非节点连接有限元理论及其软件实现
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摘要
在土木工程中,由于基体材料的性能不足,大量使用各种加筋材料来增强和改善结构的力学性能。普通钢筋混凝土、预应力钢筋混凝土、纤维增强混凝土、型钢混凝土、钢管混凝土以及岩体用锚杆进行锚固等多种复合材料和结构形式被广泛采用。结构的重要性对分析计算提出了更高的要求,建立恰当的有限元模型对加筋结构进行准确模拟,是进行有限元分析首先需要解决的难题。目前,加筋结构的有限元建模方法主要有分离式、组合式和整体式三种。但这三种方法各有缺陷,难以模拟工程实际中的复杂情况。
针对目前土木工程加筋结构有限元分析缺乏恰当有限元模型的问题,本文开展了以下方面的研究并取得了相应的成果:
1.在考察现有各种加筋结构中加筋构件力学行为的基础上,指出加筋构件的合理单元模型可以采用杆单元和梁单元,其中梁单元具有综合的力学性能,适用范围更为广泛。
2.在考察各种加筋构件与基体材料的连接方式基础上,提出了自由度层次的非节点连接方法。基于该方法,加筋单元的各节点可位于一个或多个其它单元内部,节点的自由度也无需全部与母单元的位移场一致,既有别于分离式模型通过节点实现单元连接,又有别于组合式模型要求加筋单元整体位于一个母单元。自由度层次的非节点连接方法建模灵活方便,能准确考虑加筋构件的复杂布置;通过引入节点坐标系,又可自然模拟加筋构件与基体材料之间的位移不连续性(如粘结滑移、无粘结和体外布置等)。
3.对于采用非节点连接方法建立的有限元模型,为实现其有限元分析,对平面和空间梁单元、平面和空间等参元的含转动位移场进行了深入的研究。将单元内一点的转动分为一点平均转动和特定方向线元转动两种形式,给出了两种形式的具体计算方法。一点转动的深入研究,为带转动自由度的节点与其它单元连接提供了理论基础,使得用梁单元模拟加筋构件得以成功实现。
4.建立了非节点连接有限元分析方法。将内节点在节点坐标系下的自由度分为一致的自由度和独立的自由度,分别用于模拟内节点与母单元位移场的连续性和不连续性,建立了内节点与母单元之间的粘结弹簧单元用于模拟内节点与母单元之间的粘结滑移。针对多种单元类型,对非节点连接情况下的有限元计算公式进行了系统推证,建立了非节点连接情况下的单元组装方法。
5.研究和解决了非节点连接情况下的带宽优化问题。带宽优化技术可极大地提高有限元软件的计算速度,是一个实用有限元软件不可缺少的功能。本文研究了非节点连接方式下的自由度情况,对现有图论带宽优化算法提出了新的改进措施,将带宽优化方法深入到自由度层次,解决了节点自由度在整体自由度向量中参与个数不一致的复杂带宽问题,实现了非节点连接情况下的带宽优化。
6.开发了非节点连接有限元软件。基于面向对象技术,在Windows系统平台采用Visual C++语言开发一个具有3D图形用户界面的有限元软件RCF,对非节点连接有限元理论进行了实现。该软件构造了有限元分析的各种类层次,具体实现了16种单元类型,可在平面和空间两种情况下对各种加筋结构进行非节点连接有限元建模和分析。
7.利用RCF软件对多个典型问题进行了计算,验证和展示了非节点连接方法及其有限元软件的正确性和可行性。
非节点连接有限元理论打破了传统的单元之间通过节点连接的限制,为有限元理论注入了新的内涵。单元种类和连接方法的灵活应用,为结构的单元离散方案提供更多的选择空间,为多种土木工程加筋结构提供了单元层次的模型支持,能够更准确地揭示局部关键部位的受力状态和结构整体力学行为,提高这些结构的有限元分析水平,在土木工程中具有广阔的应用前景。
In civil engineering, as the matrix material performance deficiencies, the reinforced materials are used largely to enhance and improve the mechanical properties of the structure. Ordinary reinforced concrete, pre-stressed reinforced concrete, and fiber reinforced concrete, steel reinforced concrete, concrete-filled steel tube, rock with a bolt anchoring, and other composite materials and forms of structural are widely used. Due to the structure’s importance, the high accurate analysis is needed. To accurately analyze the reinforced structure with finite element method, the primary problem to be solved is to establish the appropriate finite element models. At present, the finite element models of the reinforced structure have three types: the discrete model, the embedded model and smeared model. However, for flaws in the three types, it is difficult to simulate the complicated reinforced structures in engineering.
Lacking appropriate finite element analysis model of the reinforced structures in the civil engineering, the following aspects of the study are carried out and the corresponding results are obtained:
1. Based on inspecting the mechanical behavior of reinforced components in the existing reinforced structure, that the reinforcement component can be meshed with truss elements and beam elements is indicated, and beam element has a wider scope of application as its integrated mechanical properties.
2. On the basis of inspecting the various types of connectivity between reinforced components and the matrix material, the non-nodal connectivity method at the level of degrees of freedom is raised. Based on this method, the nodes of a reinforced element can be embedded into one or more other elements; while the node’s DOF does not need to match the parent element’s displacement field. This method is different from the discrete model in which the node-node connectivity is needed, and distinct from the embedded model in which the nodes of a reinforced element are all placed in the same parent element. Using the non-nodal connectivity method at the level of degrees of freedom, the reinforcement and its parent material can be meshed independently and expediently, furthermore the complex layout of reinforcements can be accurately considered. By introducing node coordinates, the displacement discontinuity between reinforced components and matrix materials, such as bond-slip, non-bonded, and disposed external, etc., can be simulated naturally.
3. In order to carry out the finite element analysis while the reinforcing element’s nodes are embedded in one or more other elements, an in-depth study is conducted about the displacement field including rotational DOF in plane beam elements, space beam elements, plane isoparametric elements and space isoparametric elements. The rotational displacements are divided into two types of the point average rotation and the infinitesimal line’s rotation, and their calculation formulas are given. The analysis of rotational displacements provides a theoretical basis to connect beam elements’nodes which have rotational DOFs with other elements, particularly when nodes are embedded in parent elements. Since nodes of a beam element can connect to their parent elements, the reinforcement can be meshed by beam elements successfully.
4. The non-nodal connectivity finite element analysis method is established. An inner node’s DOFs are divided into two types of dependence and independence. Under the coordinates at an inner node, the inner node’s dependent DOFs are equal to the displacements accounted through its parent element, and the independent the others. A bond spring element between the inner node and its parent element is raised to simulate the bond-slip between the reinforcement and the matrix material. After the finite element formulas for multi types of elements are derived, the element assemble technique in the non-nodal connectivity state is worked out.
5. The problem of bandwidth optimization in the non-nodal connectivity state is investigated. The bandwidth optimization technique can greatly improve the calculation speed of a FEM software, so is an indispensable function of a practical FEM software. Regarding the complicated circumstances of DOFs in the non-nodal connectivity state, some improvements in algorithm of graph theory are made to decrease the bandwidth; the bandwidth optimization algorithm is implemented at the level of DOF. This method resolves not only the complex issue of bandwidth optimization in the non-nodal connectivity state, but also the problem induced by that the DOFs numbers of various types of node taking part in the structure’s whole DOFs are not equal.
6. Using the object-oriented technology, a FEM software named RCF is developed to realize the non-nodal connectivity method, which provides a 3D graphical user interface and programmed in visual C++ language under the Windows platform. The levels of FEM classes are constructed in RCF. Making use of RCF and its 10 types of elements, the reinforced structures under plane and space circumstances can be meshed and analyzed by means of non-nodal connectivity method.
7. A number of typical problems are calculated by RCF; their numerical results validate and demonstrate the correctness and feasibility of the software and the non-nodal connectivity finite element method.
The non-nodal connectivity finite element method breaks the traditional restrictions of node-node connecting and adds new content to existing finite element theory. The flexible application of connecting technology and element types provides more choices to mesh a structure and afford new models at element level for a variety of reinforced structures in civil engineering, and can reveal more accurately the mechanical behavior of a structure at overall and its partial key parts, and therefore improve the finite element analysis for reinforced structures. The method has broad application prospects in civil engineering.
针对目前土木工程加筋结构有限元分析缺乏恰当有限元模型的问题,本文开展了以下方面的研究并取得了相应的成果:
1.在考察现有各种加筋结构中加筋构件力学行为的基础上,指出加筋构件的合理单元模型可以采用杆单元和梁单元,其中梁单元具有综合的力学性能,适用范围更为广泛。
2.在考察各种加筋构件与基体材料的连接方式基础上,提出了自由度层次的非节点连接方法。基于该方法,加筋单元的各节点可位于一个或多个其它单元内部,节点的自由度也无需全部与母单元的位移场一致,既有别于分离式模型通过节点实现单元连接,又有别于组合式模型要求加筋单元整体位于一个母单元。自由度层次的非节点连接方法建模灵活方便,能准确考虑加筋构件的复杂布置;通过引入节点坐标系,又可自然模拟加筋构件与基体材料之间的位移不连续性(如粘结滑移、无粘结和体外布置等)。
3.对于采用非节点连接方法建立的有限元模型,为实现其有限元分析,对平面和空间梁单元、平面和空间等参元的含转动位移场进行了深入的研究。将单元内一点的转动分为一点平均转动和特定方向线元转动两种形式,给出了两种形式的具体计算方法。一点转动的深入研究,为带转动自由度的节点与其它单元连接提供了理论基础,使得用梁单元模拟加筋构件得以成功实现。
4.建立了非节点连接有限元分析方法。将内节点在节点坐标系下的自由度分为一致的自由度和独立的自由度,分别用于模拟内节点与母单元位移场的连续性和不连续性,建立了内节点与母单元之间的粘结弹簧单元用于模拟内节点与母单元之间的粘结滑移。针对多种单元类型,对非节点连接情况下的有限元计算公式进行了系统推证,建立了非节点连接情况下的单元组装方法。
5.研究和解决了非节点连接情况下的带宽优化问题。带宽优化技术可极大地提高有限元软件的计算速度,是一个实用有限元软件不可缺少的功能。本文研究了非节点连接方式下的自由度情况,对现有图论带宽优化算法提出了新的改进措施,将带宽优化方法深入到自由度层次,解决了节点自由度在整体自由度向量中参与个数不一致的复杂带宽问题,实现了非节点连接情况下的带宽优化。
6.开发了非节点连接有限元软件。基于面向对象技术,在Windows系统平台采用Visual C++语言开发一个具有3D图形用户界面的有限元软件RCF,对非节点连接有限元理论进行了实现。该软件构造了有限元分析的各种类层次,具体实现了16种单元类型,可在平面和空间两种情况下对各种加筋结构进行非节点连接有限元建模和分析。
7.利用RCF软件对多个典型问题进行了计算,验证和展示了非节点连接方法及其有限元软件的正确性和可行性。
非节点连接有限元理论打破了传统的单元之间通过节点连接的限制,为有限元理论注入了新的内涵。单元种类和连接方法的灵活应用,为结构的单元离散方案提供更多的选择空间,为多种土木工程加筋结构提供了单元层次的模型支持,能够更准确地揭示局部关键部位的受力状态和结构整体力学行为,提高这些结构的有限元分析水平,在土木工程中具有广阔的应用前景。
In civil engineering, as the matrix material performance deficiencies, the reinforced materials are used largely to enhance and improve the mechanical properties of the structure. Ordinary reinforced concrete, pre-stressed reinforced concrete, and fiber reinforced concrete, steel reinforced concrete, concrete-filled steel tube, rock with a bolt anchoring, and other composite materials and forms of structural are widely used. Due to the structure’s importance, the high accurate analysis is needed. To accurately analyze the reinforced structure with finite element method, the primary problem to be solved is to establish the appropriate finite element models. At present, the finite element models of the reinforced structure have three types: the discrete model, the embedded model and smeared model. However, for flaws in the three types, it is difficult to simulate the complicated reinforced structures in engineering.
Lacking appropriate finite element analysis model of the reinforced structures in the civil engineering, the following aspects of the study are carried out and the corresponding results are obtained:
1. Based on inspecting the mechanical behavior of reinforced components in the existing reinforced structure, that the reinforcement component can be meshed with truss elements and beam elements is indicated, and beam element has a wider scope of application as its integrated mechanical properties.
2. On the basis of inspecting the various types of connectivity between reinforced components and the matrix material, the non-nodal connectivity method at the level of degrees of freedom is raised. Based on this method, the nodes of a reinforced element can be embedded into one or more other elements; while the node’s DOF does not need to match the parent element’s displacement field. This method is different from the discrete model in which the node-node connectivity is needed, and distinct from the embedded model in which the nodes of a reinforced element are all placed in the same parent element. Using the non-nodal connectivity method at the level of degrees of freedom, the reinforcement and its parent material can be meshed independently and expediently, furthermore the complex layout of reinforcements can be accurately considered. By introducing node coordinates, the displacement discontinuity between reinforced components and matrix materials, such as bond-slip, non-bonded, and disposed external, etc., can be simulated naturally.
3. In order to carry out the finite element analysis while the reinforcing element’s nodes are embedded in one or more other elements, an in-depth study is conducted about the displacement field including rotational DOF in plane beam elements, space beam elements, plane isoparametric elements and space isoparametric elements. The rotational displacements are divided into two types of the point average rotation and the infinitesimal line’s rotation, and their calculation formulas are given. The analysis of rotational displacements provides a theoretical basis to connect beam elements’nodes which have rotational DOFs with other elements, particularly when nodes are embedded in parent elements. Since nodes of a beam element can connect to their parent elements, the reinforcement can be meshed by beam elements successfully.
4. The non-nodal connectivity finite element analysis method is established. An inner node’s DOFs are divided into two types of dependence and independence. Under the coordinates at an inner node, the inner node’s dependent DOFs are equal to the displacements accounted through its parent element, and the independent the others. A bond spring element between the inner node and its parent element is raised to simulate the bond-slip between the reinforcement and the matrix material. After the finite element formulas for multi types of elements are derived, the element assemble technique in the non-nodal connectivity state is worked out.
5. The problem of bandwidth optimization in the non-nodal connectivity state is investigated. The bandwidth optimization technique can greatly improve the calculation speed of a FEM software, so is an indispensable function of a practical FEM software. Regarding the complicated circumstances of DOFs in the non-nodal connectivity state, some improvements in algorithm of graph theory are made to decrease the bandwidth; the bandwidth optimization algorithm is implemented at the level of DOF. This method resolves not only the complex issue of bandwidth optimization in the non-nodal connectivity state, but also the problem induced by that the DOFs numbers of various types of node taking part in the structure’s whole DOFs are not equal.
6. Using the object-oriented technology, a FEM software named RCF is developed to realize the non-nodal connectivity method, which provides a 3D graphical user interface and programmed in visual C++ language under the Windows platform. The levels of FEM classes are constructed in RCF. Making use of RCF and its 10 types of elements, the reinforced structures under plane and space circumstances can be meshed and analyzed by means of non-nodal connectivity method.
7. A number of typical problems are calculated by RCF; their numerical results validate and demonstrate the correctness and feasibility of the software and the non-nodal connectivity finite element method.
The non-nodal connectivity finite element method breaks the traditional restrictions of node-node connecting and adds new content to existing finite element theory. The flexible application of connecting technology and element types provides more choices to mesh a structure and afford new models at element level for a variety of reinforced structures in civil engineering, and can reveal more accurately the mechanical behavior of a structure at overall and its partial key parts, and therefore improve the finite element analysis for reinforced structures. The method has broad application prospects in civil engineering.
引文
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