Static and seismic bearing capacity of shallow strip footings

Soil Dynamics and Earthquake Engineering 84 (2016) 204–223

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Static and seismic bearing capacity of shallow strip footings Ernesto Cascone n, Orazio Casablanca Dipartimento di Ingegneria, University of Messina, Contrada di Dio, 98166 S. Agata Messina, Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 31 October 2015 Received in revised form 10 February 2016 Accepted 13 February 2016

In this study, the evaluation of static and seismic bearing capacity factors for a shallow strip footing was carried out by using the method of characteristics, which was extended to the seismic condition by means of the pseudo-static approach. The results, for both smooth and rough foundations, were checked against those obtained through finite element analyses. Under seismic conditions the three bearing capacity problems for Nc, Nq and Nγ were solved independently and the seismic bearing capacity factors were evaluated accounting separately for the effect of horizontal and vertical inertia forces arising in the soil, in the lateral surcharge and in the superstructure. Empirical formulae approximating the extensive numerical results are proposed to compute the static values of Nγ and the corrective coefficients that can be introduced in the well-known Terzaghi's formula of the bearing capacity to extend its applicability to seismic design of foundations. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Bearing capacity Shallow foundations Earthquakes Soil inertia Method of characteristics Pseudo-static analysis

1. Introduction Past earthquakes provided evidence of the susceptibility of shallow foundations to undergo large deformations and failure also in situations where soil liquefaction was not triggered by the ground motion (e.g. [1–5]). In fact, due to rapid changes in shaking direction and amplitude during earthquake loading, the available soil shear strength under a foundation may be repeatedly and momentarily attained inducing several instantaneous failures resulting in the accumulation of permanent settlements rather than in a gross spectacular bearing capacity failure; this makes harder the identification of the actual phenomenon. Moreover, the recognition of foundation failures is sometimes hindered by the failure of the superstructure. Also experimental evidence of foundation behaviour and bearing capacity failures under seismic loading has been provided by a number of studies reporting the results of shaking table [6–11] and dynamic centrifuge tests [12–14]. Table 1 lists some of these studies along with some details concerning the features and the results of the tests. The results of cyclic and dynamic tests pointed out the crucial influence of the accumulated permanent rotation of the foundation on the degradation of bearing capacity due to the reduction of footing-soil contact area. Rocking is the prevailing mode of deformation, the ratio of the maximum settlement smax to the width B of the foundation being usually in the range 15–30%. However, it is worth noting that in many cases the models were either too high or the cyclic horizontal force was n

Corresponding author. Tel.: þ 39 090 3977162. E-mail address: [email protected] (E. Cascone).

http://dx.doi.org/10.1016/j.soildyn.2016.02.010 0267-7261/& 2016 Elsevier Ltd. All rights reserved.

applied well above the foundation and thus they did not represent a simple foundation but rather reproduced small structures. Accordingly, the tests included in the overall behaviour of the model the inertial effects arising both in the soil mass, due to the propagation of seismic waves, and in the superstructure, due to its dynamic response and, in some cases, severe tilting or even toppling of slender models (H/B 41.75, where H is the height of the model) was observed, consistently with field evidence after strong earthquakes [15]. Seismic design of foundations requires, in principle, rigorous modelling of the seismic soil-structure interaction, capable to reproduce non-linear soil behaviour under dynamic loading. Such a general approach is, however, costly and time-consuming and is suitable only for important projects. Also the macro-element approach, though appears as a promising tool for performancebased design of foundations (e.g. [16]), still is far from being an established method of analysis. In routine analyses the evaluation of bearing capacity and the seismic response of the superstructure are decoupled. The seismic bearing capacity of a foundation can be represented by means of a bounding surface in the space of loading parameters as proposed by Pecker [17] or, alternatively, it can be evaluated using the formula introduced by Terzaghi [18] for a strip footing resting on a homogeneous dry soil subjected to a vertical and uniformly distributed load: 1 qult ¼ cN c þ qN q þ γ B Nγ 2

ð1Þ

In Eq. (1) qult represents the ultimate load that the soil can sustain under the assumption of rigid plastic behaviour; Nc, Nq and

Table 1 Experimental studies on seismic bearing capacity. Experimental device

Model dimensions

Soil

Static load

Dynamic loading

Results and notes

Taylor and Crewe [6]

Shaking table EERC Laboratory Bristol University

B ¼ 0.4 m L ¼ 0.95 m H ¼ 0.4 m D ¼ 0 m

Leighton Buzzard sand Dr ¼ 58%

W ¼ 3.65 kN V ¼ 30 kN

1976 Friuli earthquake scaled Gemona record compressed in time amax ¼ 0.44–1.5 g

smax ¼ 97 mm ϑμαξ ¼ 5°

Maugeri et al. [7, 8]

Shaking table EERC Laboratory Bristol University

B ¼ 0.4 m L ¼ 0.95 m H ¼ 0.4 m D ¼ 0.1 m

Leighton Buzzard sand γd ¼ 15.4 kN/m3 emax ¼ 0.79 emin ¼ 0.49 Dr ¼ 48.5% [7] Dr ¼ 53.34% [8]

W ¼ 3.65 kN V ¼ 30 kN [7] V ¼ 20 kN; e ¼ 0.05 m [8]

Sine dwell, f ¼ 5 Hz amax ¼ 0.15–0.665 g [7] amax ¼ 0.10–0.35 g [8]

smin ¼ 55 mm; smax ¼ 73 mm [7] dmax ¼ 4.7 mm; ϑmax ¼ 2.6°[7] smin ¼  128 mm; smax ¼ 66 mm; ϑmax ¼ 25.8° [8] Detection of the slip surface

Al Karni and Budhu [9]

Shaking table

B ¼ 0.102 m L ¼ 0.102 m H ¼ 0.178 m D ¼ 0 m

Silica sand D50 ¼ 0.55 mm Cu ¼ 2.5 emax ¼ 0.95 emin ¼ 0. 58 Dr ¼ 677 5%

W ¼ 0.205 kN

Irregular shaking amax ¼ 0–1.05 g f ¼ 3 Hz

smax ¼ 50 mm; dmax ¼ 60 mm; ϑmax ¼ 55° Detection of the slip surface

Knappet et al. [10]

Shaking table Cambridge University

B ¼ 0.05 m L ¼ 0.3 m H ¼ 0.1 m D/B ¼ 0–0.5

Silica sand D50 ¼ 0.9 mm emax ¼ 0.82 emin ¼ 0.495 Dr ¼ 67%

q ¼ 8.42 kPa

Sinusoidal shaking amax ¼ 0.16-0.31 g f ¼ 2.28-3.6 Hz

save ¼ 1.35–13.41 mm Model toppling Detection of the deformation pattern

Shirato et al. [11]

Shaking table PWRI Tsukuba

B ¼ 0.5 m L ¼ 0.5 m

Toyoura sand γd ¼ 15.7 kN/m3

W ¼ 8.39 kN

smax ¼ 6–17.6 mm dmax ¼ 3.5–101.6 mm

H ¼ 0.25 m D ¼ 0–0.05 m

Dr ¼ 80%

1993 Hokkaido Nansei Oki earthq. Schichiho Bridge record 1995 Kobe earthquake NS JMA record

ϑμαξ ¼0.57°  9.22°  toppling Pressure beneath the foundation recorded Reduction of footing–soil contact area

Zeng and Steedman [12]

Centrifuge test (50g) Cambridge Geotechnical Centrifuge Centre

prototype B ¼ 1.67 m H ¼ 5 m D ¼ 0.5 m

Hostun sand dry or saturated D50 ¼ 0.35 mm emax ¼ 0.967 emin ¼ 0.607 Dr ¼ 55–63%

prototype q ¼ 383 kPa

Irregular shaking amax ¼ 0.19–0.45 g

Dry model (prototype): smax ¼ 0.3 m; ϑmax ¼ 6.3° soil heave 0.15 m Saturated model (prototype): smax ¼ 0.5 m; ϑmax ¼ 12° soil heave 0.10 m Sudden failure was observed

Garnier and Pecker [13]

Centrifuge test (100g) LCPC Nantes Centre

prototype R ¼ 30 m

Antirion clay with vertical inclusions

W ¼ 8.9–9.3 kN

T ¼ 7 5 to 7 35 MN (applied at 11.8 m from foundation) M ¼ 7 70 to 7 170 MNm 5–10 loading cycles

smax ¼ 0.6 m dmax ¼ 1.8 m

Gajan et al. [14]

Centrifuge test (20g)

prototype B ¼ 0.4–1 m L ¼ 2.5–4 m D ¼ 0–0.7 m

Nevada sand

various models with different weight

Vertical slow cyclic tests

The settlement in dynamic tests is larger than the settlement in horizontal slow cyclic tests

Center of Geotechnical Modeling University of California, Davis

D50 ¼ 0.17 mm Cu ¼ 1.6 emax ¼ 0.881 emin ¼ 0. 536 Dr ¼ 60% and 80% San Francisco Bay mud LL ¼ 90% PL ¼ 38% Su ¼ 100 kPa

Horizontal slow cyclic tests Dynamic irregular shaking

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Reference

Reduction of footing–soil contact area and rounding of soil surface beneath the footing observed in slow cyclic horizontal loading No significant uplift observed in dynamic tests

205

Key: B,L,R,H,D: width, length, radius and height of the foundation, and depth of embedment; γd, emax, emin, Dr, D50, Cu: soil dry density, maximum and minimum void ratio, relative density, average soil particle diameter, uniformity coefficient; LL, PL, Su: soil liquid and plastic limit, undrained shear strength; W, V, e, q: weight, vertical force, eccentricity, applied pressure; T, M: applied cyclic horizontal force and moment; amax, f: maximum acceleration and frequency of dynamic horizontal loading; g: gravity acceleration; smin, smax, save, dmax, ϑmax: minimum, maximum and average permanent settlement (positive downwards), maximum horizontal permanent displacement, maximum permanent rotation.

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Nγ are the bearing capacity factors, depending on the friction angle φ of the soil; γ and c are the unit weight and the cohesion of the soil, respectively; B is the width of the foundation and q is the vertical pressure (lateral surcharge) acting on the ground surface aside the foundation. Expressions of corrective factors have been proposed over the years to account for the shape and the depth of the foundation, the inclination of the ground and of the bearing surface, the inclination and the eccentricity of the applied load (e.g. [19–22]). These solutions still form the basis of current state of practice in design of shallow foundations and, in seismic prone regions, the load inclination factors are commonly used to embody the effects of earthquake-induced inertia forces on bearing capacity. Modern studies on seismic bearing capacity were initiated by Richards et al. [23] and by Sarma and Iossifelis [24]. Since then, a number of studies were carried out approaching the problem mostly by means of the three classical methods of stability analysis, namely, limit equilibrium, limit analysis and the method of characteristics and focusing on the assessment of the influence of soil and superstructure inertia on bearing capacity factors. Table 2 lists some of these studies [1,17,24–50] along with some details concerning the method of analysis. Other solutions are available in the literature dealing with the problem of seismic bearing capacity of shallow foundations resting on sloping ground (e.g. [51–54]) in which the case of horizontal ground is treated as a particular case. In most studies the earthquake effect on the foundation is introduced through the pseudo-static approach considering horizontal inertia forces acting on the foundation, as transmitted by the superstructure, and in the soil mass involved in the plastic mechanism; in some cases the inertia forces arising in the lateral surcharge are also considered. In limit equilibrium and limit analysis solutions different shapes are assumed for the slip surface bounding the plastic volume, ranging from simplified two-wedge surface to modified log-spiral Prandtl-type surface accounting for foundation uplifting, whilst in the analyses carried out using the method of characteristics the sliding surface is derived from the fulfilment of equilibrium conditions and failure criterion. With the exception of a few studies [17,26,32,33,36–39,45], the majority of the available solutions do not allow decoupling the two, soil and superstructure, inertia effects; moreover, horizontal accelerations acting in the soil and on the superstructure are unrealistically assumed to be in phase and have the same amplitude, despite the soil and the superstructure are likely to exhibit different responses to a given seismic excitation. Other studies [31,48] are based on the pseudo-dynamic approach, originally proposed by Steedman and Zeng [55], that fails to satisfy the stress-free boundary condition of wave motion at the ground surface. In this study, the evaluation of seismic bearing capacity factors is carried out for a shallow strip foundation on a Mohr–Coulomb material under drained conditions. The method of characteristics, extended to the seismic condition through the pseudo-static approach, is adopted and the differential equilibrium equations are solved numerically via a finite difference formulation. The effects of inertia forces in the soil, in the lateral surcharge and in the superstructure are distinguished in the analysis allowing to consider different seismic acceleration coefficients in the evaluation of the inertia forces in the soil and in the lateral surcharge (kh) and in the superstructure (khi). The effect of seismic vertical acceleration is also taken into account in the assessment of the soil and superstructure inertia effects on bearing capacity reduction. In the analyses both smooth and rough foundations are considered and the effect of seismic acceleration on the plastic mechanism is pointed out. All the solutions are checked against results obtained through finite element (FE) analyses. Under static conditions

(kh ¼khi ¼0) the values of the bearing capacity factor Nγ are also obtained. The features mentioned above, taken altogether, represent a novel contribution to the problem of seismic bearing capacity of shallow foundations. In fact, to the Authors' knowledge, there are no comprehensive solutions available that account, globally, for the effects (evaluated separately) of soil, lateral surcharge and superstructure inertia, for the effect of vertical acceleration on soil inertia and for the roughness of the foundation and that have been checked through FE analyses for large ranges of relevant parameters. Furthermore, most of the available solutions are provided in graphical form for a few values of the friction angle and narrow ranges of the seismic horizontal acceleration, so that their use in practice is limited. Conversely, in this study, useful empirical formulae approximating the extensive numerical results (reported in tables and shown in figures) are proposed to compute the static values of Nγ and the corrective coefficients of static bearing capacity factors that can be introduced in current expression of the bearing capacity (Eq. (1)) to extend its applicability to seismic design of foundations.

2. Method of analysis The approach used in deriving bearing capacity factors is the method of characteristics, widely employed in static analysis (e.g. [56–59]) as well as in pseudo-static analysis of shallow foundations (e.g. [26,34,36,39]). The basic assumptions of the method are that soil behaves as a rigid-plastic material, obeys the Mohr–Coulomb yield criterion and in a soil mass at incipient failure, equilibrium equations and plastic condition must be satisfied. The resulting set of equations can be solved to obtain the ultimate load on a foundation. Under plane strain conditions the differential equations of equilibrium are:

τ

∂ xy ∂σ x ∂x þ ∂y

¼X

∂τxy ∂σ y þ ¼Y ∂x ∂y

ð2Þ

where x and y are the horizontal and vertical directions, σi and τij are normal and shear stresses acting on horizontal and vertical planes and the terms X and Y are the horizontal and vertical body forces per unit volume. The plastic condition is expressed by the following equation: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ  σ 2 σx þ σy x y sin φ þ c cos φ þ τ2xy ¼ ð3Þ 2 2 where c and φ are the cohesion and the friction angle of the soil, respectively. Under pseudo-static conditions the body forces can be expressed as: X ¼ γ  sin ϑ Y ¼ γ  cos ϑ

ð4Þ

where: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ  ¼ γ k2h þ ð1  kv Þ2

ð5Þ

γ is the soil unit weight and ϑ ¼ tan  1



kh 1  kv



ð6Þ

is the inclination of γn to the vertical direction, kh and kv being the horizontal and vertical seismic acceleration coefficients (obtained dividing corresponding seismic accelerations by gravity acceleration), the latter assumed positive in the upward direction.

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207

Table 2 Theoretical studies on seismic bearing capacity. Reference

Method

Seismic effects

Seismic acceleration Plastic mechanism

Sarma and Iossifelis [24]

LE

H: SO þLSþ ST

kh ¼khi

Log-spiral shear zone between active and passive wedge Focus of the log-spiral at the edge of the footing Based on slope stability analysis

Budhu and Al-Karni [25]

LE

H&V: SO þ LS þST kh ¼khi kv ¼ kvi

Log-spiral shear zone between active and passive wedge Focus of the log-spiral at the edge of the footing

Richards et al. [1]

LE

H&V: SO H:LSþ ST

kh ¼ khi kv

Two-wedge simplified Coulomb mechanism

Shi and Richards [26]

MC LE

H: SO þLSþ ST

kh khi ¼ f  kh

MC: derived from equilibrium conditions LE: two-wedge simplified Coulomb mechanism

Zhu [27]

LE

H&V: SOþ ST

kh ¼khi kv ¼ kvi

One-sided Prandtl-type mechanism consisting of rigid triangular blocks

Fishman et al. [28]

LE

H: SOþ LSþ ST

kh khi ¼ n  tanf

Two-wedge simplified Coulomb mechanism

Choudhury and Subba Rao [29] LE

H&V: SOþ LSþ ST

kh ¼khi kv ¼ kvi

Log-spiral shear zone between active and passive wedge Critical position of the focus of the log-spiral

Merlos and Romo [30]

LE

H: ST

khi

Logarithmic spiral

Ghosh and Choudhury [31]

LE

H&V: SOþ LSþ ST

kh ¼khi kv ¼ kvi

Two-wedge simplified Coulomb mechanism

Castelli and Motta [32,33]

LE

H&V: SO

kh kv khi

Circular, based on slope stability analysis

H: LSþST Kumar and Mohan Rao [34,35]

MC

H&V: SO þ LS þST kh ¼khi

Derived from equilibrium and yield conditions

Maugeri and Novità [36]

MC

H: SO þLSþ ST

kh khi ¼ f  kh

Derived from equilibrium and yield conditions

Cascone et al. [37,38]

MC

H: SO þLSþ ST

kh khi

Derived from equilibrium and yield conditions

Caputo et al. [39]

MC

H: SO þLSþ ST

kh khi

Derived from equilibrium and yield conditions

Ghahramani and Berrill [40]

ZEL

H: SO þLSþ ST

kh ¼khi

Goursat radial shear zone between active Coulomb and passive Rankine wedges

Pecker and Salençon [41]

LA

H: SOþ ST

kh khi

One-sided Prandtl-type mechanisms

Dormieux and Pecker [42]

LA

H: SOþ ST

kh ¼khi

One-sided Prandtl-type mechanism

Conte [43]

LA

H: SOþ ST

kh khi

One-sided Prandtl mechanism

Paolucci and Pecker [44]

LA

H: SOþ ST

kh khi

3D modified one-sided Prandtl-type mechanism accounting for possible uplifting

Paolucci and Pecker [45]

LA

H: SOþ LSþ ST

kh khi

One-sided Prandtl-type mechanism Modified one-sided Prandtl-type mechanism accounting for possible uplifting

Pecker [17]

LA

H: SOþ ST

kh khi

Modified one-sided Prandtl-type mechanism accounting for possible uplifting

Soubra [46]

LA

H: SOþ LSþ ST

kh ¼khi

One-sided mechanism

Chatzigogos et al. [47]

LA

H: SOþ ST

kh khi

3D modified one-sided Prandtl-type mechanism

Ghosh [48]

LA

H&V: SO þ ST

kh ¼khi kv ¼ kvi

Two-wedge mechanism

Shafiee and Jahanandish [49]

FEM (Plaxis)

H: SOþ LSþ ST

kh ¼khi

Not shown

Pane et al. [50]

FDM (FLAC)

H: SOþ LSþ ST

kh khi

Not shown

LE: Limit Equilibrium; MC: Method of Characteristics; LA: Limit analysis (kinematic approach); ZEL: Zero Extension Line; FEM: Finite Element Method; FDM: Finite Difference Method; H, V: Horizontal, Vertical; SO,LS,ST: Soil inertia, Lateral surcharge inertia, Structure inertia.

Under static conditions kh ¼kv ¼ 0 and the body forces result X ¼0 Y ¼γ

ð7Þ

Substituting the plastic condition into the differential equilibrium equations a set of two iperbolic partial derivative equations is obtained that, starting from the boundary conditions, can be integrated to determine the stress state and the associated characteristic lines network.

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Three boundary-value problems must in turn be solved: the Cauchy problem, the Riemann problem and the Goursat problem, each of them concerning one zone of the soil mass in the condition of plastic equilibrium, as shown in Fig. 1a for the Hill mechanism representing the case of a perfectly smooth foundation and in Fig. 1b for the Prandtl mechanism, relevant to the case of a perfectly rough foundation. Further numerical finite element (FE) analyses were carried out by using the commercial code Plaxis [60]. The FE analyses were carried out under plane strain conditions using triangular 15-noded elements with 12-point Gaussian integration and a fourth-order polynomial interpolation for displacements. Vertical boundaries were moved away from the domain of interest and restrained in the horizontal direction. In the analyses an elasto-plastic model with an associated flow rule was employed to describe soil behaviour. In both static and seismic FE analyses for the evaluation of the factor Nγ, to avoid the singularity at the edges of the smooth foundation and prevent violation of the yield criterion a triangular stress distribution was applied on the ground surface, having maximum amplitude at the centre of the foundation and being zero at the edges. To simulate a rough foundation a rigid plate element was introduced without any interface element and a uniform stress distribution was applied on the plate. In the FE analyses of both smooth and rough foundations the amplitude of the applied stresses was increased until an abrupt increment in the vertical displacement of the centre of the foundation was attained, implying that the ultimate bearing capacity was reached. The solution of the Nc and Nq does not involve any particular numerical difficulty in both static and seismic conditions. Conversely, to prevent numerical instabilities in the Nγ problem it was necessary to introduce a very small value of the lateral surcharge q (q/γ ¼ 10  7) in the analyses carried out by using the method of characteristics and a value of the cohesion as small as 10  3 kPa in the finite element analyses. These additional contributions to the actual value of the bearing capacity factor are fairly negligible.

For rough foundations (δ/φ ¼1) Fig. 2 shows that present results are more conservative than those by Meyerhof [19], Vesic [21] and Bolton and Lau [58] and less conservative than those proposed by Brinch-Hansen [20]. Finally, for both smooth and rough foundations, the values of Nγ were found to be coincident with those computed by Martin [59] which are claimed to be exact values since it was shown that, in the realm of flow associativeness, through an accurate refinement of the mesh of characteristics it is possible to achieve the coincidence of the stress and velocity calculations as well as to extend the stress field beyond the plastic volume. Thus, the use of the values of Nγ given in Table 3 and Fig. 2 (or, equivalently, those provided by Eq. (10), introduced below) is recommended in static bearing capacity analyses [59]. In the range φ ¼ 15°C45° the numerical values of Nγ (Table 3) are satisfactorily fitted by the expression       1  n3 Nγ ¼ N q  1 tan 1:34φ n þ ð10Þ 2 where n ¼ tan δ= tan φ

ð11Þ

For a perfectly rough foundation (n ¼1) Eq. (10) approximates with satisfactory accuracy the numerical values Nγ with maximum error of about 9% for φ ¼15°, when Nγ is small; for a smooth foundation (n ¼0) the maximum error is as large as 23% for φ ¼15°. The error is slightly larger for 0 on o1, attaining maximum values of about 15% for n ¼1/3 and φ ¼45°. The values of Nγ for perfectly smooth and perfectly rough foundations were obtained also by performing finite element analyses. The results of the FE analyses are shown in Table 3 and Fig. 2 and are in a very good agreement with those obtained using the method of characteristics. The values of the static bearing capacity factors obtained through the method of characteristics will be used in the following sections to evaluate consistent ratios of seismic to static bearing capacity factors.

3. Static bearing capacity factors For a weightless soil (γ ¼0) under static conditions the method of characteristics allows evaluating the exact solutions of the bearing capacity factors Nc [61] and Nq [62] in closed algebraic form: π φ eπ tan φ N q ¼ tan 2 þ ð8Þ 4 2   ð9Þ N c ¼ Nq  1 cot φ that depend only on the angle φ and are independent of the roughness of the foundation. If the self-weight of the soil is considered (γ a0) the values of the bearing capacity factor Nγ can be obtained through the method of characteristics only by numerical integration of the equilibrium equations. Unlike Nc and Nq, Nγ depends also on the roughness of the soilfoundation interface, which can be expressed by the soil-foundation friction angle δ, and on the dilation angle ψ, assumed herein equal to φ. Table 3 lists the numerical results obtained for Nγ as a function of φ for different values of the ratio δ/φ. In Fig. 2 the results are compared with some of the solutions available for the bearing capacity factor Nγ. For the case of smooth foundations (δ/φ ¼ 0) the results obtained in this study using the method of characteristics are in a perfect agreement with literature results obtained using the same method of analysis, and also in a very good agreement with the results of the finite element analyses presented herein.

4. Seismic bearing capacity factors The seismic bearing capacity qultE can be written as: 1 qultE ¼ cN cE þ qN qE þ γ B N γE 2

ð12Þ

that is the seismic counterpart of the static formula (Eq. 1) where NcE, NqE and NγE are the bearing capacity factors under seismic conditions that, as previously mentioned, depend on the inertia forces arising in the soil involved in the plastic mechanism and on the inertia forces transmitted by the superstructure onto the foundation. Fig. 3 shows the sketches of the problems considered in the analyses. Specifically, Figs. 3a, b, and c show the loading schemes adopted for the evaluation of the seismic bearing capacity factors NscE , N sqE and N sγ E , accounting for the effect of soil inertia (denoted by superscript s) while Figs. 3d, e and f are relevant to the evass ss luation of the seismic bearing capacity factors N ss cE , N qE and N γ E accounting for the effect of the inertia of the superstructure (denoted by superscript ss). 4.1. Effect of soil inertia on bearing capacity factors To determine the bearing capacity factors the characteristic equations where integrated for values of the angle of φ in the range 15 45°, values of kh in the range 0-tanφ and different values of the ratio kv/kh, varying from 1 to 1. In the analyses φ was varied with a

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209

Fig. 1. Goursat, Riemann and Cauchy domains of the plastic mechanism of (a) a smooth and (b) a rough foundation. Fig. 2. Static bearing capacity factor Nγ for smooth and rough foundations. Table 3 Static bearing capacity factor Nγ. φ

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

Nγ (Method of characteristics)

Nγ (FEM)

δ/φ

δ/φ

0

1/3

1/2

2/3

1

0.70 0.83 0.98 1.15 1.35 1.58 1.85 2.16 2.53 2.96 3.46 4.05 4.74 5.55 6.52 7.65 9.00 10.61 12.52 14.82 17.58 20.90 24.94 29.84 35.83 43.19 52.27 63.53 77.59 95.25 117.56 145.97 182.40 229.50 290.90 371.68

0.90 1.08 1.29 1.54 1.82 2.16 2.56 3.03 3.59 4.25 5.02 5.94 7.03 8.33 9.88 11.73 13.95 16.62 19.83 23.71 28.41 34.14 41.14 49.73 60.32 73.41 89.70 110.07 135.69 168.10 209.35 262.21 330.45 419.22 535.55 689.61

0.99 1.19 1.43 1.70 2.03 2.41 2.86 3.39 4.01 4.75 5.62 6.65 7.87 9.33 11.05 13.12 15.58 18.54 22.09 26.37 31.55 37.84 45.49 54.84 66.33 80.50 98.05 119.90 147.29 181.80 225.54 281.34 353.13 446.16 567.63 727.91

1.07 1.29 1.54 1.84 2.19 2.60 3.09 3.66 4.33 5.12 6.05 7.15 8.46 10.00 11.83 14.01 16.61 19.71 23.43 27.90 33.29 39.82 47.74 57.40 69.22 83.78 101.75 124.10 152.02 187.13 231.54 288.16 360.81 454.79 577.36 738.83

1.18 1.42 1.69 2.02 2.40 2.84 3.36 3.96 4.68 5.51 6.49 7.64 9.00 10.61 12.51 14.75 17.43 20.61 24.42 28.99 34.48 41.11 49.15 58.93 70.89 85.58 103.72 126.23 154.34 189.66 234.32 291.20 364.21 458.70 582.04 744.84

0

1 0.75

1.26

1.61

2.98

3.49

6.75

7.68

15.25

17.68

35.97

43.83

87.03 Fig. 3. Sketches of loading conditions considered in the analyses.

120.75

239.98

step of 1° while kh was varied with a step of 0.01, or even 0.001 when approaching the limit condition tanϑlim ¼tanφ (or kh,lim ¼tanφ when kv ¼ 0), representing, according to Richards et al. [23], the fluidization condition.

For the evaluation of N scE and N sqE a weightless medium is considered. This implies that no inertia effects develop in the soil underneath the foundation (Fig. 3a and b), however inertial effects can arise in the lateral surcharge q acting on the ground surface aside the foundation, usually due to foundation embedment (Fig. 3b). As a result, the seismic bearing capacity factor N scE , evaluated assuming q¼0, coincides with its homologous static factor Nc, while values of N sqE , evaluated assuming c¼ 0, can be computed by inclining the lateral surcharge q to account for the effect of a horizontal inertia shear stress component khq. Table 4 lists some of the

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Table 4 Seismic bearing capacity factor N sqE . kh

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

φ 15

20

25

30

35

40

45

3.94 3.78 3.59 3.36 3.07 2.64

6.40 6.16 5.90 5.61 5.27 4.89 4.40 3.66

10.66 10.31 9.93 9.52 9.07 8.57 8.02 7.38 6.60 5.46

18.40 17.85 17.27 16.66 16.00 15.30 14.55 13.73 12.82 11.80 10.58 8.91

33.30 32.41 31.48 30.51 29.49 28.42 27.29 26.09 24.81 23.43 21.94 20.28 18.36 15.98

64.20 62.67 61.09 59.45 57.75 55.98 54.13 52.21 50.19 48.08 45.84 43.46 40.91 38.14 35.05 31.46 26.86

134.88 132.05 129.13 126.12 123.02 119.83 116.53 113.12 109.59 105.93 102.13 98.17 94.02 89.66 85.05 80.13 74.80 68.92 62.19

Fig. 5. Seismic bearing capacity factor N sγE for smooth and rough foundations: (a) effect of horizontal and (b) vertical seismic acceleration and comparison with FE analysis results for φ¼ 30°.

Table 5 Seismic bearing capacity factor N sγE for smooth foundations. kh

Fig. 4. Seismic bearing capacity factor N sqE : (a) effect of horizontal and (b) of vertical seismic acceleration and comparison with FE analysis results for φ¼ 30°.

values of NsqE evaluated using the method of characteristics (MC) for kv ¼0 and different values of the angle of φ. Fig. 4a shows the plots of N sqE against kh. For increasing kh the bearing capacity factor decreases and abruptly drops to zero when the limit value kh,lim ¼tanφ is attained, meaning that the soil shear strength is fully mobilized to resist seismic induced shear stresses in the lateral surcharge and the footing has no bearing capacity.

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

φ 15

20

25

30

35

40

45

0.70 0.61 0.52 0.42 0.30 0.17

1.58 1.44 1.30 1.15 0.99 0.81 0.62 0.36

3.46 3.23 3.01 2.77 2.52 2.26 1.98 1.67 1.33 0.87

7.65 7.27 6.88 6.48 6.06 5.63 5.17 4.70 4.19 3.64 3.01 2.21

17.58 16.88 16.16 15.43 14.67 13.90 13.09 12.26 11.39 10.47 9.50 8.45 7.28 5.88 3.26

43.19 41.78 40.35 38.88 37.38 35.84 34.26 32.64 30.96 29.23 27.43 25.54 23.55 21.42 19.11 16.48 13.26

117.56 114.38 111.14 107.85 104.48 101.05 97.54 93.96 90.28 86.51 82.64 78.64 74.50 70.20 65.71 60.97 55.91 50.42 44.37

The effect of the vertical component of the seismic acceleration is shown in Fig. 4b for φ ¼30°. Positive (upward) values of kv dramatically affect N sqE since they reduce the vertical component of the lateral surcharge giving rise to larger inclinations (angle ϑ in Eq. (6)) of the resultant lateral surcharge at the foundation level. Conversely, negative (downward) values of kv lead to an increase of the bearing capacity if compared to the case kv ¼0.

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Table 6 Seismic bearing capacity factor N sγE for rough foundations. φ

kh

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

15

20

25

30

35

40

45

1.18 1.06 0.92 0.78 0.62 0.40

2.84 2.63 2.41 2.19 1.94 1.67 1.36 0.92

6.49 6.14 5.77 5.39 4.96 4.53 4.07 3.57 2.96 2.14

14.75 14.11 13.44 12.75 12.03 11.27 10.48 9.65 8.75 7.76 6.62 5.15

34.48 33.23 31.94 30.61 29.25 27.83 26.37 24.84 23.24 21.56 19.76 17.80 15.61 12.97 7.91

85.58 82.95 80.26 77.51 74.69 71.79 68.81 65.74 62.57 59.28 55.85 52.25 48.45 44.38 39.93 34.87 28.57

234.32 228.18 221.93 215.56 209.05 202.41 195.62 188.67 181.54 174.22 166.69 158.92 150.87 142.49 133.73 124.48 114.61 103.86 91.77

The evaluation of the bearing capacity factor N sγ E was carried out assuming c¼ 0, γ a0 and q ffi 0 (Fig. 3c). Fig. 5a shows the curves of N sγ E versus the horizontal seismic coefficient in the soil kh for different values of φ, kv ¼0, and for the two cases of perfectly smooth (δ/φ ¼0) and perfectly rough (δ/φ ¼ 1) foundation. As for the case of N sqE , also N sγ E vanishes when the limit condition kh,lim ¼tanφ is reached. Fig. 5b shows the effect of the vertical acceleration on the values of N sγ E obtained for smooth and rough foundation and φ ¼30°. N sγ E decreases for increasing kh and for positive values of kv. Tables 5 and 6 list some of the values of N sγ E for both smooth and rough foundations, respectively. In Figs. 4b and 5b the results obtained through finite element analyses are plotted together with those obtained using the method of characteristics. It can be observed that there is a satisfactory agreement among results obtained using the two methods of analysis. Seismic bearing capacity factors may be conveniently normalized with respect to their homologous static factors to point out the reduction in bearing capacity due to seismic induced soil inertia effects and to obtain corrective coefficients that can be applied to the usual bearing capacity formula (Eq. (1)) in order to account for this effect. As discussed above, owing to the absence of self-weight, for any value of φ and kh it is NscE ¼ Nc, thus the corrective coefficient for Nc is: escE ¼

NscE ¼1 Nc

ð13Þ

The corrective coefficients for NsqE and Nsγ E are: esqE ¼ s

eγ E ¼

N sqE Nq

N sγ E Nγ

Fig. 6. Corrective coefficients to account for soil inertia effect for kv ¼ 0: (a) esqE , (b) esγE .

Table 7 Coefficients of Eqs. (15) and (16). A

The curves obtained for esqE and esγ E , the latter for both cases of smooth and rough foundation, are plotted in Fig. 6 for the case kv ¼0: it is evident that soil inertia has a significant effect on bearing capacity. For example, for kh ¼0.3, the corrective coefficients are as small as esqE ¼0.82 and esγ E ¼0.74 for smooth andesγ E ¼ 0.76 for rough foundation when φ ¼ 35°, and reduce to esqE ¼0.75 and esγ E ¼0.57 for smooth and esγ E ¼ 0.63 for rough foundation when φ ¼25°, showing that the effect of soil inertia

b2

b3

esqE

0.92

0

0.511

0.118

esγE (δ/φ¼0)

0.92

0.290

 0.277

0.716

esγE (δ/φ¼1)

0.92

0.198

 0.014

0.528

cannot be neglected in bearing capacity analysis, especially for lower values of φ. Empirical expressions were developed to fit the numerical results obtained through the method of characteristics that allow expressing the corrective coefficients for the effect of soil inertia in the following form:  B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NsjE k 2 2 esjE ¼ kh þ ð1 kv Þ ð15Þ ¼ 1  A h cot φ Nj 1  kv where j ¼q or γ according to bearing capacity factor under consideration, A is independent of the bearing capacity factor and of the roughness δ of the soil-foundation interface, and B is a function of tanφ and has the general form: B ¼ b1 tan 2 φ þb2 tan φ þb3

ð14Þ

b1

ð16Þ

The values of the coefficients A, b1, b2 and b3 are given in Table 7 for the different cases considered in the analyses. Eq. (15) can be used for any value of φ 40 and of kh and kv provided that the condition tanϑ otanφ is fulfilled. The value of the horizontal acceleration coefficient kh to be introduced in Eq. (15) depends on the peak ground acceleration (PGA) expected at the site and on deformations (permanent settlements and rotations) considered admissible for the footing. In principle, if no deformation is admitted, then kh ¼ PGA/g, where g is the gravity acceleration; conversely, accounting for the pseudo-static

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Fig. 7. Effect of kv on the corrective coefficients and comparison with Eq. (15) for φ¼ 30°: (a) esqE , (b) esγE for smooth foundations and (c) esγE for rough foundations.

displacement analyses, as suggested by Richards et al. [1] for shallow foundations and by Biondi et al. [63] for retaining walls. For φ ¼ 30°, Fig. 7 shows the effect of the vertical acceleration on the corrective coefficients and also that the values of esqE (Fig. 7a) and esγ E (Fig. 7b for δ/φ ¼ 0 and Fig. 7c for δ/φ ¼1) computed through Eq. (15) closely match the numerical results provided by the method of characteristics. The plastic mechanisms obtained under static and seismic conditions are shown in Fig. 8 for the NsqE problem and in Fig. 9 for the N sγ E problem for the cases of δ/φ ¼0 and δ/φ ¼1. All the figures are relevant to φ ¼ 30°, kv ¼0 and kh ¼0, 0.1, 0.3 and 0.5. In the N sqE problem the characteristic lines in the Cauchy zone are straight lines and their directions depend on the boundary condition represented by the inclination of the lateral surcharge (i.e. on kh and kv); the characteristic lines in the Goursat zone span the full width of the foundation and are also straight lines but their directions are not affected by kh and kv since the boundary condition under the foundation consists in a vertical uniform pressure. Finally, the characteristic lines are log–spiral curves in the Riemann zone. In Fig. 8 it is apparent that increasing kh the Cauchy zone is extending and the Riemann zone is reducing, the overall effect on the plastic volume is that its length increases while its maximum depth is not affected by the value of the seismic acceleration. In the N sγ E problem the effect of soil inertia produces a change in the inclination of the characteristic lines with respect to the static case, mainly in the Cauchy zone, and, to a minor extent, also in the Goursat zone. As it is known, for γ a 0 the Riemann zone degenerates (e.g., [57]). At the ground level, under the foundation, the inclination of the characteristic lines does not change since it depends on the boundary condition represented by the vertical ultimate load. For increasing kh the Goursat zone reduces and the Cauchy zone increases, while the depth of the mechanism remains constant. As it could be expected the plastic mechanisms relevant to the smooth foundation (Fig. 9a) are much smaller and shallower than the corresponding mechanisms obtained for the case of the rough foundation (Fig. 9b). In both cases the plastic volume involves half of the foundation width. In Fig. 10 the characteristic lines are superimposed to the contours of incremental displacements provided by the FEM analyses for the N sqE (Fig. 10a) and N sγ E (Fig. 10b for δ/φ ¼0 and Fig. 10c for δ/φ ¼1) problems for the case φ ¼30° and kh ¼0.3. It can be observed that the stress field and the displacement field obtained using the two different methods of analysis are practically coincident. 4.2. Effect of superstructure inertia on bearing capacity factors According to the schemes of Fig. 3d, e and f, to determine the ss ss bearing capacity factors N ss cE , N qE and N γ E the load acting on the foundation was inclined at an angle ϑi   khi ð17Þ ϑi ¼ tan  1 1  kvi where khi and kvi are respectively the ratio of the horizontal and vertical inertia forces, HE and VE, transmitted onto the foundation by the superstructure in seismic condition to the static vertical force Q:

Fig. 8. Characteristic lines network for the N sqE problem for φ¼ 30°, kh ¼ 0–0.5, kv ¼ 0.

nature of the proposed approach and if small deformations are admitted, then a fraction βf of the PGA may be used: kh ¼ βf PGA/g. A proper choice of the reduction coefficient βf should be based on

khi ¼ H E =Q and kvi ¼ V E =Q

ð18Þ

and depend on the seismic loading and on the dynamic response of the structure. Both the method of characteristics and the finite element method were used to evaluate the seismic bearing capacity factors, however, using the method of characteristics with γ a 0 it was not possible to establish correct boundary conditions for the case of

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213

Fig. 9. Characteristic lines network for the N sγE problem for φ¼30°, kh ¼0–0.5, kv ¼0: (a) smooth and (b) rough foundations.

Fig. 10. Comparison between the characteristic lines network and the total incremental displacement contours obtained through FE analyses for φ¼30°, kh ¼ 0.3, kv ¼0 for (a) the N sqE problem, (b) the N sγE problem of a smooth foundation and (c) the N sγE problem of a rough foundation.

rough foundation, thus this case (Nss γ E with δ/φ ¼1) was analysed only through FE method. ss ss In Fig. 11 N ss cE , N qE and N γ E are plotted against khi for different values of the angle of φ. For increasing values of khi the bearing capacity factors decrease until a limit value for khi (i.e. a limit inclination of the applied load) is attained, implying that the foundation fails by sliding. This limit value can be expressed as  

  exp π =2  φ tan φ 1 þ sin φ  

  tan ϑi;lim ¼ tan φ ð19Þ exp π =2  φ tan φ 1 þ sin φ  1 for the bearing capacity factors N ss cE (Fig. 11a), as derived in closed form by Kezdi [64] and, similarly to the case of soil inertia effect, as tan ϑi;lim ¼ tan φ

ð20Þ Nss qE

N ss γE

for the bearing capacity factors (Fig. 11b) and (Fig. 11c and d). When the limit conditions expressed by Eqs. (19) and (20) are ss ss achieved, N ss γ E abruptly drops to zero, whereas N cE andN qE attain ss limit but positive values. For NcE this minimum value is

approximately constant and equal to about 3 (Fig. 11a), while for Nss qE it varies between about 1.8 and 4 when φ varies in the range 15  45° (Fig. 11b). Tables 8–11 list the values of the bearing capacity factors obtained for φ ¼ 15°C45° and for several values of the acceleration coefficient khi. The effect of the vertical component of the seismic acceleration is shown in Fig. 12 for φ ¼30° and for different values of the ratio kvi/khi. Changes in kvi/khi result in changes of the inclination of the resultant load acting on the foundation, leading to a reduction or an increase of the seismic bearing capacity factors, with respect to the condition kvi ¼0, according to either positive or negative sign of kvi. The results obtained by means of the finite element analyses are plotted in Fig. 12 together with those computed by using the method of characteristics: the agreement among results is very satisfactory. The reduction of bearing capacity factors due to inertia effects in the superstructure can be expressed through ratios of seismic to

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ss ss ss Fig. 11. Effect of superstructure horizontal inertia force on seismic bearing capacity factors: (a) N ss cE , (b) N qE , (c) N γE for smooth and (d) N γE for rough foundations.

Table 8 Seismic bearing capacity factor N ss cE .

Table 9 Seismic bearing capacity factor N ss qE .

φ

khi

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

khi

15

20

25

30

35

40

45

10.98 10.24 9.48 8.72 7.95 7.20

14.83 13.76 12.67 11.59 10.53 9.50 8.52 7.60

20.72 19.09 17.47 15.88 14.35 12.89 11.52 10.24 9.07 7.99

30.14 27.55 25.02 22.59 20.28 18.11 16.10 14.25 12.56 11.04 9.67 8.45

46.12 41.77 37.60 33.66 29.97 26.57 23.46 20.63 18.10 15.83 13.81 12.03 10.47 9.09 7.89

75.31 67.45 60.07 53.22 46.94 41.22 36.08 31.48 27.41 23.81 20.66 17.91 15.51 13.43 11.63 10.07 8.72

133.88 118.31 104.00 90.98 79.26 68.81 59.56 51.44 44.34 38.19 32.86 28.27 24.33 20.95 18.04 15.56 13.43 11.61 10.04

static bearing capacity factors: ess cE ¼ ess qE ¼ ess γE ¼

N ss cE Nc Nss qE Nq N ss γE Nγ

ð21Þ

that can be introduced as corrective coefficients into the bearing capacity formula (Eq. (1)).

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

φ 15

20

25

30

35

40

45

3.94 3.67 3.37 3.03 2.66 2.18

6.40 5.93 5.43 4.92 4.39 3.85 3.27 2.55

10.66 9.82 8.96 8.10 7.25 6.42 5.61 4.80 3.99 3.06

18.40 16.81 15.24 13.72 12.25 10.85 9.53 8.29 7.12 6.02 4.95 3.83

33.30 30.15 27.11 24.22 21.51 18.98 16.64 14.50 12.55 10.78 9.17 7.70 6.33 5.01 3.17

64.20 57.49 51.17 45.29 39.87 34.92 30.45 26.43 22.85 19.66 16.84 14.35 12.15 10.19 8.44 6.84 5.28

134.88 119.20 104.75 91.58 79.71 69.09 59.68 51.39 44.13 37.81 32.32 27.57 23.46 19.90 16.83 14.16 11.83 9.78 7.93

ss ss The coefficients ess cE , eqE and eγ E have the same meaning of inclination coefficients and can be used even under static conditions assuming ϑi as the inclination of the static load applied onto the foundation. In general, if a static force HS is applied onto the foundation, then an equivalent value of khi ¼(HS þ HE)/Q should be considered to take into account both static and seismic effects of the superstructure on the evaluation of bearing capacity. ss ss The curves obtained for ess cE , eqE and eγ E are plotted in Fig. 13 for the case kvi ¼0 and show that the inertia of the superstructure has a dramatic effect on the seismic reduction of bearing capacity. The ss coefficients ess cE and eqE are noticeably affected by both khi and φ

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Table 10 Seismic bearing capacity factor N ss γE for smooth foundations. khi

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

φ 15

20

25

30

35

40

45

0.70 0.58 0.47 0.36 0.27 0.17

1.58 1.32 1.08 0.86 0.67 0.50 0.35 0.22

3.46 2.90 2.39 1.93 1.53 1.19 0.90 0.65 0.45 0.28

7.65 6.40 5.28 4.28 3.43 2.70 2.09 1.58 1.17 0.84 0.58 0.36

17.58 14.64 12.03 9.76 7.82 6.19 4.83 3.72 2.83 2.12 1.56 1.12 0.78 0.51 0.24

43.19 35.65 29.09 23.47 18.74 14.81 11.59 8.98 6.90 5.25 3.95 2.94 2.16 1.56 1.11 0.75 0.48

117.56 95.79 77.25 61.70 48.84 38.34 29.87 23.10 17.75 13.55 10.29 7.76 5.82 4.34 3.21 2.35 1.70 1.20 0.83

Table 11 Seismic bearing capacity factor N ss γE for rough foundations (FEM). khi

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

φ 15

20

25

30

35

40

45

1.26 1.10 0.95 0.77 0.56 0.33

3.02 2.65 2.28 1.90 1.52 1.14 0.77 0.42

6.75 5.97 5.16 4.34 3.56 2.83 2.16 1.55 1.00 0.55

15.23 13.47 11.56 9.73 8.05 6.50 5.12 3.92 2.93 2.06 1.33 0.70

35.97 31.06 26.23 21.89 18.26 14.90 11.84 9.42 7.27 5.51 4.08 2.90 2.03 1.31

87.03 75.64 65.03 53.40 44.60 34.72 28.35 23.44 17.60 13.76 10.44 7.71 5.66 4.01 2.85 1.82 1.20

239.98 202.87 171.55 147.70 118.80 93.05 73.30 55.63 43.20 34.51 26.74 19.78 15.55 11.32 8.42 6.19 4.51 3.11 2.15

(Fig. 13a and b), whereas the coefficient and ess γ E (Fig. 13c and d) rapidly decreases for increasing khi, but is less influenced by the angle φ if values of khi far from khi,lim are considered. The numerical results obtained through the method of characteristics and, for ess γ E and δ/φ ¼ 1, through finite element analyses were interpolated by empirical equations that allow expressing the corrective coefficients for the effect of superstructure inertia in the following form:  D N ss khi jE ¼ ¼ 1  C cot φ ð22Þ ess jE Nj 1  kvi where j ¼c, q or γ according to bearing capacity factor under consideration, C is a constant coefficient and D is a quadratic function of tanφ: D ¼ d1 tan 2 φ þ d2 tan φ þ d3

ð23Þ

215

The values of the coefficients C, d1, d2 and d3 are given in Table 12 for the different cases considered in the analyses. Eq. (22) reproduces satisfactorily the numerical results, as it is shown in Fig. 14 for φ ¼ 30°, including also the effect of the vertical acceleration coefficient, but its use is limited by the condition tanϑi otanϑi,lim, expressed by Eqs. (19) and (20). Moreover, Eq. (22), though representing an empirical approximation, allows an accurate assessment of ess qE , in good agreement with the analytical closed-form solution provided by De Simone [22]. ss Finally, coefficient ess cE can be evaluated as a function of eqE ; in fact, by using Caquot's theorem of corresponding states it is possible to show that: ess cE ¼

N q ess qE  1 Nq  1

ð24Þ

The plastic mechanisms obtained using the method of characteristics under static and seismic conditions are shown in ss Fig. 15a and b for the N ss cE and N qE problems for φ ¼30°, kvi ¼0 and khi ¼0, 0.1, 0.3 and 0.5. Under static conditions (khi ¼0, i.e. vertical load) the characteristic networks for the Nc and Nq problem are identical and symmetrical with respect to the vertical of the foundation (only part of the plastic mechanism is shown in Fig. 15). Under seismic ss conditions (khi 40) for both N ss cE (Fig. 15a) and N qE (Fig. 15b) problems the plastic mechanism is asymmetric. The characteristic lines in the Cauchy zone are straight lines and their directions depend on the boundary condition consisting in a free field condition for the case of N ss cE and in a vertical uniform pressure for the case of Nss qE . The characteristic lines in the Goursat zone span the full width of the foundation and are also straight lines whose direction depends on khi (and kvi). The characteristic lines in the Riemann zone are log–spiral curves. For increasing values of khi the length of the Cauchy zone and the depth of the plastic volume decrease, these reductions being more remarkable for the case of Nss qE (Fig. 15b). A similar effect of khi on the plastic volume is observed in Fig. 16a, concerning the N ss γ E problem of a smooth foundation (δ/φ ¼0, φ ¼ 30°, khi ¼0–0.5, kvi ¼0). In this case, however, the characteristic lines in the Goursat domain are curves and their slope under the foundation depends on the load inclination. Fig. 16b shows, for the same set of parameters, the incremental displacement contours for the N ss γ E problem for the case of a rough foundation (δ/φ ¼1), as obtained via finite element analyses. Apart from the already observed reduction of the plastic volume, it is evident that even for small values of khi the plastic mechanism becomes significantly asymmetric with a negligible volume of soil involved close to the edge of the foundation opposite to the load inclination. As it could be expected, and similarly to the results obtained while analysing the effect of soil inertia, the plastic mechanisms of the smooth foundation (Fig. 16a) are much smaller and shallower than the corresponding mechanisms obtained for the case of the rough foundation (Fig. 16b). ss In Fig. 17, for the three problems of Nss cE (Fig. 17a), N qE (Fig. 17b) and N ss ( δ / φ ¼0) (Fig. 17c), the characteristic lines are superγE imposed to the contours of incremental displacements obtained performing FE analyses, showing an almost perfect coincidence in the plastic mechanisms provided by the two different methods of analysis.

5. Discussion of results Comparing results in Figs. 6 and 13 it is apparent that for a given value of kh ¼khi the effect of the inertia of the superstructure involves a more remarkable reduction of bearing capacity than the effect of the inertia of the soil. However, if, as it could be expected,

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ss ss ss Fig. 12. Effect of superstructure vertical inertia force on seismic bearing capacity factors (φ¼30°): (a) N ss cE , (b) N qE , (c) N γE for smooth and (d) N γE for rough foundations.

the superstructure, owing to its intrinsic capability to dissipate energy, responds with a lower acceleration with respect to soil, then the effect of soil inertia may result of comparable or even larger relevance than the effect of the superstructure inertia. Curves in Fig. 18 represents the loci for which the conditions esqE s ss ¼ ess qE (Fig. 18a) and eγ E ¼ eγ E (Fig. 18b) are satisfied for different values of the angle φ. For a given φ, a point lying below the corresponding curve is relative to a pair (tanϑ, tanϑi) for which the effect of soil inertia is more important than the effect of superstructure inertia; it is apparent that the effect of soil inertia is not negligible when the ratio tanϑ/tanϑi is large or φ is small. Figs. 8 and 9 and Figs. 15 and 16 show that the plastic volume under the foundation obtained in the two cases of soil and superstructure inertia. For increasing kh a larger inertia force in the soil mass is to be balanced and therefore a longer slip surface should develop; conversely, for increasing khi, the effect of superstructure inertia is that of a larger inclination of the applied load which makes the slip mechanism shallower and more localized. Soil and superstructure inertia effects have been evaluated separately; however, it was verified that superposition applies, that is the reduction in bearing capacity factors due to the overall seismic (soil and superstructure inertia) effect can be obtained as the product of the corrective coefficients relative to soil inertia and superstructure inertia. Fig. 19 shows the numerical results obtained for φ ¼30°, kh ¼ khi and kv ¼ kvi ¼0 in the three cases: soil inertia only (curve A), superstructure inertia only (curve B) and combined effect (curve C), relatively to the Nq (Fig. 19a) and Nγ (Fig. 19b) problems. The combined effect is very well approximated by the product (curve D) of results obtained in the two separate analyses, accounting each for one effect only: in the Nq problem curves C and D are practically coincident, in the Nγ problem maximum error is of about 1%. The superposition is obviously verified also for the Nc problem that is not affected by soil inertia. Finally, it can be proved that the superposition holds also for the general case kh akhi and kv akvi.

The corrective coefficients, as obtained through Eqs. (15) and (22), were compared with some of the solutions available in literature. The cases of soil inertia only (Fig. 20), superstructure inertia only (Fig. 21) and combined soil and superstructure effect for kh ¼khi (Fig. 22) were considered, assuming φ ¼30°. The coefficient esqE (Fig. 20a) is almost coincident with that proposed by Shi and Richards [26] who used both the method of characteristics and the method of limit equilibrium on a Coulombtype failure mechanism. A good agreement, up to about kh ¼0.4 was found also with the equations proposed by Paolucci and Pecker [45] and by Pane et al. [50], derived using the upper bound theorem of limit analysis and a numerical finite difference approach, respectively. The coefficients esγ E (Fig. 20b) for smooth and rough foundations bracket the results by Shi and Richards [26], are in a fair agreement with the results obtained by Conte [43] using upper bound theorem of limit analysis, and are slightly more conservative than those proposed by Paolucci and Pecker [45] and by Pane et al. [50]. The equation proposed by Cascone et al. [37], derived for smooth foundations and kv ¼ 0, provides a good estimate of esγ E , but is closer to the results presented herein for rough foundations. ss ss The coefficients ess cE , eqE and eγ E are plotted in Fig. 21 together with the classical solutions for corrective inclination factors and also with the solution recently proposed by Pane et al. [50]. Differences among various solutions are noticeable, especially for ess cE and ess qE (Fig. 21a and b), the solution obtained herein through the method of the characteristics resting approximately in the middle. A smaller scattering is observed for ess γ E (Fig. 21c) for which the solution relative to smooth foundation is comprised between Meyerhof [19] and Brinch-Hansen [20] formulas, whereas the solution relative to rough foundation is very close to Vesic [21] formula for values of khi up to 0.3, and approximately lies in the middle of the available solutions. The bearing capacity factors NqE and NγE, accounting for both soil and superstructure inertia for kh ¼khi are shown in Fig. 22a and b along with other results available in the literature. The

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curves for NqE (Fig. 22a) show a moderate scattering; the proposed solution matches that by Kumar and Mohan Rao [34], based on the method of characteristics, and represents an upper bound of the results. The curves for NγE (Fig. 22b) exhibit significant differences, especially in the static values (kh ¼ khi ¼0) and in the range of smaller acceleration coefficients (kh ¼khi o0.2). The results obtained for smooth foundation represent a lower bound of the available solutions while those relative to the case of rough foundation plot approximately in the middle of the solutions. When seismic bearing capacity factors are normalized with respect to their static values, providing the corrective coefficients eqE ¼ esqE U ess qE eγ E ¼ esγ E U ess γE

ð25Þ

that account for both soil and superstructure inertia effect, differences among solutions are, to a certain extent, damped (Fig. 22c,d). Most results relative to eqE are approximately coincident (Fig. 22c) and several solutions relative to eγE, namely those by Shi and Richards [26], Sarma and Iossifelis [24] and Soubra [46] derived using different methods of analysis, fall within the range defined by the present results evaluated for smooth and rough foundations (Fig. 22d), and seem to be suitable for the evaluation of seismic bearing capacity factors. However, a close scrutiny of Fig. 22a–d suggests that great care should be taken in the choice of the static value of the bearing capacity factors to be multiplied by the seismic corrective coefficients. In this sense, the use of Eqs. (8)–(10) for the static bearing capacity factors, of Eqs. (14)–(16) for the corrective coefficients accounting for the soil inertia effect and of Eqs. (22) and (23) for the corrective coefficients accounting for the superstructure inertia effect leads to a consistent evaluation of seismic bearing capacity of shallow foundations. Thus, the seismic bearing capacity factors may expressed in the following form: N cE ¼ N c U escE Uess cE N qE ¼

Nq U esqE Uess qE

217

Nγ E ¼ Nγ Uesγ E Uess γE

ð26Þ

and seismic bearing capacity can be evaluated through Eq. (12).

6. Conclusions In this study, the evaluation of static and seismic bearing capacity factors for a shallow strip footing was carried out by using the method of characteristics, which was extended to the seismic condition by means of the pseudo-static approach. The results were checked against those obtained by employing a finite element code. In the analyses both smooth and rough foundations were considered. The values of Nγ obtained under static conditions using the method of characteristics were found in a perfect agreement with literature results that may be regarded as exact [59]. The computed values of Nγ were fitted by an empirical equation that can be used in static bearing capacity analyses. Under seismic conditions the three bearing capacity problems for Nc, Nq and Nγ were solved independently and the seismic bearing capacity factors were evaluated accounting separately for the effect of inertia forces arising in the soil, in the lateral surcharge and in the superstructure and also considering the effect of both horizontal and vertical components of seismic acceleration.

Table 12 Coefficients of Eqs. (22) and (23).

ess cE ess qE ess γE ess γE

C

d1

d2

d3

0.4 0.65

3.894 1.780

2.326 1.727

0.019 0.004

(δ/φ¼0)

0.65

3.056

2.683

0.562

(δ/φ¼1)

0.9

2.005

1.452

0.191

ss ss ss Fig. 13. Corrective coefficients to account for superstructure inertia effect for kvi ¼ 0: (a) ess cE , (b) eqE , (c) eγE for smooth foundations, and (d) eγE for rough foundations.

218

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ss ss Fig. 14. Effect of kvi on the corrective coefficients for superstructure inertia effect and comparison with Eq. (22) for φ¼ 30°: (a) ess cE , (b) eqE , (c) eγE for smooth foundations and (d) ess γE for rough foundations.

ss Fig. 15. Characteristic lines networks for (a) the N ss cE and (b) N qE problem for φ¼ 30°, khi ¼0–0.5, kvi ¼ 0.

Soil inertia significantly affects bearing capacity factors N sqE and N sγ E while it has no effect on N scE since this factor is evaluated under the assumption of weightless soil and hence no inertia forces may arise. Superstructure inertia has the effect of inclining the applied load and dramatically reduces bearing capacity factors

ss ss s s Nss cE , N qE and N γ E . The factors N qE and N γ E vanish when the limit condition tanϑlim ¼tanφ is reached, meaning that soil shear strength is fully mobilized by the seismic excitation; similarly a limit value tanϑi,lim may be reached (Eqs. 19 and 20) implying that the foundation fails by sliding. Compared to the case of horizontal

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219

Fig. 16. (a) Characteristic lines networks for the N ss γE problem of a smooth foundation and (b) total incremental displacement contours obtained through FE analyses for the N ss γE problem of a rough foundation (φ ¼30°, khi ¼ 0–0.5, kvi ¼0).

Fig. 17. Comparison between the characteristic lines networks and the total incremental displacement contours obtained through FE analyses for a) the N ss cE ss problem, b) the N ss qE problem and c) the N γE problem of a smooth foundation (φ¼30°, khi ¼ 0.3, kvi ¼ 0).

acceleration only, positive (upward) vertical acceleration induces a significant reduction of bearing capacity factors; conversely, negative (downward) vertical acceleration leads to an increase of bearing capacity. As it could be expected, larger values of the factors N sγ E and N ss γE were obtained when a perfectly rough foundation was considered. Corrective coefficients were defined as the ratios of seismic to static bearing capacity factors to point out the reduction in bearing capacity due to seismic effects; empirical formulae of corrective coefficients, fitting the extensive numerical results, were developed that allow accounting for soil and superstructure inertia effects in bearing capacity of shallow foundations through the usual bearing capacity equation. The corrective coefficients obtained in this study are in a good agreement with solutions available in literature. However, the evaluation of bearing capacity under seismic conditions through the introduction of the

Fig. 18. Comparison between soil and superstructure inertia effect: (a) esqE ¼ ess qE curves, (b) esγE ¼ ess γE curves for smooth and rough foundations.

corrective coefficients requires a proper choice of the static bearing capacity factors. Soil and superstructure inertia effects were evaluated separately; however, it was verified that the reduction in bearing capacity factors due to the overall seismic (soil and superstructure inertia) effect can be obtained as the product of the corrective coefficients relative to soil inertia and superstructure inertia. If the same acceleration is assumed to act both in the soil and on the superstructure, the effect of the inertia of the superstructure prevails on the effect of soil inertia. However, it may be expected that the seismic-induced shear force at the base of a structure can be reduced owing to the capability of the structure to dissipate energy, the extreme case being

220

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Fig. 19. Superposition of soil and superstructure inertia effect for (a) eqE (b) eγE of a smooth foundation and (c) eγE of a rough foundation (φ¼ 30°, khi ¼0.3, kvi ¼ 0).

Fig. 20. Soil inertia effect: comparison with literature results.

represented by base isolated structures. Thus, the superstructure may respond with a lower acceleration with respect to the seismic acceleration in the soil, and the two inertial effects may then result of comparable relevance. Soil and superstructure inertia have opposite effects on the extension of the plastic volume. In fact, increasing soil inertia forces

lead to a larger Cauchy zone while the depth of the mechanism keeps constant; increasing superstructure inertia forces makes the mechanism smaller and shallower. The characteristic lines and the contours of total incremental displacements obtained through FE analyses were found in a very good agreement.

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Fig. 21. Superstructure inertia effect: comparison with literature results.

Fig. 22. Combined soil and superstructure inertia effects (kh ¼ khi): comparison with literature results.

221

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List of symbols B width of the foundation c soil cohesion escE , esqE , esγ E corrective factors accounting for the effect of soil inertia ss ss ess cE , eqE , eγ E corrective factors accounting for the effect of soil inertia HE, VE horizontal and vertical inertia forces transmitted onto the foundation by the superstructure in seismic condition HS horizontal force transmitted onto the foundation by the superstructure in static condition kh, kv horizontal and vertical seismic acceleration coefficients in the evaluation of the inertia forces in the soil and in the lateral surcharge khi, kvi horizontal and vertical seismic acceleration coefficients in the evaluation of the inertia forces in the superstructure n tanδ/tanφ Nc, Nq, Nγ bearing capacity factors under static conditions NcE, NqE, NγE bearing capacity factors under seismic conditions N scE , N sqE , N sγ E bearing capacity factors accounting for the effect of soil inertia ss ss N ss cE , N qE ,N γ E , bearing capacity factors accounting for the effect of superstructure inertia Q vertical force transmitted onto the foundation by the superstructure in static condition q vertical pressure (lateral surcharge) acting on the ground surface aside the foundation qult ultimate load of the foundation X, Y horizontal and vertical body forces per unit volume x, y horizontal and vertical directions δ soil-foundation friction angle σx, σy normal stresses acting on vertical and horizontal planes τxy, τxy shear stresses acting on vertical and horizontal and the terms ϑ inclination of the resultant of body forces in the soil under seismic conditions respect to the vertical direction ϑi inclination of the resultant of forces on the foundation under seismic conditions respect to the vertical direction γ soil unit weight φ soil friction angle

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