竖井地基固结解析理论与有限元分析
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摘要
在含水量高、压缩性高、抗剪强度低的软土地基上设置竖井并结合堆载或真空负压,能加速地基排水固结、提高土体强度、减少构筑物工后沉降和差异沉降。该技术施工简便、效果可靠且造价低廉,已在公路、铁路、港口、机场等大型基础设施建设中广泛应用。为了合理预测竖井地基的固结发展,学者们在竖井固结解析解和数值解方面开展了大量研究且已取得大量成果。但同时应看到,目前各种计算方法得到的理论预测值与实测值还有一定差距,对可能的工后沉降和沉降差预测不准。另外,近些年来出现的竖井处理深度加大和工期要求加紧等情况,均给竖井地基的设计计算方法提出了挑战。为此,本文从解析理论和有限元法两方面开展了竖井地基固结问题的研究。
(1)首先对以往研究成果从理论方法和参数取值两方面进行了综述,结果发现：目前被国内外广泛接受的单井地基径向固结度简化计算公式均可表述为一个统一的公式,不同方法在涂抹效应和井阻效应的考虑上略有差异；单井固结理论近些年的新进展主要包括非线性理论、非达西渗流理论、考虑涂抹区渗透性渐变理论以及针对真空预压法的固结理论等；竖井地基固结分析参数取值方面已经开展了大量研究,但得到的参数取值范围较大,这将给计算结果的可靠性带来影响。
(2)通过对现有塑料排水板通水量特性研究进行综述发现,排水板通水量一方面随侧向压力增大而减小,另一方面在地基固结过程中会因为排水板的弯折和淤积堵塞等影响而逐渐减小。基于这一认识,假定通水量随地基深度线性减小、随时间呈指数函数减小,获得了描述井阻空间属性和时间属性的变井阻效应数学模型。然后,仿效Hansbo和谢康和竖井固结经典理论推导方法,建立了瞬时加荷条件下考虑变井阻效应的固结理论。计算分析表明：相对于采用短期通水量恒定值进行计算的竖井固结理论,变井阻固结理论得到的固结速率减缓；当考虑通水量随地基深度减小时,地基深部土层的超静孔压消散变得更加缓慢；当竖井渗透性减小到与井周土体相当,此时竖井不再起排水通道作用,径向固结将停止。另外,将变井阻固结理论应用于室内大型模型试验和工程实例分析,结果发现变井阻理论模拟的结果较传统理论预测结果与实际情况更为接近。
(3)通过引入Hansbo非达西渗流假定和变渗透系数假定,得到渗流双重非线性模型。然后,在Biot固结理论研究基础上,应用加权残数法推导了考虑该模型的有限元方程,并自编了相应的计算程序。通过开展参数影响分析得到：考虑渗透系数随时间减小后,地基固结速率减缓趋势明显；增大非达西渗流参数m或il,固结发展速率减缓,但该影响在10%以内；考虑非达西渗流后,地基固结速率随荷载减小而减缓。
(4)进一步引入土体分段线性e-lgp模型,得到了土体压缩非线性和渗流双重非线性模型。然后,结合宁波土样固结渗流联合测试试验,介绍了该模型的参数获取过程。在给出相应的控制方程和有限元方程基础之上,详细阐述了非线性有限元法编程实现过程。最后,通过算例验证了程序的可靠性,并对考虑土体多重非线性模型时竖井地基的固结性状进行了较为详细的分析,获得了一些有益认识。
(5)对目前广泛应用的竖井地基平面应变二维等效方法进行了评述。然后,以成层土竖井地基算例和未打穿竖井地基算例对六种二维等效方法进行了考量,结果发现：地基土成层性对二维等效方法计算结果的可靠性有较显著影响,而竖井未打穿情况的影响不明显；在本文算例参数取值情况下,各种方法的误差情况差别较大,Chai(2001)方法和Tran(2008)方法得到的结果相对可靠。此外,基于等效竖向渗透系数思想,建立了以单元体流量等效的一维变形、一维渗流简化分析方法。通过算例检验发现,该法计算效率大大提高,但相比于Chai法计算精度并未改善,且计算过程相对复杂。
(6)考虑到竖井地基三维有限元分析计算量大、前处理困难,提出了将井与涂抹区在地基横截面内划分为一个网格的复合单元法。复合单元的未知场由10结点单元和线性插值函数来刻画。复合单元内部全局坐标结点自由度和局部坐标结点自由度之间的联系则通过引入经典单井固结解析理论得到建立。之后,基于加权残数法详细推导了复合单元法有限元方程,并编制了相应的计算程序。通过对单井地基和群井地基算例进行验证分析,发现复合单元法计算精度较高、计算效率也有所改善。
(7)基于大型商业软件和自编程序开展了某电厂海堤下竖井地基固结发展过程的数值模拟。分析过程中,采用Chai(2001)二维等效方法和平面应变空间渗流分析方法分别进行了计算,结果发现：两种数值方法模拟得到的地基变形发展规律近似一致；后者考虑空间渗流得到的沉降略大、侧向位移略小；两种方法的孔压预测结果差异较为明显。数值模拟结果还表明：海堤下软土地基经过塑料排水板处理后,地基固结速率较快,地基中超静孔压在各级堆载间歇期均有明显回落；在给定的堆载方案下,软土地基变形发展速率均在控制范围内。
Vertical drain combined with preloading offer significant benefits in improving sites with thick soft soil deposits by accelerating the consolidation process and it has been used extensively in the large-scale infrastructure construction for highways, railways, ports, airports and so on. A lot of achievements have been made both in theoretical approaches and numerical methods for the reliable simulation of the consolidation process by vertical drains. It was also found that a certain gap exists between the measured values and calculation results by different methods. The predictions of the post construction settlement and the differential settlement are usually in poor accuracy. Additionally, the situations of the long depth of a vertical drain for deep soil deposits and limited construction period are challenging the design and calculation procedure of vertical drain consolidation. To this end, this paper carried out the study for the simulation of the consolidation process by vertical drains from both the analytic theory and the finite element method.
1. The previous researches for both theoretical methods and parameters are summarized at first. It was found that the widely accepted formulas for the calculation of the radial average degree of consolidation can be expressed as a uniform expression. Different approaches have a slight difference in the consideration of the smear effect and the well resistance. The latest developments of the consolidation theory for vertical drained ground consist of non-linear theory, non-Darcy flow theory, the variety of soil permeability within the smear zone and the consolidation theory for the vacuum preloading method. A large number of studies have also been carried out for parameters in the analysis of the vertical drained foundation, but the value ranges of these parameters are large and it will reduce the reliability of the calculation results.
2. The discharge capacity qn has been investigated by several studies in recent years. General conclusions have been obtained by laboratory tests that the discharge capacity qw can be reduced by several factors throughout the consolidation process in field, such as the deformation of the drain, lateral stress, siltation, hydraulic gradient and so on. Based on this consideration, this paper assumed qw varies linearly with depth z and decreases exponentially with tim t and formed a mathematical model for varied well resistance effect. Then, a series of closed-form solutions following with the approaches of Hansbo's approximate solution and Xie's rigorous solution were developed with the instant loading condition, respectively. The impact of variable discharge capacity on the development of consolidation was investigated. And the following conclusions have been obtained that the existing unit cell method with a constant short time discharge capacity will over predict the consolidation rate. The development of the radial degree of consolidation may be totally interrupted when the permeability of the drain is reduced to that of the surrounding soil. The varied well resistance apparently affects the patterns of the distribution of average excess pore water pressure along the depth. Besides, the proposed method is applied in the simulation of a large indoor model test and an engineering practice. By comparing the results of the new method with classical solutions, it was found that the present method with varied well resistance could predict the consolidation process better since the behavior of PVDs is more accurately described.
3. A non-linear flow relationship, which assumes that the fluid flow in the soil skeleton obeyed Hansbo's non-Darcian flow and the coefficient of permeability decreased with the consolidation time, was incorporated into Biot's general consolidation theory for the consolidation simulation of soft ground with vertical drains. Governing equations with the coupled non-linear flow model were presented and the finite element (FE) formulations were derived based on the weighted residual method. The effect of the coupled non-linear flow on the development of consolidation was investigated. It was found that the consolidation rate became slow when the non-Darcian flow with varied permeability was considered. The retardation on the consolidation process was taken on an accelerating tendency when increasing the values of the non-linear flow parameters. The loading size also had significant effect on the consolidation behavior in the non-linear analysis of the coupled consolidation problem.
4. With the introduction of a piecewise linear e-lgp assumption, this paper further presents an FE analysis which takes into account one-dimensional non-linear property of compression and permeability. Combined with the consolidation-seepage joint test for Ningbo soil samples, the acquisition process for parameters of the one-dimensional nonlinear model is presented. Governing equations with the coupled non-linear flow model were presented firstly for force equilibrium condition and continuity condition, respectively. Based on the weighted residual method, the FE formulations were then derived and an existing FE program was modified considering the non-linear flow model. Comparative analyses with established theoretical solutions and numerical solutions were carried out and the results were satisfactory. On this basis, the effect of the coupled non-linear flow on the development of consolidation was investigated, and some useful conclusions were obtained correspondingly.
5. The two-dimensional equivalent methods are widely used in the finite element analysis of vertical drained ground. Summaries and comments about the equivalent methods are firstly carried out based on the existing researches. Then, six equivalent methods are examined for two numerical examples for layered soil foundation and partially penetrated vertical foundation. The results show that the parameter values for layered soil have significant impact on the reliability of equivalent methods, while the length of the drain has no significant effect. For the parameter values applied in the example, errors for the degree of consolidation for different methods vary considerably. Relatively, the calculation results of Chai's (2001) method and Tran's (2008) method are reliable and the largest errors for the two methods are within10%. Chai's method is worth promoting since the operation of the method is rigorous and with high computational efficient. Additionally, aims to simplify the pre-process of three dimensional analysis of vertical drain foundation and to improve the computation efficiency, the one dimensional equivalent method with1D deformation and1D seepage element was proposed based on Chai's equivalent vertical seepage idea. Numerical results were compared with traditional FEM results for a series of calculation conditions to verify the accuracy of the simplified method. And the results showed that the computation efficiency is greatly improved. However, compared with Chai's equivalent method, the accuracy of the newly proposed method is not improved and the calculation process is relatively complex.
6. Since drains are small both in spacing and size, resulting in enormous computing costs for a traditional3D FE analysis, a new spatial element that contains the drain well and its neighboring smear zone was presented. This new combined element was depicted by10nodes element, which contains8global independent nodes and2local dependent nodes. A classical analytical theory was introduced to set up the relationship between the two kinds of nodes. Since permeability diversity between the drain and the smear zone was considered, both the effects of smearing and well resistance were taken into account in the composite element method (CEM). A detailed derivation of the CEM was performed using the weighted residual method. To verify the accuracy of the CEM. numerical results were compared with analytical solutions and with traditional FEM results for two cases. These comparative analyses demonstrated that acceptable results were obtained by the CEM. Additionally, the proposed method saves1/4-1/2mesh elements and avoids slender elements for the full-scale FE analysis of vertical drain foundation.
7. Finally, numerical simulation for a vertical drain foundation under a power plant seawall was carried out based on a large-scale commercial software and by a self-coding program. The theoretical bases for the two approaches are Chai's two-dimensional equivalent method and Xie's PDSS model. It was found by the comparison of the two numerical methods that the deformation simulations of the ground by the two methods are roughly the same as each other; the settlement of the foundation predicted by the later method is relatively bigger than Chai's method since the PDSS model took into account the spatial seepage. Conclusions also obtained by the numerical simulation that the development of consolidation for the ground is obvious:the excess pore water pressure are dissipated significantly in the intermittent period of loading; the development rates of ground consolidation are within the security value.
(1)首先对以往研究成果从理论方法和参数取值两方面进行了综述,结果发现：目前被国内外广泛接受的单井地基径向固结度简化计算公式均可表述为一个统一的公式,不同方法在涂抹效应和井阻效应的考虑上略有差异；单井固结理论近些年的新进展主要包括非线性理论、非达西渗流理论、考虑涂抹区渗透性渐变理论以及针对真空预压法的固结理论等；竖井地基固结分析参数取值方面已经开展了大量研究,但得到的参数取值范围较大,这将给计算结果的可靠性带来影响。
(2)通过对现有塑料排水板通水量特性研究进行综述发现,排水板通水量一方面随侧向压力增大而减小,另一方面在地基固结过程中会因为排水板的弯折和淤积堵塞等影响而逐渐减小。基于这一认识,假定通水量随地基深度线性减小、随时间呈指数函数减小,获得了描述井阻空间属性和时间属性的变井阻效应数学模型。然后,仿效Hansbo和谢康和竖井固结经典理论推导方法,建立了瞬时加荷条件下考虑变井阻效应的固结理论。计算分析表明：相对于采用短期通水量恒定值进行计算的竖井固结理论,变井阻固结理论得到的固结速率减缓；当考虑通水量随地基深度减小时,地基深部土层的超静孔压消散变得更加缓慢；当竖井渗透性减小到与井周土体相当,此时竖井不再起排水通道作用,径向固结将停止。另外,将变井阻固结理论应用于室内大型模型试验和工程实例分析,结果发现变井阻理论模拟的结果较传统理论预测结果与实际情况更为接近。
(3)通过引入Hansbo非达西渗流假定和变渗透系数假定,得到渗流双重非线性模型。然后,在Biot固结理论研究基础上,应用加权残数法推导了考虑该模型的有限元方程,并自编了相应的计算程序。通过开展参数影响分析得到：考虑渗透系数随时间减小后,地基固结速率减缓趋势明显；增大非达西渗流参数m或il,固结发展速率减缓,但该影响在10%以内；考虑非达西渗流后,地基固结速率随荷载减小而减缓。
(4)进一步引入土体分段线性e-lgp模型,得到了土体压缩非线性和渗流双重非线性模型。然后,结合宁波土样固结渗流联合测试试验,介绍了该模型的参数获取过程。在给出相应的控制方程和有限元方程基础之上,详细阐述了非线性有限元法编程实现过程。最后,通过算例验证了程序的可靠性,并对考虑土体多重非线性模型时竖井地基的固结性状进行了较为详细的分析,获得了一些有益认识。
(5)对目前广泛应用的竖井地基平面应变二维等效方法进行了评述。然后,以成层土竖井地基算例和未打穿竖井地基算例对六种二维等效方法进行了考量,结果发现：地基土成层性对二维等效方法计算结果的可靠性有较显著影响,而竖井未打穿情况的影响不明显；在本文算例参数取值情况下,各种方法的误差情况差别较大,Chai(2001)方法和Tran(2008)方法得到的结果相对可靠。此外,基于等效竖向渗透系数思想,建立了以单元体流量等效的一维变形、一维渗流简化分析方法。通过算例检验发现,该法计算效率大大提高,但相比于Chai法计算精度并未改善,且计算过程相对复杂。
(6)考虑到竖井地基三维有限元分析计算量大、前处理困难,提出了将井与涂抹区在地基横截面内划分为一个网格的复合单元法。复合单元的未知场由10结点单元和线性插值函数来刻画。复合单元内部全局坐标结点自由度和局部坐标结点自由度之间的联系则通过引入经典单井固结解析理论得到建立。之后,基于加权残数法详细推导了复合单元法有限元方程,并编制了相应的计算程序。通过对单井地基和群井地基算例进行验证分析,发现复合单元法计算精度较高、计算效率也有所改善。
(7)基于大型商业软件和自编程序开展了某电厂海堤下竖井地基固结发展过程的数值模拟。分析过程中,采用Chai(2001)二维等效方法和平面应变空间渗流分析方法分别进行了计算,结果发现：两种数值方法模拟得到的地基变形发展规律近似一致；后者考虑空间渗流得到的沉降略大、侧向位移略小；两种方法的孔压预测结果差异较为明显。数值模拟结果还表明：海堤下软土地基经过塑料排水板处理后,地基固结速率较快,地基中超静孔压在各级堆载间歇期均有明显回落；在给定的堆载方案下,软土地基变形发展速率均在控制范围内。
Vertical drain combined with preloading offer significant benefits in improving sites with thick soft soil deposits by accelerating the consolidation process and it has been used extensively in the large-scale infrastructure construction for highways, railways, ports, airports and so on. A lot of achievements have been made both in theoretical approaches and numerical methods for the reliable simulation of the consolidation process by vertical drains. It was also found that a certain gap exists between the measured values and calculation results by different methods. The predictions of the post construction settlement and the differential settlement are usually in poor accuracy. Additionally, the situations of the long depth of a vertical drain for deep soil deposits and limited construction period are challenging the design and calculation procedure of vertical drain consolidation. To this end, this paper carried out the study for the simulation of the consolidation process by vertical drains from both the analytic theory and the finite element method.
1. The previous researches for both theoretical methods and parameters are summarized at first. It was found that the widely accepted formulas for the calculation of the radial average degree of consolidation can be expressed as a uniform expression. Different approaches have a slight difference in the consideration of the smear effect and the well resistance. The latest developments of the consolidation theory for vertical drained ground consist of non-linear theory, non-Darcy flow theory, the variety of soil permeability within the smear zone and the consolidation theory for the vacuum preloading method. A large number of studies have also been carried out for parameters in the analysis of the vertical drained foundation, but the value ranges of these parameters are large and it will reduce the reliability of the calculation results.
2. The discharge capacity qn has been investigated by several studies in recent years. General conclusions have been obtained by laboratory tests that the discharge capacity qw can be reduced by several factors throughout the consolidation process in field, such as the deformation of the drain, lateral stress, siltation, hydraulic gradient and so on. Based on this consideration, this paper assumed qw varies linearly with depth z and decreases exponentially with tim t and formed a mathematical model for varied well resistance effect. Then, a series of closed-form solutions following with the approaches of Hansbo's approximate solution and Xie's rigorous solution were developed with the instant loading condition, respectively. The impact of variable discharge capacity on the development of consolidation was investigated. And the following conclusions have been obtained that the existing unit cell method with a constant short time discharge capacity will over predict the consolidation rate. The development of the radial degree of consolidation may be totally interrupted when the permeability of the drain is reduced to that of the surrounding soil. The varied well resistance apparently affects the patterns of the distribution of average excess pore water pressure along the depth. Besides, the proposed method is applied in the simulation of a large indoor model test and an engineering practice. By comparing the results of the new method with classical solutions, it was found that the present method with varied well resistance could predict the consolidation process better since the behavior of PVDs is more accurately described.
3. A non-linear flow relationship, which assumes that the fluid flow in the soil skeleton obeyed Hansbo's non-Darcian flow and the coefficient of permeability decreased with the consolidation time, was incorporated into Biot's general consolidation theory for the consolidation simulation of soft ground with vertical drains. Governing equations with the coupled non-linear flow model were presented and the finite element (FE) formulations were derived based on the weighted residual method. The effect of the coupled non-linear flow on the development of consolidation was investigated. It was found that the consolidation rate became slow when the non-Darcian flow with varied permeability was considered. The retardation on the consolidation process was taken on an accelerating tendency when increasing the values of the non-linear flow parameters. The loading size also had significant effect on the consolidation behavior in the non-linear analysis of the coupled consolidation problem.
4. With the introduction of a piecewise linear e-lgp assumption, this paper further presents an FE analysis which takes into account one-dimensional non-linear property of compression and permeability. Combined with the consolidation-seepage joint test for Ningbo soil samples, the acquisition process for parameters of the one-dimensional nonlinear model is presented. Governing equations with the coupled non-linear flow model were presented firstly for force equilibrium condition and continuity condition, respectively. Based on the weighted residual method, the FE formulations were then derived and an existing FE program was modified considering the non-linear flow model. Comparative analyses with established theoretical solutions and numerical solutions were carried out and the results were satisfactory. On this basis, the effect of the coupled non-linear flow on the development of consolidation was investigated, and some useful conclusions were obtained correspondingly.
5. The two-dimensional equivalent methods are widely used in the finite element analysis of vertical drained ground. Summaries and comments about the equivalent methods are firstly carried out based on the existing researches. Then, six equivalent methods are examined for two numerical examples for layered soil foundation and partially penetrated vertical foundation. The results show that the parameter values for layered soil have significant impact on the reliability of equivalent methods, while the length of the drain has no significant effect. For the parameter values applied in the example, errors for the degree of consolidation for different methods vary considerably. Relatively, the calculation results of Chai's (2001) method and Tran's (2008) method are reliable and the largest errors for the two methods are within10%. Chai's method is worth promoting since the operation of the method is rigorous and with high computational efficient. Additionally, aims to simplify the pre-process of three dimensional analysis of vertical drain foundation and to improve the computation efficiency, the one dimensional equivalent method with1D deformation and1D seepage element was proposed based on Chai's equivalent vertical seepage idea. Numerical results were compared with traditional FEM results for a series of calculation conditions to verify the accuracy of the simplified method. And the results showed that the computation efficiency is greatly improved. However, compared with Chai's equivalent method, the accuracy of the newly proposed method is not improved and the calculation process is relatively complex.
6. Since drains are small both in spacing and size, resulting in enormous computing costs for a traditional3D FE analysis, a new spatial element that contains the drain well and its neighboring smear zone was presented. This new combined element was depicted by10nodes element, which contains8global independent nodes and2local dependent nodes. A classical analytical theory was introduced to set up the relationship between the two kinds of nodes. Since permeability diversity between the drain and the smear zone was considered, both the effects of smearing and well resistance were taken into account in the composite element method (CEM). A detailed derivation of the CEM was performed using the weighted residual method. To verify the accuracy of the CEM. numerical results were compared with analytical solutions and with traditional FEM results for two cases. These comparative analyses demonstrated that acceptable results were obtained by the CEM. Additionally, the proposed method saves1/4-1/2mesh elements and avoids slender elements for the full-scale FE analysis of vertical drain foundation.
7. Finally, numerical simulation for a vertical drain foundation under a power plant seawall was carried out based on a large-scale commercial software and by a self-coding program. The theoretical bases for the two approaches are Chai's two-dimensional equivalent method and Xie's PDSS model. It was found by the comparison of the two numerical methods that the deformation simulations of the ground by the two methods are roughly the same as each other; the settlement of the foundation predicted by the later method is relatively bigger than Chai's method since the PDSS model took into account the spatial seepage. Conclusions also obtained by the numerical simulation that the development of consolidation for the ground is obvious:the excess pore water pressure are dissipated significantly in the intermittent period of loading; the development rates of ground consolidation are within the security value.
引文
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