Again, it was found that buckling is strongly influenced by the response of the tank rocking mode. The oscillation of the tank is modeled by a rotary spring placed at the bottom of the tank.
Design Procedures
Objective and Organization of the Present Investigation
The third chapter is devoted to the derivation of a set of equations describing the behavior of the coupled fluid/structure/soil system. Particular attention is paid to the formulation of base heave, but effects due to soil and shell flexibility and fluid scour are also included.
Introduction
Mathematical Formulation of the Fluid Behavior
Solution of the General Fluid Response Problem
Fluid Response for Some Special Cases
It is clear that the case c=l corresponds to the rigid body rotation of the tank through an angle 0(t) about the y-axis. Also, the flushing effects become less important as the tank gets higher and as the order of the flushing mode increases.
Hydrodynamic Loads at the Tank Base
The Q, .i, and ^,s appearing in these base load expressions are functions of the H/R ratio and the corresponding tank mode. First, downscaling effects are generally only significant for relatively small values of the H/R ratio.
Simplified Mechanical Model for a Rigid Tank
Then, if ys(t) is the displacement of the sth oscillator relative to the tank wall, its equation of motion is. Now, select the parameters (masses, moment of inertia, heights, springs) of the considered mechanical system from the equations.
Introduction
Definition of System Parameters
Part of the bottom plate can also come loose and lose contact with the foundation. It is assumed that this happens in such a way that the lower circumference of the cylindrical part rotates an additional small angle φu(t) around the y-axis and is simultaneously translated with RΦu(t) along the z-axis.
Discussion of the Mathematical Formulation of the Coupled System
To complete the mathematical formulation of the problem, the various matching, boundary and initial conditions must also be considered. According to the above, the behavior of the system is generally strictly determined by a nonlinear system of coupled partial differential equations.
Choice and Outline of Method of Approach
The fundamental difference between finite element and other commonly used energy methods lies in the form and use of functions for geometric discretes. This pressure is expressed as a function of time-dependent functions of structural displacements.
Behavior of the Structurai Components
However, the formulation of the general soil/structure interaction problem is still associated with considerable difficulties and uncertainties. In the present study, the soil/structure interaction is modeled by placing two springs at point G of the foundation.
Equations of Motion for an Unanchored Fluid-Filled Tank
Lack of knowledge of the total displacements prevents explicit calculation of the virtual strain energy of the bottom plate. In that calculation, non-linear strain-displacement relationships must be used for the behavior of the base plate.
Governing Equations for the Coupled System
Sloshing Formulation
Choice of Damping Matrix
Introduction
Free Vibration of Anchored Tanks
In Table 4.2 a comparison is made between the results of the current analysis (Λ∕==5) and those of reference [24], again for different H/L values and for the wide tank. It is noted that a favorable comparison between the results of other studies is also reported in [24]. Moreover, these equations indicate that the assumed spatial modes of the shell (modes of a transverse beam of the same length L, as the tank) are probably very close to the exact modes, at least for the case of anchored tanks.
Finally, the data from Tables 4.2 and 4.3 also demonstrate another standard feature of anchored liquid-filled tanks.
Sloshing∕Slructural and SIoshing∕Sloshing Coupling
Numerical results are presented in Table 4.4 for the tall tank case reviewed in the previous section. The second column shows the values of the first four splash frequencies obtained from (4.2). The last two columns represent the coupled system frequencies obtained using this analysis and presented in [24], respectively.
Since the more structural degrees of freedom are involved, the higher the maximum eigenvalues, the code used in this work to extract the eigenvalues presents problems after the inclusion of a critical number of structural modes.
Roof Effects
Also, a problem arises from the fact that often. the amplitudes of the tilt modes are very small compared to the squat frequencies. Because the more structural degrees of freedom included, the higher the eigenvalues, the code used in this work to extract the eigenvalues presented problems after including a critical number of structural modes.
Linear Ground/Structure Interaction
As a result, only the first frequency is expected to be calculated correctly, so the frequencies of the higher modes are studied in the qualification. If we look at the variation of the fluctuating component of the eigenvectors during the change, we notice the following points. For the ith mode (i>l), the rocking component increases very slightly as it increases from 0 to Then it increases rapidly in the transition region √, to ~ ; at higher values it starts to decrease again.
Since the first two modes are the most important in determining the response of the structure, the soil flexibility will clearly affect the response of the system significantly if ⅛γ < x2.
This can then cause local buckling of the tank under sufficiently large ground acceleration. Therefore, an eigenvalue problem is posed to determine the bifurcation load in the form. Between the free surface and the top of the tank, the pressure is zero because there is no liquid there.
In Figure 4.5.b, the impulsive components of the hydrodynamic pressure are plotted in the same way.
Nonlinear Ground/Structure Interaction
Therefore, the system response is obtained according to the following mixed analytical-numerical procedure. It should be noted that the character of the long-standing solution gradually changed. Some features of the solution such as non-periodic long-term response and high sensitivity.
All these observations provide evidence that the observed strange behavior of the nonlinear system is something inherent in the system itself.
Introduction
The first is the determination of the response and developed loads, the second deals with the calculation of the resulting stresses in the structure, and the third problem is related to the selection of the appropriate failure criterion. In this chapter, the analysis developed in the second and third chapters of this thesis is applied to cases that have previously been studied experimentally. In the next section, static tilting test results are obtained to derive the characteristics of a rotary spring that models the uplift behavior of unanchored tanks as presented in Section 3.6.
In the last two sections, analytical results are obtained and compared with available experimental data, for unanchored tanks with harmonic or transient base excitation.
Static Tilt Tests with Unanchored Tanks
Then, the dependence of the dynamics of an unanchored reservoir on the stiffness of this source is examined parametrically. It is clear that for small values, a large change in the fundamental frequency of the beam can result from changing the spring stiffness. These observations prompted the development of the current analytical model for the behavior of unmoored tanks.
For the purposes of this work and to obtain analytical results for the dynamics of the Πb2 reservoir, static tilt tests were performed for this specific reservoir.
Uplift Spring and Tank Dynamics
The frequencies of the new system - including buoyancy - are denoted w] (i=1 to 5) and the results of the parametric study are shown in Figure 5.4. The transition in the values of the first three frequencies is similar to what is observed in Figure 5.1 and is very different from what occurs in Figure 5.1. At least for the system under investigation, the true values of rotational spring stiffness (30-500 Nm) appear to be in the range where the fundamental structural frequency is drastically affected by changes in the stiffness ku.
One can thus expect that this has an important effect on the dynamics of the system, for excita.
Shell Flexibility Effects in Unanchored Tanks
The most important properties of the system are the dramatic dependence of the lowest frequency on ku, for relatively small values of ku and also the fact that → .√, as ku ~→ oc. To further investigate the validity of the above findings, the response of the same tank to harmonic base excitation with amplitude ao and forcing frequency w was investigated. For forcing frequencies typically expected, it was again found that shell flexibility has no significant effect on the dynamics of the unanchored tank.
Values of the relative acceleration associated with uplift, soil flexibility and the first mode of the radial shell, obtained for the critical damping ratio of 0.05 and normalized by ao, are shown in this table, for different water levels.
Sloshing and Structural Coupling
Harmonic Buckling Tests
It also becomes clear that the behavior of the system for w > 4 Hz is controlled by the lifting and not the sloshing effects. Examining the response history of the system components in more detail, it is found that the coil effects are important for ,√ = 2.5 to 5 Hz. The phase levels for this case are shown in Figure 5.7.e-f, while the moment at the bottom of the tank is shown in Figure 5.7.g.
One possible reason for the difference between the experimentally and analytically determined critical base accelerations could be the different stress concentrations developed at the bottom of the tank for various forcing frequencies.
Transient Buckling Tests
In order to properly design such structures, it is necessary to know the response of the tank and its contents to the base excitation. System nonlinearities due to loss of contact and large displacements that develop at the bottom of the dur tank. For the unanchored tank, it is found - for the areas of the considered forcing frequencies - that in addition to pres.
For such tanks, apart from the consequences of the dramatic reduction of the frequency of the predom.
Fujita, K., "A Seismic Response Analysis of a Cylindrical Liquid Storage Tank Including the Effect of Scourging," Bull. Schmitt, A.F., "Forced Oscillations of a Fluid in a Cylindrical Tank Undergoing Both Translation and Rotation," Convair Astronautics, Report ZU. Bushnell, D., "BOSOR5 - Program for Cracking Elastic-Plastic Complex Shells of Revolution Including Large Deflections and Creep," Computers and Structures, Vol.
Appendix A
General Solution
Before attempting to determine the unknowns in the expression for the general solution Φ„, as given by (A.2), some appropriate changes are first made. In order to associate all the sloshing effects with one term in the solution, it has been found convenient to replace z with (z-H) in the argument for sinh. Otherwise, there would be a problem in determining a„(f), since z has its own Fourier components in {cos(λi.
Appendix B
Appendix C
Appendix D
ELEMENTS OF MATRIX EQUATION (3.19)
The subscript f indicates quantities evaluated by letting c=l, while the subscript u indicates quantities evaluated for arbitrary c. In general, the notation used in the above and following expressions is consistent with that adopted in Chapters 2 and 3. It is clear that the assumed spatial functions (‰, vm, um) appear only in the expressions of those elements in the equations of motion associated with the corresponding degrees of freedom √,m, ξm and ςm.
Appendix E
Appendix F
