Seismic bearing capacity of strip footing on partially saturated soil using moda

Abhijit Anand and Rajib Sarkar

Department of Civil Engineering, IIT (ISM) Dhanbad, Dhanbad 826004, India

Abstract: The present study proposes a novel and simplified methodology to assess the seismic bearing capacity (SBC) of a shallow strip footing by incorporating strength non-linearity arising due to partial saturation of a soil matrix.Furthermore,developed methodology incorporates the modal response analysis of soil layers to assess SBC.A constant matric suction distribution profile has been considered throughout the depth of the soil.The Van Genuchten equation and corresponding fitting parameters have been considered to quantify matric suction in the analysis.SBC has been obtained for three different geomaterials; viz.sand, fly ash and clay, based on their predominant grain size and diverse soil water characteristics curve(SWCC) attributes.Variation of SBC with different modes of vibration and damping ratio are reported for ranges of matric suction pertinent to the geomaterials considered in the study.The relative significance of matric suction on SBC has been reported for suction values within the transition zone of each geomaterial.It is observed that the SBC of sand is drastically reduced, with matric suction reaching beyond the residual suction value.The SBC of fly ash remains constant beyond the residual suction value, whereas the SBC of clay shows an increasing trend toward the practical range of matric suction values.

Keywords: seismic bearing capacity; modal analysis; matric suction

1 Introductio n

Shallow foundations are often considered as one of the most cost-effective solutions to safely transfer the load from the superstructure to the underlying ground.A reasonable prediction of the bearing capacity of shallow foundations has therefore been the most important issue for geotechnical engineers during the past several years.Several researchers, basing their studies on the limit equilibrium approach (Terzaghi, 1943; Meyerhof, 1951,1963; Hansen, 1970; Vesic, 1973) and limit analysis(Michalowski, 1997; Chakraborty and Kumar, 2014)have attempted to solve the problem of bearing capacity.However, one inherent limitation for all their solutions is that in all the analyses the soil was assumed to exist either in dry or fully saturated conditions, thus neglecting the influence of matric suction on the overall bearing capacity response of a foundation.Ignoring matric suction may, therefore, yield an overly-conservative and costly design with regard to footing.

Shallow foundations often are built in arid or semiarid regions, above the ground water table, which is generally partially saturated due to the capillary action of water (vadose zone).Pressure distribution that is due to the external load on a shallow foundation often is localized within the vadose zone, and hence matric suction plays a substantial role in the bearing capacity of the footing.The partially saturated zone above the ground water table may be predominantly sub-divided into two regions: an active zone and a steady zone, as shown in Fig.1.Within the active zone, saturation, and,consequently, the matric suction of soil, varies due to various environmental factors, such as relative humidity,precipitation, evapotranspiration, etc.As a result, matric suction and its distribution varies significantly due to the variation in environmental factors, such as rainfall,temperature, etc.However, beneath the active zone and above the ground water table, matric suction and its distribution are comparatively time-independent (Lu and Likos, 2004; Vahedifard and Robinson, 2015).Shallow foundations are generally located at some depth below the active zone, and the pressure bulb is mostly confined within the steady zone.

Fig.1 Different saturation zones and influencing environmental factors (Adopted from Fredlund et al., 2012)

It may be added here that the influence of matric suction of partially saturated geomaterials on seismic bearing capacity has not been exclusively reported.It is well-established in the literature that matric suction affects the shear strength of partially saturated geomaterials and consequently, this significantly affects bearing capacity.Various researchers (Oloo, 1994; Vanapallietal., 1996; Costaet al., 2003; Abed and Vermeer, 2006;Leet al., 2013; Oh and Vanapalli, 2013, 2018; Vanapalli and Mohamed, 2013; Vahedifard and Robinson, 2015)have investigated the influence of matric suction on the bearing capacity behavior of geomaterials.Suction stress, developed due to the partial saturation of soil,may be quantified using parameters obtained from the Soil Water Characteristics Curve (SWCC).SWCC is a graphical representation of the relationship between the water content (gravimetric or volumetric) and suction (or negative pore water pressure) induced in geomaterials.Ignoring the effect of suction stress while designing a foundation under a seismic condition may prove to be erroneous.

In addition, to guarantee a safe design of a foundation situated in earthquake-prone regions, it becomes imperative to predict the behavior of the footing not only under static stress due to the load of a superstructure but also due to the seismic stresses generated because of earthquakes.In one of the earliest attempts to solve the bearing capacity problem under seismic stresses, Meyerhof (1953)assumed a horizontal inertia force acting on the footing and proposed a closed form solution to the problem.Later, several researchers (Sarma and Iossifelis, 1990;Richardset al., 1993; Budhu and Al-Karni, 1993;Kumar and Kumar, 2003; Choudhury and Subba Rao,2005; Castelli and Motta, 2012) developed various solutions for seismic bearing capacity in which the stresses generated due to earthquakes were quantified by pseudo-static inertia force by considering the inertia of soil and structural inertia.Based on the results obtained from a pseudo static-based analysis, a substantial reduction of bearing capacity due to seismic stresses was established (Sarma and Iossifelis, 1990; Budhu and Al-Karni, 1993; Richardset al., 1993; Dormieux and Pecker, 1995; Salencon and Pecker, 1995; Paolucci and Pecker, 1997; Soubra, 1997, 1999; Zhu, 2000; Kumar and Kumar, 2003; Maedaet al., 2004; Choudhury and Subba Rao, 2005; Merlos and Romo, 2006; Chatzigogoset al., 2007; Ghosh, 2008; Castelli and Motta, 2012).Several researchers, based on numerical techniques such as finite element methods, obtained the seismic performance of soil deposits (Zeghalet al., 2006; Quanet al., 2018; Demir and Ozener, 2019; Pistolaset al.,2020).However, the influence of matric suction was not given due consideration.

Various codes recommend an increase in the allowable bearing capacity by a factor of 1.25 to accommodate the seismic design of a footing(Chowdhury and Dasgupta, 2017).However, the codal recommendation of increase in allowable bearing capacity by a factor of 1.25 should be carefully adopted,considering the physical characteristics of geomaterials and the adequacy of the method of assorted analyses.In view of this, it is noteworthy that the pseudo-static approach has the inherent limitation of considering the constant effect of inertia throughout the depth of soil layers.Moreover, this approach ignores the effect of the depth of bedrock and the dynamic characteristics of geomaterials; consequently, it is mostly applicable for one dominant mode shape.

Therefore, the present study aims to develop a seismic bearing capacity formulation for unsaturated or partially saturated geomaterials through modal response analyses.This investigation has been carried out for three different geomaterials, viz.sand, fly ash, and clay materials, to conform a large range of suction.In this regard, it may be mentioned that fly ash is considered as a geomaterial in addition to natural soils, such as sand and clay, because the utilization of fly ash in applications of filling that takes place in low-lying areas, construction of embankments,and mine filling may provide a sustainable solution for safe and efficient disposal (DiGioia and Nuzzo, 1972;Leonards and Bailey, 1982; Tothet al., 1988; Kim and Chun, 1994; Horiuchiet al., 2000; Kaniraj and Havangi,2001; Kaniraj and Gayathri, 2003; Ghosh and Subba Rao,2006; Bhardwaj and Mandal, 2008; Nadaf and Mandal,2017; Bhattet al., 2019; Bhatia and Kumar, 2020).In the present study, methodology adopted by Zhanget al.(2019) has been extended to incorporate seismic stresses and suction-induced strength parameters to obtain the bearing capacity of the footing by considering its merits compared to its limitations.A constant matric suction distribution profile has been considered to integrate the effect of matric suction.Since the depth of bedrock plays a significant role in the overall seismic-bearing capacity of a footing (Chowdhury and Tilak, 2012), the influence of the depth of bedrock on seismic bearing capacity also has been incorporated.

This study attempts to provide a simplified comprehensive solution for the evaluation of the seismic bearing capacity of a strip footing on various unsaturated or partially saturated geomaterials by incorporating effects of modes of vibration, the damping ratio of the materials, and the depth of bedrock, along with their respective variations.The influence of matric suction of partially or unsaturated geomaterials on the seismic bearing behavior of strip footing is highlighted.A set of parametric studies also is presented to highlight the influence of various governing parameters on overall seismic-bearing capacity.

2 Statement of the problem and methodology

Figure 2 shows the schematic of the problem that is considered in the present study.A rigid and rough strip footing of widthBoverlying the surface of a homogeneous, partially-saturated soil medium is considered for investigation.The footing is assumed to be subjected to external pressurepdue to superstructure load and earthquake load at the bedrock level.The bedrock (shear wave velocity,Vs≥ 760 m/s) is assumed to be at a depth ofHfrom the ground surface, as shown in Fig.2.

The earthquake wave is assumed to propagatefrom bedrock to the surface, and in this process the overlying partially-saturated soil modifies ground motion characteristics.Harmonic horizontal bedrock motion generating vertically propagating shear waves is considered for the analyses carried out in the present study.

2.1 Evaluation of static bearing capacity

The present section discusses the formulation developed for the evaluation of the static bearing capacity of the footing.Under the static loading condition, the net stress component at any point A subtending an angleβwith the footing (refer Fig.2) would provide the summation of stresses occurring from geostatic overburden pressure due to the self-weight of the soil,as well as external pressure,p, from the superstructure.Stresses due to the self-weight of the soil may be expressed as follows:

Fig.2 Schematics of the problem considered in the study

where,σ′z,γ=vertical effective stress,σ′x,γ=lateral effective stress;τxz,γ=shear stress,γ=bulk unit weight of the soil, andk=coefficient of the lateral earth pressure at rest.

Additional stresses at any depthzfrom the ground surface due to pressurepcan be expressed as (Das,2015):

Hence, the total stress at any point A has developed due to external pressurep, and the soil overburden pressure may be obtained by adding the components as follows:

wherekpis the coefficient of passive earth pressure.Also,due to uniform loading on the strip footing, and owing to the symmetry of the problem geometry, the highest stress increments would be expected to occur beneath the footing, along the centerline.At the centerline of the footing, principal stresses beneath the centerline of the footing can be obtained by substitutingθ=0 andβ1=-β2(refer Fig, 2) in Eqs.(1)-(3).The principal stresses beneath the footing along the centerline can therefore be expressed as:

As the point beneath the footing reaches its limit equilibrium due to stresses from the external load, at the limiting external load point, the principal stresses must satisfy the following conditions as per the Mohr-Coulomb′s failure theory:

Substituting expressions for principal stresses from Eq.(4) to Eq.(5) and, after rearranging the terms, the expression for limiting load ‘p’can be expressed as:

where,c′tis the total effective cohesion (true + apparent)andφ′ is the effective angle of internal friction.Equation (6)has two variables, i.e.,βandz,and therefore, some additional information is required to obtain a nontrivial solution for the limiting external load.Zhanget al.(2019) obtained the value ofβcorresponding to maximum depth (zmax) by partially differentiating ∂z/∂βand consequently, substituting the derivative as 0 (zero).Following a similar approach in the present study, the value ofβcorresponding tozmaxis obtained as:

Substituting Eq.(7) into Eq.(6) yields the ultimate limiting load (pu) beneath the footing.In the present study, the depth (z) is assumed to be equal to a quarter of the footing width by considering the maximum depth of the influence zone of footing that corresponds to the maximum plastic distance beneath the shallow foundation (Bejan and Teodoru, 2016; Zhuet al., 2017;Zhanget al., 2018, 2019).Considering this factor, the final expression ofpmay be expressed as:

2.2 Evaluation of seismic bearing capacity

Several researchers (Sarma and Iossifelis, 1990;Budhu and Al-Karni, 1993; Richardset al., 1993;Choudhury and Subba Rao, 2005) have attempted to solve the problem of seismic bearing capacity by adopting a pseudo-static approach.In the pseudo-static approach, however, the seismic acceleration co-efficient is assumed to be constant throughout a given depth, and no emphasis on response of a soil profile is considered.Soil profile, existing between bedrock and a ground surface, substantially affects seismic wave propagation under strong ground motion and hence affects bearing capacity behavior.In this regard, depth of bedrock is a significant factor influencing induced seismic forces(Chowdhury and Tilak, 2012).Assuming shear wave velocity to be constant throughout depth, the soil will undergo free-field vibration, and the period of vibration can be expressed as (Kramer, 1996) given below:

where,His the depth of bedrock from ground level,Vsis the shear wave velocity at the foundation level,andnis the mode number (n=0, 1, 2, 3, …, ∞).The corresponding Eigen-function can be expressed as(Chowdhury and Dasgupta, 2017):

The amplitude of vibration for thenth mode can be expressed as:

Therefore, acceleration due to seismic force can be expressed as:

Hence, the horizontal coefficient of seismic acceleration can be expressed as:

where, g=acceleration due to gravity.The vertical seismic acceleration coefficient is considered to be twothirds ofkh(Kramer, 1996), and therefore the expression ofkvcan be expressed as follows:

The depth (z) in Eqs.(14) and (15) corresponds to the maximum depth of the influence zone of the footing,which may be determined from the maximum plastic distance beneath a shallow foundation.It and is assumed to be a quarter of the footing width (Bejan and Teodoru,2016; Zhuet al., 2017; Zhanget al., 2018, 2019).The expressions obtained for horizontal and vertical seismic acceleration coefficients (as presented in Eqs.(14) and(15)), are adopted in the subsequent formulation of seismic bearing capacity (Eqs.(16)-(23)).

The expression for seismically induced pseudo-static stresses was adopted by following the methodology proposed by Budhu and Al-Karni (1993) while considering the applicability of that methodology for the present problem.Therefore, the stress state of the soil due to external load and seismic loads, becomes modified (Fig.2).

At the centerline of the footing, total stress due to additional loadp,seismic stresses, and soil overburden may be obtained by adding the components as follows:

where,kpEis the coefficient of passive earth pressure under a seismic condition.

The principal stresses, can therefore be expressed as:

Beneath the foundation, the load stemming from the superstructure (external load) and the load due to the self-weight of the soil would be dominant as compared to horizontal seismic load.Let us assume

As it is evident that a magnitude of ‘a’ would be substantially higher than a magnitude of ‘b’(stresses due to external loadpwould be much high as compared to self-weight).Therefore, under this condition, the mathematical approximation formulais valid for a situation which the relative error is less than 5% whena＞ 5b＞ 0 (Zhanget al., 2019; Wang, 1985).

Hence, the expression for principal stresses can be simplified as:

Substituting the expressions for principal stresses as obtained from Eqs.(21) and (22), by taking Eq.(5)and rearranging the terms, we obtain the expression for external load as:

Additionally, by substituting the expressions ofkhandkv(from Eqs.(14) and (15)) in Eq.(23) yields the bearing capacity of the footing under seismic stresses.For subsequent analyses, bedrock depth,H=30 m;Z=0.24 (corresponding to Zone-IV as per IS 1893 (Part-1) (2016)) andI/R=0.5 are adopted unless mentioned specifically (assuming an importance factorI=1.5 for an important structure and a response reduction factor considering footing for a building withR=3.0 as per IS 1893 (Part-1)(2016).The expression for a coefficient of passive earth pressure under a seismic condition (kpE)in Eq.(23) has been adopted from Lancellotta (2007) as given below:

where,δ=the friction angle at the soil wall interface(δ=φ′ in the present condition).It could be seen that for a smooth wall face (δ=0) assumption and in a static condition (ω=0), Eq.(24) simplifies aswhich is same as the expression of Rankine′s (1857)passive earth pressure coefficient for a smooth wall in a static condition.Puttingδ=φ′ in Eqs.(24) and (26)yields the value ofkpEas:

2.3 Matric suction of unsaturated geomaterials

A substantial effort has been put forth by researchers to model the unsaturated behavior of a soil medium.Fredlundet al.(1978) extended the well-established Mohr-Coulomb′s criterion to quantify the shear strength of unsaturated soils by adopting two independent stress state variables, as shown in Eq.(29):

where,c′=effective cohesion;σ=total normal stress;ua=pore-air pressure;uw=pore water pressure;φ′=the effective friction angle; andφb=the suction angle,which represents the contribution of matric suction.Zhanget al.(2019) adopted the aforementioned bilinear shear strength model to yield a closed form formulation of the allowable bearing capacity of a shallow strip footing resting on a homogeneous half-space.However,the assumption of a linear function between shear strength and matric suction, as shown in Eq.(29), does not satisfactorily model the unsaturated behavior of geomaterials for a large range of matric suction, as there has been substantial experimental evidence showing that the suction angle (φb) is significantly non-linear with matric suction within a transition zone of matric suction(Vanapalliet al., 1996).

To model the nonlinear variation of shear strength with matric suction, several researchers (Vahedifard and Robinson, 2015; Vanapalliet al., 1996; Lu and Likos,2006; Prakashet al., 2019 etc.) have adopted the fitting parameters (van Genuchten, 1980) obtained from the soil water characteristics curve (SWCC) to model unsaturated or partially saturated behavior of geomaterials.

In this study, a constant matric suction (ψ) distribution profile has been considered throughout the depth of soil or the geomaterial layer.Unsaturated behavior of geomaterials has been quantified by adopting the van Genuchten (vG) (1980) model and fitting parameters obtained from the SWCC of the soil.The development of suction-based strength parameters are discussed below.

Under an unsaturated framework, the effective stress in soil can be represented as given below (Lu and Likos,2006):

where,σ′=Effective stress,σ=total stress,ua=poreair pressure (=atmospheric pressure assumed in the present study), andσs=suction stress.The expression for suction stress (σs) can be presented by following equations, which were proposed by Luet al.(2010):

here,Serepresents the effective degree of saturation,which can be expressed as below (Lu and Likos, 2004):

where,Sris the residual saturation obtained from SWCC andSis the pore-water degree of saturation of the soil.The effective degree of saturation can be expressed in terms of matric suction, as follows (Lu and Griffiths,2004):

where,ψ=matric suction=(ua-uw),αvandnv=van Genuchten (1980) fitting parameters,mv=1-1/nv.Considering a constant matric suction in soil throughout the entire depth of the layer, an apparent cohesion (capp)is considered to be induced in the soil, which can be represented by the following analytical expression(Prakashet al., 2019; Lu and Griffiths, 2004):

where,φ′=the effective friction angle of the soil.The apparent cohesion of the geomaterial due to the presence of matric suction has been augmented in the analyses to incorporate the role of matric suction.As can be seen from Eq.(34), to quantify the shear strength of partially saturated geomaterials, two additional fitting parameters (αvandnv) are required and may be obtained from an SWCC, as previously discussed.The SWCC of a geomaterial is a graphic representation of negative pore-water pressure (or suction) with reference to its degree of saturation (or gravimetric/volumetric water content) (Prakashet al., 2019).A typical SWCC of a geomaterial is presented in Fig.3.The SWCC exhibits two distinct boundaries, and within the boundaries the matric suction may be characterized as the boundary effect zone, the transition zone, and the residual zone of matric suction.Determination of water content (or the degree of saturation) of the in-situ soil sample (or a sample prepared in the laboratory) and its corresponding negative pore-water pressure (or matric suction) may be obtained from a set of laboratory apparatuses, such as a tensiometer, a matric potential sensor, an equitensiometer,etc.A prescribed parametric model is usually fitted to a limited number of test data using the best fit technique(Liet al., 2019), and fitting parameters (αvandnv) are obtained.To arrive at the fitting parameters (αvandnv), a prescribed parametric model often is used (for example,van Genuchten′s 1980 model in the present study, as shown in Fig.3).The van Genuchten (1980) model may be represented as:

Fig.3 Typical SWCC depicting the different zones of matric suction and air-entry value

The above parametric model is fitted to scattered test input data, and the best fit may be obtained by using the SoilVision software package, which adopts a non-linear regression algorithm based on the quasi-Newton method(Deka and Sreedeep, 2015).Parameterαvis related with the inverse of air-entry pressure value and parameternvis related with the slope of the curve at the inflection point (rate of desaturation).It may be mentioned here that for the instantaneous nature of earthquake loading,the influence of environmental factors on the matric suction distribution profile is ignored.

3 Validation of the developed solution

3.1 Static validation

The bearing capacity formulation developed for static loading was validated with published results in the literature, and a close agreement was obtained.These results were compared regarding the bearing capacity factor (Nγ) for a rigid and rough surface stripfooting placed on a cohesionless soil for a range of friction angles.Comparison of the bearing capacity factor is presented in Fig.4.For static validation, the bearing capacity factor (Nγ) has been obtained from pertinent studies in the literature (Ukritchonet al., 2003;Hansen and Christensen, 1969; Caquot and Kerisel,1953; Booker, 1969; Bolton and Lau, 1993; Meyerhof,1963; Chen, 1975; Soubra, 1999; Michalowski, 1997;Terzaghi, 1943; Vesic, 1973; Frydman and Burd, 1997;Griffiths, 1982).

Fig.4 Comparison of bearing capacity factor (Nγ) for a rigid and rough surface strip footing

3.2 Seismic bearing capacity

The solution developed for the seismic bearing capacity based on the pseudo-static approach was validated considering a rigid, rough surface footing placed on a cohesionless soil (φ′=30°).The results are compared in terms of a dimensionless form of seismic bearing capacity ratio (SBCR) as defined in Eq.(36):

where,pu,s=the ultimate seismic bearing capacity(USBC) of the foundation or the limiting load on the foundation prior to shear failure, andγ=the unit weight of soil, whileB=footing width.

The comparison of SBCR with earlier studies is presented in Fig.5.The comparison of results obtained from the present study shows good agreement in the seismic bearing capacity ratio for a wide range of seismic coefficients.For validation, seismic bearing capacity values reported in the established literature have been adopted (Rajet al., 2018; Sarma and Iossifelis, 1990;Zhu, 2000; Soubra, 1999; Kumar and Kumar, 2003;Kumar and Rao, 2003; Kumar and Ghosh, 2006;Chakraborty and Kumar, 2013).

Fig.5 Comparison of SBCR for a rigid and rough surface strip footing under pseudo-static stresses

4 Properties of the materials

Three different geomaterials viz.sand, fly ash, and clay, are considered in the present study, based on their inherent range of transition zone of matric suction.Properties of the geomaterials are presented in Table 1.To model the unsaturated behavior of sand and clay, SWCC curve-fitting parameters have been adopted from values reported by Vahedifard and Robinson (2015).

Table 1 Material properties of geomaterials

For fly ash, geotechnical properties are adopted from the mean value reported by Pandian (2004), and the van Genuchten (1980) fitting parameters have been adoptedas the mean value reported by Prakashet al.(2019),based on a set of the large number of experimental results.The dynamic properties (shear modulus,Gand the damping ratio,ξ) of the geomaterials may be obtained by employing laboratory tests such cyclic triaxial and resonant column testing.The shear modulus(G)and the damping ratio (ξ) of the sand and the fly ash are in compliance with Chattaraj and Sengupta (2017),whereas the dynamic properties of clay comply with Villacreseset al.(2020).The expression for shear wave velocity has been obtained from pertinent literature(Konaiet al., 2018; Zhenget al., 2020).In the present study, a homogeneous soil with a uniform shear modulus with depth (G=constant) has been assumed for all the geomaterials.However, the shear modulus (G) value may also be considered as varying with depth and, for mathematical simplicity, the shear modulus may be considered to be linearly varying with depth (Liuet al.,2019; Basuet al., 2019) for granular geomaterial such as sand or fly ash.

To quantitatively describe the variation in the maximum value of the horizontal coefficient of seismic acceleration (kh) due to the variation in the profile ofG,two different profiles, viz.i)G=constant with depth and ii) Linear variation ofGwith depth [G=G0(z/H)](Chowdhury and Dasgupta, 2017; Chowdhuryet al.,2015) have been considered.The results are presented in Table 2.Here,G0refers to the shear modulus at a depth of (z=H).It may be observed from Table 2 that the horizontal coefficient of seismic acceleration(kh) calculated for the two shear modulus profiles yields similar values for all the modes of vibration,for both sand and fly ash.It may be highlighted here that although the fundamental periods obtained by considering the two profiles of shear modulus vary(especially for lower modes), the periods are mostly in the range of constant spectral acceleration (for IS 1893(Part-1) (2016): 0.10-0.67 s (soft soil site); for Euro Code (EN 1998-1) (CEN, 2004): 0.20-0.80 (for ground type D)) of the design response spectrum.This results in similar values of seismic acceleration coefficients for the two shear modulus profiles.Moreover, since the study is conducted for shallow footing with an influence zone that is not very deep into the ground, the effect of the shear modulus profile would not be distinct, unlike the foundations resting on deeper ground.Hence, it is assumed that variation in the shear modulus profile would not make much difference in the seismic bearing capacity, and uniform shear modulus with depth is considered throughout the study.

5 Results and discussions

The results are presented in dimensionless form as a seismic bearing capacity ratio (SBCR) as defined in Eq.(36).For an evaluation of the seismic-bearing capacity, the modal participation factor (κn) is adopted from Chowdhury and Dasgupta (2017) for each mode of the vibration of soil.The following sections discuss the various aspects of the seismic bearing capacity of various geomaterials by considering SWCC fitting parameters.Although the results are reported for a constant value of maximum plastic distance, the results would vary for different assumptions of maximum plastic distance values (Zhanget al., 2019).

5.1 Influence of mode number (n) on SBCR

The seismic bearing capacity of the foundation was obtained for various modes of vibrations for all threegeomaterials considered viz.sand, clay and fly ash.The results are presented for a constant matric suction (ψ=10 kPa) distribution, with the parameters presented in Table 1 for each geomaterial.Variation in SBCR with mode number (n) is presented in Fig.6 for various damping ratios of materials.A comparison of the curves obtained for all the three geomaterials substantiates the fact that the bearing capacity is lowest for the first mode or the fundamental mode of vibration (or natural frequency).For higher modes of vibrations, bearing capacity increases and tends to attain a constant value.In addition, the effect of the damping ratio on SBCR of partially saturated geomaterials is observed to be nominal.However, at lesser modes of vibrations, the influence of damping is higher and it gradually becomes reduced for higher modes.SBCR is higher for large values of damping ratios.Similar observations are obtained for all the geomaterials considered.

Fig.6 Variation of SBCR with mode number for (a) sand;(b) fly ash; and (c) clay for various damping ratios

Table 2 Variation in the value of T and kh for sand and fly ash materials, assuming two different shear modulus profiles

A comparison of SBCR obtained for all geomaterials reveals that the variation of SBCR with mode of vibration is highest for clay, followed by fly ash and sand.For sand withξ=8%, SBCR increases by 45.82% when the mode of vibration varies from 1 to 4, while for fly ash and clay, the increases in SBCR are 50.65% and 66.71%for damping ratios of 4% and 5%, respectively.It may be therefore inferred that for the fundamental mode of vibration, the percentage reduction of SBCR is highest for fine-grained soils.

The reduction in SBCR due to seismic stresses induced in geomaterials has been quantified by use of the seismic reduction factor (SRF), defined below as:

In the above expression, static bearing capacity may be obtained from Eq.(8).For a comprehensive investigation, a total of 6 modes of vibrations are considered.Values obtained forSRFfor all geomaterials are presented in Table 3 for all 6 modes of vibrations after considering different damping ratios.It is observed that with an increase in mode number (n),SRFtends to attain unity, i.e., the seismic bearing capacity value tends to attain a static bearing capacity value.The variation ofSRFwith mode is observed to be substantial for finegrained soils.

5.2 Influence of matric suction (ψ) on SBCR

As discussed earlier, a constant matric suction distribution profile has been assumed for the investigation.SWCC fitting parameters (αvandnv) adopted for each geomaterial are presented in Table 1.To incorporate the influence of matric suction on the SBCR of the footing,a range of suitable matric suction values for each of the geomaterials has been considered and the results are presented in this section.

Table 3 SRF for various modes of vibrations for geomaterials with different damping ratios

Variation of SBCR with matric suction for sand is presented in Fig.7(a), for a damping ratio (ξ) of 8% and for four different modes of vibration (n).For comparison, a static bearing capacity (SBC) ratio also has been shown.It is observed that the SBCR for sand increases up to a particular value of matric suction.Thereafter, SBCR gradually becomes reduced with an increase in matric suction.The SBCR of sand tends to attain a residual value for a higher range of matric suction.Similar observations have been reported by various other researchers based on their experimental investigations on coarse-grained soil under a static loading condition (Mohamed and Vanapalli, 2006;Oh and Vanapalli, 2013).This phenomenon may be explained by the inherent characteristics of partially saturated coarse-grained soils obtained from the SWCC.For coarse-grained soils, the magnitude of the residualsuction range is very small; consequently, the transition zone has a very narrow range of matric suction.The shear strength of granular soils shows an increasing trend until matric suction reaches a residual value.The shear strength is significantly reduced with an increase in matric suction beyond the residual matric suction value (Vanapalliet al., 1996; Oh and Vanapalli, 2013;Fredlundet al., 2012).For granular soil, such as the sand considered in the present study, the residual suction value lies in the proximity of 10 kPa (Vahedifard and Robinson, 2015).It may be noted that the SBCR of the sand considered in the present study increases to a matric suction value of 10 kPa, which is close to the residual suction value of the granular material.A similar trend is observed for all the modes of vibration.Therefore,it is inferred that the bearing capacity is substantially reduced in the post-residual suction zone.This inference may be given sufficient consideration when designing a foundation by examining the unsaturated behavior of granular soils.

Fig.7 Variation of SBCR with matric suction for various modes: (a) sand (ξ=8%); (b) fly ash (ξ=4%); (c) clay(ξ=5%)

Variation of SBCR with matric suction for fly ash is presented in Fig.7(b) for a damping ratio (ξ) of 4%.It may be noted that the variation of SBCR with matric suction exhibits a different pattern than does the sand.As evidence, within the lower range of matric suction values, SBCR rises gradually and reaches a peak value.Subsequently, the SBCR attains an almost constant residual value for a higher range of matric suction.For silty soils, through the use of laboratory measurements,it has been established that shear strength increases with a rise in matric suction up to the residual suction point for the soil.Beyond the residual zone, matric suction does not substantially contribute to shear strength;hence, the soil attains a residual strength (Fredlundet al., 2012).Since fly ash consists of particles in the range of silt to sand (Bhattet al., 2019; Pandian, 2004; Cokca and Yilmaz, 2004), the observed SBCR behaviour obtained for fly ash may be considered to be in direct compliance with the laboratory observation.However,as evident from Fig.7(b), SBCR is substantially reduced as the magnitude of matric suction lessens, and tends to approach the air-entry value of the suction.Therefore, a design engineer should lend utmost importance to this reduction in SBCR with a fluctuation in matric suction while designing a foundation over fly ash deposit by considering unsaturated behavior (Anand and Sarkar,2020).As matric suction is further reduced and tends to approach a zero value (corresponding to either dry or fully saturated state), the SBCR of fly ash is minimal.Hence, a proper ground improvement technique should be sought in a fly ash fill while also designing a shallow foundation for a load bearing structure, particularly in seismically active zones where frequent changes in weather may affect the saturation of a fly ash deposit.

For fine-grained soils such as clay, a variation in SBCR with matric suction is shown in Fig.7(c).It is observed that the SBCR for clay does not attain a residual value that is within the range of matric suction considered in the present study.It may be noted that for fine-grained soils such as clay, the residual suction value lies at a very high value of matric suction when compared to coarse-grained soils (Fredlundet al., 2012).Hence,matric suction seems to enhance the SBCR of clay-like materials for the practical range of matric suction values considered in the present study.The variation in SBCR with matric suction for different modes of vibration also is observed to be marginal when compared to static bearing capacity.

5.3 Influence of the depth of bedrock (H) on SBCR

Classical bearing capacity theories based on thepseudo-static approach do not place an emphasis on the influence of the depth of bedrock, and a constant value of horizontal or vertical seismic acceleration coefficient is generally adopted in these analyses.Ignoring the depth of bedrock and its subsequent impact on the modal vibration of a soil system may not yield a reliable solution for the seismic bearing capacity of the footing.The present section discusses the impact of the depth of bedrock on the SBCR of the footing for all three geomaterials considered in this study.To quantify the influence of the depth of bedrock on the SBCR, depth of bedrock (H) is varied in a range between 5 m and 30 m.

The influence of the depth of bedrock on SBCR for an unsaturated sand deposit has been presented in Fig.8 for different vibration modes (n).As evidenced from Fig.8, SBCR is greater when the bedrock is at a shallow depth.With an increase in the depth of bedrock, SBCR decreases gradually and tends to attain a constant value for a large depth of bedrock.For a fundamental mode of vibration (n=1), a sharp reduction in SBCR is observed up to a bedrock depth of 10 m.With an increase in bedrock depth, no substantial variation in SBCR is observed.For higher modes of vibrations (n=2, 3 and 4),a gradual reduction in SBCR with an increase in bedrock depth is observed.It also is noted that the influence of damping ratio (ξ) on SBCR gradually causes a reductionfor higher modes of vibrations of the unsaturated sand deposit.

Fig.8 Variation of SBCR with depth of bedrock (H) and damping ratio (ξ) for sand (a) n =1; (b) n =2; (c) n =3; and (d) n =4

Similarly, for fly ash, the influence of depth of bedrock on SBCR for three different damping ratios (ξ=4%, 6% and 8%) is presented in Fig.9.A smaller damping ratio for fly ash has been assumed based on experimental results obtained from Chattaraj and Sengupta (2017).For a fundamental mode of vibration (n=1) SBCR is minimal for all depths of bedrocks considered in the present study, and the variation of SBCR with depth of bedrock is insubstantial.For all other frequencies,however, a reduction in SBCR is observed for all modes of vibrations with an increase in the depth of bedrock.For fly ash, the influence of depth of bedrock on SBCR is more prominent for the second natural frequency (n=2)for all the values of damping ratios that were considered.As the mode of vibration increases, the influence of the damping ratio on SBCR gradually diminishes.

Fig.9 Variation of SBCR with depth of bedrock (H) and damping ratio (ξ) for fly ash (a) n =1; (b) n =2; (c) n =3; and (d) n =4

For fine-grained soils, plots of SBCR with a depth of bedrock for various modes of vibrations and damping ratio are presented in Fig.10.For the fundamental mode of vibration, SBCR is reduced, with an increase in the depth of bedrock.Further, it attains a residual value for bedrock depth of more than 10 m.For higher modes of vibrations, the residual value is reached for a large depth of bedrock.

Fig.10 Variation of SBCR with depth of bedrock (H) and damping ratio (ξ) for clay (a) n =1; (b) n =2; (c) n =3; and (d) n =4

5.4 Influence of strength parameters of geomaterials

5.4.1 Effects of friction angle (φ) for sand and fly ash

Shear strength of non-plastic geomaterials, such as sand and fly ash, is quantified based on its angle of internal friction or angle of shearing resistance.The present section discusses the influence of the angle of internal friction of geomaterials on the SBCR of the footing under an unsaturated framework.Pandian(2004) reported a range of effective friction angle of fly ash obtained from various thermal power plants in India.Similarly, various other researchers, such as Bhattet al.(2019) reported on the angle of internal friction as taken from various sources across the globe and noted that the angle of internal friction varies from 25° to 40°.Similarly, for sand, the angle of internal friction may range from 27° to 45° for practical considerations (Das and Yudhbir, 2004; Bhattet al., 2019).Therefore, a range of values for the angle of internal friction for sand and fly ash has been considered.For sand, five different values of friction angle (φ=20°, 25°, 30°, 35° and 40°) have been assumed for a constant matric distribution profile.The analyses were carried out for three damping ratios.However, no significant influence of damping ratio on SBCR of unsaturated sand was obtained and therefore,for brevity, results are presented for a single value ofdamping ratio (ξ=10%), as presented in Fig.11.

Fig.11 Variation of SBCR with friction angle (φ) of sand for various modes of vibration (for damping ratio,ξ=10%)

The SBCR of unsaturated sand is found to increase nonlinearly with an increase in friction angle.As previously discussed, SBCR is the least for the fundamental mode of vibration.Results are shown for a constant matric suction value of 10 kPa.Under dry or saturated conditions, when matric suction is absent,the bearing capacity will be different and is discussed in much detail in subsequent sections.It also may be noted that results are presented for the zone factorZ=0.24 corresponding to a higher seismic zone (Zone-IV), as per the Indian Standard.

Similarly, for fly ash, the range of values for the angle of internal friction has been adopted based on recommendations of Bhattet al.(2019).The angle of internal friction has been adopted to include a range of 25° to 40°.The variation in SBCR with friction angle is presented in Fig.12 for four different modes of vibration.For brevity, results are presented for a single damping ratio of fly ash (ξ=6%).For the analysis, a matric suction(ψ) value of 60 kPa has been assumed.From these results,it may be inferred that a fly ash deposit, if compacted at higher density, may exhibit satisfactory bearing capacity behavior, even in an unsaturated condition.

Fig.12 Variation of SBCR with friction angle (φ) of fly ash for various mode of vibrations (for damping ratio,ξ=6%)

The combined influence of matric suction and the friction angle on the SBCR of sand has been obtained and is presented in Fig.13 forξ=10%.It has been well established in the literature that the overall contribution of suction in shear strength can be a summation of the contribution up to the air-entry value, as well as from the air-entry value to the residual suction range, beyond which the effect of suction on shear strength ceases to exist.Results of the present study are in agreement with where shear strength is observed to increase up to the residual suction value (i.e., up to 10 kPa for the sand considered in the study).Hence, it could be deduced that within the transition zone, the SBCR of coarse-grained soils increases with an increase in both matric suctionand the angle of internal friction.With an increase in the magnitude of matric suction beyond the residual suction value, bearing capacity reduces and attains residual values for all modes of vibration considered in the study.However, the variation of SBCR with matric suction is not significant.Similar to earlier observations, it is seen that the SBCR of sand is least for the fundamental mode of vibration.

Fig.13 Variation of SBCR with matric suction (ψ) and friction angle (φ) for sand (damping ratio, ξ=10%) for (a) n=1; (b) n=2;(c) n=3; and (d) n=4

Figure 14 shows the variation of SBCR with matric suction and a friction angle for fly ash by consideringξ=6%.It may be observed that for each friction angle,SBCR is least for the fundamental mode of vibration.For the unsaturated parameters considered in the study of fly ash, it is evident from Fig.14 that beyond a matric suction value of 60 kPa, the influence of suction stress on the overall bearing capacity becomes reduced, and the fly ash deposit assumes a residual value for all the friction angles that are considered.This may be due to the fact that fly ash attains a residual suction value within a range of 60 kPa, as reported by Prakashet al.(2018).It may be noticed that the residual suction value of fly ash materials is significantly greater than that for sand due to the physical and chemical characteristics of fly ash.Hence, the SBCR of unsaturated fly ash tends to predict higher values than is the case for sand materials.Beyond the residual suction range, the contribution of suction in shear strength becomes reduced, and subsequently SBCR attains residual values.However, the pattern of variation of SBCR for fly ash with a friction angle and matric suction is starkly different than that of sand beyond the residual suction value.It is observed that sand significantly loses bearing capacity beyond the residual suction value, whereas fly ash seems to retain its bearing capacity beyond the residual suction value for all modes of vibration.

Fig.14 Variation of SBCR with matric suction (ψ) and friction angle (φ) for fly ash (damping ratio, ξ=6%) for (a) n=1; (b) n=2;(c) n=3; and (d) n=4

5.4.2 Influence of true effective cohesion (c′) for clay

For fine-grained soils such as clay, the influence of true effective cohesion (c′) on SBCR has been investigated.Results are presented in dimensionless forms such asc′/γB, referred to as the ‘true cohesion factor(TCF)’.The capillary cohesion generated due to the partial saturation of geomaterials has been considered separately.To distinguish the two types of cohesion induced in the soil, few researchers prefer to designate true effective cohesion as ‘cementation’.However,to avoid ambiguity, in the present paper the term true cohesion factor (TCF) refers to the contribution of fine-grained soil due to inherent true effective cohesion present in fine-grained geomaterials.Figure 15 presents the variation of SBCR for four different TCF values(e.g., 0.5, 1.0, 1.5 and 2.0) of fine-grained soils forξ=8%.

A linear increase in SBCR with an increase in TCF is observed for all the modes of vibration for unsaturated clay material, as evidenced in Fig.15.Classical bearing capacity theories proposed in the literature also suggest a linear increase in the bearing capacity of a shallow strip footing with true effective cohesion.Figure 16 shows the variation of SBCR with matric suction and the TCF of clayey material by considering an 8% damping ratio.It is observed that SBCR for clayey materials tends to increase with an increase in the matric suction value,even for a large value of suction.Hence, it may be stated that the effect of matric suction on the SBCR of clayey materials will be on the conservative side.Besides, it is also noticed that the trend in the variation of SBCR is uniform for all modal vibrations for the range of matric suction values considered, as well as the true cohesion of the clayey material.Hence, it is inferred that, for fine-grained unsaturated geomaterials, SBCR is not significantly affected by modes of vibration for the practical range of matric suction and true cohesion.

Fig.15 Variation of SBCR with TCF of clay for various mode numbers (n)

Fig.16 Variation of SBCR with matric suction (ψ) and TCF for clay (damping ratio, ξ=8%) for (a) n=1; (b) n=2; (c) n=3;and (d) n=4

6 Summary and conclusions

A novel methodology has been adopted to investigate the seismic bearing capacity of a shallow strip footing resting on partially saturated geomaterials.This methodology has been developed to incorporate critical parameters such as modes of vibration, damping ratio, depth of bedrock, and matric suction of partially saturated geomaterials in the evaluation of seismic bearing capacity.Strength nonlinearity, arising due to the partial saturation of the soil matrix, has been considered.The present study comprehensively quantifies the role of matric suction on seismic bearing capacity.Furthermore,to cater to a wide range of matric suction characteristics,results are reported for three different geomaterials(viz.sand, fly ash, and clay) for their inherently different suction stress-based characteristics.The major conclusions of the present study may be summarized as follows:

· Matric suction of different unsaturated geomaterials variously influences the seismic bearing capacity of a shallow strip footing.It is observed that seismic bearing capacity that considers matric suction reduces to an extent of 40% compared to the static bearing capacity for various geomaterials taken into account in the study.Hence, an adequate consideration of unsaturation parameters is highly warranted during the evaluation of the seismic bearing capacity of footings that rest on unsaturated geomaterials.

· From the modal analyses presented in this paper,it may be concluded that the seismic bearing capacity of a strip footing on various unsaturated geomaterials is significantly influenced by its first mode of vibration.The reduction in the bearing capacity of geomaterials under unsaturated condition ranges from 20%-40% with respect to the static bearing capacity in the first mode of vibration.With an increase in mode number, the seismic bearing capacity gradually increases and tends to attain a value of static bearing capacity.

· The damping ratio of the geomaterials has a marginal effect on the seismic bearing capacity of a strip footing.However, it is observed that seismic bearing capacity has displays an increasing trend with regard to the damping ratio of geomaterials.

· Alhough classical bearing capacity theories ignore the effect of bedrock, it is observed that the depth of bedrock has a reasonable effect on seismic bearing capacity.With a shallower depth of bedrock, seismic bearing capacity shows an increasing trend for the geomaterials considered in the study.

· The seismic bearing capacity of sand is drastically reduced, with matric suction reaching beyond the residual suction value.For fly ash materials, seismic bearing capacity remains constant beyond the residualsuction value, whereas for clay the bearing capacity shows an increasing trend in the practical range of matric suction values.