Linear and nonlinear elastic analysis of closely spaced strip foundations using Pasternak model

Abstract

In this study, an attempt is made to determine the interaction effect of two closely spaced strip footings using Pasternak model. The study considers small strain problem and has been performed using linear as well as nonlinear elastic analysis to determine the interaction effect of two nearby strip footings. The hyperbolic stress-strain relationship has been considered for the nonlinear elastic analysis. The linear elastic analysis has been carried out by deriving the equations for the interference effect of the footings in the framework of Pasternak model equation; whereas, the nonlinear elastic analysis has been performed using the finite difference method to solve the second order nonlinear differential equation evolved from Pasternak model with proper boundary conditions. Results obtained from the linear and the nonlinear elastic analysis are presented in terms of non-dimensional interaction factors by varying different parameters like width of the foundation, load on the foundation and the depth of the rigid base. Results are suitably compared with the existing values in the literature.

Keywords

bearing capacity / linear and non-linear elasticity / foundation / interaction effect / numerical modeling / Pasternak model

Introduction

The classical theories of shallow foundations which are widely used in practice, are valid as long as footings are isolated and there exists no other neighboring footing in the close proximity. In many situations for instance, scarcity of land, restrictions in property lines, architecture of the buildings or to accommodate structural requirements and with increased infrastructural development in the recent past and limited availability of space, buildings are constructed very close to each other. Therefore quite often, designed foundations or the group of foundations are placed in close proximity of each other. In such situations, the stress or failure zone of individual footings may interfere with those of the neighboring footings. Such interference of footings affects the characteristic behavior (bearing capacity, settlement, failure mechanism, etc.) of the individual footings occurring in a group. Stuart [] was the first to investigate the interference effect on the ultimate bearing capacity of two closely spaced rigid strip footings resting on the surface of cohesionless soil medium using limit equilibrium method. West and Stuart [] conducted experiments in the laboratory and analyzed the problem using method of stress characteristics and concluded that the interference affects significantly the ultimate bearing capacity of the interacting footings revealing the necessity of studies on the interference phenomenon on the characteristic behavior of footings. After hibernation of a long period, in the recent past a number of theoretical works are reported on the subject using method of stress characteristics [,], analytical approach [], probabilistic approach [], upper bound limit analysis [–], lower bound limit analysis [], finite difference method [,], finite element method [,]. Besides these theoretical analysis mentioned above, a number of experimental studies on the interacting footings have been performed by various researchers to investigate the load-deformation response of two footings resting on cohesionless soil medium [–].

The review exhibits that except the analysis of Nainegali et al. [], no significant work has yet been conducted on asymmetric footings. However, very often situations may arise near a property line where two strip footings of different size are subjected to unequal loading. Even if the footings are identical in shape and size, the loading may be asymmetric. Thus, a need is felt to study the interference of such footings and symmetric cases are then the sub-sets of the general problem. In this analysis, an effort is made to explore the interference effect of two closely spaced horizontal strip footings resting on dry homogenous soil deposit. The analysis is performed using Pasternak soil model, which considers a shear beam below the loaded area to take care of the continuity of the settlement distribution below the footing. Though a number of numerical techniques [–] available in the literature to address similar class of problem, Pasternak soil model is chosen in this study for its easy understanding and robust implementation capability. The soil is assumed to obey both linear and nonlinear elasticity. The settlement response of an isolated footing subjected to a uniformly distributed load is obtained from the settlement response of an isolated footing subjected to line load. The interference effect of two closely spaced footings is obtained using the principle of superposition. The interaction factors (x_L and x_R) for the interacting footings with respect to the settlement at different clear spacing, S are reported here, where x_L and x_R are defined later.

Definition of problem

Two asymmetric perfectly rough rigid strip footings designated as left and right footing are placed close to each other at a clear spacing, S on the surface of a homogeneous, isotropic but linearly as well as nonlinearly elastic soil layer (Fig. 1). There exists a rigid base at a depth, H from the ground surface. P_L and P_R (P_R = nP_L and n: non-dimensional parameter) are the loads that are applied on the left and right footings of width b_L and b_R (b_R = ab_L and a: non-dimensional parameter) respectively. The modulus of elasticity of the soil deposit is assumed as 30 MPa. The objective is to institute a parametric study to find the effect of interference on the settlement of the rigid strip footings. The width and load intensity of the left footing are kept constant wherein the width and the load intensity of the right footing are varied. The depth of the rigid base is varied as well. The different parameters and the range within which those are varied are given in Table 1. The study considers the loading within the working range, settlement being within the permissible limit for which the soil may be considered to be linearly or nonlinearly elastic without any appreciable deviation from the actual behavior observed for soils experiencing relatively small strain [].

Mathematical formulation

The settlement response of the proposed model has been arrived by applying uniformly distributed load with intensity q, over a length of B directly on the top of the soil layer. Figure 2 shows the actual and the idealized conditions of the soil-foundation system considered for the analysis. The soil under the footing has been modeled using Pasternak model and solved for both linear and nonlinear elastic approaches. Analysis is carried out considering a strip footing; hence, plane strain condition is adopted. Formulation developed for isolated footing will be used to understand the influence of interference effect of two closely spaced footings by adopting principle of superposition.

Linear elastic analysis

The general equation of Pasternak [] model is given as:

(1)q(x) = kw(x)-G_{\mathrm{p}}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}w}{\mathrm{d}{x}^{\mathrm{2}}}},

where q(x) is the load intensity applied on the soil (N/m²), k is the stiffness constant of the spring elements (N/m³), G_p is the shear modulus of the shear layer (N/m²) and w(x) is the settlement profile at the ground surface (m).

Vlazov and Leontiev [] approach is used to determine the settlement profile at any given depth in the soil medium. The settlement profile of the soil at any depth is given as:

(2)w(x,z)= w(x)·h(z),

where h(z) describes the pattern of variation in the z direction. Several variations of h(z) have been proposed by Vlazov and Leontiev [] including the linear and exponential variations. In every variation, the function h(z) varies from “1” at z=0 to “0” at z= H (H is the depth of the rigid base from the ground surface). In this study, the linear variation of h(z) has been adopted as given in Eq. (3):

(3)h(z)= \mathrm{1} -{\displaystyle \frac{z}{H}}.

According to the Pasternak model, as the shear layer is vertically incompressible and does not contribute to any settlement except transferring the loads laterally, the function h(z) does not vary along the depth of the shear layer. Hence, the value of the function h(z) remains “1” throughout the depth of the shear layer right from the top of the ground surface i.e., from z= 0 to z= 1. From z= 1 to z= H, the function h(z) varies linearly from “1” to “0”. Hence the function h(z) given by Eq. (3) is modified to suit the Pasternak model as:

(4)

Isolated strip footing

The total load from the superstructure is assumed to be transferred to the strip footing through a line load acting at the center of the footing, which in turn, transmits the load to the soil through a uniformly distributed load of length equal to that of the width of the footing. In this study, the settlement response of the footing is determined using the settlement response obtained due to a line load through influence factors.

For the line load formulation, the Eq. (1) reduces to

(5)-G_{\mathrm{p}}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}w}{\mathrm{d}{x}^{\mathrm{2}}}}+k\cdot w(x)=\mathrm{0.}

The solution to the second order differential equation is:

(6)w(x)=A_{\mathrm{1}}{e}^{-\left(\sqrt{\frac{k}{G_{\mathrm{p}}}}\right)x}+A_{\mathrm{2}}{e}^{\left(\sqrt{\frac{k}{G_{\mathrm{p}}}}\right)x},

where A₁ and A₂ are constants which can been determined by applying suitable boundary conditions.

At a distance very far from the application of the line load on the ground surface, there cannot be any deformation due to the line load; hence, w(-∞) = 0, and w(∞) =0. The total shear force N(x) of a footing resting on the soil medium is the sum of the shear force due to the beam effect of the footing V(x) and the shear force due to the soil medium i.e., due to the shear layer, G_{\mathrm{p}}{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}x}}. Since the beam effect due to the footing is not considered in this analysis, V(x) is zero. Hence, shear force due to the given loading condition (-P/2 at x=0) is given as:

(7)-P/\mathrm{2}=G_{\mathrm{p}}{{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}x}}}_{\mathrm{a}\mathrm{t}\hspace{0.17em}x=\mathrm{0}},

rearranging,

(8){{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}x}}}_{\mathrm{a}\mathrm{t}\hspace{0.17em}x=\mathrm{0}}={\displaystyle \frac{-P}{\mathrm{2}{G}_{\mathrm{p}}}}\mathrm{.}

These boundary conditions can be incorporated by dividing the soil medium into two zones as shown in Fig. 3(a). Thus,

(9a)

(9b)\hspace{1em}\hspace{1em}={\displaystyle \frac{-P}{\mathrm{2}{G}_{\mathrm{p}}\beta }}{e}^{-\beta x}[x>\mathrm{0}],

where b = \sqrt{{\displaystyle \frac{k}{G_{\mathrm{p}}}}}.

The settlement response of an isolated footing subjected to a line load is used as an influence factor to estimate the settlement response of an isolated footing due to uniformly distributed load. The settlement response due to an arbitrary load q(m) (Fig. 3(b)) at any arbitrary distance “x” from the origin can be obtained by dividing the soil medium in to three zone (Fig. 3(c)) as follows:

(10a)

(10b)

(10c)

The settlement response of an isolated strip footing subjected to a uniformly distributed load, at any depth can be obtained from Eqs. (4) and 8:

(11a)

(11b)

(11c)

Interaction of two closely spaced strip footings

Figure 3(d) shows two footings with left footing having a width b_L and right footing having a width b_R with a clear spacing S between them. The principle of superposition is used to get interference effect of two closely spaced footings. The settlement response due to the interference effect of two closely spaced footings has been determined by dividing the soil medium into five zones and the respective equations for each zone are:

For-∞ <x<- (S/2+b_L):

(12a)w(x,z)=(q_{\mathrm{L}}/\mathrm{2}k)(e^{\beta (x+S/\mathrm{2})}-e^{\beta (x+(b_{\mathrm{L}}+S/\mathrm{2}))})h(z)+(q_{\mathrm{R}}/\mathrm{2}k)(e^{\beta (x-(b_{\mathrm{R}}+S/\mathrm{2})}-e^{\beta (x-S/\mathrm{2})})h(z).

For – (S/2+ b_L)<x<- S/2:

(12b)w(x,z)=(q_{\mathrm{L}}/\mathrm{2}k)(-(\mathrm{2}-e^{\beta (x+S/\mathrm{2})}-e^{-\beta (x+(b_{\mathrm{L}}+S/\mathrm{2}))}))h(z)+(q_{\mathrm{R}}/\mathrm{2}k)(e^{\beta (x-(b_{\mathrm{R}}+S/\mathrm{2})}-e^{\beta (x-S/\mathrm{2})})h(z).

For-S/2<x<S/2:

(12c)w(x,z)=(q_{\mathrm{L}}/\mathrm{2}k)(-(e^{-\beta (x+S/\mathrm{2})}-e^{-\beta (x+(b_{\mathrm{L}}+S/\mathrm{2}))}))h(z)+(q_{\mathrm{R}}/\mathrm{2}k)(e^{\beta (x-(b_{\mathrm{R}}+S/\mathrm{2})}-e^{\beta (x-S/\mathrm{2})})h(z).

For S/2<x<S/2+b_R:

(12d)w(x,z)=(q_{\mathrm{L}}/\mathrm{2}k)(-(e^{-\beta (x+S/\mathrm{2})}-e^{-\beta (x+(b_{\mathrm{L}}+S/\mathrm{2}))}))h(z)-(q_{\mathrm{R}}/\mathrm{2}k)(\mathrm{2}-e^{-\beta (x-(b_{\mathrm{R}}+S/\mathrm{2})}-e^{-\beta (x-S/\mathrm{2})})h(z).

For S/2 + b_R<x< ∞:

(12e)w(x,z)=(q_{\mathrm{L}}/\mathrm{2}k)(-(e^{-\beta (x+S/\mathrm{2})}-e^{-\beta (x+(b_{\mathrm{L}}+S/\mathrm{2}))}))-(q_{\mathrm{R}}/\mathrm{2}k)(e^{-\beta (x-(b_{\mathrm{R}}+S/\mathrm{2})}-e^{-\beta (x-S/\mathrm{2})})h(z).

Nonlinear elastic analysis

The nonlinearity in the Pasternak model is introduced by adopting Konder and Zelasko [] hyperbolic stress-displacement relationship. According to Konder and Zelasko [] the relationship between the load and the deformation is given as follows:

(13)q={\displaystyle \frac{k_{\mathrm{0}}w}{\mathrm{1}+\left({\displaystyle \frac{k_{\mathrm{0}}}{q_{\mathrm{u}}}}\right)w}},

where q is the load intensity, q_u is the ultimate bearing resistance of soil, k₀ is the initial stiffness constant of the spring elements and w is the settlement.

Introducing nonlinearity in the general equation of Pasternak model works out to be:

(14)q={\displaystyle \frac{k_{\mathrm{0}}w}{\mathrm{1}+\left({\displaystyle \frac{k_{\mathrm{0}}}{q_{\mathrm{u}}}}\right)w}}-G_{\mathrm{p}}{\displaystyle \frac{d^{\mathrm{2}}w}{d{x}^{\mathrm{2}}}}.

The second order differential equation can be converted into two first order equations as follows:

(15a){\displaystyle \frac{\mathrm{d}w}{\mathrm{d}x}}=n^{\prime },

(15b){\displaystyle \frac{\mathrm{d}{n}^{\prime }}{\mathrm{d}x}}=\left({\displaystyle \frac{\mathrm{1}}{G_{\mathrm{p}}}}\right)\left({\displaystyle \frac{k_{\mathrm{0}}w(\mathrm{1})}{\mathrm{1}+({\displaystyle \frac{k_{\mathrm{0}}}{q_{\mathrm{u}}}}w(\mathrm{1}))}}\right)-q\left({\displaystyle \frac{\mathrm{1}}{G_{\mathrm{p}}}}\right).

Built-in subroutine available in Matlab employing finite difference scheme has been used to solve the differential equations by adopting suitable boundary conditions.

Isolated strip footing

As shown in the Fig. 3(c), the soil medium is divided into three zones. Boundary conditions as adopted in linear elastic analysis holds good for nonlinear elastic analysis as well. The equation for each zone is as follows:

(16a)

(16b)

(16c)

Even though the settlement response is different for each zone, curves must satisfy displacement continuity and slope continuity. The middle zone (2) is subdivided in to two zones as zone (2a) and zone (2b) to satisfy relevant boundary conditions. The boundary condition given in the Eq. (16b) is at the middle of the zone (2), which becomes the right boundary condition for the zone (2a) and the left boundary condition for the zone (2b). Following additional boundary conditions are adopted for solving the differential equations:

(17){{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}x}}}_{(x=-b/\mathrm{2})}=c_{\mathrm{1}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}x}}}_{(x=b/\mathrm{2})}=c_{\mathrm{2}}

where c₁ andc₂ are constants that need to be determined through iteration procedure. A Matlab code is written such that an iterative process with iterations stopping on reaching a value of c₁ for which the settlement at x= -b/2 due to the zone (1) settlement curve is equal to the settlement due to the zone (2a) settlement curve. A similar iteration process is also run for finding the value of c₂ such that, at x= b/2, the deflection due to zone (2b) deflection curve is equal to the deflection due to deflection curve of zone (3).

Interaction of two closely spaced strip footings

The nonlinear elastic analysis of the interference effect of two closely spaced footings has been studied by applying the principle of superposition. Sensitivity analysis has also been performed for interacting foundations at different spacing by varying different parameters like depth of the rigid base (H/b_L = 2.0, 4.0), width of the footings (b_R/b_L = 1.0, 2.0) and load intensities applied (q_L=q_R= 0.2 MPa, 0.4 MPa). In all the cases it was observed that the change in the results of settlement response is insignificant beyond the domain sizes of extreme zones (zone (1) and zone (5)) lx = 10b. So, an optimum domain size of lx= 10b is chosen for zone (1) and zone (5).

Material property estimation

The values of stiffness constant k and Pasternak shear modulus G_p required for the determination of the settlement response w(x) in the Pasternak model have been derived from the Elastic modulus E and the Poisson’s ratio µ of the soil. While deriving the values of k and G_p, it is assumed that only the top unit depth of the soil from the ground surface mainly transforms the deformation in the horizontal direction and the soil beneath it, will be responsible for vertical deformation making the soil medium similar to that of Pasternak model.

The stress-strain relationship for plane strain condition [] is given as:

(18a){\sigma }_{xx}={\displaystyle \frac{E_{\mathrm{0}}}{(\mathrm{1}-v_{\mathrm{0}}^{\mathrm{2}})}}w(x){\displaystyle \frac{\mathrm{d}h(z)}{\mathrm{d}z}}{v}_{\mathrm{0}},

(18b){\sigma }_{zz}={\displaystyle \frac{E_{\mathrm{0}}}{(\mathrm{1}-v_{\mathrm{0}}^{\mathrm{2}})}}w(x){\displaystyle \frac{\mathrm{d}h(z)}{\mathrm{d}z}},

(18c){\tau }_{xz}={\displaystyle \frac{E_{\mathrm{0}}}{\mathrm{2}(\mathrm{1}+v_{\mathrm{0}})}}{\displaystyle \frac{\mathrm{d}w(x)}{\mathrm{d}x}}h(z),

where {\sigma }_{xx} and {\sigma }_{zz} is the axial stress in the x and z direction respectively, {\tau }_{xz} is shear stress in the xz plane.

(19a)E_{\mathrm{0}}={\displaystyle \frac{E}{(\mathrm{1}-{\mu }^{\mathrm{2}})}},

(19b){\mu }_{\mathrm{0}}=\mu (\mathrm{1}-\mu ).

The strain energy W_{\sigma } of the soil due to the axial stresses in the element of width “dx” per unit length is given by:

(20)W_{\sigma }={\displaystyle {\int }_{\mathrm{1}}^H({\displaystyle \frac{\mathrm{1}}{\mathrm{2}{E}_{\mathrm{0}}}}({\sigma }_{xx}^{\mathrm{2}}+{\sigma }_{zz}^{\mathrm{2}})-{\displaystyle \frac{v_{\mathrm{0}}}{E_{\mathrm{0}}}}{\sigma }_{xx}{\sigma }_{zz})}\mathrm{d}x\mathrm{d}z.

Figure 4 shows the stress acting on the shear layer. Due to shear stresses, the strain energy W_{\tau } of the soil in width “dx” per unit length of soil is:

(21)W_{\tau }={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}{({\displaystyle \frac{\mathrm{1}}{\mathrm{2}G}}\cdot {\tau }_{xz}^{\mathrm{2}})}^{\mathrm{2}}}\mathrm{d}x\mathrm{d}z.

From the Pasternak model, the strain energy in the spring element per unit length is []:

(22)W_{\sigma \mathrm{p}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}k{(w(x))}^{\mathrm{2}}\mathrm{d}x.

And the strain energy in the shear element per unit length is []:

(23)W_{\tau \mathrm{p}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{G}_{\mathrm{p}}{\left({\displaystyle \frac{\mathrm{d}w}{\mathrm{d}x}}\right)}^{\mathrm{2}}\mathrm{d}x\mathrm{.}

Equating Eq. (20) with (22) and (21) with (23), the expressions for k and G_p can be obtained as:

(24)k={{\displaystyle {\int }_1^H{\displaystyle \frac{E_{\mathrm{0}}}{(\mathrm{1}-v_{\mathrm{0}}^{\mathrm{2}})}}\left({\displaystyle \frac{\mathrm{d}h(z)}{\mathrm{d}z}}\right)}}^{\mathrm{2}}\mathrm{d}z,

(25)G_{\mathrm{p}}={{\int }_{\mathrm{0}}^{\mathrm{1}}{\displaystyle \frac{E_{\mathrm{0}}}{\mathrm{2}(\mathrm{1}+v_{\mathrm{0}})}}(h(z))}^{\mathrm{2}}\mathrm{d}z.

On integration, equations simplify to:

(26)k={\displaystyle \frac{E_{\mathrm{0}}}{(\mathrm{1}-v_{\mathrm{0}}^{\mathrm{2}})}}{\displaystyle \frac{\mathrm{1}}{(H-\mathrm{1})}},

(27)G_{\mathrm{p}}={\displaystyle \frac{E_{\mathrm{0}}}{\mathrm{2}(\mathrm{1}+v_{\mathrm{0}})}}.

By knowing the values of elastic modulus and the Poisson’s ratio of the soil, the material properties of the proposed model can be estimated using the Eqs. (26) and (27).

Results and discussion

The Pasternak model parameters (k and G_p) are determined based on the parameters given in Table 1, and Eqs. (26) and 27. The interaction effect between the footings is reported in terms of non-dimensional interaction factors (x_L and x_R) with respect to the settlement of the footing. The interaction factor for the left (x_L) as well as for the right (x_R) footings can be defined as

\hspace{0.17em}{\xi }_{\mathrm{L}}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{Average\hspace{0.17em}\hspace{0.17em}settlement\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}left\hspace{0.17em}footing\hspace{0.17em}in\hspace{0.17em}presence\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}right\hspace{0.17em}footing}{Average\hspace{0.17em}settlement\hspace{0.17em}of\hspace{0.17em}an\hspace{0.17em}\hspace{0.17em}isolated\hspace{0.17em}footing\hspace{0.17em}of\hspace{0.17em}width\hspace{0.17em}and\hspace{0.17em}load\hspace{0.17em}\mathrm{int}ensity\hspace{0.17em}same\hspace{0.17em}as\hspace{0.17em}that\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}left\hspace{0.17em}footing}},

{\xi }_{\mathrm{R}}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{Average\hspace{0.17em}\hspace{0.17em}settlement\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}right\hspace{0.17em}footing\hspace{0.17em}in\hspace{0.17em}presence\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}left\hspace{0.17em}footing}{Average\hspace{0.17em}settlement\hspace{0.17em}of\hspace{0.17em}an\hspace{0.17em}\hspace{0.17em}isolated\hspace{0.17em}footing\hspace{0.17em}of\hspace{0.17em}width\hspace{0.17em}and\hspace{0.17em}load\hspace{0.17em}\mathrm{int}ensity\hspace{0.17em}same\hspace{0.17em}as\hspace{0.17em}that\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}right\hspace{0.17em}footing}}.

It is worth mentioning that in this study, the physical dimensions and loading condition of the right footing have been varied using two parameters a and n by keeping the dimensions and loading condition constant for the left footing.

Linear elastic analysis

The variation of interaction factors (x_L and x_R) with S/b_L for symmetric (a = 1.0) as well as asymmetric (a = 2.0) footings for different values of n/a and H/b_L ratio is shown in Figs. 5 and 6, respectively. It is worth noting that the symmetric arrangement of footings considers the left and the right footings to be symmetric in terms of width of the footings whereas in terms of loading, the ratio n/a is varied with the ratio n/a = 1 being the symmetric case of loading. On the contrary, the asymmetric arrangement of footings considers the left and the right footings to be asymmetrical in terms of width of the footings whereas in terms of loading, the ratio n/a is varied with the ratio n/a = 1 being the symmetric case of loading. It can be seen that the magnitude of interaction factors for both left and right footings decreases with the increase in the S/b_L ratio. In all the cases, the interaction factors for the left and the right footings attains a value of 1 at S/b_L value equal to approximately 5.0 and 7.5 for H/b_L = 2.0 and 4.0 respectively. It can be also noted that the interference effect increases with increase in the depth of the rigid base (H).

Nonlinear elastic analysis

The nonlinear elastic analysis of the interference effect of the closely spaced footings is carried out based on the parameters mentioned in Table 1. The magnitude of initial tangent modulus (E_i), Poisson’s ratio (µ) and the corresponding angle of internal friction (f) is considered as 30 MPa, 0.3 and 35° as proposed by Das [] for medium dense sand. The unit weight of the medium dense sand is considered as 20 kN/m³. As mentioned earlier, the additional parameter required in the nonlinear elastic analysis is the ultimate bearing capacity of the foundation, q_u (Eq. (13)). The ultimate bearing capacity of single isolated surface strip footing resting on sand can be determined using Terzaghis’s classical equation, whereas the ultimate bearing capacity of the interacting foundations is different from that of an isolated foundation for the obvious reason. The magnitude of the ultimate bearing capacity of symmetric (a = 1.0) interacting foundations, {q^{\prime }}_{\mathrm{u}}can be determined using the efficiency factor as proposed by Kumar and Ghosh [] at different spacing between the foundations. In Eq. (14), it is assumed that the magnitude of the ultimate bearing capacity of the symmetric interacting foundations is same as that of the asymmetric interacting foundations under similar condition, which does not cause any serious lapse as the asymmetric interacting foundations usually exhibit higher ultimate bearing capacity than that of the symmetric interacting foundations. Therefore, for the sake of simplicity, the ultimate bearing capacity of the asymmetric foundations (a = 2.0) at different spacing is assumed to be the same as that of the symmetric (a = 1.0) interacting foundations at the corresponding spacing.

The interaction factors for the left and the right footings are determined at different S/b_L ratios varying from 0.5 to 10 with rigid base at a depth restricted to H/b_L ratio equal to 2.0 and 4.0 and plotted in Figs. 7-8. In Fig. 7, it is considered that the left and the right footings to be symmetric in terms of width; whereas in terms of loading, the footings are asymmetrical with different n/α ratios except n/α = 1, which belongs to symmetric case of loading. In Fig. 8, it is considered that the left and the right footings to be asymmetrical with respect to the width of the footings. It can be observed that the interaction factors for the left and the right footings decreases as the value of S/b_L increases. For both symmetric and asymmetrical arrangements, the interaction factor attains a value of 1 at S/b_L ratio approximately equal to 5.0 and 8.5 for H/b_L = 2.0 and 4.0 respectively. It can also be noted that the interference effect increases with increase in the depth of the rigid base (H).

Comparison

In Table 2, the values of interaction factors (x_L and x_R) obtained from this study are compared with those reported by Nainegali [] using finite element method for H/b_L = 4.0. It can be noted from Table 2 that the present values of the interaction factors are in close agreement with those provided by Nainegali []. It can be seen that the maximum percentage difference between the present values and those provided earlier by Nainegali [] is approximately 6.8% for the case compared.

The variation of interaction factors with S/b_L obtained from the linear elastic analysis is compared with those obtained from the nonlinear elastic analysis. Figure 9 shows the comparison of interaction factors with the variation of S/b_L ratio for the symmetric footings with different H/b_L ratios, whereas Fig. 10 presents the comparison of interaction factors for the asymmetric footings with different H/b_L ratios. In all the cases, two different loading conditions are considered i.e., symmetric (n/a = 1.0) and asymmetric (n/a = 2.0) loading. It can be observed that the interaction factors obtained from the linear elastic analysis are found to be marginally higher than those determined from the nonlinear elastic analysis for n/a = 1.0. However, at n/a = 2.0, a reverse trend i.e., the results obtained from the nonlinear elastic analysis are found to be higher than those determined from the linear elastic analysis.

Conclusions

The interference effect of two nearby strip footings has been studied using Pasternak model. The study is performed using both linear and nonlinear elastic analysis and the effect of interaction is presented in terms of dimensionless interaction factors. It is observed that the settlement response of the interacting footings is more compared to that of an isolated footing. At lower spacing between the footings, it is observed that the interference effect is high and as the spacing increases, the interference effect reduces and finally at larger spacing, the footings tend to behave as isolated one. In case of two asymmetric interacting footings, it is noted that the interference effect is more on the footing of smaller width. In case of two footings with asymmetric loading, the interference effect is more for the footing with smaller load. On increasing the depth of the rigid base, the interference effect is increased and continued for larger spacing between the footings. In case of asymmetric loading it is observed that as the load on one of the footings is increased, there is higher increase in the values of interaction factors for the nonlinear elastic analysis compared to those obtained from the linear elastic analysis. As the depth of the rigid base increases, it is observed that the difference in the values of interaction factors between the linear and the nonlinear elastic analysis decreases.

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