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Previous studies on gabled hyperbolic paraboloid shells (gabled hypar) revealed that shell loads are transferred to the supports mainly through diagonal arch action, and the contribution of edge beams, which have been traditionally included based on the assumptions of membrane theory, is actually very limited. This finding introduced a new shape of gabled hypars in which the edge beams are removed. This paper investigated the behaviors of gabled hypars with and without edge beams for various cases considering the effects of rise-to-span ratio (RSR) and lateral support movement. The FE analyses results indicated that, when the RSR was low, distribution of shell stress showed large variations. Lateral support movement caused an increase of tensile stresses, a decrease of compressive stresses, and intensified stress variation. When edge beams were not used, deflections were increased substantially, and local fluctuation of stress in the vicinity of the supports was intensified. Such behaviors were aggravated when RSR was low and proper constraints against the lateral support movement were not provided, thus resulting in inefficient systems. As such, gabled hypars without edge beams should be designed with caution. Keywords: gabled hypar; hyperbolic paraboloid shell; finite element analysis; membrane theory 1. Introduction are referred to as edge beams, and beams located along Shell structures have been used historically in the straight lines where the shell panels meet are called large spatial structures (Park, 2005; Lainez et al. ridge or crown beams (Fig.1.b). 2009) and the concept of the form-resistant-structure The rationale for locating beams at the edges of the is continuously adopted in current architectural shell panels stems from the membrane theory. In the forms (Kim and Shin, 2011). The gabled hyperbolic membrane theory, it is assumed that the gravity load paraboloid shell (referred to as gabled hypar) is one of acting on the shell panels is resisted by the in-plane the most common types of hyperbolic paraboloid shell shear stress of the shell, and the shearing forces are structures. The basic unit of the gabled hypar consists transformed into tensile and compressive forces by of four shell panels, which have straight edges and four concave or convex arch action and then transferred to columns as supports. Sometimes multiples of the basic the perimeter (i.e., edge beams and ridge beams) as units are combined together to form a shell cluster. The axial forces. Since the membrane theory is based on a behavior of a single gabled hypar is largely influenced simplified assumption that the external force is resisted by the support conditions. Usually four corners of a only by the in-plane shearing forces, the effects of the single gabled hypar are confined using tie rods or tie flexural stress and out-of-plane shearing force of the beams in order to restrain excessive lateral movements. shell panels cannot be considered (Billington, 1982). Shell panels have zero curvature along the direction Numerical analysis using the finite element method of the perimeter but have convex or concave surfaces is capable of considering the flexural moment, out- along the diagonal direction (Fig.1.a). In general, of-plane shearing forces, and their interactions with beams are located at the boundaries of shell panels. The the in-plane shearing forces. The results from the beams located along the sloping lines of shell panels refined finite element analysis show that shell loads are transferred to the supports through a different mechanism, which cannot be explained using the *Contact Author: Eunjong Yu, Assistant Professor, membrane theory. That is, the shell loads are directly Department of Architectural Engineering, Hanyang University, transferred to corner supports by the convex arch 222 Wangsimni-ro, Seongdong-gu, Seoul 133-791, Korea action along the diagonal direction of the shell panel Tel: +82-2-2220-0312 Fax: +82-2-2220-1945 (from the crown to the corner supports), and the E-mail: firstname.lastname@example.org tensile forces created by concave arch action in the ( Received October 1, 2013 ; accepted October 15, 2014 ) Journal of Asian Architecture and Building Engineering/January 2015/166 159 perpendicular direction are comparatively insignificant Billington (1995) removed the edge beams since the (Shaaban and Ketchum, 1976; Simmonds, 1989). tensile action between end points of the ridge became As such, stresses on perimeter beams are reduced; insignificant. Based on the FE analysis results, they however, shell panels, especially those in the vicinity asserted that a gabled hypar can be designed without of the supports, are subjected to very high compressive edge beams if there is sufficient shell thickness in stress. Reduced significance of the perimeter beams the vicinity of the supports in order to resist high resulted in the proposal of a new and simple gabled compressive stress. However, the gabled hypar model hypar shape, in which the edge beams along the slant that Jadik and Billington (1995) used for their study edge line were removed (Jadik and Billington, 1995). was very limited. The model used in their study was In this paper, behaviors of gabled hypars without a 24 m x 24 m shell structure composed of four 12 m edge beams were investigated extensively and x 12 m shell panels. The rise-to-span ratio was fixed compared with those of traditional gabled hypars. at 0.1 (i.e., the height of the ridge was 2.4 m), and all Specifically, the influences of the rise ratio (the ratio supports were assumed to be completely restrained. of the height to width of a gabled hypar) on the global However, the geometry, especially the rise ratio, may behavior of the structure, including buckling behavior, affect the global behavior of the gabled hypar, and were investigated. In addition, the effects of lateral the effects of lateral movement of supports should movement between the supports were also investigated. be considered. Thus, in this study, the effects of such factors were examined using FE analyses for the 2. Analytical Model cases when the edge beams were and were not used, Fig.1. shows a typical configuration of the traditional respectively. gabled hypar, which includes ridge beams and edge Fig.2. and Table 1. show the geometry of the gabled beams. Shaaban and Ketchum (1976) analyzed the hypars and dimensions of members used in this study, structural behavior of the traditional gabled hypar which are basically identical to previous studies except subjected to gravity load when support movements that various rise ratios ranging from 0.05 to 0.25 were were restrained. They revealed that the shell loads used. Each shell panel, which consists of one-quarter were transferred mainly by the compressive force due of a roof shell, has a dimension of 12 m x 12 m and to the arch action from the crown to corner supports. was divided into 40 elements with a grid of 300 mm Tensile forces due to the concave arch action in the spacing in each direction. The dimensions of ridge perpendicular direction diminished due to the inward beams are 600 mm wide and 200 mm deep, including displacement of the perimeter beams. Jadik and the shell thickness. (a) (b) Fig.1. Gabled Hyperbolic Paraboloid Shell (a) (b) (c) Fig.2. Geometry of the Gabled Hypar Model 160 JAABE vol.14 no.1 January 2015 Chang-Soon Rha The size of edge beams, if included, is 300 mm x Eq. (1). 400 mm. In the case of hypars without edge beams, the thickness of the shell around the corners was increased = pz/2kt (1) as shown in Fig.2.c. The numbers inside the elements in Fig.2.c denote the shell thickness in millimeters. where pz is the uniform vertical pressure, k = c/ab is the The dots in Fig.2.c indicate the supported nodes by twist of the surface, a and b are projected dimensions the columns, which were distributed to avoid stress in each direction, and c and t are the rise and thickness concentration. The shell and beams were modeled as of the shell, respectively. Thus, the magnitude of homogenous concrete with a unit weight of 24 kN/m , principal stress is inversely proportional to the RSR. a modulus of elasticity of 21 GPa, and a Poisson's ratio In the case of principal compressive stress, when of 0.15. the RSR is high and edge beams are used, an almost uniform compressive stress is observed. On the Table 1. Member Size of the Gabled HP Shell Model contrary, when the RSR is low and edge beams are Component Sizes not used, compressive stress is concentrated at the Shell 24 m x 24 m middle (i.e., diagonal line from the crown to edge), as Thickness 75 mm can be seen in Fig.3.c. Note that Fig.3.c was plotted Ridge Beam 600 mm x 200 mm along the diagonal direction shown in Fig.4.c, which Edge Beam 300 mm x 400 mm is perpendicular to the direction shown in Fig.4.b. Height 1.2 m (0.05), 2.4 m (0.1), 3.6 m (0.15), The magnitude of principal compressive stresses (Rise ratio) 4.8 m (0.2), 6.0 m (0.25) was higher than predicted by the membrane theory Modeling and analysis were carried out using the regardless of RSR (Fig.3.d), which implies that the finite element analysis software package, SAP2000 load is transferred to the edge through convex arch (CSI, 2005). Shell panels were modeled using four- action (i.e., compressive arch action). node quadrilateral shell elements with conventional From the results above, authors can conclude that Kirchhoff's thin-plate theory, which neglects the out- the load resisting capacity of hypar becomes superior of-plane shear effects. Edge beams were modeled as when the RSR is high and edge beams are used, since a typical frame element. Ridge beams combined with lower but evenly distributed stress was developed, a shell, as shown in Fig.2.b, were modeled using a which is an indication of an efficient structural system. layered shell element, which is a built-in element in The behavior of each hypar can be presumed from the SAP2000 element library. This element is useful the deflection of shell panels. The graphs in Fig.5. when the centerline of the shell is not aligned with the indicate the overall deflected shape of a panel (Fig.5.a), centerline of the beams and yields reasonable results the maximum deflection of shell panels in a parallel (Kim et al., 2012). direction (Fig.5.b), in a perpendicular direction (Fig.5.c) of the edge line, and in a vertical direction (Fig.5.d). 3. Analysis Results It can be seen that the displacement of shell panels is 3.1 Effects of Rise-to-Span Ratio larger in all directions when edge beams are not used. The effects of rise-to-span ratio (RSR) on the Fig.5.c suggests a plausible indication of tensile stress behavior of the hypars were investigated in this section. distributions in Fig.3.a (note the smaller tensile regions All corners of the hypars were supported as shown in when edge beams were not used) since the larger Fig.2.c, and no lateral movements were considered. lateral (inward) deflection of edge beam or edge line Fig.3.a shows typical distributions of principal caused a decrease of tensile arch action. stress in a concave direction measured along the Fig.6. shows shear force distribution along the edge convex direction as indicated in Fig.4.b. Unlike the beams and maximum shear force with respect to the assumptions in the membrane theory, the tensile stress, RSR. In membrane theory, the edge beams are required which is the indication of concave arch action, is to resist shear force, which is developed due to shell observed only in less than half of the diagonal. When stress via tensile and compressive stresses given the edge beams were not used, the tensile stress region by Eq. (1). However, FE analysis indicated that the decreased, which is believed to be the result of weak magnitudes of shear force at the edge beams are very lateral restraints. small regardless of RSR. The FE analyses also showed Fig.3.b, which is a plot of the maximum tensile that fluctuations in the vicinity of support, which is stress magnitudes with respect to the RSR, indicates believed to be a consequence of stress concentration, that the peak stress magnitudes were similar to those increase with a decrease of RSR (Fig.6.b). Thus, when obtained by the membrane theory when the RSR was the RSR is low, edge beams may be required to resist relatively high. However, when the RSR decreased, high peak shear force. Local reinforcement of the shell the magnitude of peak tensile stress was much smaller near the supports also may be required as the shell than that expected by the membrane theory. In stress shows large variation (Fig.3.a). membrane theory, the magnitudes of principal tensile Linear buckling analyses were performed to and compressive stress are identical and are given by investigate the buckling loads which cause instability JAABE vol.14 no.1 January 2015 Chang-Soon Rha 161 Table 2. Buckling Factors of the structure. The obtained buckling loads are expressed in Table 2. in terms of buckling factor, R/S Ratio (RSR) w/edge beams w/o edge beams which is the ratio of buckling load to the self-weight of 0.05 2.15 0.92 the structure. Generally, lower buckling factors were 0.10 6.28 2.29 obtained when the RSR decreased and when edge 0.15 12.22 3.96 beams were not used. It should be noted that, in the case when RSR was 0.05 and edge beams were not 0.20 19.08 5.70 used, a buckling factor of less than 1.0 was obtained, 0.25 26.09 7.26 which indicates that the structure cannot support even its own weight. (a) (b) (d) (c) Fig.3. Principal Shell Stresses; (a) Distribution of Shell Stress in the Concave Direction (Along Convex Diagonal), (b) Maximum Tensile Stress, (c) Distribution of Shell Stress in the Convex Direction (Along Concave Diagonal), and (d) Maximum Compressive Stress (a) (b) (c) Fig.4. Locations and Directions of Presented Stresses; (a) Observed Shell Panel, (b) Direction of Diagonal and Shell Stress Associated with Concave Arch Action, and (c) Direction of Diagonal and Shell Stress Associated with Convex Arch Action 162 JAABE vol.14 no.1 January 2015 Chang-Soon Rha (a) (b) (d) (c) Fig.5. Maximum Displacement of Edge Beam or Edge Line with Respect to RSR; (a) Typical Deformed Shape, (b) in a Parallel Direction to the Edge Beam, (c) Perpendicular Direction to Edge Beam, and (d) Vertical Displacement (a) (b) Fig.6. Shear Forces at the Edge Beam; (a) for RSR = 0.05, 0.20, and (b) Maximum Shear Force 3.2 Effects of Lateral Support Movement rods were modeled using the truss element with 200 Lateral support movements have detrimental effects GPa of elastic modulus. Supports were modeled with on the load-resisting capacity of hypars since they a two-way roller and a pin at opposite supports of a decrease the arching action. Thus, usually four corners diagonal, while the other corners were modeled with of single gabled hypars are supported with very stiff one-way rollers. Columns were not considered in the columns or confined using tie rods or tie beams in order models. to restrain excessive lateral movements. In this section, The graphs in Fig.7. show behaviors of hypars the effect of lateral support movement was investigated with tie rods compared with those of fully restrained by analyzing hypars in which supports were restrained hypars. As expected, displacements of hypars caused by tie rods. Two different tie rod sizes (53 mm and 76 by roof load increased with the decrease of the RSR mm) were used to change the degree of restraint. Tie and size of tie rods. When the RSR exceeded 0.15, JAABE vol.14 no.1 January 2015 Chang-Soon Rha 163 existence of edge beams did not affect the magnitude lateral movement of supports was allowed. When of displacements. The largest lateral support movement the lateral movement was restrained, the diagonals occurred in the case when both RSR and the diameter from the crown to edge were subjected to the largest of tie rods were smallest and the edge beams were compressive stress. However, as the degree of restraint not used. (Fig.7.b) However, the magnitude of lateral decreased, compressive stress along the diagonal line support movement in such a case was about 70 mm, as also decreased, which indicates that the mechanism small as 3% of the distance between the supports. of compression arch was diminished. As a result, Figs.7.c and 7.d indicate that, when the supports variations in magnitude of compressive stress were are not completely restrained, inward displacement increased. and downward deflection of the edge beam or edge Peak tensile and compressive principal stresses line drastically increased. For example, when lateral were plotted with the RSR shown in Figs.5.b and 5.d, movements were completely restrained, the inward respectively. When lateral movements were restrained, displacement of the edge line was about 1.5 mm, as the magnitudes of principal tensile stress were similar can be seen in Fig.5.c (RSR=0.15 and edge beams to or less than those predicted by membrane theory, were not used). However, when the supports were while principal compressive stresses were about partially restrained, the displacement increased almost twice as large as those suggested by membrane theory 20-fold (Fig.7.c). Downward deflection of the edge (Fig.5.). When lateral support movements were allowed, showed a similar result. It should be noted that, when the magnitude of peak tensile stresses increased, and the RSR is low (less than 0.10), the edge beams have the magnitude of peak compressive stresses decreased. some contribution in decreasing the displacement, as This tendency is clear when the RSR is reduced. mentioned above. Specifically, for the hypars with RSR=0.05 not using Figs.8.a and 8.c show distributions of principal edge beams, the shell is subjected to very large tensile tensile and compressive shell stress, respectively, stress and relatively large compressive stress, which is depending on the degree of lateral restraints. The believed to be the consequence of a large fluctuation tensile region increased when lateral supports were of stress distribution. However, this tendency was allowed to move. Hence, it seems that the edge beam considerably reduced when edge beams were used. does not alter the principal tensile stress distribution. Fig.9. is the contour plot of the principal shell stress On the contrary, the distribution of stress in the for RSR=0.1. It is clearly shown that, when lateral compressive diagonal differed greatly when the movements were allowed, the area of tensile stress in (a) (b) (c) (d) Fig.7. Maximum Displacement of Edge Beam or Edge Line with Respect to RSR; (a) Typical Deformed Shape (b) in a Parallel Direction to the Edge Beam, (c) Perpendicular to the Edge Beam, and (d) Vertical Displacement 164 JAABE vol.14 no.1 January 2015 Chang-Soon Rha (a) (b) (c) (d) Fig.8. Principal Shell Stresses when Subjected to Partial Lateral Constraint with 53 mm and 76 mm Tie Rods; (a) Tensile Stresses for RSR = 0.10, (b) Maximum Tensile Stress, (c) Compressive Stresses for RSR = 0.10, and (d) Maximum Compressive Stress the concave direction increased (Figs.9.a and 9.b), and variation at the shell, which seemed to be reflective of the area of compressive stress in the convex direction a weakened compressive arch action. decreased (Figs.9.c and 9.d). Removal of the edge beams decreased the area of In summary, when the lateral movements of supports tensile stress at the shell due to the inward deflection were not completely restrained, hypar displacement of the edge line. However, the peak values of the increased drastically. In addition, greater tensile stress shell stresses were almost the same as those when the and a large stress fluctuation were observed in the shell edge beams were used. In addition, the magnitudes of stress, which indicates that the contribution of diagonal shear stress carried by the edge beams were always compressive arching action was reduced. These small irrespective of the RSR, which was the reason tendencies were apparent when the RSR decreased. for removing the edge beams. However, when the When the RSR was greater than 0.15, the use of edge edge beams were not used, deflections increased beams did not make significant differences. However, substantially and local fluctuation of stress at the shell when the RSR was low, edge beams were effective in panel and edge beam in the vicinity of the supports was decreasing the displacement and peak shell stress. intensified. These behaviors were aggravated when the RSR was low, and strong constraints against the lateral 4. Conclusion support movement were not provided. Thus, although This paper investigated the behaviors of gabled removing the edge beam can provide flexibility in hypars with and without edge beams for various cases design, gabled hypars without edge beams in such considering the effects of rise-to-span ratio (RSR) and cases may result in inefficient structural systems from lateral support movement. The FE analyses performed the viewpoint of structural design. in this study indicated that, when the RSR is low, distribution of shell stress showed large variations and References 1) Billington D.P. (1982) Thin Shell Concrete Structures. 2nd Ed. the critical buckling load was reduced. Lateral support McGraw-Hill Book Company. movement tends to affect the stability of the structure. 2) CSI Analysis Reference Manual. (2005) Such movement caused a considerable increase of shell 3) Jadik T. and Billington D.P. (1995) Gabled Hyperbolic Paraboloid deflection, an increase of the tensile stresses, a decrease Roofs without Edge Beams. 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(2011) Finite Element Analysis Shells. Journal of the Structural Division, 102(ST11), pp.2151- of Inverted Umbrella-type Hyperbolic Paraboloid Shell. Journal of the Korean Association for Spatial Structures, 11(1), pp.87-95. 12) Simmonds S.H., (1989) Effect of Support Movement on 8) Lainez, J.M.C., Verdejo, J.R.J., Macias, B.S.M., and Calero, J.I.P. Hyperbolic Paraboloid Shells. Journal of Structural Engineering, (2009) The Key-role of Eladio Dieste, Spain and the Americas in 115(1), pp.19-31. the Evolution from Brickwork to Architectural Form. Journal of Asian Architecture and Building Engineering, 8(2), pp.355-362. (a) (b) (c) (d) Fig.9. Principal Shell Stress Contours for RSR=0.10; (a) Stresses in the Concave Direction when Fully Constrained, (b) Stresses in the Concave Direction when Restrained with 53 mm Tie Rods, (c) Stresses in the Convex Direction when Fully Constrained, and (d) Stresses in the Convex Direction when Restrained with 53 mm Tie Rods 166 JAABE vol.14 no.1 January 2015 Chang-Soon Rha
Journal of Asian Architecture and Building Engineering – Taylor & Francis
Published: Jan 1, 2015
Keywords: gabled hypar; hyperbolic paraboloid shell; finite element analysis; membrane theory
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