation of the shear stiffness module, the influence of the static shear stress, generation of pore water pressure and evolution with the number of cycles. Kumar et al. [27] studied cyclic softening subjected to cyclic loading by stages. Their results indicated that the reduction of the shear module and the damping ratio were not affected by changes in the initial dry density and water content. However, Chávez et al. [28] performed the reproduction of the collapse of a coastal structure due to cyclic softening under small-scale laboratory conditions and under controlled conditions. Eslami et al [29] worked with soil and the same plasticity index with fresh water and saline water. The results indicated that the plasticity index is an insufficient indicator of the procedure that evaluates the potential of cyclic softening.

induced cyclic loading. Tsai et al. [16] argue that cyclic softening and loss of strength in saturated clays are fundamental problems in engineering, such as slope stability, dam safety or bearing capacity of the foundation. Due to this problem, there is a keen interest to study the calculation of bearing capacity of a shallow foundation under cyclic loading in cohesive soil. From the cyclic simple shear tests carried out by Patiño [17], an analysis of the samples from the port of Barcelona at Prat pier are carried out. From this analysis, a formulation of pore water pressure generation was obtained from the cyclic simple shear tests, and it depends on the effective vertical stress “in situ” (σ′ov), static shear stress (τo), cyclic shear stress (τc), and the void ratio(e0). This formulation was implemented to the finite difference software FLAC2D [18] through the programming code FISH, that can determine the bearing capacity of foundations and estimate the maximum cyclic load (CL) that a footing in cohesive soil can resist before failing by cyclic softening.

2. Background 2.1. Cyclic softening Seed et al. [15] denominate “classic” cyclic liquefaction to significant loss of strength and stiffness due to the generation of cyclic pore water pressure. A reduction of the stiffness and resistance of the soil due to a repeated cyclic load is what is called the cyclic softening of clays [19]. However, Boulanger and Idriss [20]; Chu et al. [21] recommend that the term liquefaction be used to describe the development of significant deformations or the loss of resistance as in the case of sands, while for the clays the term of failure by cyclic softening. The cyclic pore water pressure is the one that induces a cyclic softening and a loss of resistance in the soil and indicates that the maximum pore-water pressure in clay is typically less than in sand for the purpose of soil failure [21]. Bray and Sancio [13] observations and results of cyclic tests show that fine-grained soils (clay and silty soils) underwent cyclic softening during the earthquakes in 1994 Northridge, 1999 Kocaeli, and 1999 Chi-Chi. However, one of the first reports of cyclic softening was during the 1964 Alaska earthquake. The landslide on the Fourth Avenue, downtown Anchorage, Alaska, ranged from several centimeters to more than 5 m were produced in the almost normally consolidated clay [22,23]. According to the susceptibility analysis proposed by Bray and Sancio [13], which indicates that soil with plasticity index (PI) < 12 were susceptible to cyclic softening, loose soils with 12 < PI < 18 were systematically more resistant and soils with PI > 18 were not susceptible to cyclic softening. Boulanger and Idriss [20,22] presented a new criterion to evaluate the silts and clays to the susceptibility of the cyclic softening, this criterion was based on the mechanical behavior of stress-strain. Subsequently, empirical relationships were derived to determine the cyclic resistance ratio (CRR = τc / Su ) . Tsai et al. [16] propose a simplified procedure to evaluate the cyclic softening, that estimates the amplitude of the cyclic shear stress and after the equivalent number of cycles. Several authors investigated factors that most affect cyclic

2.2. Pore water pressure generation The development of pore water pressure and the shear stress for a soil element subjected to cyclic loading without drainage is shown in Fig. 1 from the studies of Andersen [30]. In part a) the load cycles are observed with cyclic shear stress (τc ) for a constant shear stress (τa ), in our study cases τa = τ0 ; where (τ0 ) was the static shear stress. Part b) shows how the cyclic shear stress generates pore water pressure (u ) that can be divided into two components, the first one is permanent (up) and the second component was a cyclic component (uc ). This increase in pore water pressure reduced the effective stresses of the soil, which is seen with an increased shear strains as shown in part c); where there was a permanent component (γp ) and a cyclical component (γc ). There are several empirical formulas to generate pore water pressure, one of which is based on energetic methods such as [31]. Booker et al. [32] proposed a simplified alternative to the equation of Martin et al. [33] that was obtained experimentally from data of liquefiable soils. Otherwise, Martin et al. [33], Byrne [34] or in the case of Matasovíc and Vucetic [35] developed a model called the generalized degradation-pore water pressure generation, which expresses the cyclic strength (cyclic softening) with stiffness degradation and cyclic pore water pressure, with the concept of volumetric threshold shear strain. Hyde and Ward [36] proposed a power model for pore water pressure generation for silty clay adjusted to cyclic triaxial tests by monotonic strain-controlled and low frequency. The model can estimate pore water pressure in the soils of an offshore foundation, both during and at 2

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and robustness in the bearing capacity calculation of the foundations. Chu et al. [21] performed a bearing capacity analysis using resistance reduction to take cyclic softening into account by comparing safety factors during the earthquake versus bearing capacity failure in the Wufeng region, Taiwan. Cascone and Casablanca [48] use the characteristics method to evaluate static and seismic bearing capacity for shallow strip footing and for rough and smooth foundations under a pseudo static approach. Likewise, they proposed a correction of coefficients of the Terzaghi formula to apply to the seismic design of foundations. Pane et al. [50] worked on the basis of the pseudostatic approach in the finite difference method (FDM) to evaluate the seismic effects on the bearing capacity of shallow strip foundations. Conti [51] worked on the seismic bearing capacity of strip footing based on a pseudostatic approach and limit analysis in cohesive-frictional and purely cohesive soils. Results give a reduction of the coefficients of the Terzaghi equation for vertical bearing capacity and the failure envelope of the normalized load variables. However, the effects of pore water pressure or reduction of the shear strength due to seismic effects were not taken into account. Cinicioglu and Erkli [52] argued that when calculating the seismic bearing capacity in cohesive soils this would have limited practical use if the influence of all parameters was not taken into account. They considered a pseudostatic coefficient to define the seismic load but did not take into account the pore water pressure.

the end of a design storm. Later, Green at al. [37] developed a model called GMP, that relates the generation of residual pore water pressure with the energy dissipated per unit volume of the soil. Nie et al. [38] proposed a model to predict the evolution of pore water pressure under cyclic shear loading. Galindo Aires et al. [39–41], through an evaluation of the degradative damage, proposed to solve the evolution of the cycles and the generation of pore water pressure under cyclic loads. Chian [42] adjusts a simplified exponential equation for the dissipation of pore water pressure over time for liquefiable sands. This equation was applied to dynamic centrifugal tests and shows an adequate adjustment in the history of time. Nie et al. [43] presented a constitutive model for soil cyclic loading predicting the behavior of soft clays under undrained cyclic triaxial load. This model simulated cyclic triaxial tests in kaolin and these predictions were generally in accordance with measured pore water pressures and axial stresses. Ren et al. [44] developed a hyperbolic model to predict the development of undrained pore water pressure. This model is a hyperbolic relationship between the accumulated plastic stress and pore water pressure which was verified by experimental data. Shi et al. [45] proposed a constitutive law to simulate cyclic strength degradation of natural clays as a result of the loss of structure and excess pore water pressure accumulation. During the pore water pressure dissipation, severe displacements and settlements can be generated. These problems can lead to the inclination of the buildings, displacements, and settlements of the crowning of a dam, lateral expansion of the wall of a bridge, etc. [42]. From the results of cyclic simple shear, Martinez et al. [26] indicated that the pore water pressure during cyclic loading controls the risk of sudden failure and the development of permanent deformations.

3. Description of the tests Laboratory data used in the present work were carried out by Patiño Nieto [17] at the Universidad Politécnica de Madrid, subsoil samples from the Port of Barcelona at the Prat pier. The soil deposit is of alluvial origin that dates back to the recent Quaternary or more specific to the Holocene era. The samples analyzed belong to a sedimentary environment classified as silty clays and clayey silts of medium to low plasticity. Alonso and Gens [53] describe the area with four strata, the first stratum of silts and clays with a certain content of organic matter, in the upper part of the stratum intercalations of sandy sands and silts, with a thickness of 50 m. The second stratum was 7 m of gravel and sand with traces of silt. The third layer corresponds to clays similar to stratum one, but with a higher density, this stratum can reach 14 m. And, finally, the fourth stratum was gravel and sand interspersed with clay, the thickness of this stratum can exceed 40 m. The tests carried out come from undisturbed samples obtained from two exploratory borings, SA-1 and SA-2 located between 30 and 52 m depth from sea level, in the studied area the bottom of the sea is at a depth of 29 m. Patiño et al. [24] and Martinez et al. [26] describe the stress–strain behavior of this soil as typically plastic, with positive pore water pressure that shows a contractive behavior, so it is a deposit of normally consolidated soil or soil with a low degree of preconsolidation. The tests performed on these samples were classification, index properties, cyclic simple shear tests which are presented in Tables 1 and 2. From the tests carried out, it can see in Table 1 that the average natural moisture was 29.24%, from a range of 24–37 with a standard deviation of 3.126 and a coefficient of variation of 0.107. The apparent density had an average of 1.97 g/cm3 with a standard deviation of 0.039, a coefficient of variation of 0.02 and a range of 1.85–2.08 g/cm3. The content of fines has an average value of 98.66% between 87.26 and 99.99, with a standard deviation of 2.621 and a coefficient of variation of 0.027. The percentage of particles with a diameter smaller than 2 µm was between 14 and 41. The plasticity index had an average value of 18.15% between 12.3 and 19.90 with a standard deviation of 2.11 and a coefficient of variation 0.119. If it takes into account, the classification of Bray and Sancio [13] our soil under study contains a plasticity index between 12.30 and 19.90; so it is inside a soil susceptible to cyclic softening. The samples for the cyclic simple shear tests were obtained through

2.3. Bearing capacity The first solutions for bearing capacity were presented by Prandtl [1] and Terzaghi [2], followed by Meyerhof [3,4], Brinch Hansen [5,6] and Vesic [7]. These solutions depend on factors such as cohesion (c ), friction angle (φ ), dilation angle (ψ ), soil unit weight (γ ), effective load outside the foundation (q ) and width of footing (B ). However, the bearing capacity under cyclic loads has not been prolifically studied. According to studies from triaxial tests by Ishihara [19], the MohrCoulomb failure criterion can be compared for a dynamic load with a conventional static load condition. The principal difference is the cohesion, where the dynamic is between 1.6 and 2.4 higher than static. Dakoulas and Gazetas [46] carried out a dynamic analysis based on the elasto-plastic constitutive model Pastor-Zienkiewicz, on the caissontype quay walls in the port of Kobe due to an earthquake in 1995. During the earthquake, it was possible to observe the evolution of the lateral displacements, the plastic strains and the increase of the pore water pressure. Xiao et al. [47] implemented a soil strain softening and rate model in a finite element software to evaluate failures in design coastal and offshore shallow foundations after the action of the cyclic load considering the degradation of the soil. A structure such as a foundation subjected to cyclic loading produces several stress conditions in the ground below it. Fig. 2 shows in a simplified way the conditions of stresses develop along a potential failure surface. The stress conditions were also observed that can be simulated in the laboratory with cyclic simple shear test (case 1 and 3) or triaxial test, compression (case 2) or extension (case 4). In the present paper it focused on cases 1 and 3 because there are laboratory data of cyclic simple shear tests. Cascone and Casablanca [48] mention the different calculation methods to analyze bearing capacity using numerical methods such as limit equilibrium, limit analysis, characteristics method, zero extension line, finite element method, and finite difference method. Other authors, such as Frydman and Burd [49] compared FLAC2D [18] with a finite element code and conclude that FLAC2D has greater efficiency 3

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Fig. 2. Hypothetical fault scheme to represent stress conditions that can be simulated with cyclic simple shear tests and triaxial tests [24].

- Undisturbed samples. - The sample size always the same, 7 cm in diameter and 1.91 cm in height. - Consolidation stress equal to the effective vertical stress “in situ” ' (σv0 ). - Undrained conditions and evaluation of the generated pore water pressure. - Finalization of the test for permanent shear strains (γp ) of 15% or cyclic strains (γc ) of 15%, number of cycles (N) of 1300 or pore water pressure of 95% of the consolidation stress. - Stress controlled during the cyclic stage. - Sinusoidal wave with an amplitude equal to the cyclic shear stress (τc ) and a period of 10 s.

Table 1 Summary of index properties. Parameter

Unit

No. of data

Average

Range

Natural moisture Natural unit weight Content of fines Percentage D < 2 μ Liquid limit Plastic limit Plasticity index Specific gravity

% g/cm3 % % % % % –

78 78 78 14 16 16 16 16

29.24 1.97 98.66 30.21 40.25 22.58 17.67 2.78

24–37 1.85–2.08 87.26–99.99 14–41 30.50–45.30 18.20–25.50 12.30–19.90 2.74–2.80

an unconventional methodology. A conventional commercial extractor to removed the sample from the “shelby” tube generate a pressure during the extraction process which compresses the sample significantly and consequently disturbs it. The new methodology proposed by Patiño et al. [24] consisted in using a pipe cutter and pipe vise. First, the shelby tube was placed on the vise so that it did not turn, then the tube cutter was clamped at the required distance from the sample. The cutter was rotated as many times as necessary until the tube was cut without damaging the sample.

The main influence parameters of these tests are: the initial shear stress (τ0 ) , voids ratio (e0 ) at the initial of test and the amplitude of the applied cyclic shear stress (±τc ) . A large number of tests were performed for different values of (τ0 ) and (τc ) that are shown in Table 2. Data were normalized relative to the effective vertical consolidation stress “in ' ) were the initial shear stress ( τ'0 ), cyclic shear stress ( τ'c ) and situ” (σv0 pore water pressure (

u ' σv0

σv 0

σv0

). Pore water pressure values are those taken at

the end of the test when it reaches the critical state. The evaluation and interpretation of the cyclic simple shear test data shown in Table 2, will center at obtaining a model of pore water pressure generation under cyclic loading. Later, this model to be implemented in a numerical model to calculate the bearing capacity of shallow foundation in cohesive soil under cyclic loading.

3.1. Cyclic simple shear Data provided by the laboratory were made with cyclic simple shear equipment. The equipment has the capacity to carry out tests under undrained (constant volume) and drained conditions (constant axial load). Under the criterion of Bjerrum and Landva [54], cyclic simple shear tests with a constant volume are equivalent to undrained tests. Change in vertical stress applied to the sample is equivalent to the change in pore water pressure that would have occurred if the sample had been prevented from draining by a constant applied vertical stress condition. Therefore, a change in the effective normal stress required to maintain a constant volume in the sample will be the change in pore water pressure. To minimize the factors that can affect the behavior of the soil during the test, the following conditions were established:

4. Pore water pressure generation 4.1. Pore water pressure generation under cyclic shear load A pore water pressure analysis was performed according to the various parameters measured in the tests, voids ratio, initial shear stress, and cyclic shear stress. Part a) of Fig. 3 shows the variation of the normalized pore water pressure with normalized initial shear stress, the pore water pressure decreases as the initial shear stress increases. The maximum pore water pressure value occurs for a zero initial shear stress and varies from a 4

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Table 2 Summary of results of cyclic simple shear tests. Test No.

Sample Ident.

e0

τfailure (kPa)

' (kPa) σv0

τ0 (kPa)

τc (kPa)

τ0/ σv' 0

' τc / σv0

' u/ σv0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

SA-1-M1 SA-1-M1 SA-1-M1 SA-1-M2 SA-1-M4 SA-1-M4 SA-1-M6 SA-1-M6 SA-1-M6 SA-1-M6 SA-1-M6 SA-1-M6 SA-1-M7 SA-1-M7 SA-1-M8 SA-1-M9 SA-1-M9 SA-1-M9 SA-1-M9 SA-1-M10 SA-1-M10 SA-1-M10 SA-1-M10 SA-1-M10 SA-1-M10 SA-1-M10 SA-1-M10 SA-2-M1 SA-2-M1 SA-2-M1 SA-2-M1 SA-2-M1 SA-2-M2 SA-2-M2 SA-2-M3 SA-2-M3 SA-2-M3 SA-2-M3 SA-2-M3 SA-2-M4 SA-2-M4 SA-2-M4 SA-2-M4 SA-2-M4 SA-2-M4 SA-2-M5 SA-2-M5 SA-2-M5 SA-2-M5 SA-2-M5 SA-2-M5 SA-2-M5 SA-2-M5 SA-2-M5 SA-2-M6 SA-2-M6 SA-2-M6 SA-2-M6 SA-2-M6 SA-2-M6 SA-2-M7 SA-2-M7 SA-2-M7 SA-2-M7 SA-2-M7 SA-2-M7 SA-2-M7 SA-2-M7 SA-2-M7 SA-2-M8 SA-2-M8 SA-2-M8 SA-2-M8 SA-2-M8

0.90 1.04 1.05 1.02 0.92 0.92 0.74 0.79 0.85 0.89 0.72 0.79 0.88 0.88 0.68 0.73 0.80 0.95 0.98 0.76 0.78 0.80 0.81 0.76 0.81 0.80 0.77 0.88 0.83 0.91 0.79 0.75 0.99 0.90 0.84 0.81 0.83 0.75 0.85 0.77 0.77 0.81 0.69 0.89 0.75 0.91 0.76 0.72 0.85 0.82 0.84 0.75 0.79 0.79 0.85 0.85 0.83 0.81 0.74 0.73 0.89 0.90 0.79 0.82 0.77 0.76 0.81 0.81 0.81 0.75 0.80 0.78 0.77 0.82

85.5 85.5 85.5 80.3 93.54 93.54 108.62 108.62 108.62 108.62 108.62 108.62 111.73 111.73 130.7 145.00 145.00 145.00 145.00 142.67 142.67 142.67 142.67 142.67 142.67 142.67 142.67 98.22 98.22 98.22 98.22 98.22 99.78 99.78 116.93 116.93 116.93 116.93 116.93 165.78 165.78 165.78 165.78 165.78 165.78 102.12 102.12 102.12 102.12 102.12 102.12 102.12 102.12 102.12 110.17 110.17 110.17 110.17 110.17 110.17 101.08 101.08 101.08 101.08 101.08 101.08 101.08 101.08 101.08 147.59 147.59 147.59 147.59 147.59

277 277 277 283 311 311 349 349 349 349 349 349 366 366 389 401 401 401 401 413 413 413 413 413 413 413 413 294 294 294 294 294 315 315 328 328 328 328 328 343 343 343 343 343 343 347 347 347 347 347 347 347 347 347 364 364 364 364 364 364 373 373 373 373 373 373 373 373 373 384 384 384 384 384

0 41.55 55 57 31 62 0 0 35 52 52 70 0 0 0 0 20 60 80 0 21 21 41 41 62 83 83 0 15 29 44 44 47 63 0 16 33 66 66 17 51 51 69 69 86 0 0 17 35 52 52 69 69 87 0 36 55 55 73 73 0 19 37 37 56 56 75 75 93 0 19 19 38 58

27.7 27.7 14 28 62 31 70 87 52 35 52 35 55 73 58 80 80 60 40 83 62 83 62 83 41 21 41 44 59 59 29 44 47 32 66 82 66 16 33 69 34 51 17 34 17 69 87 69 69 35 52 17 34 17 91 55 36 55 18 36 75 75 56 75 37 56 19 37 19 77 77 96 77 38

0.00 0.15 0.20 0.20 0.10 0.20 0.00 0.00 0.10 0.15 0.15 0.20 0.00 0.00 0.00 0.00 0.05 0.15 0.20 0.00 0.05 0.05 0.10 0.10 0.15 0.20 0.20 0.00 0.05 0.10 0.15 0.15 0.15 0.20 0.00 0.05 0.10 0.20 0.20 0.05 0.15 0.15 0.20 0.20 0.25 0.00 0.00 0.05 0.10 0.15 0.15 0.20 0.20 0.25 0.00 0.10 0.15 0.15 0.20 0.20 0.00 0.05 0.10 0.10 0.15 0.15 0.20 0.20 0.25 0.00 0.05 0.05 0.10 0.15

0.10 0.10 0.05 0.10 0.20 0.10 0.20 0.25 0.15 0.10 0.15 0.10 0.15 0.20 0.15 0.20 0.20 0.15 0.10 0.20 0.15 0.20 0.15 0.20 0.10 0.05 0.10 0.15 0.20 0.20 0.10 0.15 0.15 0.10 0.20 0.25 0.20 0.05 0.10 0.20 0.10 0.15 0.05 0.10 0.05 0.20 0.25 0.20 0.20 0.10 0.15 0.05 0.10 0.05 0.25 0.15 0.10 0.15 0.05 0.10 0.20 0.20 0.15 0.20 0.10 0.15 0.05 0.10 0.05 0.20 0.20 0.25 0.20 0.10

0.65 0.78 0.45 0.7 0.75 0.7 0.85 0.7 0.75 0.65 0.75 0.6 0.62 0.8 0.9 0.85 0.8 0.68 0.62 0.8 0.7 0.8 0.8 0.75 0.65 0.42 0.55 0.6 0.8 0.8 0.65 0.75 0.7 0.65 0.8 0.75 0.75 0.35 0.6 0.82 0.5 0.7 0.35 0.65 0.45 0.75 0.7 0.8 0.66 0.55 0.6 0.42 0.65 0.43 0.75 0.7 0.65 0.68 0.35 0.5 0.75 0.78 0.8 0.75 0.6 0.71 0.45 0.65 0.45 0.8 0.8 0.78 0.75 0.58

(continued on next page) 5

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Table 2 (continued) Test No.

Sample Ident.

e0

τfailure (kPa)

' (kPa) σv0

τ0 (kPa)

τc (kPa)

τ0/ σv' 0

' τc / σv0

' u/ σv0

75 76 77 78

SA-2-M8 SA-2-M8 SA-2-M8 SA-2-M8

0.75 0.77 0.79 0.81

147.59 147.59 147.59 147.59

384 384 384 384

58 77 77 96

58 19 38 19

0.15 0.20 0.20 0.25

0.15 0.05 0.10 0.05

0.68 0.4 0.6 0.42

value of 0.60 to 0.90. Part b), shows the variation of the pore water pressure normalized with the normalized cyclic shear stress. Cyclic shear stress with the null value would mean a static case, so it goes from a value of 0.05 to 0.25. The lowest pore water pressure values occur for the lowest cyclic shear stress, the highest values being for a value of 0.20 cyclic shear stress. For the maximum value of cyclic shear stress of 0.25, pore water pressure varies from a value of 0.70 to 0.78, which shows that pore water pressure does not reach a value of 1, as would be the case of granular material (sands). If it analyzes the dispersion of the initial voids ratio in Table 2, it can see that the average was 0.82 with a standard deviation of 0.08. This means taking values of the void ratio as a minimum of 0.74 and a maximum of 0.90 for an analysis based on this parameter. As a first approximation, the adjustment corresponding pore water pressure normalized with cyclic shear stress was used to introduce the variables of initial shear stress and void ratio. A formulation consisting of two parts is proposed (Δu = A + B ); part A depends on the initial shear stress, void ratio, and cyclic shear stress. Part B is the first adjustment between pore water pressure and the cyclic shear stress. The introduction of the variable of the void ratio in the structure of the formulation significantly increases the correlation. So the formulation is as follows,

Δu ' σvo

τ τ = a1 ⎛⎜ 0' − 'c ⎞⎟ (e0 − e¯) + σ σ vo ⎠ ⎝ vo

⎛ ⎜β + 0 ⎜⎜ e⎝

Fig. 4. The fit between estimated and actual pore water pressure generation data.

From the data available in Table 2, the best fit was obtained for β0 = −0.063 andβ1 = −0.041, a1 = 2.61 and e¯ = 0.82 with a correlation R2 = 85%. In Fig. 4 it is observed how the influence of cyclic shear stress on the pore water generation is in itself very significant, regardless of the other two variables. Although Eq. (1) allows us to the other two variables: initial shear stress and voids ratio. For the adjusted formulation, it should be mentioned that the number of load cycles is not taken into account. This is because it focused on the critical point that the cyclic softening of the soil will occur and thus it did not consider intermediate behaviors. The values of pore water pressure ( u' ) in Fig. 4 represent the generation of pore water

⎞ β1 ⎟ τc ⎟ ' ⎟ σvo ⎠

(1)

Where:

Δu pore water pressure generation ' effective vertical stress “in situ” σvo τ0 initial shear stress τc cyclic shear stress e0 voids ratio e¯ average voids ratio a1, β0, β1 empirical constants

σv0

pressure in the critical state. As a result of the behavior of soft undrained soils subjected to cyclic loading, it can say: - The decrement of the effective stress does not become null for the

Fig. 3. Variation of pore water pressure with normalized initial shear stress and normalized cyclic shear stress. 6

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taken into account. This zone has a horizontal distance estimated at (6 * B) measured from the left contour; this area maintains the established mesh size and then increases the mesh width by a value of 1.01 until the contour. The vertical load on the footing was applied as a controlled downward velocity on the nodes of the footing surface. In turn, the displacement of the footing is calculated as the integral of the velocity in each calculation step. Therefore, the application of velocity has to be very small to reduce the effect of the initial velocity on the results.

soft soil to fail. - The proposed formulation depends on the stress state of soil and 3 adjustment parameters. - The proposed formulation is only for the generation of maximum pore water pressure, which leads to the critical state and soil failure. 5. Numerical methodology (Finite Difference Method-FLAC2D) A finite difference computational code called FLAC2D was used, which is described as an explicit finite difference program for the solution of differential equations [55]. The general calculation sequence embodied in FLAC2D first invokes the equations of motion to derive new velocities and displacements from stresses and forces. Then, strain rates are derived from velocities, and new stresses from strain rates. FLAC2D uses the Wilkins method [56] also called the finite volume method. This method deriving difference equations for elements of any shape, and boundaries can be any shape, and any element can have any property value as in the finite element method. This software includes an internal programming option (FISH), which allows us to obtain or calculate the desired variables and control the analysis process. Therefore, it was able to insert the pore water pressure formulation developed in the present work and run the model in the program. The bearing capacity of footing was modeled with the FLAC2D program in plane strain using small strains. The soil was modeled under the criterion failure of Mohr-Coulomb and flow rule not associated with dilation angle equal to zero.

5.1.1. Convergence analysis To determine the mesh size and velocity of load application on the nodes of footing, a convergence analysis was made to guarantee the precision for the solution and calculation time optimization. From Fig. 6 part a), it can be seen that from a mesh size of 0.5 m the values of static bearing capacity (Phe) have a little variation. Further, in part b) from a velocity of application on the nodes of the footing of 2.5e-7 the values of static bearing capacity (Phe) tend to stabilize. These values are those adopted for later work calculations. Otherwise, Frydman and Burd [49] argued that more precise solutions can be achieved considering that unbalanced force is as small as possible and not simply decreasing the mesh size. 5.2. Properties of soil The soil used for the modeling in FLAC2D is a silty clay of low plasticity. It first decided to make an analysis of the modulus of elasticity (E ) and the Poisson's coefficient (ν ) parameters. First, a modulus of elasticity of 5 MPa and a Poisson coefficient of 0.25 was chosen following the study of the soil of the Port of Barcelona in [57]. These values like a volumetric module (K ) 3.333 MPa and shear module (S ) 2 MPa were modified by ± 10 times the value of the modules. For our analysis these two parameters did not influence the result, so they were not taken into account. Further, assuming a null value of the soil weight, the width B of the footing did not influence the calculations made. The parameters of cohesion and angle of internal friction were chosen from an analysis based on the work developed by Patiño et al. [57] for this same soil of the Port of Barcelona as shown in Fig. 7. This analysis consisted of determining the range of values, which, in our

5.1. Numerical model The grid of de model is divided by the user into a finite-difference mesh composed of quadrilateral elements, but FLAC internally subdivides each element into two overlaid sets of constant-strain triangular elements [18]. For the size of the model, pore water pressure and the slip surface were taken into account so that the contours do not interfere with the calculation of bearing capacity. Regarding the size of the model, an adequate size of (20 * B) wide was determined by (12 * B) high, with B being the width of footing as shown in Fig. 5. In order to optimize and reduce the calculation time of the model, an area of influence of pore water pressure and the slip surface was

Fig. 5. Model and mesh used in FLAC2D. 7

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Fig. 6. Convergence to determine mesh size and distributed load velocity in the application width.

5.3. Calculation process in FLAC2D

case, used large strains, that were considered to have reached a critical state. According to Fig. 7, the internal friction angle values from 20 to 25 were chosen, corresponding to shear strains from 7% to 15%. Cohesion went from a value of 10 kPa to 50 kPa, a range of shear strains from 3% to more than 15%. Thus the maximum allowable strain was limited to a value and worked with the cohesion and internal friction angle made-up or equivalent parameters in Fig. 7.

A calculation process is proposed that has been structured in 6 stages and is shown in Fig. 8. The first corresponds to the creation of the mesh, designation of soil and water properties, own weight and boundary conditions. The second stage is the permanent load (PL) on the footing and the effective load outside the foundation (q ). The third stage is the cyclic load (CL) on the footing. The fourth corresponds to the generation of pore water pressure. The fifth is the equilibrium balance between pore water pressures and soil stresses. And, finally, the sixth stage corresponds to the calculation of bearing capacity. This process is iterative until finding the maximum value of cyclic load in the footing, which leads to a sixth stage that meets two criteria. The first is that bearing capacity to be zero or close to zero, and the second that meets the Mohr Coulomb failure-criterion. If the two criteria are not met, return to step three to change the cyclic load to a greater or lesser value, as appropriate to carry out the calculations again.

5.2.1. Dilatancy Pane et al. [50] show that values of friction angle less than 25 working with non-associated flow rule has an insignificant effect in the calculation of bearing capacity, but higher values become a significant effect. Loukidis and Salgado [58] indicate that the results of bearing capacity by finite element method can be very similar to the analytical methods, characteristics and limit analysis. However, for this result, the mesh must discretize very fine and work with the associated flow rule. The effect of non-associated flow rule that have significant results of bearing capacity to those calculated with associated flow rule is also considered. Therefore, bearing capacity solutions in sands with associated flow rule are less conservative than those not associated. For this work the parameter of dilatancy was taken as a null value.

5.3.1. Calculation cases For the analysis of the calculation of bearing capacity under cyclic load in a cohesive soil, it was decided to carry out a total of 324 cyclic cases that corresponded to combinations for different properties of soils (c, φ ), effective load outside the foundation (q ), void ratios (e0 ) and

Fig. 7. Variation of the angle of internal friction and apparent cohesion, as a function of the shear strains, Patiño et al. [57]. 8

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effect. Later, the application of this load was plotted with the displacement produced and bearing capacity of the soil was obtained when this load tends to be asymptotic at a constant value. 5.3.4. Security factor The safety factor is a ratio of parameters calculated under certain conditions that can lead to failure. The calculation of the safety factor in FLAC is based on the strength reduction method (SRM). This method has several advantages over other methods, the most important one being that it can automatically find the critical surface of failure [59]. It was taken into account two ways of calculating the safety factor for this work: the first is the strength, the minimum value of the average shear strength of the soil between the average shear stress developed along the most critical potential failure surface [60]. The second way is the critical load, a relationship between the minimum load to generate failure and the actual load. In this case, a reduction was made only in the cohesion of the soil. Dawson et al. [59], established that the safety factors obtained with the strength reduction method were slightly higher than those determined by the limit analysis. Considering that both solutions were obtained by different methods, the method of reduction is valid under the consideration that more refined the mesh the more the solution approaches the other method. Dawson et al. [59] used a plastic flow rule for comparison and determined that an associated flow rule ignores the effects of the elastic constants, the initial stress, and the stress-path. Chen [61] says that these effects have no effect on the collapse load of the associated material. However, for more realistic soil models, such as those with nonassociated flow rules, these factors cannot be ignored and must be taken into account. 5.3.5. Description of calculation methods The scheme of possible ways of calculating bearing capacity under cyclic load is shown in Fig. 9. This scheme was initially divided into two parts by considering the type of failure, that can be a general shear failure or a local shear failure. Afterwards, the calculation method was selected, this can be bearing capacity (Ph) or safety factor (FS ). After determining the method of calculation it was continued with the level of strains, which can be large or small strains. For the present work, the general shear failure was established by the bearing capacity method with small strains. However, a compare was made in some cases with the safety factor method implemented in FLAC. A disadvantage at the calculations was the edge of the footing. The formulation generates large shear stresses in this area, therefore, the generation of pore water pressure is very large and creates a point of instability in the model. To solve this problem, it was imposed for the model to have pore water pressure at the surface with a null value and fix pore pressure contour. For a better follow-up of the evaluation of the bearing capacity in the 6th stage of Fig. 8, the bearing capacity method (Ph) was chosen as a calculation method. This method was less sensitive than the safety factor (FS) method, which is better when analyzing values when approaching cyclic softening. Fig. 10 shows: the generation of pore water pressure, the difference between local shear failure and general shear failure as well as the envelope of the Mohr-Coulomb failure criterion. Part a) and b) correspond to the generation of pore water pressure due to a cyclic load. In part c) the local shear failure was observed, where the maximum displacement was generated only at the edge of the footing and did not exist slip surface failure. However, part d) was observed as a general shear failure and how the slip surface failure was formed. Parts e) and f) shows the points of the area of influence of failure mechanism for types of local and general shear failure.

Fig. 8. Scheme of the calculation process in FLAC2D.

Table 3 Numerical calculation cases in FLAC. c (kPa)

φ (°)

q (% Phe)*

e0

PL (% Phe)*

10, 25, 50

20, 25

0; 10; 20;40

0.74; 0.82; 0.90

1; 10; 25; 50; 75; 90

* The values of (q ) and (PL) are expressed as a percent of (Phe).

permanent load (PL) as can be seen in Table 3. 5.3.2. Calculation methodology in FLAC Using established models, an analysis was carried out under two possible methodologies to calculate the bearing capacity in FLAC2D. The first methodology corresponds to applying velocity in the nodes of the footing (load with settlement) and the second was the safety factor with the SRM technique (Strength Reduction Method). The last methodology can be implemented in two ways, one applied to cohesion and friction, and the other with only the cohesion parameter. 5.3.3. Bearing capacity To calculate bearing capacity in FLAC2D models, it was applied vertical downward velocity along the width of the footing. This velocity was applied to the model in the footing nodes, which is controlled and quantified as the vertical displacement for each calculation step that is performed and must be sufficiently small to minimize any inertial

6. Analysis and discussion of results The analysis of the general failure results applying the bearing 9

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Fig. 9. Scheme of calculation of footing with pore water pressure.

was almost zero, which means that there would be no influence of void ratios, the same as when PL/Phe = 1. The influence of the void ratio was in the range between PL/Phe = 0.25 and 0.90; the maximum values for a value of q = 0 of 3%, for q = 0.10*Phe the variation was 2%, for q = 0.20*Phe of 3% and for q = 0.40*Phe the variation was 2%. This maximum variation shows the values in global terms and for PL/ Phe = 0.75. In all cases, it has shown that the most critical void ratio is the highest and the least critical a smaller void ratio. This is consistent with the theory, since for a higher void ratio value is expected to be more susceptible to cyclic softening. Based on this analysis, the charts in Fig. 11 are presented only for the highest void ratio, which is e0 = 0.90 and will be the most critical or susceptible to cyclic softening.

capacity methodology was based on the generation of the proposed design charts (Fig. 11). These charts are shown in Fig. 11 and correspond to cohesion (c ) of 10, 25 and 50 kPa, friction angle (φ ) of 20 and 25° and effective load outside the foundation (q ) of 0, 0.1, 0.2 and 0.40 of the static bearing capacity (Phe). In addition, the void ratios (e0 ) were varied to show their influence. 6.1. Full charts Part a) in Fig. 11 corresponds to the cohesion of 10 kPa and a friction angle of 20°. In this chart, it can be observed that curves for values greater than 50% of PL/Phe have the same trend. The difference between values of q = 0 and q = 0.40*Phe is 15% compared in global terms but if expressed in relative terms of the load this can increase up to 40%. Part b) has a cohesion of 10 kPa and friction angle of 25°, this chart shows the same tendency for PL/Phe values greater than 50% and for minor values a considerable variability of 15% in global terms for a value close to zero of PL/Phe. Analogously, the charts for other combinations of cohesion and an angle of friction, show a similar tendency as observed for the first chart.

6.3. Influence of an effective load outside the foundation An analysis was made of the variation of the curves as a function of an effective load outside the foundation for soil with voids ratios e0 = 0.74, 0.82, 0.90 and shown in Fig. 13. It is appreciated that for values of more than 50% of PL/Phe have the same curves tendency so it can say that there is little influence of an effective load outside the foundation. However, for values lower than 50% there is a considerable difference between the maximum and minimum values of 13% in global terms. Fig. 13 shows the variation based on the void ratio as mentioned above is very small, so it only focused on analyzing part c. The most critical case is when there is a greater effective load outside the foundation and the most favorable case is when this load is zero. Moreover, moving from a value of q = 0 to a value of q = 0.10 for the worst case of PL/Phe = 0.01 is 9% in global terms and this variation decreases as a value of the permanent load.

6.2. Influence of the void ratio The influence of the void ratio was analyzed for the first soil varying an effective load outside the foundation of q = 0, 0.10, 0.20 and 0.40 * Phe and for three void ratios e1 = 0.74; e2 = 0.82 and e3 = 0.90 as shown in Fig. 12. Due to the small influence of the void ratio, only the best fit for each data group is presented. The maximum variation due to the void ratio was 3% in global terms and occurs when the variation of voids ratios goes from e0 = 0.74 to e0 = 0.90. For small values of PL/Phe from 0.01 to 0.25 the variation 10

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Fig. 10. General shear failure and local shear failure.

and the least favorable was q = 0.40*Phe: when comparing q = 0*Phe with q = 0.40*Phe in part a); the maximum variation of Phd/Phe was 14% in global terms for a 0.01 PL/Phe. When the angle of friction is changed to 25°, the variation in part b) of Fig. 14 becomes 2% in global terms. All the values in general decrease relative to Fig. 14 part b), with the maximum variation of 11% in global terms when comparing for q = 0 and q = 0.40*Phe with 0.01 of PL/Phe.

6.4. Influence of cohesion For our analysis, three soils were taken with the same value of internal friction angle to demonstrate the influence of the cohesion. It was analyzed the curves for a void ratio (e0) of 0.90 and (q) that goes from 0, 0.10, 0.20 and 0.40 of PL/Phe. Fig. 14 shows little variation when the cohesion was increased from 10 to 25 and 50 kPa with an internal friction angle of 20°. Case of q = 0 is the most favorable case was q = 0 11

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Fig. 11. Design charts proposed for different types of terrain (c, φ ), overload (q ) and permanent load (PL ). 1.0

9% in global terms. For soil that increases cohesion to 25 kPa in the part b) of Fig. 11, the variation is smaller with a maximum of 12% and a minimum of 8% in global terms. When increasing the cohesion to 50 kPa in the part c) of Fig. 11, the variation was 11% as a maximum value and the minimum about 9% in global terms for 0.01 of PL/Phe. Based on the calculations represented in Figs. 11–14, the following conclusions can be clearly observed. The internal friction angle has an influence of up to 10% in global terms when the permanent load is low in relation to the static bearing capacity, decreasing as said permanent load increases (Fig. 11). The void ratio for the range of values considered (0.74, 0.82 and 0.90) has a negligible influence on the evaluation of the dynamic bearing capacity (Fig. 12). The influence of the effective load outside the foundations represents an influential parameter for the design of footing when the permanent load has a very low value in relation to the static bearing capacity (Fig. 13). In the case of cohesion, for ranges studied (10, 25 and 50 kPa) a much lower influence is observed with maximum variation values of only 1% in global terms for the same effective load outside the foundation (Fig. 14). It is worth mentioning that the use of charts (Figs. 11–14) within the range of data in which the analysis was performed is valid, although a similar trend is expected for soils with geomechanical characteristics (cohesion and friction angle) that are outside the analyzed range.

0.9 0.8

Phd/Phe

0.7

0.6 0.5 0.4

q=0*Phe

(Best Įt line)

0.3

q=0.10*Phe (Best Įt line)

0.2

q=0.20*Phe (Best Įt line) q=0.40*Phe (Best Įt line)

0.1

0.0

0.2

0.4

0.6

0.8

1.0

PL/Phe Fig. 12. Influence of void ratio for soil with c = 10 kPa and φ = 20°, with q = 0; 0,10; 0,20; 0,40*Phe.

6.5. Influence of friction angle The influence of the friction angle was analyzed based on the two soils, for the cohesion of 10 kPa, 25 kPa and 50 kPa that can be seen in Fig. 11. These graphs show that there is a variation in almost all the development of the curves from a value of 0.90 of PL/Phe to 0. In Fig. 11 part a), the maximum value of 14% is presented in global terms, giving values of greater (Phd/ Phe ) for the cohesion of 20° and lower values for a cohesion value of 25°. This variation decrease, for q = 0.10*Phe to 10%, for q = 0.20*Phe to 8% and for q = 0.40*Phe to

7. Case study Carrefour Shopping Center 7.1. Description As an example of the application of the formulation and 12

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Fig. 13. Influence of an effective load outside the foundation for soil with c = 10 kPa and φ = 20°, with e0 = 0.74, 0.82, 0.90.

increase the bearing capacity of shallow foundations and wick drains at a depth of 20 m to precharge soft soils. The stratigraphic profile is shown in Fig. 15: a) consists of marine sediments with alternate layers of clay, silt, and sand with the water table at 2 m deep. The soil profile has 2 m of medium dense fill of gravelly clay (GC) called “old fill”; the following 5.2 m are saturated fine grain soils (ML/CL) of soft to firm resistance and low plasticity; subsequently there is a 1.2 m layer of loose to medium dense silty sand (SP/SM). This sand layer is on 0.9 m of soil (ML/CL) followed by highplasticity clay (CH) of medium-high stiffness that extends to a depth

methodology presented here, it is solved for the historical case of Carrefour Shopping Center [12,16,22], located in northwestern Turkey during the Kocaeli earthquake in 1999. This earthquake had a magnitude of 7.4 that generated a maximum acceleration of 0.24 g in the analysis area. This area was used as an auxiliary parking area with planned improvements to the soils for future construction. At the time of the earthquake, it was in one of the phases of the improvement of the soil, so there were improved areas and others that were not improved. These improvements consisted of a surcharge fill of 3.3 m called “new fill”, high-modulus columns by jet grouting of 60 cm in diameter to

Fig. 14. Influence of cohesion for soils with φ = 20° and φ = 25°. 13

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Fig. 15. (a) Soil profile Carrefour Shopping Center near SE2, the positions of settlement rings and soil layers A-E, (adapted from [12]). (b) Settlements measured by extensometer [12].

internal friction angle (φ ) of 20° and 25°, it was analyzed whether the cyclic softening is possible for the earthquake produced in the study area. It should be remembered that the cohesion and friction angle values are not the geotechnical values of the soil, but correspond to the made-up values that allow us to obtain certain shear strain at each point of soil when subjected to cyclic loading. The calculation of the shear stress produced by the earthquake is based on the simplified representation of the distribution of shear stress with depth in a soil column in one dimension that was proposed by Seed and Idriss [62].

Table 4 Soil properties for the numerical model. Soil layer

Depth (m)

PI

Su (kN/m3)

γ (kN/m3)

Existing Fill (GC) Limos arcillosos (ML/CL) Arena limosa (SM) Limos arcillosos (ML/CL) Arcilla de alta plasticidad (CH)

0–2 2–7.2 7.2–8.4 8.4–9.3 9.3–35

– 10 – 11 37

– 25 – 31 35

20 18.6 20 18.6 16.4

Note: the overload of the new fill of 3.3 m has a total unit weight of 20 kN/m3.

τeff = 0.65

greater than 35 m [22]. Geomechanical properties are shown in Table 4. Settlements measured by the extensometer (SE-2) after the earthquake are shown in Fig. 15, where the vertical movements of each layer of soil located between the measurement rings for layers A through E. Jet-grouting columns were effective at mitigating damage by reducing cyclic shear stress and pore water pressures in the improved areas [12]. However, the sections that did not have soil treatment experienced settlements of almost 12 cm (Fig. 15b) produced by cyclic softening in clay soils. A fine-grain soil with a plasticity index greater than or equal to 7 may be susceptible to cyclic softening [22]. Tsai et al. [16] assessed the bearing capacity with a global pseudostatic stability approach before and during the earthquake. The safety factor before the earthquake was 3, during the earthquake without strength reduction was 1.3 and during the earthquake with a 20% strength reduction in soil was 0.9.

amax σv0 rd g

(2)

Where,

τeff = effective cyclic shear stress amax = peak ground acceleration g = acceleration of gravity σv0 = in situ vertical total stress rd= stress reduction factor According to Tsai et al. [16] and Fig. 15b, the depth of affection at which the collapse occurs is 10 m under the existing fill. By means of (2), an average shear stress value (τav ) was obtained in the affected zone of 19.7 kPa. The external load value (CL ) that produces average shear stress of 19.7 kPa in the 10 m of influence can be calculated in the first approximation applying the theory of elasticity and distributed in the application width is 56 kN/m. This is the value of cyclic surface load that produces the same increase in shear stress in the area of influence as the earthquake and can be compared with the values in Fig. 15b obtained from the proposed charts of Fig. 11. Comparing the value of cyclic surface load obtained with Table 5, it can see that it is a cyclic load value that fits within charts 1 and 2. This corresponds to a reasonable value to produce cyclic softening in this type of soil. In turn, it corresponds to made-up soil values with the cohesion of 10 kPa and the internal friction angle of 20–25°. The results indicate that the type of soil studied due to the earthquake is susceptible to cyclic softening. Besides, the procedure together with the chart can be used in a practical and simple way as a first approach to detect the phenomenon of cyclic softening.

7.2. Case study development The resolution of the case study can be approached in a simplified manner using the methodology outlined in this investigation. In the simplified problem shown in Fig. 16 it can be assumed, that there is an effective load outside foundation (q ), distributed in the application width, equal to 40 kN/m, which corresponds to the old fill. A permanent load (PL) distributed in the application width of the new embankment equal to 106 kN/m, which corresponds to the sum of old fill and new fill. A single cohesive soil was considered to underlie the model capable of producing cyclic softening due to the accumulation of pore water pressures produced by the cyclical load of the earthquake. Within the geomechanical properties used for the elastic numerical model is the modulus of elasticity (E ) with a value of 5 MPa, a Poisson coefficient (ν ) of 0.25, and unit weight (γ ) 19.62 kN/m3. The geometric characteristics of the model are shown in Fig. 16. For different values of properties the cohesion (c ) 10, 25, 50 kPa and

8. Conclusions It was possible to evaluate the influence of pore-water pressure in the calculation of the bearing capacity of shallow foundations in 14

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Fig. 16. Equivalent model for the Carrefour case.

Table 5 Cyclic load values obtained from charts (Fig. 11). Chart of Fig. 11

Cohesion c [kPa]

Friction angle ϕ [°]

Cyclic Load CL [kN/ m]

1 2 3 4 5 6

10 10 25 25 50 50

20 25 20 25 20 25

52.54 71.77 117.71 126.39 238.61 245.26

-

Note: the cyclic load (CL) for this case study is distributed in the application width.

-

cohesive soils under cyclic load. An equation of pore-water pressure generation from laboratory data is proposed. This formulation was implemented in FLAC2D, which allowed generating charts of design. These charts of design within the range of data in which the analysis was performed is valid, although a similar trend is expected for soils with geomechanical characteristics (cohesion and friction angle) that are outside the analyzed range. The internal friction angle and the effective load outside are the parameters that has the greatest influence on the dynamic bearing capacity (Phd). On the other hand, the cohesion and void ratio have hardly any influence on the evaluation of this dynamic bearing capacity for the ranges of values considered (c = 10, 25 y 50 kPa) y (e0 = 0.74, 0.82 and 0.90). The main aspects and conclusions are the following:

-

-

-

- From extensive laboratory testing carried out in cyclic simple shear tests, it was observed that the generation of pore water pressure under cyclic load depends on variables such as initial shear stress (τ0) , cyclic shear stress (τc ), effective vertical consolidation stress ' ) and initial voids ratio (e0 ) . Due to this number of variables, (σv0 multivariable nonlinear regression methods were used to make the fit more robustly and to consider the interdependence between all the variables. The fit value for the new formulation was 85% for the experimental data. - The formulation was implemented in the FLAC2D program employing FISH code to apply it to the particular case of a footing by varying the properties of the soil (c, φ ), effective load outside the foundation (q ), void ratio (eo) and permanent load (PL). A

-

15

convergence analysis was made to guarantee the precision for the solution and calculation time optimization. The methodology (Fig. 8) for the numerical calculation includes 6 stages that allow considering the different stress states that the soil acquires in each phase of the analysis, in this way it is possible to considers the generation of pore water pressure produced by the cyclic load in susceptible soils of cyclic softening. Charts (Fig. 11) were presented to analyze the bearing capacity before cyclic loading with a cohesion property of 10, 25 and 50 kPa; and of an internal friction angle of 20 and 25° that allows limiting the strains of the soil according to Fig. 7. The influence of the void ratio (e0 ) had a small variation of 2–3% in global terms. Since this variation was not significant, it was adopted to take the most critical value of 0.90 for subsequent calculations. The variation of the effective load outside the foundation (q ) was up to 13% in global terms for small values of the permanent load (PL), decreasing such variation as the value of the permanent load rises, until it becomes almost null. The influence of cohesion taking into account the same effective load outside the foundation (q ) and the same void ratio (e0 ) was of the order of 1% in global terms. But when comparing the values for effective load outside the foundation (q = 0) with (q = 0.40*Phe) the variation becomes 11% in global terms. This indicates an increase in cyclic softening resistance with increased cohesion. The friction angle is the parameter that has the greatest influence along with the charts in the dynamic bearing capacity (Phd) with values of 8, 9 and 10% for permanent load (PL) of 0.01 comparing values of friction angle of 20° and 25°. That is, the greater the angle of internal friction, the greater the susceptibility to cyclic softening. The formulation and methodology presented here were applied to the documented real case (Carrefour Shopping Center [12,16,22]), where the vertical movements of each layer of soil are measured by extensometers placed at different depths. In this case, the susceptibility of the soil to cyclic softening was demonstrated. Results showed that the procedure together with the chart can be used in a practical and simple way as a first approach to detect the phenomenon of cyclic softening.

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Declaration of Competing Interest

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