Analysis of Rigid Short Caisson with Granular Core Incorporating Nonlinear Interface and Base Responses

Abstract

It is difficult to construct a conventional shallow foundation in alluvial lowlands because of soft soils and high ground water table. A rigid short caisson foundation with granular core is being proposed for alluvial lowlands. The proposed foundation is analyzed using non-linear hyperbolic stress–displacement responses of homogeneous alluvial deposits. Extensive parametric studies are carried out to study the effects of length ratio (L/d₀), diameter ratio (d/d₀) of granular core with respect to casing, relative stiffness of shaft (α_τ), relative casing base stiffness (α_b), and friction angle of granular material (ϕ_gp) on the load sharing and the settlement of the proposed foundation.

Appendices

Appendix A: Sinking and Installation of Proposed Foundation

In order to help in sinking of the proposed foundation, material (soil) is dredged or excavated and removed from the inside of well casing (steining). If its own weight is not sufficient, it may be possible to put on the well an additional weight and kentledge to overcome side friction and effect sinking. Sinking can be expedited by excavating deeper than cutting edge and making a sump, a few centimeters deep, in the well. A sump of greater depth is dangerous as it may cause the well to sink with a jerk. After being sunk to its final stage, granular material is filled in and compacted to achieve maximum dry density. Proposed foundation after sinking and installation is shown in Fig. 21.

Fig. 21

Proposed foundation after sinking and installation

Appendix B: Analysis

The loads shared by the well casing, Q_st, and the granular core, Q_gp, are controlled by the stiffness and the geometry of the foundation and the interfacial shear stresses. The vertical force equilibrium for the composite foundation with granular core inside is expressed as

$$ \hbox{Q}=\hbox{Q}_{\rm st}+\hbox{Q}_{\rm gp} $$

(B.1)

or

$$ \hbox{Q}=\hbox{Q}_{\rm st,s} +\hbox{Q}_{\rm st,b}+\hbox{Q}_{\rm gp,L} $$

(B.2)

where Q_st,s and Q_st,b are respectively the loads shared by the shaft surface and the base of the casing and Q_gp,L the load at the base of granular core (i.e. at the level of the foundation) (Fig. 4). Equation B.2 in terms of stresses becomes

$$\hbox{q}\,\cdot\,\pi{\left(\frac{\hbox{d}_0^{2}}{4}\right)}=\pi \hbox{d}_0\hbox{L}\tau+\pi \left( {\frac{\hbox{d}_0^{2}-\hbox{d}^{2}}{4}} \right)\hbox{q}_{\rm st,b}+\pi\left( {\frac{\hbox{d}^{2}}{4}} \right) \hbox{q}_{\rm gp,L} $$

(B.3)

or

$$ \hbox{q}=4\left({\frac{\hbox{L}}{\hbox{d}_0}}\right) \tau+\left({1-\left( {\frac{\hbox{d}^{2}}{\hbox{d}_0^{2}}} \right)} \right)\hbox{q}_{\rm st,b}+\left( {\frac{\hbox{d}^{2}}{\hbox{d}_0^{2}}}\right) \hbox{q}_{\rm gp,L} $$

(B.4)

where q, τ, q_st,b and q_gp,L are the average stresses acting on the foundation, outer surface of the casing and the bases of the casing and the granular core respectively.

The vertical equilibrium of an element of the granular core (Fig. 5), neglecting its weight, is

$$ (\sigma_{\rm z}+\Updelta\sigma_{\rm z})\left( {\frac{\pi }{4}} \right)\hbox{d}^{2}-\sigma_{\rm z} \left( {\frac{\pi }{4}} \right)\hbox{d}^{2}-\tau_{\rm gp}( \pi \hbox{ d})=0 $$

(B.5)

or

$$ \left({\frac{\hbox{d}\sigma\hbox{z}}{\hbox{dz}}} \right)- \left( {\frac{4}{\hbox{d}}} \right)\tau_{\rm gp}=0 $$

(B.6)

where σ_z and τ_gp are the vertical and shear stresses on the surface of granular core respectively.

Assuming full mobilization of interface shear resistance between the granular core and the inner surface of the casing, i.e. τ_gp = σ_h·tan δ = K σ_z tan δ where K is the coefficient of lateral earth pressure and δ is the wall friction angle. Substituting for τ_gp in Eq. B.6, the solution of the above differential equation is obtained as

$$ \sigma_{\rm z} = \hbox{c}_{0}\,\hbox{exp}\,(\hbox{c}_{1}\hbox{z}) $$

(B.7)

where c₁ = (4/d) K tan δ and c₀ is a constant. At the top of granular core i.e. z = 0, $\sigma_{\rm z}=\hbox{q}_{\rm gp}=\frac{\hbox{Q}\hbox{gp}}{\pi\left({\frac{\hbox{d}^{2}}{4}}\right)}.$ Hence, c₀ = q_gp. The stress transferred by the granular core, q_{gp, L}, to the soil below becomes

$$ \hbox{q}_{\rm gp,L}=\hbox{q}_{\rm gp}\hbox{R}_{\rm gp} $$

(B.8)

where R_gp = exp (c₁L).

Substituting Eq. B.8 in Eq. B.4, one gets

$$ \hbox{q}=4\left( {\frac{\hbox{L}}{\hbox{d}_0}} \right) \tau+\left( {1-\left( {\frac{\hbox{d}^{2}}{\hbox{d}_0^{2}}} \right)} \right)\hbox{q}_{\rm st,b}+\left( {\frac{\hbox{d}^{2}}{\hbox{d}_0^{2}}} \right) \hbox{q}_{\rm gp }\hbox{R}_{\rm gp} $$

(B.9)

Settlement of soil below granular core, w_sL (Poulos and Davis 1981; Scott 1981), for linear stress–settlement response is

$$ \hbox{w}_{\rm sL}=\sigma_{\rm z,L} \left( {\frac{\hbox{d }\left( { 1-\nu_{\rm s}^2} \right)}{\hbox{E}_{\rm s}}} \right)=\left( {\frac{{\hbox{q}_{\rm gp}}_{\rm ,L}}{\hbox{k}_{\rm sL}}} \right) $$

(B.10)

where I_f = an influence factor = constant and $\hbox{k}_{\rm sL}=\left( {{\hbox{E}_{\rm s}}/{\hbox{d}\left({1-\nu_{\rm s}^2}\right)\hbox{I}_{\rm f}}}\right)$—modulus of subgrade reaction of the soil below the granular core.

Duncan and Chang (1970) proposed a hyperbolic relationship for stress, σ, strain, ɛ, relationship that is non-linear. The hyperbolic relation is of the form

$$ \Updelta\sigma=\frac{\varepsilon}{\left({\hbox{a}+\hbox{b}\,\varepsilon} \right)} $$

(B.11)

where a and b are constants.

Considering the non-linear stress–settlement behavior of the alluvial soil and using the above mentioned hyperbolic relationship, one may write

$$ \hbox{q}_{\rm gp,L}=\frac{\hbox{w}_{\rm sL}}{\left( {\frac{1}{\hbox{k}_{\rm sL}}}\right)+\left( {\frac{1}{\hbox{q}_{\rm ult}}}\right)\hbox{w}_{\rm sL}} $$

(B.12)

or

$$ \hbox{q}_{\rm gp,L }=\left({\frac{\hbox{k}_{\rm sL}\hbox{ w}_{\rm sL} }{\left( {1+\left( {\frac{\hbox{k}_{\rm sL}} {\hbox{q}_{\rm ult}}} \right)\hbox{w}\hbox{sL}} \right)}} \right). $$

(B.13)

Substituting the value of q_gp,L from Eq. B.8, one gets

$$ \hbox{q}_{\rm gp} \hbox{R}_{\rm gp}=\left( {\frac{\hbox{k}_{\rm sL} \hbox{w}_{\rm sL} }{\left( {1+\left( {\frac{\hbox{k}_{\rm sL}} {\hbox{q}_{\rm ult} }} \right)\hbox{w}_{\rm sL}}\right)}} \right) $$

(B.14)

for the settlement, w_sL at the base of the granular core in terms of q_gp, as

$$ \hbox{w}_{\rm sL}=\frac{\hbox{q}_{\rm gp}\hbox{R}_{\rm gp} }{\left( {\hbox{k}_{\rm sL}\left({1-\left( {\frac{\hbox{q}_{\rm gp} \hbox{R}_{\rm gp} }{\hbox{q}_{\rm ult}}} \right)} \right)} \right)} $$

(B.15)

The granular core is under K₀ condition and its compression, Δw_gp, is evaluated by integrating the one dimensional compression equation for an element as

$$ \Updelta \hbox{w}_{\rm gp}=\int\limits_0^L \frac{\sigma_{\rm z}}{\hbox{D}_{\rm gp}} \hbox{dz} $$

(B.16)

$$ \Updelta \hbox{w}_{\rm gp}=\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{D}_{\rm gp}} } \right)\left[ {\frac{\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right)\hbox{d}_0 (\hbox{R}_{\rm gp} -1)}{\hbox{t}}} \right] $$

(B.17)

where R_gp = exp (t(L/d₀)/(d/d₀)), t = 4 K_o tan δ, D_gp = the constrained modulus = [E_gp(1−ν_gp)/(1 + ν_gp)(1−2ν_gp)] = β E_gp and β  =  (1−ν_gp)/[(1 + ν_gp)(1−2 ν_gp)].

The settlement at the top of the granular core, w_sL,0, i.e. at z = 0, is the sum of the compression of the core and the settlement of the alluvial soil below, and is given as

$$ \hbox{w}_{\rm st} =\hbox{w}_{\rm sL,0}=\hbox{w}_{\rm sL} +\Updelta \hbox{w}_{\rm sL} $$

(B.18)

$$ \hbox{w}_{\rm st}=\frac{\hbox{q}_{\rm gp}\hbox{R}_{\rm gp} }{\left({\hbox{k}_{\rm sL}\left( {1-\left( {\frac{\hbox{q}_{\rm gp}\hbox{R}_{\rm gp} }{\hbox{q}_{\rm ult}}} \right)} \right)} \right)}+\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{D}_{\rm gp} }} \right)\left[ {\frac{\left({\frac{\hbox{d}}{\hbox{d}_0 }} \right)\hbox{d}_0 (\hbox{R}_{\rm gp}-1)}{\hbox{t}}} \right] $$

(B.19)

$$ \hbox{w}_{\rm st} =\hbox{q}_{\rm gp}\hbox{f}_1 $$

(B.20)

where

$$ \hbox{f}_{1}=\left( {\frac{R_{\rm gp} }{\hbox{k}_{\rm sL} \left( {1-\left( {\frac{\hbox{q}_{\rm gp}\hbox{R}_{\rm gp} }{\hbox{q}_{\rm ult}}} \right)} \right)}} \right)+\left( {\frac{\hbox{d}\left( {R_{\rm gp} -1} \right)}{\hbox{D}_{\rm gp} \hbox{t}}} \right). $$

(B.21)

The casing assumed to be rigid settles by w_st. Compatibility of displacements of the caisson and the top of the granular core requires

$$ \hbox{w}_{\rm st} = \hbox{w}_{\rm gp,0} $$

(B.22)

The non-linear shaft stress–shaft displacement and casing base pressure–base settlement (Fig. 6) relationships (Scott 1981) are

$$ \tau=\left({\frac{\hbox{k}_{\rm sts}\hbox{w}_{\rm st}}{\left( {1+\left( {\frac{\hbox{k}_{\rm sts}}{\tau _{\rm ult}}} \right)\hbox{w}_{\rm st}} \right)}} \right) $$

(B.23)

and

$$ \hbox{q}_{\rm st,b}=\left( {\frac{\hbox{k}_{\rm stb} \hbox{w}_{\rm st} }{\left( {1+\left( {\frac{\hbox{k}_{\rm stb} }{\hbox{q}_{\rm ult} }} \right)\hbox{w}_{\rm st}} \right)}} \right) $$

(B.24)

where τ_ult, q_ult, k_st,s and k_st,b are the ultimate shear, ultimate bearing stress, casing shaft–soil and casing base–soil stiffness (subgrade) constants respectively. The stiffness, k_st,s for casing shaft–soil stiffness is related to k_st,b, the casing base–soil stiffness (Scott 1981) as

$$ \hbox{k}_{\rm st,s}=\alpha_{\tau}\hbox{k}_{\rm st,b} $$

(B.25)

while k_st,b in turn is related to the subgrade constant k_sL as

$$ \hbox{k}_{\rm st,b}=\alpha_{\rm b} \hbox{k}_{\rm sL} $$

(B.26)

where α_τ and α_b are constants of proportionality.

Substituting Eqs. B.23 and B.24 in Eq. B.9, one gets

$$ \hbox{q}=4\left({\frac{\hbox{L}}{\hbox{d}_0}} \right) \left( {\frac{\hbox{k}_{\rm sts} \hbox{w}_{\rm st} }{\left( {1+\left( {\frac{\hbox{k}_{\rm sts} }{\tau _{\rm ult} }} \right)\hbox{w}_{\rm st} } \right)}} \right)+\left( {1-\left( {\frac{\hbox{d}^{2}}{\hbox{d}_0^{2}}} \right)} \right)\left( {\frac{\hbox{k}_{\rm stb} \hbox{w}_{\rm st} }{\left( {1+\left( {\frac{\hbox{k}_{\rm stb} }{\hbox{q}_{\rm ult} }} \right)\hbox{w}_{\rm st} } \right)}} \right)+\left( {\frac{\hbox{d}^{2}}{\hbox{d}_0^{2}}} \right) \hbox{q}_{\rm gp }\hbox{R}_{\rm gp}. $$

(B.27)

Substituting the values of k_st,s and k_st,b from Eq. B.25 and Eq. B.26 respectively into Eq. B.27, one gets

$$ \hbox{q} =\left( {\frac{\hbox{w}_{\rm st} }{\hbox{d}_0 }} \right)\left[ {\left( {\frac{4\alpha_\tau \alpha_{\rm b}\left( {\frac{\hbox{L}}{\hbox{d}_0}} \right)\hbox{k}_{\rm s,L} \hbox{d}_0}{1+\alpha_\tau \alpha_{\rm b} \left( {\frac{\hbox{k}_{\rm sL} \hbox{ d}_0 }{\tau _{\rm ult} }} \right)\left( {\frac{\hbox{w}_{\rm st} }{\hbox{d}_0 }} \right)}} \right)+\left( {1-\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right)^2}\right)\left( {\frac{\alpha_{\rm b} \hbox{k}_{\rm sL} \hbox{d}_0 }{1+\alpha_{\rm b} \left( {\frac{\hbox{k}_{\rm sL} \hbox{d}_0}{\hbox{q}_{\rm ult} }} \right)\left( {\frac{\hbox{w}_{\rm st}}{\hbox{d}_0}} \right)}} \right)} \right]+\left( {\frac{\hbox{d}}{\hbox{d}_0}} \right)^{2}\hbox{q}_{\rm gp}R_{\rm gp}. $$

(B.28)

Terms of Eq. B.20 may be re-arranged as

$$ \hbox{w}_{\rm st}=\hbox{q}_{\rm gp} \hbox{f}_{\rm 1}=\hbox{q}_{\rm gp} \left[ {\left( {\frac{\hbox{R}_{\rm gp} }{\hbox{k}_{\rm sL} \left( {1-\left( {\frac{\hbox{q}_{\rm gp} \hbox{R}_{\rm gp} }{\hbox{q}_{\rm ult}}}\right)} \right)}} \right)+\left( {\frac{\left( {\frac{\hbox{d}}{\hbox{d}_0}} \right)\hbox{d}_0 \left( {\hbox{R}_{\rm gp} -1} \right)}{\hbox{D}_{\rm gp} \hbox{t}}}\right)}\right] $$

(B.29)

Normalizing w_st with respect to d₀, one gets

$$ \left( {\frac{\hbox{w}_{\rm st} }{\hbox{d}_0 }} \right)=\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{d}_0}} \right) \left[ {\left( {\frac{\hbox{R}_{\rm gp} }{\hbox{k}_{\rm sL} \left( {1-\left( {\frac{\hbox{q}_{\rm gp} \hbox{R}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)} \right)}} \right)+\left( {\frac{\left( {\frac{\hbox{d}}{\hbox{d}_0}} \right)\hbox{d}_0 \left( {\hbox{R}_{\rm gp} -1} \right)}{\hbox{D}_{\rm gp} \hbox{t}}} \right)} \right] $$

(B.30)

$$ \left( {\frac{\hbox{w}_{\rm st} }{\hbox{d}_0 }} \right) = \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{d}_0 }} \right) \left( {\frac{1}{\hbox{k}_{\rm sL} }} \right)\left( {\left( {\frac{\hbox{R}_{\rm gp} }{\left( {1-\left( {\frac{\hbox{q}_{\rm gp} \hbox{R}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)} \right)}} \right)+\left( {\frac{\left( {\hbox{R}_{\rm gp} -1} \right)}{\left( {\frac{\hbox{D}_{\rm gp} }{\hbox{k}_{\rm sL} \hbox{d}}} \right)\hbox{t}}} \right)} \right) $$

(B.31)

$$ \left( {\frac{\hbox{w}_{\rm st} }{\hbox{d}_0}} \right)= \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{k}_{\rm sL} \hbox{d}}} \right)\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right) \left[ {\left( {\frac{\hbox{R}_{\rm gp}}{\left( {1-\left( {\frac{\hbox{q}_{\rm gp} \hbox{R}_{\rm gp} }{\hbox{q}_{\rm ult}}} \right)} \right)}} \right)+\left( {\frac{\left( {\hbox{R}_{\rm gp} -1} \right)}{\hbox{R t}}} \right)} \right]=\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{k}_{\rm sL} \hbox{ d}}} \right)\left( {\frac{\hbox{d}}{\hbox{d}_0 }}\right) \hbox{f}_{2} $$

(B.32)

where

$$ \hbox{R}= \hbox{relative granular core stiffness} =(\hbox{D}_{\rm gp}/\hbox{k}_{\rm sL}\hbox{d}). $$

(B.33)

Substituting the value of K_sL from Eq. B.12 and value of D_gp into above equation, one gets

$$ \hbox{R}=\left( {\frac{\hbox{D}_{\rm gp} }{\hbox{k}_{\rm sL} \hbox{d}}} \right) $$

(B.34)

$$ =\left( {\frac{\beta \hbox{E}_{\rm gp} }{\left( {\frac{\hbox{E}_{\rm s} }{\hbox{d}(1-\upsilon_{\rm s}^2 )\hbox{I}_{\rm f}}} \right)\hbox{d}}} \right) $$

(B.35)

$$ =\beta^\ast \left( {\frac{\hbox{E}_{\rm gp} }{\hbox{E}_{\rm s}}} \right)=\beta^\ast \hbox{M}_{\rm R} $$

(B.36)

where $\beta^\ast= (1- \nu_{\rm s}^{2}) I_{\rm f}\,\cdot\,\beta, \beta\,=\,(1 - \nu_{\rm gp}) / [(1+ \nu_{\rm gp}) (1- 2\,\nu_{\rm gp})]$ and

$$ \hbox{f}_{2}=\left[ {\left( {\frac{\hbox{R}_{\rm gp} }{\left( {1-\left( {\frac{\hbox{q}_{\rm gp} \hbox{R}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)} \right)}} \right)+\left( {\frac{\left( {\hbox{R}_{\rm gp} -\hbox{1}} \right)}{\hbox{R t}}} \right)} \right]. $$

(B.37)

Substituting Eq. B.31 in Eq. B.28, one gets

$$ \hbox{q}=\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{k}_{\rm sL} \hbox{ d}_0 }} \right)\hbox{f}_2 \left[ {\left( {\frac{4\alpha_\tau \alpha_{\rm b}\left( {\frac{\hbox{L}}{\hbox{d}_0 }} \right)\hbox{ k}_{\rm sL} \hbox{d}_0 }{1+\alpha_\tau \alpha_{\rm b} \left( {\frac{\hbox{k}_{\rm sL} \hbox{ d}}{\tau_{\rm ult} }} \right)\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{k}_{\rm sL} \hbox{ d}}} \right)\hbox{f}_2 }} \right)+\left( {1-\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right)^{2}} \right)\left( {\frac{\alpha_{\rm b} \hbox{k}_{\rm sL} \hbox{d}_0 }{1+\alpha_{\rm b} \left( {\frac{\hbox{k}_{\rm sL} \hbox{d}}{\hbox{q}_{\rm ult} }} \right)\left( {\frac{\hbox{q}_{\rm gp}}{\hbox{k}_{\rm sL} \hbox{ d}}} \right)\hbox{ f}_2 }} \right)} \right] + \left( {\frac{\hbox{d}}{\hbox{d}_0}} \right)^{2}\hbox{q}_{\rm gp} \hbox{R}_{\rm gp} $$

(B.38)

Dividing both sides by q_ult,

$$ \left( {\frac{\hbox{q}}{\hbox{q}_{\rm ult} }} \right)= \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\hbox{ f}_2 \left[ {\left( {\frac{4\alpha_\tau \alpha_{\rm b} \left( {\frac{\hbox{L}}{\hbox{d}_0 }} \right)}{1+\alpha_\tau \alpha_{\rm b} \left( {\frac{\hbox{k}_{\rm sL} \hbox{ d}}{\tau_{\rm ult} }} \right)\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\left( {\frac{\hbox{q}_{\rm ult} }{\hbox{k}_{\rm sL} \hbox{d}}} \right)\hbox{f}_2 }} \right)+\left( {1-\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right)^{2}} \right)\left( {\frac{\alpha_{\rm b} }{1+\alpha_{\rm b} \left( {\frac{\hbox{k}_{\rm sL} \hbox{ d}}{\hbox{q}_{\rm ult} }} \right)\left( {\frac{\hbox{q}_{\rm gp}}{\hbox{q}_{\rm ult} }} \right)\left( {\frac{\hbox{q}_{\rm ult} }{\hbox{k}_{\rm sL} \hbox{ d}}} \right)\hbox{f}_2 }} \right)} \right] + \left( {\frac{\hbox{d}}{\hbox{d}_0}} \right)^{2} \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\hbox{R}_{\rm gp} $$

(B.39)

$$ \left( {\frac{\hbox{q}}{\hbox{q}_{\rm ult} }} \right)= \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\hbox{ f}_2 \left[ {\left( {\frac{4\alpha_\tau \alpha_{\rm b}\left( {\frac{\hbox{L}}{\hbox{d}_0 }} \right)}{1+\alpha_\tau \alpha_{\rm b} \left( {\frac{\hbox{k}_{\rm sL} \hbox{ d}}{\tau_{\rm ult} }} \right)\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\left( {\frac{\hbox{q}_{\rm ult} }{\hbox{k}_{\rm sL} \hbox{d}}} \right)\hbox{f}_2 }} \right)+\left( {1-\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right)^{2}} \right)\left( {\frac{\alpha_{\rm b} }{1+\alpha_{\rm b} \left( {\frac{\hbox{q}_{\rm gp}}{\hbox{q}_{\rm ult} }} \right)\hbox{ f}_2 }} \right)} \right] + \left( {\frac{\hbox{d}}{\hbox{d}_0}} \right)^{2} \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right) \hbox{R}_{\rm gp} $$

(B.40)

$$ \left( {\frac{\hbox{q}}{\hbox{q}_{\rm ult} }} \right)= \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\hbox{F} $$

(B.41)

or

$$ \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)=\left( {\frac{\hbox{q}}{\hbox{q}_{\rm ult} }} \right)\left( {\frac{1}{\hbox{F}}} \right) $$

(B.42)

or

$$ \left( {\frac{\hbox{Q}_{\rm gp} }{\hbox{Q}_{\rm ult} }} \right)=\left( {\frac{\hbox{q}}{\hbox{q}_{\rm ult} }} \right)\left( {\frac{(\hbox{d/d}_0)^{2}}{\hbox{F}}} \right) $$

(B.43)

or

$$ \left( {\frac{\hbox{Q}_{\rm st} }{\hbox{Q}_{\rm ult} }} \right)=\left( {1-\left( {\frac{\hbox{Q}_{\rm gp} }{\hbox{Q}_{\rm ult} }} \right)} \right)\ast 100 $$

(B.44)

where

$$ \hbox{F} = \left[ {\hbox{f}_2 \left( {\frac{4\alpha_\tau \alpha_{\rm b} \left( {\frac{\hbox{L}}{\hbox{d}_0 }} \right)}{1+\alpha_\tau \alpha_{\rm b} \left( {\frac{\hbox{A}_{\rm st} }{\hbox{B}_{\rm st} }} \right)\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\hbox{f}_2 }} \right)+\left( {1-\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right)^{2}} \right)\left( {\frac{\alpha_{\rm b} }{1+\alpha_{\rm b} \left( {\frac{\hbox{q}_{\rm gp}}{\hbox{q}_{\rm ult} }} \right)\hbox{ f}_2 }} \right)} \right]+\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right)^{2} \hbox{R}_{{\rm gp}} $$

(B.45)

where A_st = (k_sL  d/τ_ult), B_st = (k_sL  d/q_ult)

From Eq. B.32, one gets

$$ \left( {\frac{\hbox{w}_{\rm st} }{\hbox{d}_0 }} \right)= \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{k}_{\rm sL} \hbox{ d}}} \right)\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right) \hbox{f}_{2}=\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\left( {\frac{\hbox{q}_{\rm ult} }{\hbox{k}_{\rm sL} \hbox{ d}}} \right)\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right)\hbox{f}_2 =\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult}}} \right)\left( {\frac{1}{\hbox{B}_{\rm st} }} \right)\left( {\frac{\hbox{d}}{\hbox{d}_0}} \right) \hbox{f}_{2}. $$

(B.46)

Similarly from Eq. B.15, the settlement, w_sL of granular core at the base is

$$ \hbox{w}_{\rm sL}=\frac{\hbox{q}_{\rm gp} \hbox{R}_{{\rm gp}}}{\left( {\hbox{k}_{\rm s,L} \left( {1-\left( {\frac{\hbox{q}_{\rm gp} \hbox{R}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)} \right)} \right)} $$

(B.47)

$$ \left( {\frac{\hbox{w}_{\rm sL} }{\hbox{d}_0 }} \right)= \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\left( {\frac{\hbox{q}_{\rm ult} }{\hbox{k}_{\rm sL} \hbox{ d}}} \right)\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right)\frac{\hbox{R}_{\rm gp} }{\left( {\left( {1-\left( {\frac{\hbox{q}_{\rm gp} \hbox{R}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)} \right)} \right)} $$

(B.48)

$$ \left( {\frac{\hbox{w}_{\rm SL}}{\hbox{d}_0 }} \right)= \left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\left( {\frac{1}{\hbox{B}_{\rm st} }} \right)\left( {\frac{\hbox{d}}{\hbox{d}_0 }} \right)\hbox{J} $$

(B.49)

where

$$ \hbox{J}=\left( {\frac{\hbox{R}_{\rm gp}}{\left( {1-\left( {\frac{\hbox{q}_{\rm gp} }{\hbox{q}_{\rm ult} }} \right)\hbox{R}_{\rm gp} } \right)}} \right) $$

(B.50)

is the settlement influence coefficient.

Appendix C: Numerical Example

Consider a composite foundation consisting of circular short open caisson of length 1.5 m, outer diameter (d₀) 1.0 m, and inner diameter (d) 0.85 m sinked in alluvial soil deposit and granular material is filled in and compacted to achieve maximum dry density. The density and frictional angle (ϕ) of soil and granular core material are 18.5 kN/m³ and 30° and 24.0 kN/m³ and 40° respectively. The length to diameter ratio (L/d₀) and diameter ratio (d/d₀) are 1.5 and 0.85 respectively.

Hatanaka and Uchida (1996) provided a simple correlation between ϕ and SPT N_cor for granular material, which may be expressed as

$$ \phi =\sqrt{20{\hbox {N}}_{\rm cor} }+20 $$

(C.1)

and

$$ \hbox{N}_{\rm cor}= \hbox{C}_{\rm N} \hbox{N}_{\rm F} $$

(C.2)

where

$$ \hbox{C}_{\rm N}= \hbox{Overburden Correction factor} =0.77 \log_{10 }\frac{2000}{\sigma_v^{\prime}} $$

(C.3)

and σ ^′ _v  = Effective overburden pressure in kN/m² and N _F —Field SPT Value.

Several investigators have correlated the values of modulus of elasticity, E_s, with field SPT value, N_F. Schmertmann (1970) proposed the following correlation between modulus, E_s, of granular material and N_F.

$$ \hbox{E} (\hbox{kN}/\hbox{m}^{2})=766\,\hbox{N}_{\rm F } $$

(C.4)

Using Eqs. C.1 to C.4, one gets E_s = 3,830 kN/m² and E_gp =  19,150 kN/m² and hence, ratio (E_gp/E_s) = 2.67

Referring to Eq. 6 as reproduced below:

$$ \hbox{q}_{\rm ult} = 4\left( {\frac{\hbox{L}}{\hbox{d}_0}} \right) \tau+\left( {1-\left( {\frac{\hbox{d}^{2}}{\hbox{d}_0^{2}}} \right)} \right)\hbox{q}_{\rm st,b}+\left( {\frac{\hbox{d}^{2}}{\hbox{d}_0^{2}}} \right) \hbox{q}_{\rm gp }\hbox{R}_{\rm gp} $$

(C.5)

where$\tau=\hbox{K}\,\sigma^{\prime}_{\nu}\,\,\hbox{tan}\,\,\Updelta=3.68\,\hbox{kN/m}^{2};$ $\hbox{q}_{\rm st,b}=\sigma^{\prime}_{\nu}\hbox{N}_{\rm q}=234.6\,\hbox{kN/m}^{2};$ $\hbox{ q}_{\rm gp}=\sigma^{\prime}_{\nu_{\rm gp}} \hbox{N}_{\rm{q}_{\rm gp}}=1251.9\,\hbox{kN/m}^{2};$ c₁ = 4.0 K tan δ = 1.15; and R_gp = exp (c₁(L/d₀) = 5.65.

Substituting the above values in Eq. C.5, one gets

$$ \hbox{q}_{\rm ult}=314.6\,\hbox{kN/m}^{2}. $$

Using Eq. 16, one gets

$$ \hbox{R} = 3.2 $$

Using Eqs. 10, 21–25, and solving numerically, one gets

• Ratio of percent load carried by casing/steining $\left({{\hbox{Q}_{\rm st} }/{\hbox{Q}_{\rm ult}}} \right)=93.47\%;$

• Normalized settlement of composite foundation $\left( {{\hbox{w}_{\rm st} }/{\hbox{d}_0 }} \right)= 0.0385;$

• Normalized settlement of soil below granular core $\left( {{\hbox{w}_{\rm sL} }/{\hbox{d}_0 }} \right)= 0.0328.$