Numerical Attached. ical R/C. to their thin. Open access under CC BY-NC-ND license. Available online at Abstract.

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1 Available online at Procedia Engineering 1 (011) The Twelfth East Asia-Pacific Conference on Structural Engineering and Construction Numerical Evaluation of the Effects of Edge Beams Attached to the Cylindri ical R/C Shell StructureS es T. HARA a Departmentt of Civil Engineering and Architecture, Tokuyama College of Technology, Japan Abstract In Reinforced concrete (R/C) shell structures, the free edges of them are not usually continuous depending on thee design and the supporting purposes. To strengthen thesee edges, edge beams are placed on such cutting edges. Too obtain the sufficient strength of such shell structures, s an appropriate size of edge beams must be applied. In this paper, the numerical analysis of the R/C cylindrical shell with edge beams by using the nonlinear finite element method. Inn numerical analysis, the degenerate shell elements e with layered approach are adopted. Both shells and beams aree modeled by these elements. To represents the nonlinearr behavior of R/C shell elements, combinedd material andd geometric nonlinearities are taken into account. From the numerical analyses, the stiffness of the edge beams plays ann important role to show the deformation characteristics and the ultimate strength of R/CC shell structures. Consideringg the effects of edge beams on these phenomena, the minimum size of the edge beamm to represent the maximumm ultimate strength of cylindrical R/C shell is represented under several conditions. 011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and/or peer-review under responsibility of [name organizer] Keywords: R/C Shell, FEM, edge beam, reinforcement ratio, ultimate strength 1. Introduction Reinforced concrete (R/C) shell structures have been applied to the civil and industrial structures duee to their thin and elegant shapes. They also show a continuous configuration and transfer ann applied load to a supporting structure smoothly. In structural s characteristics, R/C cylindrical shells show a progressivee failure due to their high redundancy even e under severe conditions. Therefore, R/C shell structures aree suitable to utilize as a public spaces. However, in general, the free edges of them do not show mechanically continuous structure depending on the design and the supporting purposes. These cutting edges are prone to show a weakness of R/C shell. Therefore, to strengthen these edges, edge beams aree a Corresponding author: Published by Elsevier Ltd. doi:.1/j.proeng Open access under CC BY-NC-ND license.

2 71 T. HARA / Procedia Engineering 1 (011) placed on such cutting edges. To obtain the sufficient strength of such shell structures, an appropriate size of edge beams must be applied. In this paper, the numerical analysis of the R/C cylindrical shell with edge beams by using the nonlinear finite element method. In numerical analysis, the degenerate shell elements with layered approach are adopted. Both shells and beams are modeled by these elements. To represents the nonlinear behavior of R/C shell elements, combined material and geometric nonlinearities are taken into account. From the numerical analyses, the stiffness of the edge beams plays an important role to shows the deformation characteristics and the ultimate strength of R/C shell structures. Considering the effects of edge beams on these phenomena, the minimum size of the edge beam to represent the maximum ultimate strength of cylindrical R/C shell is represented under several reinforcing conditions.. Numerical procedure.1. Specimen Figure 1 shows the R/C cylindrical panel supported by edge beams on meridional direction. The specimen shows the cylindrical shape with 90mm x 90mm plan and has 88.75mm radius and mm thickness considering the possibility of experimental analyses. The size of the edge beams is the numerical parameter in this analysis. In both the cylindrical shell and edge beams, 0.75mm stainless wires are used as the reinforcements and are placed in the middle of the shell thickness in both meridional and hoop direction. The reindorcement of the edge beam is placed on both top and bottom surfaces. They are placed in equi-distance of 5mm. The specimen is made by use of the steel mold to avoid the geometric imperfections. The micro concrete with aggregate size.5mm will be used. The material properties are shown in Tables 1 and defined from the material tests. 950 R= x x 950 y z z y 190 Loading Point Figure 1: Geometric dimensions of R/C cylindrical shell (mm)

3 T. HARA / Procedia Engineering 1 (011) Specimens are assumed to be pin-supported at the ends of meridional edgess (see Figure ). Also, both meridional and hoop directional edges are free without these ends. The height and the width of the edge beam are the numerical parameters and define the rigidity of thee supporting beam. Also, the reinforcing ratio in the edge beams is the numericall parameter. The specimenss are subjected to a uniformly distributedd lateral load. z y point loads x Support Figure : Loading and supporting of R/C cylindrical shell Table 1: Material properties of concrete Table : Materiall properties of steel Compressive Strength (MPa) 38. Yield Stress(MPa) 35 Tensile Strength(MPa) 3.8 Tensile Stress(MPa) 9 Young s Modulus(GPa) 3. Young s Modulus(GPa) 0 Poisson s Ratio 0.0 Tangential Modulus(GPa) 1 However, in numerical analysis, considering c the comparison with the experimental data, pointt concentrate loads are applied (see Figure 1)... Numerical model Figure 3: FE Mesh In numerical analyses, the finite element procedure is applied. Figure 3 shows the FE mesh of thiss analysis. The full model is adopted. In the cylindrical portion of the model is divided d into 3 elements in

4 718 T. HARA / Procedia Engineering 1 (011) meridional and hoop directions, respectively. Each supporting beam is also divided into x3 elements. Both shell and beam elements are divided into 8 concrete layers and steel layers based on the layered approach (Hara 008, 009, Hinton et al. 198). Boundary conditions are pin supported at both ends of the edge beam on both supporting meridian. The supporting beams are connected to cylindrical shell on both meridional edges. Both hoop edges of cylindrical shell are free..3. Numerical assumptions In numerical analysis, the degenerate shell element is adopted and the geometric and material nonlinearities are taken into account. 9 nodes Heterosis element is used and reduced integration is performed to avoid the numerical problems. The numerical simulation is performed under the displacement incremental scheme. The yield condition of concrete is defined as the Dracker-Prager type, which is assumed that concrete yields when the equivalent stress based on mean stress and second deviatoric stress invariants reaches uniaxial compressive strength (Hinton et al. 198). The crushing condition is controlled by strains. The ultimate compressive strain of concrete is assumed as by Kupfer s experiment (Kupfer et al. 199). Also, after cracking of concrete, the tension stiffening parameters accounting for the tensile strength of concrete are introduced. The material nonlinearities of steel are assumed to be bilinear stress-strain relation for the reinforcement. 3. Numerical Results 3.1. Cylindrical shell without edge beam Load(kN) Figure : Load deflection relation of R/C cylindrical shell without edge beam Displacement(cm)

5 T. HARA / Procedia Engineering 1 (011) Figure shows the load deflection behaviour of R/C shell without edge beamm (Hara 0) ). R/C shell iss supported at only corners in verticall hoop directions. The reinforcing ratioss of the shell and the edgee beams are 0.% respectively. The deflection shows the vertical deflection at the centre of R/C cylindricall shell. The shell deformedd downward with increasing the load. Then, the diagonal cracks appeared at eachh corner of thee shell and thee load carrying capacity decreased. Figure 5: Deformation pattern of R/C cylindricall shell without edge beam Figure 5 shows the deformation patterns of R/C cylindrical shell at the maximum deformation pattern represents the failure mechanisms appropriately. loading. Thee 7 Load(kN) x1 x1 x1 8x1 x Displacement(cm) Figure : Load deflection relation of R/C cylindrical shell with 1cm meridional edge beams

6 70 T. HARA / Procedia Engineering 1 (011) Cylindrical shell with edge beam on both meridional edges To find the effectiveness of the edge beams, R/C cylindrical shell with edge beams on meridional edges is investigated. The numerical parameter is the size of the beam. The width of the beam changes from cm to cm and the thickness of it changes from 1cm to 5cm for each beam width. The reinforcing ratio is 0.%. Figure shows the load deflection curve for R/C cylindrical shell with meridional edge beams. The thickness of the beam is 1cm. The deflections are measured at the centre of shell surface. From the figure, the strength of R/C cylindrical shell grows with the width of the edge beams. However, the load carrying capacity decreases after peak strength. Figure 7: Deformation pattern of R/C cylindrical shell with xcm meridional edge beams 1 1 Ultimate Load(kN) 8 b=cm b=cm b=cm b=8cm b=cm Beam thickness(cm) Figure 8: Ultimate strength of R/C cylindrical shell with meridional edge beams of several size

7 T. HARA / Procedia Engineering 1 (011) Figure 7 shows the deformation pattern of R/C cylindrical shell with meridional edge beams of the size of cm width and cm thickness Figure 8 shows the ultimate strength of R/C cylindrical shell with edge beams on both meridional edges. Numerical parameter is the width of the edge beam. From the figure, the thicker the beam thickness is the larger the ultimate strength. Also the wider the beam width is the larger the ultimate strength is. The reinforcing ratio is 0.% The effects of the reinforcing ratio on ultimate strength of R/C shell in edge beams The effects of reinforcing ratio in the edge beams concerning the ultimate strength of R/C shell is investigated by using several models. Figure 9 shows the load deflection relation of R/C shell with edge beams with 0.% and 1.0% reinforcing ratio, respectively. In Figure 9(a), the edge beam has cm width and cm thickness. Load(kN Load(kN % 1.0% 0.% 1.0% (a) Edge beam of cmxcm Displacement(cm) (b) Edge beam of cmxcm Displacement(cm) Figure 9: Load deflection relation of R/C shell with edge beam In Figure 9(b), the edge beam has cm width and cm thickness. In Figure 9(a), the higher the reinforcing ratio, the larger the ultimate strength is. However, in Figure 9(b), both the ultimate strengths show same results. Therefore, the ultimate strength depends on both the shape of the edge beam and the reinforcing ratio. Figure shows the relation between the ultimate strength and the reinforcing ratio concerning the several edge beam dimensions. The ultimate strength grows with the growth of the reinforcing ratio except cmxcm edge beam.

8 7 T. HARA / Procedia Engineering 1 (011) Load(kN) cmxcm cmxcm cmxcm cmxcm Figure : The relation between the ultimate strength of R/C shell and the reinforcing ratio with edge beams. Conclusions Reinforcing retio(%) In this paper, the numerical analysis of the R/C cylindrical shell with edge beams was performed by using the nonlinear finite element method. The R/C cylindrical shell was supported at four corners and was subjected to a uniformly distributed load. To represent the nonlinear behaviour of R/C shell elements, combined material and geometric nonlinearities were taken into account. From the numerical analyses, following conclusions are obtained 1) The edge beam placed on the meridional edges improves the ultimate strength of R/C shell having free edges under uniformly distributed load. ) The thickness and the width of the edge beams predominantly influence the ultimate strength of R/C cylindrical shell. 3) The reinforcing ratio in the edge beam influences the ultimate strength of R/C shell under uniformly distributed load. Especially in the thin edge beams, the strength grows with the growth of reinforcing ratio. ) For thick edge beam, the growing beam width is not effective. However, for thin edge beams, it is effective for edge supporting shells to improve their ultimate strength. 5) The growth of the width and the thickness of the edge beam provides totally heavier structure. The ratio of the strength to the weight will be an important factor. In this analysis, 3cm beam thickness will be recommended. In this paper, the numerical evaluations are presented. These phenomena should be confirmed by experimental tests considering the several factors and will be provided as the design data.

9 T. HARA / Procedia Engineering 1 (011) Acknowledgments This research work was done under the supports of Grants-in-Aid for Scientific Research, Japan Ministry of Education, Culture, Sports, Science and Technology (No ). References [1] Hara T. Numerical and experimental evaluation of R/C shell. Proceedings of International Conference on Advances in Structural Engineering and Mechanics (ASEM 08), Cheju; 008, pp [] Hara T. Application of computational technologies to R/C structural analysis. The First International Conference on Computational Technologies in Concrete Structures (CTCS 09), Cheju; 009, pp [3] Hara T. Ultimate strength of R/C cylindrical shell with edge beam. Proceedings of International Association for Shell and Spatial Structures (IASS0), Shanghai (in published); 0. [] Hinton E and Owen DJR. Finite element software for plates and shells. Prineridge Press, Swansea; 198 [5] Kupfer H and Hilsdorf KH. Behaviour of concrete under biaxial stress. ACI Journal (8); 199, pp. 5-.