Consolidation by vertical drains under time‐dependent loading

A solution for the consolidation by vertical drains under time‐dependent loading is presented in this paper. Considering the well resistance and the smear action, the simultaneous basic partial differential equations of the consolidation by vertical drains are obtained for the arbitrary loading method. However, the impulse function method cannot be directly applied to obtain the solution. The partial differential equations and the solution conditions that satisfy the impulse function method are obtained after some mathematical processing. The solution for the consolidation by vertical drains under time‐dependent loading is obtained by virtue of the impulse function method and the solution under instantaneous loading. The solutions under single ramp loading and multi‐ramp loading are obtained and the feasibility of Carrillo's method under time‐dependent loading is discussed. Further, the characteristics of the consolidation by vertical drains under instantaneous loading and time‐dependent loading are discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

Consolidation of viscoelastic soil by vertical drains incorporating fractional-derivative model and time-dependent loading

This paper presents a general solution to the consolidation system of viscoelastic soil by vertical drains incorporating a fractional-derivative model and arbitrary time-dependent loading. The fractional-derivative Merchant model is introduced to describe the viscoelastic behavior of saturated soil around the vertical drains. Based on this model, the governing partial differential equation of a consolidation system incorporating vertical and horizontal drainage is obtained for the equal strain condition. Then, a general solution to the consolidation system involving arbitrary time-dependent loading is derived using the Laplace transform technique and eigenfunction expansion method. Further, two comparisons are presented to verify the exactness of the proposed solution, and the consolidation behavior involving four time-dependent loadings is illustrated and discussed.     

竖向排水井地基工后沉降预测

预测工后沉降是排水固结预压加固软土地基的关键问题之一。根据等应变竖向排水井地基固结理论,提出了沉降曲线的近似计算方法。通过实测沉降资料与近似理论曲线的拟合,获得竖向排水井地基的最终沉降量和平均固结度,从而预估其工后沉降量。将该方法分别应用于真空预压和堆载预压处理的竖井地基,进行工后沉降预测,结果表明该方法是可行的     

成层竖向排水井地基固结分析

实际工程中竖井地基具有成层性,有时竖井也并未完全打穿软土层。在竖井打设区满足竖井等应变固结理论、下卧层满足一维太沙基固结理论假设的基础上,将现有竖井地基固结理论推广到成层未打穿竖井地基情况。利用边界条件和竖直向连续条件,确定该系统的正交关系,并给出了其固结解答,该解具有广泛的适用性。通过对竖井打设区和下卧层层数的变化,即可获得现有关于简单未打穿竖井地基的固结解答。将该解答编制成应用程序,对一算例进行了分析。结果表明,平均固结度按孔压定义和按变形定义是不相同的,硬表层的存在会加快其下土层的固结。     

Consolidation of sensitive clay with vertical drain

The coefficient of consolidation is one of the most important parameters that control the rate of consolidation. Conventional consolidation theories assume that the coefficient of consolidation is constant during the whole consolidation process. In the case of sensitive clay, the coefficient of consolidation is strongly dependent on the level of effective stress of clay. With the dissipation of pore water pressure and the increase of effective stress, the soil structure of the upper subsoil is gradually destroyed downwards and its coefficient of consolidation becomes smaller. At the same time, the coefficient of permeability of the vertical drains drops down because of the kinking or bending effect. The destructured upper subsoil and the kinking of the vertical drain hinder the dissipation of the pore pressure in the lower subsoil. This paper presents a model to describe the above important phenomena during the consolidation of sensitive clay with vertical drain. The solution for proposed model can be obtained by extending the closed‐form solution of the consolidation of double‐layered ground with vertical drain by the interactive method introducing the concept of the moving boundary and the reduction of discharge capacity of vertical drain. The numerical results obtained from the finite element method package PLAXIS (Ver. 7.2) are adopted to compare those obtained from the present algorithm. Moreover, the rationality of the moving boundary is explained by the distributions of the excess pore water pressure in natural soil zone along the radial direction. Wenzhou airport project is taken as a case study in this paper. The results for the sensitive soil with decaying sand drain agree very well with the in situ measured data. The calculated results can properly explain two general phenomena observed in the consolidation of soft sensitive soil ground with vertical drains: one is that the time to achieve the same consolidation degree is much longer under heavy loading than that under light loading; the other is that the consolidation speed is much slower in the lower subsoil than in the upper subsoil. Finally, it is indicated that the vertical drains can decrease the hindrance effect of the destructured subsoil. Copyright © 2007 John Wiley & Sons, Ltd.     

双层竖井地基半透水边界固结分析

在现有双层竖井地基固结理论基础上,对双面半透水边界下的固结问题进行了研究,给出了相应的固结解答。将该解答编制成应用程序,对一算例进行了参数分析,以了解双层竖井地基半透水边界的固结特性。算例分析表明,半透水边界对竖井地基的固结影响较大：半透水参数(Rt,Rb)越大,则整个系统的孔压消散越快;半透水参数对与其相邻的土层的固结影响要比远离其边界的土层的固结影响大得多;按变形定义和按孔压定义的平均固结度两者的差别受半透水边界参数的影响。     

一种竖井地基竖墙化等效计算方法

考虑井阻作用,推导了等应变条件下竖墙地基水平向固结解析解,并与等应变条件下的竖井地基径向固结解析解进行比较,得到竖墙地基平面应变问题和竖井地基轴对称问题之间的等效方法。该方法转换公式简单,且能保证两种情形下同一深度处平均孔压在任一时刻相等,确保了转换为平面应变有限元分析的准确性。     

考虑幂函数渗透压缩变化的双层砂井地基固结性状

采用Barron轴对称固结及大变形固结问题的某些简化与假定,推导建立了砂井地基大变形固结控制方程,利用建立的双层砂井地基大变形固结方程及编制的计算程序,通过引入软土渗透系数、有效应力与孔隙比之间的幂函数关系k =ced与e=a( )b,对瞬时加载下双层砂井地基固结性状进行算例计算。结果表明：（1）双层软土幂函数渗透关系及压缩关系中诸参数对双层砂井地基固结性状有重要影响：随着两层软土幂函数渗透关系中参数c1、c2的增加（渗透性增加）、或幂函数压缩关系中参数a1、a2的增加,各土层水平径向与竖向孔隙比减小更快,沉降发展速率与超静孔压消散速率也相应增加,且沉降发展速率快于孔压消散速率。（2）两层土在分界面处的孔隙比及平均超静孔压均出现明显的突变,将沿深度分布曲线分成形状不同的两段,表现出不同的固结性状。     

Analysis of soil consolidation by vertical drains with double porosity model

The soil around a drain well is traditionally divided into smeared zone and undisturbed zone with constant hydraulic conductivity. In reality, hydraulic conductivity of the soil changes continuously and it may not be always appropriate to approximate its distribution with two zones. In this study, the horizontal hydraulic conductivity of the soil is described by an arbitrary function of radial distance. The horizontal flow under equal strain condition is analysed for a soil–drain system with a circular or regular polygonal boundary. It is found that the horizontal flow can be generally characterized with a linear equation in which the flow rate of water through soil–drain interface is proportional to the difference between the average excess pore pressure in the soil and the excess pore pressure in the drain well. The water exchange between the drain and the soil is analogous to that between fractures and matrix in a double porosity system, a popular conceptual model of fracture rocks. On the basis of this characterization, a simplified approach to analyse soil–drain systems is developed with one‐dimensional double porosity model (DPM). Analytical solutions for both fully and partially penetrating drains are derived. The solution for partially penetrating drains is compared with both numerical and approximate analytical results in literature. Copyright © 2004 John Wiley & Sons, Ltd.     

连续渗透边界下非饱和土竖井地基固结解析解EI北大核心CSCD

基于非饱和土轴对称固结理论和等应变假设,引入连续渗透边界条件,采用边界条件齐次化、本征函数法,推导得到瞬时均布荷载下非饱和土竖井地基三维固结解析解。通过与双面完全渗透边界条件下已有解析解进行对比,验证了所得解析解的正确性。对所得解进行算例分析发现:通过设置合理上、下界面参数,所得解可用于模拟实际上、下边界透水透气任意分布的情况,弥补了目前无法考虑上、下边界渗透性介于渗透与不渗透之间或不对称分布的不足;在井径比及井深适当的前提下,当径、竖向渗透系数比大于2时,竖向渗流对于超孔隙压力消散的影响较小;当考虑竖向渗流时,上述影响随着上、下边界渗透性能的增强而愈加明显。     

Probabilistic analyses of soil consolidation by prefabricated vertical drains for single‐drain and multi‐drain systems

Natural soils are one of the most inherently variables in the ground. Although the significance of inherent soil variability in relation to reliable predictions of consolidation rates of soil deposits has long been realized, there have been few studies that addressed the issue of soil variability for the problem of ground improvement by prefabricated vertical drains. Despite showing valuable insights into the impact of soil spatial variability on soil consolidation by prefabricated vertical drains, available stochastic works on this subject are based on a single‐drain (or unit cell) analyses. However, how the idealized unit cell solution can be a supplement to the complex multi‐drain systems for spatially variable soils has never been addressed in the literature. In this study, a rigorous stochastic finite elements modeling approach that allows the true nature of soil spatial variability to be considered in a reliable and quantifiable manner, both for the single‐drain and multi‐drain systems, is presented. The feasibility of performing an analysis based on the unit cell concept as compared with the multi‐drain analysis is assessed in a probabilistic context. It is shown that with proper input statistics representative of a particular domain of interest, both the single‐drain and multi‐drain analyses yield almost identical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Analytical models for consolidation of combined composite ground improved by impervious columns and vertical drains

In recent years, a new technique of ground improvement, which involves the combined use of impervious column and vertical drains, has been proposed and utilized in many field projects to accelerate consolidation and increase bearing capacity of soft soil ground. To cover the possible distribution patterns of impervious columns and vertical drains, 2 analytical models, including Model A with outward flow and Model B with inward flow within the soils, are proposed to predict the consolidation of combined composite ground by considering the following factors: (1) disturbance effects of both impervious columns and vertical drains, (2) the well resistance of vertical drains, and (3) time‐variant loadings. The average degrees of consolidation predicted by the proposed analytical models are compared with several existing solutions and then against the measured data in the literature. The consolidation behavior of a combined composite ground is investigated by the proposed analytical solutions. The results show that the combined use of impervious columns and vertical drains can remarkably accelerate the consolidation rate of soft soils compared with the single use of either of them. The average degrees of consolidation predicted by both analytical models agree well with the measured data. Compared with Model B, Model A usually predicts a faster consolidation rate because of a shorter drainage path. Many factors can influence consolidation behavior of combined composite ground, such as loading scheme, distribution patterns and the disturbance effects of impervious columns and vertical drains, and compression modulus ratio of impervious column to soil.     

A new simplified Hypothesis B method for calculating the consolidation settlement of ground improved by vertical drains

Vertical drains are widely used in soft ground improvements to accelerate the consolidation process. This paper develops a new simplified Hypothesis B method for calculating the consolidation settlement of a soil layer improved by vertical drains under the instant and ramp loadings. As a comparison, the traditional Hypothesis A method is also used to calculate the settlement. Then, a fully coupled finite element consolidation analysis is utilized to examine and verify this simplified method and Hypothesis A method. For the instant loading, Carrillo‐Barron method and Zhu‐Yin method are used to obtain the average degree of consolidation for vertical drain system. Typical parameters, such as over‐consolidation ratio (OCR), smear zone, and space ratio of vertical drains, are considered. It is found that the calculation results from the new simplified method in this study agree well with finite element simulations, and relative errors are in the range of 0.1% to 12.3%. Comparatively, there are obvious differences between the calculated results from Hypothesis A method and finite element results. Carrillo‐Olson method and Zhu‐Yin method are utilized to obtain the average degree of consolidation for the vertical drain system to consider the ramp loading. Equivalent time is determined from half of the construction period to calculate the creep compression under the ramp loading. The accuracy of this simplified Hypothesis B method using both Carrillo‐Olson method and Zhu‐Yin method is acceptable with the relative errors less than 9.4%.     

Nonlinear radial consolidation of vertical drains under a general time‐variable loading

By incorporating the nonlinear variation of a soil's compressibility and permeability during the process of consolidation, an analytical solution for the radial consolidation of vertical drains has been developed for a general time‐variable loading. The general solution was verified for the cases of instantaneous loading and ramp loading. Detailed solutions were further derived for two special loading schemes: multistage loading and preloading–unloading–reloading. The nonlinear consolidation behavior of a vertical drain subjected to these two types of loading schemes was then investigated by a parametric study. The results show that the loading rate, the ratio of the compressibility index to the permeability index (C_c/C_k), and the initial stress state have a significant influence on the consolidation rate. A smaller value of C_c/C_k, a larger initial stress, or a fast loading rate always leads to a rapid consolidation rate. During the unloading period, a negative excess pore water pressure may occur, and a slower unloading rate may reduce this negative value. Copyright © 2014 John Wiley & Sons, Ltd.     

任意附加应力下竖井径竖向耦合固结等应变解

竖向排水体包括砂井和塑料排水板已广泛应用于加速软黏土地基的固结。不同形式的轴对称条件下含竖向排水体均质各向同性地基的固结解析解答也相继被提出,而各种解答均假定地基中附加应力在竖向排水体的深度范围内是均匀分布的。显然,当竖向排水体深度与地面堆载区宽度的比值相对较高,或竖向排水体的影响区域靠近地面堆载区的边缘时,这个假定并不适用。为此,在现有径竖向耦合固结等应变解的基础上,提出任意分布的地基附加应力条件下的解答,并分析附加应力分布形式和线性加载速率对地基固结的影响。结果表明,均匀分布的附加应力条件下的解答低估了地基的固结速率,而瞬时加载条件下的解答则高估了地基的固结速率。     

变荷载下考虑井阻随时空变化的竖井地基固结解

竖井排水固结法中井阻随时空演变（即由淤堵和弯折所引起的竖井排水能力下降）的现象已引起广泛关注,且变井阻对竖井地基固结速率的影响不容忽略。但目前能同时考虑变荷载及井阻随时间和空间变化的固结解析解还鲜有报道。考虑井阻随时空演变过程,引入实际中广泛采用的单级或多级加载模式,建立了竖井地基固结模型,并应用分离变量法获得固结模型的解析解答。通过与已有的解析解、有限差分解及工程实测值进行对比分析,充分验证了该模型的正确性。通过大量的计算,分析变井阻参数对竖井地基固结性状的影响。结果表明：竖井地基固结速率随竖井最终排水能力的增大而加快,随深度井阻参数及时间井阻参数的增大而减缓,且时间井阻参数的影响更为显著。     

Nonlinear consolidation of vertical drains with coupled radial-vertical flow considering time and depth dependent vacuum pressure

The system of vacuum pressure combined with vertical drains to accelerate soil consolidation is one of the most effective ground improvement methods. The consolidation theories of soft soil improved by vertical drains including void ratio–dependent compressibility and permeability have been widely applied in practice to predict the consolidation behavior. In this paper, analytical solutions of the consolidation of vertical drains are derived incorporating the loss and propagating stage of vacuum pressure. In addition, special solutions are obtained for the cases of instantaneous surcharge loading and staged surcharge loading, based on the general solution. The solution is verified by ignoring the propagating stage of vacuum pressure formation and comparing it with an existing solution. The effects of vacuum pressure loss and propagating stage combined with other parameters are investigated through the ratio between excess pore water pressure and surcharge loading.     

塑料排水板处理地基非线性固结计算研究

针对目前国内外竖井地基非线性固结计算大多均采用近似等效法求解的不足,根据软土固结理论及数学理论推导,基于塑料排水板椭圆柱假定,推导了考虑地基土水平渗透系数、压缩模量随固结过程非线性变化影响下的排水板处理地基非线性固结控制方程,并利用数学中非线性微分方程理论,给出了非线性固结解析解。通过与Indraratna等非线性固结近似解对比分析,认为理想井下预压荷载越大（Nu越大）,则Indraratna等近似法成果误差越大;且在Cc/Ck = 1.8、Nu = 5时Indraratna等近似解与该理想井非线性固结精确解析解误差高达16%,而解析解与精确数值解基本一致。最后通过Indraratna等典型工程案例验证分析,认为所得理论更符合工程实际,具有较好的工程应用价值。     

Consolidation analysis for soft soil with non-Darcian flow considering variable discharge capacity of prefabricated vertical drain

Ground improvement is a complex issue, and accurately predicting the consolidation and settlement of soft soil with prefabricated vertical drain (PVD) presents a significant challenge. Recent laboratory and field tests have highlighted the influence of the variable discharge capacity of PVD and the non-Darcian flow behavior of soft soil on consolidation. However, existing theories have not yet considered these two factors simultaneously. To address this gap, a numerical solution for consolidation analysis incorporating non-Darcian flow and variable discharge capacity of PVD was developed and applied in a test area. The results of this study demonstrate the significant impact of both non-Darcian flow and variable discharge capacity on the consolidation rate. A comprehensive comparison between the findings of this numerical solution, the degenerate solution, and monitored data reveals clear differences. Notably, the non-Darcian exponent (s) and discharge capacity parameters (A₃) were found to exert a greater influence on consolidation behavior compared to other factors. These findings provide valuable guidance for the design and implementation of following airport ground improvement strategies.     

考虑井阻时空变化的竖井地基非线性固结解析解

软土固结过程中展现出明显的非线性压缩和渗透特性,同时竖井的淤堵效应常导致井阻在固结过程中随深度和时间不断演化,但目前能考虑井阻随时空演化的竖井地基非线性固结解析解还很鲜见。通过引入孔隙比与有效应力及孔隙比与渗透系数间的半对数模型描述了土体的非线性固结特性,建立了能同时考虑井阻随时空变化和涂抹影响的竖井地基非线性固结模型,并采用分离变量法获得了固结模型的解析解。将特定参数下固结解的计算结果与实测数据、已有的竖井地基固结解答进行了对比分析以验证其可靠性。最后,对竖井地基的非线性固结性状开展了大量计算分析。结果表明：竖井渗透系数随深度线性衰减越明显则地基固结速率越慢;外荷载一定时,随着软土压缩指数c_c与渗透率指数c_k之比的增大,竖井地基固结速度减慢;在c_c /c_k值不变的情况下,外荷载增加,地基固结速率加快。在涂抹区的3种径向渗透系数变化模式中,抛物线变化模式下的地基固结速度最快,线性变化模式下的地基固结速度次之,恒定模式下的地基固结速度最慢,且这种性状并不因为考虑井阻变化或土体非线性固结特性而发生改变。