杆系结构几何非线性动静态分析方法及其在塔机中的应用
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摘要
以大型动臂式塔式起重机金属结构为代表的复杂杆系结构的应用日趋广泛,对其进行准确和高效的几何非线性动静态分析受到越来越多的关注。本文在国家十一五科技支撑计划项目(2006BAJ12B03)资助下,以D6560/50t大型建筑主体施工动臂变幅大吨位起重机为应用背景,对杆系结构的整体稳定性、几何非线性大位移分析、轴力对动态刚度的影响和动载荷作用下瞬态动力学分析的关键问题进行了深入的探讨和研究。
从变截面Bernoulli-Euler梁静态挠度微分方程出发,推导出转角位移方程,并列写为有限元格式,给出惯性矩二次变化和四次变化变截面梁单元精确刚度阵。应用该精确刚度阵分析其稳定性问题时只需将每个杆件划分为一个单元即可得到数值精确解。通过微分方程法得到了这两种变截面梁单元的静态精确形函数,从而可通过此精确形函数由经典有限元方法推导出相应变截面梁的精确刚度阵。建立了使用精确形函数表达变截面Bernoulli-Euler梁精确刚度阵的积分和微分格式,其中微分格式更为简洁和易用。将具有弹性支撑的非共线链式分支子结构当作一个超单元,使用传递矩阵法联系两端载荷和位移关系,提出了梁杆结构稳定性分析的传递矩阵模型缩减法,该方法在保证精度的同时使得系统刚度阵的阶数得到极大降低。
为了准确的分析变截面梁结构大位移、大转动小应变问题,提出了一种基于更新拉格朗日(UL)格式和随动坐标法的Bernoulli-Euler梁单元计算方法。考虑了弯曲变形引起轴向长度变化的非线性,分别由转角位移方程和精确形函数这两种方法推导了计入弓形效应的附加刚度,修正了变截面梁单元计及二阶效应的切线刚度阵。结合随动坐标法,在变形后位形上建立了简支梁式的单元随动坐标系,得到变截面梁单元的大位移全量平衡方程。使用Newton-Raphson法进行多载荷步数值迭代求解,迭代过程中不断修正由于轴力变化以及位形变化导致结构刚度的改变。
与动力刚度法(Dynamic Stiffness Method)推导等截面梁自由振动分析的动态刚度阵不同,本文首先获得承受常轴力的Bernoulli-Euler梁横向自由振动微分方程的通解,并通过位移边界条件消去待定常数,得到精确形函数;使用有限元方法,建立了使用精确形函数表达等截面Bernoulli-Euler梁动态刚度阵的微分格式,该微分格式精确刚度阵同样适用于等截面梁静态刚度阵。运用虚功原理完整地证明了该微分格式对于自由振动问题和静态问题的正确性和适用性。仿照静态挠度的Timoshenko放大系数,提出了Bernoulli-Euler梁横向振动固有频率的轴力影响系数近似公式,结合Wittrick-Williams算法和动态刚度阵证明了当轴力在±0.5倍第一阶欧拉临界力之间变化时,该近似公式最大误差不超过2%。
针对动态刚度阵不能分析杆系结构的瞬态动力学问题,通过梁单元横向和纵向自由振动的精确形函数推导了完整描述等截面Bernoulli-Euler梁的横向和纵向位移场,使用有限元方法分别推导了质量阵和刚度阵,质量阵和刚度阵各元素均为固有频率和轴力的超越函数。刚度阵考虑了二阶效应的影响;质量阵考虑了截面自身旋转惯性影响。建立了用于杆系结构瞬态动力学分析的动力平衡方程,并给出了稳定和高效的求解方案。
在上述理论研究中,均通过经典算例验证了方法的正确性和有效性。最后,以上述理论研究为基础,对D6560/50t塔机梭形变截面动臂以及整机进行了几何非线性动静态分析。由稳定性分析结果可知,随着幅度的增加,动臂整体欧拉临界力单调递减,而该动臂所能承受的极限起重力矩则不断增长,并在75.5123m幅度时达到峰值;随着吊臂幅度的增加,整机的稳定性安全系数是不断增加的。由整机的动刚度分析结果可知,其第一阶固有频率为塔身前后摆动引起吊臂的点头运动,当动臂俯仰角增大而幅度减小时,其第一阶固有频率不断减小,考虑和不考虑轴力引起的几何非线性导致固有频率的误差不断增大,当动臂俯仰角超过57°时,两者误差超过工程允许误差5%,此时必须考虑轴力对于固有频率的影响。由整机的大位移分析结果可知,当动臂俯仰角小于63°时,可以使用二阶效应分析代替大位移分析,此时最大相对误差小于工程允许误差5%;而当动臂俯仰角大于63°时,此时必须使用大位移分析才能得到准确的计算结果。由此,本文的理论研究为D6560/50t动臂式塔机的研发提供了有力的支持。
The complicated beam structures, which is represented by large luffing jib tower crane's metal skeletal structure, gets more and more abroad attention on its accurate and efficient geometric nonlinear static and dynamic analysis. Sponsored by the National Key Technology Research and Development Program (Grant No. 2006BAJ12B03) and the application background of D6560/50t—the large tonnage luffing jib tower crane for the main construction of large buildings , several theories and technologies for geometric nonlinear static and dynamic analysis of beam structures, such as global stability analysis, the large displacement analysis, free vibration analysis and transient dynamic analysis, are discussed in this dissertation.
From the governing differential equation of lateral deflection of the tapered Bernoulli-Euler beam, the slope-deflexion equations are derived and transformed into the finite element formulation. The exact stiffness matrix of the tapered beam are proposed whose inertia moment are quadratic and quartic, respectively. The proposed exact stiffness matrix will lead to the exact numerical solution by modelling each member by only one element in the buckling analysis. The exact static shape functions of the two tapered beam are presented, and can be used in developing the exact stiffness matrix through classic finite element method. The integral and differential formulation of exact stiffness matrix of the tapered beam are proposed expressed in the exact shape function, and the differential formulation is more concise and effective. The model reduction technique of transfer matrix in the stability analysis of beam strcutures is presented. The noncollinear branch chain substructure with elastic supports is modelled as a super element, and the force-displacement relation between both ends is developed by the transfer matrix method, so the order of the model is greatly reduced and guarantee the accuracy of the computation.
A Bernoulli-Euler beam mechanism for static analysis of large displacement, large rotation but small strain planar tapered beam structures is proposed using the Updated Lagrangian formulation and the moving coordinate method. The nonlinear effect of the bending distortions on the axial action is considered to manifest itself as an axial change in length. The aforementioned stiffness matrix with second-order effects is amended, by developing the auxiliary stiffness of bowing effect through the slope-deflexion equations and the exact shape fuctions, respectively. The moving coordinate method is employed for obtaining the large displacement total equilibrium equations, and the hinged-hinged moving coordinate system is constructed at the last updated configuration. The multipe load steps Newton-Raphson scheme is adopted for the solution of the nonlinear equations, and the global stiffness is modified due to the variation of axial load and configuration in each iteration.
Unlike the dynamic stiffness method(DSM) to develop the dynamic stiffness matrix for free vibration of the uniform beam, the exact solutions of the differential equation governing the lateral vibration of an axially loaded uniform beam are found, and then the dynamic exact shape function are obtained by eliminating the intergal constants through the displacement boudary conditions. The differential formulation of dynamic stiffness matrix of the beam are proposed expressed in the dynamic exact shape function, and the differential formulation can be used to obtain the static exact stiffness matrix if the the static exact shape function is introduced. The principle of virtual work is adopted to elaborate the validity of the generalized differential formulation. To follow the Timoshenko magnification factor of lateral deflection, the approximated formula to compute axially loaded influence factor of the natrual frequencies for lateral vibration of Bernoulli-Euler beam is proposed, and the Wittrick-Williams algorithm and the dynamic stiffness matrix are used to prove that the maximum relative error of the proposed approximate formula is smaller than 2%, when axial load is between the postive and negative half of the first order Euler critical load.
Due to the dynamic stiffness matrix cannot be used in transient dynamic analysis of beam structures, the lateral and axial displacement field are derived from the dynamic exact shape function of free lateral and axial vibration of the uniform beam, the mass matrix and stiffness matrix are developed by finite element method. Each element of the mass matrix and stiffness matrix is the transcendental function of the natrual frequencies and axial load. The second order effect is considered in the stiffness matrix while the self rotate inertia of the section is considered in the mass matrix. The dynamic equilibrium equations are presented for the transient dynamic analysis of beam structures, meanwhile the stable and efficient solution scheme is proposed to solve the equations.
The validity and efficiency of the proposed theories and technologies are shown by solving various numerical examples found in the literatures. Finally, based on the former theories and technologies, the geometric nonlinear static and dynamic analysis of the D6560/50t's shuttle-type tapered luffing jib and overall structrues are implemented. From the result of the stability analysis, as the lifting range increases, the global Euler critical load of the luffing jib is monotonously decreasing, while the extreme lifting moment of the jib is increasing, and reach the peak value at the lifting range 75.5123m. The buckling safety factor of the overall structures is increasing as the lifting range increases. From the result of the dynamic stiffness analysis of the overall structures, the first frequency is dive motion of the luffing jib induced by the swinging of the crane shaft. when the luffing angle inceases, that means lifting range decreases, the first frequency is monotonously decreasing, and the frequency error due to the second order effect is increasing. When the luffing angle exceeds 57degree, the frequency error exceeds the engineering allowable error 5%, so it should consider the impact on the frequencies due to the axial load. From the result of the large displacement analysis of the overall structures, if the luffing angle is less than 63degree, the large displacement analysis can be replaced by the second-order effect analysis, the maximum relative error is smaller than the engineering allowable error 5%, while the luffing angle exceeds 63degree, only the large displacement analysis lead to the more accurate result. So the proposed theories and technologies provide strong support for the research and development of the D6560/50t luffing jib tower crane.
从变截面Bernoulli-Euler梁静态挠度微分方程出发,推导出转角位移方程,并列写为有限元格式,给出惯性矩二次变化和四次变化变截面梁单元精确刚度阵。应用该精确刚度阵分析其稳定性问题时只需将每个杆件划分为一个单元即可得到数值精确解。通过微分方程法得到了这两种变截面梁单元的静态精确形函数,从而可通过此精确形函数由经典有限元方法推导出相应变截面梁的精确刚度阵。建立了使用精确形函数表达变截面Bernoulli-Euler梁精确刚度阵的积分和微分格式,其中微分格式更为简洁和易用。将具有弹性支撑的非共线链式分支子结构当作一个超单元,使用传递矩阵法联系两端载荷和位移关系,提出了梁杆结构稳定性分析的传递矩阵模型缩减法,该方法在保证精度的同时使得系统刚度阵的阶数得到极大降低。
为了准确的分析变截面梁结构大位移、大转动小应变问题,提出了一种基于更新拉格朗日(UL)格式和随动坐标法的Bernoulli-Euler梁单元计算方法。考虑了弯曲变形引起轴向长度变化的非线性,分别由转角位移方程和精确形函数这两种方法推导了计入弓形效应的附加刚度,修正了变截面梁单元计及二阶效应的切线刚度阵。结合随动坐标法,在变形后位形上建立了简支梁式的单元随动坐标系,得到变截面梁单元的大位移全量平衡方程。使用Newton-Raphson法进行多载荷步数值迭代求解,迭代过程中不断修正由于轴力变化以及位形变化导致结构刚度的改变。
与动力刚度法(Dynamic Stiffness Method)推导等截面梁自由振动分析的动态刚度阵不同,本文首先获得承受常轴力的Bernoulli-Euler梁横向自由振动微分方程的通解,并通过位移边界条件消去待定常数,得到精确形函数;使用有限元方法,建立了使用精确形函数表达等截面Bernoulli-Euler梁动态刚度阵的微分格式,该微分格式精确刚度阵同样适用于等截面梁静态刚度阵。运用虚功原理完整地证明了该微分格式对于自由振动问题和静态问题的正确性和适用性。仿照静态挠度的Timoshenko放大系数,提出了Bernoulli-Euler梁横向振动固有频率的轴力影响系数近似公式,结合Wittrick-Williams算法和动态刚度阵证明了当轴力在±0.5倍第一阶欧拉临界力之间变化时,该近似公式最大误差不超过2%。
针对动态刚度阵不能分析杆系结构的瞬态动力学问题,通过梁单元横向和纵向自由振动的精确形函数推导了完整描述等截面Bernoulli-Euler梁的横向和纵向位移场,使用有限元方法分别推导了质量阵和刚度阵,质量阵和刚度阵各元素均为固有频率和轴力的超越函数。刚度阵考虑了二阶效应的影响;质量阵考虑了截面自身旋转惯性影响。建立了用于杆系结构瞬态动力学分析的动力平衡方程,并给出了稳定和高效的求解方案。
在上述理论研究中,均通过经典算例验证了方法的正确性和有效性。最后,以上述理论研究为基础,对D6560/50t塔机梭形变截面动臂以及整机进行了几何非线性动静态分析。由稳定性分析结果可知,随着幅度的增加,动臂整体欧拉临界力单调递减,而该动臂所能承受的极限起重力矩则不断增长,并在75.5123m幅度时达到峰值;随着吊臂幅度的增加,整机的稳定性安全系数是不断增加的。由整机的动刚度分析结果可知,其第一阶固有频率为塔身前后摆动引起吊臂的点头运动,当动臂俯仰角增大而幅度减小时,其第一阶固有频率不断减小,考虑和不考虑轴力引起的几何非线性导致固有频率的误差不断增大,当动臂俯仰角超过57°时,两者误差超过工程允许误差5%,此时必须考虑轴力对于固有频率的影响。由整机的大位移分析结果可知,当动臂俯仰角小于63°时,可以使用二阶效应分析代替大位移分析,此时最大相对误差小于工程允许误差5%;而当动臂俯仰角大于63°时,此时必须使用大位移分析才能得到准确的计算结果。由此,本文的理论研究为D6560/50t动臂式塔机的研发提供了有力的支持。
The complicated beam structures, which is represented by large luffing jib tower crane's metal skeletal structure, gets more and more abroad attention on its accurate and efficient geometric nonlinear static and dynamic analysis. Sponsored by the National Key Technology Research and Development Program (Grant No. 2006BAJ12B03) and the application background of D6560/50t—the large tonnage luffing jib tower crane for the main construction of large buildings , several theories and technologies for geometric nonlinear static and dynamic analysis of beam structures, such as global stability analysis, the large displacement analysis, free vibration analysis and transient dynamic analysis, are discussed in this dissertation.
From the governing differential equation of lateral deflection of the tapered Bernoulli-Euler beam, the slope-deflexion equations are derived and transformed into the finite element formulation. The exact stiffness matrix of the tapered beam are proposed whose inertia moment are quadratic and quartic, respectively. The proposed exact stiffness matrix will lead to the exact numerical solution by modelling each member by only one element in the buckling analysis. The exact static shape functions of the two tapered beam are presented, and can be used in developing the exact stiffness matrix through classic finite element method. The integral and differential formulation of exact stiffness matrix of the tapered beam are proposed expressed in the exact shape function, and the differential formulation is more concise and effective. The model reduction technique of transfer matrix in the stability analysis of beam strcutures is presented. The noncollinear branch chain substructure with elastic supports is modelled as a super element, and the force-displacement relation between both ends is developed by the transfer matrix method, so the order of the model is greatly reduced and guarantee the accuracy of the computation.
A Bernoulli-Euler beam mechanism for static analysis of large displacement, large rotation but small strain planar tapered beam structures is proposed using the Updated Lagrangian formulation and the moving coordinate method. The nonlinear effect of the bending distortions on the axial action is considered to manifest itself as an axial change in length. The aforementioned stiffness matrix with second-order effects is amended, by developing the auxiliary stiffness of bowing effect through the slope-deflexion equations and the exact shape fuctions, respectively. The moving coordinate method is employed for obtaining the large displacement total equilibrium equations, and the hinged-hinged moving coordinate system is constructed at the last updated configuration. The multipe load steps Newton-Raphson scheme is adopted for the solution of the nonlinear equations, and the global stiffness is modified due to the variation of axial load and configuration in each iteration.
Unlike the dynamic stiffness method(DSM) to develop the dynamic stiffness matrix for free vibration of the uniform beam, the exact solutions of the differential equation governing the lateral vibration of an axially loaded uniform beam are found, and then the dynamic exact shape function are obtained by eliminating the intergal constants through the displacement boudary conditions. The differential formulation of dynamic stiffness matrix of the beam are proposed expressed in the dynamic exact shape function, and the differential formulation can be used to obtain the static exact stiffness matrix if the the static exact shape function is introduced. The principle of virtual work is adopted to elaborate the validity of the generalized differential formulation. To follow the Timoshenko magnification factor of lateral deflection, the approximated formula to compute axially loaded influence factor of the natrual frequencies for lateral vibration of Bernoulli-Euler beam is proposed, and the Wittrick-Williams algorithm and the dynamic stiffness matrix are used to prove that the maximum relative error of the proposed approximate formula is smaller than 2%, when axial load is between the postive and negative half of the first order Euler critical load.
Due to the dynamic stiffness matrix cannot be used in transient dynamic analysis of beam structures, the lateral and axial displacement field are derived from the dynamic exact shape function of free lateral and axial vibration of the uniform beam, the mass matrix and stiffness matrix are developed by finite element method. Each element of the mass matrix and stiffness matrix is the transcendental function of the natrual frequencies and axial load. The second order effect is considered in the stiffness matrix while the self rotate inertia of the section is considered in the mass matrix. The dynamic equilibrium equations are presented for the transient dynamic analysis of beam structures, meanwhile the stable and efficient solution scheme is proposed to solve the equations.
The validity and efficiency of the proposed theories and technologies are shown by solving various numerical examples found in the literatures. Finally, based on the former theories and technologies, the geometric nonlinear static and dynamic analysis of the D6560/50t's shuttle-type tapered luffing jib and overall structrues are implemented. From the result of the stability analysis, as the lifting range increases, the global Euler critical load of the luffing jib is monotonously decreasing, while the extreme lifting moment of the jib is increasing, and reach the peak value at the lifting range 75.5123m. The buckling safety factor of the overall structures is increasing as the lifting range increases. From the result of the dynamic stiffness analysis of the overall structures, the first frequency is dive motion of the luffing jib induced by the swinging of the crane shaft. when the luffing angle inceases, that means lifting range decreases, the first frequency is monotonously decreasing, and the frequency error due to the second order effect is increasing. When the luffing angle exceeds 57degree, the frequency error exceeds the engineering allowable error 5%, so it should consider the impact on the frequencies due to the axial load. From the result of the large displacement analysis of the overall structures, if the luffing angle is less than 63degree, the large displacement analysis can be replaced by the second-order effect analysis, the maximum relative error is smaller than the engineering allowable error 5%, while the luffing angle exceeds 63degree, only the large displacement analysis lead to the more accurate result. So the proposed theories and technologies provide strong support for the research and development of the D6560/50t luffing jib tower crane.
引文
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