Soil spring stiffnesses for laterally loaded large diameter rigid caissons

Abstract

Laterally loaded large diameter caissons that exhibit rigid body motion are often analyzed by modeling the soil as equivalent springs. Equivalent soil spring stiffness is usually developed empirically. In this study, the soil spring stiffnesses are developed rigorously considering three-dimensional caisson-soil interaction based on a continuum-based analytical framework. A laterally loaded rigid circular caisson embedded in elastic soil is analyzed using the principle of virtual work. The soil displacement field is chosen based on the geometry and kinematics of the caisson, and the vertical displacement of the soil caused by the large-diameter caisson rotation is explicitly considered in the analysis, which has been neglected in previous studies. An iterative algorithm is used to obtain the solutions. It is shown that the caisson-soil interaction can be represented by a five-spring model along the shaft and at the base of the caisson. The equations of these spring stiffnesses are mathematically derived from the analysis without recourse to empiricism. Similarly, the equivalent spring stiffnesses of the caisson-soil system replacing the caisson and soil continuum are derived analytically from the solution. Based on the detailed parametric studies, fitted algebraic equations for these spring stiffnesses are obtained, which can be readily used by practicing engineers.

Introduction

Caissons are a type of deep foundations used for supporting heavy structures like bridges, high rise towers, heliostats, and silos^1,2. These are rigid and stubby foundations with large diameters and low aspect (length to diameter) or slenderness ratio ranging between 2 and 5². Well foundations are a variant of caissons with the slenderness ratio also varying between 2 and 5 that are used as bridge foundations in India. Several important bridges like the Tagus bridge in Portugal, San Francisco–Oakland Bay and Brooklyn bridges in the USA, Port Island and Nishinomiya-ko bridges in Japan, and Howrah and Ganga bridges in India have been constructed on caissons or well foundations. Design of caissons and well foundations against the ultimate limit state can be performed by calculating the load bearing capacity using the limit equilibrium-based analysis³ or continuum-based analysis^4,5,6,7. For design against the serviceability limit states, vertical and lateral displacements of caissons are checked against the corresponding allowable values.

Lateral loads and moments acting at the caisson head arise from the actions of wind, wave, traffic or earthquake on the superstructure it supports, and accurate analysis under these lateral loads and moments is required for design of caissons against the serviceability limit states. The traditional analysis of laterally loaded slender piles with translational Winkler soil springs is not applicable for caissons⁸ because of their low slenderness ratios. The large diameters of the caissons generate non-negligible vertical displacements at the caisson-soil interface as the caisson rotates under external lateral loads and moments. Thus, the effect of vertical soil displacement (and the corresponding resistance of soil) should be taken into account using rotational soil springs. However, there is no analytical method available in the literature for determination of the translational and rotational spring stiffnesses for analysis of stubby caissons.

In previous studies, Gerolymos & Gazetas, (2006)⁹ and Varun et al., (2009)² analyzed laterally loaded caissons using four-spring models in which distributed translational and rotational soil springs along the side of the caisson and concentrated shear translational and rotational springs at the base of the caisson were used. Chiou et al., (2012)¹⁰, based on the recommendations of the Japan Road Association (1990)¹¹, analyzed a bridge caisson in gravelly soils using a six-spring model comprising compressive and shear springs along the shaft in the front and at the sides, and at the base of the caisson. These apart, continuum-based finite element (FE) analysis have been used for analysis of laterally loaded caissons^4,5,12,13. While spring-based analyses are convenient, computationally efficient, and easy to perform¹⁴, determining the spring stiffnesses is often not easy¹⁵ and involves empirical methods leading to errors in the final results. In fact, the rotational and translational spring stiffnesses for caissons are not available in the literature for a wide range of soils. Continuum-based analyses, on the other hand, capture the three-dimensional (3D) interaction between the caisson and soil more accurately than the soil-spring approach, but these methods are usually computationally intensive, require 3D numerical methods to obtain solutions, and are not frequently used in practice. The simplified continuum approach, in which assumptions are made regarding the stress or displacement fields in the soil to obtain solutions, has been successfully used in soil-structure interaction problems^16,17. The simplified continuum approach has the advantage that the 3D soil-structure interaction is accounted for without very high computational time and effort. Recently, a simplified continuum-based semi-analytical method was developed for the analysis of laterally loaded rigid poles^18,19,20. However, the vertical soil displacement was assumed to be zero in this method, which introduces artificial restraints in the system and makes the analysis inaccurate for large diameter caissons with slenderness ratio 2–5.

In this study, a continuum-based analysis is developed for determining the displacement and translational and rotational spring stiffnesses of laterally loaded cylindrical caissons in multi-layered elastic soils. The three-dimensional (3D) interaction between the caisson and surrounding soil continuum is taken into account by defining an appropriate soil displacement field which explicitly accounts for the rotation of large diameter caissons and includes nonzero vertical soil displacement adjacent to the caissons. The principle of virtual work is used to obtain the equations governing caisson and soil displacements. Solutions to the equations are obtained analytically and numerically following an iterative algorithm. The analysis shows that the soil continuum surrounding the caisson can be naturally represented by a set of five soil springs, both rotational and translational, attached to the caisson. Equations quantifying the five soil-spring stiffnesses are obtained as part of the solution without recourse to empiricism. Systematic parametric studies are performed to investigate the effect of soil elastic constants and layering on the spring stiffnesses based on which fitted algebraic equations are developed for the five springs that can be used in a soil spring-based analysis of laterally loaded caissons. Additionally, the equivalent translational and rotational foundation stiffnesses have been determined that can be used to replace the caisson at the base of the superstructure for structural analysis of the superstructure without considering the caisson foundation underneath. Example problems are provided to demonstrate how the obtained spring stiffnesses can be used to calculate caisson displacements that can be used for caisson design against serviceability limit states.

Theoretical development

Problem description

A cylindrical rigid caisson of radius r_C (diameter D_C = 2r_C) and length L_C embedded in a soil continuum consisting of n horizontal layers and subjected to a concentrated lateral force F_a and a moment M_a at the head is considered, as shown in Fig. 1. The layered soil deposit is assumed to extend to large radial distances horizontally. The depths of the soil layers are finite except for the bottom (n^th) layer, which extends down vertically to a very large depth. The depth of any soil layer i from the ground level is H_i (H₀ = 0). The soil is assumed to be linear elastic, isotropic, and homogeneous within each layer, and is characterized by Lame’s constants λ_S and G_S. The caisson head is flushed with the horizontal ground surface, and the base rests on top of the n^th layer. There is no slippage or separation between the caisson and soil. A right-handed cylindrical (r, θ, z) coordinate is used for the analysis as shown in Fig. 1. The origin of the coordinate system is at the centre of the caisson head and the z axis points downward along the axis of the caisson.

Fig. 1

Laterally loaded rigid caisson in multi-layer soils.

Displacements in caisson and soil

The caisson is assumed to behave as a rigid body undergoing lateral translation and rotation, as shown in Fig. 2.

Fig. 2

Deflected shape of caisson.

Hence, the caisson displacement w(z) at any depth z from the ground surface can be expressed as

$$w\left( z \right)={w_t} - \Theta z$$

(1)

where w_t is the caisson head displacement at the ground level (w_t = w(0)), and Θ is the angle of rotation of the caisson axis in the vertical plane. Θ is a constant and does not change with depth because of the rigid behavior of the caisson. Equation (1) is valid for the domain 0 ≤ z ≤ L_C.

Considering the kinematics of the rigid caisson as shown in Fig. 3, The displacements in the soil surrounding the caisson is defined as:

$${u_r}\left( {r,\theta ,z} \right)={u_{rC}}\left( {\theta ,z} \right){\phi _r}\left( r \right)=w\left( z \right)\cos \theta \,{\phi _r}\left( r \right)$$

(2a)

$${u_\theta }\left( {r,\theta ,z} \right)={u_{\theta C}}\left( {\theta ,z} \right){\phi _\theta }\left( r \right)= - w\left( z \right)\sin \theta \,{\phi _\theta }\left( r \right)$$

(2b)

$$\begin{gathered} {u_z}\left( {r,\theta ,z} \right)={u_{zC}}\left( {\theta ,z} \right){\phi _z}(r)=\left[ {{u_{zC1}}\left( z \right)+{u_{zC2}}\left( {\theta ,z} \right)} \right]{\phi _z}(r) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=w(z)\alpha {\phi _z}(r)+{r_C}\Theta \cos \theta {\text{ }}{\phi _z}(r)\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \approx {r_C}\Theta \cos \theta {\text{ }}{\phi _z}(r) \hfill \\ \end{gathered}$$

(2c)

For domain L_C ≤ z < ∞, u_r and u_θ are identical to those given by equations (2a)-(2b) and u_z is given by

$$\begin{gathered} {u_z}\left( {r,\theta ,z} \right)={u_{zC}}\left( {\theta ,z} \right){\phi _z}(r)=\left[ {{u_{zC1}}\left( z \right)+{u_{zC2}}\left( {\theta ,z} \right)} \right]{\phi _z}(r) \hfill \\ \,\,\,\,\,\,=w(z)\alpha (z){\phi _z}(r)+{r_C}\frac{{dw\left( z \right)}}{{dz}}\cos \theta {\text{ }}{\phi _z}(r) \hfill \\ \,\,\,\,\,\, \approx {r_C}\frac{{dw\left( z \right)}}{{dz}}\cos \theta {\text{ }}{\phi _z}(r) \hfill \\ \end{gathered}$$

(2d)

In equations (2a)-(2d), w(z) is the displacement of the caisson axis described in Eq. (1) for domain 0 ≤ z ≤ L_C, and is the displacement of the flexible soil column axis underneath the caisson for the domain L_C ≤ z < ∞; u_r, u_θ, and u_z are the soil displacements in the radial, tangential, and vertical directions, respectively; u_rC and u_θC are the radial and tangential components of horizontal caisson displacement at the caisson-soil interface (i.e., u_rC = u_r(r_C, θ, z) and u_θC = u_θ(r_C, θ, z)); u_zC1 and u_zC2 are the two parts of vertical caisson displacement u_zC at the caisson-soil interface (u_zC = u_zC1 + u_zC2 = u_z(r_C, θ, z)); and φ_r, φ_θ, and φ_z are dimensionless displacement attenuation functions of soil associated with u_r, u_θ, and u_z, respectively. The functions φ_r, φ_θ, and φ_z vary along the radial distance r with their values equal to 1 at the caisson-soil interface (i.e., φ_r = φ_θ = φ_z = 1 at r = r_C) and 0 at large horizontal distances away from the caisson (i.e., φ_r = φ_θ = φ_z = 0 at r = ∞). The vertical displacement u_zC1 caused by the rotatory motion of the caisson is usually small because, in practical problems, Θ is small (sinΘ ≈ Θ) and this makes the angle α (see Fig. 3) also small (with sinα ≈ α) and u_zC1 → 0. However, the vertical displacement u_zC2 arising from the caisson radius r_C cannot be neglected because r_C is large for caissons. The assumed soil displacement field expressed in equations (2a)-(2d) ensures the continuity and compatibility of displacements and strains in soil in the radial, tangential, and vertical directions taking account of the rigid translation and rotation of the caisson, and also ensures that the soil displacements and strains decrease with an increase in radial distance from the caisson.

Fig. 3

Soil displacements at the caisson-soil interface caused by caisson movement.

Equilibrium of caisson-soil system and principle of virtual work

Under the action of F_a and M_a, reactive forces from the soil develop in the form of a distributed force p(z) and a distributed moment m(z) acting along the caisson shaft, and in the form of a concentrated shear force S_b and concentrated moment M_b acting at the base of the caisson. The forces acting are shown in the free body diagram of the caisson in Fig. 4. The reactive forces, in turn, act on the soil, loaded by the caisson, as evident from the free body diagram of the soil.

Fig. 4

Free body diagrams (FBDs) of caisson and soil.

For a virtual displacement δw of the caisson, the virtual displacement and rotation at the caisson head are δw_t and δΘ, respectively. Therefore, considering the forces and moments acting on the caisson (Fig. 4) and applying the principle of virtual work results in

$$\left[ {{F_a} - \int\limits_{0}^{{{L_C}}} {p\left( z \right)dz} - {S_b}} \right]\delta {w_t}+\left[ {{M_a}+\int\limits_{0}^{{{L_C}}} {p\left( z \right)zdz} - \int\limits_{0}^{{{L_C}}} {m\left( z \right)dz} - {M_b}+{S_b}{L_C}} \right]\delta \Theta =0$$

(3)

As δw_t and δΘ are arbitrary and non-zero, Eq. (3) results in

$${F_a}=\int\limits_{0}^{{{L_C}}} {p\left( z \right)dz} +{S_b}$$

(4a)

$${M_a}= - \int\limits_{0}^{{{L_C}}} {p\left( z \right)zdz} +\int\limits_{0}^{{{L_C}}} {m\left( z \right)dz} +{M_b} - {S_b}{L_C}$$

(4b)

Equation (4a) and (4b) describe the equilibrium of the caisson.

The virtual displacement δw of the caisson generates virtual strains in the adjacent soil. Thus, the principle of virtual work can be applied on the soil considering the forces acting on the soil (Fig. 4), and this results in

$$\begin{gathered} \int\limits_{0}^{{{L_C}}} {p\left( z \right)\delta wdz} +\int\limits_{0}^{{{L_C}}} {m\left( z \right)\delta \Theta dz} +{S_b}\delta {w_b}+{M_b}\delta \Theta \hfill \\ \quad \quad \quad \quad \quad \quad - \int\limits_{0}^{\infty } {\int\limits_{0}^{{2\pi }} {\int\limits_{{{r_C}}}^{\infty } {{\sigma _{lq}}\delta {\varepsilon _{lq}}rdrd\theta dz} } } - \int\limits_{{{L_C}}}^{\infty } {\int\limits_{0}^{{2\pi }} {\int\limits_{0}^{{{r_C}}} {{\sigma _{lq}}\delta {\varepsilon _{lq}}rdrd\theta dz} } } =0 \hfill \\ \end{gathered}$$

(5)

where δε_lq is the virtual soil strain tensor caused by the virtual caisson displacement δw, σ_lq is the corresponding soil stress tensor (suffixes l and q are the free indices used to represent the second order strain and stress tensors), and w_b is the displacement of caisson base at z = L_C, which is calculated as w_b = w_t − ΘL_C. The first four terms on the left-hand side of Eq. (5) are the components of the external virtual work and the remaining two terms represent the internal virtual work of the soil.

The stresses and strains in soil are related by linear elasticity as σ_lq = λ_Sδ_lqε_mm + 2G_Sε_lq (suffix m is a dummy index). Expressing the stresses in terms of strains and the Lame’s constants, expressing the strains in terms of the displacement functions (defined in equations (2a)-(2d)), and considering explicit soil layering as shown in Fig. 1 (suffix i denoting the i^th layer with i = 1, 2, … n), Eq. (5) can be rewritten as a variational equation in the form $\sum\limits_{{i=1}}^{n} {A({w_i})\delta {w_i}}$ + B(Θ)δΘ + C(φ_r)δφ_r + D(φ_θ)δφ_θ + F(φ_z)δφ_z = 0, from which algebraic or differential equations (and corresponding boundary conditions) of the functions w, Θ, φ_r, φ_θ, and φ_z describing the equilibrium of the caisson-soil system can be obtained in the form A(w_i) = 0, B(Θ) = 0, C(φ_r) = 0, D(φ_θ) = 0, and F(φ_z) = 0.

Forces and moments on caisson and corresponding soil spring stiffnesses

Considering the variations of w(z) over the domains H_i−1 ≤ z ≤ H_i (with H_i ≤ L_C) and L_C ≤ z < ∞, the equations of the distributed soil reaction force p(z) [N/m], distributed soil reaction moment m(z) [N-m/m], and concentrated shear force S_b [N] and moment M_b [N-m] arising from soil reaction at the caisson base are obtained as

$${p_i}={k_{sh,i}}\,{w_i}$$

(6a)

$${m_i}={k_{s\Theta ,i}}\,\Theta$$

(6b)

$${S_b}={K_{b,hh}}{w_b}+{K_{b,h\Theta }}\Theta$$

(6c)

$${M_b}={K_{b,\Theta h}}{w_b}+{K_{b,\Theta \Theta }}\Theta$$

(6d)

where p_i and m_i are respectively the functions p(z) and m(z) in the i^th layer. The parameters k_{sh, i} [N/m²] and k_sΘ,i [N] in equations (6a)-(6b) are given by

$$\begin{aligned} {k_{sh,i}}=\pi \int\limits_{{{r_C}}}^{\infty } {\left[ {\left( {{\lambda _{Si}}+2{G_{Si}}} \right)r{{\left( {\frac{{d{\phi _r}}}{{dr}}} \right)}^2}+2{\lambda _{Si}}\left( {{\phi _r} - {\phi _\theta }} \right)\frac{{d{\phi _r}}}{{dr}}+\left( {{\lambda _{Si}}+3{G_{Si}}} \right)\frac{1}{r}{{\left( {{\phi _r} - {\phi _\theta }} \right)}^2}} \right.} \\ \left. {+2{G_{Si}}\left( {{\phi _r} - {\phi _\theta }} \right)\frac{{d{\phi _\theta }}}{{dr}}+{G_{Si}}r{{\left( {\frac{{d{\phi _\theta }}}{{dr}}} \right)}^2}} \right]dr \\ \end{aligned}$$

(7a)

$${k_{s{\kern 1pt} \Theta ,i}}=\pi {G_{Si}}\int\limits_{{{r_C}}}^{\infty } {\left[ {{{\left( {{r_C}\frac{{d{\phi _z}}}{{dr}}+{\phi _r}} \right)}^2}+{{\left( {\frac{{{r_C}}}{r}{\phi _z}+{\phi _\theta }} \right)}^2}} \right]} rdr$$

(7b)

The parameters K_b,hh [N/m], K_b,hΘ [N], K_b,Θh [N], and K_b,ΘΘ [N-m] in equations (6c)-(6d) are given by

$${K_{b,hh}}={\kappa _{n1}}{m_1}{m_2}\left( {{m_1}+{m_2}} \right)$$

(8a)

$${K_{b,h\Theta }}={\kappa _{n1}}\left( {m_{1}^{2}+m_{2}^{2}+{m_1}{m_2}} \right)+\left[ {{\kappa _{n2}} - \pi {\lambda _{Sn}}\int\limits_{{{r_C}}}^{\infty } {{r_C}\left\{ {{\phi _z}\frac{{d{\phi _r}}}{{dr}}r+\left( {{\phi _r} - {\phi _\theta }} \right){\phi _z}} \right\}} dr} \right]$$

(8b)

$${K_{b,\Theta h}}= - \left[ {{\kappa _{n1}}{m_1}{m_2}+\pi {\lambda _{Sn}}\int\limits_{{{r_C}}}^{\infty } {{r_C}\left\{ {{\phi _z}\frac{{d{\phi _r}}}{{dr}}r+\left( {{\phi _r} - {\phi _\theta }} \right){\phi _z}} \right\}} dr} \right]$$

(8c)

$${K_{b,\Theta \Theta }}= - \left[ {{\kappa _{n1}}\left( {{m_1}+{m_2}} \right)+2\pi {L_C}{\lambda _{Sn}}\int\limits_{{{r_C}}}^{\infty } {{r_C}\left\{ {{\phi _z}\frac{{d{\phi _r}}}{{dr}}r+\left( {{\phi _r} - {\phi _\theta }} \right){\phi _z}} \right\}} dr} \right]$$

(8d)

where

$${m_{1,2}}=\sqrt {\frac{{ - {\kappa _{n2}} \pm \sqrt {\kappa _{{n2}}^{2} - 4{\kappa _{n1}}{k_{sh,n}}} }}{{2{\kappa _{n1}}}}}$$

(9a)

$${\kappa _{n1}}=\pi \left( {{\lambda _{Sn}}+2{G_{Sn}}} \right)\left[ {{r_C}^{2}\int\limits_{{{r_C}}}^{\infty } {{\phi _z}^{2}rdr} +\frac{{{r_C}^{4}}}{2}} \right]$$

(9b)

$$\begin{gathered} {\kappa _{n2}}=\pi \left[ {2{\lambda _{Sn}}\int\limits_{{{r_C}}}^{\infty } {{r_C}\left\{ {{\phi _z}\frac{{d{\phi _r}}}{{dr}}r+\left( {{\phi _r} - {\phi _\theta }} \right){\phi _z}} \right\}} dr} - {{G_{Sn}}\int\limits_{{{r_C}}}^{\infty } {\left\{ {{{\left( {{r_C}\frac{{d{\phi _z}}}{{dr}}+{\phi _r}} \right)}^2}+{{\left( {\frac{{{r_C}}}{r}{\phi _z}+{\phi _\theta }} \right)}^2}} \right\}} rdr - {G_{Sn}}{r_C}^{2}\left( {3+\mathop {\lim }\limits_{{a \to 0}} \int\limits_{a}^{{{r_C}}} {\frac{1}{r}dr} } \right)} \right] \\ \end{gathered}$$

(9c)

The parameters k_{sh, i} and k_sΘ,i (equations (6a)-(6b) and (7a)-(7b)) are analogous to the stiffnesses of translational and rotational soil springs, respectively, attached to the side of the caisson shaft for soil layer i, as shown in Fig. 5(a). Similarly, the parameters K_{b, hh}, K_b,ΘΘ, and K_b,Θh or K_{b, hΘ} (equations (8a)-(8d)) represent respectively the translational, rotational, and coupled stiffnesses of soil springs attached to the base of the caisson. Thus, the present continuum-based analysis shows that a 5-spring discrete model, shown in Fig. 5(a), can represent the resistances offered by the soil continuum surrounding a caisson. In fact, equations quantifying the five soil-spring stiffnesses have been derived without recourse to empiricism (equations (7a)-(7b), (8a)-(8d)).

Substituting p, m, S_b, and M_b obtained from equations (6a)-(6d) in Eq. (4a) and (4b), the caisson head displacement w_t and rotation Θ can be expressed as:

$$\left\{ {\begin{array}{*{20}{c}} {{w_t}} \\ \Theta \end{array}} \right\}=\frac{1}{{{K_{t,hh}}{K_{t,\Theta \Theta }} - {K_{t,h\Theta }}{K_{t,\Theta h}}}}\left[ {\begin{array}{*{20}{c}} {{K_{t,\Theta \Theta }}}&{ - {K_{t,h\Theta }}} \\ { - {K_{t,\Theta h}}}&{{K_{t,hh}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{F_a}} \\ {{M_a}} \end{array}} \right\}$$

(10)

where

$${K_{t,hh}}={K_{b,hh}}+\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{k_{sh,i}}dz} }$$

(11a)

$${K_{t,h\Theta }}={K_{b,h\Theta }} - {K_{b,hh}}{L_C} - \sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{k_{sh,i}}zdz} }$$

(11b)

$${K_{t,\Theta h}}={K_{b,\Theta h}} - {K_{b,hh}}{L_C} - \sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{k_{sh,i}}zdz} }$$

(11c)

$${K_{t,\Theta \Theta }}={K_{b,\Theta \Theta }}+{K_{b,hh}}L_{C}^{2} - ({K_{b,h\Theta }}+{K_{b,\Theta h}}){L_C}+\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{k_{sh,i}}{z^2}dz} +\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{k_{s{\kern 1pt} \Theta ,i}}dz} } }$$

(11d)

Equation (10) represents a set of algebraic equations that can be used to calculate the head displacement and rotation of the caisson. The parameters K_{t, hh} [N/m], K_t,ΘΘ [N-m], and K_t,Θh [N] or K_{t, hΘ} [N] in equations (11a)-(11d) are the stiffnesses of equivalent translational, rotational, and coupled springs that can be attached at the base of a superstructure replacing the caisson-soil system, as shown in Fig. 5(b). Equation (10) can be used by structural engineers to perform superstructure analysis with the foundation system replaced by the three springs K_{t, hh}, K_t,ΘΘ, and K_t,Θh shown in Fig. 5(b). Again, no empiricism is required to determine the spring stiffnesses K_{t, hh}, K_t,ΘΘ, and K_t,Θh or K_{t, hΘ}.

Fig. 5

(a) Rigid caisson with translational and rotational soil springs at the side and base under applied load and moment and (b) equivalent caisson-soil springs at the ground level.

Soil displacement functions

Quantification of the soil displacement attenuation functions φ_r, φ_θ and φ_z is necessary because these functions are required for the determination of the spring stiffnesses k_sh, k_sΘ, K_{b, hh}, K_b,ΘΘ, K_b,Θh, K_{b, hΘ}, K_{t, hh}, K_t,ΘΘ, K_t,Θh, and K_{t, hΘ}. These functions are obtained from the following differential equations, which were derived by considering independently the variations of φ_r(r), φ_θ(r), and φ_z(r):

$$\frac{{{d^2}{\phi _r}}}{{d{r^2}}}+\frac{1}{r}\frac{{d{\phi _r}}}{{dr}} - \left( {\frac{{1+{\gamma _1}}}{{{r^2}}}+{\gamma _2}} \right){\phi _r}=\frac{{{\gamma _3}+{\gamma _1}}}{r} \cdot \frac{{d{\phi _\theta }}}{{dr}} - \left( {\frac{{1+{\gamma _1}}}{{{r^2}}}} \right){\phi _\theta } - {r_C}\left( {{\gamma _4} - {\gamma _2}} \right)\frac{{d{\phi _z}}}{{dr}}$$

(12a)

$$\frac{{{d^2}{\phi _\theta }}}{{d{r^2}}}+\frac{1}{r}\frac{{d{\phi _\theta }}}{{dr}} - \left( {\frac{1}{{{r^2}}} \cdot \frac{{1+{\gamma _1}}}{{{\gamma _1}}}+{\gamma _5}} \right){\phi _\theta }= - \left( {\frac{{1+{\gamma _7}}}{r}} \right)\frac{{d{\phi _r}}}{{dr}} - \left( {\frac{1}{{{r^2}}} \cdot \frac{{1+{\gamma _1}}}{{{\gamma _1}}}} \right){\phi _r} - \frac{{{r_C}}}{r}\left( {{\gamma _6} - {\gamma _5}} \right){\phi _z}$$

(12b)

$$\frac{{{d^2}{\phi _z}}}{{d{r^2}}}+\frac{1}{r}\frac{{d{\phi _z}}}{{dr}} - \left( {\frac{1}{{{r^2}}}+{\gamma _8}} \right){\phi _z}= - \left( {\frac{{1 - {\gamma _9}}}{{{r_C}}}} \right)\left\{ {\frac{{d{\phi _r}}}{{dr}}+\frac{1}{r}\left( {{\phi _r} - {\phi _\theta }} \right)} \right\}$$

(12c)

along with the boundary conditions φ_r = φ_θ = φ_z = 1 at r = r_C and φ_r = φ_θ = φ_z = 0 at r = ∞. The dimensionless parameters γ₁-γ₉ in equations (12a)-(12c) are given by

$$\begin{gathered} {\gamma _1}=\frac{{\int\limits_{0}^{\infty } {{G_S}{w^2}dz} }}{{\int\limits_{0}^{\infty } {\left( {{\lambda _S}+2{G_S}} \right){w^2}dz} }} \\ =\frac{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{G_{Si}}{{\left( {{w_t} - \Theta {\kern 1pt} z} \right)}^2}dz} } +{G_{Sn}}\left( {\frac{{c_{1}^{2}}}{{2{m_1}}}+\frac{{c_{2}^{2}}}{{2{m_2}}}+\frac{{2{c_1}{c_2}}}{{{m_1}+{m_2}}}} \right)}}{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {\left( {{\lambda _{Si}}+2{G_{Si}}} \right){{\left( {{w_t} - \Theta {\kern 1pt} z} \right)}^2}dz} +} \left( {{\lambda _{Sn}}+2{G_{Sn}}} \right)\left( {\frac{{c_{1}^{2}}}{{2{m_1}}}+\frac{{c_{2}^{2}}}{{2{m_2}}}+\frac{{2{c_1}{c_2}}}{{{m_1}+{m_2}}}} \right)}} \\ \end{gathered}$$

(13a)

$$\begin{gathered} {\gamma _2}=\frac{{\int\limits_{0}^{\infty } {{G_S}{{\left( {\frac{{dw}}{{dz}}} \right)}^2}dz} }}{{\int\limits_{0}^{\infty } {\left( {{\lambda _S}+2{G_S}} \right){w^2}dz} }} \\ =\frac{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{G_{Si}}{\Theta ^2}dz} } +{G_{Sn}}\left( {\frac{{c_{1}^{2}{m_1}}}{2}+\frac{{c_{2}^{2}{m_2}}}{2}+\frac{{2{c_1}{c_2}{m_1}{m_2}}}{{{m_1}+{m_2}}}} \right)}}{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {\left( {{\lambda _{Si}}+2{G_{Si}}} \right){{\left( {{w_t} - \Theta {\kern 1pt} z} \right)}^2}dz} +} \left( {{\lambda _{Sn}}+2{G_{Sn}}} \right)\left( {\frac{{c_{1}^{2}}}{{2{m_1}}}+\frac{{c_{2}^{2}}}{{2{m_2}}}+\frac{{2{c_1}{c_2}}}{{{m_1}+{m_2}}}} \right)}} \\ \end{gathered}$$

(13b)

$$\begin{gathered} {\gamma _3}=\frac{{\int\limits_{0}^{\infty } {{\lambda _S}{w^2}dz} }}{{\int\limits_{0}^{\infty } {\left( {{\lambda _S}+2{G_S}} \right){w^2}dz} }} \\ =\frac{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{\lambda _{Si}}{{\left( {{w_t} - \Theta {\kern 1pt} z} \right)}^2}dz} } +{\lambda _{Sn}}\left( {\frac{{c_{1}^{2}}}{{2{m_1}}}+\frac{{c_{2}^{2}}}{{2{m_2}}}+\frac{{2{c_2}{c_4}}}{{{m_1}+{m_2}}}} \right)}}{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {\left( {{\lambda _{Si}}+2{G_{Si}}} \right){{\left( {{w_t} - \Theta {\kern 1pt} z} \right)}^2}dz} +} \left( {{\lambda _{Sn}}+2{G_{Sn}}} \right)\left( {\frac{{c_{1}^{2}}}{{2{m_1}}}+\frac{{c_{2}^{2}}}{{2{m_2}}}+\frac{{2{c_2}{c_4}}}{{{m_1}+{m_2}}}} \right)}} \\ \end{gathered}$$

(13c)

$$\begin{gathered} {\gamma _4}=\frac{{\int\limits_{{{H_{n - 1}}}}^{\infty } {{\lambda _{Sn}}{w_n}\frac{{{d^2}{w_n}}}{{d{z^2}}}dz} }}{{\int\limits_{0}^{\infty } {\left( {{\lambda _S}+2{G_S}} \right){w^2}dz} }} \\ =\frac{{{\lambda _{Sn}}\left\{ {\frac{{c_{1}^{2}{m_1}}}{2}+\frac{{c_{2}^{2}{m_2}}}{2}+\frac{{{c_1}{c_2}\left( {m_{1}^{2}+m_{2}^{2}} \right)}}{{{m_1}+{m_2}}}} \right\}}}{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {\left( {{\lambda _{Si}}+2{G_{Si}}} \right){{\left( {{w_t} - \Theta {\kern 1pt} z} \right)}^2}dz} +} \left( {{\lambda _{Sn}}+2{G_{Sn}}} \right)\left( {\frac{{c_{1}^{2}}}{{2{m_1}}}+\frac{{c_{2}^{2}}}{{2{m_2}}}+\frac{{2{c_2}{c_4}}}{{{m_1}+{m_2}}}} \right)}} \\ \end{gathered}$$

(13d)

$${\gamma _5}=\frac{{\int\limits_{0}^{\infty } {{G_S}{{\left( {\frac{{dw}}{{dz}}} \right)}^2}dz} }}{{\int\limits_{0}^{\infty } {{G_S}{w^2}dz} }}=\frac{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{G_{Si}}{\Theta ^2}dz} } +{G_{Sn}}\left( {\frac{{c_{1}^{2}{m_1}}}{2}+\frac{{c_{2}^{2}{m_2}}}{2}+\frac{{2{c_1}{c_2}{m_1}{m_2}}}{{{m_1}+{m_2}}}} \right)}}{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{G_{Si}}{{\left( {{w_t} - \Theta {\kern 1pt} z} \right)}^2}dz} } +{G_{Sn}}\left( {\frac{{c_{1}^{2}}}{{2{m_1}}}+\frac{{c_{1}^{2}}}{{2{m_2}}}+\frac{{2{c_1}{c_2}}}{{{m_1}+{m_2}}}} \right)}}$$

(13e)

$${\gamma _6}=\frac{{\int\limits_{{{H_{n - 1}}}}^{\infty } {{\lambda _{Sn}}{w_n}\frac{{{d^2}{w_n}}}{{d{z^2}}}dz} }}{{\int\limits_{0}^{\infty } {{G_S}{w^2}dz} }}=\frac{{{\lambda _{Sn}}\left\{ {\frac{{c_{1}^{2}{m_1}}}{2}+\frac{{c_{2}^{2}{m_2}}}{2}+\frac{{{c_1}{c_2}\left( {m_{1}^{2}+m_{2}^{2}} \right)}}{{{m_1}+{m_2}}}} \right\}}}{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{G_{Si}}{{\left( {{w_t} - \Theta {\kern 1pt} z} \right)}^2}dz} } +{G_{Sn}}\left( {\frac{{c_{1}^{2}}}{{2{m_1}}}+\frac{{c_{1}^{2}}}{{2{m_2}}}+\frac{{2{c_1}{c_2}}}{{{m_1}+{m_2}}}} \right)}}$$

(13f)

$${\gamma _7}\quad =\quad \frac{{\int\limits_{0}^{\infty } {{\lambda _S}{w^2}dz} }}{{\int\limits_{0}^{\infty } {{G_S}{w^2}dz} }}\quad =\quad \frac{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{\lambda _{Si}}{{\left( {{w_t} - \Theta {\kern 1pt} z} \right)}^2}dz} } +{\lambda _{Sn}}\left( {\frac{{c_{1}^{2}}}{{2{m_1}}}+\frac{{c_{2}^{2}}}{{2{m_2}}}+\frac{{2{c_1}{c_2}}}{{{m_1}+{m_2}}}} \right)}}{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{G_{Si}}{{\left( {{w_t} - \Theta {\kern 1pt} z} \right)}^2}dz} } +{G_{Sn}}\left( {\frac{{c_{1}^{2}}}{{2{m_1}}}+\frac{{c_{2}^{2}}}{{2{m_2}}}+\frac{{2{c_1}{c_2}}}{{{m_1}+{m_2}}}} \right)}}$$

(13g)

$${\gamma _8}=\frac{{\int\limits_{{{H_{n - 1}}}}^{\infty } {\left( {{\lambda _{Sn}}+2{G_{Sn}}} \right){{\left( {\frac{{{d^2}{w_n}}}{{d{z^2}}}} \right)}^2}dz} }}{{\int\limits_{0}^{\infty } {{G_S}{{\left( {\frac{{dw}}{{dz}}} \right)}^2}dz} }}=\frac{{\left( {{\lambda _{Sn}}+2{G_{Sn}}} \right)\left( {\frac{{c_{1}^{2}m_{1}^{3}}}{2}+\frac{{c_{2}^{2}m_{2}^{3}}}{2}+\frac{{2{c_1}{c_2}m_{1}^{2}m_{2}^{2}}}{{{m_1}+{m_2}}}} \right)}}{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{G_{Si}}{\Theta ^2}dz} } +{G_{Sn}}\left( {\frac{{c_{1}^{2}{m_1}}}{2}+\frac{{c_{2}^{2}{m_2}}}{2}+\frac{{2{c_1}{c_2}{m_1}{m_2}}}{{{m_1}+{m_2}}}} \right)}}$$

(13h)

$${\gamma _9}=\frac{{\int\limits_{{{H_{i - 1}}}}^{\infty } {{\lambda _{Sn}}{w_n}\frac{{{d^2}{w_n}}}{{d{z^2}}}dz} }}{{\int\limits_{0}^{\infty } {{G_S}{{\left( {\frac{{dw}}{{dz}}} \right)}^2}dz} }}=\frac{{{\lambda _{Sn}}\left\{ {\frac{{c_{1}^{2}{m_1}}}{2}+\frac{{c_{2}^{2}{m_2}}}{2}+\frac{{{c_1}{c_2}\left( {m_{1}^{2}+m_{2}^{2}} \right)}}{{{m_1}+{m_2}}}} \right\}}}{{\sum\limits_{{i=1}}^{{n - 1}} {\int\limits_{{{H_{i - 1}}}}^{{{H_i}}} {{G_{Si}}{\Theta ^2}dz} } +{G_{Sn}}\left( {\frac{{c_{1}^{2}{m_1}}}{2}+\frac{{c_{2}^{2}{m_2}}}{2}+\frac{{2{c_1}{c_2}{m_1}{m_2}}}{{{m_1}+{m_2}}}} \right)}}$$

(13i)

where

$${c_1}= - \frac{{{m_2}{w_t}}}{{\left( {{m_1} - {m_2}} \right)}} - \frac{{\left( {1 - {m_2}{L_C}} \right)\Theta }}{{\left( {{m_1} - {m_2}} \right)}}$$

(14a)

$${c_2}=\frac{{{m_1}{w_t}}}{{\left( {{m_1} - {m_2}} \right)}}+\frac{{\left( {1 - {m_1}{L_C}} \right)\Theta }}{{\left( {{m_1} - {m_2}} \right)}}$$

(14b)

The differential equations of φ_r, φ_θ, and φ_z given by equations (12a)-(12c) are solved using the one-dimensional finite difference (FD) method.

Equation (12a)-(12c) are interdependent (the functions φ_r, φ_θ, and φ_z are present in all the three differential equations). The interdependence of φ_r, φ_θ, and φ_z shows that the analysis captures the inherent connection of the different components of soil displacements. In fact, the caisson displacement and slope are functionally related to φ_r, φ_θ, and φ_z as evident from the equations of γ₁-γ₉ and of k_sh, k_sΘ, K_{b, hh}, K_{b, hΘ}, K_b,Θh, K_b,ΘΘ, K_{t, hh}, K_{t, hΘ}, K_t,Θh, and K_t,ΘΘ. Thus, the developed analysis framework explicitly captures the true soil-structure (caisson-soil) interaction in which the interrelationships between the caisson and soil displacements are mathematically established.

Solution algorithm

The input data required to perform the calculations are caisson diameter D_C, caisson length L_C, details of soil stratification n, H₁, H₂, … H_n−1, and soil elastic constants λ_Si and G_Si. First, the values of γ₁-γ₉ are assumed to be equal to 1 and the discrete nodal values of φ_θ and φ_z (at the FD nodes) are assumed to vary linearly from 1 at node 1 (first node at the caisson-soil interface) to 0 at node l (last node). With these assumed quantities, the nodal values of φ_r are calculated. Next, with the calculated nodal φ_r values and assumed nodal φ_z values, the nodal φ_θ values are calculated, and subsequently, with the calculated φ_r and φ_θ values, the nodal φ_z are calculated. After the first set of calculations, a convergence check on φ_r, φ_θ, and φ_z is performed with the criteria $\frac{1}{l}\sum\limits_{{j=1}}^{l} {\left| {\phi _{r}^{{(j),{\text{previous}}}} - \phi _{r}^{{(j),{\text{new}}}}} \right|} \leqslant {10^{ - 8}}$, $\frac{1}{l}\sum\limits_{{j=1}}^{l} {\left| {\phi _{\theta }^{{(j),{\text{previous}}}} - \phi _{\theta }^{{(j),{\text{new}}}}} \right|} \leqslant {10^{ - 8}}$ and $\frac{1}{l}\sum\limits_{{j=1}}^{l} {\left| {\phi _{z}^{{(j),{\text{previous}}}} - \phi _{z}^{{(j),{\text{new}}}}} \right|} \leqslant {10^{ - 8}}$ where φ^(j),previous and φ^(j),new are the values of the functions φ_r^(j), φ_θ^(j), and φ_z^(j) at the previous and current iterations, respectively. Iterations are continued until convergence on φ_r, φ_θ, and φ_z are achieved.

With the converged values of φ_r, φ_θ, and φ_z and their numerical derivatives, the parameters k_sh, k_sΘ, κ_n1, κ_n2, K_{b, hh}, K_{b, hΘ}, K_b,Θh, K_b,ΘΘ, K_{t, hh}, K_{t, hΘ}, K_t,Θh, and K_t,ΘΘ are calculated with which w and Θ are calculated using Eqs. (10) and (1). The calculated values of w and Θ are used to calculate γ₁-γ₉ using equations (13a)-(13i). Subsequently, the newly calculated γ₁-γ₉ are used to iteratively recalculate φ_r, φ_θ, and φ_z with the final values of φ_r^(j), φ_θ^(j), and φ_z^(j) obtained from the previous iteration as the initial starting values. With the newly calculated and converged values of φ_r, φ_θ, and φ_z, the parameters k_sh, k_sΘ, κ_n1, κ_n2, K_{b, hh}, K_{b, hΘ}, K_b,Θh, K_b,ΘΘ, K_{t, hh}, K_{t, hΘ}, K_t,Θh, and K_t,ΘΘ; w and Θ; and γ₁-γ₉ are sequentially recalculated. A convergence criterion on γ_q (q = 1, 2, …, 9) is set as ${{\left| {\gamma _{q}^{{{\text{previous}}}} - \gamma _{q}^{{{\text{new}}}}} \right|} \mathord{\left/ {\vphantom {{\left| {\gamma _{q}^{{{\text{previous}}}} - \gamma _{q}^{{{\text{new}}}}} \right|} {\gamma _{q}^{{{\text{previous}}}}}}} \right. \kern-0pt} {\gamma _{q}^{{{\text{previous}}}}}} \leqslant 0.001$. Iterations on γ₁-γ₉ (with the inner iteration on φ_r, φ_θ, and φ_z) are continued until convergence is reached at which point the calculated w and Θ are reported as the final values of caisson displacement and rotation. The convergence on γ_q is shown in Fig. 6 where the L² norm of ${{\left| {\gamma _{q}^{{{\text{previous}}}} - \gamma _{q}^{{{\text{new}}}}} \right|} \mathord{\left/ {\vphantom {{\left| {\gamma _{q}^{{{\text{previous}}}} - \gamma _{q}^{{{\text{new}}}}} \right|} {\gamma _{q}^{{{\text{previous}}}}}}} \right. \kern-0pt} {\gamma _{q}^{{{\text{previous}}}}}}$ is plotted against the number of iterarations for Problems 1–4 (details of the problems are given in the following section).

Fig. 6

Convergence of γ_q with the number of iterations.

Accuracy and nature of caisson responses

Verification with 3D finite element analysis

The accuracy of the present analysis is first verified against equivalent 3D elastic FE analysis performed using the software ABAQUS²¹. Two caissons with L_C/D_C = 2 and 5 embedded in homogenous (Problem 1) and 4-layer (Problem 2) soil deposits, respectively, are considered for the verification study, and the displacement profiles of these caissons obtained from present analysis and 3D FE analysis are shown in Fig. 7(a). The figure also shows the inputs used for the analysis of the two caissons. The FE models in ABAQUS are made by considering the caisson and the soil as a single part to ensure no separation or slippage between the caisson and the soil. The vertical curved boundary of the domain is taken at a distance of 25r_C from the caisson-soil interface and the bottom horizontal boundary is considered at a distance of 2L_C from the bottom of the caisson to minimize the boundary effect by performing convergence studies. A rigid body constraint is applied to the caissons and 8-noded brick elements with reduced integration (C3D8R) are used to model the caisson and the soil. The global mesh sizes considered in the analysis are in the range 0.3–0.5 m. Edge seeding in the radial and vertical direction from the caisson-soil interface are done with mesh size of β−5β where β varies from 0.3 to 0.5 m. The differences between head displacements obtained from the present analysis and FE analysis are 7.1% and 6.7% for the caissons with L_C/D_C = 2 and 5, respectively. Displacement profiles for Problems 1 and 2 that are obtained from the beam-spring model, where the caisson is modeled as rigid beam and the soil spring constants are calculated per Gerolymos and Gazetas (2006)⁹ and Gazetas (1991)²² are also plotted in Fig. 7(a). Because of the rigid body behavior, material properties of caissons and their thicknesses are not required in the present analysis (rigid body implies no deformation internal to the body). However, for the requirement of numerical analysis, material properties of the caissons are required by the software and are provided in Fig. 7(a).

In Fig. 7(b), the resultant soil displacement u ($u=\sqrt {u_{r}^{2}+u_{\theta }^{2}+u_{z}^{2}}$) at the ground surface (i.e., at z = 0) is plotted as a function of radial distance r for θ = 0 (i.e., in the direction along which the applied force F_a acts) for both the caissons with L_C/D_C = 2 and 5. The match between the displacement u versus normalized distance r/r_C plots obtained from the present analysis and FE analysis is good. The figure shows that displacements in soil become negligible at radial distances of about 20 times the caisson radius. The soil displacement u along the caisson-soil interface is also calculated for different values of θ (with r = r_C) at the ground surface (z = 0) and plotted in Fig. 7(c). The match between the u versus θ plots obtained from the present analysis and FE analysis is quite well.

Fig. 7

(a) Lateral displacement profiles of caissons in homogeneous (Problem 1) and 4-layer soils (Problem 2), (b) soil displacement at the surface in the direction of applied load, and (c) surface soil displacement at caisson-soil interface along the perimeter of the caisson.

For the second set of verification study, the variation of normalized caisson head displacement w_tG_S^*D_C/F_a and w_tG_S^*D_C²/M_a in homogeneous soils, arising respectively from applied force F_a and applied moment M_a, with caisson slenderness ratio L_C/D_C obtained from the present analysis and from the Fourier FE analysis of Higgins et al., (2013)²³ are compared in Fig. 8. The figure shows that the caisson displacements obtained from the present analysis match very well with those obtained by Higgins et al., (2013)²³.

Fig. 8

Normalized displacement of caisson head in homogeneous soil.

In order to systematically quantify the difference between the caisson response obtained from the present analysis and 3D FE analysis, 50 laterally loaded caisson simulations are performed using the present analysis and 3D FE analysis for a variety of caisson aspect ratios in the range 2–5 and with a variety of strong and weak soils (with and without multiple layers). The caisson head displacements w_t,ANA and w_t,FE obtained respectively from the present analytical and 3D FE solutions are plotted in Fig. 9. The scatter around the 1:1 is an indication of the present analysis. It is observed that w_t,_ANA and w_t,FE are quite comparable with w_t,ANA about 7% less than the corresponding w_t,FE on the average. The root mean square difference of the scatter in Fig. 9 is 0.97 mm and the fitted trend line with R² = 0.999 shows that w_t,ANA = 0.93w_t,FE. While the exact reason of the 7% deviation cannot be firmly ascertained, the likely cause of this deviation is because of the assumption of the soil displacement field in equations (2a)-(2d)), as given in the manuscript, which imposes some restraint in the soil deformation in the present analysis which is absent in the FE analysis.

Fig. 9

Head displacement w_t,FE of caissons obtained from 3D FE analysis versus corresponding head displacements w_t,ANA obtained from present analysis.

Also plotted in Fig. 9 are the caisson head displacements for the same cases obtained from the simplified continuum analysis of Gupta & Basu, (2016b)²⁰ without any modulus adjustment in which the vertical soil displacement u_z was assumed to be zero. As evident, the analysis of Gupta & Basu, (2016b)²⁰ do not produce accurate results for caissons with L_C/D_C = 2–5 and deviates significantly and inconsistently from the 3D FE analysis. This further proves the necessity of the present analysis in which the vertical soil displacement and rotatory soil resistance are considered. It is important to note that other simplified continuum analyses, such as that of Gupta & Basu, (2016b)²⁰, often take recourse of artificial adjustment of soil moduli to obtain accurate results. No such artificial modulus adjustments are required for the present analysis.

Validation with field test

The results of two field tests conducted by Li et al., (2022)²⁴ (Problem 3) and Yang & Liang, (2006)²⁵ (Problem 4) are used for the validation of the present analysis. The caisson displacement profiles obtained from the field test of Li et al., (2022)²⁴ and from the corresponding simulation using the present method are shown in Fig. 10 along with the details of input parameters (the input parameters are given by Li et al., (2022)²⁴. The caisson head displacements obtained from present analysis is 2.56 mm and the average caisson head displacement obtained from the field test is 2.75 mm, with the difference being 6.7%. The head displacement field test data used in the validation are obtained from multiple displacement profiles provided by Li et al., (2022)²⁴ based on measurements at different locations of the caisson cross-section.

Fig. 10

Comparison of lateral displacement of caisson obtained from present analysis and field test of Li et al. (2022).

The field test of Yang & Liang, (2006)²⁵ was performed on a free-head drilled shaft of diameter 2.59 m and length 20.12 m (Young’s modulus of the caisson material E_c = 25 GPa and Poisson’s ratio of the caisson material υ_c = 0.2) in a four-layer soil and subjected to a lateral load of 400 kN with no applied moment. The elastic soil properties at the site were calculated by Gupta & Basu, (2016)²⁰ as: E_S1 = 10 MPa, E_S2 = 44.56 MPa, E_S3 = 4.51 MPa, and E_S4 = 6.58 MPa; υ_S1 = υ_S2 = υ_S3 = υ_S4 = 0.25; and H₁ = 7.93 m, H₂ = 15.70 m, and H₃ = 20.10 m (E_Si and υ_Si are respectively the Young’s modulus and Poisson’s ratio of the i^th soil layer; E_Si = G_Si(3λ_Si + 2G_Si)/(λ_Si + G_Si) and υ_Si = 0.5λ_Si/(λ_Si + G_Si)). The test of Yang & Liang, (2006)²⁵ is simulated using the present analysis, which produced a head displacement of 3 mm which is about 5% less than the field value of 3.14 mm. The field result is expected to be somewhat greater than the calculated value because the slenderness ratio L_C/D_C of the drilled shaft is 7.8, which is greater than the range 2–5 considered in this analysis.

Soil reaction forces and moments and caisson bending moment

For the caissons described in Fig. 7(a), the distributed soil reaction force p (p = k_shw) and distributed soil reaction moment m (m = k_sΘΘ) with normalized depth are shown in Figs. 11(a) and 11(b), respectively. The soil reaction force p exhibits a linear variation with depth (because p is proportional to caisson displacement w, which varies linearly with depth) while the soil reaction moment m remains constant with depth (because m is proportional to caisson rotation Θ, which is independent of depth). Also plotted in these figures are the corresponding side spring stiffnesses k_sh and k_sΘ. It is important to note that the side spring stiffnesses are constants for a particular soil layer.

Soil reaction forces and moments along with the applied force and moment are used to calculate the bending moment (BM) in the caisson. Although, the caisson behaves as a rigid body, bending moment in the caisson is required for the structural design of caissons. Therefore, the bending moment is calculated externally by calculating the sum of all the moments either to the left or to the right of the section. The bending moment profiles for the caissons described in Fig. 7(a) are shown in Fig. 11(c). The maximum bending moment can be obtained from the figure and used for the purpose of structural design. Note that, when a distributed moment m(z) acts, the maximum bending moment occurs at a location where the shear force equals the distributed moment.

Fig. 11

Variation of (a) soil reaction force and lateral soil spring stiffness, (b) soil reaction moment and rotational soil spring stiffness, and (c) bending moment along the caissons shaft with depth for homogeneous and 4-layer soils.

Characterization of soil spring stiffnesses

Parametric studies are conducted to identify the important soil and caisson parameters that influence the spring stiffnesses. For the purpose of the parametric studies, the soil stiffness is represented by a single parameter G_S^* (instead of soil elastic constant pairs λ_S and G_S or E_S and υ_S), which is an equivalent soil shear modulus defined by Randolph, (1981)²⁶ as G_S^* = G_S(1 + 0.75υ_S). This was done based on the observation by Randolph, (1981)²⁶ that soil Poisson’s ratio υ_S does not have any significant impact on laterally loaded pile response and the minor effect of υ_S can be taken into account through G_S^*. Thus, normalizations of the spring stiffnesses (k_sh, K_{b, hh}, K_b,ΘΘ, K_b,Θh, K_{b, hΘ} K_{t, hh}, K_{t, hΘ}, K_t,Θh, and K_t,ΘΘ) are done in terms of G_S^* except for the rotational stiffness k_sΘ, which is normalized with respect to the actual soil shear modulus G_S because the analytical equation of k_sΘ (Eq. (11b)) is independent of the soil Poisson’s ratio and contains only G_S.

For cases with layered soil, the side spring stiffnesses k_sh,i and k_sΘ,i of the i^th layer are normalized with respect to the equivalent and actual shear moduli G_Si^* and G_Si of the corresponding layer. The normalizations of the base spring stiffnesses K_{b, hh}, K_b,ΘΘ, K_b,Θh, and K_{b, hΘ} are done with respect to G_Sn^* of the n^th layer in which the corresponding springs lie. For the top spring stiffnesses K_{t, hh}, K_{t, hΘ}, K_t,Θh, and K_t,ΘΘ, the normalizations are done with respect to G_S1^* of the top layer.

The practical cases investigated comprise two types of soil profiles: (1) homogeneous soil with a single soil layer and (2) two-layer soil profiles with H₁/L_C = 0.1–0.9 and G_S2^*/G_S1^* = 0.1–5.0.1.0. For these cases, the applied load eccentricity M_a/(F_aD_C) is varied over a range of 0–2. The caisson aspect ratio L_C/D_C values considered for the parametric studies lie in the range of 2–5.

Effect of caisson diameter

Figures 12(a)-(c) show the variations of the normalized spring stiffnesses with caisson diameter for homogeneous (single-layer) and two-layer soil profiles with H₁/L_C = 0.3 and G_S2^*/G_S1^* = 1.5 and for L_C/D_C = 4 and M_a/(F_aD_C) = 1. Similar plots were obtained for other cases as well. The plots show that caisson diameter has a negligible effect on the spring stiffnesses.

Fig. 12

(a) Dimensionless translational and rotational side spring stiffnesses, (b) dimensionless translational, coupled, and rotational base spring stiffnesses, and (c) dimensionless equivalent lateral, rotational, and coupled caisson-soil spring stiffnesses in homogeneous and two-layer soils as functions of caisson diameter.

Effect of caisson aspect ratio

Figures 13(a)-(c) show the variations of the normalized spring stiffnesses with caisson aspect ratio L_C/D_C for homogeneous (single-layer) and two-layer soil profiles with H₁/L_C = 0.3 and G_S2^*/G_S1^* = 1.5 and for L_C/D_C = 4 and M_a/(F_aD_C) = 1. Similar plots were obtained for other cases as well. It is evident that L_C/D_C has a large impact on the spring stiffnesses. The normalized translational spring stiffnesses k_sh and K_{b, hh} decrease with an increase in L_C/D_C while the remaining spring stiffnesses increase with an increase in L_C/D_C.

Fig. 13

(a) Dimensionless translational and rotational side spring stiffnesses, (b) dimensionless translational, coupled, and rotational base spring stiffnesses, and (c) dimensionless equivalent lateral, rotational, and coupled caisson-soil spring stiffnesses in homogeneous and two-layer soils as functions of caisson aspect ratio.

Effect of load eccentricity

Figures 14(a)-(c) show the variations of the normalized spring stiffnesses with load eccentricity M_a/(F_aD_C) for homogeneous (single-layer) and two-layer soil profiles with H₁/L_C = 0.3 and G_S2^*/G_S1^* = 1.5 and for L_C/D_C = 4. Similar plots were obtained for other cases as well. The load eccentricity impacts the soil spring stiffnesses. All the spring stiffnesses except k_sh increases with an increase of load eccentricity while k_sh exhibits an opposite trend.

Fig. 14

(a) Dimensionless translational and rotational side spring stiffnesses, (b) dimensionless translational, coupled, and rotational base spring stiffnesses, and (c) dimensionless equivalent lateral, rotational, and coupled caisson-soil spring stiffnesses in homogeneous and two-layer soils as functions of applied load eccentricity.

Design equations

Based on the observations made in Figs. 12, 13 and 14, fitted algebraic equations are developed that can be used by designers for quick calculation of the soil spring stiffnesses. For single-layer soil deposits, the parameters that affect the spring stiffness are L_C/D_C and M_a/(F_aD_C). For two-layer soil deposits, additional parameters that affect spring stiffnesses are the relative stiffness G_S2^*/G_S1^* of soil in the two layers and the ratio of top layer thickness to caisson length H₁/L_C. These parameters are considered in developing the fitted equations obtained by minimizing the least square error using the inbuilt genetic algorithm solver and fmincon command in MATLAB based on 200 cases for single-layer soils and 7500 cases for two-layer soils. The values of the parameters are considered over their practical range with small increments to generate the 200 and 7500 cases. These equations are given below.

Spring stiffnesses for Caisson in 1-layer homogeneous soil

For cases with M_a = 0:

$${k_{sh,1}}=6.53{G_{S1}}^{*}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{ - 0.41}}$$

(15a)

$${k_{s\Theta ,1}}=0.74{G_{S1}}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{1.07}}$$

(15b)

$${K_{b,hh}}=7.97{G_{S1}}^{*}{D_C}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{ - 0.72}}$$

(15c)

$${K_{b,h\Theta }}={K_{b,\Theta h}}=1.13{G_{S1}}^{*}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.28}}$$

(15d)

$${K_{b,\Theta \Theta }}=1.13{G_{S1}}^{*}{D_C}^{3}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.76}}$$

(15e)

$${K_{t,hh}}=13.42{G_{S1}}^{*}{D_C}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.35}}$$

(15f)

$${K_{t,h\Theta }}={K_{t,\Theta h}}=9.38{G_{S1}}^{*}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{1.3}}$$

(15g)

$${K_{t,\Theta \Theta }}=8.74{G_{S1}}^{*}{D_C}^{3}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{2.25}}$$

(15h)

For cases with M_a ≠ 0:

$${k_{sh,1}}=6.53{G_{S1}}^{*}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{ - 0.41}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.03}}$$

(16a)

$${k_{s\Theta ,1}}=0.74{G_{S1}}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{1.07}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{ - 0.1}}$$

(16b)

$${K_{b,hh}}=7.97{G_{S1}}^{*}{D_C}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{ - 0.72}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.01}}$$

(16c)

$${K_{b,h\Theta }}={K_{b,\Theta h}}=1.13{G_{S1}}^{*}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.28}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.08}}$$

(16d)

$${K_{b,\Theta \Theta }}=1.13{G_{S1}}^{*}{D_C}^{3}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.76}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.08}}$$

(16e)

$${K_{t,hh}}=13.42{G_{S1}}^{*}{D_C}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.35}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.03}}$$

(16f)

$${K_{t,h\Theta }}={K_{t,\Theta h}}=9.38{G_{S1}}^{*}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{1.3}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.02}}$$

(16g)

$${K_{t,\Theta \Theta }}=8.74{G_{S1}}^{*}{D_C}^{3}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{2.25}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.01}}$$

(16h)

Spring stiffnesses for caisson in 2-layer soil

For cases with M_a = 0:

$${k_{sh,i}}=5.86{G_{Si}}^{*}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{ - 0.34}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{0.04}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{0.06}}$$

(17a)

$${k_{s\Theta ,i}}=1.12G_{{Si}}^{{}}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.7}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{ - 0.23}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{ - 0.19}}$$

(17b)

$${K_{b,hh}}=7.41{G_{S2}}^{*}{D_C}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.01}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{0.04}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{ - 0.04}}$$

(17c)

$${K_{b,h\Theta }}={K_{b,\Theta h}}=0.35{G_{S2}}^{*}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{1.34}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{0.69}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{ - 0.59}}$$

(17d)

$${K_{b,\Theta \Theta }}=0.19{G_{S2}}^{*}{D_C}^{3}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{2.15}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{0.84}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{ - 0.86}}$$

(17e)

$${K_{t,hh}}=12.06{G_{S1}}^{*}{D_C}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.29}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{ - 0.17}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{0.84}}$$

(17f)

$${K_{t,h\Theta }}={K_{t,\Theta h}}=8.72{G_{S1}}^{*}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{1.26}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{ - 0.12}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{0.94}}$$

(17g)

$${K_{t,\Theta \Theta }}=8.39{G_{S1}}^{*}{D_C}^{3}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{2.21}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{ - 0.09}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{0.95}}$$

(17h)

For cases with M_a ≠ 0:

$${k_{sh,i}}=6.95{G_{Si}}^{*}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{ - 0.42}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.04}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{ - 0.03}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{0.01}}$$

(18a)

$${k_{s\Theta ,i}}=0.64G_{{Si}}^{{}}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{1.03}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{ - 0.13}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{ - 0.11}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{ - 0.01}}$$

(18b)

$${K_{b,hh}}=7.76{G_{S2}}^{*}{D_C}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.01}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.02}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{0.04}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{ - 0.06}}$$

(18c)

$${K_{b,h\Theta }}={K_{b,\Theta h}}=0.41{G_{S2}}^{*}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{1.28}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.09}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{0.36}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{ - 0.56}}$$

(18d)

$${K_{b,\Theta \Theta }}=0.25{G_{S2}}^{*}{D_C}^{3}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{2.09}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.06}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{0.45}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{ - 0.76}}$$

(18e)

$${K_{t,hh}}=13.06{G_{S1}}^{*}{D_C}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{0.27}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.02}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{ - 0.18}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{0.82}}$$

(18f)

$${K_{t,h\Theta }}={K_{t,\Theta h}}=9.26{G_{S1}}^{*}{D_C}^{2}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{1.23}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.01}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{ - 0.12}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{0.93}}$$

(18g)

$${K_{t,\Theta \Theta }}=8.63{G_{S1}}^{*}{D_C}^{3}{\left( {\frac{{{L_C}}}{{{D_C}}}} \right)^{2.2}}{\left( {\frac{{{M_a}}}{{{F_a}{D_C}}}} \right)^{0.006}}{\left( {\frac{{{H_1}}}{{{L_C}}}} \right)^{ - 0.09}}{\left( {\frac{{G_{{s2}}^{*}}}{{G_{{s1}}^{*}}}} \right)^{0.95}}$$

(18h)

Normalized spring stiffnesses calculated using the fitted equations are compared with the corresponding stiffness values obtained analytically by solving equations (7a)-(7b), (10a)-(10d), and (11a)-(11d). It is found that the predictions made using the fitted equations are quite well. For example, the spring constants k_sh and k_sΘ obtained analytically (denoted with subscript ‘ana’) and using the fitted equations are plotted (denoted with subscript ‘fitted’) are respectively plotted along with the 45° (1:1) line in Fig. 15. The scatter around the 1:1 line and the corresponding root mean square difference (= 0.23 and 0.46 for k_sh and k_sΘ, respectively) indicate the accuracy of the fitted equations. Similar scatters are obtained for K_t,hh, K_t,ΘΘ, K_t,hΘ, K_b,hh, K_b,ΘΘ, and K_b,hΘ as well with the corresponding root mean square difference of the scatters being 5.26, 31.21, 10.80, 0.56, 1.26, and 0.61, respectively. Caisson head displacement obtained using the fitted equations should be increased by 7% based on the results shown in Fig. 9.

Fig. 15

Scatters of analytically calculated and fitted normalized values of k_sh and k_sΘ around the 1:1 line.

Numerical examples

Two example problems are analyzed using the fitted algebraic equations given above to demonstrate how designers can use the equations. For the two problems, two rigid caissons, one embedded in homogeneous soil and one in two-layer soil, are selected.

For the first problem, a caisson with D_C = 5 m and L_C = 10 m embedded in a homogeneous soil with G_S = 12.5 MPa and υ_S = 0.4 is selected with F_a = 3 MN and M_a = 25 MN-m. Therefore, G_S^* = 12.5 × (1 + 0.75 × 0.4) = 16.25 MPa, L_C/D_C = 10/5 = 2 and M_a/(F_aD_C) = 25/(3 × 5) = 1.67. Using equations (16a)-(16 h), the spring stiffnesses are calculated as: k_sh = 8.1 × 10⁷ N/m², k_sΘ = 4.60 × 10⁸ N, K_b,hh = 6.2 × 10⁸ N/m, K_b,hΘ = 5.82 × 10⁸ N, K_b,ΘΘ = 4.08 × 10⁹ N-m, K_t,hh = 1.41 × 10⁹ N/m, K_t,hΘ = 9.48 × 10⁹ N, and K_t,ΘΘ = 8.49 × 10¹⁰ N-m. The caisson head displacement is calculated from Eq. (10) using K_t,hh, K_t,hΘ, or K_{t, Θh} and K_t,ΘΘ as w_t = (K_t,ΘΘ×F_a−K_t,hΘ×M_a)/(K_t,hh×K_t,ΘΘ−K_t,hΘ×K_t,Θh) = 16.4 mm, which increased by 7% gives w_t = 17.6 mm. The same problem is solved using 3D FE (ABAQUS) analysis, and the caisson head displacements w_t,FE = 17.2 mm. Alternatively, k_sh, k_sΘ, K_b,hh, K_b,hΘ, and K_b,ΘΘ can be used to calculate the caisson displacement using a stick-spring model shown in Fig. 5(a) (solution can be obtained using the one-dimensional FE or FD method). The advantage of this second approach is that the caisson bending moment and shear force can also be calculated externally. The head displacement obtained using the second approach, after increasing by 7%, is w_t, = 17.1 mm.

For the second problem, a caisson with D_C = 7 m and L_C = 30 m (L_C/D_C = 30/7 = 4.28) embedded in a two-layer soil with elastic constants G_S1 = 9.26 MPa, G_S2 = 16 MPa, υ_S1 = 0.2, and υ_S2 = 0.3 such that the top layer thickness H₁ = 17 m and the second layer extends to great depth. The caisson is subjected to F_a = 8 MN and M_a = 50 MN-m. For this problem, G_S1^* = 11.69 MPa, G_S2^* = 19 MPa, G_S2^*/G_S1^* = 1.73, H₁/L_C = 0.57, and M_a/(F_aD_C) = 0.89. The spring stiffnesses for layers 1 and 2 are calculated from equations (18a)-(18 h) as: k_sh,1 = 4.43 × 10⁷ N/m², k_sh,2 = 6.97 × 10⁷ N/m², and k_sΘ,1 = 1.83 × 10⁹ N, k_sΘ,2 = 2.87 × 10⁹ N, K_b,hh = 9.84 × 10⁸ N/m, K_b,hΘ = 1.52 × 10⁹ N, K_b,ΘΘ = 1.6 × 10¹⁰ N-m, K_t,hh = 2.35 × 10⁹ N/m, K_t,hΘ = 5.36 × 10¹⁰ N, and K_t,ΘΘ = 1.42 × 10¹² N-m. The caisson head displacement is calculated, after 7% increase, as 19.5 mm from Eq. (10) (w_t = (K_t,ΘΘ×F_a−K_t,hΘ×M_a)/(K_t,hh×K_t,ΘΘ−K_t,hΘ×K_t,Θh)) using K_t,hh, K_t,hΘ, or K_{t, Θh} and K_t,ΘΘ. The head displacement of the same caisson from 3D FE analysis is obtained w_t,FE = 20.6 mm. Subsequently, k_sh, k_sΘ, K_b,hh, K_b,hΘ, and K_b,ΘΘ are used to calculate the caisson displacement using a stick-spring model as shown in Fig. 5(a), and the head displacement obtained using the stick-spring model, after increasing by 7%, is w_t, = 19.9 mm.

The numerical models of the example problems are constructed by considering the caisson as a rigid body. Because of the rigid body assumption, the thickness of caisson is not required in the present analysis and FE analysis. In ABAQUS, however, the caisson material properties are required; the Young’s modulus and Poisson’s ratio of the caisson material for both the problems are 25 GPa and 0.2, respectively.

Conclusions

A continuum-based analytical framework is developed for analysis of laterally loaded rigid caissons with aspect ratio 2–5 embedded in layered elastic soil. The interaction between the caisson and surrounding soil continuum is considered by assuming a compatible soil displacement field that explicitly takes into account the vertical soil displacement adjacent to the caisson caused by caisson rotation. The principle of virtual work is used to obtain the equations that describe the caisson and soil displacements. One-dimensional finite difference method and closed form analytical method are used in the solution procedure, which follows an iterative algorithm. The methodology provides accurate results without any artificial soil modulus adjustment and it is also computationally efficient not requiring any elaborate numerical meshing or 3D numerical analysis.

It is shown mathematically that the caisson-soil interaction can be represented by a 5-spring model that includes both translational and rotational soil side and base springs with spring constants k_sh, k_sΘ, K_b,hh, K_b,ΘΘ, and K_b,hΘ or K_b,Θh. The equations of these spring stiffnesses are determined rigorously as part of the developed framework considering three-dimensional caisson-structure interaction. Equivalent caisson-soil springs are often used by structural engineers to replace foundations for analysis of superstructures. Such spring constants K_t,hh, K_t,ΘΘ, and K_t,hΘ or K_t,Θh are also derived in this analysis, and these springs can replace the caisson-soil system.

Parametric studies are performed based on which fitted algebraic equations for these spring stiffnesses are obtained for single (homogeneous) and two-layer soil profiles that can be readily used by the practitioners for calculation of caisson displacement and maximum bending moment. Two numerical problems are solved that demonstrate the use of the developed fitted equations.

It is to be mentioned that, although the present model can predict lateral caisson displacement accurately, the model in its current form can not capture the nonlinear behavior of soil. Further, the developed model underpredicts the caisson displacement by 7%. Considering a nonlinear soil constitutive model should be able to eliminate this limitation.