A new proportioning method for member sections of single layer reticulated domes subjected to uniform and non-uniform loads
Introduction
Buckling is one of the most important parameters in the structural design of single layer reticulated domes because these domes are apt to buckle and abruptly lose their load bearing capacities after buckling. It is affected by various structural parameters: geometrical shape of domes, support conditions, rigidities at connection between members, members slenderness ratio, half-subtended angle for members, initial imperfection, etc. Also, it is strongly affected by loading conditions [1], [2]. Some of the studies on these effects have been conducted by several researchers [3], [4], [5], [7], [8] and these effects on buckling as well as on the structural behaviors have been made more clear by the results. Also, complicated buckling characteristics of single layer reticulated domes can be revealed clearly and accurately by complete geometrical and material non-linear analysis. Sometimes, the structural designers are still puzzled in dealing with buckling problems of single layer reticulated domes, so a simple and convenient design procedure with the recent theoretical advances has been in demand.
Kato et al. [9], [10] proposed a proportioning method for member sections of a single layer reticulated dome subjected to uniform load and discussed the validity and effectiveness of the method which was based on the second-order elastic analysis.
However, they were restricted to uniform loads. Unfortunately, the domes subjected to non-uniform loads have never particularly been the subject of researches on how the buckling loads on the surface of domes are expressed by the generalized slenderness ratio of members.
This paper begins with a proposal for an alternative method to upgrade the previous method based on linear buckling stress and a knock down factor [6], [7], [11], [12], [13] which expresses the effects of shell-like buckling, since the linear buckling analysis is more familiar and easier to use than the second-order elastic analysis which is required for nonlinear analysis. And several domes with different member slenderness ratios and different shallowness are designed for the required ultimate design load, which is assumed to be a uniform load.
In the second part of the paper, the proportioned domes are analyzed to investigate if they have load bearing capacity or not to the required ultimate design load, and also the buckling loads obtained by the analyses are compared with the buckling loads estimated by axial buckling stress by the generalized slenderness ratio.
In the third part, the load bearing capacities of the proportioned domes under non-uniform load are checked. And also how the buckling stresses expressed by buckling loads can be used in the design of the single layer reticulated domes subjected to non-uniform loads is discussed.
Finally, how a systematic structural design of domes with shell-like buckling characteristics can be formed by using the knock down factors, an amplification factor for bending moments, and buckling stress of members is discussed.
Section snippets
Column under compressive force
An elastic straight column is subjected to the compressive force as shown in Fig. 1. When the compressive force reaches a critical value, the column starts to buckle and the lateral displacement appears distinct. The load corresponding to the critical point is elastic buckling load, and the axial buckling force Ncrlin of the column is calculated from equilibrium. If the elastic column includes some amount of geometric initial imperfections, the equilibrium path changes like a curve given by the
Geometry of the study model
The study model is a single layer reticulated dome as shown in Fig. 5. The dome is assumed to be on rollers at the supports in the tension ring, and fixed in the vertical and circumferential directions but free in the radial direction. In general, the magnitude of a dome is represented by the number of segments n on the rib lines (AOD, BOE, COF in Fig. 5). The number of segments n is twelve in this study which is for ordinary reticulated domes, although the number of segments may affect
Proportioning method
In references [9], [10], a proportioning method for member sections of single layer reticulated domes was proposed, which is based on the second-order elastic analysis.
A new alternative member proportioning method based on the linear buckling analysis is proposed in this study. The linear buckling analysis is more convenient than the second-order elastic analysis. Only a linear buckling analysis is required to estimate the linear buckling axial force Ncrlin which is chiefly utilized to assume
Distribution of thickness and diameter of member
An example of proportioning member sections of a dome is illustrated in Fig. 8. In this figure, the circles with various diameters represent the ratio pipe thickness of each member to maximum pipe thickness among the members except the tension ring. The pipe thickness of the tension ring is described also in Fig. 8. As discussed in the paper [9], the members on the rib lines (AO, OB, OC, OD, OE, OF of Fig. 5) are thicker than the others. The circles with small inner circles indicate that the
Buckling load of the dome subjected to non-uniformly distributed load
It can be expected that there is a difference between the buckling capacities of domes subjected to two different types of loads, uniform and non-uniform load, because they are designed against only uniform load. To investigate this problem, a series of analyses with consideration of asymmetric loads are done and then some parameters based on the results are investigated. One of the investigated parameters is the knock down factor α0 for uniform and non-uniform loads. Another is the buckling
Conclusions
In this study, a method to proportion member sections and to estimate the elasto–plastic buckling loads of single layer reticulated domes was proposed. It is based on the linear buckling analysis and the concept of the generalized slenderness ratio. The study was performed considering both uniform and non-uniform loads on the domes with the half subtended angle between 2.0° to 3.0° and the member slenderness ratio from 40 to 100. The considered ranges of the half subtended angle and the member
Acknowledgements
This study was a part of the projects entitled “Study on buckling resistant capacity of single layer reticulated domes” of Toyohahshi University of Technology and partially supported by Mokasho. The authors would like to express thanks to co-researchers Dr Yasuhiko Hangai, Dr Makoto Takayama, Dr Seisi Yamada, Dr Hideyuki Takashima and Dr Katsuo Tanaka for their invaluable directions. This work was supported in part as a program of National Science Foundation under the grant No. (C)(2)14550565
References (16)
- et al.
Effects of member buckling and yielding on ultimate strengths of space trusses
Engineering Structures
(1997) Collapse of semi-rigidly jointed reticulated domes with initial geometric imperfections
Journal of Constructional Steel Research
(1998)- The Working Group on Spatial Structures, International Association for Shell and Spatial Structures: Analysis, Design...
Buckling of reticulated shells: state-of-the-art
International Journal of Space Structures
(1995)- Kato S, Ishikawa K. On elastic–plastic buckling of a spherical reticular dome of single layer on a hexagonal plan,...
- Kato S, Yamada S, Takashima H, Shibata R. Buckling stress of a member in a rigidly joined single-layer reticular dome....
- Kato S, Shibata R, Takashima H, Ueki T. A computational procedure to evaluate buckling strength of reticular domes...
- Kato S, Muto I, Shoumura M. Effect of joint rigidity on buckling strength of single layer lattice domes. Proceedings of...
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