Influence of the size of the upper ring on the stressed state of the ribbed-ring metal dome

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1. Introduction Domes are used as coverings of buildings and structures both because of the expressiveness of their geometric shape and reliability of their structural systems. Due to their spatial rigidity and cost-effective metal consumption, they occupy a leading position among spatial coverings [1-3]. Geometric schemes of metal dome frames are diverse, depending on the overlapping spans and the purpose of the structure [4; 5]. Also, there can be different geometrical schemes of frames in ribbed-ring domes, for example, related to the number of sectors or tiers. One of the essential features of this difference is the size of the upper ring. Although the size of the upper ring often depends on the aesthetic or technological concept of a designer. Many publications considered various aspects of ribbed-ring domes by means of various computer programs. For example, stress state of the dome frame was analyzed with changing parameters of its geometric scheme [6], with different dome height-to-diameter ratios for different spans [7], when brick filling of the frame cells between steel ribs and rings [8], with different dome height ratios and different cross-sections of core elements [9], and with different heights compared to the span of the dome frame with connections [10]. However, there are no publications devoted to the study of ribbed-ring domes with relatively different sizes of the upper ring. Asymmetric load has the greatest influence on the stress-strain state of the frames of spatial structures. This load includes the second variant of the snow load on the dome cover[1]. It determines the required cross-sections of the main bearing elements of the dome frames. Six frames of spherical ribbed-ring domes with a curvature radius of 23 m, a span of 39.3 m and a height of 11.0 m (ℎ/ = 0.28) were considered as objects of the research. Domes’ frames consisted of 24 ribs, had 6 tiers, and differed from each other by the size of the upper ring. The smallest ring had a diameter of 4 m (dmin ), the largest one had a diameter of 14 m (dmax ), and the step of changing the diameter of the upper ring was 2 m. Due to the increase in the diameter of the upper ring, the lengths of the meridional ribs elements between the rings varied from 3.42 m to 2.75 m. The length of the upper ring elements varied from 0.78 m to 1.81 m. All elements of the dome frames were made of steel I-beams: the ribs are made of I 50Ш1, the upper ring was made of I 50Ш4, the remaining rings were made of I 20Ш1. The cross sections of the frame elements were oriented normally to the dome’s surfaces, with the exception of the upper ring, a cross section of which was oriented vertically (Figure 1). It was assumed that the conjugations of the upper ring to the meridional ribs, as well as conjugations of the other rings to the ribs, are rigid in the normal direction and hinged in the tangential direction. The frames are pivotally mounted on the supporting nodes at the level of the lower ring. Figure 1. Investigated models of dome frames 2. Methodology The study of the stress state of the ribbed-ring dome’s frame was carried out in the SCAD program [11] using computer models of spatial rod systems [12]. The study of dome type rod systems is usually carried out by scientists in various computer programs. Previously, the author performed a comparative study of ribbed-ring domes with various mounting methods [13; 14], as well as on the number of installed connections between meridional ribs [15]. The material presented in this paper has been obtained by the author in full compliance with the generally accepted principles of research for similar states in spatial rod systems. Dome frames are spatial rod systems pivotally mounted on supports under the meridional ribs at the level of the lower ring. It is with such computational models that they were presented in these computer studies. All computer models of domes differed from each other only in the size of the upper ring. The same nodal loads were applied to all dome frames: from the weight of the enclosing structures, which was symmetrical relative to the axis of the domes; from the weight of snow, located only on one side of the domes and being asymmetrical (Figure 2). Figure 2. Asymmetric snow load on the dome frames The combined effect of these loads creates a stress-strain state of the dome frames, which determine their bearing capacity. Despite the fact that the loads are the same, the stress-strain states in the elements of the dome frames with different sizes of the upper ring of the domes differ from each other. The outlines of deformed schemes of dome frames with different sizes of the upper rings do not differ from each other, but the values of deformations are not the same. If the maximum deflection in the most loaded part of the dome with the smallest size of the upper ring (dmin ) is equal to f = 9.5 cm , then it increases by 1.6 times in the dome with the largest size of the upper ring (dmax ) (Figure 3). The graphs of longitudinal forces N in the elements of dome frames with different sizes of the upper rings differ from each other insignificantly. And the diagrams of bending moments in the elements of dome frames with different sizes of upper rings show significant differences. These differences are associated with a change in the ratio between the values of moments Mx in the upper rings and those at the meridional ribs (Figure 4). Figure 3. The ratio of the largest deflections fdi / fdmin in the frame Figure 4. Diagrams of bending moments Mx in elements of dome frames 3. Example With an increase in the size of the upper rings, the values of bending moments Mx and longitudinal forces N in the meridional ribs change, but these changes are different and insignificant in magnitude (Figure 5): first increases within 11% of the initial value, then decreases; N first decreases, then stabilizes within 19%. Figure 5. The ratio of forces Ndi / Ndmin and Mdi / Mdmin in the meridional ribs Figure 6. The ratio of forces N Ndi / dmin and M Mdi / dmin in the upper rings at β=0 A different pattern of changes in bending moments and longitudinal forces , as well as in moments , is observed in the upper ring. At the same time, the internal forces along the axis of the one-sided asymmetric snow load (β = 0 ) and in the direction perpendicular to this axis (β = 90 ) differ from each other. For β = 0 as the size of the upper rings increases, the bending moments Mx increase significantly, reaching the value that is 3.1 times more than the moments at dmax than the moments at dmin (Figure 6). At the same time, the longitudinal forces N and moments My decrease, reaching values of 0.38N and 0.26My compared to the forcesat dmin . For β = 90 with an increase in the size of the upper rings, the bending moments Mx also increase significantly, reaching 2.7 times higher values at dmax compared to the moments at dmin (Figure 7). At the same time, the longitudinal forces N and moments decrease, reaching values at dmax of 0.46 N and 0.38My compared to the forces at . Figure 7. The ratio of forces N Ndi / dmin and M Mdi / dmin in the upper rings at β=90∘ Since the internal forces N , Mx , My in the meridional ribs and in the upper rings of the dome frames change differently when the size of the rings is changed, the studies of the normal stresses σ in them were carried out. The stress values were determined by the formula: = Ni M σi ± x,i ± Ai Wx i, M y i, Wy i, . With an increase in the size of the upper rings in the dome frames, the initial stresses ( 12.51 kN/cm2 ) at dmin in the meridional ribs change insignificantly, within 9% with no tendency to rise (Figure 8). The initial stresses ( 15.05 kN/cm2 ) in the upper rings at dmin change significantly, within 26% and have a tendency to increase (see Figure 8). Figure 8. The ratio of the greatest stresses σdi / σdmin in the elements of the dome frames 4. Conclusions 1. As a result of the research, it has been shown that with an increase in the size of the upper rings in the frames of ribbed-ring domes, the bending moments in the vertical plane in these rings increase several times, but the longitudinal forces and moments from the vertical plane decrease. 2. The normal stresses in the meridional ribs increase slightly at first, then decrease without reaching the initial values. The normal stresses in the upper rings increase significantly with an increase in their size, with a tendency to further increase. 3. With increasing the size of the upper rings in metal ribbed-ring domes, it is necessary to increase their cross-section. The reliability of metal domes mainly depends on calculations for a combination of loads considering asymmetric force influences.

About the authors

Evgeny V. Lebed

Moscow State University of Civil Engineering

Author for correspondence.
Email: evglebed@mail.ru
ORCID iD: 0000-0003-3926-8701

PhD in Engineering, Associate Professor in the Department of Metal and Wooden Structures

Moscow, Russian Federation

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