In this study, non-linear static analysis, commonly known as pushover analysis, is used to obtain the load-carrying capacity of the conical tanks. The non-linear static analysis is carried out by increasing the load value incrementally till reaching failure, which is in the form of buckling of the steel vessel. The load increase is achieved using an increasing load factor, which is multiplied by the applied hydrodynamic pressure load pattern. The non- linearity in the analysis comes from the inclusion of both geometric and material non- linearity in the finite element model previously discussed. To include both the hydrostatic and hydrodynamic pressure in the analysis, two load factors are used. The first is PHS,
which corresponds to the hydrostatic pressure while the second is PHD, which corresponds
to the hydrodynamic pressure. The analysis starts with a value of the load factor PHD equals
to zero and then the load factor PHS is increased incrementally until it reaches the actual
value of the hydrostatic pressure acting on the tank. After this stage, the value of PHS is
kept constant and the value of PHD begins at zero and increases until failure occurs. At the
end of the analysis, the value of PHD at failure is recorded along with the deformations,
forces, and stresses corresponding to the failure value. It should be noted that the load increments near the failure are reduced in order to better capture the failure load factor. This is achieved by doing more than one trial for each analysis.
A group of 75 tanks of practical dimensions are chosen for this study with Rb ranging from
4.0m to 6.0m, h from 5.0m to 9.0m, and θv = 30o, 45o, 60o withsteel yield stress of 300
MPa. The tanks are preliminary designed under hydrostatic pressure based on the simplified method proposed by Sweedan and El Damatty (2009) assuming good tanks regarding the level of geometric imperfection. As yielding of the tank usually precedes buckling, i.e., inelastic buckling is expected, the main idea of this simplified design procedure was to prevent the tank shell from reaching the yielding state at any point under hydrostatic pressure. The concept was based on developing a stress magnification factor that relates the maximum stresses that occur in the walls of the tanks resulting from membrane, bending and geometric imperfection effects, to the theoretical values obtained from membrane behaviour.
Based on the distribution of different pressure modes shown in Fig. 2-4, it is clear that the total base shear will result from cos θ mode only, i.e., n=1. Regarding the vertical distribution of the pressure modes, only the first vertical mode, i.e., i=1, is included in the analysis of calculating the conical tank capacities.
The analysis starts with applying the axi-symmetric hydrostatic pressure on the tank walls. A typical plot for the radial deformations of the tank walls at different angles θ is shown in Fig. 2-6. As it is expected due to the inclination of the walls, a buckling wave is noticed near the base of the tank due to the high compressive stresses at that location. It is worth mentioning that although this buckling wave occurred, the tank is still able to resist more loads due to hydrodynamic pressure. This is the case since the tanks are designed to sustain a value of PHS exceeding the actual hydrostatic pressure. After reaching this stage, the
hydrodynamic pressure with the cos θ pattern is applied on the walls of the tank and increased incrementally while the hydrostatic pressure is kept constant.
By comparing the radial deformations of the tank walls at failure shown in Fig. 2-6 to those just after the hydrostatic pressure phase, it is noticed that along the line θ = 0o the radial
displacement peak location is shifted up and the buckling wave near the base of the tank becomes more clear. At θ=90o, the distribution and the values of the radial displacement
are nearly the same. Finally at θ=180o, the radial displacements at failure are with negative
sign, i.e., inwards the tank and the buckling wave is reduced. A typical circumferential distribution of the tank deformations is shown in Fig. 2-7.
Fig. 2-6 Radial deformations of the tank walls at the end of each phase
Fig. 2-7 Deformed shape at the end of each phase of loading
In order to check the assumption of excluding the base rocking motion effects on the hydrodynamic pressure when obtaining the capacities of steel conical tanks, two sets of time history analyses were performed; one with allowing the tank base to undergo rocking motion while the other preventing such motion. The results of the analyses showed that the difference between the base shear values recorded at buckling for the two cases to be less than 5% for different tank configurations, i.e., inclination angle and level of imperfections. As such, the curves developed in this study for estimating the seismic capacity of steel conical tanks can be applied to both ground and elevated tanks.
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 Z (m ) Radial displacement (m) hydrostatic HS+Hd (0) HS+Hd (90) HS+Hd (180) Phase 1 Phase 2 (θ=0o) Phase 2 (θ=90o) Phase 2 (θ=180o) Unloaded tank Phase 1 Phase 2