Theories and methods for undrained strength and deformation of saturated soils
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摘要: 饱和土不排水强度及变形计算是岩土工程中颇具难度的复杂课题，至今业内对其多方面的问题仍存争议。结合笔者多年的相关学习研究，对此课题进行了较系统深入的探究。首先讨论了饱和土不排水计算的总应力法与有效应力法及各自的局限，并从有限元基本方程阐述了孔隙水压的处理，指出不管采用何种方法均应注意区分两类孔隙水压和两种总应力，即“土水分算”。随后，审视了Skempton-Henkel超静水压计算公式对不同应力路径的计算准确性，将它与MC强度准则相结合构建了饱和土不排水强度模型，对等向固结与不等向固结下饱和土不排水强度特性进行了分析，可为实际工程计算中强度参数选取提供参考。建议实际不排水计算采用广义Tresca强度准则，针对采用MC强度准则按有效应力法进行不排水计算会有较大误差的问题，除指出应直接输入不排水强度外，还给出一种采用等代强度参数的处理方法。在上述讨论基础上，剖析了固结不排水强度指标（CU指标）的缺陷，指出仅当土中总应力路径与测定CU指标的试验中相同时，用此指标直接计算才能给出准确的结果，特别是对地基承载力问题直接用此指标计算会给出严重偏危险的结果。还对基坑工程中主动、被动土水压力采用CU指标并土水合算的问题进行了分析，建议了合理计算方法及相应公式；对发生不排水极限土压时的滑移面倾角这一业内多人深感困惑的问题也进行了探讨。最后，针对饱和地基承载力及短期沉降计算，剖析了业内现行方法的问题，在深入解读有关因素影响机理的基础上给出新的计算方法。所给承载力计算公式，可较准确统一计算排水和不排水条件下的承载力，能更好保证设计的安全经济；所给沉降计算新方法能较准确计算饱和地基的短期沉降，对地基的排水沉降也可望取得较好结果，值得进一步发展完善。Abstract: The calculation of undrained strength and deformation of saturated soils is a complex and difficult issue in geotechnical engineering, and there are still contradictory opinions regarding its many aspects. In this study, a systematic and in-depth exploration of this topic is carried out according to the author's years of relevant studies. Firstly, the total stress method and effective stress method for undrained analysis as well as their respective limitations are discussed. The treatment of pore water pressure is explained from the basic equation of finite elements, and it is pointed out that regardless of the method used, attention should be paid to distinguishing between the two types of the pore water pressures and those of the total stresses. Then, the accuracy of Skempton-Henkel's formula for calculating the excess pore pressure under different stress paths is examined, and then it is combined with the MC strength criterion to form an undrained strength model for saturated soils. The undrained strength characteristics of saturated soils under isotropic and anisotropic consolidations are analyzed, which can provide reference for the selection of strength parameters for engineering calculations. It is suggested that the generalized Tresca strength criterion should be used for practical undrained calculations. To prevent computation errors in undrained analysis using the MC criterion together with the effective stress method, a method of using the equivalent strength parameters is proposed, which is a better alternative to the direct inputting of the undrained strength. Based on the above discussions, the shortcomings of the consolidated undrained strength index (CU index) are analyzed, and it is pointed out that only when the total stress path in the soils is the same as that in the tests for determining the CU index, this index can be directly used for calculation to give accurate results. Especially for the calculation of bearing capacity of foundations, direct use of the CU index will give serious erroneous results. The drawbacks of using the CU index for calculating the active and passive soil water pressures in excavation engineering are analyzed, and the reasonable method and corresponding formula are proposed. The issue of the inclination angle of the sliding surface when the undrained ultimate soil pressures occur, puzzled to many, is also discussed in depth. Finally, regarding the calculation of bearing capacity and short-term settlements of saturated foundation, the deficiencies of the conventional methods are analyzed, and new methods are proposed based on the deep understanding of the influencing mechanisms of the relevant factors. The proposed formula for bearing capality can more accurately calculate the bearing capacity under both drained and undrained conditions, which can thus better ensure the safety and economy of the design. The new method for settlement can effectively calculate the short-term settlement of saturated soils, and also shows good prospective for drained conditions.
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0. 引言
离心机振动台是当前海洋、水利、土木等领域岩土抗震研究的重要科学试验设施之一，其有效性、先进性已为国际共识[1-7]，可较真实地模拟地震荷载下岩土体变形和岩土构筑物破坏的过程。但是，由于离心机振动台运行在高g值离心力场中且与吊篮存在严重的运动学和动力学耦合，又受众多非线性因素影响。因此，如何提高离心机振动台的控制性能是学者们关注和亟需解答的一个重要课题。
目前国内现役离心机振动台仅有9套，分别归属于南京水利科学研究院、中国水利水电科学研究院、清华大学、香港科技大学、同济大学、浙江大学、成都理工大学、交通运输部天津水运工程科学研究所和中国地震局工程力学研究所。其中，张建民等[4]介绍了清华大学离心机振动台的性能指标和功能特色，并基于开展的性能检测试验评价了离心机振动台的系统性能。陈云敏等[5]介绍了浙江大学的离心机振动台的性能指标和结构组成，并基于开展的饱和砂土地基震陷试验对离心机振动台系统的性能进行了初步检验。顾行文等[6]介绍了南京水利科学研究院的NS-2型离心机振动台的性能指标和关键技术，并基于某沥青混凝土墙砂砾石坝试验对离心机振动台系统性能指标进行了初步验证。但这些成果，对迭代算法和柔性结构带给离心机振动台控制性能的影响，鲜见讨论。
本文以中国地震局工程力学研究所1500 kg离心机振动台为样本，介绍了离心机振动台的总体设计和基本组成，通过50g离心加速度下多个典型地震波的输入/输出对比测试，探讨了迭代算法、柔性结构对离心机振动台控制性能的影响，同时检验了不同负载下动力离心试验地震荷载的可重复效果，以期为正在建设和规划的离心机振动台研制提供参考。
1. 离心机振动台系统整体设计
1.1 试验设备与性能指标
中国地震局工程力学研究所与哈尔滨工业大学联合研制的DCIEM-40-300水平单向离心机振动台整体采用吊篮与振动台一体化的方法，如图 1所示。其主要性能指标：离心加速度100g，有效负载3000 kg，最大半径5.5 m，有效吊篮净空1.8 m×1.6 m×1.0 m；振动加速度30g，振动速度1 m/s，振动位移±10 mm，振动负载1500 kg，振动频宽10~300 Hz，平台有效尺寸1.6 m×0.8 m×0.8 m。除此之外，还配备了160 ch动态数据采集仪和自主研发了包括孔压、土压、加速度的各种宽频带高分辨力测量传感器。
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图 1 DCIEM-40-300离心机振动台系统Figure 1. Shaking table test system of DCIEM-40-300 centrifuge1.2 一般问题和设计方案
铰接装置作为柔性连接结构在许多离心机振动台上得到应用，以减小离心力荷载下台面和吊篮变形对作动器安全性带来的危险。但是，铰接装置本身存在间隙，且现有离心机振动台多数采用多阀单缸作动器[4-6]，在往返过程中会涉及到力换向的问题，换向过程间隙的存在可能对离心机振动台控制精度和稳定性产生影响。
中国地震局工程力学研究所的负载1500 kg离心机振动台，采用了双推力作动器，结构示意图如图 2所示，每个作动器由两条柱塞缸和一个三级伺服阀集成，且柱塞缸内部增设弹簧以使球头、球窝、摩擦板间始终紧密配合形成零间隙连接，避免了柔性结构。振动台总体设计如图 3所示，两个双推力作动器对称布置在运动平台两侧，以使作用线更接近岩土模型质心。运动平台由28个圆形弹性橡胶剪切轴承支撑。每个作动器由安装在配油板上的28 MPa高压蓄能器供油，并通过1.3 MPa低压蓄能器回油。蓄能器主要用作储油和稳压，可满足离心机不停机情况下连续激发多次振动试验的需求。
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图 3 负载1500 kg离心机振动台设计方案与结构组成Figure 3. Design scheme and structural composition of centrifugal shaking table with a load of 1500 kg1.3 试验设计与评价指标
本次试验选用的地震波有El Centro波、ChiChi波和Kobe波，其输入波时程和频谱如图 4所示，均为50g离心加速度下的压缩输入波。迭代算法是通过多次重复激振试验系统来生成正式试验的驱动信号，旨在使振动台的输出信号尽可能接近输入信号，根据最新研究成果，本文研究采用适于高频液压作动器的时域波形复现的迭代算法。此外，由于中国地震局工程力学研究所负载1500 kg离心机振动台作动器与平台间为无间隙连接，需要通过在作动器球窝与平台摩擦板间加入橡胶叠，利用橡胶叠来模拟柔性结构。
2. 迭代算法的影响
2.1 地震动峰值
振动台负载1200 kg时，离心加速度50g下，采用迭代算法前后10g El Centro波、10g ChiChi波的输出波时程曲线如图 5，6所示。经计算采用迭代算法后的El Centro波输出波最大峰值误差为5.3%，ChiChi波输出波最大峰值误差为6.5%，平均峰值误差为5.9%；而未采用迭代算法的El Centro波的实测输出波最大峰值误差为7.8%，ChiChi波输出波最大峰值误差为4.5%，平均峰值误差6.15%。迭代算法前后的平均峰值误差相差不明显，即迭代算法基本不影响再现地震动的最大峰值误差、平均峰值误差。
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图 5 迭代算法对10g El Centro波加速度时程及频谱影响Figure 5. Effects of iterative algorithm on time history of acceleration and frequency spectrum of 10g El Centro waves![]()
图 6 迭代算法对10g ChiChi波加速度时程及频谱影响Figure 6. Effects of iterative algorithm on time history of acceleration and frequency spectrum of 10g ChiChi waves2.2 地震动频宽
图 5和图 6给出了振动台负载1200 kg时，离心加速度50g下，采用迭代算法前后10g El Centro波、10g ChiChi波的输出波频谱。经计算采用迭代算法后的El Centro波输出波谱面积误差为6.06%，ChiChi波输出波谱面积误差为3.97%，平均谱面积误差为5.02%；而未采用迭代算法的El Centro波的实测输出波谱面积误差为5.98%，ChiChi波输出波谱面积为3.88%，平均谱面积误差4.94%。通过对比前后数据，发现在20~200 Hz频宽变化条件下，迭代算法对谱面积误差影响可忽略。
3. 柔性结构的影响
3.1 地震动峰值
图 7，8给出了振动台负载1200 kg时，离心加速度50g下，在有无柔性结构时7.5g Kobe波、20g El Centro波的输出波时程曲线。经计算柔性结构工况下的7.5g Kobe波输出波最大峰值误差为7.11%，20g El Centro波输出波最大峰值误差为3.63%，平均峰值误差为5.32%；而采用无柔性结构即球头、球窝与摩擦板零间隙连接时Kobe波输出波最大峰值误差为3.51%，El Centro波输出波最大峰值误差为3.07%，平均峰值误差为3.29%；柔性结构对峰值误差影响较小。但在加速度时程图中可明显看出柔性结构在7.5g Kobe波下基本复现不出持时在0.2 s后加速度峰值≤2.5g的波形，而在20g El Centro波下对0.65 s后加速度峰值≤2.5g的波形的复现效果虽比7.5gKobe波下有所改善，但与无柔性结构相比仍存在一定的差距。
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图 7 柔性结构对7.5g Kobe波加速度时程及频谱影响Figure 7. Effects of flexible structure on time history of acceleration and frequency spectrum of 7.5g Kobe waves![]()
图 8 柔性结构对20g El Centro波加速度时程及频谱影响Figure 8. Effects of flexible structure on time history of acceleration and frequency spectrum of 20g El Centro waves3.2 地震动频宽
振动台负载1200 kg时，离心加速度50g下，在有无柔性结构时7.5g Kobe波、20g El Centro波的输出波频谱图对比，如图 7，8所示。经计算柔性结构工况下7.5g Kobe波输出波谱面积误差为8.37%，20g El Centro波输出波谱面积误差为7.95%，平均谱面积误差为8.16%；而采用无柔性结构即无间隙连接时Kobe波输出波谱面积误差为4.89%，El Centro波输出波谱面积误差为8.09%，平均谱面积误差为6.49%；柔性结构对谱面积误差影响较小。但在频谱图中可明显看出柔性结构对于7.5gKobe地震波的复现精度有显著影响，尤其是80~150 Hz的高频段；对于20g El Centro地震波的复现精度影响在频宽20~120 Hz可忽略，但对于120~170 Hz频段的小幅的复现能力影响不容忽略，逊于无柔性结构条件。
4. 不同负载下重复性
此外，探讨了无柔性结构不同负载下离心振动台的波形重复性。更换不同重量的负载即1200 kg和600 kg，先后输入了两次同一幅值20g El Centro波，加速度时程曲线及频谱如图 9所示。负载1200 kg时，输出波峰值误差为3.07%，谱面积误差为8.09%；负载600 kg时，输出波峰值误差为3.2%，谱面积误差为5.56%。虽然不同负载在120~200 Hz频宽下的振动有所差异，但是从峰值误差和谱面积误差来看影响较小，可以忽略。对比前后两次输出波的峰值误差、谱面积误差，发现其均无明显差异，证明了离心机振动台具有良好的重复性和稳定性。
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图 9 不同负载下20g El Centro波加速度时程及频谱对比Figure 9. Comparison of time history of acceleration of 20g El Centro waves and frequency spectrum with different loads5. 结论
（1）20~200 Hz频宽变化条件下，迭代算法对最大峰值误差、平均峰值误差、平均谱面积误差的影响可忽略。采用迭代算法前后平均峰值误差分别为5.90%，6.15%，平均谱面积误差分别为5.02%，4.94%。
（2）柔性结构对于加速度峰值≤7.5g地震波的复现精度有显著影响，尤其是80~150 Hz的高频段；对于加速度峰值≥20g地震波的复现精度影响在频宽20~120 Hz可忽略，但对于120~170 Hz频段的小幅值成分的复现能力影响不容忽略，逊于无柔性结构条件。
（3）同时，在无柔性结构条件下对比不同负载即1200 kg和600 kg下同一输入荷载的实测台面输出波，发现两输出波的峰值误差与谱面积误差均无明显差异，证明了离心机振动台具有良好的重复性和稳定性，为开展平行模型试验和检验不同抗震设计方法提供了有力条件。
致谢: 感谢土力学及岩土工程界各位同仁的信任，使本人有机会作黄文熙讲座；感谢清华大学岩土工程及相关学科的众多同事对本人研究工作多年的支持和帮助。本文撰写过程中与李广信教授、龚晓南院士、张建民院士等多位同行专家的诸多讨论交流使本人受益良多；刘华北教授团队协助开展了饱和土不排水强度唯一性的部分试验验证工作；吊装换填地基的有关研究及现场实测得到中石化重型起重运输工程公司与北京中岩大地科技股份有限公司的大力支持与协作；博士生林世杰、付浩承担了文中大部分算例的计算。在此一并致谢。 -
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图 2 不同剪胀角计算的地基荷载沉降曲线[5]
Figure 2. Load-displacement curves calculated under different dilation angles
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图 3 MC模型计算应力路径与实际应力路径的差异
Figure 3. Differences between actual stress path and that calculated by MC model
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图 5 由MC模型按有效应力法计算的弹性有效应力路径
Figure 5. Elastic effective stress path calculated by MC model
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图 6 条基承载力问题用两种方法计算结果的对比
Figure 6. Comparison of bearing capacities of strip footing calculated by two methods
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图 8 三轴加载应力路径及相应强度指标
Figure 8. Stress paths and corresponding strength indices under triaxial loading
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图 9 两种参数计算的浅基础荷载-位移曲线
Figure 9. Load-displacement curves calculated by using different types of parameters
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图 12 K0固结土发生不排水极限土压时的总应力路径
Figure 12. Total stress paths in K0-consolidated soil at undrained ultimate earth pressure
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图 13 4种方法计算的总主动土水压力比较
Figure 13. Active soil water pressures calculated by four methods
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图 14 4种方法计算的总被动土水压力比较
Figure 14. Passive soil water pressures calculated by four methods
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图 16 有效应力法有限元计算的被动土压滑移面倾角
Figure 16. Slip surface at passive earth pressure calculated by effective stress FEM (=, , c=10 kPa)
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图 19 圆形均布荷载下布氏地基的沉降计算对比
Figure 19. Calculated settlements of Boussinesq foundation under circular uniform loads
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图 20 有限压缩层上圆形均布荷载中心点沉降计算对比
Figure 20. Calculated settlements of finite compressive layers under circular uniform loads
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图 21 矩形均布荷载下布氏地基的计算沉降对比
Figure 21. Calculated settlements of Boussinesq foundation under rectangular uniform loads
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图 23 各区块下卧层顶附加压力系数
Figure 23. Load pressure coefficients on underlying layer in each sub-area
表 1 地层参数
Table 1 Soil parameters
序号 地层名称 层厚/m 重度γ/(kN·m-3) 黏聚力c/kPa 内摩擦角φ/(°) 压缩模量Es1-2/MPa ① 填土 3 19.0 13.6 17.2 — ⑥1 粉土 2 19.9 11.6 26.8 6.5 ⑥2 黏土 11 17.8 16.0 14.1 2 粉质黏土 9 19.5 16.0 16.0 3.5~10 
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