Buckling Analysis and Study of Non-Linear Behaviour of Framed Domes

Buckling Analysis and Study of Non-Linear Behaviour of Framed Domes

1Sreekuttan N.H, 2Neethi B.

1 Q/QC Engineer, Galfar Engineering & Contracting, Muscat, Oman

2 Consultant Structural Engineer, Ram Associates, Trivandrum, Kerala

Abstract Dome structures are among the most preferred types of structures in Civil Engineering field because of its special features such as, they can enclose maximum amount of space with a minimum quantity of material also maintains a good architectural lineage from the early stage of construction. and they have been constructed from mud, stone, wood, brick, concrete, metal, glass, and at last plastic over the centuries. The way a framed dome works depends on many factors including the configuration of the members. General buckling, local buckling and individual member buckling are the types of instability that must be checked in design of a dome. In this study aimed to find out the performance of different framed dome in nonlinear buckling and concerned with the finite element software ANSYS 14.5. Spherical domes with rigid joints are considered, the joint system provides sufficient rigidity for the connections and that the elements are designed for resisting bending and shear in addition to the axial forces. Otherwise, the structures will be susceptible to snap-through buckling. Buckling is a sudden failure occurs to the structure when it reaches a critical load, which is the maximum load which a member can support before it becomes unstable. In the present study buckling load of varies framed domes ribbed dome, Schwedler dome, Geodesic dome are calculated and the results are compared, to understand the effect of arrangements of the elements of the dome during buckling analysis

Index Terms Framed Dome,Ribbed Dome,Schwedler Dome,Geodesic Dome,Snap Through Buckling,Member Buckling,Buckling

1. INTRODUCTION

   Dome are the structural system that consists of one or more layers of elements that are arched in all directions. The surface of a dome may be s i n g l e layer surface such as a sphere or a paraboloid, or it may consist of of different layers. Domes are of special interest to engineers and architects as they enclose a maximum amount of space with minimum constructional materials. Domes are also exceptionally suitable for covering sports stadiums, assembly halls, exhibition centers, swimming pools and industrial buildings in which large unobstructed areas are essential and where minimum interference from internal supports is required. The provision of unobstructed sight- lines for large numbers of people is the primary requirement in sports halls and can easily be satisfied through the adoption of a dome shape. In the last 25 years construction, steel sections have largely replaced reinforced concrete. Framed steel dome structures have been widely used all over the world during last few decades. In most instances the joints of a steel dome are better described as rigid, rather than pin connected, in actual practice joints of the dome

   must be welded or riveted connection so the joints does not allow 100 % rotation. Hence for the present study the joints are considered as rigid joint with framed members

2. FRAMED DOME

   General

   Present study is Focusing on non linear buckling behavior of single- layer framed domes. According to Makowskis classification [6], some commonly used configurations are shown in Fig.1

   Geometric and Material Parameters

   In this paper Ribbed spherical dome, Schwedler dome and Geodesic dome with rigid joints are considered. The spans (D) of the dome considered for the domes are 5m and rise is limited to 2.5m. Total number of rings in the dome is selected as 5 and it is equally spaced, the members are considered as framed sections having a cross sectional dimension of 5cmx5cm. In most instances the joints of a steel dome are better described as rigid, rather than pin connected, which raises a question about the accuracy of the static analysis. a more sophisticated analysis should be used to properly analyze such a structure. Currently, only limited experimental and theoretical information is available concerning the response and analysis of rigid steel framed domes.

   The dome was modeled as linear isotropic material and the Indian standard steel sections. Properties of the material used for the analysis are tabulated.

   Modulus of elasticity	2.1 x 10^11N/m2
   Poisson ratio	0.3
   Density	7800 kg/m3

   Modulus of elasticity	2.1 x 10^11N/m2
   Poisson ratio	0.3
   Density	7800 kg/m3

   Table 1 Material Properties of Steel

3. STRUCTURAL PLANNING AND MODELING

   Description of Software Used

   Finite element method is considered to be the best tool for analyzing the structures. Recently many software uses this method for analyzing and designing. The most popular and the accurate software is ANSYS 14.5 software. It is a general purpose finite element modeling package for numerically solving a wide variety of mechanical problems. All users, from designers to advanced experts, can benefit directions, and rotations about the x, y, and z axes.

   Finite Element Modeling

   The finite element analysis (FEA) of the beam using Ansys

   14.5 software was done. The elements of the domes are modeled as 2 noded 188 beam element.

   Ribbed dome consists of a number of intersecting ribs and rings. ribbed dome will not be structurally stable unless it is designed as rigidly-jointed system, since it does not have diagonal elements[1],

   Schwedler dome consists of meridional ribs connected together to number of horizontal polygonal rings. To stiffen the resulting structure so that it will be able to resist unsymmetrical loads, each trapezium formed by intersecting meridional ribs with horizontal rings is subdivided into two triangles by introducing a diagonal member.

   A geodesic dome configuration is obtained by mapping patterns on the faces of a polyhedron and projecting the resulting configuration onto a curved surface. Geodesic dome is obtained by mapping a triangulated pattern on five neighboring faces of an icosahedron and projecting the result onto a sphere which is concentric with the icosahedron.

   Boundary Conditions

   The supporting conditions of the domes was assumed that the domes are rigidly attached to a ring beam at the bottom, fixed

   free boundary condition is applied i.e. dome is fixed at the base and free at the top.

   Meshing

   To achieve high accuracy, the meshing of the element should be fine as possible. The results heavily depend upon the quality of mesh.

   3.4 Loading

   For the study of time history characteristics of the domes a concentrated load is applied at the apex of the dome. Loading will increased by defined number of sub steps and the buckling load at which the sudden failure to the structure is calculated.

4. METHODOLOGY

   The BEAM188 element is suitable for analyzing slender to moderately stubby/thick beam structures. BEAM188 is a linear (2- node) beam element in 3-D with six degrees of freedom at each node. The degrees of freedom at each node

   include translations in x, y, and z directions, and rotations about the x, y, and z directions. Warping of cross sections is assumed to be unrestrained. The beam elements are well- suited for linear, large rotation, and/or large train nonlinear applications. It is applicable to most beam structures and Support linear as well as nonlinear analyses, including plasticity, large deformation, and nonlinear collapse BEAM 188 easy to use, both in preprocessing and post processing phase

5. FINETE ELEMENT ANALYSIS

   Large displacement non-linear buckling analysis is considered in the study. Linear analysis assumes a linear stress strain relationship, and together with information about the material behaviour can be used to check for local member or joint failure. However, instability effects beyond the point at which the material yields require non- linear analysis to account for member effects such as plastic yielding. Non-linear methods are generally more technical and more computationally exhaustive, this technique employs a non-linear static analysis with gradually increasing loads to seek the load level at which your structure becomes unstable.

   1. The Instability Phenomenon

      To summarize, one major characteristic of non-linear buckling, as opposed to Eigen value buckling, is that non-linear buckling phenomenon includes a region of instability in the post-buckling region, whereas eigen value buckling only involves linear, pre- buckling behavior up to the bifurcation (critical loading) point. This behavior is shown graphically in Fig 2 [1]

      General buckling, local buckling and individual member buckling are the types of instability that must be checked in the design. [2]

      Member buckling is the instability of one member in dome can imply the collapse of the structure. Member buckling can be avoided by ensuring an adequate bending stiffness of the members. [2]

      Node instability, will occurs when the combined axial forces in all of the members attached to a joint cannot balance the external load. When this happens the node experiences a much larger displacement than the neighboring nodes. The dynamic loads involved when the node leaps from one position to a more distant position are very harmful for the whole structure. [2]

      Line instability, which appears when all the nodes and members in a ring are involved in the loss of stability [2] General instability, where the loss of stability simultaneously appears at several nodes. [2]

      An elastic structure with non-linear unstiffening characteristics may lose its stability in one of two ways. First, the unstiffening associated with the initial mode of deformation may, by gradual reduction, reach the stage where the stiffness is lost completely. It is then said that the load-deflection equilibrium path of the structure has reached a "limit point' and a dynamic jump occurs to a highly deformed configuration. This process is referred to as snap through or snap buckling and is typical of the symmetric deformation of shallow arches and domes[1].

      Fig. 2.Non linear vs Eigen value buckling behaviour

      Fig. 4 Load deflection graph (Ribbed dome)

      Fig. 2.Non linear vs Eigen value buckling behaviour

      Fig. 4 Load deflection graph (Ribbed dome)

6. RESULT ANDDISCUSSION

   For the study of non linear behaviour of Ribbed dome, Schwedler dome and Geodesic dome a concentrated load is applied at the crown of the dome and the load is incremented by number of sub steps up to the failure of the model. Then the buckling load is found out from the load deflection graph. When buckling occurs the structure will shows a sudden large deflection without any increase in the load and then reaching to new equilibrium geometry.

   1. Ribbed Dome

      Buckling analysis of ribbed dome shows that it is very much week against buckling it is consist with number of ribs and rings. The finite element model of ribbed dome shown in fig 3

      Fig. 3.Finete element model of Ribbed dome (front view)

      Fig. 3.Finete element model of Ribbed dome (front view)

      The dome shows instability during the buckling analysis at which 15.6kN.load deflection curve of ribbed dome is shown in fig 4.

   2. Schwedler Dome

      Buckling analysis of Schwedler dome shows that it performs better than ribbed dome, may be due to the bracings provided to the geometry. The finite element model of Schwedler dome shown in fig 5

      Fig. 5 Finite element model of Schwedler dome (top view)

      Fig. 5 Finite element model of Schwedler dome (top view)

      Fig. 6 Load deflection graph(Schwedler Dome)

      Fig. 6 Load deflection graph(Schwedler Dome)

      The buckling load of Schwedler dome was found at 26.11kN.load deflection curve of Schwedler dome is shown in fig 6.

   3. Geodesic Dome

   Fig. 7 Finete element model of Geodesic dome

   Fig. 7 Finete element model of Geodesic dome

   Geodesic dome shows the best performance rather than ribbed dome and Schwedler dome .from the stress distribution diagram it is shows that the load transfer mechanism of geodesic dome is absolutely good.fig 7 and 8 shows the finite element model and load deflection graph of the dome.

   7

   8

   7

   8

   Fig. 8 Load Deflection graph of geodesic dome

   Fig. 8 Load Deflection graph of geodesic dome

   The buckling load of Geodesic dome was found at 34.5kN.load deflection curve of Geodesic dome is shown in fig 8.

7. CONCLUSION

The Buckling analysis of framed domes having different configuration has done. From the results it is concluded that the arrangements of element has significant role in the load carrying mechanism of the framed domes. Geodesic domes will performs better than ribbed dome and Schwedler dome even though it consumes lesser quantity of material .it is due to the stability of the geometry.

REFERENCES

1. E.S. Kameshki and M.P. Saka, Optimum geometry design of nonlinear braced domes using genetic algorithm, Computers and Structures, vol 85, 2007,pp 7179.

2. Tien T. Lan. Handbook of Structural Engineering, Institute of Building Structures,

3. Ronaldo C. Battista, Miche`le S. Pfeil, Eduardo M. Batista, Strengthening A Reticulated Spherical Dome Against Local Instabilities, Journal of Constructional Steel Research,vol57,2001,pp1528.

4. Ehab Hamed etal, Nonlinear long-term behaviour of spherical shallow thin- walled concrete shells of revolution, International Journal of Solids and Structures,vol 47, 2010, pp 204215

5. Peter N.Khakina Buckling load of thin spherical shells based on the theorem of work energy, International Journal of Engineering and Technology,vol 05, 2013, pp 392-395

6. Makowski ZsAnalysis, Design and Construction of Braced Domes,Nichols Pub. Co., 1984, New York.