Static shear buckling of cylindrical liquid tanks

Abstract

The usage of tanks has been quite common throughout the industry during the last decades. They fall into the category of shell structures and this is how they need to be studied. The focus of this thesis is concentrated on ground based thin cylindrical tanks filled with water made out of steel. Thin shells are susceptible to buckling, a phenomenon that takes several forms and calls for extra attention. The one which is investigated is shear buckling through the application of a static horizontal ground earthquake force on a clamped at the top tank. Such a problem is quite common nowadays to be tackled by engineers with the aid of FEM programs for practical purposes, something which on the other hand deprives the engineer of getting a deep understanding and insight of the phenomenon. For that reason, an analytic approach will be followed, in which shear buckling is studied within the boundaries of elasticity for a nonlinear shell theory in order, in that manner, to simulate reality in the best possible way. The first chapter is devoted to help the reader understand the basic aspects of buckling found in the literature, which are vital for someone who desires to investigate buckling in general. Also, a synopsis of the modern structural codes and standards given regarding the matter, showing the lack of information on shear buckling for liquid filled tanks. The next chapter starts by demonstrating the initial step towards formulating the problem which is no other than the choice of an appropriate displacement field along with a suitable middle surface strain theory. After the most common shell theories are discussed, it is concluded that a deep nonlinear thin shell theory would fit the case under study. For educational purposes, the applied theory is developed from scratch, despite the fact that popular shell theories, which meet the aforementioned criteria are thoroughly described. Moving on, the equations of motion are derived and a discussion is held on how a solution can be reached in order to acquire the critical-buckling load, revealing all the different methods found in the literature regardless if they were used or not. In the third chapter, the strong form of the initial problem is converted into a weak formulation due to the high complexity of the nonlinear partial differential equations of motion. For this reason, the perturbation method is utilized, an approximation technique which by expressing the displacements in terms of a very small perturbation parameter ε, it allows the breakdown of the nonlinear problem into an infinite number of linear sub-problems. The simplest equations that could describe the problem are employed and the solution is divided into two different cases based on the order of ε; the linear and the nonlinear one. At first, the solution of the 1st order in terms of ε linear problem is determined, in relation to the unknown buckling force, which serves as a tool for the 2nd order problem. The resulted EoMs, despite the linearization process that has taken place, are still hard to tackle due to the existent variable coefficients. For that purpose, calculation of the buckling load is searched numerically, a procedure which requires the final system of EoMs for the desired number of modes to be converted into a system of 1st order odes. Due to the high computational cost, the numerical approach has been implemented solely for a beam clamped onto the ground under the application of a distributed load along its length and a concentrated force on the top, as an example to demonstrate the methodology. The steps of the numerical procedure that should be followed for the case of the cylindrical tank filled with water are then extensively described and illustrated. In that sense, it is crystal clear of the method in which instability due to shear can be investigated in complex problems such that of a shell tank. After this thesis, the future researcher is provided with all those tools that will allow him to follow the correct path in order to extract some quite interesting results like; the effect of the tank’s geometric characteristics, the various boundary conditions, the different levels of water as well as the effect of the miscellaneous existing non-linear shell theories on shear buckling. Such conclusions will definitely fill the gap in the current structural codes and standards and will contribute largely in the field of research regarding the cylindrical liquid tanks.