Research On The Second-Order Effect Of Reinforced Concrete Bent-Frames And The Effective Length Of Bent-Columns

Further research on the second-order effect of reinforced concrete variable bentcolumns is required among problems of the structural second-order effect. Thesecond-order effect of pin-jointed bent columns has hitherto been included only inSoviet and Chinese design codes. From1970s to1990s, Shao-long Hu and Chuan-zhiZhao et al conducted the overall instability analysis of first-order variable bent columnsutilizing the initial parameter method and provided the suggested effective lengthcoefficient calculated by the initial parameter method. However, this coefficient was notappropriate due to their improper judgment on the concept of the research. In thebeginning of the21st century, Zhan-you Zhu and Yi Liu from the research team towhich the author belongs one after another conducted a new series of systematicresearch and analyses on the second-order effect of bent frame and accomplished manysignificant research achievements. While after taking all factors into consideration, therevision group of Chinese Standard,“Code for Design of Concrete Structure”（vision2010）, think that these existing research finding still need to be justified and perfectedand hence the former effective length of bent column remains. Henceforth, provisionsfor the effective length of pin-jointed bent column in the current Chinese “Code forDesign of Concrete Structure” have been applied for more than forty years without anyrevisions, which lack analytical evidences.Under this background, this paper conducts a further analyses and research on lawsof the second-order effect of bent frame structures and the quantification of the effectivelength of bent columns on the basis of the former research and the findings are asfollows:①In the regard of the structural analysis of the second order effect, the η-l₀method, the finite element method based on the geometric stiffness （also named thedummy column method） and the finite element iterative method based on equivalentlater force are summarized and introduced here. This paper conducts a theoreticalderivation and a detail introduction of the implementation process of applying the finiteelement method based on the geometric stiffness to resolve the P-△effect ofstructures, and a prismatic cantilever is utilized to display this process. Meanwhile,expanding the application of this method, we propose and demonstrate that this methodcan be utilized to consider both the P-△effect and the P-δ effect of structures by means of subdividing the element. Besides, it is further demonstrated that the acquiredanalytical results from applying the iterative method and the virtual column method toconsider the second order effect of structures are uniform. And we use the iterativemethod to analyze the detail principle of the virtual column method.②Taking the bent span, the crane tonnage, the magnitude of wind load and thelength ratio β of the upper bent column to the lower bent column as variables, wesystematically study the impact of these factors on laws of the second order effect ofbent frame structures and the quantification of the effective length coefficient of bentcolumns through the elastic first order and second order analyses on a series of bentframe structure cases with single floor, singe span and the crane. At the same time, lawsof the second order effect of bent frame structures and the quantification of the effectivelength coefficient of bent columns are analyzed, on the basis of which we propose asuggestion of the quantification of the effective length coefficient of bent columns onthe bent direction for bent frame structures with single floor, singe span and the craneand theoretically analyze the reason why the effective length coefficient of bentcolumns is greater than2.0.③It is justified that the P-△effect does not increase the interior shear of bentcolumns by the bet frame structure cases with single floor, singe span and the crane andthe reason is that the P-△effect is caused by the vertical load and the horizontaldisplacement and that technically it does not effect the equilibrium conditions of thestructures with the effect of lateral forces.④Laws of the second order effect of single-floor bent frame structures of industrialbuildings which have two equal spans and cranes and the quantification of the effectivelength coefficient of bent columns are preliminarily studied and we recognize that theimpact of the length ratio of the upper bent column to the lower bent column onlaws of the second order effect of this kind of bent frame structures and thequantification of the effective length coefficient of bent columns is uniform with thatimpact for bent frame structures with single floor and single span. Simultaneously, incomparison with bent frame structures with single floor and single span, laws of thesecond order effect of this kind of bent frame structures and the quantification of theeffective length coefficient of bent columns are more complicated. The effective lengthcoefficient of the peripheral bent columns can be either greater or small than that of thebent frame structures with single floor and single span. In contrast, the effective lengthcoefficient of interior columns is usually smaller, and the value is less than1.0in most cases.