框架结构动力稳定性分析的新方法研究
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摘要
结构的稳定性问题一直是结构分析的重点问题之一,非线性静力稳定性问题的研究已经比较深入,研究的成果已经能够指导实际工程的静力设计。相对于静力稳定,动力稳定的研究难度较大,目前还不能很好的解决结构非线性动力稳定性问题。其中的一个难点是动力失稳的判断准则,其与静力失稳的有很大的区别,除了判定切线刚度矩阵的正定性外,还必须结合结构的动力响应的敛散性来综合确定。所以,要研究结构的动力稳定性,必须对结构进行非线性动力时程分析,在时程分析的过程中判定结构的动力稳定性。结构动力稳定性研究的基本过程如下：逐渐增大动力荷载,每一级荷载下都要进行一次非线性动力时程分析,常常需要试算多次才可找到动力失稳临界荷载的大致范围,再通过细化荷载增量,最终得到满足一定精度要求的动力失稳临界荷载值。
动力稳定性分析涉及到大量的非线性时程计算,而每一次非线性时程分析,需要的计算工作量也很巨大,必须进行多次迭代,方能收敛,尤其是在接近动力失稳临界荷载时,需要的迭代次数会增加很多。即使结构比较简单,需要的计算时间也会较长,略微复杂的结构需要的计算时间会很长,甚至由于迭代次数增加导致误差的积累过大,最后得到错误的结果,所以研究高效率的结构动力稳定性分析的新方法是很有必要的。
由于动力稳定性研究更具实际意义,所以有必要对其进行系统研究,虽然研究的难度很大,但可以采用逐步深入的研究方法,不断积累研究成果,为最终解决结构动力稳定性问题打下基础。所以,本文以工程中常见的框架结构为研究对象,利用样条函数建立了框架结构的QR法分析模型,以动力非线性时程分析为主要手段,研究框架结构动力稳定性,提出了一些新的分析模型和算法。
论文先从样条函数入手,讨论了三次及五次样条函数的基本概念及基本特性,并重点研究了本文后续章节中需要用到的主要知识,为框架结构QR法离散化模型及动力时程分析中样条加权残数法的建立奠定了基础。为了将动力稳定性与静力稳定性做对比并且探求动力失稳临界荷载与静力失稳临界荷载的关系,论文专门利用QR法建立了框架结构静力非线性稳定分析的模型,提出相应的算法,并编制计算程序,通过算例分析,得到了具体结构的静力稳定临界荷载。结果表明,框架结构非线性静力稳定性分析的QR法计算结果与有限元法的计算结果十分接近,但QR法在计算效率上更具有优势。由于动力非线性稳定性计算工作量十分巨大,利用QR法建立动力稳定模型,将会大大减少未知量数目,减小动力方程阶数,在非线性动力稳定性分析中的优势会更加明显。
在非线性静力问题研究的基础上,重点研究框架结构非线性动力稳定性。非线性动力稳定性需要涉及到非线性时程分析,而时程分析中算法的效率直接决定着动力稳定性分析的效率,为此本文利用样条加权残数法建立了框架结构非线性动力时程分析的新算法,并编制了相应的计算程序,通过算例分析表明,本文建立的非线性时程分析算法的计算精度和计算效率均较高,尤其是其计算效率明显优于有限元法,能在较短的时间内完成框架结构的非线性时程分析。同时结合QR法建立框架结构动力分析模型,大大降低了非线性动力方程组中未知量的数目,使得迭代计算速度加快,提高了非线性时程分析的效率。
在解决了非线性动力时程分析的问题之后,利用QR法和样条加权残数法建立了框架结构动力稳定性分析的新模型,提出了动力失稳判定准则和具体的判定方法,即：逐渐增大动力荷载,对于每一级动力荷载均进行一次非线性动力时程分析,在时程分析过程中,观察框架结构的等效刚度矩阵的正定性,同时结合结构的动力位移响应的敛散性来判定结构的动力稳定性。具体的判定标准是：若结构在某一级别的动力荷载作用下等效刚度矩阵非正定,且位移响应发散,此时即可判定结构已经发生动力失稳。为了得到结构动力稳定的临界荷载,可以减小荷载增量继续进行时程分析,当很小的荷载增量,导致了结构等效刚度矩阵非正定且结构位移响应的突然增加(发散)时,此时的结构动力失稳临界荷载即可确定。利用以上判定方法,在结构非线性静力稳定性分析的QR法程序的基础上,编制了框架结构动力稳定分析的计算程序,利用程序计算了大量的算例,得到了不同条件下框架结构的动力稳定临界荷载值。通过比较分析,探讨了影响动力稳定临界荷载的因素及动力稳定性的规律,得到以下结论：
1)本文提出的框架结构动力稳定性分析的方法,能够满足框架结构稳定性分析的精度和计算速度的要求,尤其是其在计算效率上的优势使得框架结构动力稳定性分析成为可能,大大减小了利用有限元法进行框架结构动力稳定性分析的难度,能够在有限的时间内完成结构的框架结构的动力稳定性分析,得到结构动力失稳的临界荷载值。这是本文能在较短的时间内(学位论文阶段)完成大量算例的关键。
2)水平阶跃荷载作用下,框架结构的动力稳定临界荷载与静力失稳临界荷载相比较,有较大幅度的降低,本文的大量算例分析表明,降低的幅度大致在30%-40%之间,实际工程中可以将静力稳定的临界荷载按照这一比例折减,作为结构动力稳定临界荷载指导结构设计,即将框架结构动力稳定性的临界荷载取为静力稳定临界荷载的60-70%。对于实际的框架结构,可以避免复杂的动力稳定性分析,直接进行静力稳定性分析,由静力失稳临界荷载值乘以一个折减系数(0.6～0.7)即可估算动力失稳临界荷载。若需精确的动力失稳临界荷载值,可以参考估算值,在估算值附近进行动力稳定性分析,避免了动力稳定性分析中荷载取值的盲目性,从而提高了计算效率。
3)框架结构在水平阶跃荷载作用下,动力荷载持时与动力失稳临界荷载的关系可以概括为：当荷载持时小于结构基本周期的0.5倍时候,荷载持时越短,动力稳定临界荷载幅值越大,荷载持时与临界荷载关系密切；当荷载持时大于结构基本周期的0.5倍以后,荷载持时长短对结构动力稳定临界荷载的影响不大,此时,临界荷载变化幅度很小,即荷载持时与临界荷载幅值关系不大。在实际工程的设计中,需要考虑动力稳定性时可以取较长的荷载持时得到的临界荷载值作为结构动力失稳临界荷载,这样做可以保证结构动力稳定。
4)在水平阶跃荷载作用下,多层框架的层数与框架结构动力稳定性的关系可以初步概况如下：随着框架层数的增加,框架结构动力稳定临界荷载与静力稳定临界荷载的比值有逐渐减小的趋势,对于一般的多层(10层以下)框架结构,这一比值基本大致为0.67～0.72.随着框架结构层数的增加,框架各层在水平方向的动力临界荷载值之和逐渐减小,说明随着框架层数的增加,框架越容易发生动力失稳。这就表明,对于高层框架,很有必要对其进行动力稳定性分析。
5)水平动力荷载作用下,框架结构的竖向荷载对结构的动力稳定性影响较大,随着竖向荷载的增加,水平动力荷载临界值逐渐降低,在实际工程中,必须考虑框架结构的竖向荷载对其稳定性的影响。当然,在实际框架结构的动力稳定性分析中,除了考虑竖向荷载之外,还要考虑竖向荷载(尤其是竖向恒载)对结构动力特性的影响,在计算地震荷载作用下框架结构的动力稳定性时,必须考虑产生竖向恒载构件的质量方可进行动力稳定性分析,得到的动力失稳临界荷载才更加接近实际,对实际工程具有指导意义。
The stability of structure is one of the most important problems in structure analysis. Nolinear static stability problem has solved and the results has applied in the static design of structure.The dynamic stability problems of structure have not solved and can not be applied in structure design. One of the most difficult problems of dynamic stability is the dynamic stability criterion, which is different with static stability. The criteria of dynamic stability of structure include the judgements of both the positive definiteness of effective stiffness matrix and the dynamic response of structure. To study the dynamic stability of structure, the time history analysis must be done, and the stability of structure can be judged during time history analysis. Generally the time history analysis must be done at every level of dynamic load and the rough range of critical dynamic load can obtained by several times of the time history analysis. Then the exact critical load of structure under dynamic loads can be got by minishing the increment of dynamic loads.
The dynamic stability analysis involve large numbers of nonlinear time history analyses, and in every analysis massive calculation and multiple iterative computations must be done for convergence. The number of times for iterative computations increases substantially when the dynamic loads are close to the critical load. Even simple frame structure, the time spent in dynamic stability analysis is lengthy. If complex structure is minor complex, the computational time will much lengthy, and the Increased iteration error accumulation may lead to wrong results. So, it is necessary to study the more efficient methods of dynamic stability analysis of frame structural.
Although its very difficult to study dynamic stability of structure, the dynamic stability must be thoroughly studied because of its practical significance. The most feasible way is to study the problem step by step, and begin with a simple subject. So in this paper frame structure is selected as the subject investigated, and the calculation model of QR-method of dynamic stability is established by spline function. The non-linear time history analysis is used in the dynamic stability of frame structure as the main principal method.
Spline function is the basis of QR-method, so the basic property and concept of Spline function are studied at first. The fundamental of conception that may be used in this paper on QR-method is discussed as emphasis, so that the QR discretization model of calculation and the spline weighted residual method in time history analysis of dynamic stability can be obtain from it. In order to establish the model and calculation method of dynamic stability, the static stability model and calculation method is established at first which can also be used for contrast of dynamic stability and static stability to show differences. An example of static stability of frame structure is studied by QR-method and finite element method respectively, and the critical loads of the frame is obtained. The results reveal that the critical loads of the frame structure by QR-method differ litter from that by finite element method, but QR-method shows higher efficiency in calculation speed and accuracy than finite element method. In view of above reasons and the huge amount of calculation in dynamic stability analysis, QR-method is adopted in the establishmet of analysis model to increase of calculation efficiency and coped with the problems in nonlinear analysis.
The nonlinear dynamic stability of frame structure is studied after the nonlinear static stability of frame is solved by QR-method. Nonlinear dynamic time history analysis is the basis of nonlinear dynamic stability problems, so it is emphasized and the high efficiency method in nonlinear dynamic time history analysis must be established. The time history analysis model and calculation method is established by spline weighted residual method, then the calculation program is designed. An example revealed that nonlinear dynamic time history analysis methods established by spline weighted residual method have more accurate and efficient comparing with of other time history analysis methods of finite element method. The calculation model established by QR-method which has less number of unknown variables can accelerate the speed of iterative calculation and increase the efficiency of nonlinear dynamic time history analysis greatly.
The dynamic stability analysis model is accomplished by QR method and spline weighted residual method afer the nonlinear dynamic time history analysis method was settled. The criterion for dynamic instability criteria and the concrete methods to judge dynamic instability of frame structure are presented detailedly. As the dynamic loads is increased step by step, nonlinear dynamic time history analysis is done at every level of dynamic loads. During every time history analysis, the positive definiteness of effective stiffness matrix and the convergence property of dynamic response are investigated at every time history step to judge if the frame structure is in state of dynamic instability. When the effective stiffness matrix becomes nonpositive definite and the dynamic response becomes misconvergence, the frame structure has lose dynamic stability and becomes unstable. In order to get the critical load of dynamic stability, load increment is minish and time history analysis is repeated. The above course proceeds if the load increment which induces structure unstable is in the range of acceptable precision. The calculation program is designed for above method to solve dynamic stability problems of frame structure on the basis of static dynamic program of QR method. Large numbers of calculation examples for different frames with different conditions are studied by using the dynamic stability program, and the critical load of dynamic stability of frame structure is obtained. The results of these examples reveal some of the rules of dynamic stability of frame structures:
1. The new method presented in this paper can be used to analyze the stability of frame structure with satisfactory accuracy and speed, and the high efficiency of the new method makes it possible to study the dynamic stability of frame structure and reduces the difficulty of dynamic stability of frame structure using finite element method. The new method of this paper can complete dynamic stability analysis of frame structure and obtain the critical load, which is the key reason for that many calculating examples can be accomplished during the degree thesis stage.
2. The critical dynamic loads of frame structure under step load in horizontal direction have a great reduction compared with the static critical loads. The examples reveal that there is A30～40%drop when the critical loads of dynamic and static contrast. So, the critical load of dynamic stability can be estimated through multiplying the value of static critical load by a parameter about0.6to0.7. When the ccuracy is not required too high, the complex dynamic stability analysis can be avoided in the design of frame structure by reducing the value of static critical load30～40%. If the high precision of dynamic critical loads is required, the efficiency of dynamic stability analysis can be improved considering the estimated dynamic critical load stability, and the blindness in searching dynamic critical loads can be avoided.
3. The relationship between dynamic critical loads and the duration of dynamic loads for frame structure under step dynamic load in horizontal direction can be summarized as follows. When load acting time is short, usually less than half of fundamental period of the frame structure, the critical loads of frame structure under step load decrease obviously as the load acting time increases. The critical loads of frame structure under step loads vary scarcely when load acting time is greater than half of fundamental period of the structure. That is to say, the critical loads of frame structure are always remains the same. So in the practice, we can analysis the dynamic stability of frame structure at a long load acting time (offten larger than half of fundamental period) and have not to calculate critical loads at every load acting time, and the critical loads from that can be used in frame structure design with great confidence.
4. The vertical loads, usually self weight of walls and floor slab, has great influence on the critical load of frame structure, and the critical load decreases substantially when the vertical loads increase. So it is emphasized that the vertical loads must be considered in calculating dynamic stability of frame structure for actual engineering design. Of course, in the dynamic stability analysis for actual frame structure, in addition to the vertical load, considering the influence of the vertical load (especially for vertical constant load) on the dynamic characteristics of frame structure in the calculation of the seismic critical load of frame structure is very necessary, because the mass of the beams, slabs, and other elements can substantially influence both the dynamic characteristics of frame structures and the dynamic stability analysis of the structure. The dynamic critical load of frame structure considering the vertical load and the mass of the vertical laod elements have more practical significance for practical engineering because it more close to reality.
5. The influence of the stories of frame structure on the critical loads of dynamic stability can be be summarized as follows. The ratio of dynamic critical loads and static critical loads for frame structure decreases slightly as the floors of frame increase, and that ratio is between0.67and0.72when the floors of frame belows15. The total dynamic loads in horizontal direction decrease when the floors of frame increases, so frame structure tends to be more easily unstable when the foors increase and it is necessary to analysis the dynamic stability of high frame structure.
动力稳定性分析涉及到大量的非线性时程计算,而每一次非线性时程分析,需要的计算工作量也很巨大,必须进行多次迭代,方能收敛,尤其是在接近动力失稳临界荷载时,需要的迭代次数会增加很多。即使结构比较简单,需要的计算时间也会较长,略微复杂的结构需要的计算时间会很长,甚至由于迭代次数增加导致误差的积累过大,最后得到错误的结果,所以研究高效率的结构动力稳定性分析的新方法是很有必要的。
由于动力稳定性研究更具实际意义,所以有必要对其进行系统研究,虽然研究的难度很大,但可以采用逐步深入的研究方法,不断积累研究成果,为最终解决结构动力稳定性问题打下基础。所以,本文以工程中常见的框架结构为研究对象,利用样条函数建立了框架结构的QR法分析模型,以动力非线性时程分析为主要手段,研究框架结构动力稳定性,提出了一些新的分析模型和算法。
论文先从样条函数入手,讨论了三次及五次样条函数的基本概念及基本特性,并重点研究了本文后续章节中需要用到的主要知识,为框架结构QR法离散化模型及动力时程分析中样条加权残数法的建立奠定了基础。为了将动力稳定性与静力稳定性做对比并且探求动力失稳临界荷载与静力失稳临界荷载的关系,论文专门利用QR法建立了框架结构静力非线性稳定分析的模型,提出相应的算法,并编制计算程序,通过算例分析,得到了具体结构的静力稳定临界荷载。结果表明,框架结构非线性静力稳定性分析的QR法计算结果与有限元法的计算结果十分接近,但QR法在计算效率上更具有优势。由于动力非线性稳定性计算工作量十分巨大,利用QR法建立动力稳定模型,将会大大减少未知量数目,减小动力方程阶数,在非线性动力稳定性分析中的优势会更加明显。
在非线性静力问题研究的基础上,重点研究框架结构非线性动力稳定性。非线性动力稳定性需要涉及到非线性时程分析,而时程分析中算法的效率直接决定着动力稳定性分析的效率,为此本文利用样条加权残数法建立了框架结构非线性动力时程分析的新算法,并编制了相应的计算程序,通过算例分析表明,本文建立的非线性时程分析算法的计算精度和计算效率均较高,尤其是其计算效率明显优于有限元法,能在较短的时间内完成框架结构的非线性时程分析。同时结合QR法建立框架结构动力分析模型,大大降低了非线性动力方程组中未知量的数目,使得迭代计算速度加快,提高了非线性时程分析的效率。
在解决了非线性动力时程分析的问题之后,利用QR法和样条加权残数法建立了框架结构动力稳定性分析的新模型,提出了动力失稳判定准则和具体的判定方法,即：逐渐增大动力荷载,对于每一级动力荷载均进行一次非线性动力时程分析,在时程分析过程中,观察框架结构的等效刚度矩阵的正定性,同时结合结构的动力位移响应的敛散性来判定结构的动力稳定性。具体的判定标准是：若结构在某一级别的动力荷载作用下等效刚度矩阵非正定,且位移响应发散,此时即可判定结构已经发生动力失稳。为了得到结构动力稳定的临界荷载,可以减小荷载增量继续进行时程分析,当很小的荷载增量,导致了结构等效刚度矩阵非正定且结构位移响应的突然增加(发散)时,此时的结构动力失稳临界荷载即可确定。利用以上判定方法,在结构非线性静力稳定性分析的QR法程序的基础上,编制了框架结构动力稳定分析的计算程序,利用程序计算了大量的算例,得到了不同条件下框架结构的动力稳定临界荷载值。通过比较分析,探讨了影响动力稳定临界荷载的因素及动力稳定性的规律,得到以下结论：
1)本文提出的框架结构动力稳定性分析的方法,能够满足框架结构稳定性分析的精度和计算速度的要求,尤其是其在计算效率上的优势使得框架结构动力稳定性分析成为可能,大大减小了利用有限元法进行框架结构动力稳定性分析的难度,能够在有限的时间内完成结构的框架结构的动力稳定性分析,得到结构动力失稳的临界荷载值。这是本文能在较短的时间内(学位论文阶段)完成大量算例的关键。
2)水平阶跃荷载作用下,框架结构的动力稳定临界荷载与静力失稳临界荷载相比较,有较大幅度的降低,本文的大量算例分析表明,降低的幅度大致在30%-40%之间,实际工程中可以将静力稳定的临界荷载按照这一比例折减,作为结构动力稳定临界荷载指导结构设计,即将框架结构动力稳定性的临界荷载取为静力稳定临界荷载的60-70%。对于实际的框架结构,可以避免复杂的动力稳定性分析,直接进行静力稳定性分析,由静力失稳临界荷载值乘以一个折减系数(0.6～0.7)即可估算动力失稳临界荷载。若需精确的动力失稳临界荷载值,可以参考估算值,在估算值附近进行动力稳定性分析,避免了动力稳定性分析中荷载取值的盲目性,从而提高了计算效率。
3)框架结构在水平阶跃荷载作用下,动力荷载持时与动力失稳临界荷载的关系可以概括为：当荷载持时小于结构基本周期的0.5倍时候,荷载持时越短,动力稳定临界荷载幅值越大,荷载持时与临界荷载关系密切；当荷载持时大于结构基本周期的0.5倍以后,荷载持时长短对结构动力稳定临界荷载的影响不大,此时,临界荷载变化幅度很小,即荷载持时与临界荷载幅值关系不大。在实际工程的设计中,需要考虑动力稳定性时可以取较长的荷载持时得到的临界荷载值作为结构动力失稳临界荷载,这样做可以保证结构动力稳定。
4)在水平阶跃荷载作用下,多层框架的层数与框架结构动力稳定性的关系可以初步概况如下：随着框架层数的增加,框架结构动力稳定临界荷载与静力稳定临界荷载的比值有逐渐减小的趋势,对于一般的多层(10层以下)框架结构,这一比值基本大致为0.67～0.72.随着框架结构层数的增加,框架各层在水平方向的动力临界荷载值之和逐渐减小,说明随着框架层数的增加,框架越容易发生动力失稳。这就表明,对于高层框架,很有必要对其进行动力稳定性分析。
5)水平动力荷载作用下,框架结构的竖向荷载对结构的动力稳定性影响较大,随着竖向荷载的增加,水平动力荷载临界值逐渐降低,在实际工程中,必须考虑框架结构的竖向荷载对其稳定性的影响。当然,在实际框架结构的动力稳定性分析中,除了考虑竖向荷载之外,还要考虑竖向荷载(尤其是竖向恒载)对结构动力特性的影响,在计算地震荷载作用下框架结构的动力稳定性时,必须考虑产生竖向恒载构件的质量方可进行动力稳定性分析,得到的动力失稳临界荷载才更加接近实际,对实际工程具有指导意义。
The stability of structure is one of the most important problems in structure analysis. Nolinear static stability problem has solved and the results has applied in the static design of structure.The dynamic stability problems of structure have not solved and can not be applied in structure design. One of the most difficult problems of dynamic stability is the dynamic stability criterion, which is different with static stability. The criteria of dynamic stability of structure include the judgements of both the positive definiteness of effective stiffness matrix and the dynamic response of structure. To study the dynamic stability of structure, the time history analysis must be done, and the stability of structure can be judged during time history analysis. Generally the time history analysis must be done at every level of dynamic load and the rough range of critical dynamic load can obtained by several times of the time history analysis. Then the exact critical load of structure under dynamic loads can be got by minishing the increment of dynamic loads.
The dynamic stability analysis involve large numbers of nonlinear time history analyses, and in every analysis massive calculation and multiple iterative computations must be done for convergence. The number of times for iterative computations increases substantially when the dynamic loads are close to the critical load. Even simple frame structure, the time spent in dynamic stability analysis is lengthy. If complex structure is minor complex, the computational time will much lengthy, and the Increased iteration error accumulation may lead to wrong results. So, it is necessary to study the more efficient methods of dynamic stability analysis of frame structural.
Although its very difficult to study dynamic stability of structure, the dynamic stability must be thoroughly studied because of its practical significance. The most feasible way is to study the problem step by step, and begin with a simple subject. So in this paper frame structure is selected as the subject investigated, and the calculation model of QR-method of dynamic stability is established by spline function. The non-linear time history analysis is used in the dynamic stability of frame structure as the main principal method.
Spline function is the basis of QR-method, so the basic property and concept of Spline function are studied at first. The fundamental of conception that may be used in this paper on QR-method is discussed as emphasis, so that the QR discretization model of calculation and the spline weighted residual method in time history analysis of dynamic stability can be obtain from it. In order to establish the model and calculation method of dynamic stability, the static stability model and calculation method is established at first which can also be used for contrast of dynamic stability and static stability to show differences. An example of static stability of frame structure is studied by QR-method and finite element method respectively, and the critical loads of the frame is obtained. The results reveal that the critical loads of the frame structure by QR-method differ litter from that by finite element method, but QR-method shows higher efficiency in calculation speed and accuracy than finite element method. In view of above reasons and the huge amount of calculation in dynamic stability analysis, QR-method is adopted in the establishmet of analysis model to increase of calculation efficiency and coped with the problems in nonlinear analysis.
The nonlinear dynamic stability of frame structure is studied after the nonlinear static stability of frame is solved by QR-method. Nonlinear dynamic time history analysis is the basis of nonlinear dynamic stability problems, so it is emphasized and the high efficiency method in nonlinear dynamic time history analysis must be established. The time history analysis model and calculation method is established by spline weighted residual method, then the calculation program is designed. An example revealed that nonlinear dynamic time history analysis methods established by spline weighted residual method have more accurate and efficient comparing with of other time history analysis methods of finite element method. The calculation model established by QR-method which has less number of unknown variables can accelerate the speed of iterative calculation and increase the efficiency of nonlinear dynamic time history analysis greatly.
The dynamic stability analysis model is accomplished by QR method and spline weighted residual method afer the nonlinear dynamic time history analysis method was settled. The criterion for dynamic instability criteria and the concrete methods to judge dynamic instability of frame structure are presented detailedly. As the dynamic loads is increased step by step, nonlinear dynamic time history analysis is done at every level of dynamic loads. During every time history analysis, the positive definiteness of effective stiffness matrix and the convergence property of dynamic response are investigated at every time history step to judge if the frame structure is in state of dynamic instability. When the effective stiffness matrix becomes nonpositive definite and the dynamic response becomes misconvergence, the frame structure has lose dynamic stability and becomes unstable. In order to get the critical load of dynamic stability, load increment is minish and time history analysis is repeated. The above course proceeds if the load increment which induces structure unstable is in the range of acceptable precision. The calculation program is designed for above method to solve dynamic stability problems of frame structure on the basis of static dynamic program of QR method. Large numbers of calculation examples for different frames with different conditions are studied by using the dynamic stability program, and the critical load of dynamic stability of frame structure is obtained. The results of these examples reveal some of the rules of dynamic stability of frame structures:
1. The new method presented in this paper can be used to analyze the stability of frame structure with satisfactory accuracy and speed, and the high efficiency of the new method makes it possible to study the dynamic stability of frame structure and reduces the difficulty of dynamic stability of frame structure using finite element method. The new method of this paper can complete dynamic stability analysis of frame structure and obtain the critical load, which is the key reason for that many calculating examples can be accomplished during the degree thesis stage.
2. The critical dynamic loads of frame structure under step load in horizontal direction have a great reduction compared with the static critical loads. The examples reveal that there is A30～40%drop when the critical loads of dynamic and static contrast. So, the critical load of dynamic stability can be estimated through multiplying the value of static critical load by a parameter about0.6to0.7. When the ccuracy is not required too high, the complex dynamic stability analysis can be avoided in the design of frame structure by reducing the value of static critical load30～40%. If the high precision of dynamic critical loads is required, the efficiency of dynamic stability analysis can be improved considering the estimated dynamic critical load stability, and the blindness in searching dynamic critical loads can be avoided.
3. The relationship between dynamic critical loads and the duration of dynamic loads for frame structure under step dynamic load in horizontal direction can be summarized as follows. When load acting time is short, usually less than half of fundamental period of the frame structure, the critical loads of frame structure under step load decrease obviously as the load acting time increases. The critical loads of frame structure under step loads vary scarcely when load acting time is greater than half of fundamental period of the structure. That is to say, the critical loads of frame structure are always remains the same. So in the practice, we can analysis the dynamic stability of frame structure at a long load acting time (offten larger than half of fundamental period) and have not to calculate critical loads at every load acting time, and the critical loads from that can be used in frame structure design with great confidence.
4. The vertical loads, usually self weight of walls and floor slab, has great influence on the critical load of frame structure, and the critical load decreases substantially when the vertical loads increase. So it is emphasized that the vertical loads must be considered in calculating dynamic stability of frame structure for actual engineering design. Of course, in the dynamic stability analysis for actual frame structure, in addition to the vertical load, considering the influence of the vertical load (especially for vertical constant load) on the dynamic characteristics of frame structure in the calculation of the seismic critical load of frame structure is very necessary, because the mass of the beams, slabs, and other elements can substantially influence both the dynamic characteristics of frame structures and the dynamic stability analysis of the structure. The dynamic critical load of frame structure considering the vertical load and the mass of the vertical laod elements have more practical significance for practical engineering because it more close to reality.
5. The influence of the stories of frame structure on the critical loads of dynamic stability can be be summarized as follows. The ratio of dynamic critical loads and static critical loads for frame structure decreases slightly as the floors of frame increase, and that ratio is between0.67and0.72when the floors of frame belows15. The total dynamic loads in horizontal direction decrease when the floors of frame increases, so frame structure tends to be more easily unstable when the foors increase and it is necessary to analysis the dynamic stability of high frame structure.
引文
[1]秦荣.高层与超高层建筑[M],北京：科学技术出版社,2007.
[2]秦荣.工程结构非线性[M],北京：科学技术出版社,2006.
[3]韩强,张善元,杨桂通.结构静动力屈曲问题研究进展[J],力学进展,1998,28(3)：943-953.
[4]符华.鲍络金.弹性体系的动力稳定性[M].林砚田,译.北京：高等教育出版社,1990.
[5]曲乃泗,张斌.任意空间杆系结构的动力稳定性分析[J].工业建筑,1990,(5)：24-28.
[6]Bernard Budiansky. Theory of Buckling and Post-Buckling Behavior of Elastic Structures[J]. Advances in Applied Mechanics, Volume 14,1974:1-65.
[7]B.Budiansky, R.S.Roth. Axisymmetric Dynamic Buckling of Clamped Shallow Spherical Shells. Collect-ed Papers on Instability of Shells Structures. NASA TND-1510, Washington D.C,1962:597-606.
[8]Simitses G J. Dynamic stability of suddenly loaded structures. New York:Springer-Verlag,1990.
[9]Bong Jo Ryu, Kazuo Katayama, Yoshihiko Sugiyama.Dynamic stability of Timoshenko columns subjected to subtangential forces[J]. Computers and Structures 68 (1998) 499-512.
[10]Bong Jo Ryu,Yoshihiko Sugiyama, Kyung Bin Yim, Gyu Seop Lee. Dynamic stability of an elastically restrained column subjected to triangulary distributed subtangential forces[J]. Computers and Structures 76 (2000) 611-619.
[11]L.Briseghella, C.E.Majorana, C.Pellegrino. Dynamic stability of elastic structures:a finite element approach. Computers and Structures 69 (1998) 11-25.
[12]K.D.Hjelmstad, E.B.Williamson. Dynamic stability of structural systems subjected to base excitation. Engineering Structures,1998,20(1998):425-432.
[13]E.B.Williamson, J.Rungamornrat. Numerical analysis of dynamic stability under random excitation. Engineering Structures 24 (2002):479-490.
[14]M. Darabi, M.Darvizeh, A.Darvizeh. Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading[J], Composite Structures,2007.
[15]M.Javidruzi,A.Vafai,J.F.Chen,J.C.Chilton. Vibration, buckling and dynamic stability of cracked cylindrical shells. Thin-Walled Structures 42 (2004) 79-99.
[16]Dankov EmilT. Dynamic buckling of longth inelastic plate under rapidly applied shear loading. AIAA Journal,1995,34(8):1752～1755
[17]Tanchum Weller. Dynamic buckling of beams and plates subjected to axial impact[J]. Computers and Structures,1989,32 (3):835～851.
[18]Alejandro T. Brewer, Luis A. Godoy. Dynamic buckling of discrete structural systems under combined step and static loads[J]. Nonlinear Dynamics 9:249-264,1996.
[19]叶继红,沈祖炎.单层网壳结构在简单荷载下动力稳定分析[J].哈尔滨工业大学学报,1997,30(1)：24-31.
[20]叶继红,沈祖炎.结构阻尼对网壳结构动力稳定性能的影响[J].工业建筑,1997,27(3)：33-37.
[21]叶继红,沈祖炎.初始缺陷对网壳结构动力稳定性能的影响[J].士木工程学报,1997,30(1)：37-42.
[22]沈祖炎,叶继红.运动稳定性理论在结构动力分析中的应用[J].工程力学,1997,14(3)：21-28.
[23]沈世钊,陈昕.网壳结构稳定性[M].北京：科学出版社,1999.
[24]王策,沈世钊.球面网壳阶跃荷载作用动力稳定性[D],建筑结构学报,2001,22(1)：62-68.
[25]郭海山.单层球面网壳结构动力稳定性及抗震性能研究[D].哈尔滨工业大学,2002.
[26]郭海山,钱宏亮,沈世钊.地震作用下单层球面网壳结构的动力稳定性[J].地震工程与工程振动,2002,23(1)：31-37.
[27]郭海山,王林安,沈世钊.水平阶跃荷载作用下单层球面网壳结构的动力稳定性[J].哈尔滨工业大学学报,2002,35(5)：17-22.
[28]郭海山,沈世钊.单层网壳结构动力稳定性分析方法[J].建筑结构学报,2003,24(3)：1-9.
[29]孙建恒,夏亨熹.网壳结构非线性动力稳定分析.空间结构,1994,1(1)：25-31.
[30]赵淑丽,孙建恒,孙超.点支承两向叉筒单层网壳结构非线性动力稳定分析.空间结构,2007,13(2)：1-5.
[31]姚洪涛,靳路明,孙建恒等.单层柱面正交异型网壳的非线性动力稳定全过程分析[J].河北农业大学学报,2002,29(4)：11.
[32]孙强.直杆的动力稳定性(Ⅰ-Ⅲ)[J],安徽建筑工业学院学报(自然科学版),1996,4(1-3).
[33]孙强.斜杆的动力稳定性[J],力学与实践,1997,19(5)：63-65.
[34]夏鹭平.截锥形直杆的动力稳定性研究[J],合肥工业大学学报(自然科学版),2001,24(2)：194-197.
[35]殷学纲.梁的动力稳定性分析的有限元方法[J].应用力学学报,1995,12(4)：82-87.
[36]戴仰山,彭诚洋,闫发锁.梁在轴向流固冲击作用下的弹塑性动力屈曲[J].哈尔滨工程大学学报,2006,27(2)：6-9.
[37]冯志华,胡海岩.内共振条件下直线运动梁的动力稳定性[J].力学学报,2002,34(3)：389-400.
[38]肖刚,李四平.直杆动力屈曲的计算机模拟[J],上海交通大学学报,2008,42(6)：5001-8001.
[39]何艳丽.桅杆结构的动力稳定性分析[D],同济大学博士学位论文,1999.
[40]何艳丽,王增春,董石麟等.桅杆结构的动力稳定性分析[J],四川建筑科学研究,2001,Vol.27(3)：1-4
[41]何艳丽.王肇民等.桅杆结构的动力稳定性分析[J],四川建筑科学研究,2001,27(3)：1-3.
[42]张其林,Udo Peil任意激励下弹性结构的稳定分析[J].土木工程学报,1998,31(1).：26-32.
[43]刘迎春,单层钢框架动力稳定性与防倒塌设计理论研究[D].大庆石油学院,2006.
[44]计静,地震作用下单层钢框架动力稳定性研究[D].大庆石油学院,2003.
[45]刘迎春,张文福,计静.单层框架的动力稳定性判据.油气田地面工程,2006,25(12)：31.
[46]曲淑英.大型空间框架结构的动力稳定性分析[J],世界地震工程.1998,14(3)：44-48.
[47]曲乃泗,曲淑英.空间框架结构动力不稳定区域的确定[J].应用力学学报,1999,16(2)：108-111.
[48]秦荣.大型复杂结构非线性分析的新理论新方法[M],北京：科学技术出版社,2011.
[49]Paul Dierckx, Eduard Smets, Robert Piessens. A new method of cubature using spline functions[J], Ocean Dynamics,1981,Vol.34(2):61-79.
[50]Christian H. Reinsch.Smoothing by spline functions. II[J], Numerische Mathematik,1971, Vol.16(5):, 451-454.
[51]李岳生,齐东旭.样条函数方法[M],北京：科学出版社,1979.
[52]王省富等编著,样条函数及其应用[M],西安：西北：工业大学出版社,1989.9
[53]秦荣.结构力学的样条函数方法[M],南宁：广西人民出版社,1985.
[54]秦荣,样条有限点法[J],数值计算与计算机应用,1981,2(2)
[55]江世永,薛顺应.结构分析的样条函数方法[J],力学进展,Vol.25,No.2,1990,206-210.
[56]秦荣,陈祥福,梁福崇.结构力学的样条函数方法及其最新进展[J],工程学,1991,vol.8,No.1：36-43.
[57]Tan Fu-qi. A simplified method for the construction of the cubic spline function [J], Applied Mathematics and Mechanics,1985,Vol.6,No.6:599:604.
[58]Giovanna Pittaluga, Laura Sacripante. Quintic spline interpolation on uniform meshes[J],Acta Mathematica Hungarica,1996,Vol.72,No.3:167-175.
[59]Fernanda Patricio. Cubic spline functions and initial value problems [J], BIT Numerical Mathematics, 1978,Vol18.(3):342-347.
[60]侯云山,刘宏兵.三次样条函数的计算机求解[J].中州大学学报,2004,No.3
[61]谭福期.一个构造三次样条函数的简便方法[J],应用数学和力学,1985,vol.6,No.6：567-572.
[62]衣娜.构造B样条函数的积分方法及其具体运算[J],广西民族大学学报(自然科版),2007,vol.13,No.1：66-69.
[63]秦荣.计算结构力学[M],北京：科学技术出版社,2001.
[64]Gerhard Opfer, Hans Joachim Oberle.The derivation of cubic splines with obstacles by methods of optimization and optimal control [J],Numerische Mathematik,1987,Vol.52,No.1:17-31.
[65]唐建国,蒋九林.五次样条函数的构造[J],郑州大学学报(理学版),2005,vol.37,No.3：15-18.
[66]蒙世奎.用插值多项式构造五次、三次样条函数[J],广西民族学院学报,1996,vol.2.No.1：49-55.
[67]秦荣.弹性力学问题的QR法[J],广西力学学会学术报告,1984.
[68]Qin Rong, Spline Finite-Point Method in Cylindrical Shell Analysis[J], Pro. of International Conference on Computer-Aided Analysis and Design of Concrete Structures,17th-21st, September, 1984, Split, Yugoslavia.
[69]Y.K.Cheung, S.Swaddiwudhipong. Analysis of tall frame structure using finite strip method[J], International Journal of Structure,1981, Vol.1(4).
[70]秦荣.高层建筑结构分析的新方法[J].工程力学,1991,8(4),P41-50.
[71]秦荣,梁爱明.高层框架分析的新方法[J],工程力学,1988,5(2).
[72]秦荣.结构分析的QR法[J],《解析与数值结合法的理论及其应用》,长沙：湖南大学出版社,1989.
[73]赵艳林.结构分析的样条函数方法[J],广西大学学报(自然科学版),1993,vol.18,No.3：52-57.
[74]秦荣.超限高层建筑结构分析的QR法[M].北京：科学技术出版社,2010.
[75]刘建秀.框架等结构极限分析的样条函数法[J],郑州轻工业学院学报,1994,vol.9,No.1：39-44.
[76]李秀梅.高层钢框架结构分析的新方法研究[D],广西大学博士学位论文(导师秦荣),2009.
[77]秦荣,李秀梅,高层框架双重非线性分析的QR法[J],计算力学学报,1997,14(增刊)
[78]秦荣,高层建筑结构弹塑性分析的新方法[J],土木工程学报,1994,27(6)
[79]王晓洁.高层框架结构几何非线性稳定分析的QR法[D].广西大学硕士学位论文,2005.
[80]倪骁慧.考虑初始几何缺陷的高层框架结构非线性静力稳定性分析的QR法[D].广西大学硕士学位论文,2006.5.
[81]钟咏梅,高日.钢框架结构非线性力学分析模型综述[J],钢结构,2005,Vol.20(78)：1-4.
[82]郑廷银,赵惠麟.空间钢框架结构的改进双重非线性分析[J],工程力学,2003,Vol.20(6)：202-208.
[83]赵金城,沈祖炎.钢框架弹塑性非线性有限元分析方法的改进[J],钢结构,1994,Vol.9(26)：248-252.
[84]张文元,张耀春.高层钢结构双重非线性分析的塑性铰法[J],哈尔滨建筑大学学报,2000,Vol.33(6)：1-7.
[85]Donald W. White.Plastic-Hinge methods for advanced analysis of steel frames[J],Journal of Constructional Steel Research,1993,Vol.24(2):121-152.
[86]G. Cocchetti, G. Maier. Elastic-plastic and limit-state analyses of frames with softening plastic-hinge models by mathematical programming[J],International Journal of Solids and Structures, 2003,Vol.40(25):7219-7244.
[87]Maria Eugenia Marante, Ricardo Picon, Julio Florez-Lopez. Analysis of localization in frame members with plastic hinges [J], International Journal of Solids and Structures,2004, Vol.41(14):3961-3975.
[88]Cuong Ngo-Huu, Seung-Eock Kim, Jung-Ryul Oh. Nonlinear analysis of space steel frames using fiber plastic hinge concept[J], Engineering Structures,2007,Vol.29(4):649-657.
[89]闫月梅.多高层钢结构框架的整体稳定分析[J].西安科技学院学报,2003,Vol.22(2)：382-385.
[90]程鹏,童根树.单跨两层框架的稳定性[J].轻钢结构,2001,Vol.22(2)：15-18.
[91]陈骥.刚架平面稳定的整体设计法[J].钢结构,2003,Vol.18(4)：46-50.
[92]李国强,沈祖炎,钢结构框架体系弹性及弹塑性分析与计算理论[M],上海：上海科学技术出版社,1998.
[93]张其林,沈祖炎.钢杆件极限承载全过程的样条函数解法[J],建筑结构学报,1996,vol.17,No.1：46-51
[94]Geng-shu Tong, Lei Zhang, Guo-ran Xing.Inelastic storey-buckling factor of steel frames[J],Journal of Constructional Steel Research,2009,Vol.65(2):443-451.
[95]Donald W. White, Jerome F. Hajjar. Stability of steel frames:the cases for simple elastic and rigorous inelastic analysis/design procedures [J], Engineering Structures,2000,Vol.22(2):155-167.
[96]Siu Lai Chan. Inelastic post-buckling analysis of tubular beam-columns and frames[J],Engineering Structures,1989,Vol.11(1):23-30.
[97]蒋友谅.非线性有限元法[M].北京：北京工业学院出版社,1988.
[98]李庆扬,关治,白峰杉.数值计算原理[M].北京：清华大学出版社,2000.
[99]R.克拉夫,J.彭津,王光远等泽校.结构动力学第二版(修订版)[M],高等教育出版社,2007.
[100]包世华.结构动力学[M],武汉理工大学出版社,2007.
[101]陈伶莉.工程结构动力分析数值方法[M].西安：西安交通大学出版社,2006.
[102]孔向东,钟万勰.非线性动力系统刚性方程精细时程积分法[J],大连理工大学学报,2002,06.
[103]梁启智,李逊.框架结构非线性动力时程分析的一个方法[J],工程力学,1990,Vol.7,No.2：62-70.
[104]曲大庄,邹经湘,黄文虎等.用杜哈密几分积分的样条函数方法求解结构动力响应[J],振动工程学报,1992,Vo5(4)：367-371.
[105]梁启智,李逊.框架结构非线性动力时程分析的一个方法[J],工程力学,1990,02.
[106]王焕定,张永山,王伟等.非线性结构时程分析的高阶单步法[J],地震工程与工程振动,1996,03.
[107]H. Antes, M. Schanz, S. Alvermann. Dynamic analyses of plane frames by integral equations for bars and Timoshenko beams[J],Journal of Sound and Vibration,2004,Vol.276(3-5):807-836.
[108]Mario Paz著,李裕澈,刘勇生等译.结构动力学理论与计算[M],北京：地震出版社,1993.
[109]李秀梅,秦荣,王建军.结构地震响应分析的高斯精细时程积分法[J].四川建筑科学研究,2009,(02).
[110]王海波,余志武,陈伯望.非线性动力分析避免状态矩阵求逆的精细积分多步法[J],振动与冲击,2008,vo1.27,No.4：105-107.
[111]付朝江.结构非线性动力分析混合时间积分并行算法[J],力学与实践,2010,vo1.32,No.3：96-100.
[112]储德文,王元丰.精细直接积分法的积分方法选择[J],[程力学,2002,vol.19,No.6：115-119.
[113]张琳,聂孟喜,仝辉.三次和五次B样条函数在动力响应分析中的应用[J].清华大学学报(自然科学版),2006,No.3.
[114]殷学纲,李宾.样条函数——加权残数直接积分法[J],振动工程学报,1988,vol.1,No.4：82-87.
[115]周亮,谢习华.求解结构物动力响应的样条函数加权残值法[J],长沙铁道学院学报,2003,vo1.21,No.3：90-95.
[116]Qin Rong, Spline Function Method for Dynamic Response[J], First World Congress on Computational Mechanics,1986, U.S.
[117]秦荣.结构动力问题的样条函数方法[J],广西大学学报(自然科学版),1983,No.1：91-102.
[118]黄绍派.高层建筑结构弹塑性动力分析的QR法及其工程应用法[D].广西大学硕士学位论文,2002.
[119]Wang Gang, Cheng-Tzu, Hsu Thomas. Static and dynamic analysis of arbitrary quadrilateral flexural plates by B-spline functions[J],International Journal of Solids and Structures,1994, Vol.31(5): 657-667.
[120]秦荣.求解结构动力反应的样条函数法[J],工程力学,1985,vol.2,No.2：52-63.
[121]秦荣.计算结构动力学[M],桂林：广西师范大学出版社,1997.
[122]贺拥军,董石麟.网架结构竖向地震反应的动态样条加权残数法分析[J],工业建筑,2001,03.
[123]陈淮.样条加权残数法求解非线性结构动力响应问题[J],铁道科学与工程学报,1993,01.
[124]陈淮.求解偏微分方程的样条加权残数法[J],铁道科学与工程学报,1990,02.
[125]殷学纲,李宾.样条函数一加权残数直接积分法[J],振动工程学报,1988,04.
[126]秦荣.无条件稳定的动态样条加权残数法[J],工程力学,1990,01.
[127]李君,张耀春.高层钢结构的非线性动力全过程分析方法[J],哈尔滨建筑大学学报,2000,Vo133(2)：16-19.
[128]刘锋,吕西林.冲击载荷作用下框架结构的非线性动力响应[J],振动工程学报,2008年,02期.
[129]L. Bousshine, A. Chaaba, G. De Saxce. Plastic limit load of plane frames with frictional contact supports Original Research Article[J],International Journal of Mechanical Sciences,2002, Vol.44(11):2189-2216.
[130]孙文俊.考虑大变形影响的框架极限荷载计算[J],河海大学学报,1992,Vol.20,No.5：6-13.
[131]余平.商用多层框架结构按弹塑性分析的极限荷载[J],郑州粮食学院学报,1995,Vo1.16,No.2：70-77.
[132]张昕,蒋通.框架结构极限荷载分析的便携解法[J],力学季刊,2001,Vol.22,No.2：252-257.
[133]Claudia C. Marin-Artieda, Gary F. Dargush. Approximate limit load evaluation of structural frames using linear elastic analysis[J],Engineering Structures,2007,Vol.29(3):296-304.
[134]沈祖炎.钢结构稳定、抗震与非线性分析理论[M].北京：中国建筑工业出版社,2005.
[135]沈世钊,陈昕.网壳结构稳定性[M].北京：科学出版社,1999.
[136]王策,沈世钊.单层球面网壳动力稳定性分析[J].土木工程学报,2000,33(6)：17-24.
[137]李忠学,沈祖炎,邓长根.杆系钢结构非线性动力稳定性识别与判定准则[J],同济大学学报(自然科学版),2000,Wol.28 No.2：148-151.
[138]李双蓓,刘立国,倪骁慧.结构稳定性研究的现状和新方法的探索.广西大学学报(自然科学版),2004,29(增)：107-110.
[139]李忠学,李元齐,严慧,季渊.结构非线性动力稳定性研究中的关键问题探讨[J].空间结构,2000,(04)
[140]韩强,张善元,杨桂通.结构静动力屈曲问题研究进展[J],力学进展,1998,28(3)：943-953.
[141]彭英,杨平.结构动力屈曲的研究进展[J],武汉理工大学学报(交通科学与工程版),2007,31(2)：328-332.
[142]刘迎春,张文福,计静.单层框架的动力稳定性判据.油气田地面工程,2006,25(12)：31.
[143]曲淑英.大型空间框架结构的动力稳定性分析[J],世界地震工程.1998,14(3)：44-48.
[144]卓曙君,周科健.结构动力稳定性的若干问题[J],国防科技大学学报,1988,10(3)：100-105.
[145]梁建文,何玉敖.一个实用的结构动力失稳判别准则[J],工程力学,1995,12(2)
[146]吴玉华,亓兴军,郭剑飞.拱结构动力稳定实用判别方法[J],地震工程与工程振动,2010,Vo1.30(4)：31-36.
[147]L.Briseghella, C.E.Majorana, C.Pellegrino. Dynamic stability of elastic structures:a finite element approach. Computers and Structures 69 (1998) 11-25.
[148]Anthony N. Kounadis. Dynamic buckling of simple two-bar frames using catastrophe theory[J],International Journal of Non-Linear Mechanics,2002,Vol.37(7):1249-1259.
[149]李忠学,沈祖炎,邓长根.广义位移控制法在动力稳问题中的应用[J].同济大学学报,1998,26(6)：609-612.
[150]王策,沈世钊.球面网壳阶跃荷载作用动力稳定性[D],建筑结构学报,2001,22(1)：62-68.
[151]Li Qing-ning. Dynamic buckling mode of an elastic bar[J].Applied Mathematics and Mechanics((English Edition,).1990,11(1):63-68.
[152]赵洪金,刘超,董宁娟.周期荷载作用下组合压杆动力稳定性分析[J],建筑结构,2011,Vo1.41(2)：68-70.
[153]黄友钦,顾明,周咂毅.积雪漂移对空间结构动力稳定性的影响[J],振动与冲击,2011,Vo1.30(2)：124-128.
[154]赵洪金,刘超,孟昭博等.周期径向荷载作用下圆弧格构拱动力稳定性分析[J],地震工程与工程振动,2011,Vo1.31(6)：73-77.
[155]谌磊.弧形钢闸门空间框架的动力稳定性分析[D].西北农林科技大学硕士学位论文,2006.
[156]杨帆,杜文风.输电铁塔结构非线性动力稳定研究的探讨[J],电力建设,2003,(12).
[157]曲乃泅,张斌.任意空间杆系结构的动力稳定性分析[J].工业建筑,1990,(05).
[158]白海洋韩志军涂安全等.阶跃载荷作用下弹塑性杆的动力屈曲[J].科学技术与工程,2008,8(8)：6512-7512.
[159]龚良贵.冲击荷载作用下弹性直杆动力屈曲问题研究[J],南昌大学学报(工科版).2001,23(2)：30-34.
[160]戴平.多层厂房刚架结构动力稳定性的试验研究[J].武汉工业大学学报,1997,19(3)：116-118.
[161]刘迎春,单层钢框架动力稳定性与防倒塌设计理论研究[D].大庆石油学院,2006.
[162]计静,地震作用下单层钢框架动力稳定性研究[D].大庆石油学院,2003.
[2]秦荣.工程结构非线性[M],北京：科学技术出版社,2006.
[3]韩强,张善元,杨桂通.结构静动力屈曲问题研究进展[J],力学进展,1998,28(3)：943-953.
[4]符华.鲍络金.弹性体系的动力稳定性[M].林砚田,译.北京：高等教育出版社,1990.
[5]曲乃泗,张斌.任意空间杆系结构的动力稳定性分析[J].工业建筑,1990,(5)：24-28.
[6]Bernard Budiansky. Theory of Buckling and Post-Buckling Behavior of Elastic Structures[J]. Advances in Applied Mechanics, Volume 14,1974:1-65.
[7]B.Budiansky, R.S.Roth. Axisymmetric Dynamic Buckling of Clamped Shallow Spherical Shells. Collect-ed Papers on Instability of Shells Structures. NASA TND-1510, Washington D.C,1962:597-606.
[8]Simitses G J. Dynamic stability of suddenly loaded structures. New York:Springer-Verlag,1990.
[9]Bong Jo Ryu, Kazuo Katayama, Yoshihiko Sugiyama.Dynamic stability of Timoshenko columns subjected to subtangential forces[J]. Computers and Structures 68 (1998) 499-512.
[10]Bong Jo Ryu,Yoshihiko Sugiyama, Kyung Bin Yim, Gyu Seop Lee. Dynamic stability of an elastically restrained column subjected to triangulary distributed subtangential forces[J]. Computers and Structures 76 (2000) 611-619.
[11]L.Briseghella, C.E.Majorana, C.Pellegrino. Dynamic stability of elastic structures:a finite element approach. Computers and Structures 69 (1998) 11-25.
[12]K.D.Hjelmstad, E.B.Williamson. Dynamic stability of structural systems subjected to base excitation. Engineering Structures,1998,20(1998):425-432.
[13]E.B.Williamson, J.Rungamornrat. Numerical analysis of dynamic stability under random excitation. Engineering Structures 24 (2002):479-490.
[14]M. Darabi, M.Darvizeh, A.Darvizeh. Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading[J], Composite Structures,2007.
[15]M.Javidruzi,A.Vafai,J.F.Chen,J.C.Chilton. Vibration, buckling and dynamic stability of cracked cylindrical shells. Thin-Walled Structures 42 (2004) 79-99.
[16]Dankov EmilT. Dynamic buckling of longth inelastic plate under rapidly applied shear loading. AIAA Journal,1995,34(8):1752～1755
[17]Tanchum Weller. Dynamic buckling of beams and plates subjected to axial impact[J]. Computers and Structures,1989,32 (3):835～851.
[18]Alejandro T. Brewer, Luis A. Godoy. Dynamic buckling of discrete structural systems under combined step and static loads[J]. Nonlinear Dynamics 9:249-264,1996.
[19]叶继红,沈祖炎.单层网壳结构在简单荷载下动力稳定分析[J].哈尔滨工业大学学报,1997,30(1)：24-31.
[20]叶继红,沈祖炎.结构阻尼对网壳结构动力稳定性能的影响[J].工业建筑,1997,27(3)：33-37.
[21]叶继红,沈祖炎.初始缺陷对网壳结构动力稳定性能的影响[J].士木工程学报,1997,30(1)：37-42.
[22]沈祖炎,叶继红.运动稳定性理论在结构动力分析中的应用[J].工程力学,1997,14(3)：21-28.
[23]沈世钊,陈昕.网壳结构稳定性[M].北京：科学出版社,1999.
[24]王策,沈世钊.球面网壳阶跃荷载作用动力稳定性[D],建筑结构学报,2001,22(1)：62-68.
[25]郭海山.单层球面网壳结构动力稳定性及抗震性能研究[D].哈尔滨工业大学,2002.
[26]郭海山,钱宏亮,沈世钊.地震作用下单层球面网壳结构的动力稳定性[J].地震工程与工程振动,2002,23(1)：31-37.
[27]郭海山,王林安,沈世钊.水平阶跃荷载作用下单层球面网壳结构的动力稳定性[J].哈尔滨工业大学学报,2002,35(5)：17-22.
[28]郭海山,沈世钊.单层网壳结构动力稳定性分析方法[J].建筑结构学报,2003,24(3)：1-9.
[29]孙建恒,夏亨熹.网壳结构非线性动力稳定分析.空间结构,1994,1(1)：25-31.
[30]赵淑丽,孙建恒,孙超.点支承两向叉筒单层网壳结构非线性动力稳定分析.空间结构,2007,13(2)：1-5.
[31]姚洪涛,靳路明,孙建恒等.单层柱面正交异型网壳的非线性动力稳定全过程分析[J].河北农业大学学报,2002,29(4)：11.
[32]孙强.直杆的动力稳定性(Ⅰ-Ⅲ)[J],安徽建筑工业学院学报(自然科学版),1996,4(1-3).
[33]孙强.斜杆的动力稳定性[J],力学与实践,1997,19(5)：63-65.
[34]夏鹭平.截锥形直杆的动力稳定性研究[J],合肥工业大学学报(自然科学版),2001,24(2)：194-197.
[35]殷学纲.梁的动力稳定性分析的有限元方法[J].应用力学学报,1995,12(4)：82-87.
[36]戴仰山,彭诚洋,闫发锁.梁在轴向流固冲击作用下的弹塑性动力屈曲[J].哈尔滨工程大学学报,2006,27(2)：6-9.
[37]冯志华,胡海岩.内共振条件下直线运动梁的动力稳定性[J].力学学报,2002,34(3)：389-400.
[38]肖刚,李四平.直杆动力屈曲的计算机模拟[J],上海交通大学学报,2008,42(6)：5001-8001.
[39]何艳丽.桅杆结构的动力稳定性分析[D],同济大学博士学位论文,1999.
[40]何艳丽,王增春,董石麟等.桅杆结构的动力稳定性分析[J],四川建筑科学研究,2001,Vol.27(3)：1-4
[41]何艳丽.王肇民等.桅杆结构的动力稳定性分析[J],四川建筑科学研究,2001,27(3)：1-3.
[42]张其林,Udo Peil任意激励下弹性结构的稳定分析[J].土木工程学报,1998,31(1).：26-32.
[43]刘迎春,单层钢框架动力稳定性与防倒塌设计理论研究[D].大庆石油学院,2006.
[44]计静,地震作用下单层钢框架动力稳定性研究[D].大庆石油学院,2003.
[45]刘迎春,张文福,计静.单层框架的动力稳定性判据.油气田地面工程,2006,25(12)：31.
[46]曲淑英.大型空间框架结构的动力稳定性分析[J],世界地震工程.1998,14(3)：44-48.
[47]曲乃泗,曲淑英.空间框架结构动力不稳定区域的确定[J].应用力学学报,1999,16(2)：108-111.
[48]秦荣.大型复杂结构非线性分析的新理论新方法[M],北京：科学技术出版社,2011.
[49]Paul Dierckx, Eduard Smets, Robert Piessens. A new method of cubature using spline functions[J], Ocean Dynamics,1981,Vol.34(2):61-79.
[50]Christian H. Reinsch.Smoothing by spline functions. II[J], Numerische Mathematik,1971, Vol.16(5):, 451-454.
[51]李岳生,齐东旭.样条函数方法[M],北京：科学出版社,1979.
[52]王省富等编著,样条函数及其应用[M],西安：西北：工业大学出版社,1989.9
[53]秦荣.结构力学的样条函数方法[M],南宁：广西人民出版社,1985.
[54]秦荣,样条有限点法[J],数值计算与计算机应用,1981,2(2)
[55]江世永,薛顺应.结构分析的样条函数方法[J],力学进展,Vol.25,No.2,1990,206-210.
[56]秦荣,陈祥福,梁福崇.结构力学的样条函数方法及其最新进展[J],工程学,1991,vol.8,No.1：36-43.
[57]Tan Fu-qi. A simplified method for the construction of the cubic spline function [J], Applied Mathematics and Mechanics,1985,Vol.6,No.6:599:604.
[58]Giovanna Pittaluga, Laura Sacripante. Quintic spline interpolation on uniform meshes[J],Acta Mathematica Hungarica,1996,Vol.72,No.3:167-175.
[59]Fernanda Patricio. Cubic spline functions and initial value problems [J], BIT Numerical Mathematics, 1978,Vol18.(3):342-347.
[60]侯云山,刘宏兵.三次样条函数的计算机求解[J].中州大学学报,2004,No.3
[61]谭福期.一个构造三次样条函数的简便方法[J],应用数学和力学,1985,vol.6,No.6：567-572.
[62]衣娜.构造B样条函数的积分方法及其具体运算[J],广西民族大学学报(自然科版),2007,vol.13,No.1：66-69.
[63]秦荣.计算结构力学[M],北京：科学技术出版社,2001.
[64]Gerhard Opfer, Hans Joachim Oberle.The derivation of cubic splines with obstacles by methods of optimization and optimal control [J],Numerische Mathematik,1987,Vol.52,No.1:17-31.
[65]唐建国,蒋九林.五次样条函数的构造[J],郑州大学学报(理学版),2005,vol.37,No.3：15-18.
[66]蒙世奎.用插值多项式构造五次、三次样条函数[J],广西民族学院学报,1996,vol.2.No.1：49-55.
[67]秦荣.弹性力学问题的QR法[J],广西力学学会学术报告,1984.
[68]Qin Rong, Spline Finite-Point Method in Cylindrical Shell Analysis[J], Pro. of International Conference on Computer-Aided Analysis and Design of Concrete Structures,17th-21st, September, 1984, Split, Yugoslavia.
[69]Y.K.Cheung, S.Swaddiwudhipong. Analysis of tall frame structure using finite strip method[J], International Journal of Structure,1981, Vol.1(4).
[70]秦荣.高层建筑结构分析的新方法[J].工程力学,1991,8(4),P41-50.
[71]秦荣,梁爱明.高层框架分析的新方法[J],工程力学,1988,5(2).
[72]秦荣.结构分析的QR法[J],《解析与数值结合法的理论及其应用》,长沙：湖南大学出版社,1989.
[73]赵艳林.结构分析的样条函数方法[J],广西大学学报(自然科学版),1993,vol.18,No.3：52-57.
[74]秦荣.超限高层建筑结构分析的QR法[M].北京：科学技术出版社,2010.
[75]刘建秀.框架等结构极限分析的样条函数法[J],郑州轻工业学院学报,1994,vol.9,No.1：39-44.
[76]李秀梅.高层钢框架结构分析的新方法研究[D],广西大学博士学位论文(导师秦荣),2009.
[77]秦荣,李秀梅,高层框架双重非线性分析的QR法[J],计算力学学报,1997,14(增刊)
[78]秦荣,高层建筑结构弹塑性分析的新方法[J],土木工程学报,1994,27(6)
[79]王晓洁.高层框架结构几何非线性稳定分析的QR法[D].广西大学硕士学位论文,2005.
[80]倪骁慧.考虑初始几何缺陷的高层框架结构非线性静力稳定性分析的QR法[D].广西大学硕士学位论文,2006.5.
[81]钟咏梅,高日.钢框架结构非线性力学分析模型综述[J],钢结构,2005,Vol.20(78)：1-4.
[82]郑廷银,赵惠麟.空间钢框架结构的改进双重非线性分析[J],工程力学,2003,Vol.20(6)：202-208.
[83]赵金城,沈祖炎.钢框架弹塑性非线性有限元分析方法的改进[J],钢结构,1994,Vol.9(26)：248-252.
[84]张文元,张耀春.高层钢结构双重非线性分析的塑性铰法[J],哈尔滨建筑大学学报,2000,Vol.33(6)：1-7.
[85]Donald W. White.Plastic-Hinge methods for advanced analysis of steel frames[J],Journal of Constructional Steel Research,1993,Vol.24(2):121-152.
[86]G. Cocchetti, G. Maier. Elastic-plastic and limit-state analyses of frames with softening plastic-hinge models by mathematical programming[J],International Journal of Solids and Structures, 2003,Vol.40(25):7219-7244.
[87]Maria Eugenia Marante, Ricardo Picon, Julio Florez-Lopez. Analysis of localization in frame members with plastic hinges [J], International Journal of Solids and Structures,2004, Vol.41(14):3961-3975.
[88]Cuong Ngo-Huu, Seung-Eock Kim, Jung-Ryul Oh. Nonlinear analysis of space steel frames using fiber plastic hinge concept[J], Engineering Structures,2007,Vol.29(4):649-657.
[89]闫月梅.多高层钢结构框架的整体稳定分析[J].西安科技学院学报,2003,Vol.22(2)：382-385.
[90]程鹏,童根树.单跨两层框架的稳定性[J].轻钢结构,2001,Vol.22(2)：15-18.
[91]陈骥.刚架平面稳定的整体设计法[J].钢结构,2003,Vol.18(4)：46-50.
[92]李国强,沈祖炎,钢结构框架体系弹性及弹塑性分析与计算理论[M],上海：上海科学技术出版社,1998.
[93]张其林,沈祖炎.钢杆件极限承载全过程的样条函数解法[J],建筑结构学报,1996,vol.17,No.1：46-51
[94]Geng-shu Tong, Lei Zhang, Guo-ran Xing.Inelastic storey-buckling factor of steel frames[J],Journal of Constructional Steel Research,2009,Vol.65(2):443-451.
[95]Donald W. White, Jerome F. Hajjar. Stability of steel frames:the cases for simple elastic and rigorous inelastic analysis/design procedures [J], Engineering Structures,2000,Vol.22(2):155-167.
[96]Siu Lai Chan. Inelastic post-buckling analysis of tubular beam-columns and frames[J],Engineering Structures,1989,Vol.11(1):23-30.
[97]蒋友谅.非线性有限元法[M].北京：北京工业学院出版社,1988.
[98]李庆扬,关治,白峰杉.数值计算原理[M].北京：清华大学出版社,2000.
[99]R.克拉夫,J.彭津,王光远等泽校.结构动力学第二版(修订版)[M],高等教育出版社,2007.
[100]包世华.结构动力学[M],武汉理工大学出版社,2007.
[101]陈伶莉.工程结构动力分析数值方法[M].西安：西安交通大学出版社,2006.
[102]孔向东,钟万勰.非线性动力系统刚性方程精细时程积分法[J],大连理工大学学报,2002,06.
[103]梁启智,李逊.框架结构非线性动力时程分析的一个方法[J],工程力学,1990,Vol.7,No.2：62-70.
[104]曲大庄,邹经湘,黄文虎等.用杜哈密几分积分的样条函数方法求解结构动力响应[J],振动工程学报,1992,Vo5(4)：367-371.
[105]梁启智,李逊.框架结构非线性动力时程分析的一个方法[J],工程力学,1990,02.
[106]王焕定,张永山,王伟等.非线性结构时程分析的高阶单步法[J],地震工程与工程振动,1996,03.
[107]H. Antes, M. Schanz, S. Alvermann. Dynamic analyses of plane frames by integral equations for bars and Timoshenko beams[J],Journal of Sound and Vibration,2004,Vol.276(3-5):807-836.
[108]Mario Paz著,李裕澈,刘勇生等译.结构动力学理论与计算[M],北京：地震出版社,1993.
[109]李秀梅,秦荣,王建军.结构地震响应分析的高斯精细时程积分法[J].四川建筑科学研究,2009,(02).
[110]王海波,余志武,陈伯望.非线性动力分析避免状态矩阵求逆的精细积分多步法[J],振动与冲击,2008,vo1.27,No.4：105-107.
[111]付朝江.结构非线性动力分析混合时间积分并行算法[J],力学与实践,2010,vo1.32,No.3：96-100.
[112]储德文,王元丰.精细直接积分法的积分方法选择[J],[程力学,2002,vol.19,No.6：115-119.
[113]张琳,聂孟喜,仝辉.三次和五次B样条函数在动力响应分析中的应用[J].清华大学学报(自然科学版),2006,No.3.
[114]殷学纲,李宾.样条函数——加权残数直接积分法[J],振动工程学报,1988,vol.1,No.4：82-87.
[115]周亮,谢习华.求解结构物动力响应的样条函数加权残值法[J],长沙铁道学院学报,2003,vo1.21,No.3：90-95.
[116]Qin Rong, Spline Function Method for Dynamic Response[J], First World Congress on Computational Mechanics,1986, U.S.
[117]秦荣.结构动力问题的样条函数方法[J],广西大学学报(自然科学版),1983,No.1：91-102.
[118]黄绍派.高层建筑结构弹塑性动力分析的QR法及其工程应用法[D].广西大学硕士学位论文,2002.
[119]Wang Gang, Cheng-Tzu, Hsu Thomas. Static and dynamic analysis of arbitrary quadrilateral flexural plates by B-spline functions[J],International Journal of Solids and Structures,1994, Vol.31(5): 657-667.
[120]秦荣.求解结构动力反应的样条函数法[J],工程力学,1985,vol.2,No.2：52-63.
[121]秦荣.计算结构动力学[M],桂林：广西师范大学出版社,1997.
[122]贺拥军,董石麟.网架结构竖向地震反应的动态样条加权残数法分析[J],工业建筑,2001,03.
[123]陈淮.样条加权残数法求解非线性结构动力响应问题[J],铁道科学与工程学报,1993,01.
[124]陈淮.求解偏微分方程的样条加权残数法[J],铁道科学与工程学报,1990,02.
[125]殷学纲,李宾.样条函数一加权残数直接积分法[J],振动工程学报,1988,04.
[126]秦荣.无条件稳定的动态样条加权残数法[J],工程力学,1990,01.
[127]李君,张耀春.高层钢结构的非线性动力全过程分析方法[J],哈尔滨建筑大学学报,2000,Vo133(2)：16-19.
[128]刘锋,吕西林.冲击载荷作用下框架结构的非线性动力响应[J],振动工程学报,2008年,02期.
[129]L. Bousshine, A. Chaaba, G. De Saxce. Plastic limit load of plane frames with frictional contact supports Original Research Article[J],International Journal of Mechanical Sciences,2002, Vol.44(11):2189-2216.
[130]孙文俊.考虑大变形影响的框架极限荷载计算[J],河海大学学报,1992,Vol.20,No.5：6-13.
[131]余平.商用多层框架结构按弹塑性分析的极限荷载[J],郑州粮食学院学报,1995,Vo1.16,No.2：70-77.
[132]张昕,蒋通.框架结构极限荷载分析的便携解法[J],力学季刊,2001,Vol.22,No.2：252-257.
[133]Claudia C. Marin-Artieda, Gary F. Dargush. Approximate limit load evaluation of structural frames using linear elastic analysis[J],Engineering Structures,2007,Vol.29(3):296-304.
[134]沈祖炎.钢结构稳定、抗震与非线性分析理论[M].北京：中国建筑工业出版社,2005.
[135]沈世钊,陈昕.网壳结构稳定性[M].北京：科学出版社,1999.
[136]王策,沈世钊.单层球面网壳动力稳定性分析[J].土木工程学报,2000,33(6)：17-24.
[137]李忠学,沈祖炎,邓长根.杆系钢结构非线性动力稳定性识别与判定准则[J],同济大学学报(自然科学版),2000,Wol.28 No.2：148-151.
[138]李双蓓,刘立国,倪骁慧.结构稳定性研究的现状和新方法的探索.广西大学学报(自然科学版),2004,29(增)：107-110.
[139]李忠学,李元齐,严慧,季渊.结构非线性动力稳定性研究中的关键问题探讨[J].空间结构,2000,(04)
[140]韩强,张善元,杨桂通.结构静动力屈曲问题研究进展[J],力学进展,1998,28(3)：943-953.
[141]彭英,杨平.结构动力屈曲的研究进展[J],武汉理工大学学报(交通科学与工程版),2007,31(2)：328-332.
[142]刘迎春,张文福,计静.单层框架的动力稳定性判据.油气田地面工程,2006,25(12)：31.
[143]曲淑英.大型空间框架结构的动力稳定性分析[J],世界地震工程.1998,14(3)：44-48.
[144]卓曙君,周科健.结构动力稳定性的若干问题[J],国防科技大学学报,1988,10(3)：100-105.
[145]梁建文,何玉敖.一个实用的结构动力失稳判别准则[J],工程力学,1995,12(2)
[146]吴玉华,亓兴军,郭剑飞.拱结构动力稳定实用判别方法[J],地震工程与工程振动,2010,Vo1.30(4)：31-36.
[147]L.Briseghella, C.E.Majorana, C.Pellegrino. Dynamic stability of elastic structures:a finite element approach. Computers and Structures 69 (1998) 11-25.
[148]Anthony N. Kounadis. Dynamic buckling of simple two-bar frames using catastrophe theory[J],International Journal of Non-Linear Mechanics,2002,Vol.37(7):1249-1259.
[149]李忠学,沈祖炎,邓长根.广义位移控制法在动力稳问题中的应用[J].同济大学学报,1998,26(6)：609-612.
[150]王策,沈世钊.球面网壳阶跃荷载作用动力稳定性[D],建筑结构学报,2001,22(1)：62-68.
[151]Li Qing-ning. Dynamic buckling mode of an elastic bar[J].Applied Mathematics and Mechanics((English Edition,).1990,11(1):63-68.
[152]赵洪金,刘超,董宁娟.周期荷载作用下组合压杆动力稳定性分析[J],建筑结构,2011,Vo1.41(2)：68-70.
[153]黄友钦,顾明,周咂毅.积雪漂移对空间结构动力稳定性的影响[J],振动与冲击,2011,Vo1.30(2)：124-128.
[154]赵洪金,刘超,孟昭博等.周期径向荷载作用下圆弧格构拱动力稳定性分析[J],地震工程与工程振动,2011,Vo1.31(6)：73-77.
[155]谌磊.弧形钢闸门空间框架的动力稳定性分析[D].西北农林科技大学硕士学位论文,2006.
[156]杨帆,杜文风.输电铁塔结构非线性动力稳定研究的探讨[J],电力建设,2003,(12).
[157]曲乃泅,张斌.任意空间杆系结构的动力稳定性分析[J].工业建筑,1990,(05).
[158]白海洋韩志军涂安全等.阶跃载荷作用下弹塑性杆的动力屈曲[J].科学技术与工程,2008,8(8)：6512-7512.
[159]龚良贵.冲击荷载作用下弹性直杆动力屈曲问题研究[J],南昌大学学报(工科版).2001,23(2)：30-34.
[160]戴平.多层厂房刚架结构动力稳定性的试验研究[J].武汉工业大学学报,1997,19(3)：116-118.
[161]刘迎春,单层钢框架动力稳定性与防倒塌设计理论研究[D].大庆石油学院,2006.
[162]计静,地震作用下单层钢框架动力稳定性研究[D].大庆石油学院,2003.