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Fundamental modes of tank-liquid systems under horizontal motions
Engineering Structures 28 (2006) 1450–1461 www.elsevier.com/locate/engstruct
Fundamental modes of tank-liquid systems under horizontal motions Juan C. Virella, Luis A. Godoy ∗ , Luis E. Su´arez Department of Civil Engineering and Surveying, University of Puerto Rico Mayag¨uez, PR 00681-9041, Puerto Rico Received 24 February 2005; received in revised form 12 September 2005; accepted 15 December 2005 Available online 29 March 2006
Abstract This paper reports results on the fundamental impulsive modes of vibration of cylindrical tank-liquid systems anchored to the foundation under horizontal motion. The analyses are performed using a general purpose finite element (FE) program, and the influence of the hydrostatic pressure and the self-weight on the natural periods and modes is considered. The roof and walls are represented with shell elements and the liquid is modeled using two techniques: the added mass formulation and acoustic finite elements. Tank height to diameter ratios from 0.40 to 0.95 were used, with a liquid level at 90% of the height of the cylinder. The effect of the geometry on the fundamental modes for the tank-liquid systems is studied using eigenvalue and harmonic response analyses. Similar fundamental periods and mode shapes were found from these two approaches. The fundamental modes of tank models with aspect ratios (H/D) larger than 0.63 were very similar to the first mode of a cantilever beam. For the shortest tank (H/D = 0.40), the fundamental mode was a bending mode with a circumferential wave n = 1 and an axial half-wave (m) characterized by a bulge formed near the mid-height of the cylinder. c 2006 Elsevier Ltd. All rights reserved. Keywords: Added mass; Dynamics; Finite elements; Horizontal motion; Hydrostatic pressure; Tanks; Shells
1. Introduction Many above-ground steel tanks have suffered significant damage during past earthquakes, and this has motivated great interest in understanding and predicting the seismic behavior of tanks. In the early 1960s, Housner [1] considered cylindrical rigid tanks anchored to the foundations and subjected to horizontal translation, and decomposed the hydrodynamic response as the contribution of an impulsive and a sloshing component. The impulsive component was attributed to the part of the liquid that moves with the tank, while the sloshing component, which was characterized by long-period oscillations, was formed by the liquid near the free surface. To model these effects, Housner [1] developed equations to compute the impulsive and sloshing liquid masses, along with their location above the tank base. The fundamental impulsive mode consisted of a cantilever beam type mode. Veletsos and Yang [2] and Haroun and Housner [3] found that the pressure distributions due to the liquid for rigid and flexible anchored tanks were similar, but the magnitude was highly dependent on the flexibility of the wall. Housner [1], ∗ Corresponding author. Tel.: +1787 265 3815; fax: +1787 833 8260.
E-mail address: [email protected] (L.A. Godoy). c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2005.12.016
Haroun and Housner [2], and Veletsos and co-workers [2,4, 5], adopted the assumption that a cylindrical tank containing liquid predominantly develops a cantilever beam type mode in response to a horizontal base motion. For the fundamental cantilever beam mode, Veletsos [4] included the tank flexibility by replacing the pseudo-acceleration function instead of the ground acceleration in the relevant response equations. Malhotra and Veletsos [5] stated that, because of the large differences in the natural periods of the impulsive and sloshing responses, these two actions can be considered uncoupled, even though most of the response is affected by the motion of the liquid due to the impulsive component. Other researchers were not convinced by this cantilever mode and explored other more complex modes as possible fundamental modes under horizontal excitation. Natchigall et al. [6] proposed to use shell modal forms to model tankliquid systems, stating that the cantilever beam type mode adopted by previous researchers [1–5] was obsolete. They concluded that the fundamental modes of steel cylindrical tanks subjected to earthquake excitations are not associated with the first fundamental modes of a cantilever beam, but rather should be modeled using a circumferential wavy pattern with cos(nθ ), where n is the circumferential wave number, with 5 < n < 25.
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Design codes for steel tanks, such as API 650 [7] and AWWA-D100 [8], based their seismic standards for anchored flexible tanks on the recommendations given by Veletsos [4], together with a response spectrum analysis, in which cantilever beam modes are assumed to dominate the response of the tank-liquid systems to horizontal excitation. To calculate the actual seismic behavior of tank-liquid systems it is essential to accurately represent the predominant modes of vibrations, so a detailed study is performed in this paper to determine which modes are dominant in predicting the response. The natural frequencies and mode shapes of cylindrical tanks partially filled with liquid have been studied in the technical literature by means of experiments [9], analytical and semi-analytical approaches [10,11], and finite element (FE) techniques [9,12]. However, those studies have not been able to establish the fundamental modes of vibration for the prediction of the response of the tank-liquid systems to a horizontal motion. Barton and Parker [13] used finite element models of tankliquid systems to study the seismic response of anchored and unanchored tanks, in which the tank was modeled with shell elements and the liquid was represented with liquid finite elements and added liquid mass. For cylindrical tanks with height/diameter larger than 0.5 under horizontal excitation, Barton and Parker [13] stated that those modes involving deformations of the cylinder with the form cos(nθ ) and n > 1 have very small participation factors, and are not important in predicting the response. Thus, only the cantilever beam mode (i.e. n = 1) would be fundamental in predicting the horizontal seismic response for tanks with height/diameter > 0.5. This conclusion is in disagreement with the findings reported by Natchigall et al. [6]. Because of these conflicting views on which modes are dominant in the seismic response of thinwalled tanks, there is a need to investigate the fundamental modes of tank-liquid systems subjected to a horizontal base acceleration for the range of shell dimensions of interest in the oil industry, and to determine the influence of geometrical parameters of the tanks on their behavior. 2. Tank models The tanks considered in this paper have clamped or pinned condition at the base, with a cone roof partially supported by a set of radial beams and columns, as shown in Figs. 1 and 2. The radial beams are connected at their ends directly to the tank cylinder. In the finite element model the interior supporting columns are represented by linear springs that take into account the axial stiffness of the columns. The springs are connected to the ring beams and to the bottom of the tank. Since only anchored tanks are considered, the bottom of the tank is not included in the model. The rafters stiffen the roof to such an extent that predominantly cylinder modes result for the empty tank, regardless of the roof geometry (Virella [14], Virella et al. [15]). The geometries considered in this work have aspect ratios H /D = 0.40 (Model A), H /D = 0.63 (Model B), and H /D = 0.95 (Model C), where H and D are indicated in Fig. 1. In all computations, the liquid height was assumed to
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Fig. 1. Section of a typical anchored tank with roof rafters.
Fig. 2. Typical model of cone roof tank with rafters.
be HL = 0.9H . For the three models considered, the tapered thicknesses of the shells were designed using the API650 [7] provisions. The geometry of each of the three types of tanks considered is shown in Fig. 3. The finite element package ABAQUS [16] was used to perform the computations using S3R triangular elements and S4R quadrilateral elements. The S4R is a four-node, doubly curved shell element with reduced integration, hour-glass control, and finite membrane strain formulation. The S3R is a three-node, degenerated version of the S4R, with finite membrane strain formulation. The element S3R has constant bending and membrane strain approximations, therefore a high mesh refinement is required to model pure bending situations. Both elements S3R and S4R are discussed in more detail in Hibbit et al. [17]. The number of elements used in the finite element meshes for Models A, B, and C are listed in Table 1. 3. Tank-liquid models The primary interest of this study is to evaluate the natural periods, mode shapes and dynamic response to horizontal ground motions of cylindrical tanks partially filled with a liquid.
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Fig. 3. Tank models with cone roof supported by rafters: t = shell thickness; tr = roof thickness; tr = 0.00635 m (roof with rafters). (a) Tank with H /D = 0.95; (b) tank with H /D = 0.63; (c) tank with H /D = 0.40. Table 1 Number of finite element elements in the models with added liquid mass Model
H /D
Number of shell elements
A B C
0.40 0.63 0.95
10,439 12,767 15,871
The recommendation by Housner [1] on the convenience of separating the impulsive and convective actions to characterize the hydrodynamic response of horizontally excited tank-liquid systems is adopted here. Only those modes corresponding to the impulsive mode in which there is a coupling action between the tank and liquid are considered in this paper. The liquid is represented by means of an added mass approach, and with acoustic finite elements. A density ρ = 983 kg/m3 and a bulk modulus K = 2.07 GPa (i.e. the properties of water) are used in the computations. 3.1. Model with added liquid mass The finite element meshes for the structures of Models A, B, and C with added liquid mass are indicated in Table 1 and Fig. 4. Convergence studies were carried out and the finite element meshes listed in Table 1 were found to provide good results for the purpose of the present study. The bottom plate of the tank was not included in the model, since this study only
Fig. 4. Finite element mesh for Model A, with added liquid mass.
covers anchored tanks, with emphasis on the behavior of the cylindrical shell and not on the foundation. The model with added mass liquid was previously presented by Virella et al. [18], and is summarized here. The added mass is obtained from a pressure distribution for the impulsive mode of the tank-liquid system due to Veletsos and Shivakumar [19], which has a cosine distribution along the tank cylinder (see Fig. 5). This pressure distribution for the rigid body horizontal motion of a rigid tank-liquid system is described as Pi (η, θ, t) = ci (η)ρ R x¨g (t) cos θ
(1)
where Pi is the impulsive pressure; η is a non-dimensional vertical coordinate = z/HL; z is the vertical coordinate
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Fig. 6. Impulsive pressures for the tank-liquid systems with χg = 1 m/s2 .
Fig. 5. Impulsive pressure distribution around the tank circumference: (a) circumferential distribution; (b) pressure components in the direction of the excitation.
measured from the tank bottom; R is the tank radius; x¨g (t) is the ground acceleration; and t is the time. The function ci (η) defines the impulsive pressure distribution along the cylinder height, and is computed as ci (η) = 1 −
∞
ccn (η)
(2)
n=1
where ccn (η) is the function that defines the convective pressure distribution along the cylinder height, computed as ccn (η) =
2 cosh[λn (H /R)η] . λ2n − 1 cosh[λn (H /R)]
(3)
The parameter λn is the nth root of the first derivative of the Bessel function of the first kind and first order. The first three roots are λ1 = 1.841, λ2 = 5.311, and λ3 = 8.536. The function ci (η) converges rapidly as the number of terms in the summation increase, and hence it is sufficient to include three coefficients ccn in Eq. (2). The pressure distributions defined in Eq. (1) for each of the tank-liquid systems considered in this paper and for θ = 0 are presented in Fig. 6. The added mass of the liquid on the tank shell is calculated from the pressure distribution in Eq. (1). To obtain the lumped mass m i at each node, the height of the cylinder is divided into several segments and the area of the pressure below the curve for the segment is divided by the normal acceleration of the rigid wall x¨g . The lumped masses obtained from the added mass have the same vertical variation as the impulsive pressure
Fig. 7. Model with normal mass along the cylinder height.
distribution from which they are derived (see Fig. 7), and they have a uniform distribution around the circumference of the tank. Because this added mass is the liquid mass that moves together with the tank, it acts normal to the cylindrical shell. For a radial section of the tank (Fig. 7), the lumped mass m i at each location is computed using the rectangular rule. For an interior node at the tank shell, the lumped mass m i is computed as Pi h (4) mi = an where Pi is the pressure at node i , h is the constant distance between nodes, and an is the reference normal acceleration. Note that an becomes the amplitude of x¨g for θ = 0. For nodes at the liquid surface and at the bottom of the tank, the lumped mass is ml =
Pl h 2 an
(5)
where m l is the mass at a boundary node l and Pl is the pressure at node l. Because the added masses are determined from the impulsive pressure which is normal to the shell surface, they must be added in such a way that they only add inertia in this direction. For this reason, the added masses are sometimes referred to as the “normal masses”.
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For each of the tank-liquid systems considered in this study, the mass distributions calculated as described previously were verified by comparing them with the impulsive mass ratios proposed by Housner [1]. The impulsive mass ratio Mir for the particular tank-liquid system is defined as
Table 2 Comparisons of impulsive mass ratios calculated with the present study and with the Housner equation (11) Model
H /D
Mir
Housner Mi/Mt
Difference (%)
Mi , (6) Mt where Mi is the total impulsive mass and Mt is the total liquid mass. For our case, the total impulsive mass Mi can be calculated by first obtaining a mass resultant m res for the tank meridian in the direction of the ground excitation, and then integrating around the circumference to compute the total impulsive mass of the tank. The mass resultant m res is calculated as the sum of the individual masses m i at the different nodes along the θ = 0 meridian. The added mass components in the direction of the excitation are directly proportional to the impulsive pressure which varies with a cosine distribution around the tank circumference (see Fig. 5(a)). Hence, to integrate the radial mass in order to obtain the impulsive mass, we must project the mass m res along the direction of the excitation. Moreover, m res was calculated for the θ = 0 meridian. To calculate m res for θ > 0, it must be multiplied by cos θ , in the same way as the pressure was defined (see Fig. 5(b)). At the arc that forms an angle θ with respect to the horizontal, the horizontal component of the impulsive mass component in the direction of the excitation is
A B C
0.40 0.63 0.95
0.4057 0.5710 0.7040
0.406 0.587 0.736
0.07 2.73 4.35
Mir =
dm i = (m res cos θ ) cos θ (R dθ ).
(7)
The total impulsive mass Mi is then π/2 cos2 (θ ) dθ = π Rm res . Mi = 4Rm res
(8)
0
The total liquid mass is calculated using the following expression Mt = π R 2 HLγL /g,
(9)
where γL is the liquid unit weight and g is the acceleration of gravity. The impulsive mass ratio defined in Eq. (6) is m res g . (10) Mir = R HL γ L
Fig. 8. Model with normal mass around the circumference.
scheme needs to be introduced in order to represent this added mass. It must be pointed out, though, that some structural analysis programs allow for the addition of lumped masses along three global Cartesian axes, but it is not common to have the option of adding masses in cylindrical or polar coordinates. The added liquid mass in lumped form is attached to the shell nodes by means of rigid, massless links with small lengths (see Fig. 8). The links are rigid truss elements with supports oriented in the local axes of the truss elements. The supports must permit the motion of the nodal masses only in the direction normal to the shell. Hence, the motion of the support is restricted in the global tangential direction (perpendicular to the element axis) and in the vertical direction, but it is free to move in the global radial direction (i.e. the local axial direction). The total impulsive mass Mi in a specific direction calculated with Eq. (8) is twice the impulsive mass computed using the methodology of Ref. [1]. However, as the masses can only move in the radial direction, it can be shown that half of this total impulsive mass is excited in a specific direction.
The impulsive mass ratio Mir can also be calculated using the expression proposed by Housner [1]:
3.2. Model with liquid finite elements
Mi tanh(1.7R/HL) = . Mt 1.7R/HL
A typical finite element mesh for the tank (Model A in this case) in which the liquid is modeled with finite elements is shown in Fig. 9(a). Acoustic three-dimensional finite elements based on linear wave theory were used in this part of the study to represent the liquid. Such a model represents the dilatational motion of the liquid, allowing a wave to be described as a single pressure degree of freedom at each point in space. The formulation is based on the Laplace equation in the pressure domain and, although viscosity effects can be accounted for, an inviscid liquid was considered for the present study. The formulation of the acoustic elements is described in
(11)
Table 2 shows that the differences between the impulsive mass ratios computed with the present analysis and with the recommendation by Housner [1] are smaller than 5%. Since the added mass comes from the liquid pressure that acts in the direction normal to the shell, the mass should be assigned in the direction normal to the cylinder. However, because in most FE programs the same magnitude of the nodal mass is assigned to all the degrees of freedom, a sui generis
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Fig. 9. Typical finite element mesh for Model A with liquid acoustic elements: (a) tank structure mesh; (b) liquid mesh. Table 3 Number of finite elements in the models where the liquid is represented with acoustic elements Model A B C
H /D 0.40 0.63 0.95
Number of elements Shell
Acoustic
10,536 12,216 12,216
39,312 56,784 56,784
Refs. [16,17]. The liquid mesh has 39,312 elements for Model A and 56,784 elements for Models B and C (see Table 3). The elements for the liquid are identified in ABAQUS [16] as AC2D4, which are solid, eight-node brick acoustic elements with bilinear interpolation and with only one pressure unknown per node. The finite element mesh for the liquid of Model A is shown in Fig. 9(b). For the finite element model of the tank, triangular shell elements (S3R) are used in the roof and quadrilateral and triangular elements are used for the cylinder. The number of elements for each model is listed in Table 3. It was mentioned before that it was not necessary to include the bottom of the tank in the tank models, because this study emphasizes the behavior of the cylindrical shell. However, the bottom of the tank was discretized with finite elements in the model with liquid acoustic elements in order to specify the boundary condition (i.e. contact interaction) between the bottom surface of the liquid and the tank. Quadrilateral (S4R) and triangular (S3R) shell elements were used for the bottom of the tank, and pinned supports were placed at each node of the tank bottom, to represent an anchored tank resting on a rigid base. The location of each node on the constrained surfaces of the liquid corresponds exactly to the location of a node on the tank cylinder or base. Surface tied normal contact was considered between the surfaces of the liquid and the tank walls and the bottom. This contact formulation is based on a master–slave approach, in which both surfaces remain in contact throughout the simulation, allowing the transmission of normal forces between them. The formulation for normal contact is described in Refs. [16,17]. No sloshing waves were considered in this study, and thus no pressure was applied to the nodes at the free liquid surface. The boundary conditions specified at the interfaces of the liquid model are illustrated in Fig. 10.
Fig. 10. Conditions assumed for the three-dimensional tank-liquid finite element model. Table 4 Results of the free vibration analyses using the normal mass models Model
H /D
Tmax (s)
Tmin (s)
α3
Number of modes
A B C
0.40 0.63 0.95
0.644 0.859 1.057
0.189 0.199 0.205
0.74 0.81 0.79
891 796 794
4. Fundamental periods and mode shapes 4.1. Free vibration analyses To compute the natural periods and mode shapes for the normal mass models of the tank-liquid systems, the hydrostatic pressure and the self-weight of the tank were initially applied, i.e. they were modeled as a pre-stress state. The natural periods and mode shapes were computed until the ratio α3 of the horizontal component M3 (i.e. global direction 3) of the accumulated effective mass to the total mass of the system (α3 = M3 /(Mi + Mshell )) was larger than 70% (see Table 4). The analysis uses the Lanczos solver [16], in which all the natural frequencies and mode shapes are found within a specified range of frequencies. The fundamental modes were identified as those with the largest participation factors in the translational directions. Table 4 displays the range of the natural periods and the total number of modes found from the free vibration analyses of the tank-liquid systems. For Model A, 891 modes with natural
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Fig. 11. Variation of natural period with the normalized modal participation factor in the horizontal axis 3 for Model A.
periods between 0.64 s and 0.19 s were found with the condition α3 ≥ 0.7. The natural periods ranged from 0.86 s and 0.20 s (796 modes) for Model B and between 1.0 s and 0.20 s (794 modes) for Model C. Typical cylinder modes (see Ref. [15]) characterized by circumferential wave numbers (n) and axial half-wave numbers (m) resulted for all the modes of each of the tank-liquid systems. Fig. 11 illustrates, for Model A, the variation of the natural periods with the participation factor P3 (in absolute value) along the axis 3 horizontal direction. The participation factor is normalized with respect to its maximum value. Fig. 11 shows that the participation factor (P3 ) is much larger for a certain mode, the fundamental mode, indicating that the response of the system to a horizontal motion will be predominantly determined by this mode. Similar results were obtained for Models B and C, and thus the same approach described before was used to determine the fundamental modes for these tankliquid systems. Fig. 12 illustrates the variation of the natural period with the circumferential wave number n, for modes with axial halfwave number m = 1. The figure also shows the fundamental period calculated previously for each of the three models. The modes with the largest natural periods in Fig. 12 had small participation factors P3 , and therefore their contribution to the response of the tank-liquid systems to horizontal motions can be neglected. The modes for Models A, B, and C with the three largest participation factors P3 are listed in Table 5. This table shows that, for Models B and C, there are modes (the second and third) that will contribute to the response to horizontal ground motions that are not cantilever beam modes (i.e. n > 1). However, the fundamental mode still dominates the response, as α3 for the first mode of Model A is 94% of the total α3 obtained adding those of the three modes considered. For other models, α3 for the first mode is even higher: 98% for Model B and 97% for Model C. Therefore, for practical purposes, only the fundamental mode needs to be considered in order to predict the response of cylindrical tank-liquid systems subjected to horizontal base motions. These results compared well with the recommendations by Barton and Parker [13], who stated that, for cylindrical tank-liquid systems with aspect ratios H /D larger than 0.5, the modes characterized by circumferential wave numbers n > 1 and axial half-wave numbers m ≥ 1 are not important for determining the response to horizontal base motions. The present work showed that even for tanks
Fig. 12. Variation in the natural period T with n for m = 1 for Models A, B and C.
Fig. 13. Damping ratios for the two frequency ranges considered in the dynamic analyses: (a) frequency range 1; (b) frequency range 2.
with an aspect ratio H /D = 0.40, which falls outside the range considered by Barton and Parker [13], modes with n > 1 and m ≥ 1 have very small participation factors in the direction of the horizontal motion. 4.2. Steady state harmonic response analyses A linear, steady state harmonic analysis of the tank-liquid systems under uniaxial horizontal motion was carried out to find the periods and deformed shapes that will be excited by this input. To account for the geometric non-linearity, the hydrostatic pressure and the self-weight of the tank were included in an initial static step; in this way, the stiffness of the system was modified prior to the dynamic analysis. The tank-liquid system was next loaded with the pressure distribution given by Eq. (1), and a ground acceleration of 1g (g = 9.81 m/s2 ) was considered. The linear, steady state harmonic response to a horizontal harmonic motion in one
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Table 5 Modes of the tank-liquid systems relevant for the response to horizontal ground motions Mode 1 2 3
Model A T (s)
α3
n
m
Model B T (s)
α3
n
m
Model C T (s)
α3
n
m
0.2116 0.2021 0.1957
0.64 0.02 0.02
1a 1 1
1 5 4
0.2395 0.2362 0.2046
0.79 0.01 0.01
1a 28 1
1 7 7
0.3001 0.2960 0.2998
0.77 0.01 0.01
1a 20 25
1 6 6
a Fundamental mode.
Fig. 14. Variation of the normalized radial displacement with the excitation period for Model A with added mass: –•– Node A; –– Node B (see Fig. 15).
direction was obtained by direct integration. The tank-liquid systems with the liquid accounted for by added masses were excited along the axis 3 horizontal direction, while the models with acoustic liquid elements were excited along the axis 1 horizontal direction. The dynamic response as a function of the excitation period was obtained. The periods of the tank-liquid systems excited by the ground motion will be those associated with peaks in a plot of a maximum response versus excitation period. The radial displacement response (U-radial) yields a deformed shape associated with a mode of vibration of the tankliquid system. To introduce damping to the models, a Rayleigh damping matrix [C] was used: [C] = β[K ] + α[M]
(12)
where β is the stiffness proportional coefficient and α is the mass proportional coefficient. The Rayleigh damping coefficients were computed by considering two ranges of frequencies. The frequency ranges are shown in Fig. 13(a) and (b). The values of β are 0.003351 for the frequency range 1 and 0.001273 for the frequency range 2. The coefficient α is 0.3175 for frequency range 1 and 0.8042 for frequency range 2. By using the two frequency bands, the damping ratio of the system was kept from 3% to 4% for the ranges of the excitation frequencies considered in the dynamic analyses, as can be seen in Fig. 13. The variation in the radial displacement (U-radial) with excitation period for Model A with the added mass approach is shown in Fig. 14. The displacements shown are those at two nodes of the cylinder where maximum displacements occurred. These nodes, identified as A and B, are shown in Fig. 15. The highest peaks in Fig. 14 occur at the excitation period associated with the fundamental mode. A similar behavior was
obtained for Models B and C, and the fundamental modes were also identified from the excitation period with the highest radial displacement peak. The fundamental modes resulted in bending modes (n = 1). Similar fundamental modes were obtained from the free vibration and from the harmonic response analyses for the added mass models as shown in Table 6, with differences of less than 3% in all cases. The fundamental modes for each of the tank-liquid system, in which the liquid was represented with added masses, are presented in Figs. 15–17. The maximum radial displacements at the meridian in the direction of the excitation did not occur near the top of the tank for any model, as is the case for the fundamental mode of a cantilever beam. As shown in Fig. 15(b), for Model A (the shortest tank with H /D = 0.40) the fundamental mode is a bending mode (n = 1) with an axial half-wave (m) characterized by a bulge near the mid-height of the cylinder. However, as shown in Figs. 16(b) and 17(b), for Models B and C which have aspect ratios larger than 0.63, the fundamental modes tend to the first mode of a cantilever beam, even though the maximum radial displacements do not occur at the top of the tank. In the cases studied so far, the liquid in the tank was accounted for by means of the added mass formulation. In the sequel, the analyses are repeated using acoustic elements to represent the liquid. The variation in the radial displacement with the excitation period for Model A, using acoustic elements, is presented in Fig. 18. The mode that contributes the most to the response of the tank-liquid system is identified by the peak in Fig. 18. Notice from Figs. 14 and 18 that the fundamental period for Model A obtained with the added mass approach and with the models with acoustic elements are quite similar; in fact, the differences are smaller than 3% for the three tank models. Evidently, the response to the ground motion with periods equal to the fundamental modes (indicated in Figs. 14 and 18) are much larger than that computed for other excitation periods. Thus, for both models (with liquid acoustic elements and added mass) and regardless of the H /D ratio, the response to a horizontal harmonic excitation is dominated by the fundamental mode. The deformed shapes associated with the fundamental modes for the models with acoustic elements were bending modes (n = 1) that tend to the first mode of a cantilever beam, similar to those presented in Figs. 15–17 for the added mass models. For instance, the deformed shape shown in Fig. 19 for the fundamental mode of Model C computed using acoustic elements is similar to that computed with the added mass model and displayed in Fig. 17.
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Fig. 15. Fundamental mode for Model A with liquid added mass: (a) 3D view; (b) deformed shape in the meridian with maximum displacements.
Fig. 16. Fundamental mode for Model B with liquid added mass: (a) 3D view; (b) deformed shape in the meridian with maximum displacements.
Fig. 17. Fundamental mode for Model C with liquid added mass: (a) 3D view; (b) deformed shape in the meridian with maximum displacements. Table 6 Fundamental periods for the tank-liquid systems obtained by using two formulations to represent the liquid Model
H /D
Periods with added mass formulation Free vibration (s) Harmonic response (s)
Periods with acoustic FE Harmonic response (s)
A B C
0.40 0.63 0.95
0.212 0.239 0.300
0.210 0.244 0.296
0.212 0.240 0.303
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impulsive pressure distributions for the fundamental modes for flexible and rigid tanks increased with their aspect ratios H /D. However, even for the tallest tank (Model C, where differences are larger), the impulsive pressures for the fundamental mode of the flexible and rigid tanks have similar shapes. 4.3. Comparisons with previous studies
Fig. 18. Variation of the normalized radial displacement with the excitation period for Model A with liquid acoustic elements: –•– Node A; –– Node B (see Fig. 15).
The fundamental periods obtained from the steady state harmonic response analyses and from the free vibration (eigenvalue) analyses using different formulations to represent the liquid are presented in Table 6. The results are very similar for the three models of the tanks: in all cases, the differences are smaller than 3%. The variation in the impulsive pressure with the period of the ground motion for tank models with liquid acoustic elements was obtained. The pressure was normalized and it is calculated at the point of the cylinder where the maximum pressure occurred. In all cases, the pressure showed a pronounced peak at the same period than for the maximum radial displacements (see Fig. 18); therefore, both responses are in phase. The impulsive hydrodynamic pressure for the fundamental mode follows the meridianal distribution shown in Fig. 20, with a cosine circumferential variation similar to that illustrated in Fig. 5. The impulsive hydrodynamic pressure distribution for Models A and C is illustrated in Fig. 20. The pressure is shown in the direction of the excitation and for the fundamental mode along the meridian at which the maximum response occurs. The pressure distribution for an anchored rigid tank obtained from Eq. (1) is also shown in Fig. 20. The two distributions are similar in shape, with the smallest differences occurring for Model A, and increasing successively for Models B and C. These results indicate that the difference between the
The fundamental periods obtained in this paper for the impulsive modes of cylindrical tanks under horizontal translation were compared with those proposed by Veletsos and Shivakumar [19] and Sakai et al. [20]. Both groups of researchers identified cantilever beam fundamental modes. The equation for the fundamental impulsive mode by Veletsos and Shivakumar [19] considers a constant shell thickness. For the tanks considered in this paper, which have variable shell thickness (see Fig. 3), the average thickness of the shell courses below the liquid surface was computed and used in the equation of Veletsos and Shivakumar [19]. For tanks with variable shell thickness, Sakai et al. [20] used the thickness at an elevation of 1/3 of the liquid height measured from the bottom of the tank to compute the fundamental impulsive period. A comparison between the impulsive fundamental periods computed from the free vibration analyses in this paper and those from Refs. [19] and [20] is displayed in Fig. 21. The fundamental periods from this study are very similar to those predicted by Veletsos and Shivakumar [19], with the maximum difference being smaller than 3.5%. Notice that even for Model A, in which the fundamental mode is not quite similar to a cantilever beam type mode, the fundamental period compared extremely well (0.15% difference) with that proposed by Veletsos and Shivakumar. Smaller fundamental periods are predicted by the formulation of Sakai et al. [20], with differences between 9% and 14%. 5. Conclusions The conclusions stated in this paper apply for cylindrical tank-liquid systems with height/diameter ratios from 0.40 to 0.95 with roof, anchored to the foundation, with a perfect or as-designed geometry, in which the influence of the hydrostatic pressure and the self-weight is considered.
Fig. 19. Fundamental mode for Model C with acoustic liquid elements: (a) 3D view; (b) side view.
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Fig. 21. Variation in fundamental period (TD ) with liquid height to tank diameter ratio (HL /D).
Fig. 20. Pressure distribution at the meridian with maximum displacements for the fundamental mode: (a) Model A; (b) Model C.
It was verified that the response of a tank-liquid system subjected to a horizontal ground motion can be accurately predicted by considering just the fundamental mode. The fundamental mode for the tank-liquid systems is a bending mode (n = 1), regardless of the height-to-tank diameter ratios (H /D) considered in this study. This conclusion agrees with that stated previously by Barton and Parker [13]. The largest natural periods are associated with cylinder modes characterized by circumferential waves n > 1 and axial halfwaves m = 1. However, these modes are not the fundamental modes for predicting the response to a horizontal base motion, in the sense that they do not have the largest modal participation factors. It was proved that, for the range of shell dimensions of interest for the oil industry, the fundamental modes are not those associated with circumferential wavy patterns, as proposed by Natchigall et al. [6]. Nevertheless, it must be pointed out that this study considers tanks with perfect geometry, i.e. the effects of imperfections in the shell were not taken into account. Similar fundamental periods and mode shapes were found from the free vibration (eigenvalue) analyses and from the harmonic response analyses, using the added mass formulation and the model with liquid acoustic finite elements. The differences were smaller than 3% in all cases. Thus, the added mass models can provide a good approximation for calculating the response of tanks filled with liquid, as the results compare very well with the more sophisticated models in which the
liquid is represented by acoustic finite elements. This is true regardless of the aspect ratio (H /D) of the tank. The fundamental modes of tanks with aspect ratios H /D larger than 0.63 are similar to the first mode of a cantilever beam. For the shortest tank (H /D = 0.40), the fundamental mode is a bending mode (n = 1) with an axial half-wave (m) characterized by a bulge formed near the mid-height of the cylinder. The differences between the impulsive pressure distributions for the fundamental modes calculated considering flexible and rigid tanks increase with the aspect ratio (H /D) of the tanks. However, even for the tank with the largest aspect ratio considered here (Model C, H /D = 0.95) where the differences are largest, the impulsive pressures of flexible and rigid tanks have similar shapes. The fundamental periods obtained with the detailed finite element models used in this paper compared well with the recommendations of Veletsos and Shivakumar [19], with differences smaller than 3.5% in all cases. The formulation for the fundamental period due to Sakai et al. [20] predicts smaller fundamental periods than those found in this paper, resulting in differences between 9% and 14%. Acknowledgments J.C. Virella was supported by a PR-EPSCoR post-doctoral fellowship grant EPS-0223152 for this research. The authors thank Dr. C.A. Prato (National University of C´ordoba, Argentina) for his contribution during the early stages of this work. Partial support from Mid America Earthquake Center to this research is gratefully acknowledged. References [1] Housner GW. The dynamic behavior of water tanks. Bulletin of the Seismological Society of America 1963;53(2):381–9. [2] Veletsos AS, Yang JY. Earthquake response of liquid storage tanksadvances in civil engineering through mechanics. In: Proceedings of the second engineering mechanics specialty conference. Raleigh (NC): ASCE; 1977. p. 1–24. [3] Haroun MA, Housner GW. Earthquake response of deformable liquid storage tanks. Journal of Applied Mechanics 1981;48(2):411–8.
J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461 [4] Veletsos AS. Seismic response and design of liquid storage tanks. Guidelines for the seismic design of oil and gas pipeline systems. In: Technical council on lifeline earthquake engineering. New York: ASCE; 1984. p. 255–370, 443–61. [5] Malhotra P, Veletsos AS. Uplifting response of unanchored liquid storage tanks. Journal of Structural Engineering 1994;120(12):3525–47. [6] Natchigall I, Gebbeken N, Urrutia-Galicia JL. On the analysis of vertical circular cylindrical tanks under earthquake excitation at its base. Engineering Structures 2003;25:201–13. [7] American Petroleum Institute. API Standard 650. Steel tanks for oil storage. 8th ed. 1988. [8] American Water Works Association. AWWA Standard for welded steel tanks for water storage. AWWA D100. 1984. [9] Maz´uch T, Hor´acek J, Trnka J, Vesely J. Natural modes and frequencies of a thin clamped-free steel cylindrical storage tank partially filled with water: FEM and measurement. Journal of Sound and Vibration 1996; 193(3):669–90. [10] Gupta RK. Free vibrations of partially filled cylindrical tanks. Department of Engineering, Launceston (Australia): University of Tasmania; 1994. [11] Han RPS, Liu JD. Free vibration analysis of a fluid-loaded variable thickness cylindrical tank. Journal of Sound and Vibration 1994;176(2): 235–53. [12] Goncalves PB, Ramos NRSS. Free vibration analysis of cylindrical tanks partially filled with liquid. Journal of Sound and Vibration 1996;195(3):
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429–44. [13] Barton DC, Parker JV. Finite element analysis of the seismic response of anchored and unanchored liquid storage tanks. Earthquake Engineering and Structural Dynamics 1987;15(3):299–322. [14] Virella JC. Buckling of steel tanks subject to earthquake loadings. Ph.D. thesis. Department of Civil Engineering and Surveying, University of Puerto Rico at Mayag¨uez; 2004. [15] Virella JC, Godoy LA, Su´arez LE. Influence of the roof on the natural periods of empty tanks. Engineering Structures 2003;25:877–87. [16] Hibbit HD, Karlsson BI, Sorensen P. ABAQUS standard user manual, version 6.4. Rhode Island: Hibbit, Karlsson and Sorensen, Inc.; 2002. [17] Hibbit HD, Karlsson BI, Sorensen P. ABAQUS theory manual, version 6.4. Rhode Island: Hibbit, Karlsson and Sorensen, Inc.; 2002. [18] Virella JC, Godoy LA, Su´arez LE. Effect of pre-stress states on the impulsive modes of vibration of cylindrical tank-liquid systems under horizontal motion. Journal of Vibration and Control 2005;11(9): 1195–220. [19] Veletsos AS, Shivakumar P. Tanks containing liquids or solids. In: Beskos DE, Poulos A, editors. Computer analysis and design of earthquake resistant structures: A Handbook, 3; 1997. p. 725–73 [chapter 15]. [20] Sakai F, Ogawa H, Isoe A. Horizontal, vertical, rocking fluid-elastic response and design of cylindrical liquid storage tanks. In: Proceedings of the 8th world conference on earthquake engineering. vol. V. 1984. p. 263–70.
Fundamental modes of tank-liquid systems under horizontal motions Juan C. Virella, Luis A. Godoy ∗ , Luis E. Su´arez Department of Civil Engineering and Surveying, University of Puerto Rico Mayag¨uez, PR 00681-9041, Puerto Rico Received 24 February 2005; received in revised form 12 September 2005; accepted 15 December 2005 Available online 29 March 2006
Abstract This paper reports results on the fundamental impulsive modes of vibration of cylindrical tank-liquid systems anchored to the foundation under horizontal motion. The analyses are performed using a general purpose finite element (FE) program, and the influence of the hydrostatic pressure and the self-weight on the natural periods and modes is considered. The roof and walls are represented with shell elements and the liquid is modeled using two techniques: the added mass formulation and acoustic finite elements. Tank height to diameter ratios from 0.40 to 0.95 were used, with a liquid level at 90% of the height of the cylinder. The effect of the geometry on the fundamental modes for the tank-liquid systems is studied using eigenvalue and harmonic response analyses. Similar fundamental periods and mode shapes were found from these two approaches. The fundamental modes of tank models with aspect ratios (H/D) larger than 0.63 were very similar to the first mode of a cantilever beam. For the shortest tank (H/D = 0.40), the fundamental mode was a bending mode with a circumferential wave n = 1 and an axial half-wave (m) characterized by a bulge formed near the mid-height of the cylinder. c 2006 Elsevier Ltd. All rights reserved. Keywords: Added mass; Dynamics; Finite elements; Horizontal motion; Hydrostatic pressure; Tanks; Shells
1. Introduction Many above-ground steel tanks have suffered significant damage during past earthquakes, and this has motivated great interest in understanding and predicting the seismic behavior of tanks. In the early 1960s, Housner [1] considered cylindrical rigid tanks anchored to the foundations and subjected to horizontal translation, and decomposed the hydrodynamic response as the contribution of an impulsive and a sloshing component. The impulsive component was attributed to the part of the liquid that moves with the tank, while the sloshing component, which was characterized by long-period oscillations, was formed by the liquid near the free surface. To model these effects, Housner [1] developed equations to compute the impulsive and sloshing liquid masses, along with their location above the tank base. The fundamental impulsive mode consisted of a cantilever beam type mode. Veletsos and Yang [2] and Haroun and Housner [3] found that the pressure distributions due to the liquid for rigid and flexible anchored tanks were similar, but the magnitude was highly dependent on the flexibility of the wall. Housner [1], ∗ Corresponding author. Tel.: +1787 265 3815; fax: +1787 833 8260.
E-mail address: [email protected] (L.A. Godoy). c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2005.12.016
Haroun and Housner [2], and Veletsos and co-workers [2,4, 5], adopted the assumption that a cylindrical tank containing liquid predominantly develops a cantilever beam type mode in response to a horizontal base motion. For the fundamental cantilever beam mode, Veletsos [4] included the tank flexibility by replacing the pseudo-acceleration function instead of the ground acceleration in the relevant response equations. Malhotra and Veletsos [5] stated that, because of the large differences in the natural periods of the impulsive and sloshing responses, these two actions can be considered uncoupled, even though most of the response is affected by the motion of the liquid due to the impulsive component. Other researchers were not convinced by this cantilever mode and explored other more complex modes as possible fundamental modes under horizontal excitation. Natchigall et al. [6] proposed to use shell modal forms to model tankliquid systems, stating that the cantilever beam type mode adopted by previous researchers [1–5] was obsolete. They concluded that the fundamental modes of steel cylindrical tanks subjected to earthquake excitations are not associated with the first fundamental modes of a cantilever beam, but rather should be modeled using a circumferential wavy pattern with cos(nθ ), where n is the circumferential wave number, with 5 < n < 25.
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Design codes for steel tanks, such as API 650 [7] and AWWA-D100 [8], based their seismic standards for anchored flexible tanks on the recommendations given by Veletsos [4], together with a response spectrum analysis, in which cantilever beam modes are assumed to dominate the response of the tank-liquid systems to horizontal excitation. To calculate the actual seismic behavior of tank-liquid systems it is essential to accurately represent the predominant modes of vibrations, so a detailed study is performed in this paper to determine which modes are dominant in predicting the response. The natural frequencies and mode shapes of cylindrical tanks partially filled with liquid have been studied in the technical literature by means of experiments [9], analytical and semi-analytical approaches [10,11], and finite element (FE) techniques [9,12]. However, those studies have not been able to establish the fundamental modes of vibration for the prediction of the response of the tank-liquid systems to a horizontal motion. Barton and Parker [13] used finite element models of tankliquid systems to study the seismic response of anchored and unanchored tanks, in which the tank was modeled with shell elements and the liquid was represented with liquid finite elements and added liquid mass. For cylindrical tanks with height/diameter larger than 0.5 under horizontal excitation, Barton and Parker [13] stated that those modes involving deformations of the cylinder with the form cos(nθ ) and n > 1 have very small participation factors, and are not important in predicting the response. Thus, only the cantilever beam mode (i.e. n = 1) would be fundamental in predicting the horizontal seismic response for tanks with height/diameter > 0.5. This conclusion is in disagreement with the findings reported by Natchigall et al. [6]. Because of these conflicting views on which modes are dominant in the seismic response of thinwalled tanks, there is a need to investigate the fundamental modes of tank-liquid systems subjected to a horizontal base acceleration for the range of shell dimensions of interest in the oil industry, and to determine the influence of geometrical parameters of the tanks on their behavior. 2. Tank models The tanks considered in this paper have clamped or pinned condition at the base, with a cone roof partially supported by a set of radial beams and columns, as shown in Figs. 1 and 2. The radial beams are connected at their ends directly to the tank cylinder. In the finite element model the interior supporting columns are represented by linear springs that take into account the axial stiffness of the columns. The springs are connected to the ring beams and to the bottom of the tank. Since only anchored tanks are considered, the bottom of the tank is not included in the model. The rafters stiffen the roof to such an extent that predominantly cylinder modes result for the empty tank, regardless of the roof geometry (Virella [14], Virella et al. [15]). The geometries considered in this work have aspect ratios H /D = 0.40 (Model A), H /D = 0.63 (Model B), and H /D = 0.95 (Model C), where H and D are indicated in Fig. 1. In all computations, the liquid height was assumed to
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Fig. 1. Section of a typical anchored tank with roof rafters.
Fig. 2. Typical model of cone roof tank with rafters.
be HL = 0.9H . For the three models considered, the tapered thicknesses of the shells were designed using the API650 [7] provisions. The geometry of each of the three types of tanks considered is shown in Fig. 3. The finite element package ABAQUS [16] was used to perform the computations using S3R triangular elements and S4R quadrilateral elements. The S4R is a four-node, doubly curved shell element with reduced integration, hour-glass control, and finite membrane strain formulation. The S3R is a three-node, degenerated version of the S4R, with finite membrane strain formulation. The element S3R has constant bending and membrane strain approximations, therefore a high mesh refinement is required to model pure bending situations. Both elements S3R and S4R are discussed in more detail in Hibbit et al. [17]. The number of elements used in the finite element meshes for Models A, B, and C are listed in Table 1. 3. Tank-liquid models The primary interest of this study is to evaluate the natural periods, mode shapes and dynamic response to horizontal ground motions of cylindrical tanks partially filled with a liquid.
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Fig. 3. Tank models with cone roof supported by rafters: t = shell thickness; tr = roof thickness; tr = 0.00635 m (roof with rafters). (a) Tank with H /D = 0.95; (b) tank with H /D = 0.63; (c) tank with H /D = 0.40. Table 1 Number of finite element elements in the models with added liquid mass Model
H /D
Number of shell elements
A B C
0.40 0.63 0.95
10,439 12,767 15,871
The recommendation by Housner [1] on the convenience of separating the impulsive and convective actions to characterize the hydrodynamic response of horizontally excited tank-liquid systems is adopted here. Only those modes corresponding to the impulsive mode in which there is a coupling action between the tank and liquid are considered in this paper. The liquid is represented by means of an added mass approach, and with acoustic finite elements. A density ρ = 983 kg/m3 and a bulk modulus K = 2.07 GPa (i.e. the properties of water) are used in the computations. 3.1. Model with added liquid mass The finite element meshes for the structures of Models A, B, and C with added liquid mass are indicated in Table 1 and Fig. 4. Convergence studies were carried out and the finite element meshes listed in Table 1 were found to provide good results for the purpose of the present study. The bottom plate of the tank was not included in the model, since this study only
Fig. 4. Finite element mesh for Model A, with added liquid mass.
covers anchored tanks, with emphasis on the behavior of the cylindrical shell and not on the foundation. The model with added mass liquid was previously presented by Virella et al. [18], and is summarized here. The added mass is obtained from a pressure distribution for the impulsive mode of the tank-liquid system due to Veletsos and Shivakumar [19], which has a cosine distribution along the tank cylinder (see Fig. 5). This pressure distribution for the rigid body horizontal motion of a rigid tank-liquid system is described as Pi (η, θ, t) = ci (η)ρ R x¨g (t) cos θ
(1)
where Pi is the impulsive pressure; η is a non-dimensional vertical coordinate = z/HL; z is the vertical coordinate
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Fig. 6. Impulsive pressures for the tank-liquid systems with χg = 1 m/s2 .
Fig. 5. Impulsive pressure distribution around the tank circumference: (a) circumferential distribution; (b) pressure components in the direction of the excitation.
measured from the tank bottom; R is the tank radius; x¨g (t) is the ground acceleration; and t is the time. The function ci (η) defines the impulsive pressure distribution along the cylinder height, and is computed as ci (η) = 1 −
∞
ccn (η)
(2)
n=1
where ccn (η) is the function that defines the convective pressure distribution along the cylinder height, computed as ccn (η) =
2 cosh[λn (H /R)η] . λ2n − 1 cosh[λn (H /R)]
(3)
The parameter λn is the nth root of the first derivative of the Bessel function of the first kind and first order. The first three roots are λ1 = 1.841, λ2 = 5.311, and λ3 = 8.536. The function ci (η) converges rapidly as the number of terms in the summation increase, and hence it is sufficient to include three coefficients ccn in Eq. (2). The pressure distributions defined in Eq. (1) for each of the tank-liquid systems considered in this paper and for θ = 0 are presented in Fig. 6. The added mass of the liquid on the tank shell is calculated from the pressure distribution in Eq. (1). To obtain the lumped mass m i at each node, the height of the cylinder is divided into several segments and the area of the pressure below the curve for the segment is divided by the normal acceleration of the rigid wall x¨g . The lumped masses obtained from the added mass have the same vertical variation as the impulsive pressure
Fig. 7. Model with normal mass along the cylinder height.
distribution from which they are derived (see Fig. 7), and they have a uniform distribution around the circumference of the tank. Because this added mass is the liquid mass that moves together with the tank, it acts normal to the cylindrical shell. For a radial section of the tank (Fig. 7), the lumped mass m i at each location is computed using the rectangular rule. For an interior node at the tank shell, the lumped mass m i is computed as Pi h (4) mi = an where Pi is the pressure at node i , h is the constant distance between nodes, and an is the reference normal acceleration. Note that an becomes the amplitude of x¨g for θ = 0. For nodes at the liquid surface and at the bottom of the tank, the lumped mass is ml =
Pl h 2 an
(5)
where m l is the mass at a boundary node l and Pl is the pressure at node l. Because the added masses are determined from the impulsive pressure which is normal to the shell surface, they must be added in such a way that they only add inertia in this direction. For this reason, the added masses are sometimes referred to as the “normal masses”.
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For each of the tank-liquid systems considered in this study, the mass distributions calculated as described previously were verified by comparing them with the impulsive mass ratios proposed by Housner [1]. The impulsive mass ratio Mir for the particular tank-liquid system is defined as
Table 2 Comparisons of impulsive mass ratios calculated with the present study and with the Housner equation (11) Model
H /D
Mir
Housner Mi/Mt
Difference (%)
Mi , (6) Mt where Mi is the total impulsive mass and Mt is the total liquid mass. For our case, the total impulsive mass Mi can be calculated by first obtaining a mass resultant m res for the tank meridian in the direction of the ground excitation, and then integrating around the circumference to compute the total impulsive mass of the tank. The mass resultant m res is calculated as the sum of the individual masses m i at the different nodes along the θ = 0 meridian. The added mass components in the direction of the excitation are directly proportional to the impulsive pressure which varies with a cosine distribution around the tank circumference (see Fig. 5(a)). Hence, to integrate the radial mass in order to obtain the impulsive mass, we must project the mass m res along the direction of the excitation. Moreover, m res was calculated for the θ = 0 meridian. To calculate m res for θ > 0, it must be multiplied by cos θ , in the same way as the pressure was defined (see Fig. 5(b)). At the arc that forms an angle θ with respect to the horizontal, the horizontal component of the impulsive mass component in the direction of the excitation is
A B C
0.40 0.63 0.95
0.4057 0.5710 0.7040
0.406 0.587 0.736
0.07 2.73 4.35
Mir =
dm i = (m res cos θ ) cos θ (R dθ ).
(7)
The total impulsive mass Mi is then π/2 cos2 (θ ) dθ = π Rm res . Mi = 4Rm res
(8)
0
The total liquid mass is calculated using the following expression Mt = π R 2 HLγL /g,
(9)
where γL is the liquid unit weight and g is the acceleration of gravity. The impulsive mass ratio defined in Eq. (6) is m res g . (10) Mir = R HL γ L
Fig. 8. Model with normal mass around the circumference.
scheme needs to be introduced in order to represent this added mass. It must be pointed out, though, that some structural analysis programs allow for the addition of lumped masses along three global Cartesian axes, but it is not common to have the option of adding masses in cylindrical or polar coordinates. The added liquid mass in lumped form is attached to the shell nodes by means of rigid, massless links with small lengths (see Fig. 8). The links are rigid truss elements with supports oriented in the local axes of the truss elements. The supports must permit the motion of the nodal masses only in the direction normal to the shell. Hence, the motion of the support is restricted in the global tangential direction (perpendicular to the element axis) and in the vertical direction, but it is free to move in the global radial direction (i.e. the local axial direction). The total impulsive mass Mi in a specific direction calculated with Eq. (8) is twice the impulsive mass computed using the methodology of Ref. [1]. However, as the masses can only move in the radial direction, it can be shown that half of this total impulsive mass is excited in a specific direction.
The impulsive mass ratio Mir can also be calculated using the expression proposed by Housner [1]:
3.2. Model with liquid finite elements
Mi tanh(1.7R/HL) = . Mt 1.7R/HL
A typical finite element mesh for the tank (Model A in this case) in which the liquid is modeled with finite elements is shown in Fig. 9(a). Acoustic three-dimensional finite elements based on linear wave theory were used in this part of the study to represent the liquid. Such a model represents the dilatational motion of the liquid, allowing a wave to be described as a single pressure degree of freedom at each point in space. The formulation is based on the Laplace equation in the pressure domain and, although viscosity effects can be accounted for, an inviscid liquid was considered for the present study. The formulation of the acoustic elements is described in
(11)
Table 2 shows that the differences between the impulsive mass ratios computed with the present analysis and with the recommendation by Housner [1] are smaller than 5%. Since the added mass comes from the liquid pressure that acts in the direction normal to the shell, the mass should be assigned in the direction normal to the cylinder. However, because in most FE programs the same magnitude of the nodal mass is assigned to all the degrees of freedom, a sui generis
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Fig. 9. Typical finite element mesh for Model A with liquid acoustic elements: (a) tank structure mesh; (b) liquid mesh. Table 3 Number of finite elements in the models where the liquid is represented with acoustic elements Model A B C
H /D 0.40 0.63 0.95
Number of elements Shell
Acoustic
10,536 12,216 12,216
39,312 56,784 56,784
Refs. [16,17]. The liquid mesh has 39,312 elements for Model A and 56,784 elements for Models B and C (see Table 3). The elements for the liquid are identified in ABAQUS [16] as AC2D4, which are solid, eight-node brick acoustic elements with bilinear interpolation and with only one pressure unknown per node. The finite element mesh for the liquid of Model A is shown in Fig. 9(b). For the finite element model of the tank, triangular shell elements (S3R) are used in the roof and quadrilateral and triangular elements are used for the cylinder. The number of elements for each model is listed in Table 3. It was mentioned before that it was not necessary to include the bottom of the tank in the tank models, because this study emphasizes the behavior of the cylindrical shell. However, the bottom of the tank was discretized with finite elements in the model with liquid acoustic elements in order to specify the boundary condition (i.e. contact interaction) between the bottom surface of the liquid and the tank. Quadrilateral (S4R) and triangular (S3R) shell elements were used for the bottom of the tank, and pinned supports were placed at each node of the tank bottom, to represent an anchored tank resting on a rigid base. The location of each node on the constrained surfaces of the liquid corresponds exactly to the location of a node on the tank cylinder or base. Surface tied normal contact was considered between the surfaces of the liquid and the tank walls and the bottom. This contact formulation is based on a master–slave approach, in which both surfaces remain in contact throughout the simulation, allowing the transmission of normal forces between them. The formulation for normal contact is described in Refs. [16,17]. No sloshing waves were considered in this study, and thus no pressure was applied to the nodes at the free liquid surface. The boundary conditions specified at the interfaces of the liquid model are illustrated in Fig. 10.
Fig. 10. Conditions assumed for the three-dimensional tank-liquid finite element model. Table 4 Results of the free vibration analyses using the normal mass models Model
H /D
Tmax (s)
Tmin (s)
α3
Number of modes
A B C
0.40 0.63 0.95
0.644 0.859 1.057
0.189 0.199 0.205
0.74 0.81 0.79
891 796 794
4. Fundamental periods and mode shapes 4.1. Free vibration analyses To compute the natural periods and mode shapes for the normal mass models of the tank-liquid systems, the hydrostatic pressure and the self-weight of the tank were initially applied, i.e. they were modeled as a pre-stress state. The natural periods and mode shapes were computed until the ratio α3 of the horizontal component M3 (i.e. global direction 3) of the accumulated effective mass to the total mass of the system (α3 = M3 /(Mi + Mshell )) was larger than 70% (see Table 4). The analysis uses the Lanczos solver [16], in which all the natural frequencies and mode shapes are found within a specified range of frequencies. The fundamental modes were identified as those with the largest participation factors in the translational directions. Table 4 displays the range of the natural periods and the total number of modes found from the free vibration analyses of the tank-liquid systems. For Model A, 891 modes with natural
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Fig. 11. Variation of natural period with the normalized modal participation factor in the horizontal axis 3 for Model A.
periods between 0.64 s and 0.19 s were found with the condition α3 ≥ 0.7. The natural periods ranged from 0.86 s and 0.20 s (796 modes) for Model B and between 1.0 s and 0.20 s (794 modes) for Model C. Typical cylinder modes (see Ref. [15]) characterized by circumferential wave numbers (n) and axial half-wave numbers (m) resulted for all the modes of each of the tank-liquid systems. Fig. 11 illustrates, for Model A, the variation of the natural periods with the participation factor P3 (in absolute value) along the axis 3 horizontal direction. The participation factor is normalized with respect to its maximum value. Fig. 11 shows that the participation factor (P3 ) is much larger for a certain mode, the fundamental mode, indicating that the response of the system to a horizontal motion will be predominantly determined by this mode. Similar results were obtained for Models B and C, and thus the same approach described before was used to determine the fundamental modes for these tankliquid systems. Fig. 12 illustrates the variation of the natural period with the circumferential wave number n, for modes with axial halfwave number m = 1. The figure also shows the fundamental period calculated previously for each of the three models. The modes with the largest natural periods in Fig. 12 had small participation factors P3 , and therefore their contribution to the response of the tank-liquid systems to horizontal motions can be neglected. The modes for Models A, B, and C with the three largest participation factors P3 are listed in Table 5. This table shows that, for Models B and C, there are modes (the second and third) that will contribute to the response to horizontal ground motions that are not cantilever beam modes (i.e. n > 1). However, the fundamental mode still dominates the response, as α3 for the first mode of Model A is 94% of the total α3 obtained adding those of the three modes considered. For other models, α3 for the first mode is even higher: 98% for Model B and 97% for Model C. Therefore, for practical purposes, only the fundamental mode needs to be considered in order to predict the response of cylindrical tank-liquid systems subjected to horizontal base motions. These results compared well with the recommendations by Barton and Parker [13], who stated that, for cylindrical tank-liquid systems with aspect ratios H /D larger than 0.5, the modes characterized by circumferential wave numbers n > 1 and axial half-wave numbers m ≥ 1 are not important for determining the response to horizontal base motions. The present work showed that even for tanks
Fig. 12. Variation in the natural period T with n for m = 1 for Models A, B and C.
Fig. 13. Damping ratios for the two frequency ranges considered in the dynamic analyses: (a) frequency range 1; (b) frequency range 2.
with an aspect ratio H /D = 0.40, which falls outside the range considered by Barton and Parker [13], modes with n > 1 and m ≥ 1 have very small participation factors in the direction of the horizontal motion. 4.2. Steady state harmonic response analyses A linear, steady state harmonic analysis of the tank-liquid systems under uniaxial horizontal motion was carried out to find the periods and deformed shapes that will be excited by this input. To account for the geometric non-linearity, the hydrostatic pressure and the self-weight of the tank were included in an initial static step; in this way, the stiffness of the system was modified prior to the dynamic analysis. The tank-liquid system was next loaded with the pressure distribution given by Eq. (1), and a ground acceleration of 1g (g = 9.81 m/s2 ) was considered. The linear, steady state harmonic response to a horizontal harmonic motion in one
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Table 5 Modes of the tank-liquid systems relevant for the response to horizontal ground motions Mode 1 2 3
Model A T (s)
α3
n
m
Model B T (s)
α3
n
m
Model C T (s)
α3
n
m
0.2116 0.2021 0.1957
0.64 0.02 0.02
1a 1 1
1 5 4
0.2395 0.2362 0.2046
0.79 0.01 0.01
1a 28 1
1 7 7
0.3001 0.2960 0.2998
0.77 0.01 0.01
1a 20 25
1 6 6
a Fundamental mode.
Fig. 14. Variation of the normalized radial displacement with the excitation period for Model A with added mass: –•– Node A; –– Node B (see Fig. 15).
direction was obtained by direct integration. The tank-liquid systems with the liquid accounted for by added masses were excited along the axis 3 horizontal direction, while the models with acoustic liquid elements were excited along the axis 1 horizontal direction. The dynamic response as a function of the excitation period was obtained. The periods of the tank-liquid systems excited by the ground motion will be those associated with peaks in a plot of a maximum response versus excitation period. The radial displacement response (U-radial) yields a deformed shape associated with a mode of vibration of the tankliquid system. To introduce damping to the models, a Rayleigh damping matrix [C] was used: [C] = β[K ] + α[M]
(12)
where β is the stiffness proportional coefficient and α is the mass proportional coefficient. The Rayleigh damping coefficients were computed by considering two ranges of frequencies. The frequency ranges are shown in Fig. 13(a) and (b). The values of β are 0.003351 for the frequency range 1 and 0.001273 for the frequency range 2. The coefficient α is 0.3175 for frequency range 1 and 0.8042 for frequency range 2. By using the two frequency bands, the damping ratio of the system was kept from 3% to 4% for the ranges of the excitation frequencies considered in the dynamic analyses, as can be seen in Fig. 13. The variation in the radial displacement (U-radial) with excitation period for Model A with the added mass approach is shown in Fig. 14. The displacements shown are those at two nodes of the cylinder where maximum displacements occurred. These nodes, identified as A and B, are shown in Fig. 15. The highest peaks in Fig. 14 occur at the excitation period associated with the fundamental mode. A similar behavior was
obtained for Models B and C, and the fundamental modes were also identified from the excitation period with the highest radial displacement peak. The fundamental modes resulted in bending modes (n = 1). Similar fundamental modes were obtained from the free vibration and from the harmonic response analyses for the added mass models as shown in Table 6, with differences of less than 3% in all cases. The fundamental modes for each of the tank-liquid system, in which the liquid was represented with added masses, are presented in Figs. 15–17. The maximum radial displacements at the meridian in the direction of the excitation did not occur near the top of the tank for any model, as is the case for the fundamental mode of a cantilever beam. As shown in Fig. 15(b), for Model A (the shortest tank with H /D = 0.40) the fundamental mode is a bending mode (n = 1) with an axial half-wave (m) characterized by a bulge near the mid-height of the cylinder. However, as shown in Figs. 16(b) and 17(b), for Models B and C which have aspect ratios larger than 0.63, the fundamental modes tend to the first mode of a cantilever beam, even though the maximum radial displacements do not occur at the top of the tank. In the cases studied so far, the liquid in the tank was accounted for by means of the added mass formulation. In the sequel, the analyses are repeated using acoustic elements to represent the liquid. The variation in the radial displacement with the excitation period for Model A, using acoustic elements, is presented in Fig. 18. The mode that contributes the most to the response of the tank-liquid system is identified by the peak in Fig. 18. Notice from Figs. 14 and 18 that the fundamental period for Model A obtained with the added mass approach and with the models with acoustic elements are quite similar; in fact, the differences are smaller than 3% for the three tank models. Evidently, the response to the ground motion with periods equal to the fundamental modes (indicated in Figs. 14 and 18) are much larger than that computed for other excitation periods. Thus, for both models (with liquid acoustic elements and added mass) and regardless of the H /D ratio, the response to a horizontal harmonic excitation is dominated by the fundamental mode. The deformed shapes associated with the fundamental modes for the models with acoustic elements were bending modes (n = 1) that tend to the first mode of a cantilever beam, similar to those presented in Figs. 15–17 for the added mass models. For instance, the deformed shape shown in Fig. 19 for the fundamental mode of Model C computed using acoustic elements is similar to that computed with the added mass model and displayed in Fig. 17.
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Fig. 15. Fundamental mode for Model A with liquid added mass: (a) 3D view; (b) deformed shape in the meridian with maximum displacements.
Fig. 16. Fundamental mode for Model B with liquid added mass: (a) 3D view; (b) deformed shape in the meridian with maximum displacements.
Fig. 17. Fundamental mode for Model C with liquid added mass: (a) 3D view; (b) deformed shape in the meridian with maximum displacements. Table 6 Fundamental periods for the tank-liquid systems obtained by using two formulations to represent the liquid Model
H /D
Periods with added mass formulation Free vibration (s) Harmonic response (s)
Periods with acoustic FE Harmonic response (s)
A B C
0.40 0.63 0.95
0.212 0.239 0.300
0.210 0.244 0.296
0.212 0.240 0.303
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impulsive pressure distributions for the fundamental modes for flexible and rigid tanks increased with their aspect ratios H /D. However, even for the tallest tank (Model C, where differences are larger), the impulsive pressures for the fundamental mode of the flexible and rigid tanks have similar shapes. 4.3. Comparisons with previous studies
Fig. 18. Variation of the normalized radial displacement with the excitation period for Model A with liquid acoustic elements: –•– Node A; –– Node B (see Fig. 15).
The fundamental periods obtained from the steady state harmonic response analyses and from the free vibration (eigenvalue) analyses using different formulations to represent the liquid are presented in Table 6. The results are very similar for the three models of the tanks: in all cases, the differences are smaller than 3%. The variation in the impulsive pressure with the period of the ground motion for tank models with liquid acoustic elements was obtained. The pressure was normalized and it is calculated at the point of the cylinder where the maximum pressure occurred. In all cases, the pressure showed a pronounced peak at the same period than for the maximum radial displacements (see Fig. 18); therefore, both responses are in phase. The impulsive hydrodynamic pressure for the fundamental mode follows the meridianal distribution shown in Fig. 20, with a cosine circumferential variation similar to that illustrated in Fig. 5. The impulsive hydrodynamic pressure distribution for Models A and C is illustrated in Fig. 20. The pressure is shown in the direction of the excitation and for the fundamental mode along the meridian at which the maximum response occurs. The pressure distribution for an anchored rigid tank obtained from Eq. (1) is also shown in Fig. 20. The two distributions are similar in shape, with the smallest differences occurring for Model A, and increasing successively for Models B and C. These results indicate that the difference between the
The fundamental periods obtained in this paper for the impulsive modes of cylindrical tanks under horizontal translation were compared with those proposed by Veletsos and Shivakumar [19] and Sakai et al. [20]. Both groups of researchers identified cantilever beam fundamental modes. The equation for the fundamental impulsive mode by Veletsos and Shivakumar [19] considers a constant shell thickness. For the tanks considered in this paper, which have variable shell thickness (see Fig. 3), the average thickness of the shell courses below the liquid surface was computed and used in the equation of Veletsos and Shivakumar [19]. For tanks with variable shell thickness, Sakai et al. [20] used the thickness at an elevation of 1/3 of the liquid height measured from the bottom of the tank to compute the fundamental impulsive period. A comparison between the impulsive fundamental periods computed from the free vibration analyses in this paper and those from Refs. [19] and [20] is displayed in Fig. 21. The fundamental periods from this study are very similar to those predicted by Veletsos and Shivakumar [19], with the maximum difference being smaller than 3.5%. Notice that even for Model A, in which the fundamental mode is not quite similar to a cantilever beam type mode, the fundamental period compared extremely well (0.15% difference) with that proposed by Veletsos and Shivakumar. Smaller fundamental periods are predicted by the formulation of Sakai et al. [20], with differences between 9% and 14%. 5. Conclusions The conclusions stated in this paper apply for cylindrical tank-liquid systems with height/diameter ratios from 0.40 to 0.95 with roof, anchored to the foundation, with a perfect or as-designed geometry, in which the influence of the hydrostatic pressure and the self-weight is considered.
Fig. 19. Fundamental mode for Model C with acoustic liquid elements: (a) 3D view; (b) side view.
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Fig. 21. Variation in fundamental period (TD ) with liquid height to tank diameter ratio (HL /D).
Fig. 20. Pressure distribution at the meridian with maximum displacements for the fundamental mode: (a) Model A; (b) Model C.
It was verified that the response of a tank-liquid system subjected to a horizontal ground motion can be accurately predicted by considering just the fundamental mode. The fundamental mode for the tank-liquid systems is a bending mode (n = 1), regardless of the height-to-tank diameter ratios (H /D) considered in this study. This conclusion agrees with that stated previously by Barton and Parker [13]. The largest natural periods are associated with cylinder modes characterized by circumferential waves n > 1 and axial halfwaves m = 1. However, these modes are not the fundamental modes for predicting the response to a horizontal base motion, in the sense that they do not have the largest modal participation factors. It was proved that, for the range of shell dimensions of interest for the oil industry, the fundamental modes are not those associated with circumferential wavy patterns, as proposed by Natchigall et al. [6]. Nevertheless, it must be pointed out that this study considers tanks with perfect geometry, i.e. the effects of imperfections in the shell were not taken into account. Similar fundamental periods and mode shapes were found from the free vibration (eigenvalue) analyses and from the harmonic response analyses, using the added mass formulation and the model with liquid acoustic finite elements. The differences were smaller than 3% in all cases. Thus, the added mass models can provide a good approximation for calculating the response of tanks filled with liquid, as the results compare very well with the more sophisticated models in which the
liquid is represented by acoustic finite elements. This is true regardless of the aspect ratio (H /D) of the tank. The fundamental modes of tanks with aspect ratios H /D larger than 0.63 are similar to the first mode of a cantilever beam. For the shortest tank (H /D = 0.40), the fundamental mode is a bending mode (n = 1) with an axial half-wave (m) characterized by a bulge formed near the mid-height of the cylinder. The differences between the impulsive pressure distributions for the fundamental modes calculated considering flexible and rigid tanks increase with the aspect ratio (H /D) of the tanks. However, even for the tank with the largest aspect ratio considered here (Model C, H /D = 0.95) where the differences are largest, the impulsive pressures of flexible and rigid tanks have similar shapes. The fundamental periods obtained with the detailed finite element models used in this paper compared well with the recommendations of Veletsos and Shivakumar [19], with differences smaller than 3.5% in all cases. The formulation for the fundamental period due to Sakai et al. [20] predicts smaller fundamental periods than those found in this paper, resulting in differences between 9% and 14%. Acknowledgments J.C. Virella was supported by a PR-EPSCoR post-doctoral fellowship grant EPS-0223152 for this research. The authors thank Dr. C.A. Prato (National University of C´ordoba, Argentina) for his contribution during the early stages of this work. Partial support from Mid America Earthquake Center to this research is gratefully acknowledged. References [1] Housner GW. The dynamic behavior of water tanks. Bulletin of the Seismological Society of America 1963;53(2):381–9. [2] Veletsos AS, Yang JY. Earthquake response of liquid storage tanksadvances in civil engineering through mechanics. In: Proceedings of the second engineering mechanics specialty conference. Raleigh (NC): ASCE; 1977. p. 1–24. [3] Haroun MA, Housner GW. Earthquake response of deformable liquid storage tanks. Journal of Applied Mechanics 1981;48(2):411–8.
J.C. Virella et al. / Engineering Structures 28 (2006) 1450–1461 [4] Veletsos AS. Seismic response and design of liquid storage tanks. Guidelines for the seismic design of oil and gas pipeline systems. In: Technical council on lifeline earthquake engineering. New York: ASCE; 1984. p. 255–370, 443–61. [5] Malhotra P, Veletsos AS. Uplifting response of unanchored liquid storage tanks. Journal of Structural Engineering 1994;120(12):3525–47. [6] Natchigall I, Gebbeken N, Urrutia-Galicia JL. On the analysis of vertical circular cylindrical tanks under earthquake excitation at its base. Engineering Structures 2003;25:201–13. [7] American Petroleum Institute. API Standard 650. Steel tanks for oil storage. 8th ed. 1988. [8] American Water Works Association. AWWA Standard for welded steel tanks for water storage. AWWA D100. 1984. [9] Maz´uch T, Hor´acek J, Trnka J, Vesely J. Natural modes and frequencies of a thin clamped-free steel cylindrical storage tank partially filled with water: FEM and measurement. Journal of Sound and Vibration 1996; 193(3):669–90. [10] Gupta RK. Free vibrations of partially filled cylindrical tanks. Department of Engineering, Launceston (Australia): University of Tasmania; 1994. [11] Han RPS, Liu JD. Free vibration analysis of a fluid-loaded variable thickness cylindrical tank. Journal of Sound and Vibration 1994;176(2): 235–53. [12] Goncalves PB, Ramos NRSS. Free vibration analysis of cylindrical tanks partially filled with liquid. Journal of Sound and Vibration 1996;195(3):
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429–44. [13] Barton DC, Parker JV. Finite element analysis of the seismic response of anchored and unanchored liquid storage tanks. Earthquake Engineering and Structural Dynamics 1987;15(3):299–322. [14] Virella JC. Buckling of steel tanks subject to earthquake loadings. Ph.D. thesis. Department of Civil Engineering and Surveying, University of Puerto Rico at Mayag¨uez; 2004. [15] Virella JC, Godoy LA, Su´arez LE. Influence of the roof on the natural periods of empty tanks. Engineering Structures 2003;25:877–87. [16] Hibbit HD, Karlsson BI, Sorensen P. ABAQUS standard user manual, version 6.4. Rhode Island: Hibbit, Karlsson and Sorensen, Inc.; 2002. [17] Hibbit HD, Karlsson BI, Sorensen P. ABAQUS theory manual, version 6.4. Rhode Island: Hibbit, Karlsson and Sorensen, Inc.; 2002. [18] Virella JC, Godoy LA, Su´arez LE. Effect of pre-stress states on the impulsive modes of vibration of cylindrical tank-liquid systems under horizontal motion. Journal of Vibration and Control 2005;11(9): 1195–220. [19] Veletsos AS, Shivakumar P. Tanks containing liquids or solids. In: Beskos DE, Poulos A, editors. Computer analysis and design of earthquake resistant structures: A Handbook, 3; 1997. p. 725–73 [chapter 15]. [20] Sakai F, Ogawa H, Isoe A. Horizontal, vertical, rocking fluid-elastic response and design of cylindrical liquid storage tanks. In: Proceedings of the 8th world conference on earthquake engineering. vol. V. 1984. p. 263–70.