基于无网格自然单元法的超长桩水平承载力研究
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摘要
随着科学的发展，数值方法在工程上的应用越来越广泛，其中有限元方法在工程上的应用最广。但有限元在处理大变形的非线性力学问题中存在网格畸变或缠结等不足，同时有限元在求解液体振动、裂纹扩展、材料相变和成形等不定边界或可动边界问题时，需要重新划分网格，而新旧网格之间物理量转换将产生新的误差。因此，无网格方法应运而生。无网格方法不需要借助任何单元，直接利用节点构造插值函数，避免了有限元方法的以上缺陷。在多种无网格方法中自然单元法以其独特的优势获得了快速的发展。
当前，超长桩大量应用于超高层建筑、大跨桥梁和深水港口工程中。然而，超长桩现阶段的研究相对滞后，工程中仍按普通桩理论进行设计，现有理论及分析模型不能充分考虑分层土特性或横纵荷载共同作用的影响，也无法很好地反映超长桩的承载性状。特别在港口工程中，超长桩通常承受很大的水平荷载，并且在横纵荷载共同作用下，往往发生大变形，而传统有限元方法在模拟计算大变形的超长桩和土体的相互作用时存在网格畸变或缠结等问题。
针对以上问题本文在自然单元法理论基础上进一步改进了插值函数，并将其应用到成层地基中超长桩水平承载力问题上。主要的研究工作及创新点如下：
1、改进了自然单元法的插值函数，编制三维无网格自然单元法计算程序。本文采用的是Voronoi图中二阶结构的边元素作为插值变量，大幅度提高了计算效率。无网格程序不需要借助任何单元，直接利用节点构造插值函数，避免了有限元中的网格畸变等问题，弥补了有限元在处理大变形、非线性力学等问题中存在的明显缺陷，具有较高的工程实用价值和广阔的应用前景。将程序计算结果与解析解及有限元计算结果相比较，验证了本程序的正确性和合理性。
2、基于变分原理计入横纵向荷载共同作用下超长桩的P-Δ效应，推导出单元刚度修正矩阵。针对横纵向荷载共同作用下超长桩的大变形问题，建立考虑非线性大变形的无网格自然单元法计算方法。通过对实例的计算分析，验证了该方法的正确性和合理性，得到P-Δ效应特征。结果表明：由于P-Δ效应，桩的位移响应与荷载为非线性关系。当地基土质较差，桩自由长度较大时，P-Δ效应对桩身位移和内力的影响不可忽略。本方法为计算时考虑P-Δ效应提供了一种简便易行的方法，在工程设计和施工中具有一定的价值。
3、基于无网格自然单元法计算并分析了荷载大小、加载顺序、长径比、桩顶约束条件、水平荷载作用位置、桩土相对刚度比和荷载分布形式等诸多参数对超长桩水平承载力的影响，并得出相关结论。
4、当层状地基中采用p-y曲线法模拟桩周土非线性特性时，桩周土水平位移的计算误差随荷载增大而增大。针对该问题本文采用层状弹性体系理论考虑土体纵向连续性，并利用层状各向同性体的研究方法，建立了适用于桩周层状地基的水平位移系数传递矩阵解法。根据有限单元等效载荷的计算原理，推导出水平位移系数矩阵，并给出外荷载较大时，p-y曲线法桩周土体水平位移的修正式。根据所建立的模型编制程序，对某三层地基土水平位移系数矩阵进行了计算和分析，验证了该方法的正确性和合理性。由于计入了土体的纵向连续性，本文方法所得的水平位移影响系数曲线在荷载作用点的及其邻近区域较Mindlin解的曲线更平滑，位移小于Mindlin解的结果。这表明当土层间性质差异较大时，本文方法能更好地体现层状土体实际分布差异的影响和临近土层间的相互作用。
Numerical method is widely used in engineering as the development of science, inwhich the finite element has the widest applications. However, the finite element hasthe insufficiency of mesh distortion or tangles in dealing with issues of non-linearmechanics of large transformation and the problems of re-meshing and new errorsgenerated by the physical quantity conversion between the new and the old meshwhen solving issues of indefinite or movable boundaries as fluid vibration, crackpropagation, material phase change and forming, therefore the meshless method isapplied. Different from finite element method, the meshless method requires nodeinformation rather than unit or grid information, which can overcome theinconvenience and defects of applying the finite element method. In a variety ofmeshless methods, natural element method with its unique advantages obtained rapiddevelopment.
Along with the large-scale construction of high-rise buildings, long-span bridgesand deepwater ports, super-long pile has been widely used to meet the requirements ofbearing capacity and displacement control of foundation. The working mechanismand force character of super-long piles are different from normal piles with a short ormedium length. But presently, as research of super-long pile in layered soil is stillrelatively backward, the method of design and calculation of super-long piles remainsin use of the theory of normal piles. The existing theory and analytical model are notcapable to fully consider characteristics of layered soil or the combined effect oflateral-vertical loads, as well as can not reflect the working properties of super-longpiles accurately.
The following aspects have been researched on calculation method and bearingbehavior in this paper:
1. In order to improve the computational efficiency of natural element method,the new interpolating function is constructed by edge elements of Voronoifigure which replaces body elements. The natural element method can overcome the inconvenience and defects of applying the finite element methodto deal with the large deformation problem. This method’s validity andrationale can be verified by calculating and analyzing the examples.
2. According to the variation principle, this paper calculated the P-Δ effect ofsuper-long pile under the combined effects of lateral and axial loads, andobtained the modified element stiffness matrix. Faced with the largedeformation problem of super-long pile under the combined effects of lateraland axial loads, the nonlinear calculation method is established based on themeshless natural element method. The correctness and rationality of themethod can be verified by calculating and analyzing the example. Thecharacteristic of the P-Δ effect can be obtained. Because of the P-Δ effect, thereis a nonlinear relationship existed between the displacement response andexternal loads. The calculated results indicate that the P-Δ effect cannot beoverlooked where the soil of the foundation is relatively loose and where theunsupported length of pile is relatively large.
3. According to the natural element method, this paper discussed the influencefactors of horizontal bearing capacity, including load value, loading sequence,length-diameter ratio, pile top restriction, horizontal load position, relativestiffness ratio of pile-soil and load distribution form.
4. When utilizing the p-y curve to simulate the non-linear characteristics of soilsurrounding pile in layered foundations, due to having not taken into accountthe soil Mass’ longitudinal continuity, the calculation deviation of horizontaldisplacement increases with the growth of a load. This paper adopted thelayered elasticity system theory to consider the soil Mass’ longitudinalcontinuity, as well as utilizing the research method for layered isotropic bodies,assuming that the horizontal resistance is evenly distributed around theperimeter of the pile's cross-section. then an appropriate transfer matrix methodof horizontal displacement coefficient for the soil surrounding pile in layeredfoundations was established. According to the calculation principle of finite element equivalent load, the horizontal displacement coefficient matrix wasdeduced as well as providing a corrected formula for the horizontaldisplacement of soil surrounding pile through the p-y curve method when theexternal load was increased. Following the established model a program wascreated which was used for calculating and analyzing the horizontaldisplacement coefficient matrix of three-layered soil in order to verify thismethod’s validity and rationale. Where there is a relatively large discrepancy inthe soil layers’ properties, this paper’s method is able to reflect the influence onthe layered soil’s actual distributional difference as well as the nearby soillayers’ interaction.
当前，超长桩大量应用于超高层建筑、大跨桥梁和深水港口工程中。然而，超长桩现阶段的研究相对滞后，工程中仍按普通桩理论进行设计，现有理论及分析模型不能充分考虑分层土特性或横纵荷载共同作用的影响，也无法很好地反映超长桩的承载性状。特别在港口工程中，超长桩通常承受很大的水平荷载，并且在横纵荷载共同作用下，往往发生大变形，而传统有限元方法在模拟计算大变形的超长桩和土体的相互作用时存在网格畸变或缠结等问题。
针对以上问题本文在自然单元法理论基础上进一步改进了插值函数，并将其应用到成层地基中超长桩水平承载力问题上。主要的研究工作及创新点如下：
1、改进了自然单元法的插值函数，编制三维无网格自然单元法计算程序。本文采用的是Voronoi图中二阶结构的边元素作为插值变量，大幅度提高了计算效率。无网格程序不需要借助任何单元，直接利用节点构造插值函数，避免了有限元中的网格畸变等问题，弥补了有限元在处理大变形、非线性力学等问题中存在的明显缺陷，具有较高的工程实用价值和广阔的应用前景。将程序计算结果与解析解及有限元计算结果相比较，验证了本程序的正确性和合理性。
2、基于变分原理计入横纵向荷载共同作用下超长桩的P-Δ效应，推导出单元刚度修正矩阵。针对横纵向荷载共同作用下超长桩的大变形问题，建立考虑非线性大变形的无网格自然单元法计算方法。通过对实例的计算分析，验证了该方法的正确性和合理性，得到P-Δ效应特征。结果表明：由于P-Δ效应，桩的位移响应与荷载为非线性关系。当地基土质较差，桩自由长度较大时，P-Δ效应对桩身位移和内力的影响不可忽略。本方法为计算时考虑P-Δ效应提供了一种简便易行的方法，在工程设计和施工中具有一定的价值。
3、基于无网格自然单元法计算并分析了荷载大小、加载顺序、长径比、桩顶约束条件、水平荷载作用位置、桩土相对刚度比和荷载分布形式等诸多参数对超长桩水平承载力的影响，并得出相关结论。
4、当层状地基中采用p-y曲线法模拟桩周土非线性特性时，桩周土水平位移的计算误差随荷载增大而增大。针对该问题本文采用层状弹性体系理论考虑土体纵向连续性，并利用层状各向同性体的研究方法，建立了适用于桩周层状地基的水平位移系数传递矩阵解法。根据有限单元等效载荷的计算原理，推导出水平位移系数矩阵，并给出外荷载较大时，p-y曲线法桩周土体水平位移的修正式。根据所建立的模型编制程序，对某三层地基土水平位移系数矩阵进行了计算和分析，验证了该方法的正确性和合理性。由于计入了土体的纵向连续性，本文方法所得的水平位移影响系数曲线在荷载作用点的及其邻近区域较Mindlin解的曲线更平滑，位移小于Mindlin解的结果。这表明当土层间性质差异较大时，本文方法能更好地体现层状土体实际分布差异的影响和临近土层间的相互作用。
Numerical method is widely used in engineering as the development of science, inwhich the finite element has the widest applications. However, the finite element hasthe insufficiency of mesh distortion or tangles in dealing with issues of non-linearmechanics of large transformation and the problems of re-meshing and new errorsgenerated by the physical quantity conversion between the new and the old meshwhen solving issues of indefinite or movable boundaries as fluid vibration, crackpropagation, material phase change and forming, therefore the meshless method isapplied. Different from finite element method, the meshless method requires nodeinformation rather than unit or grid information, which can overcome theinconvenience and defects of applying the finite element method. In a variety ofmeshless methods, natural element method with its unique advantages obtained rapiddevelopment.
Along with the large-scale construction of high-rise buildings, long-span bridgesand deepwater ports, super-long pile has been widely used to meet the requirements ofbearing capacity and displacement control of foundation. The working mechanismand force character of super-long piles are different from normal piles with a short ormedium length. But presently, as research of super-long pile in layered soil is stillrelatively backward, the method of design and calculation of super-long piles remainsin use of the theory of normal piles. The existing theory and analytical model are notcapable to fully consider characteristics of layered soil or the combined effect oflateral-vertical loads, as well as can not reflect the working properties of super-longpiles accurately.
The following aspects have been researched on calculation method and bearingbehavior in this paper:
1. In order to improve the computational efficiency of natural element method,the new interpolating function is constructed by edge elements of Voronoifigure which replaces body elements. The natural element method can overcome the inconvenience and defects of applying the finite element methodto deal with the large deformation problem. This method’s validity andrationale can be verified by calculating and analyzing the examples.
2. According to the variation principle, this paper calculated the P-Δ effect ofsuper-long pile under the combined effects of lateral and axial loads, andobtained the modified element stiffness matrix. Faced with the largedeformation problem of super-long pile under the combined effects of lateraland axial loads, the nonlinear calculation method is established based on themeshless natural element method. The correctness and rationality of themethod can be verified by calculating and analyzing the example. Thecharacteristic of the P-Δ effect can be obtained. Because of the P-Δ effect, thereis a nonlinear relationship existed between the displacement response andexternal loads. The calculated results indicate that the P-Δ effect cannot beoverlooked where the soil of the foundation is relatively loose and where theunsupported length of pile is relatively large.
3. According to the natural element method, this paper discussed the influencefactors of horizontal bearing capacity, including load value, loading sequence,length-diameter ratio, pile top restriction, horizontal load position, relativestiffness ratio of pile-soil and load distribution form.
4. When utilizing the p-y curve to simulate the non-linear characteristics of soilsurrounding pile in layered foundations, due to having not taken into accountthe soil Mass’ longitudinal continuity, the calculation deviation of horizontaldisplacement increases with the growth of a load. This paper adopted thelayered elasticity system theory to consider the soil Mass’ longitudinalcontinuity, as well as utilizing the research method for layered isotropic bodies,assuming that the horizontal resistance is evenly distributed around theperimeter of the pile's cross-section. then an appropriate transfer matrix methodof horizontal displacement coefficient for the soil surrounding pile in layeredfoundations was established. According to the calculation principle of finite element equivalent load, the horizontal displacement coefficient matrix wasdeduced as well as providing a corrected formula for the horizontaldisplacement of soil surrounding pile through the p-y curve method when theexternal load was increased. Following the established model a program wascreated which was used for calculating and analyzing the horizontaldisplacement coefficient matrix of three-layered soil in order to verify thismethod’s validity and rationale. Where there is a relatively large discrepancy inthe soil layers’ properties, this paper’s method is able to reflect the influence onthe layered soil’s actual distributional difference as well as the nearby soillayers’ interaction.
引文
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