## Abstract

Design charts for the rate of settlement ofembankments on soft foundations aredeveloped on the basis of 2D consolidation during the construction and post-construction periods. The embankment loading is assumed toincrease linearly with time during construction and to remain constant thereafter.

The clay deposit is considered to consist of ahomogeneous layer of finite thickness and infinite lateral extent having cross-anisotropic permeability.

The hydraulic boundary conditions of permeable top and impermeable bottom are used for the most common case (a drainage blanket at the base of the embankment and a clay deposit with impermeable bottom) of 2D consolidation analysis.

The charts presented take into account the relative layer thickness of the clay layer, embankment geometry, cross-anisotropic permeability and construction period.

The paper reveals that the post-construction settlement becomes insignificant for relatively thick deposit with permeability in the horizontal direction much higher than that in the vertical direction.

The problem of 2D consolidation of a homogenous cross-anisotropic clay layer underneath instantaneously applied uniform strip loading has received attention by Davis & Poulos (1972). Similarly, Dunn & Razouki (1974) treated the problem of Terzaghi–Rendulic consolidation beneath instantaneously applied embankment loading. The problem of time-dependent loading has received attention by Terzaghi & Fröhlich (1936) and Olson (1977) but in connection with 1D consolidation only. However, the construction period for high embankments is relatively long so that to predict therate of settlement of high embankments on softfoundations, a 2D consolidation analysis under time-dependent loading is required.

Experience has shown that the dissipation of excess pore water pressure frequently proceeds at a greaterrate than is predicted from the 1D theory. In connection with embankments on soft foundation, this fact may render the need for expedients such as vertical sand drains, removal of compressible subsoil or a structural solution unnecessary.

## 2D Consolidation

It is well known that there are two different theories of consolidation namely the Terzaghi–Rendulic theory or the diffusion theory (Davis & Poulos 1972) and the Biot (1941*a*,* b*) theory.

Biot's theory of consolidation can be regarded as a true theory of consolidation while the Terzaghi–Rendulic theory is a pseudo consolidation theory as it uncouples the equilibrium of total stresses and the continuity of the soil mass.

Sills (1975) has shown that if the sum of normal total stresses is constant in time, Biot's equation can be reduced to Terzaghi–Rendulic equation.

However, Davis & Poulos (1972) have shown that the Biot theory can predict the Mandel–Cryer effect at the early stages of consolidation but this phenomenon is of little significance at the later stages of consolidation. They added that for a strip loading, the diffusion theory gives results in close agreement with those by the Biot theory for all values of Poisson's ratio of soil skeleton, so that the use of Terzaghi–Rendulic theory is quite justified for all practical problems (Murray 1978).

The 2D Terzaghi–Rendulic equation is given by (Murray 1978; Schiffman 1958):

$$mathtex$$\[\frac{{\partial}u}{{\partial}t}{\,}={\,}C_{X}\frac{{\partial}^{2}u}{{\partial}x^{2}}{\,}+{\,}C_{Z}\frac{{\partial}^{2}u}{{\partial}z^{2}}{\,}+{\,}\frac{{\partial}u_{e}}{{\partial}t}\]$$mathtex$$(1)

where

*u*(*x*,*z*,*t*)=excess pore water pressure at position (x,z) and time (t).*C*_{x}=coefficient of consolidation for horizontaldrainage.*C*_{z}=coefficient of consolidation for vertical drainage.$$mathtex$$\(\frac{{\partial}\mathit{u}_{\mathit{e}}}{{\partial}\mathit{t}}=\)$$mathtex$$rate of imposition of excess pore water pressure.

## Embankment loading

The embankment is assumed to be completely flexible and symmetrical so that the embankment loading is represented by a symmetrical trapezoidal loading as shown in Figure 1.

The load is assumed to increase linearly with time during the construction period and to remain constant thereafter (single ramp loading) as shown in Figure 1.

Accordingly,

$$mathtex$$\[p{\ast}(t){\,}={\,}\left\{\begin{array}{l}p\frac{t}{t_{c}}{\,}={\,}p\frac{Tv}{Tc}{\,}for{\,}0{\leq}t{\leq}t_{c}{\,}or{\,}{\leq}Tv{\leq}Tc\\p{\,}for{\,}t{\geq}t_{c}{\,}or{\,}Tv{\geq}Tc\end{array}\right\}\]$$mathtex$$(2)

where

*p**(*t*)=applied load at the centerline of the embankment at time t.*p*=applied load at the centerline of the embankment loading at the end of construction period.*T*_{v}=dimensionless time factor for vertical drainage (or in vertical direction)

$$mathtex$$\[T_{v}{\,}={\,}C_{z}\frac{t}{H^{2}}\]$$mathtex$$(3)

*T*_{c}=dimensionless time factor in vertical direction at the end of construction period

$$mathtex$$\[T_{c}{\,}={\,}C_{z}\frac{t_{c}}{H^{2}}\]$$mathtex$$(4)

where

*H*=total thickness of clay layer.*C*_{z}=coefficient of consolidation in vertical direction.t=actual time.

tc=construction period.

## Excess pore water pressure

The clay deposit under the embankment is assumed to be a finite clay layer of infinite lateral extent having cross-anisotropic permeability.

The excess pore water pressure in the clay layer due to the embankment loading was determined after Henkel (1960) and Scott (1965) for plane strain conditions as follows:

$$mathtex$$\[u_{e}{\,}={\,}[{\sigma}_{3}{\,}+{\,}N({\sigma}_{1}{\,}{-}{\,}{\sigma}_{3})]\]$$mathtex$$(5)

where

$$mathtex$$\[N{\,}={\,}\frac{\sqrt{3}}{2}A{\,}{-}{\,}\frac{1}{3}{\,}+{\,}\frac{1}{2}{\,}={\,}0.866A{\,}+{\,}0.211\]$$mathtex$$(6)

A=Skempton (1954) pore pressure parameter.

*σ*_{1},*σ*_{3}=major and minor principal total stressesrespectively.

It is quite obvious from Equation (5) that to arrive at the imposed excess pore water pressure due to embankment loading, the stresses in the soil mass should be determined first. For this purpose, use was made of the superposition principle together with the Boussinesq solution for triangular loading presented by Poulos & Davis (1974). The stresses due to embankment loading during construction can be written as follows:

$$mathtex$$\[\begin{array}{l}{\sigma}_{z}(t){\,}={\,}\frac{p}{(1{\,}{-}{\,}{\delta}^{}){\pi}}{\,}\frac{x}{b_{1}}({\alpha}_{1}{\,}{-}{\,}{\alpha_{2}^{{\ast}}}(t){\,}+{\,}{\beta_{2}^{{\ast}}}(t){\,}{-}{\,}{\beta}_{1}){\,}+{\,}2{\beta}_{1}{\,}{-}{\,}2{\beta_{2}^{{\ast}}}(t){\,}(1{\,}{-}{\,}\frac{t}{t_{c}}(1{\,}{-}{\,}{\delta}){)}{\,}+{\,}\frac{t}{t_{c}}(1{\,}{-}{\,}{\delta}){\,}({\alpha_{2}^{{\ast}}}(t){\,}{-}{\,}{\beta_{2}^{{\ast}}}(t))\end{array}\]$$mathtex$$(7a)

$$mathtex$$\[\begin{array}{l}{\sigma}_{x}(t){\,}={\,}\frac{p}{(1{\,}{-}{\,}{\delta}){\pi}}{\,}\frac{x}{b_{1}}({\alpha}_{1}{\,}{-}{\,}{\alpha_{2}^{{\ast}}}(t){\,}+{\,}{\beta_{2}^{{\ast}}}(t){\,}{-}{\,}{\beta}_{1}){\,}+{\,}2{\beta}_{1}{\,}{-}{\,}2{\beta_{2}^{{\ast}}}(t){\,}1{\,}{-}{\,}\frac{t}{t_{c}}(1{\,}{-}{\,}{\delta}){\,}+{\,}\frac{t}{t_{c}}(1{\,}{-}{\,}{\delta}){\,}({\alpha_{2}^{{\ast}}}(t){\,}{-}{\,}{\beta_{2}^{{\ast}}}(t)){\,}+{\,}\frac{2z}{b_{1}}ln\frac{{R_{3}^{{\ast}}}(t){R_{4}^{{\ast}}}(t)}{R_{o}R_{2}}\end{array}\]$$mathtex$$(7b)

$$mathtex$$\[{\tau}_{xz}(t){\,}={\,}\frac{pz}{(1{\,}{-}{\,}{\delta}){\pi}b_{1}}{\,}[{\alpha}_{1}{\,}{-}{\,}{\alpha_{2}^{{\ast}}}(t){\,}+{\,}{\beta_{2}^{{\ast}}}(t){\,}{-}{\,}{\beta}_{1}]\]$$mathtex$$(7c)

where

*σ*_{z}(*t*)=vertical total stress due to embankment loading during construction.*σ*_{x}(*t*)=horizontal total stress due to embankment loading during construction.*τ*_{xz}(*t*)=shear stress due to embankment loading during construction.

$$mathtex$$\[{\alpha}_{1}{\,}={\,}tan^{{\,}{-}{\,}1}{\,}\frac{x}{z}{\,}+{\,}tan^{{\,}{-}{\,}1}{\,}\frac{b_{1}{\,}{-}{\,}x}{z}\]$$mathtex$$(8a)

$$mathtex$$\[{\beta}_{1}{\,}={\,}tan^{{\,}{-}{\,}1}{\,}\frac{2b_{1}{\,}{-}{\,}x}{z}{\,}{-}{\,}tan^{{\,}{-}{\,}1}{\,}\frac{b_{1}{\,}{-}{\,}x}{z}\]$$mathtex$$(8b)

$$mathtex$$\[R_{o}{\,}={\,}\sqrt{x^{2}{\,}+{\,}z^{2}}\]$$mathtex$$(8c)

$$mathtex$$\[R_{2}{\,}={\,}\sqrt{(2b_{1}{\,}{-}{\,}x)^{2}{\,}+{\,}z^{2}}\]$$mathtex$$(8d)

$$mathtex$$\[{\alpha_{2}^{{\ast}}}(t){\,}={\,}tan^{{\,}{-}{\,}1}{\,}\frac{x{\,}{-}{\,}(b_{1}{\,}{-}{\,}{b_{2}^{{\ast}}}(t))}{z}{\,}+{\,}tan^{{\,}{-}{\,}1}{\,}\frac{b_{1}{\,}{-}{\,}x}{z}\]$$mathtex$$(8e)

$$mathtex$$\[{\beta_{2}^{{\ast}}}(t){\,}={\,}tan^{{\,}{-}{\,}1}{\,}\frac{b_{1}{\,}+{\,}{b_{2}^{{\ast}}}(t){\,}{-}{\,}x}{z}{\,}{-}{\,}tan^{{\,}{-}{\,}1}{\,}\frac{b_{1}{\,}{-}{\,}x}{z}\]$$mathtex$$(8f)

$$mathtex$$\[{R_{3}^{{\ast}}}(t){\,}={\,}\sqrt{(x{\,}{-}{\,}(b{\,}{-}{\,}{b_{2}^{{\ast}}}(t)))^{2}{\,}+{\,}z^{2}}\]$$mathtex$$(8g)

$$mathtex$$\[{R_{4}^{{\ast}}}(t){\,}={\,}\sqrt{(b_{1}{\,}+{\,}{b_{2}^{{\ast}}}(t){\,}{-}{\,}x)^{2}{\,}+{\,}z^{2}}\]$$mathtex$$(8h)

$$mathtex$$\[{\delta}{\,}={\,}\frac{b_{2}}{b_{1}}\]$$mathtex$$(8i)

*b*_{2}=half width of the upper base of the embankment loading.$$mathtex$$\({\mathit{b_{2}^{{\ast}}}(\mathit{t})=\)$$mathtex$$half width of the crest of the embankment loading at the time t during construction.

*b*_{1}=half width of the lower base of the embankment loading.*δ*=ratio of the crest width to base width.*p*=magnitude of embankment pressure at centerline at the end of construction.

For the post-construction period the upper crest width is constant (i.e. $$mathtex$$\({\mathit{b_{2}^{{\ast}}}(\mathit{t})\)$$mathtex$$=*b*_{2}), and the above equations can be used to evaluate the total principal stresses.

In order to evaluate the excess pore water pressure by making use of Equation (5), the principal stresses can be evaluated from the following equation.

$$mathtex$$\[{\sigma}_{1},{\sigma}_{3}{\,}={\,}\frac{{\sigma}_{z}{\,}+{\,}{\sigma}_{x}}{2}{\pm}\sqrt{\left(\frac{{\sigma}_{z}{\,}{-}{\,}{\sigma}_{x}}{2}\right)^{2}{\,}+{\,}{\tau_{xz}^{2}}}\]$$mathtex$$(9)

The rate of imposition of excess pore water pressure due to the time-dependent embankment loading was evaluated analytically and it can be shown easily that

$$mathtex$$\[\frac{{\partial}u_{e}}{{\partial}t}{\,}={\,}\frac{{\partial}}{{\partial}t}{\,}[{\sigma}_{3}{\,}+{\,}N({\sigma}_{1}{\,}{-}{\,}{\sigma}_{3})]{\,}={\,}\frac{p}{(1{\,}{-}{\,}{\delta}){\pi}}F(t)\frac{d{b_{2}^{{\ast}}}}{dt}\]$$mathtex$$(10)

where

$$mathtex$$\[\begin{array}{l}F(t){\,}={\,}A_{1}{\,}{-}{\,}\frac{2{\beta_{2}^{{\ast}}}(t)}{b_{1}}{\,}{-}{\,}2{\delta}^{{\ast}}(t)A_{12}{\,}+{\,}A_{2}{\,}+{\,}A_{3}{\,}+{\,}A_{4}{\,}+{\,}A_{5}{\,}+{\,}A_{43}{\,}+{\,}NA_{6}\\A_{1}{\,}={\,}\frac{x}{b_{1}}A_{11}{\,}{-}{\,}\frac{2{\beta_{2}^{{\ast}}}(t)}{b_{1}}{\,}{-}{\,}2A_{12}{\delta}^{{\ast}}(t)\\A_{11}{\,}={\,}\frac{{\,}{-}{\,}z}{z^{2}{\,}+{\,}d^{2}}{\,}+{\,}\frac{z}{z^{2}{\,}+{\,}f^{2}}\\A_{12}{\,}={\,}\frac{z}{z^{2}{\,}+{\,}f^{2}}\\A_{2}{\,}={\,}\frac{z}{b_{1}}A_{21}{\,}{-}{\,}\{1{\,}{-}{\,}{\delta}^{{\ast}}(t)\}A_{11}\\A_{21}{\,}={\,}\frac{d}{z^{2}{\,}+{\,}d^{2}}{\,}+{\,}\frac{f}{z^{2}{\,}+{\,}f^{2}}\\A_{3}{\,}={\,}\frac{{\beta_{2}^{{\ast}}}(t){\,}{-}{\,}{\alpha_{2}^{{\ast}}}(t)}{b_{1}}\\A_{4}{\,}={\,}{\,}{-}{\,}\frac{z}{b_{1}}[(A_{41})^{2}{\,}+{\,}(A_{42})^{2}]^{{\,}{-}{\,}1/2}{\,}={\,}{\,}{-}{\,}\frac{z}{b_{1}}A_{43}\\A_{41}{\,}={\,}ln\frac{{R_{3}^{{\ast}}}(t){R_{4}^{{\ast}}}(t)}{R_{1}R_{2}}\\A_{42}{\,}={\,}{\alpha}_{1}{\,}{-}{\,}{\alpha_{2}^{{\ast}}}(t){\,}+{\,}{\beta_{2}^{{\ast}}}(t){\,}{-}{\,}{\beta_{1}^{2}}\\A_{43}{\,}={\,}[(A_{41})^{2}{\,}+{\,}(A_{42})^{2}]^{{\,}{-}{\,}1/2}\\A_{5}{\,}={\,}A_{41}{\ast}A_{42}{\,}{-}{\,}A_{11}{\ast}A_{42}\\A_{6}{\,}={\,}\frac{z}{b_{1}}[(A_{41})^{2}{\,}+{\,}(A_{42})^{2}]^{{\,}{-}{\,}1/2}A_{7}\\A_{7}{\,}={\,}2A_{41}{\ast}A_{21}{\,}+{\,}2{\ast}A_{42}{\ast}A_{11}\end{array}\right\}\]$$mathtex$$(11)

where

$$mathtex$$\[{\delta}^{{\ast}}(t){\,}={\,}\frac{{b_{2}^{{\ast}}}(t)}{b_{1}}\]$$mathtex$$(11)

$$mathtex$$\[d{\,}={\,}x{\,}{-}{\,}b_{1}{\,}+{\,}{b_{2}^{{\ast}}}(t)\]$$mathtex$$(12)

$$mathtex$$\[f{\,}={\,}b_{1}{\,}+{\,}{b_{2}^{{\ast}}}(t){\,}{-}{\,}x\]$$mathtex$$(13a)

## Hydraulic boundary conditions

The consolidation problem is a boundary value problem (Wylie & Barrett 1988). The solution of the corresponding differential equation requires the knowledge of boundary conditions. In practice, it is common to usea drainage blanket underneath the embankment so that the surface of the clay layer can be considered tobe permeable. However, the most important case in practice requiring a 2D consolidation analysis is thatof a clay deposit with impermeable bottom. This is especially true for a relatively thick deposit. For this reason, this paper is devoted to this important case of hydraulic boundary condition (PTIB = permeable top and impermeable bottom), as shown in Figure 2.

Due to the symmetry of the problem, the vertical axis of symmetry can be considered as an impermeable boundary and only the right hand half of the problem needs to be considered.

The lateral boundary was taken from the centerline of the embankment at a distance 25 times the half-base width of the embankment. Just to the right of the assumed lateral boundary, a negative exponentialexcess pore water pressure distribution was adopted as recommended by Dunn & Razouki (1974).

## Consolidation analysis

The 2D consolidation analysis was carried out numerically using the ADI technique (Alternating Direction Implicit technique) together with the graded mesh as shown in Figures 2 & 3. Logarithmic time increment with ten cycles per decade was used and the systemof simultaneous algebraic equations during each ofodd and even time steps of a cycle was solved usingthe Gaussian elimination process (Wylie & Barrett 1988).

## Average degree of consolidation

During the construction period, the average degree of consolidation at the centerline of the embankment was calculated at the end of each cycle as follows:

$$mathtex$$\[U_{av}{\,}={\,}\frac{t}{t_{c}}{\,}{-}{\,}\frac{\begin{array}{l}H\\{\int}\\0\end{array}u{\,}dz}{\begin{array}{l}H\\{\int}\\0\end{array}u_{e_{tc}}dz}{\,}={\,}\frac{T_{v}}{T_{c}}{\,}{-}{\,}\frac{\begin{array}{l}H\\{\int}\\0\end{array}u{\,}dz}{\begin{array}{l}H\\{\int}\\0\end{array}u_{e_{tc}}dz}\]$$mathtex$$(13b)

For the post-construction period the average degree of consolidation at the centerline of the embankment can be written as follows:

$$mathtex$$\[U_{av}{\,}={\,}1{\,}{-}{\,}\frac{\begin{array}{l}H\\{\int}\\0\end{array}u{\,}dz}{\begin{array}{l}H\\{\int}\\0\end{array}u_{e_{tc}}dz}\]$$mathtex$$(14)

where

U

_{av}=Average degree of consolidation settlement at centerline.*ρ*t=Primary consolidation settlement at time (t).*ρ*∞=Total primary consolidation settlement (at t = ∞).U=Excess pore water pressure.

*u*_{etc}=Total imposed excess pore water pressure during the whole construction period which is equal to the instantaneous excess pore water pressure that might have been generated throughout the clay layer had the stress p been applied instantaneously.*u*_{et}=Total imposed excess pore water pressure from the start of construction till time t.

## Computer program

A computer program was written in Fortran Power Station Version (4) for the analysis of the problem.

Before using this program it was necessary to check its validity by comparing its results with the available solutions. The first available solution is that of 1D consolidation under instantaneous loading with uniform initial excess pore water pressure distribution (Das 1985). For this purpose the computer program was run for a construction period of *t*_{c}=0 and a relatively thin clay layer with *H*/*b*_{1}=0.001 (very close to zero) and *C*_{x}/*C*_{z}=0. The embankment was chosen such that *δ*=0.99 to be close to a uniform load.

As shown in Figure 4 excellent agreement wasobtained between the computer program solution and the exact one.

Another comparison was made with Davis & Poulos (1972) for the 2D consolidation under instantaneous uniform strip loading for the case of *H*/*b*_{1}=10 and *C*_{x}/*C*_{z}=1, while *δ*=0.99 was chosen for the computer solution to be close to uniform strip loading.

Figure 4 shows the good agreement between the two solutions. The slight difference between the two solutions can be attributed to the method of calculating the initial excess pore water pressure distribution.

The initial excess pore pressure distribution by Davis & Poulos (1972) was obtained for an elastic layer underlain by a rough rigid base (Poulos 1967) while the present work was based on Boussinesq solution.

Another comparison of the computer solution was made with Olson (1977) for 1D consolidation under time-dependent loading and excellent agreement was obtained, as shown in Figure 5.

## Design charts

To assist the geotechnical engineer in the design of embankments, design charts for the average degree of consolidation under time-dependent loading should be developed. As the 2D effect becomes more pronounced as the relative layer thickness *H*/*b*_{1} increases, the charts are developed for four values of *H*/*b*_{1} namely 1, 2, 5 and 10. Similarly the 2D effect becomes more pronounced as *C*_{x}/*C*_{z} increases. For this reason four values of *C*_{x}/*C*_{z}of 1, 5, 10 and 25 were adopted. To take into account the important effect of construction period, eight different values for the dimensionless time factor T_{c} = 0, 0.1, 0.2, 0.5, 1, 2, 5, and 10 were adopted. However, as the effect of the embankment geometry is of little significance (Dunn & Razouki 1974), the charts were developed for *δ*=0.25 and *δ*=0.75 only.

Figures 6 to 9 present the design charts for the caseof *δ*=0.25 and *H*/*b*_{1}=1, 2, 5 and 10 respectively, while Figures 10 to 13 present the design charts for the case of *δ*=0.75 and *H*/*b*_{1}=1, 2, 5, 10 respectively.

It is quite obvious from the design charts that the consolidation process becomes faster as each of *C*_{x}/*C*_{z} and *H*/*b*_{1} increases. However, as the constructionperiod increases, the process of consolidation during construction becomes slower.

## Conclusion

The main conclusions of this work can be summarized as follows:

The design charts developed allow the rapid determination of the average degree of consolidation at centerline of the embankment for any construction period, relative layer thickness, cross-anisotropic permeability and embankment geometry.

Excellent agreement was obtained between the solution of this work and the available exact solution of 1D consolidation under each of instantaneous and time-dependent loading.

For the case of 2D consolidation of isotropic layer subjected to instantaneous uniform strip loading, the solution of this work was in good agreement with that of Davis & Poulos (1972).

The average degree of consolidation is significantly affected by the construction period.

An increase in

*H*/*b*_{1}makes the consolidation process (at centerline of the embankment) faster.The rate of settlement becomes faster with the increase of

*C*_{x}/*C*_{z}(i.e. with the increase of anisotropic permeability).An increase in the construction period reduces to a great extent the post construction settlement. For a dimensionless time factor at the end of construction of T

_{c}= 10, a great part of the consolidation process occurs during construction.

- © 2003 The Geological Society of London