具有油缸支承的起重机箱形伸缩臂的稳定性研究
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摘要
目前对起重机伸缩臂稳定性校核主要还局限于中国起重机设计规范《GB3811/83》。其伸缩臂的计算长度是在未考虑油缸影响下采用能量方法推导而得,未免不够精确;而最近重新修订的新规范的计算模型采用的是变截面阶梯柱,在理论上更精确一些,但也没有考虑油缸的影响。含有各种形式油缸支承的伸缩臂的稳定性计算校核一直是困扰设计者的难题,吸引了众多专家学者做了大量研究,各自取得一些进展,但至今仍然没有给出完整的解决方案。
起重机伸缩式吊臂通常由多节变截面的箱形臂组成,各臂节间轴向可相对滑动,在忽略摩擦力的情况下,油缸的作用力是非保向力,箱形伸缩臂的部分或全部臂节不直接承受轴力,其轴力由布置于吊臂内部的油缸承受。此时,吊臂的轴压临界力计算与简单的受压变截面阶梯柱的计算不同,所得临界力亦有所区别。考虑油缸时,伸缩臂的临界力取决于吊臂的截面惯性矩和油缸的支承形式,油缸的支承形式不同,其临界力也不同,本文将着重讨论此问题。
本文在考虑了油缸的支承作用下,采用微分方程法和精确有限元法两种方法求起重机伸缩臂在起升平面外的欧拉临界力的精确解。微分方程法:适用于臂节数较少时(2～3节),不需要有限元知识,为设计人员提供便捷的计算公式,求解精确,但有局限性;精确有限元方法:适合于臂节数较多的情况,此时,失稳特征方程的求解较繁锁,又没有特殊规律可寻,不易采用手工计算,该方法为设计人员提供相应的计算理论,但需要有计算机软件与硬件的支持。
本文给出单个油缸支承在顶部时伸缩臂欧拉临界力精确解的递推公式;通过比较用微分方程法、弹性支座法、有限元法三种方法确定伸缩臂的临界力;并求出了伸缩臂在不同支承情况下的失稳特征方程;同时给出了伸缩臂的变截面系数;详细的阐述了采用精确有限元法求解伸缩臂的临界力。
对应《GB3811/83》规范的有关计算参数和图表,用本研究成果给出更为精确的计算参数,进而得出油缸对箱形伸缩臂整体稳定性的影响,在理论上更为严密。并将之与没考虑油缸支承作用的不同计算模型进行比较,分析结果表明,考虑油缸时伸缩臂的临界力大于不考虑油缸时伸缩臂的临界力。以往中国起重机设计规范《GB3811/83》和新规范中用忽略油缸支承作用的变截面阶梯柱力学模型计算起重机多节伸缩臂之稳定性,总体上是偏保守的。本文提出了考虑油缸之后的精确数据,相信这部分数据将为修改完善《起重机设计规范》提供重要参考依据。希望本研究所做的这些工作能够在这一领域起到某些补充以及能够对理论发展起到一定的推动作用。
At present, stability analysis for telescopic boom still limitation to the design rules for cranes《GB3811/83》national standards of People’s Republic of China.
In the original design rules for cranes, effective length of telescopic boom is confirmed by energy method without considering effect of cylinder, which is not precise; In the new revised design rules for cranes, considering the model as tapered stepped column, the result is more precise in theory,and still not considering effect of cylinder. Stability analysis and checking for telescopic boom with cylinder is still puzzling the designer, though many expert and scholar have done much research, till now, there is no accurate conclusion.
Crane telescopic boom is made up of many tapered box sections which can axially slip relatively, only the boom is subject to bending moment, while supporting cylinder in boom bears axial force, which offers a non-conservative force, and the friction between booms is not taken account, therefore, its mechanical model cannot be equivalent to tapered stepped column’s, and Euler’s critical load is different. In the precondition of stability for cylinder, loading capability of telescopic boom depends on section inertia moment of boom and supporting form. As different placement and supporting form of cylinder for various telescopic booms, effective length coefficients are different too. The capability is irrespective with inertia moment of cylinder. The paper will emphasis this problem.
The two method of differential equation and precise theoretical solution of Euler’s critical load at out-of-lifting-plane for crane multi-telescopic-boom with cylinder supporting is provided in this paper. Differential equation applicable for less pitch number(two-three),that doesn’t need finite element knowledge, provide a portable formula; precise finite element method applicable for multi-telescopic-boom, because it’s trouble to calculation and have no regularity, so this method provide the designer corresponding theory, but that need software and hardware to support.
In the precondition of stability for cylinder in the paper, deduced a recurrence formula with single cylinder supporting at the top; and the critical force of telescopic boom is calculated with three different methods, including differential equation method, elastic support method and finite element method; getting destabilizing characteristic equation with different supporting type; meanwhile, solved variable cross-section length coefficient; then detailed represent using precise finite element method to solve critical force of telescopic boom.
Corresponding to parameter and graph table about the rules for cranes GB3811-83,the paper give more accurate parameter, and then get the effect of cylinders to global stability, which is more accurate in theory. The result is compared with that of methods which did not take account of the effect of cylinders using the model of tapered stepped column. Euler’s critical load of the model considering the effect of supporting cylinder is greater than those neglecting it. In the previous design rules for cranes national standards of People’s Republic of China, the mechanical model to calculate crane multi-telescopic-boom’s stability is tapered stepped column neglecting the effect of supporting cylinder, thus the result generally incline to safety. The paper provide accurate data, and believe this can provide reference to the rules for cranes perfectly. The author really hope this can complement and motivation the theory developing in this realm.
起重机伸缩式吊臂通常由多节变截面的箱形臂组成,各臂节间轴向可相对滑动,在忽略摩擦力的情况下,油缸的作用力是非保向力,箱形伸缩臂的部分或全部臂节不直接承受轴力,其轴力由布置于吊臂内部的油缸承受。此时,吊臂的轴压临界力计算与简单的受压变截面阶梯柱的计算不同,所得临界力亦有所区别。考虑油缸时,伸缩臂的临界力取决于吊臂的截面惯性矩和油缸的支承形式,油缸的支承形式不同,其临界力也不同,本文将着重讨论此问题。
本文在考虑了油缸的支承作用下,采用微分方程法和精确有限元法两种方法求起重机伸缩臂在起升平面外的欧拉临界力的精确解。微分方程法:适用于臂节数较少时(2～3节),不需要有限元知识,为设计人员提供便捷的计算公式,求解精确,但有局限性;精确有限元方法:适合于臂节数较多的情况,此时,失稳特征方程的求解较繁锁,又没有特殊规律可寻,不易采用手工计算,该方法为设计人员提供相应的计算理论,但需要有计算机软件与硬件的支持。
本文给出单个油缸支承在顶部时伸缩臂欧拉临界力精确解的递推公式;通过比较用微分方程法、弹性支座法、有限元法三种方法确定伸缩臂的临界力;并求出了伸缩臂在不同支承情况下的失稳特征方程;同时给出了伸缩臂的变截面系数;详细的阐述了采用精确有限元法求解伸缩臂的临界力。
对应《GB3811/83》规范的有关计算参数和图表,用本研究成果给出更为精确的计算参数,进而得出油缸对箱形伸缩臂整体稳定性的影响,在理论上更为严密。并将之与没考虑油缸支承作用的不同计算模型进行比较,分析结果表明,考虑油缸时伸缩臂的临界力大于不考虑油缸时伸缩臂的临界力。以往中国起重机设计规范《GB3811/83》和新规范中用忽略油缸支承作用的变截面阶梯柱力学模型计算起重机多节伸缩臂之稳定性,总体上是偏保守的。本文提出了考虑油缸之后的精确数据,相信这部分数据将为修改完善《起重机设计规范》提供重要参考依据。希望本研究所做的这些工作能够在这一领域起到某些补充以及能够对理论发展起到一定的推动作用。
At present, stability analysis for telescopic boom still limitation to the design rules for cranes《GB3811/83》national standards of People’s Republic of China.
In the original design rules for cranes, effective length of telescopic boom is confirmed by energy method without considering effect of cylinder, which is not precise; In the new revised design rules for cranes, considering the model as tapered stepped column, the result is more precise in theory,and still not considering effect of cylinder. Stability analysis and checking for telescopic boom with cylinder is still puzzling the designer, though many expert and scholar have done much research, till now, there is no accurate conclusion.
Crane telescopic boom is made up of many tapered box sections which can axially slip relatively, only the boom is subject to bending moment, while supporting cylinder in boom bears axial force, which offers a non-conservative force, and the friction between booms is not taken account, therefore, its mechanical model cannot be equivalent to tapered stepped column’s, and Euler’s critical load is different. In the precondition of stability for cylinder, loading capability of telescopic boom depends on section inertia moment of boom and supporting form. As different placement and supporting form of cylinder for various telescopic booms, effective length coefficients are different too. The capability is irrespective with inertia moment of cylinder. The paper will emphasis this problem.
The two method of differential equation and precise theoretical solution of Euler’s critical load at out-of-lifting-plane for crane multi-telescopic-boom with cylinder supporting is provided in this paper. Differential equation applicable for less pitch number(two-three),that doesn’t need finite element knowledge, provide a portable formula; precise finite element method applicable for multi-telescopic-boom, because it’s trouble to calculation and have no regularity, so this method provide the designer corresponding theory, but that need software and hardware to support.
In the precondition of stability for cylinder in the paper, deduced a recurrence formula with single cylinder supporting at the top; and the critical force of telescopic boom is calculated with three different methods, including differential equation method, elastic support method and finite element method; getting destabilizing characteristic equation with different supporting type; meanwhile, solved variable cross-section length coefficient; then detailed represent using precise finite element method to solve critical force of telescopic boom.
Corresponding to parameter and graph table about the rules for cranes GB3811-83,the paper give more accurate parameter, and then get the effect of cylinders to global stability, which is more accurate in theory. The result is compared with that of methods which did not take account of the effect of cylinders using the model of tapered stepped column. Euler’s critical load of the model considering the effect of supporting cylinder is greater than those neglecting it. In the previous design rules for cranes national standards of People’s Republic of China, the mechanical model to calculate crane multi-telescopic-boom’s stability is tapered stepped column neglecting the effect of supporting cylinder, thus the result generally incline to safety. The paper provide accurate data, and believe this can provide reference to the rules for cranes perfectly. The author really hope this can complement and motivation the theory developing in this realm.
引文
1张魁元,王雪.汽车起重机伸缩臂的维护和护理.维修工艺. 2004,(11):28~30
2谷礼新,郑海斌,彭卫平.塔式起重机起重臂结构和稳定性有限元分析.机电工程技术. 2005, 34(8):28~36
3朱渝春,冯贤桂.伸缩臂式起重机回转侧向动载荷的计算.工程机械. 2005,(2):11~14
4陆念力,张立强,顾迪民.非保向力作用下压杆稳定计算的等效支座法及其在吊臂分析中的应用.建筑机械. 1996,(7):8~13
5纪爱敏,罗衍领.起重机伸缩吊臂截面优化设计.建筑机械化. 2006,(3):19~24
6 GB3811-83起重机设计规范.中国标准出版社, 1983:76~82
7陆念力,孟晓平,顾迪民.具有外伸臂的多跨连续附着压杆的非线性变形及整体稳定性计算.工程机械. 1996,(9):3~6
8 Loannis G. Raftoyiannis,John Ch. Ermopoulos. Stability of Tapered and Stepped steel Columns with Initial Imperfections. Engineering Structures. 2005,27(8):1248~1257
9刘木南.浅谈起重机起重臂的制造技术.工程机械. 2003,(2):23~26
10汤茂旭,刘木南.轮式起重机起重臂的发展历程.建筑机械化. 2007,(2):29~34
11 Otsuka Hiromitsu, Koga Tatsuzo. Buckling of circular cylindrical shell under beam-like bending(1st report) experiment, Transactions of the Japan Society for Aeronautical and Space Sciences. 1998,(41):38~45
12 S.P.Timoshenko, J.M.Gere .Theory of elastic stability, 2nd ed,McGraw-Hill, 1961:203~236
13 Klaus-Jürgen Bathe. Advances in Nonlinear Finite Element Analysis of Automobiles. 1997, 64(5): 881-891
14 Klaus-Jürgen Bathe. Finite Element procedures. Prentice Hall, Inc. 1996,(6):485~628
15兰天,姚卓智.桅杆结构的非线形分析.中国建筑研究院结构所. 1980,(2):2~4
16楚中毅,陆念力,楚兰英等.梁杆结构稳定性分析的一种精确有限元方法及其优化.建筑机械.2001, (9):68~73
17陆念力,兰朋.二阶理论条件下的梁杆系统精确有限元方程及应用.哈尔滨建筑大学学报. 1998, (4) :21~25
18尹刚,冯贤桂.变截面压杆的临界压力近似计算.重庆工学院学报. 2005,(11): 22~24
19陈玮璋.起重机金属结构.人民交通出版社, 1987:134~176
20孙焕纯,王跃方.对桁架结构稳定分析经典理论的讨论.计算力学学报. 2005,(3): 316~320
21楚中毅,陆念力,车仁炜.一种梁杆结构稳定性分析的精确有限元法.哈尔滨建筑大学学报. 2002, 35(4): 25~28
22顾迪民.工程起重机.第一版.中国建筑工业出版社,1981: 257~275
23王硕哲,顾迪民.伸缩油缸对箱形吊臂受力状况的影响.起重运输机械. . 1983,(11):34~37
24陆念力,兰朋,白桦.起重机箱型伸缩臂稳定性分析的精确理论解.哈尔滨建筑大学学报.2000,33(2):89~93,
25 David H. Ellis , Scott R. Swengel, George W. Archibald,etal. A sociogram for the cranes of the world. Behavioural Processes. 1998,(43): 125~151
26兰朋.二阶应力作用下梁杆精确有限元方程及其在非线性分析和稳定计算中的应用.《哈尔滨建筑大学硕士论文》.1996 :32~45
27陆念力,孟晓平,顾迪民.梁杆系统精确有限元方程及其在几何非线性分析和稳定计算中的应用.建筑机械.1996,(3) 18~21
28 S.Shimizu, M.Nakano. Strength and Behavior of the Corner Zones in Steel Rigid Frame Columns with Shifted Beams. Journal of Constructional Steel Research. 2000, 53(2): 245~263
29 J.L. Bonet, P.F. Miguel. Biaxial Bending Moment Magnifier Method. 2004, 26(13): 2007~2019
30 Yang. Y.B. Incrementally small-deformation theory fro nonlinear analysis of structural frames. Engineer Structure. 2002, 24(6): 783-798
31陈铁云,沈惠申.结构的屈曲.上海科学技术文献出版社, 1993: 1~3
32 N.A. Fallah, C.Bailey. Comparison of Finite Element and Finite Volume Methods Application in Geometrically Nonlinear Stress Analysis. Applied Mathematical Modelling. 2004, 24(7): 439~455
33 Samir Z. Al-Sadder. Exact Expressions for Stability Functions of a General Non-prismatic Beam-column Member. Journal of Constructional Steel Reaserch.2004, 60(11): 561~584
34 J.G. Teng, J. Yao. Distortional Buckling of Channel Beam-columns. Thin-walled structures. 2003, 41(7): 595~617
35 Yukio Hayase, Tadatoshi Kamiya. Wind Load for Tower Type Crawler Crane During Halting. Technical Review. 2001, (4): 61~64
36胡铁华,唐洪学,马成林.箱形伸缩吊臂设计变量对性能参数的影响线.建筑机械. 1996, (9) : 61~64
37 S.R.Bodner, M.B.Rubin. Modelling the buckling of axially compressed elastic cylindrical shells . AIAA Journal. 2005, (43) :103~110
38 Q.S.Li. Buckling analysis of multi-step non-uniform columns. Advances in Structure Engineering. 2000 , (3) :139~144
39李以申,陆念力,顾迪民.塔式起重机起重臂结构稳定计算的研究(一)——具有连续变化截面析架构件的欧拉临界力计算.建筑机械. 1996,(12): 23~26
40 Lan Peng, Liu Manlan, Lu Nianli. Out-of-plane stability of a bending beam. GongchengLixue. 2005, (22): 152~155
41 Q.S.Li. Analytical solutions for buckling of multi-step non-uniform columns with arbitrary distribution of flexural stiffness or axial distributed loading. International Journal of Mechanical Sciences. 2001 (43) :349~366
42 J.L. Bonet, P.F. Miguel. Biaxial Bending Moment Magnifier Method. 2004, 26(13): 2007~2019
43纪爱敏. QY25K型汽车起重机伸缩吊臂的有限元分析.工程机械.2003, (1): 24~29
44李以申,陆念力,顾迪民.塔式起重机起重臂结构稳定计算的研究(四)——转角位移方程法求解构件欧拉临界力.建筑机械. 1997,(2): 9~13
45 Guo-Qiang Li, Jin-Jun Li. A tapered Timoshenko–Euler beam element for analysis of steel portal frames [J]. Journal of Constructional Steel Research. 2002, 58(12): 1531~1544
46 Potrc I, Sraml M. On the Numerical Analysis of the Contact Problems at Crane Machanisms. Zeitschrift fur Angewandte Mathematik and Mechanik. 2000, 80(l2): 491~492
47李以申,陆念力,顾迪民.塔式起重机起重臂结构稳定计算的研究(三)——微分方程法求解阶梯状变截面非保向力悬臂梁构件的欧拉临界力.建筑机械. 1997,(1): 12~19
48王勖成.有限单元法.清华大学出版社,2003:55 ~77
49蒋红旗,王繁生.起重机吊臂结构有限元分析.农业机械学报. 2006,(3):20 ~23
50纪爱敏,罗衍领.起重机伸缩吊臂截面优化设计.建筑机械化.2006,(3):15 ~21
51尹刚,冯贤桂.变截面压杆的临界压力近似计算.重庆工学院学报. 2005, (11):22~24
52孟晓平,陆念力,李良.非保向力作用下起重机吊臂起升平面外整体稳定特征方程及应用.哈尔滨建筑大学学报. 1997, (4) :99~105
53 Mustafa Sabuncu, Kaan Evran. Dynamic stability of a rotating asymmetric cross-section Timoshenko beam subjected to an axial periodic force. Finite Elements in Analysis and Design. 2005, (41): 1011~1026
54 S. Naguleswaran. Transverse vibration and stability of an Euler–Bernoulli beam with step change in cross-section and in axial force [J]. Journal of Sound and Vibration. 2004, 270 (5): 1045~1055
55 Kim Moon-Young a, Kim Sung-Bo, Kim Nam-II. Spatial stability of shear deformable curved beams with non-symmetric thin-walled sections. Stability formulation and closed-form solutions. Computers and Structures. 2005,(83):58~68
56 Phani Kumar V.Nukala, Donald W. White. A Mixed Finite Element for Threedimensional Nonlinear Analysis of steel Frames. Computer Methods in Applied Mechanics and Engineering. 2004, 193(23~26): 2507~2545
2谷礼新,郑海斌,彭卫平.塔式起重机起重臂结构和稳定性有限元分析.机电工程技术. 2005, 34(8):28~36
3朱渝春,冯贤桂.伸缩臂式起重机回转侧向动载荷的计算.工程机械. 2005,(2):11~14
4陆念力,张立强,顾迪民.非保向力作用下压杆稳定计算的等效支座法及其在吊臂分析中的应用.建筑机械. 1996,(7):8~13
5纪爱敏,罗衍领.起重机伸缩吊臂截面优化设计.建筑机械化. 2006,(3):19~24
6 GB3811-83起重机设计规范.中国标准出版社, 1983:76~82
7陆念力,孟晓平,顾迪民.具有外伸臂的多跨连续附着压杆的非线性变形及整体稳定性计算.工程机械. 1996,(9):3~6
8 Loannis G. Raftoyiannis,John Ch. Ermopoulos. Stability of Tapered and Stepped steel Columns with Initial Imperfections. Engineering Structures. 2005,27(8):1248~1257
9刘木南.浅谈起重机起重臂的制造技术.工程机械. 2003,(2):23~26
10汤茂旭,刘木南.轮式起重机起重臂的发展历程.建筑机械化. 2007,(2):29~34
11 Otsuka Hiromitsu, Koga Tatsuzo. Buckling of circular cylindrical shell under beam-like bending(1st report) experiment, Transactions of the Japan Society for Aeronautical and Space Sciences. 1998,(41):38~45
12 S.P.Timoshenko, J.M.Gere .Theory of elastic stability, 2nd ed,McGraw-Hill, 1961:203~236
13 Klaus-Jürgen Bathe. Advances in Nonlinear Finite Element Analysis of Automobiles. 1997, 64(5): 881-891
14 Klaus-Jürgen Bathe. Finite Element procedures. Prentice Hall, Inc. 1996,(6):485~628
15兰天,姚卓智.桅杆结构的非线形分析.中国建筑研究院结构所. 1980,(2):2~4
16楚中毅,陆念力,楚兰英等.梁杆结构稳定性分析的一种精确有限元方法及其优化.建筑机械.2001, (9):68~73
17陆念力,兰朋.二阶理论条件下的梁杆系统精确有限元方程及应用.哈尔滨建筑大学学报. 1998, (4) :21~25
18尹刚,冯贤桂.变截面压杆的临界压力近似计算.重庆工学院学报. 2005,(11): 22~24
19陈玮璋.起重机金属结构.人民交通出版社, 1987:134~176
20孙焕纯,王跃方.对桁架结构稳定分析经典理论的讨论.计算力学学报. 2005,(3): 316~320
21楚中毅,陆念力,车仁炜.一种梁杆结构稳定性分析的精确有限元法.哈尔滨建筑大学学报. 2002, 35(4): 25~28
22顾迪民.工程起重机.第一版.中国建筑工业出版社,1981: 257~275
23王硕哲,顾迪民.伸缩油缸对箱形吊臂受力状况的影响.起重运输机械. . 1983,(11):34~37
24陆念力,兰朋,白桦.起重机箱型伸缩臂稳定性分析的精确理论解.哈尔滨建筑大学学报.2000,33(2):89~93,
25 David H. Ellis , Scott R. Swengel, George W. Archibald,etal. A sociogram for the cranes of the world. Behavioural Processes. 1998,(43): 125~151
26兰朋.二阶应力作用下梁杆精确有限元方程及其在非线性分析和稳定计算中的应用.《哈尔滨建筑大学硕士论文》.1996 :32~45
27陆念力,孟晓平,顾迪民.梁杆系统精确有限元方程及其在几何非线性分析和稳定计算中的应用.建筑机械.1996,(3) 18~21
28 S.Shimizu, M.Nakano. Strength and Behavior of the Corner Zones in Steel Rigid Frame Columns with Shifted Beams. Journal of Constructional Steel Research. 2000, 53(2): 245~263
29 J.L. Bonet, P.F. Miguel. Biaxial Bending Moment Magnifier Method. 2004, 26(13): 2007~2019
30 Yang. Y.B. Incrementally small-deformation theory fro nonlinear analysis of structural frames. Engineer Structure. 2002, 24(6): 783-798
31陈铁云,沈惠申.结构的屈曲.上海科学技术文献出版社, 1993: 1~3
32 N.A. Fallah, C.Bailey. Comparison of Finite Element and Finite Volume Methods Application in Geometrically Nonlinear Stress Analysis. Applied Mathematical Modelling. 2004, 24(7): 439~455
33 Samir Z. Al-Sadder. Exact Expressions for Stability Functions of a General Non-prismatic Beam-column Member. Journal of Constructional Steel Reaserch.2004, 60(11): 561~584
34 J.G. Teng, J. Yao. Distortional Buckling of Channel Beam-columns. Thin-walled structures. 2003, 41(7): 595~617
35 Yukio Hayase, Tadatoshi Kamiya. Wind Load for Tower Type Crawler Crane During Halting. Technical Review. 2001, (4): 61~64
36胡铁华,唐洪学,马成林.箱形伸缩吊臂设计变量对性能参数的影响线.建筑机械. 1996, (9) : 61~64
37 S.R.Bodner, M.B.Rubin. Modelling the buckling of axially compressed elastic cylindrical shells . AIAA Journal. 2005, (43) :103~110
38 Q.S.Li. Buckling analysis of multi-step non-uniform columns. Advances in Structure Engineering. 2000 , (3) :139~144
39李以申,陆念力,顾迪民.塔式起重机起重臂结构稳定计算的研究(一)——具有连续变化截面析架构件的欧拉临界力计算.建筑机械. 1996,(12): 23~26
40 Lan Peng, Liu Manlan, Lu Nianli. Out-of-plane stability of a bending beam. GongchengLixue. 2005, (22): 152~155
41 Q.S.Li. Analytical solutions for buckling of multi-step non-uniform columns with arbitrary distribution of flexural stiffness or axial distributed loading. International Journal of Mechanical Sciences. 2001 (43) :349~366
42 J.L. Bonet, P.F. Miguel. Biaxial Bending Moment Magnifier Method. 2004, 26(13): 2007~2019
43纪爱敏. QY25K型汽车起重机伸缩吊臂的有限元分析.工程机械.2003, (1): 24~29
44李以申,陆念力,顾迪民.塔式起重机起重臂结构稳定计算的研究(四)——转角位移方程法求解构件欧拉临界力.建筑机械. 1997,(2): 9~13
45 Guo-Qiang Li, Jin-Jun Li. A tapered Timoshenko–Euler beam element for analysis of steel portal frames [J]. Journal of Constructional Steel Research. 2002, 58(12): 1531~1544
46 Potrc I, Sraml M. On the Numerical Analysis of the Contact Problems at Crane Machanisms. Zeitschrift fur Angewandte Mathematik and Mechanik. 2000, 80(l2): 491~492
47李以申,陆念力,顾迪民.塔式起重机起重臂结构稳定计算的研究(三)——微分方程法求解阶梯状变截面非保向力悬臂梁构件的欧拉临界力.建筑机械. 1997,(1): 12~19
48王勖成.有限单元法.清华大学出版社,2003:55 ~77
49蒋红旗,王繁生.起重机吊臂结构有限元分析.农业机械学报. 2006,(3):20 ~23
50纪爱敏,罗衍领.起重机伸缩吊臂截面优化设计.建筑机械化.2006,(3):15 ~21
51尹刚,冯贤桂.变截面压杆的临界压力近似计算.重庆工学院学报. 2005, (11):22~24
52孟晓平,陆念力,李良.非保向力作用下起重机吊臂起升平面外整体稳定特征方程及应用.哈尔滨建筑大学学报. 1997, (4) :99~105
53 Mustafa Sabuncu, Kaan Evran. Dynamic stability of a rotating asymmetric cross-section Timoshenko beam subjected to an axial periodic force. Finite Elements in Analysis and Design. 2005, (41): 1011~1026
54 S. Naguleswaran. Transverse vibration and stability of an Euler–Bernoulli beam with step change in cross-section and in axial force [J]. Journal of Sound and Vibration. 2004, 270 (5): 1045~1055
55 Kim Moon-Young a, Kim Sung-Bo, Kim Nam-II. Spatial stability of shear deformable curved beams with non-symmetric thin-walled sections. Stability formulation and closed-form solutions. Computers and Structures. 2005,(83):58~68
56 Phani Kumar V.Nukala, Donald W. White. A Mixed Finite Element for Threedimensional Nonlinear Analysis of steel Frames. Computer Methods in Applied Mechanics and Engineering. 2004, 193(23~26): 2507~2545