Next: Anchored Tank Response Up: Nonlinear Earthquake Response of Previous: Nonlinear Earthquake Response of
Liquid motion in rigid containers reflects primarily the effect of sloshing on the response of these tanks. As a simplification for the analysis of circular cylindrical tanks, one may consider a rectangular strip in the middle of the tank and analyze it as a twodimensional problem. Alternatively, the full threedimensional model may be used. The linear fundamental period of the twodimensional model is given by
 (189) 
whereas for the threedimensional model, it is given by
 (190) 
where H is the liquid depth. Although the linear fundamental periods of both the two and the threedimensional models are close, as shown in Table (5.1), the corresponding wave heights may differ significantly. It is unconservative to use the twodimensional model for predicting the response of cylindrical liquid storage tanks. This is attributed to the difference in the mode shapes of both models. The maximum response calculated based on two and threedimensional models are presented and compared in Tables (5.2) and (5.3). Note that W denotes the total weight of the contained liquid and R is the tank radius. Results show that using the linear sloshing assumptions at the free surface underestimates its wave height. However, it predicts well the base shear exerted on the tank. Using the nonlinear sloshing assumptions shows that the positive sloshing amplitude is larger than the negative amplitude.
Table 5.1: Periods of Different Models for Liquid Sloshing in Rigid Tanks  Model Type  Tank Type  2D Model  3D Model  Broad Tank  7.74 sec  6.89 sec  Tall Tank  4.33 sec  4.00 sec  
Table 5.2: Liquid Sloshing in Rigid Containers Under El Centro Record  Wave Height (ft)    Case  Left End  Right End  OTM/WR  Base Shear/W  Broad Tank  2D  2.27  2.27  0.053  0.064  Linear Sloshing  3D  5.19  5.19  0.066  0.096  Broad Tank  2D  2.18  2.35  0.052  0.062  Nonlinear Sloshing  3D  4.77  3.61  0.067  0.103  Tall Tank  2D  3.84  3.84  0.197  0.141  Linear Sloshing  3D  2.23  2.23  0.216  0.155  Tall Tank  2D  4.24  3.56  0.196  0.143  Nonlinear Sloshing  3D  2.32  2.36  0.216  0.155  
Table 5.3: Liquid Sloshing in Rigid Containers Under the Northridge Record  Wave Height (ft)    Case  Left End  Right End  OTM/WR  Base Shear/W  Broad Tank  2D  1.62  1.62  0.094  0.113  Linear Sloshing  3D  1.52  1.52  0.086  0.125  Broad Tank  2D  1.59  1.64  0.092  0.105  Nonlinear Sloshing  3D  1.47  1.59  0.089  0.127  Tall Tank  2D  1.64  1.64  0.379  0.262  Linear Sloshing  3D  2.46  2.46  0.362  0.266  Tall Tank  2D  1.68  1.67  0.366  0.259  Nonlinear Sloshing  3D  2.43  2.59  0.378  0.266  
Figures (5.3) and (5.4) show the free surface wave heights of the two extreme opposite points on the principal diameter which parallels the earthquake excitation of the broad and tall tanks, respectively. Figures (5.5) and (5.6) present the deformed finite element mesh for both broad and tall tanks, respectively. Several ways to discretize the liquid domain were attempted to obtain the earthquake response of the liquid. It was found that the performance of an orthogonal finite element mesh yields better results and smoother free surface profile. This is attributed to the fact that the orthogonal mesh provides orthogonal mapping between the Cartesian and curvilinear coordinates, which is compatible with the stream and equipotential lines resulting from the solution of Laplace equation. Using other forms of finite element mesh in the nonlinear sloshing problem may result in a broken free surface profile and, as a result, the analysis may fail to converge.
Figure 5.3: Nonlinear Time History Response of the Wave Height in the Broad Tank, 3D Model 
Figure 5.4: Nonlinear Time History Response of the Wave Height in the Tall Tank, 3D Model 
Figure: Deformed Liquid Domain of the Broad Tank at t=43.5 Sec, Northridge Record 
Figure: Deformed Liquid Domain of the Tall Tank at t=32.5 Sec, El Centro Record

Next: Anchored Tank Response Up: Nonlinear Earthquake Response of Previous: Nonlinear Earthquake Response of A. Zeiny
20000906