dynamic response of concrete rectangular liquid ...

Fourth International Conference of Earthquake Engineering and Seismology 12-14 May 2003 Tehran, Islamic Republic of Iran

DYNAMIC RESPONSE OF CONCRETE RECTANGULAR LIQUID STORAGE TANKS By J. Z. Chen1, and M. R. Kianoush2 1. Graduate Student, Department of Civil Engineering, Ryerson University, Toronto, Ontario, Canada. E-mail: [email protected] 2. Professor, Department of Civil Engineering, Ryerson University, Toronto, Ontario, Canada. E-mail: [email protected]

ABSTRACT In most of the current design codes and standards for concrete liquid storage tanks, the hydrodynamic pressures are determined assuming that the tank walls are rigid. The Housner’s model, which assumes that the mass of liquid is lumped on the wall based on rigid wall boundary condition in the calculation hydrodynamic pressure, is widely used in practice. Previous research has indicated that the flexibility of wall can affect the distribution and magnitude of hydrodynamic pressures. In this paper, the effect of wall flexibility on hydrodynamic pressures in concrete rectangular tanks is studied using an approach referred to as the sequential method. The procedure on how to establish the impulsive hydrodynamic pressures using the sequential method based on a two-dimensional analysis is described. A comparison shows that the lumped mass approach overestimates the base shear and wall displacements.

1. INTRODUCTION Liquid storage tanks as part of environmental engineering facilities are primarily used for water and sewage treatment plants and other industrial wastes. Normally, they are constructed of reinforced concrete in the form of rectangular or circular configurations. The behavior of liquid storage tanks during earthquakes is more important than the economic values of the tanks and their contents. It is important that utility facilities remain operational following an earthquake to meet the emergency requirements such as firefighting water or meet the public demands as a source of water supply. On the other hand, the containment of hazardous waste is important to meet the public safety after a seismic event. For these reasons, serviceability becomes the prime design consideration in most of these structures. A good understanding of the seismic behaviour of these structures is necessary in order to meet safety objectives while containing construction and maintenance costs. There are currently few guidelines and standards available in North America for the design of liquid storage tanks. There is a great deal of inconsistency among these guidelines and standards for this important broad class of structures. This is mainly due to the lack of understanding of the seismic behaviour of these structures. An extensive research project is currently underway at Ryerson University to resolve some of the

issues of concern for design of these types of structure. As par of this ongoing research effort, in this paper, the dynamic response of rectangular tanks for liquid storage tanks is described. Early investigation of dynamic analysis of fluid in containers subjected to acceleration was conducted by Housner (1963). An approximate method was proposed to include the effect of hydrodynamic pressure for a two fold-symmetric-fluid container subjected to horizontal acceleration. The fluid response was represented by impulsive and convective components. The fluid was assumed to be incompressible and the container was assumed to have rigid walls. Yang and Veletsos (1976) used Flűggle’s shell theory to analyze circular tanks. It was found that for tanks with realistic flexibility, the impulsive forces are considerably higher than those in rigid wall. Veletsos et al. (1984) considered the effect of the wall flexibility on the magnitude and distribution of the hydrodynamic pressures and associated tank forces. They assumed that the tank-fluid system behaved like a single degree of freedom system and the base shear and moment were evaluated for several prescribed modes of vibration. Most of the research conducted on liquid storage tanks, as mentioned above have been of circular configurations. For rectangular tanks, Haroun (1984) gave a very detailed analysis method in the typical system of loadings. The hydrodynamic pressures were calculated by classical potential flow approach. The formula of hydrodynamic pressures only considered the rigid wall condition. Park et al. (1990) studied the dynamic behaviour of rectangular tanks using boundary element modeling for the fluid motion and finite element modeling for the solid walls. The time history analysis method was used to obtain the dynamic response of fluid storage tanks subjected earthquakes. Both impulsive and convective effects were considered. Later they presented an analytical method with formulas for the 3-D hydrodynamic pressures calculation and applied Rayleigh-Ritz method using assumed vibration modes of rectangular plate with boundary conditions as admissible functions (Kim et al., 1996). In this paper, a new procedure to determine hydrodynamic pressures for rectangular tanks is proposed. The method considers the effect of wall flexibility on impulsive pressures.

2. HYDRODYNAMIC PRESSURE Figure 1(a) shows a 3-D rectangular tank. It is assumed that the liquid storage tank is fixed to the rigid foundation and a Cartesian coordinate system (x, y, z) is used with the origin located at the center of the tank base. Furthermore, it is assumed that the width to the length ratio of the tank is so large that the unit width of tank can represent the tank and the corresponding 2-D model as shown in Figure 1(b). The fluid filled in the rectangular tank is of height, Hl above the base. The fluid is considered to be ideal, which is incompressible, invicid, and with a mass density ρl. The response of the body of fluid to an earthquake can be treated as gravity waves on its free surface, which is irrotational in most instances.

(a) 3 –D model of rectangular tank (b) 2 –D model of rectangular tank Figure 1. Schematic of Rectangular Tank

The governing equation of motion in matrix form can be expressed by:

[M ]⋅ {u&&r }+ [C ]⋅ {u& r }+ [K ]{u r } = −[M ]{u&&g }+ {P}

(1)

Where:

{u r }, {u& r }, {u&&r } : Displacement, velocity and acceleration of rectangular wall relative to the ground motion

{u&&g }: Horizontal ground acceleration in x direction

{P} : Hydrodynamic pressures on the wall surface [K ] : Stiffness matrix of rectangular tank wall [M w ] : Mass matrix of rectangular tank wall [C ] : Damping matrix of rectangular tank wall

The solution of velocity potential, which satisfies the boundary conditions, can be solved by the method of separation of variables introduced by Currie (1973) . The Hydrodynamic pressure distribution on the flexible wall related to the velocity potential can be expressed by: ∞

p=

∑ i =1

2 ⋅ ρ ⋅ tanh(λi ⋅ L ) H l x cos(λi y) ∫0 cos(λi y) ⋅ u&&(t)dy λ ⋅ Hl i

(2)

Where λi = (2i-1)π/2Hl. The detailed derivation of the above equation is discussed by Chen (2003). As the series in Eq.(2) convergence very fast, only the first term of the series may be used in practical application. For the rigid tank ü(t) =üg(t) which means that the acceleration along the height of the wall is same as the acceleration of ground motion, then Eq.(2) becomes: ∞

p=

∑ i =1

2 ⋅ (−1)i ⋅ ρl tanh(λi ⋅ Lx ) ⋅ cos(λi ⋅ y ) ⋅ u&&g (t ) λ2i ⋅ H l

(3)

This result is the same as the hydrodynamic pressure equation for the rigid wall derived by Haroun (1984).

3. THE SEQUENTIAL METHOD In order to consider effects of flexibility of the tank wall on hydrodynamic pressures in dynamic analysis, a sequential method is used in this investigation. The sequential method is a technique in which the two fields of fluid and structure are coupled by applying results from the first analysis as loads or boundary conditions for the second analysis. Basically the dynamic response of liquid storage tank must be solved by “strong” coupled method, which is that the data must be transferred or shared between at each step of the solution to maintain accuracy of dynamic response analysis. From Eq.(1), the hydrodynamic pressure can be treated as the external forces applied on the rectangular tank wall, and the boundary conditions of rectangular tank wall determines the hydrodynamic pressure in Eq.(2). Actually these two equations must be solved simultaneously because the interaction between the rectangular tank wall and the hydrodynamic pressure occurs at the same time. Since it is difficult to solve the dynamic response of wall and hydrodynamic pressure directly from Eq.(1), we can apply the sequential method to approximate it. The sequential method is carried out by the following procedure. First the dynamic response of the flexible tank wall subjected to an earthquake is analyzed at time step t=∆t. Then the hydrodynamic pressure is determined, which also includes the effect of flexibility of the tank wall. Finally the hydrodynamic pressure is applied on the tank wall at the next time step. The procedure is then repeated at each time step until the analysis is complete. Figure 2 shows in a flowchart format the procedure for analysis and Figure 3 shows how the data is transferred between rectangular tank wall and fluid.

Specify the Inertial Condition of Rectangular tank (solid wall) Specify the Inertial Condition of Fluid inside ({P}=0 at first time step) Apply the Seismic Forces and Hydrodynamic Pressure on the Rectangular Tank Wall Calculate the Structural Dynamic Response of Rectangular Tank Wall Transfer Dynamic Response of Flexible Wall to Fluid Domain Calculate the Hydrodynamic Pressure (Boundary Condition Is Defined by Dynamic Response of Wall.)

Input Ground Accerleration Record for the Next Time Step yes

No. End

Figure 2. Procedure of sequential analysis

Rectangular tank Wall

Vibration Mode of flexible wall

Fluid

Hydrodynamic pressure Figure 3. The transfer data between two fields

The sequential method is incorporated into a computer software package, SAP IV (Bathe et al. 1974) using the subroutine called HYDRO developed for this purpose. The direct step-by-step integration method is used because this method can calculate the displacement, velocity and acceleration of the flexible wall subjected to an earthquake at each time step. The impulsive hydrodynamic pressure is calculated by the subroutine HYDRO that considers the effect of flexibility of tank wall in the boundary condition.

4. ANALYSIS OF A RECTANGULAR TANK In order to demonstrate the efficiency of the sequential method as described above, and to study the effect of different modeling approaches on dynamic response of a storage tank, a rectangular tank is analyzed under a horizontal ground motion. The North-South component of El-Centro accelerogram of 1940 Imperial Valley record with a maximum peak ground acceleration of 0.33 g is used. This is the same rectangular tank that has been analyzed by Kim et al. (1996). in their investigations. It should be mentioned that after a detailed review, the writers disagree with some of the results presented in the aforementioned reference. For two-dimensional analysis, a one-meter strip of tank wall is considered. The tank wall is discretized into two and 22 elements in

the horizontal and vertical directions respectively using a 2-D plane rectangular element. The dimensions and the properties of the tank are as follows: ρw = 2300 Kg/m3 ρl = 1000 Kg/m3 E = 2.0776E10 Pa Hw =12.3 m ν = 0.17

tw = 1.2 m Lx = 9.8 m Lz = 28 m Hl =11.2 m

For the liquid storage tank, six conditions referred to as models are considered. In both models 1 and 2, the impulsive mass of liquid is determined using the procedure described by Housner (1963) . In model 1, both the impulsive mass and the inertial mass of wall is lumped at an equivalent height h, determined by: h=

M i ⋅ hi + M w ⋅ h w hi + h w

(4)

Where: hi = height from the base of the wall to the center of gravity of the impulsive lateral force hw = height from the base of the wall to the center of gravity of the tank wall Mi = equivalent mass of impulsive component of stored liquid Mw = equivalent mass of wall In model 1, the period of vibration of the tank wall is determined using the classical approach for a cantilever wall. This represents a typical model for tank wall used in most of the current codes and standards for concrete liquid containing structures. In model 2, the mass of wall is distributed over the height of the wall while the impulsive mass of the liquid is concentrated at height hi as given by Housner (1963). Model 3 is the same as model 2, but the impulsive mass is determined using the proposed method assuming rigid wall condition. Models 4, 5 and 6 are also based on the proposed method. Model 4 is the same as model 3 except that the impulsive mass is distributed over the height of the wall. In model 5, the hydrodynamic impulsive pressure is determined considering the wall flexibility. This model is expected to provide the most accurate results among the six different models. Model six is the same as model 5 except that the wall is assumed to be infinitely rigid. In this case, the hydrodynamic pressure is determined assuming a rigid wall condition. In models 1 to 4, the mode superposition method is used for dynamic analysis while the last two models are analyzed using the direct step-by-step method including the sequential procedure. Damping ratio for all models is assumed to be 5% of critical. The Rayleigh damping [C] = "[M]+ $[K] is used in the direct step-by-step integration method. The values " and $ are determined using the procedure as described by Bathe et al. (1974) . A summary of the calculation results for all models is listed in Table 1. This is in terms of the maximum base shear and top displacement during the time history analysis. Peak values corresponding to two different times at which maximum values occur are shown. From calculation results, it is observed that for model 1, the results in particular base shear is much higher than those observed in other models. This gives an indication that the procedure adopted in the codes and standards are too conservative. It is observed that the results of Model 2 and 3 are very similar. The difference in results between these two models does not exceed 10%. For model 4, although the period of first two modes are not much different from models 2 and 3, there is a major difference in the values of the base shear which is less for model 4. This effect is due to the mass

distribution. In these three models, the rigid wall boundary condition is used to calculate the impulsive pressure, which is approximated by added masses placed on the wall. The direct step-by-step integration method together with the proposed sequential analysis is applied to model 5. In this case, the effect of flexibility of wall on hydrodynamic pressure and tank wall is considered. The base shear is increased due to the flexibility of tank wall as compared to model 4 but the maximum displacement is somewhat smaller. From calculation results, it is observed that the results obtained using the Housner’s method (model 2) are higher than those obtained for models 3 and 4. However, the peak values of displacement and base shear occur almost at the same time during the time history analysis. Model 6, which is considered as a special case, represents an infinitely rigid wall. In this case, the wall moves with the ground motion. The maximum base shear is much less than that in the flexible wall (model 5).

5. CONCLUSIONS The dynamic response of liquid storage tanks is studied in this investigation. Using the proposed model, the hydrodynamic pressures are no longer required to be approximated by added mass but they can be treated as external forces. The advantage of the proposed model is that it can consider the effect of the flexibility of the wall on the calculation of hydrodynamic pressures. In order to analyze the dynamic response of liquid storage tanks, the sequential method is applied. To demonstrate the efficiency of the proposed model using the sequential method, the response of a rectangular tank is studied based on a time-history analysis. This study shows that the dynamic response of liquid storage tanks calculated using the current design codes or standards in terms of base shear is too conservative. It is concluded that the effect of the flexibility of the tank wall should be considered in the calculation of hydrodynamic pressures in concrete rectangular tanks.

6. REFERENCES Bathe, K. J., Wilson, E. L. and Peterson, F. E. (1974). SAP IV – A Structural Analysis Program For Static and Dynamic Response of Linear System, College of Engineering, University of California, Berkeley, EERC 73-11, June 1973, Revised April 1974. Chen, J. Z. (2003). Dynamic Analysis of Rectangular Liquid Storage Tanks, Thesis (draft), Department of Civil Engineering, Ryerson University, Toronto, Ontario, Canada, 2003. Currie, I. G. (1973). Fundamental Mechanics of Fluid. McGraw-Hill, Inc., New York, N.Y. Haroun, M. A. (1984). Stress Analysis of Rectangular Walls Under Seismically Induced Hydrodynamic Loads, Bulletin of the Seismological Society of America, Vol. 74, No. 3, June 1984 Housner, G. W. (1963). The Dynamic Behavior of Water Tanks, Bulletin of the Seismological Society of American, Vol.53, No.2, February, 1963. Kim, J. K., Koh, H. M. and Kwahk, I. J. (1996). Dynamic Response of Rectangular Flexible Fluid Containers, Journal of Engineering Mechanics, Vol.122, No. 9, September, 1996, ASCE, ISSN 07339399/96/0009-0807-0817. Park, J. H., Koh, H. M. and Kim, J. (1990) Liquid-Structure Interaction Analysis by Coupled Boundary Element-Finite Element Method in Time Domain, Proc., 1990 7th International Conference on Boundary Element Technology, BE-TECH/92, Computational Mechanics Publication, Southampton, England, 89-92. Veletsos, A. S. (1984). Seismic Response and Design of Liquid Storage Tanks. Guidelines for the Seismic Design of Oil and Gas Pipeline Systems, Tech, Council on Lifeline Earthquake Engineering., ASCE, New York, N.Y. Yang, J. Y. and Veletsos, A. S. (1976). Dynamic Behavior of Fluid-Tank System-Ph D Thesis, Civil engineering, Rice University, Houston, Texas March 1976.

Table 1. Summary of Dynamic Response of Liquid Storage Tank Model 1 (Housner)

Model 2 (Housner)

Model 3 (Proposed)

Impulsive component

Lumped

Lumped

Lumped

Boundary Condition

Rigid

Rigid

Rigid

Inertial Mass

Lumped

Distributed

Distributed

Wall Type

Flexible

Flexible

Flexible

Mode Superposition Method

Mode Superposition Method

Mode Superposition Method

T1 = 0.2389 T2 = 0 5%

T1 = 0.3001 T2 = 0.08201 5%

T1 = 0.2967 T2 = 0.07989 5%

Model

Fluid

Wall

Analysis Method

Schematic of Model

Period (sec) Damping Ratio tmax (sec)

2.56

4.31

2.64

4.52

2.63

4.52

Base Shear FB (KN)

806.400

395.160

429.720

356.760

401.760

300.360

33.89

25.63

Top Displacement dA (mm) 36.53 17.90 35.30 29.61 * tmax : time at which the peak values of dynamic response is reached during the time-history analysis

Table 1. Summary of Dynamic Response of Liquid Storage Tank – (continued) Model 4 (Proposed)

Model 5 (Proposed)

Model 6 (Proposed)

Impulsive Component

Distributed

Distributed

Distributed

Boundary Condition

Rigid

Flexible

Rigid

Inertial Mass

Distributed

Distributed

Distributed

Wall Type

Flexible

Flexible

Rigid

Mode Superposition Method

Direct Step-by-Step Integration Method and Sequential Method

Direct Step-by-Step Integration Method and Sequential Method

--α=1.528, β=0.0027 2.65 4.56

--α=0, β=0

Model

Fluid

Wall

Analysis Method

Schematic of Model

T1 = 0.3464 T2 = 0.06458 5%

Period (sec) Damping Ratio tmax (sec)

2.70

4.76

Base Shear FB (KN)

309.960

298.200

308.462

327.468

Top Displacement dA (mm) 33.16 31.88 23.09 26.93 (t=4.57sec) * tmax : time at which the peak values of dynamic response is reached during the time-history analysis

2.16

5.030

107.960

102.344

0

0