起重机变截面复杂梁杆系统稳定性与非线性大位移研究
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摘要
具有复杂梁杆系统的起重机械在重载吊装机械中极为典型，其结构高耸细长，承载形式复杂，对其进行准确的稳定性与几何非线性分析是保证其安全工作的重要基础。同时，为合理利用材料、减轻结构自重，起重机金属结构通常会存在变截面格构式构件及其组合结构，其特殊的边界条件和结构形式为起重机金属结构的理论分析和设计计算带来了很大困难，这一直是行业中众多技术人员关注的焦点。为此，本文在国家十二五科技支撑计划项目(2011BAJ02B01-02)资助下，以具有变截面格构式构件的起重机复杂梁杆系统为研究背景和应用对象，对起重机复杂梁杆系统的稳定性、几何非线性大位移与变截面结构的抗扭性能进行深入的研究和探讨。
基于纵横弯曲理论，对惯性矩沿轴向二次变化的变截面悬臂梁柱的侧向位移和稳定性进行了分析。运用微分方程法建立了考虑轴力影响的变截面Bernoulli-Euler梁的挠曲微分方程，推导了变截面悬臂梁柱在复合载荷作用下的挠度精确表达式和失稳特征方程，并在Timoshenko放大系数与精确理论放大系数基础上，给出了形式简单的变截面悬臂构件轴力影响系数的近似表达式。分析结果表明，在工程中常用锥度范围内，当轴力与欧拉临界力的比值小于0.6时，该近似表达式所引起的误差均小于2%，因此，惯性矩二次变化的变截面悬臂梁顶端挠度可以使用统一的近似放大系数公式乘以相应仅有横向载荷作用所产生的最大挠度表出。同时在上述研究的基础之上，讨论了计及二阶效应的情况下，弹性约束对变截面悬臂梁侧向位移与稳定性的影响，分别给出了轴力影响系数、弹性支撑影响系数和屈曲特征方程表达式。
在以上变截面梁挠曲微分方程基础上，引入剪切变形的影响，推导出计及二阶效应的惯性矩二次变化变截面Timoshenko梁单元转角位移方程，并列写为有限元格式，给出了计及剪切变形与二阶效应影响的惯性矩二次变化变截面梁单元精确切线刚度矩阵，得到一种新型的变截面梁单元。该精确梁单元能方便的实现变截面梁到等截面梁、计及剪切和不计及剪切变形之间的转化，与传统有限元方法中梁单元自然衔接过渡，便于统一建模求解应用。通过经典的算例分析表明，该精确变截面梁单元在稳定性和二阶效应分析中，只需划分一个单元即可以得到精确的数值解；对于细长梁，可以忽略剪切变形的影响，但当梁较短即杆件高度相对跨度很小，而又须考虑非线性效应时，必须计入剪切变形的影响才能达到满意的计算精度；剪切变形使结构的横向位移增大，使梁杆的屈曲能力减弱。
针对空间变截面梁杆结构抗扭刚度问题的研究，本文将空间桁架分解成平面桁架计算，根据空间桁架扭转时的等效力分配原则，将复杂的空间桁架扭转问题转化为简单变截面单片平面桁架结构的弯曲问题，最终推导出空间变截面结构扭转刚度的表达式。首先以单片变截面桁架结构为研究对象，给出了不同腹杆布置形式的变截面桁架各杆件内力与侧向位移表达式。以单片桁架结构的柔度系数为基础，全面考虑弦杆、腹杆对空间桁架结构的侧向刚度和抗扭刚度的影响，推导出适用于矩形空间桁架结构的侧向位移表达式和扭转刚度计算公式，并给出了扭矩的等效力偶分配系数计算式。最后讨论了腹杆对变截面空间梁杆结构侧向刚度的影响，当构件长细比较大时，可忽略腹杆对结构侧向刚度的影响，可将变截面空间刚架结构等效为惯性矩二次变化的实腹式变截面梁。实际算例表明，应用本文方法计算空间桁架结构的扭转刚度是正确和有效的。
基于虚位移原理与更新的拉格朗日(U.L.)格式，提出了一种可用于分析变截面Timoshenko梁单元几何非线性大位移问题的计算方法。采用转角、位移独立插值的方法，给出了考虑剪切变形影响的惯性矩二次变化变截面梁单元插值函数，并建立了同时考虑轴力、剪切、弯曲效应及其耦合项在内的平面惯性矩二次变化变截面梁柱单元几何非线性虚功增量方程，得到了变截面梁单元大位移切线刚度阵。该方法严格计入了剪切变形与变截面因素的影响，当锥度系数
1时，可很好的退化为相应的等截面梁单元，因此，本文分析方法适用于变截面梁、等截面梁以及其组合结构的大位移几何非线性分析。最后，编制了计及剪切变形的变截面梁杆系统大位移几何非线性分析计算程序，通过典型算例分析，验证了本文几何非线性大位移分析建模方法的正确性。
Hoisting machinery with complex truss structure is the typical heavy lifting machinery, which has slender structure and can bearing complex load form. The analysis of the stability and geometric nonlinear is important to ensure working safely. In order to use materials rationally and reduce the weight of structural, crane metal structures are usually fabricated as non-uniform cross-section lattice components or their composite structure. The special boundary conditions and structure types induce greater difficulties in the calculation of the theoretical analysis and design, which are the focus points of many industries and scholars. Therefore, by the supporting of the National Key Technology Research and Development Program (Grant No.2011BAJ02B01-02), this paper researches on stability, large displacement geometric nonlinear and torsional behavior with the background and application object of the tapered lattice crane’s beam-bar structures.
Based on the vertical and horizontal bending theory, the lateral displacement and stability of tapered cantilever are analyzed, whose inertia moment varies quadratic in axial. Using differential equation method, the deflection differential equation of non-uniform Bernoulli-Euler beam with axial force effects is established. The deflection exact expression and instability characteristic equation of the non-uniform cantilever are deduced under complex loads. Approximate formula of tapered cantilever member axial force influence coefficients is presented according to the Timoshenko factor and the exact factor. The results show that, while the taper range in commonly engineering, and the ratio of the axial force and Euler critical force is less than0.6, then the error of approximate expression is less than2%. Therefore, the tip deflection of the quadratic inertia moment variation cantilever can be approximated using a unified amplification factor formula multiplied by the maximum deflection caused by transverse load only. Meanwhile, based on the above research, axial force impact factor, impact factor of elastic support and buckling characteristic equation are expressed with taking into account the second-order effects and the effects on lateral displacement and stability of tapered cantilever by elastic constraints.
Based on the deflection differential equations of the non-uniform beam, considering the shear deformation effect and the second order effects, the angle displacement equation of tapered Timoshenko beam, whose moment of inertia changes quadratic, is deduced and written as finite element formulation. Precise element tangent stiffness matrix of tapered beam with secondary changes in moment of inertia, taking into account the effect of shear deformation, is obtained. The precise beam elements can realize the transformation between the tapered beam and the uniform beam, taking into account of shear deformation or not, and with good performance degradation. The result of classical example shows that the accurate numerical solution can be obtained by simply performing as one element in analysis of stability and second-order effects, using the precision tapered beam element; For slender beams, shear deformation effects can be ignored, but when the beam is relatively short, the height is small, and the nonlinear effects has to be considered, the shear deformation effect must be included for achieving satisfactory accuracy; Shear deformation increases the lateral displacement of the structure and diminished the buckling capacity of beams.
In order to study the torsional stiffness problem of space non-uniform beam-bar structure, space truss are decomposed into flat truss, and according to the equivalent force distribution principle of space truss in torsion, the complex spatial truss torsional problem is switched into simple single non-uniform cross-section truss’ bending problem in this paper, and then, the torsional stiffness expressions of the spatial tapered structure are deduced. First, taking the single-chip tapered truss structure as research object, the each member’s force and lateral displacement expressions of the tapered truss with different arrangement in the form of webs are given. Then, based on the flexibility coefficient of monolithic truss structure, considering comprehensively the effects of lateral stiffness and torsional stiffness caused by the chords, webs of space truss structure, the lateral displacement and torsional stiffness formulas are deduced for rectangular space truss structure, and the distribution coefficient formula of torque equivalent couple is also given. Final, the effect on lateral stiffness of tapered space beam caused by webs is addressed. When the member slenderness is relatively large, the effect of the lateral stiffness caused by webs can be ignored, and the non-uniform cross-section space frame structure can be equivalent to a solid-web tapered beam with moment of inertia quadratic variation. Result of examples show that the application of this method to calculate the torsional stiffness of space truss structure is absolutely correct and effective.
Based on the virtual displacement principle and the updated Lagrangian (U.L.) format, a calculation method of Timoshenko beam element to analyze the geometrically nonlinear large displacement is proposed. By using angular displacement independent interpolation method, element interpolation function of tapered beam is presented by considering the shear deformation effect. Element geometrical nonlinear incremental virtual work equation of tapered plane beam with moment of inertia secondary variation is established by taking into account the axial force, shear, bending effect and its coupling terms simultaneously. Large displacement element tangent stiffness matrix of tapered beam is obtained. The presented method strictly takes account into the effect caused by shear deformation and tapered cross-section factors. It can be well degenerated into a uniform section beam element while the taper coefficient is1. Therefore, the proposed analysis method is suitable for large displacement geometric nonlinear structural analysis of tapered beam, uniform beam and other composite forms. Finally, large displacement geometric nonlinear analysis and calculation procedures of tapered beam-bar system considering the shear deformation are programed, and the numerical results show that, the proposed geometric nonlinear large displacement analysis modeling method and programe are correct.
基于纵横弯曲理论，对惯性矩沿轴向二次变化的变截面悬臂梁柱的侧向位移和稳定性进行了分析。运用微分方程法建立了考虑轴力影响的变截面Bernoulli-Euler梁的挠曲微分方程，推导了变截面悬臂梁柱在复合载荷作用下的挠度精确表达式和失稳特征方程，并在Timoshenko放大系数与精确理论放大系数基础上，给出了形式简单的变截面悬臂构件轴力影响系数的近似表达式。分析结果表明，在工程中常用锥度范围内，当轴力与欧拉临界力的比值小于0.6时，该近似表达式所引起的误差均小于2%，因此，惯性矩二次变化的变截面悬臂梁顶端挠度可以使用统一的近似放大系数公式乘以相应仅有横向载荷作用所产生的最大挠度表出。同时在上述研究的基础之上，讨论了计及二阶效应的情况下，弹性约束对变截面悬臂梁侧向位移与稳定性的影响，分别给出了轴力影响系数、弹性支撑影响系数和屈曲特征方程表达式。
在以上变截面梁挠曲微分方程基础上，引入剪切变形的影响，推导出计及二阶效应的惯性矩二次变化变截面Timoshenko梁单元转角位移方程，并列写为有限元格式，给出了计及剪切变形与二阶效应影响的惯性矩二次变化变截面梁单元精确切线刚度矩阵，得到一种新型的变截面梁单元。该精确梁单元能方便的实现变截面梁到等截面梁、计及剪切和不计及剪切变形之间的转化，与传统有限元方法中梁单元自然衔接过渡，便于统一建模求解应用。通过经典的算例分析表明，该精确变截面梁单元在稳定性和二阶效应分析中，只需划分一个单元即可以得到精确的数值解；对于细长梁，可以忽略剪切变形的影响，但当梁较短即杆件高度相对跨度很小，而又须考虑非线性效应时，必须计入剪切变形的影响才能达到满意的计算精度；剪切变形使结构的横向位移增大，使梁杆的屈曲能力减弱。
针对空间变截面梁杆结构抗扭刚度问题的研究，本文将空间桁架分解成平面桁架计算，根据空间桁架扭转时的等效力分配原则，将复杂的空间桁架扭转问题转化为简单变截面单片平面桁架结构的弯曲问题，最终推导出空间变截面结构扭转刚度的表达式。首先以单片变截面桁架结构为研究对象，给出了不同腹杆布置形式的变截面桁架各杆件内力与侧向位移表达式。以单片桁架结构的柔度系数为基础，全面考虑弦杆、腹杆对空间桁架结构的侧向刚度和抗扭刚度的影响，推导出适用于矩形空间桁架结构的侧向位移表达式和扭转刚度计算公式，并给出了扭矩的等效力偶分配系数计算式。最后讨论了腹杆对变截面空间梁杆结构侧向刚度的影响，当构件长细比较大时，可忽略腹杆对结构侧向刚度的影响，可将变截面空间刚架结构等效为惯性矩二次变化的实腹式变截面梁。实际算例表明，应用本文方法计算空间桁架结构的扭转刚度是正确和有效的。
基于虚位移原理与更新的拉格朗日(U.L.)格式，提出了一种可用于分析变截面Timoshenko梁单元几何非线性大位移问题的计算方法。采用转角、位移独立插值的方法，给出了考虑剪切变形影响的惯性矩二次变化变截面梁单元插值函数，并建立了同时考虑轴力、剪切、弯曲效应及其耦合项在内的平面惯性矩二次变化变截面梁柱单元几何非线性虚功增量方程，得到了变截面梁单元大位移切线刚度阵。该方法严格计入了剪切变形与变截面因素的影响，当锥度系数
1时，可很好的退化为相应的等截面梁单元，因此，本文分析方法适用于变截面梁、等截面梁以及其组合结构的大位移几何非线性分析。最后，编制了计及剪切变形的变截面梁杆系统大位移几何非线性分析计算程序，通过典型算例分析，验证了本文几何非线性大位移分析建模方法的正确性。
Hoisting machinery with complex truss structure is the typical heavy lifting machinery, which has slender structure and can bearing complex load form. The analysis of the stability and geometric nonlinear is important to ensure working safely. In order to use materials rationally and reduce the weight of structural, crane metal structures are usually fabricated as non-uniform cross-section lattice components or their composite structure. The special boundary conditions and structure types induce greater difficulties in the calculation of the theoretical analysis and design, which are the focus points of many industries and scholars. Therefore, by the supporting of the National Key Technology Research and Development Program (Grant No.2011BAJ02B01-02), this paper researches on stability, large displacement geometric nonlinear and torsional behavior with the background and application object of the tapered lattice crane’s beam-bar structures.
Based on the vertical and horizontal bending theory, the lateral displacement and stability of tapered cantilever are analyzed, whose inertia moment varies quadratic in axial. Using differential equation method, the deflection differential equation of non-uniform Bernoulli-Euler beam with axial force effects is established. The deflection exact expression and instability characteristic equation of the non-uniform cantilever are deduced under complex loads. Approximate formula of tapered cantilever member axial force influence coefficients is presented according to the Timoshenko factor and the exact factor. The results show that, while the taper range in commonly engineering, and the ratio of the axial force and Euler critical force is less than0.6, then the error of approximate expression is less than2%. Therefore, the tip deflection of the quadratic inertia moment variation cantilever can be approximated using a unified amplification factor formula multiplied by the maximum deflection caused by transverse load only. Meanwhile, based on the above research, axial force impact factor, impact factor of elastic support and buckling characteristic equation are expressed with taking into account the second-order effects and the effects on lateral displacement and stability of tapered cantilever by elastic constraints.
Based on the deflection differential equations of the non-uniform beam, considering the shear deformation effect and the second order effects, the angle displacement equation of tapered Timoshenko beam, whose moment of inertia changes quadratic, is deduced and written as finite element formulation. Precise element tangent stiffness matrix of tapered beam with secondary changes in moment of inertia, taking into account the effect of shear deformation, is obtained. The precise beam elements can realize the transformation between the tapered beam and the uniform beam, taking into account of shear deformation or not, and with good performance degradation. The result of classical example shows that the accurate numerical solution can be obtained by simply performing as one element in analysis of stability and second-order effects, using the precision tapered beam element; For slender beams, shear deformation effects can be ignored, but when the beam is relatively short, the height is small, and the nonlinear effects has to be considered, the shear deformation effect must be included for achieving satisfactory accuracy; Shear deformation increases the lateral displacement of the structure and diminished the buckling capacity of beams.
In order to study the torsional stiffness problem of space non-uniform beam-bar structure, space truss are decomposed into flat truss, and according to the equivalent force distribution principle of space truss in torsion, the complex spatial truss torsional problem is switched into simple single non-uniform cross-section truss’ bending problem in this paper, and then, the torsional stiffness expressions of the spatial tapered structure are deduced. First, taking the single-chip tapered truss structure as research object, the each member’s force and lateral displacement expressions of the tapered truss with different arrangement in the form of webs are given. Then, based on the flexibility coefficient of monolithic truss structure, considering comprehensively the effects of lateral stiffness and torsional stiffness caused by the chords, webs of space truss structure, the lateral displacement and torsional stiffness formulas are deduced for rectangular space truss structure, and the distribution coefficient formula of torque equivalent couple is also given. Final, the effect on lateral stiffness of tapered space beam caused by webs is addressed. When the member slenderness is relatively large, the effect of the lateral stiffness caused by webs can be ignored, and the non-uniform cross-section space frame structure can be equivalent to a solid-web tapered beam with moment of inertia quadratic variation. Result of examples show that the application of this method to calculate the torsional stiffness of space truss structure is absolutely correct and effective.
Based on the virtual displacement principle and the updated Lagrangian (U.L.) format, a calculation method of Timoshenko beam element to analyze the geometrically nonlinear large displacement is proposed. By using angular displacement independent interpolation method, element interpolation function of tapered beam is presented by considering the shear deformation effect. Element geometrical nonlinear incremental virtual work equation of tapered plane beam with moment of inertia secondary variation is established by taking into account the axial force, shear, bending effect and its coupling terms simultaneously. Large displacement element tangent stiffness matrix of tapered beam is obtained. The presented method strictly takes account into the effect caused by shear deformation and tapered cross-section factors. It can be well degenerated into a uniform section beam element while the taper coefficient is1. Therefore, the proposed analysis method is suitable for large displacement geometric nonlinear structural analysis of tapered beam, uniform beam and other composite forms. Finally, large displacement geometric nonlinear analysis and calculation procedures of tapered beam-bar system considering the shear deformation are programed, and the numerical results show that, the proposed geometric nonlinear large displacement analysis modeling method and programe are correct.
引文
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