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CivilComp Proceedings
ISSN 17593433 CCP: 77
PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 58
Limit Analysis of Reinforced Concrete Shells of Revolution and its Application M.A. Danieli (Danielashvili)
Department of Civil Engineering, College of Judea and Samaria, Ariel, Israel M.A. Danieli (Danielashvili), "Limit Analysis of Reinforced Concrete Shells of Revolution and its Application", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", CivilComp Press, Stirlingshire, UK, Paper 58, 2003. doi:10.4203/ccp.77.58
Keywords: reinforced concrete, shell, revolution, limit analysis, discrete form, general algorithm, application.
Summary
The theory of limit analysis is one of the most developed, wellgrounded and
important chapters in the modern theory of plasticity. Three main
theorems underlie the static and kinematical methods of the theory of limit analysis.
According to the kinematical method, the
ultimate load does not exceed the load, which may be determined from consideration
of kinematically admissible distribution of displacements  kinematical mechanism.
It is assumed that with exhausting of loadcarrying capacity rigid system turns into
kinematical mechanism.
In this work axially symmetric shells of revolution of the arbitrary shape are considered, which do not contain any zones of negative Gauss curvature. Discrete continual model of the shell's body is assumed. Concrete fills constantly the shell's body, having ultimate resistance to compression, but does not resist to extension. The reinforcement in the discrete form is placed along meridians and ring lines, being under conditions of oneaxis stress. Both in extension and in compression it is ideal rigidplastic. There are no limitations for distribution of the reinforcement along meridians. The acting axis symmetric loads, both useful and fixed, are presented separately by means of vertical and horizontal constituents. They are laid in the form of evenly distributed linear loads along the ring lines, at the level of the shell's middle surface. Geometrical data of the shell  shell's thickness, radiuses and ordinates are given in the discrete form as well along the ring lines. It is assumed that the shell is fixed along its edges. On the basis of existing experimental researches, as general form of the plastic collapse mechanism the collapse scheme is assumed, which is realized as a result of the reinforcement yielding. It is a summation of rigid units, three reversed rigid plastic hinges and infinite set of meridional cracks located between them. Due to a number of the rigidplastic hinges the considered breaking scheme is called threeparameter. Conditions of analysis of the plastic mechanism's units are considered both as internal and external forces projection equations and moment equation and on the ground of application of the principle of virtual displacements to virtual mechanism in the form of equality of the work of internal and external forces. As a result, the general formula is obtained as a sum for determination of ultimate load parameter P in the nondimensional form. It is applicable also for the cases of hinged immovable and hingedmovable support. Structure of the formula does not limit in any way shapes of the meridian, load and reinforcement distribution, thickness changes, etc., allows to take into consideration breaking character of load, geometry and physical factors along the meridian. The nondimensional parameter P is a function of three nondimensional radiuses and ordinates of ring plastic hinges and is determined according to extreme principle of the theory of limit analysis: locating of plastic hinges corresponding with the real collapse scheme, minimizing of the parameter P magnitude. For realization of possibilities following the structure of obtained formula, the proper numerical computing general algorithm is developed. It is intended and adapted for realization at PC. Minimal P magnitude is determined by the method of sequential looking over all kinamatically possible states, i.e. those, for which work of external forces at virtual displacements is positive. According to the specific task, computing of P is divided into two relatively independent and sequential parts: a) determination of the magnitudes included in the formula for P determination, for specific problem on the basis of its analytical model; b) computing of P for determination of ring plastic hinges location by means of radiuses and ordinates. While preparing basic data for giving of the shell's geometry and ultimate forces in the meridional reinforcement, continualdiscrete method is accepted, and for giving of load and ultimate forces in the reinforced rings, discrete method is accepted. Such approach allows realizing all possibilities resulting from structure of the obtained formula. The developed general algorithm is widely used both for research tasks and in practical calculations in designing of new buildings and evaluation of loadcarrying capacity of existing buildings. As a practical sample the following cases are considered: a) Task of reinforced concrete shell designing; under given loads, geometry and quantitative view of reinforcement distribution the qualitative reinforcing parameter is determined. Its middle surface is a conjugated shell  truncated cone with spherical segment; thickness is variable. Results of calculation are correlated with the results of calculation according to the theory of plasticity, and for this aim the proper fields of bending movement and normal forces. Results of natural static and dynamic testing of this object are presented. Results of calculations and testing showed enough rigidity of this construction, that confirm in this case relevancy of rigidplastic analysis application. b) Task of evaluation of loadcarrying capacity of the precast concrete underground reservoir for liquids. Its middle surface is a system of conjugated shells of revolution of mixed curvature of two truncated cones and shallow spherical segment; the truncated cones are the reservoir's walls and spherical segment is an overlay. c) Application of load carrying capacity evaluation of the very shallow reinforced concrete dome of the floor in development of the project of its strengthening. The samples illustrate the wide possibilities of the developed general algorithm.
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