变截面细长压杆失稳基本理论的研究
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摘要
变截面细长压杆由于其节省材料且具有良好的受力特性,故在起重机械、桥梁结构、飞机结构中被大量使用,但是与之相关的稳定性分析理论极不完善,这主要是工程中变截面失稳问题所求得的微分方程不是常系数,求解时往往要遇到数学上的困难。常用变截面压杆稳定计算的方法有:静力法、能量法、半解析求解法、有限单元法等。文中基于小挠度理论简要分析了变截面压杆失稳问题,展开了对变截面压杆构件失稳承载力的初步研究。
本文主要的工作是:(1)分析了两种常见的变截面情况,沿杆轴线线性变化的变截面压杆在两种不同支承条件(两端铰支、一端固定一端弹性支承)的失稳及沿杆轴线阶梯形变截面压杆(两端铰支)失稳问题进行了理论分析及数值求解。文中采用静力法,根据临界状态平衡的二重性,建立了平衡微分方程,将二阶偏微分方程化为贝塞尔函数进行理论分析,采用Matlab图形法及C++编程求得几种常见截面形式临界荷载。(2)采用能量法分析了变截面压杆失稳。工程中大部分变截面压杆失稳问题的求解在数学上存在困难或过于复杂,一般采用近似方法求解。求解变截面压杆的稳定问题中对于简单的构件采用能量法较为实用且具有较好的精度,但对变截面段数较多的构件所涉及的工作量较大,文中采用能量法分析了两端铰支变截面压杆失稳问题,并与前节静力分析计算结果进行了对比,结果表明所得的临界荷载与静力分析求得的临界荷载吻合较好。(3)简要分析了变截面压杆另一种数值计算方法:有限单元法。有限单元法具有对复杂几何构形的适应性,对于各种物理问题的可应用性,建立于严格理论基础上的可靠性,适合计算机实现的高效性等。随着有限元单元法通过计算机的实现,其具有效率高,更适合工程结构的分析,在文中采用了三维退化梁单元几何非线性有限元方程,对几何非线性方程进行适当改造,得到了变截面压杆的刚度矩阵,很适合于工程结构的总体分析。(4)采用ANSYS有限元分析软件进行特征值屈曲分析。文中分析了两端铰支的变截面压杆失稳构件的屈曲变形模态和稳定承载力,并将所得的结果与静力理论分析值进行了比较,验正了静力分析结果。
The members of non-uniform cross-section are commonly used as column in the design of various structures such as building frames, cranes, aircraft manufacturing, bridge structure, masts etc for its favorable capacity of carrying compressive loading. But the theory of elastic stability for arbitrary variable cross is not perfect. Since the cross-sectional properties of the column vary along its axis, the coefficients of the governing differential equation are variable and the solutions of differential equations are difficult in practical projects. Historically, solutions of elastic stability have developed many methods, such as theoretical analysis method; power series solutions; semi-analytic method; the finite element method etc. The main purpose of this paper is to investigate buckling of the slender bars which have involved the main methods of non-uniform cross-section in the theory of small deflection.
In this paper, there are four aspects I have been done. Firstly, I introduce the buckling of the two kinds of common variable cross-section along the axis of the compressive rods, linear (the two ends hinged, the one end fixed and another end elastic bearing) and stepped (the two ends hinged) with the regular cross-sectional shapes considered. In theoretical analysis method, the differential equations are obtained by dual property of critical conditions and solved by Bessel function. Secondly, the other method of solving equations is introduced. Power series solutions have two methods which are Timoshenko power solution and Riz power solution. The differential equations in the practical project are implicated to solve in most condition and the approximate methods are applied in common condition. The power series solution is useful, but the workloads are very huge, especially when the numbers of stage of variable cross-section are increased. As an illustration, the non-uniform cross-section of the two ends hinged constrained is analyzed. Thirdly, the other method of analysis which is the finite element method is explained in brief. The finite element method based on rigorous mathematic theory is extensive adapted to the complicated geometric configurations. It is also high efficiency, especially development of the computer technology. In this paper, the finite element equation of geometric non-linearity is obtained. It is modified to obtain the non-uniform cross-section stiffness matrices and is the very useful in project analysis.
In the end, the common software of ANSYS is introduced to solve the eigenvalue problem of variable cross-section. According to the needing, the two ends hinged is analyzed by ANSYS for verification purposes only and both analytical and numerical results correlate with reasonable accuracy. The results are based on the assumption that material failure occurs for small lateral displacements.
本文主要的工作是:(1)分析了两种常见的变截面情况,沿杆轴线线性变化的变截面压杆在两种不同支承条件(两端铰支、一端固定一端弹性支承)的失稳及沿杆轴线阶梯形变截面压杆(两端铰支)失稳问题进行了理论分析及数值求解。文中采用静力法,根据临界状态平衡的二重性,建立了平衡微分方程,将二阶偏微分方程化为贝塞尔函数进行理论分析,采用Matlab图形法及C++编程求得几种常见截面形式临界荷载。(2)采用能量法分析了变截面压杆失稳。工程中大部分变截面压杆失稳问题的求解在数学上存在困难或过于复杂,一般采用近似方法求解。求解变截面压杆的稳定问题中对于简单的构件采用能量法较为实用且具有较好的精度,但对变截面段数较多的构件所涉及的工作量较大,文中采用能量法分析了两端铰支变截面压杆失稳问题,并与前节静力分析计算结果进行了对比,结果表明所得的临界荷载与静力分析求得的临界荷载吻合较好。(3)简要分析了变截面压杆另一种数值计算方法:有限单元法。有限单元法具有对复杂几何构形的适应性,对于各种物理问题的可应用性,建立于严格理论基础上的可靠性,适合计算机实现的高效性等。随着有限元单元法通过计算机的实现,其具有效率高,更适合工程结构的分析,在文中采用了三维退化梁单元几何非线性有限元方程,对几何非线性方程进行适当改造,得到了变截面压杆的刚度矩阵,很适合于工程结构的总体分析。(4)采用ANSYS有限元分析软件进行特征值屈曲分析。文中分析了两端铰支的变截面压杆失稳构件的屈曲变形模态和稳定承载力,并将所得的结果与静力理论分析值进行了比较,验正了静力分析结果。
The members of non-uniform cross-section are commonly used as column in the design of various structures such as building frames, cranes, aircraft manufacturing, bridge structure, masts etc for its favorable capacity of carrying compressive loading. But the theory of elastic stability for arbitrary variable cross is not perfect. Since the cross-sectional properties of the column vary along its axis, the coefficients of the governing differential equation are variable and the solutions of differential equations are difficult in practical projects. Historically, solutions of elastic stability have developed many methods, such as theoretical analysis method; power series solutions; semi-analytic method; the finite element method etc. The main purpose of this paper is to investigate buckling of the slender bars which have involved the main methods of non-uniform cross-section in the theory of small deflection.
In this paper, there are four aspects I have been done. Firstly, I introduce the buckling of the two kinds of common variable cross-section along the axis of the compressive rods, linear (the two ends hinged, the one end fixed and another end elastic bearing) and stepped (the two ends hinged) with the regular cross-sectional shapes considered. In theoretical analysis method, the differential equations are obtained by dual property of critical conditions and solved by Bessel function. Secondly, the other method of solving equations is introduced. Power series solutions have two methods which are Timoshenko power solution and Riz power solution. The differential equations in the practical project are implicated to solve in most condition and the approximate methods are applied in common condition. The power series solution is useful, but the workloads are very huge, especially when the numbers of stage of variable cross-section are increased. As an illustration, the non-uniform cross-section of the two ends hinged constrained is analyzed. Thirdly, the other method of analysis which is the finite element method is explained in brief. The finite element method based on rigorous mathematic theory is extensive adapted to the complicated geometric configurations. It is also high efficiency, especially development of the computer technology. In this paper, the finite element equation of geometric non-linearity is obtained. It is modified to obtain the non-uniform cross-section stiffness matrices and is the very useful in project analysis.
In the end, the common software of ANSYS is introduced to solve the eigenvalue problem of variable cross-section. According to the needing, the two ends hinged is analyzed by ANSYS for verification purposes only and both analytical and numerical results correlate with reasonable accuracy. The results are based on the assumption that material failure occurs for small lateral displacements.
引文
[1]郭英涛,任文敏.关于限失稳的研究进展[J]. 2004, 34(1):41-53.
[2]刘光栋,罗汉泉.杆系结构稳定[M].人民交通出版社,1988,9第1版.
[3]黎绍敏.稳定理论[M].人民交通出版社.
[4]何西泠,余国城.当量长度确定变截面压杆临界弯曲载荷[J].起重运输机械,1994(4):12-13.
[5]苏惠芹. H型截面轴心受压钢构件限制失稳的基本理论研究[D].硕士论文,2006,3.
[6] S.P.铁摩辛柯,J.M.盖莱.弹性稳定理论(第二版) [M].科学出版社,1965:120-123.
[7] Tompson,J.M.T.and Hunt,G.W. A general theory of elastic stability [M]. 1973.
[8]刘鸿文.高等材料力学[M].高等教育出版社,1985第1版.
[9]赵毅强.楔形杆件结构的弹性稳定分析[J].建筑结构学报,1990,11(6):58-68.
[10]吴亚平.变截面压杆稳定计算的等效刚度法[J].力学与实践,1994,01:58-60.
[11]刘庆潭.含锥形变截面压杆稳定计算的传递矩阵法[J].计算结构力学及应用,1996,13(3).
[12]戴保东,郑荣等.变截面压杆稳定计算的有限元法[J].山西机械,2000,9(3):16-18.
[13] Lee, S.J. Ohb,1. Elastica and buckling load of simple tapered columns with constant volume[J]. International Journal of Solids and Structures,37(2000):2507-2518.
[14]卞敬玲,王小岗.变截面压杆稳定计算的有限单元法[J].武汉大学学报,2002,35:102-104.
[15]楼梦麟,李建元.变截面压杆稳定问题半解析解[J].同济大学学报(自然科学版),2007,7(32):857-860.
[16] Ioannis G. Raftoyiannis, John Ch. Ermopoulos. Stability of tapered and stepped steel columns with initial imperfections[J]. Engineering Structures,27(2005) :1248 -1257.
[17]严镇军.数学物理方程[M].中国科学技术大学出版社,1999,1.
[18]王勖成.有限单元法[M].清华大学出版社,2006,6.
[19]郝文化,叶裕明等. ANSYS土木工程应用实例[M].中国水利水电出版社,2005,1.
[20]王新敏. ANSYS工程结构数值分析[M].人民交通出版社,2007,10第1版.
[21]刘文顺.弹性介质上变截面压杆的稳定分析[J].焦作大学学报,2007,1(1):91
[22]吴光宇,干涌等.三维退化梁单元在变截面梁分析中的应用[J].中国市政工程,2006,1: 78-94.
[23]张来仪,孙贤.结构力学[M].重庆大学出版社,1998,7第1版
[24] E.J.Sapountzakis, G.C.Tsiatas. Elastic flexural buckling analysis of composite beams of variable cross-section by BEM[J]. Engineering Structures ,29 (2007):675-681.
[25] Ermopoulos JC. Buckling of tapered bars under stepped axial load[J]. Journal of Structural Engineering, ASCE 1985;112:1346-53.
[26] Eisenberger M. Buckling load for variable cross-section members with variable axial forces[J]. International Journal of Solids and Structures ,1991;27:135-43.
[27] Arbabi F, Li F. Buckling of variable cross-section column: Integral equation approach. Journal of Structural Engineering[J]. ASCE 1991;117:2426-41.
[28] Katsikadelis JT.,Tsiatas GC. Buckling load optimization of beams Archive of Applied Mechanics[M]. 2005;74:790-9.
[29]曾庆元.结构稳定理论[M].长沙铁道学院,1980.
[30]夏志斌,潘有昌.杆系结构稳定理论[M].高等教育出版社出版, 1985.
[31]周承倜.弹性稳定理论[M].四川人民出版社,1981.
[32] Thompson, J.M.T. and Hunt. G.W. A general theory of elastic stability [M]. 1973.
[33] A.查杰斯著,唐家祥译.结构稳定理论原理[M].甘肃人民出版社,1982.
[34] Karam Y. Maalawi. Buckling optimization of flexible columns[J]. International Journal of Solids and Structures ,39(2002):5865-5876.
[35]邹家兴.用等效刚度法计算变截面梁的变形[J].农田水利与水电,1995,(9):36-38.
[36]龙驭球,包世华主编.结构力学(下册) [M].人民教育出版社,1981:262-268.
[37]刘古岷,张若唏,张田申编.应用结构稳定计算[M].科学出版社,2007, 7第一版.
[38]刘古岷.关于阶梯柱状构件稳定的近似计算[J].建筑机械,1994,12.
[39] F.柏拉希.金属结构的屈曲强度[M].科学出版社,1965.
[2]刘光栋,罗汉泉.杆系结构稳定[M].人民交通出版社,1988,9第1版.
[3]黎绍敏.稳定理论[M].人民交通出版社.
[4]何西泠,余国城.当量长度确定变截面压杆临界弯曲载荷[J].起重运输机械,1994(4):12-13.
[5]苏惠芹. H型截面轴心受压钢构件限制失稳的基本理论研究[D].硕士论文,2006,3.
[6] S.P.铁摩辛柯,J.M.盖莱.弹性稳定理论(第二版) [M].科学出版社,1965:120-123.
[7] Tompson,J.M.T.and Hunt,G.W. A general theory of elastic stability [M]. 1973.
[8]刘鸿文.高等材料力学[M].高等教育出版社,1985第1版.
[9]赵毅强.楔形杆件结构的弹性稳定分析[J].建筑结构学报,1990,11(6):58-68.
[10]吴亚平.变截面压杆稳定计算的等效刚度法[J].力学与实践,1994,01:58-60.
[11]刘庆潭.含锥形变截面压杆稳定计算的传递矩阵法[J].计算结构力学及应用,1996,13(3).
[12]戴保东,郑荣等.变截面压杆稳定计算的有限元法[J].山西机械,2000,9(3):16-18.
[13] Lee, S.J. Ohb,1. Elastica and buckling load of simple tapered columns with constant volume[J]. International Journal of Solids and Structures,37(2000):2507-2518.
[14]卞敬玲,王小岗.变截面压杆稳定计算的有限单元法[J].武汉大学学报,2002,35:102-104.
[15]楼梦麟,李建元.变截面压杆稳定问题半解析解[J].同济大学学报(自然科学版),2007,7(32):857-860.
[16] Ioannis G. Raftoyiannis, John Ch. Ermopoulos. Stability of tapered and stepped steel columns with initial imperfections[J]. Engineering Structures,27(2005) :1248 -1257.
[17]严镇军.数学物理方程[M].中国科学技术大学出版社,1999,1.
[18]王勖成.有限单元法[M].清华大学出版社,2006,6.
[19]郝文化,叶裕明等. ANSYS土木工程应用实例[M].中国水利水电出版社,2005,1.
[20]王新敏. ANSYS工程结构数值分析[M].人民交通出版社,2007,10第1版.
[21]刘文顺.弹性介质上变截面压杆的稳定分析[J].焦作大学学报,2007,1(1):91
[22]吴光宇,干涌等.三维退化梁单元在变截面梁分析中的应用[J].中国市政工程,2006,1: 78-94.
[23]张来仪,孙贤.结构力学[M].重庆大学出版社,1998,7第1版
[24] E.J.Sapountzakis, G.C.Tsiatas. Elastic flexural buckling analysis of composite beams of variable cross-section by BEM[J]. Engineering Structures ,29 (2007):675-681.
[25] Ermopoulos JC. Buckling of tapered bars under stepped axial load[J]. Journal of Structural Engineering, ASCE 1985;112:1346-53.
[26] Eisenberger M. Buckling load for variable cross-section members with variable axial forces[J]. International Journal of Solids and Structures ,1991;27:135-43.
[27] Arbabi F, Li F. Buckling of variable cross-section column: Integral equation approach. Journal of Structural Engineering[J]. ASCE 1991;117:2426-41.
[28] Katsikadelis JT.,Tsiatas GC. Buckling load optimization of beams Archive of Applied Mechanics[M]. 2005;74:790-9.
[29]曾庆元.结构稳定理论[M].长沙铁道学院,1980.
[30]夏志斌,潘有昌.杆系结构稳定理论[M].高等教育出版社出版, 1985.
[31]周承倜.弹性稳定理论[M].四川人民出版社,1981.
[32] Thompson, J.M.T. and Hunt. G.W. A general theory of elastic stability [M]. 1973.
[33] A.查杰斯著,唐家祥译.结构稳定理论原理[M].甘肃人民出版社,1982.
[34] Karam Y. Maalawi. Buckling optimization of flexible columns[J]. International Journal of Solids and Structures ,39(2002):5865-5876.
[35]邹家兴.用等效刚度法计算变截面梁的变形[J].农田水利与水电,1995,(9):36-38.
[36]龙驭球,包世华主编.结构力学(下册) [M].人民教育出版社,1981:262-268.
[37]刘古岷,张若唏,张田申编.应用结构稳定计算[M].科学出版社,2007, 7第一版.
[38]刘古岷.关于阶梯柱状构件稳定的近似计算[J].建筑机械,1994,12.
[39] F.柏拉希.金属结构的屈曲强度[M].科学出版社,1965.