Paper No. T.3-1.2, pp. 1-5
The 6th PSU-UNS International Conference on Engineering and Techn ol ology ogy (ICET (ICET-201 -2013), 3), Novi Sad, Serbi a, May May 15-17, 15-17, 2013 2013 University o f Novi Sad, Faculty Faculty o f Technical Sciences
A SIMPLIFIED SEISMIC ANALYSIS ANALYSIS OF CIRCULAR LIQUID STORAGE TANKS *
Vladimir Vukobratović , Đorđe Lađinović University of Novi Sad, Faculty of Technical Sciences, Novi Sad, Serbia *Authors to correspondence should be addressed via email:
[email protected]
Abstract: Experience shows that inadequately designed or detailed liquid storage tanks, which are common structures in various various facilities, facilities, suffered extensive extensive damage during past earthquakes. Therefore, the satisfactory seismic response of such structures is very important. This paper presents the results of a s implified implified analysis of the total seismic response of ground-supported concrete circular liquid storage tanks with various geometrical properties. The analysis was conducted in accordance with the provisions of Eurocode 8 and a comparative analysis of the obtained results is presented. In addition, a practice-oriented approach to the analysis of local seismic response quantities of tank structures is also presented. Key Words: circular tanks, impulsive mode, convective mode, Eurocode 8
1. INTRODUCTION
In various industrial facilities, such as petroleum or wastewater treatment facilities, ground-supported liquid storage tanks represent very common structures. The satisfactory seismic response of such structures is crucial since collapse usually results in heavy consequences (fire, spilling of liquid, etc.). During a seismic action, inertial forces are induced due to the acceleration of a tank structure and hydrodynamic forces are induced due to the acceleration of liquid. The liquid mass in the lower part of a tank behaves like a mass that is rigidly connected to the tank wall and it is termed impulsive liquid mass. It accelerates along with the wall and induces impulsive hydrodynamic pressure. The liquid mass in the upper part of a tank undergoes sloshing motion and it is termed convective liquid mass. It induces convective hydrodynamic pressure. Obviously, the response of a tank structure which is exposed to the seismic action represents a complex problem. Therefore, finding a simple, but accurate enough solution for such a problem represents a big challenge challenge for a designer. 2. A DYNAMIC MODEL USED IN THE ANALYSIS
powerful software tools. Nevertheless, the total seismic response (base shear force and overturning moments) can also be properly evaluated using some of the simplified dynamic models defined in the past. Besides simplicity, these models also provide a quick evaluation of the total response, which makes them attractive for application. The simplified dynamic model of a tank-liquid system that is included in most of the design codes is based on the model defined by Housner [4]. The dynamic analysis of a tank structure can be carried out using the concept of generalised single-degree-offreedom systems (SDOF), representing the impulsive and convective modes of vibration, as shown in Fig. 1. The seismic response of SDOF systems may be calculated independently and then combined in order to provide the total response of a structure. Only the first modes of vibration need to be considered in the analysis. It should be noted that concrete and steel tanks show different behaviour under a seismic action. In the case of concrete tanks, the wall may be taken as rigid, whereas in the case of steel tanks, the wall may be taken as flexible. In the following text, attention will be focused on concrete tanks. A dynamic model which is suitable in such cases is shown in Fig. 1. For steel tanks, a more suitable model should be adopted, which is discussed elsewhere [7].
Fig. 1. A 1. A dynamic model model of a concrete concrete circular tank In the dynamic model presented in Fig. 1, mi and mc denote impulsive and convective masses of liquid
The behaviour of a tank-liquid system exposed to a seismic action can be accurately evaluated using various
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respectively, hi is the height where the resultant of impulsive pressure on the wall is located, while hi' is the height where the resultant of impulsive pressure on the wall and the base is located. Similarly, hc is the height where the resultant of convective pressure on the wall is located, while hc' is the height where the resultant of convective pressure on the wall and the base is located. The inner radius of a tank is denoted as R. All of the heights are measured from the bottom of the tank wall. The parameters of the dynamic model depend on tank’s geometry. These parameters are different in cases of tanks with rigid and flexible walls, although the difference is not substantial. In the simplified procedure proposed by Eurocode 8, the recommended design values of dynamic parameters are originally defined by Malhotra [6]. These values are related to the first impulsive and convective modes of vibration and they depend on the ratio H / R, where H denotes the height of liquid. Additionally, ml denotes the total mass of liquid. Besides already defined quantities, the coefficients C i and C c also appear as the parameters of the dynamic model and they are related to the natural periods of the impulsive and convective modes of vibrations. This will be shown below. It should be noted that sometimes additional vertical structural elements are present inside a tank. These elements cause obstruction to the sloshing motion of liquid. Unfortunately, no study is yet available to quantify the effect of such obstructions, but it is reasonable to expect that the impulsive component will increase, while the convective component will decrease. 3. TOTAL SEISMIC RESPONSE
The total seismic response of a tank structure should be analysed in terms of natural periods of vibrations, base shear force and overturning moments. As already mentioned, these quantities will be determined in accordance with the provisions of Eurocode 8. The modal properties of a tank structure should be determined at the beginning of the analysis. The natural periods of the impulsive and the convective modes are calculated by the Eqns. 1 and 2, respectively, Ti C i
H
s / R E
Tc Cc R
(1) (2)
where s represents equivalent uniform thickness of the tank wall (in the case of a wall with constant thickness, s is equal to the wall thickness), E is the modulus of elasticity of tank material and ρ is mass density of the liquid. The total base shear force is calculated by the Eqn. 3, (3) Q mi mw mr Se Ti mc S e Tc where mw represents the mass of the tank wall, mr represents the mass of the tank roof, S e(T i) is the impulsive spectral acceleration (obtained from a 5% damped elastic response spectrum for the case of reinforced concrete tanks, for both damage limitation and ultimate limit state) and S e(T c) is the convective spectral acceleration (obtained from a 0.5% damped elastic response spectrum in all cases).
The overturning moment immediately above the base plate ( M ) is calculated by the Eqn. 4, mi hi mw hw mr hr S e Ti mc hc S e Tc (4) where hw is the height of the center of gravity of the tank wall and hr is the height of the center of gravity of the tank roof. The overturning moment immediately below the base plate ( M' ) is calculated by the Eqn. 5. '
mi hi' mw hw mr hr Se Ti mc hc' Se Tc (5)
It is obvious from Eqns. 3-5 that Eurocode 8 suggests the absolute summation rule to combine the response from impulsive and convective modes. The basis for this suggestion is that the natural period of the convective mode is usually much greater than the natural period of the impulsive mode. Therefore, it is expected that the peak response of the impulsive and convective modes will occur simultaneously, i.e. when the convective mode response is near its peak. However, some recent studies show that SRSS combination rule gives better results than the absolute summation rule. 3.1. Parametric study
The seismic response of a ground-supported circular concrete tank is primarily influenced by its geometrical properties. According to Eurocode 8 standard, the ratio of the liquid height and the inner radius of a tank ( H / R) defines the parameters of the dynamic model of a tank structure. A comparative analysis of the seismic response of tanks with various geometrical properties was conducted through a parametric study and the results are presented in the following text. All of the response quantities introduced in the previous section were taken into account in the study. In the parametric study, the following input data were used: The values of H / R from the Eurocode 8 were taken into account in the analysis in order to be consistent with the code [3]. Water was considered as liquid ( ρ=1000 kg/m³). The volume of water (V ) was taken to be a constant value for all tanks and it equaled 4000 m³, which means that the mass of water was also a constant value (ml =4000 tons). Tank material was concrete C35/45 ( E =34×10 9 N/m²). The thickness of the tank wall ( s) was taken to be constant along the wall height for all analysed tanks and it equaled 0.40 m (it is clear that in the case of tanks of great height, 0.40 m is not enough from the concrete design aspect and that wall thickness should be increased, but it does not significantly affect this analysis). The analysis considered tanks without a roof (in this way, analysis was significantly simplified since the roof mass and the height of its center of gravity did not need to be calculated). The height of a tank wall ( Lw) was always taken to be 1.0 m greater than the water height (in tanks without a roof, spilling of liquid should be prevented during a seismic action and although the provisions of Eurocode 8 define sloshing
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wave height, for the purpose of this study, 1.0 m was adopted as a conservative value which satisfied all analysed tanks). In the analysis, Eurocode 8 elastic spectrum (type 1, soil type B) was used [2], as well as: reference peak ground acceleration a gR=0.20g, importance factor for the structural class III γ I =1.20 (which gives design ground acceleration a g =0.24g), damping ratio for the impulsive mode (for the case of reinforced concrete tanks) ξ i=5% and damping ratio for the convective mode ξ c=0.5% (damping correction factor η=1.35). The choice of a constant value of water volume (mass) is quite logical, since the main idea of the study was to investigate the influence of tank geometry on its total response. This way, H and R values changed for each selected H / R ratio. Besides theoretical, this study also possesses practical significance. Therefore, realistic input data were chosen with the slight exception of the wall thickness and the sloshing wave height, which has already been explained. Based on the input data described above and H / R values defined in Eurocode 8, the dynamic properties of tanks with different H / R were calculated. Additionally, the values of Lw, hw, and mw were also calculated. The seismic response of the analysed tanks is presented in the Tab. 1 and Figs. 2 and 3. Table 1. The seismic response of the analysed tanks T i T c Q M M’ H / R [sec] [sec] [kN] [kNm] [kNm] 0.3 0.050 8.41 5837 14307 50018 0.5 0.053 6.44 8688 27556 68197 0.7 0.056 5.59 11259 43730 84042 1.0 0.061 5.00 14434 71732 105872 1.5 0.072 4.56 18476 123006 145998 2.0 0.085 4.34 21797 177814 192271 2.5 0.101 4.19 25078 237931 248973 3.0 0.117 4.05 28116 300816 309076
It is obvious from the results presented in Tab. 1 that the periods of the impulsive and the convective modes are rather distant which justifies the use of absolute summation rule for the combination of responses from impulsive and convective modes.
Fig. 3. The dependence of the base shear force (Q) and the overturning moments (M, M') on the ratio H/R All of the examined quantities except the convective period of vibrations increase almost linearly with the increase of the ratio H / R. The convective period of vibrations is practically a constant value for the H / R>1.5. Therefore, it can be concluded that the seismic response of a circular tank structure mostly depends on the response in the impulsive mode, although the convective part of the response must not be neglected in order to conduct an accurate seismic analysis. The values of Q and M should be used for the calculation of stresses and stress resultants in the tank wall and at the connection to the base. The value of M' should be used for the verification of its supporting structure, base anchors or foundation. The increase of base shear force and overturning moments controlled by the increase of the ratio H / R directly influences the detailing demands which are in certain cases hard to fulfill. It should also be noted that with the increase of the ratio H / R the difference between the overturning moments tends to decrease, which is the fact that can be of great practical s ignificance. Unfortunately, the geometry of a tank structure (i.e. the ratio H / R) is in most cases pre-defined and it mainly depends on technological demands inside the facility which means that a structural designer practically has no influence on its choice. 4. LOCAL SEISMIC RESPONSE
In order to obtain local response quantities (stresses, strains, etc.) which occur in a tank structure during a seismic action, a proper simulation of all individual actions on the a structure must be made. During lateral base excitation, the tank wall is subjected to lateral hydrodynamic pressures and the tank base is subjected to vertical hydrodynamic pressures. Pressure due to wall inertia, which is in the case of walls
Fig. 2. The dependence of the impulsive (T ) i and the convective (T c ) periods of vibrations on the ratio H/R
3
with uniform thickness constant along the wall height, should be added to the impulsive hydrodynamic pressure. Since the provisions of Eurocode 8 regarding the calculation of aforementioned pressure components are informative, in the following text, simpler and more practice-oriented expressions will be introduced. The introduced expressions are essentially quite similar to those proposed by Eurocode 8 and they are based on the provisions of IITK-GSDMA Guidelines [5]. Lateral impulsive pressure exerted by the liquid on the tank wall is defined by the Eqn. 6, Se T i piw Qiw y H cos (6) q g where γ is the specific weight of liquid, φ is a circumferential angle as shown in Fig. 4, y is a vertical distance of a point on the tank wall measured from the wall bottom, Qiw( y) is the coefficient of lateral impulsive pressure on the wall defined by the Eqn. 7 and q is the behaviour factor for the impulsive mode. The value of S e(T i) should be in units of g .
y 2 2R Qiw y 0.866 1 tanh 0.866 H H
(7)
It should be noted that Eurocode 8 defines that the behaviour factor q should be equal to 1.0 in the case of the damage limitation state. In the case of ultimate limit state verifications, the use of q factors greater than 1.5 is only allowed provided that the sources of energy dissipation are explicitly identified and quantified and that inelastic response is acceptable. Nevertheless, the convective part of the liquid response shall always be evaluated on the basis of elastic response (q=1.0). This means that no energy dissipation can be associated with the convective response of the liquid.
1 pcw Qcw y Se Tc 2 R cos cos3 g 3
(9)
where Qcw( y) is the coefficient of lateral convective pressure on the wall defined by the Eqn. 10. The value of S e(T c) should be in units of g . y cosh 3.674 2 R Qcw y 0.5625 (10) H cosh 3.674 2 R Vertical convective pressure on the tank base is defined by the Eqn. 11,
pcb Qcb x S e Tc 2 R g
(11)
where Qcb( x) is the coefficient of vertical convective pressure on the tank base defined by the Eqn. 12. 9 x
3 4 x H Qcb x sec h 3.674 8 2 R 3 2 R 2R
(12)
As already said, pressure due to wall inertia should be added to the impulsive hydrodynamic pressure. It should be calculated by the Eqn. 13, Se T i m pww s (13) q g where ρm is the mass density and γm is the specific weight of tank wall material. The value of S e(T i) should be in units of g . It is obvious from the Eqns. 6-12 that in the case of circular tanks, hydrodynamic pressure due to horizontal excitation varies around the circumference of a tank as well as along the height of the tank wall, which is presented on Figs. 5 and 6.
l’
R x
φ
R φ
Fig. 5. Idealized distribution of hydrodynamic pressure around the circumference of a tank
Fig. 4. Geometric characteristics of a circular tank Vertical impulsive pressure on the tank base on a strip of length l' is defined by the Eqn. 8, x sinh 1.732 Se T i H pib 0.866 H (8) q g l ' cosh 0.866 H
impulsive component
where x is a horizontal distance of a point on the base of a tank in the direction of seismic force, from the center of the tank, as sho wn in Fig. 4. Lateral convective pressure on the tank wall exerted by the oscillating liquid is defined by the Eqn. 9,
convective component
Fig. 6. Idealized distribution of hydrodynamic pressure along the height of the tank wall
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In the absence of a more exact analysis, an equivalent linear pressure distribution, which gives the same base shear force and bending moments at the bottom of the tank wall, can be assumed in a practical analysis. During vertical base excitation, the effective weight of liquid increases, which induces additional pressure on the tank wall. Distribution of this pressure is similar to hydrostatic pressure in the case of tanks with rigid walls. The effective fluid pressure will be increased or decreased due to effects of vertical acceleration. The natural period of vibration of vertical motion of a tank and the stored liquid can be determined by the Eqn. 14, which is proposed by ACI 350.3 Standard [1]. T v 2
GSDMA and ACI provisions. The effect of both the impulsive and the convective component was considered in the model and the influence of both horizontal and vertical seismic action was taken into account. A comparative analysis of the total seismic response of tanks with various geometrical properties was conducted through a parametric study. Since the seismic response of ground-supported concrete circular tank is primarily influenced by its geometrical properties, the ratio of the liquid height and the inner radius of a ( H / R) was the main parameter in the analysis. The study showed that the impulsive period of vibrations, base shear force and overturning moments increase almost linearly with the increase of the ratio H / R, whereas the convective period of vibrations is practically a constant value for the H / R>1.5. It was therefore concluded that the seismic response of a tank structure mainly depends on the response in the impulsive mode. In addition, it should be noted that the increase of base shear force and overturning moments controlled by the increase of the ratio H / R directly influences the detailing demands.
2
RH
gsE
(14)
Hydrodynamic pressure on the tank wall due to vertical excitation can be calculated using the Eqn. 15, Sve T v y pv H 1 (15) qv g H where S ve(T v) is a vertical spectral acceleration value (obtained from a 5% damped vertical elastic response spectrum for the case of reinforced concrete tanks), in units of g , and qv is the behaviour factor applied for the vertical direction (it should be taken up to 1.5 in all cases, unless justified through an appropriate analysis). In the case of both horizontal and vertical excitation, the maximum value of hydrodynamic pressure on the tank wall should be obtained by combining pressures due to both excitations by u sing the SRSS combination rule. Most commonly, various mechanical equipment is connected to a tank structure. Such equipment should be adequately considered in a seismic design process, but that issue is beyond the scope of this paper. Nevertheless, it should be noted that sufficient flexibility of equipment should be provided in order to prevent any damage in the tank structure during a seismic action. When a tank does not have a roof, a sufficient wall height above the liquid level must be provided in order to prevent spilling of liquid, which is particularly important in the case of tanks containing toxic liquids. The sloshing wave height is provided mainly by the first convective mode. As already mentioned, the provisions of Eurocode 8 define maximum sloshing wave height (d max), which should be calculated by the Eqn. 16. Se T c d max 0.84 R (16) g
6. ACKNOWLEDGMENTS
This research was supported by the Serbian Ministry of Science and Technology, Grant No. 36043. 7. REFERENCES
[1] ACI 350.3, “Seismic Design of Liquid-Containing Concrete Structures”, American Concrete Institute, Farmington Hill, MI, USA, 2001. [2] CEN Eurocode 8, “Design of structures for earthquake resistance. Part 1: General rules, seismic actions and rules for buildings”, European Standard EN 1998-1:2004, European Committee for Standardization, Brussels, 2004. [3] CEN Eurocode 8, “Design of structures for earthquake resistance. Part 4: Silos, tanks and pipelines”, European Standard EN 1998-4:2006, European Committee for Standardization, Brussels, 2006. [4] G. W. Housner, “The dynamic behaviour of water tanks”, Bulletin of the Seismological Society of America, 1963, Vol. 53, No. 2, pp. 381-387. [5] IITK-GSDMA, “Guidelines for Seismic Design of Liquid Storage Tanks – Provisions with Commentary and Explanatory Examples”, Indian Institute of Technology Kanpur, 2007.
5. CONCLUSIONS
A simple dynamic model for analysis of the seismic response of ground-supported concrete circular liquid storage tanks was presented. The model can be applied in analysis of the total seismic response as well as for the determination of local response quantities. All of the obligatory provisions of Eurocode 8 were fully taken into account, while some of the informative ones were substituted with similar provisions defined by IITK-
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[6] P. K. Malhotra, “Seismic response of Soil-Supported Unanchored Liquid-Storage Tanks”, ASCE, Journal of Structural Engineering , 1997, Vol. 123, No. 4, pp. 440-450. [7] M. J. N. Priestley, J. H. Wood and B. J. Davidson, “Seismic design of storage tanks”, Bulletin of the New Zealand National Society for Earthquake Engineering , 1986, Vol. 19, No. 4, pp. 272-284.
