Static and Dynamic Analysis of Storage Tanks Considering ... · PDF fileStatic and Dynamic Analysis of Storage Tanks Considering Soil ... Method B is based on modeling of the structure

International Research Journal of Applied and Basic Sciences © 2013 Available online at www.irjabs.com ISSN 2251-838X / Vol, 6 (4): 515-532 Science Explorer Publications

Static and Dynamic Analysis of Storage Tanks Considering Soil-Structure Interaction

Ali Yosefi Samangany1, Reza Naderi2, Mohammad Hosein Talebpur2Hadi Shahabi far3

1. Department of Civil Engineering, Islamic Azad University, Torbat JaamBranch, Torbat Jaam, Iran 2. Department of Civil Engineering, Shahrood University of Technology , Shahrood, Iran

3. Department of Civil Engineering, Ferdowsi University, Mashad, Iran

Corresponding Author email: [email protected]

ABSTRACT: Storage tanks are one of the important and vital structures at times of earthquake and specially after the earthquake. For this reason, seismic behaviour of storage tanks has attracted attention of many researchers. But thesoil surroundingthe tank affects behaviour of structure in seismic analysis of storage tanks. For this reason, it is necessary to for designers of storage tanks to understand interaction of soil and structure in seismic analysis of tanks. In this paper, first, static and dynamic analysis of the storage tanks has been reviewedwith considering interaction of soil and structure. Then, effect of soil type on seismic behaviour of structure has been studied and the results of static and dynamic analysis results have been compared. For this reason, three kinds of soil with different parameters (soft, middle and hard soil) have been used. In the present paper, three-dimensional cubic storage tanks modelled using Abaqus software and energy absorbentboundaries method has been used to model waves propagation inboundaries. Key words: static analysis, dynamic analysis, interaction of soil and structure, cubic storage tanks

INTRODUCTION

Earthquake force is of special importance in design of storage tanks due to volume and weight of such structures. For this reason, seismic analysis of the storage tanks has been considered by many researchers. One of the oldest researchers is Jacobson who has collected valuable information about seismic analysis of storage tanks (Abaqus Analysis) . After him, other researchers such as Lysmer and Kuhlemeyer (ADF, 2000) , Stein (Chopra, 1995) , Kausel (Epstein, 1976) etc have conducted broad studies on different fields of seismic analysis of tanks but the soil surrounding thetank and its behaviour at time of seismic modelling have been regarded as an important and relatively unknown factor in the researches for storage tanks. Haron is one of the first researchers who have studied interaction of soil and structure in seismic analysis of storage tanks and its modelling based on studies of the past researchers (Haroun and Housner, 1981; Helwany, 2007; Housner, 1963; ICG, 1998) . Other researches continued his researches on interaction of soil and structure based on finite elements method by acquiring two-dimensional (Jacobsan, 1949) and three-dimensional models (Jaiswal et al., 2003) and this issue has been considered by many researchers. On the other hands, different methods have been presented based on attitude of the designers in two general classes of static and dynamic analysis. In static analysis, earthquake force and its effect are evaluated by a static force. This force is obtained based on coefficients of the code and the related coefficients. In other words, effect of earth motion on the storage tanks is evaluated and simulated with equivalent static force while force applied to the structure is determined due to earth motion resulting from earthquake force based on dynamic reflection of the structure in dynamic analysis method. Dynamic analysis method includes spectrum analysis method and time history analysis method. In dynamic analysis method, effect of earth motion is specified as acceleration spectrum. In time history dynamic analysis method, effect of earth motion is specified as site acceleration time history. Therefore, time history analysis requires seismic geotechnical studies in the site to determine earth motion for the desired site based on earthquake risk analysis and response of earth. This trend was generally a complex process and requires different parameters to be known (Kausel, 1988) . Therefore, time history dynamic analysis process should be economically assessed. Of course, spectral dynamic analysis and time history analysis are done based on clauses of control code and code procedure. For modelling, Abaqus software which has many abilities to analyze structure based on finite elements method has been used. In modeling process, three kinds of soil with different parameters (soft, middle and hard soil) have been used to study effect of soil surroundingtank in seismic analysis. In the present paper, energy absorbent boundaries method has been used in modeling process based on damper and spring in order to prevent return of earthquake waves. At the end,

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effect of soil type has been discussed in seismic analysis by drawing diagrams of displacement, wall and tank floor stress. Loading the tanks in Construction Code Generally, loads applied to tanks include static loads and dynamic loads and static loads include dead, live load, static pressure of the fluids, static pressure of soil, uplift and forces resulting from temperature change. In order to determine static pressure of soil, Rankine relation is used. In order to design the floors which are below static level of groundwater,uplift of the groundwater should be included. Structure of the tank should be designed for changing temperature. One of the other loads applied to tank is dynamic load and there are two methods for determining dynamic loads: A- equivalent static analysis method, B- dynamic analysis method. Method B is based on modeling of the structure and fluid inside it as finite elements model. Specification of this method is acquisition of interaction of water-structure vibration. The equivalent static analysis method is studied. Static analysis of tanks Static analysis is regarded as the first step in which force resulting from mass inertia of tank, liquid and soil in static analysis is the most common method of seismic analysis. One of the most important forces is pressure resulting from fluid inside the tank and soil surroundingtank. Effect of earth gravitational acceleration causes structural response. It is very important to consider support conditions in soil boundariesfor accurate determination of the problem answer. soil static pressure Static pressure of soil which is applied to back of tank walls can be calculated based on mechanical relations of soil and with help of theoretical relations of Rankineforce which is equal to:

ap K H (1)

In the above relation,P is the lateral pressure of stimulating soil and H is height of the soil.Ka is also calculated for the granular soils as follows:

21 sin(45 )

1 sin 2aK tag

(2)

Where is relation of soil friction angle. Therefore, soil pressure is obtained through the following

relation for the granular soils.

1 sin

1 sinp H

(3)

And force applied to wall is equal to:

21 sin 1*

1 sin 2F H

(4)

Effect of this force is in centre of triangular surface of the soil pressure. In case the soil is cohesive (c

and have values), the applied pressure is equal to:

2a ap K H c K

(5)

And the applied force will be equal to:

212

2aF H c K H (6)


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The first term of this force is applied to centre of the triangularsurface and the second term is uniform and pressure resulting from cohesion is applied to centre of triangular surface. In the presence of water hydrostatic pressure is also included. Static pressure of fluid Vertical pressure of fluid which is applied to floor of the tank and lateral pressure which is applied to

walls can be calculated from p h based on fluid mechanics rules and its changes is linear with height.

Dynamic analysis of tanks In analysis of time history, Rayleigh damping matrix is obtained from mass and rigidity matrices and is obtained as follows.

[ ] [ ] [ ] [ ]Fc M k c (7)

Where (CF) is damping matrix and (M) and (k) are mass and rigidity matrices of each element. In order to extract fluid damping matrix,we use relations of stress and strain derivative which relate strain periodic changes rate to shear stress with viscosity coefficient.

In order to obtain and (Rayleigh damping matrix), the following two methods have been used.

Considering that soil mass materials are not frequency-dependent damping, lower limit of Rayleigh damping

matrix is used and in this case, coefficients of and are obtained as follows:

min

min

min min

min

min

2f

(8)

Where min is materials damping and min is angular frequency in which Rayleigh damping

coefficients are minimum, minf is system vibration frequency which is usually equal to the first frequency of

system. In this method, coefficients and are obtained from numerical analyses such as modal analysis

(19-22) .

2 2i i

(9)

By selecting two main angular frequencies of 1 and 2 from modal analysis and considering fixed

damping percent, the following two relations are obtained:

1 12 2

(10)

2 22 2

(11)

Since

1 2

2 1

2

( )

(12)

2 1

2

( )

(13)

In this paper, and were calculated with the above methods and the following similar results were

obtained:

0.5

0.004

(14)


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Soil dynamic pressure Main relation for determining dynamic pressure of soil at time of earthquake is Mononobe-Okabe relation; however, since soil pressure is not important in calculation of tank, one can use recommendation of Seed and Whitman to determine excessive dynamic pressure at time of earthquake. According to recommendation of Seed and Whitman, coefficient of excessive dynamic pressure of soil at time of earthquake is obtained from the following relation.

75.0 adK (15)

R

IA )5.2( (16)

is earthquake coefficient in relation (15) and with adK , excessive dynamic pressure of soil is obtained

from the following relation:

hPKhKP adadd 0

2 )(2/)( (17)

In relation (17), is special weight of soil, dP is resultant of excessive dynamic pressure of soil, h is height

of wall, 0P is intensity of the surcharge on soil. Effect of dP in 2 3 originates from base of the wall; excessive

dynamic pressure dP is combined with static pressure of soil (Kramer, 1996) .

Fluid pressure

In dynamic analysis, fluid compressibility is withdrawn in order to determine lateral forces resulting from vibration of fluid-structure of hydrodynamic forces in water tanks but effect of waves cannot be ignored. Wavepropagation follows the following relation:

22

2( 2 )

t

(18)

On the other hand, C , therefore, we will have:

2

2 2

2C

t

(19)

In relation (19), is velocity potential function.

Instead of solving this model, one can use a set of mass and spring according to figure 1. Mechanical models were first presented for tanks with rigid walls. Hasner (Standards of Design and Calculation of Groundwater Tanks) was the first who proposed mechanical model for circular and triangular rigid tanks. Wozniak and Michel (2008) generalised Hasner’s model for short and thin tanks. Veletsos and Yang (1997) also used a different method for achieving a similar mechanical model for rigid circular tanks. Consequently, Haroun, Hasner (1984) and Veletsos (2008) expanded mechanical models for flexible tanks and Malhotra simplified this flexible model. According to theory of Hasner, fluid dynamics model with viscosity of water which is inside the tank with hard wall is a model with two degrees of freedom.

Figure 1. description of dimensions of tank and mechanical model of tank (26)


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A mechanical model replaces fluid-tank system with mass-spring system which considerably simplifies hydrodynamic forces. In this model, fluid inside the tank is divided into two components. One of these components which moves uniformly with tank is called impact component and another one which results from fluctuating movement of water wave converts fluid inside the tank into two masses ofM0andM1.M0 is connected to the tank with rigid springs and is called hard mass while M1 can fluctuate inside the tank. The following relations have been presented by Newmark by studying vibration of this model with real tank response of Housner (Rezaei Tabrizi et al., 2002) . For cubictank, specifications of mass and springs are equal to:

0

tanh(1.7 ).

(1.7 )

L HM M

L H (20)

1

0.83 tanh(1.6 ).

(1.6 )

L HM M

L H

(21)

0

0

0.38 1 ( 1)M

H HM

(22)

22

1

1 1

1 0.33 0.63 0.28 1M L L LM

H HM H H HM

(23)

2

1

2

3gM HK

ML

(24)

In the above relations, M is fluid mass,M0is hard mass,M1 is soft mass,H0 is distance between hard mass and floor of tank,H1 is distance between soft mass and floor of tank, H is height of liquid inside the tank, g is gravitational acceleration and k is rigidity of the equivalent spring. L is half of the triangular tank length in direction of earthquake. In case effect of anchor is applied to floor of tank, but when the pressure applied to wall

of tank is considered, and are equal to zero and range of surface waves is equal to:

A x (25)

Where x is displacement of spring and coefficient of is as follows:

1

2

1

0.84

1

KL M g

x KL

L M g

(26)

Determining Lateral Forces Resulting from Hard Masses Lateral forces resulting from hard masses are obtained from the following relations:

rr WR

IAP

)5.2( (27)

WW WR

IAP

)5.2(

(28)

ii WR

IAP

)5.2(

(29)

Total horizontal force resulting from hard masses is equal to:

iWrR PPPV (30)

Anchor of the wall is equal to:


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iiWWrrR hPhPhPM (31)

Overturning moment relative to balance below the foundation is equal to:

iiWWrrRB hPhPhPM (32)

In these relations,Pr , Pw , Pi are forces resulting from vibrations of ceiling, wall and impact fluid. wh

is

distance between gravity centre of the wall and top of the foundation surface. Figure (2) shows scheme of forces resulting from impact masses.

Figure 2. forces resulting from hard masses

After determining resultant and effect of lateral force resulting from vibration of the structure and liquid inside it, these forces should be distributed properly in height of wall. In rectangular tanks, we find forcesPi and resultant and its effect and then distribute this resultant as trapezoid in height of the wall. ForcePr is distributed on top of the wall and forcePw is distributed widely based on thickness of the wall. Damping of Material and Time History Analysis The assumed damping for tank and the surrounding soil are not able to damp the returnwaves. therefore, one should use energy dissipation methods in model medium boundaries such as energy absorbent. In time history analysis, damping is an important factor in application of all nonlinear factors based on dynamic relation of the system. In the present paper, time history analysis method has been used in frequency area.Therefore, motion equation is as follows:

tUMUKUCUM g (33)

In the above relation, (M) is mass matrix, U is acceleration, (C) is damping, U

is velocity, (K) is

stiffness matrix, (U) is displacement vector and gU t is earth acceleration (Kramer, 1996) . In order to solve

motion equation, there are different solutions which usually lead to omission of damping factor but one should consider energy dissipation of system in order to simulate the model with reality. Therefore, one can mention the damping matrix of the entire system as follows (Lysmer and Kuhlemeyer, 1969) :

m

n

iCFKMC1

(34)

In the above relation, (CFn) is damping matrix of nth element of viscous fluid and m is the number of fluid elements. In order to extract fluid damping matrix, relations between periodic changes rate of shear stress and strain are used based on viscosity coefficient. In relation (34), one can generally calculate coefficients of

and based on the following relation:

j

i

ij

ij

ji

ji

11

2

22 (35)


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In this relation, i and j are frequencies of two main modes of tank and i and j are their

damping. By selecting two main frequencies of 1 and 2 , modal analysis and one can obtain the following

relations based on relation (35) and dynamic analysis relation considering fixed damping rate:

22 11 (36)

22 22 (37)

By solving the above equations, we have:

21

212

(38)

21

2

(39)

But the important point about soil of the tank is that it is not directly found in motion equation. Soil surroundingthe tank is effective on earthquake force applied to the structure and emission of earthquake waves. Therefore, one should consider material of soil surroundingthe tank and interaction of soil and structure in seismic analysis process. Theory of Soil and Structure Interaction A structure interacts with its surrounding soil and this causes changes in effect of seismic waves. In seismic analysis, interaction of structure and soil should be considered. Dynamic response of soil-structure system is a function of three factors, dynamic parameters of site, forces and excitations and dynamic model of the system when it is affected by a dynamic loading. Dynamic parameters of the site include soil modulus of elasticity,soil shearmodulus, soil Poisson coefficient, and damping in soil. Damping is also divided into two classes of internal and radiation damping. Internal damping is caused by passage of vibration waves through soil and can be regarded as factor of energy loss due to residue in soil but radiation damping causes energy loss due to emission of waves from foundation of the structure to half-space and for this reason, such geometric distribution of elastic wavesis called geometrical damping. Proper analysis of dynamic reaction of soil-structure interaction system requires recognition of different components of the system and excitations which include determination of free field motion i.e. earth motion without presence of structure and calculation of scattering of earthquake waves due to soil and structure interaction. According to principleof superposition, excitations resulting from free field and interaction with each other are added and dynamic model of the system includes dynamic model relation of the foundation environment. Many models are available for consideration and analysis of interaction. Soil and structure interaction is generally classified into two direct and sub-structure methods and each is elaborated separately. Of course, it is worth noting that a method called Hybrid Method was created for analysis of soil-structure interaction recently. In this method, structure and area close to soil and the area far from soil are modelled in the next stage and effect of the distant area is considered (Livaoglu and Dogangun, 2007) . 1- In Direct Method, structure, foundation and soil are modelled altogether and analysis is done in one step. In this case, because principleof superposition is not necessary, it is possible to do nonlinear analyses. In this method, structure and main parts of the soil are generally modelled with finite elements method and are analysed with each other and motion of free field of soil is applied in boundaries. One of the important issues in modelling soil and structure interaction is to use a model which simulates nonlinear behaviour of the soil. The most general method for modelling surrounding soilis a elastoplastic model considering interaction of soil and structure and emission of waves in the soil. This model includes nonlinear behaviour and one can consider yield criterion for soil based on yield surface in stress-strain surface. In the present paper, Drucker-Prager behavioural model has been used which is approximation of the coulomb’s criterion and was presented by Drucker-Prager in 1952. Drucker-Prager criterion yield function is defined as follows:

021 kJJF D (40)

In the above relation,J1is the first invariant of stress tensor andJ2Dis the second invariant of deviatoric stress tensor. Values of andkin relation (40) are parameters of the model which are calculated based on

cohesion and friction angle of soil as described in the following relations:


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sin33

sin2

(41)

sin33

sin6

Ck

(42)

In the above relation, andC are friction angle and internal cohesion of soil. Based on this criterion,

the given boundary surface is an approximation of Mohr-Columb law which has been included as a correction of Von Mises criterion for effect of hydrostatic stress on yield. One of the advantages of Drucker-Prager criterion is lack of dual response which is given in Mohr-Columb pattern. 2- Sub Structure Method is the most common solution of soil-structure interaction. In this method, soil-structure interaction is classified to a series of simpler and separate sub problem and each sub problem is analyzed with the most suitable method and then the obtained results are combined with each other based on principleof superposition. Considering that Sub Structure Method is a linear method of soil-structure interaction analysis, it is compulsory to assume behavioural model for structure and soil. By specifying behavioural model of the system, one can elaborate details of three-dimensional modelling of storage tanks in soil based on ABAQUS considering soil-structure interaction. After selecting the suitable model, one of the solutions is considered in thetime-frequencyzone. In order to solve motion equation relating to soil-structure interaction, two solutions are used in thetime-frequencyzone when parameters and variables are independent of frequency. In order to solve motion equations, only direct integration method will be used (solution in time-zone) in case nonlinear behaviour or boundary conditions are considered. For this reason, direct method has been used in this research for solving dynamic equations of the system (MirHosseini and Aref Poor, 2008) . Boundary Conditions For many problems, dynamic response of soil and soil-structure interaction, rigid boundaries such as bed rock especiallyin horizontal directionare considerably distant from the studied zone. As a result, energy of the waves which are propagated in that zone can be omitted permanently from that zone. Finite elements analysis can be classified into three groups of primary boundaries, local boundaries and consistent boundaries (24) . Primary Boundaries Primary boundaries can be applied for modelling ground surface as free boundary. Zero relocation or zero stress conditions are specified in primary boundaries of figure (3). In lateral or lower boundaries, full reflection of primary boundaries confines energy in the grid but it is converted to serious errors in response of ground or analysis of soil-structure interaction. If the primary boundaries are structurally far from the desired zone, the reflected waves are damped sufficiently so that their effect will be withdrawn.

Figure 3. primary boundaries

Local Boundaries Use of the viscous damper (figure 4) indicates an ordinary local boundary. It can be shown that damping coefficient necessary for absorbing energy depends on angle of the wave. Since waves with different angles may collide with the boundary, therefore, a local boundary with specified damping coefficient reflects a part of the wave energy. Additional problems appear when divergent surface waves reach the local boundary. Since their fuzzy velocity depends on frequency, therefore, a frequencyindependent damperis required for absorbing their energy. Effects of reflections of the local boundaries can be reduced with increase of distance between boundary and the desired zone (Rahimi Zadeh and Khajeh Ahmad Attari, 2003) .


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Consistent Boundaries Consistent boundaries are a model for earth island boundary which can absorb all volume and surface waves in all of angels and all frequencies. Consistent boundaries or energy absorbent boundaries can be presented with frequency-independent boundary rigidity matrix which is obtained from integral equations or boundary elements method. For example, Wolff formulated a centralised parameter model comprising of springs, masses and separate dampers which have behaviour similar to consistent boundary (Rahimi Zadeh and Khajeh Ahmad Attari, 2003) . A fully simplified example of this boundary is shown in figure (5).

Figuere 4. Local boundaris

Figure 5. Consistent boundaries

Drucker-Prager Yield criterion Mohr-Columb criterion is presented in 1773 as a simple form of linear relation between shear stress and stress acting on a surface as the following relation:

.tannc (43)

In this relation, τis shear stress, n is vertical stress and c is soil cohesion and is internal friction

angle. Based on definition of Mohr-Columb,when stress on a plane reaches relation (43), it indicates plastic deformations which have been presented as motion on line. Mohr-Columb yield criterion was obtained considering all combinations of stress which cause yield as in the previous states. In Mohr-Columb yield criterion, hydrostatic stress is effective on yield of substance unlike the previous criteria. This fact is concluded from relation (44).

1 3 1 31 3

1 3 1 3

1( )cos sin tan

2 2 2

( ) 2 cos ( )sin

c

c

(44)

In hydrostatic axis, considering relation 1 2 3 and substituting it in relation (44), it can be

concluded that cohesion of granular substances is regarded as a hydrostatic stress m which is obtained from

the following relation:

.cotm C

(45)

Yield function is defined based on Mohr-Columb yield criterion as the following relation:

'

2

1.sin (cos sin .sin ) .cos 0

3mF J c (46)


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An approximation of Columb criterion was considered byDrucker-Prager in 1952 as correction of Von-Mises criterion for effect of hydrostatic stress on yield and the yield function was presented as the following relation:

' '

23 0mF J K (47)

Where and K are defined as follows:

1/2

2sin

3 (3 sin )

(48)

'

1/ 2

6 sin

3 (3 sin )

cK

(49)

In the above relations, K is the rigidity coefficient. In this research, one can model nonlinear behaviour of soil in ABAQUS software with cohesion coefficient and internal friction angle. Selection of Suitable Input Motions

It is very important to use suitable earthquake in analysis of soil-structure interaction. The selected acceleration time history should have the desired specifications for seismic motion of rock floor (such as maximum acceleration, frequency content, vibration term, magnitude, distance from the earthquake centre and tectonic regime of the region). The best input motion is the acceleration time history which has been recorded during a real earthquake in the desired location. In case of the absence of ideal acceleration time history, one can use the acceleration time histories origin of which is concurrent with origin of the probable earthquake. For this purpose, the maximum acceleration and dominating period of the bedrock become equal. Change in dominating period of the selective acceleration time history to the desired dominating period is such that if time

intervals between values of the selected acceleration time history are 1t and its dominating periodisT1, the

related dominating periodT2will be obtained from the following relation by converting time intervals of

acceleration time history to value of 2t ( Technical Standards and Research Office of Plan and Budget

Organization) .

2

2 1

1

Tt t

T (50)

Lower Energy Absorbent Boundaries of the Soil Mass

Considering modelling of a limited part of soil surroundingthe tank in direct method, energy absorbent boundaries have been used to prevent reflection of return waves from the tank. Inherent damping for the tank and its surrounding environment is not able to lose the return waves from boundaries. This causes wave energy to be confined in the above environment and is effective on results. Therefore, one should be careful when using absorbent boundaries in order to select their parameters. For this purpose, three elements of spring–orthogonal damper have been used in three vertical directions in lateral boundaries in each node. Parameters of spring-dampers are determined based on propagation velocity of longitudinal waves (in vertical direction) and transverse waves (for orthogonal vertical directions) in soil environment. As mentioned above, energy absorbent boundaries in dynamics of structures can be replaced as mass-spring-damper simple models with frequency independent coefficients (24) . Time-dependent loads are applied directly on the structure or through stochastic waves. For harmonic excitation with frequency of , dynamic stiffness coefficients are as

ratio of the applied load range P a to the resulting displacement U a as follows:

aciaakKaS (51)

So that / sa wr c andcs is shear waves speed,K is static stiffness of the structure, K a is coefficient

without spring and c a is the related damping coefficient.


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Figure 6. boundary conditions in three-dimensional model in finite elements method (21)

Damping coefficients for soil surrounding the tank are calculated based on the following relations:

(1 )

(1 2 )(1 )P

v EV

v v

(52)

2(1 )S

EV

v

(53)

. .t sC V ab (54)

. .nC Vp ab (55)

In relations (52) and (53),VS, Vp are body wave velocity and shear wave velocity respectively.Cn andCt

are damping in body waves direction and damping in shear waves. Considering figure 7, it is found that shear wave damping acts tangential on plane of element. a and b are dimensions of the cross section in area of each damper and is ambient density. In this research, Spring Dashpot has been used to model the absorbent

boundaries. This element with some properties such as Kelvin Materials which are made of a viscous part and an elastic part can be used for this purpose. In figure (7), a simple design of these elements is shown.

Figure 7. simplified design of Spring Dashpot element in ABAQUS software Earthquake Force and Its Components Generally each structure which is inside the soil is affected by six components of earth motion. These components include two lateral components, a vertical component and three torsional components. Horizontal component of ground acceleration causes hydrodynamic pressures on wall of the tank. Hydrodynamic pressures include impact pressures and fluctuation pressures. Fluctuation pressures are caused by transferring impulse vibrations to the fluid and appear as surface waves in fluid. Hydrodynamic pressures can cause shear force and bending moment and finally considerable compressive circular stresses,tensile stresses and shear stresses in wall of tank. Seismic analysis of the storage tanks in soil largely requires analysis of earthquake risk through which one can correctly evaluate performance of structure at time of earthquake and after earthquake. Based on earthquake risk analysis, one can calculate specifications of ground motion and then necessary spectra and records (Mahin Roosta and Yaghoobi, 2008) . Selection of Seismic Bedrock Earthquake effect on bedrock is origin of motions in soil layer. Bedrock is a layer of ground which has high rigidity and has no considerable site effects. Generally, layers of the ground which have shear wave speed of above 600 to 1000 m/s are selected as seismic bedrock (Technical Standards and Research Office of Plan and Budget Organization) . Modelling soil and tank in ABAQUS software Modelling the contact surface of soil and tank is one of the important parts of soil-tank interaction modelling. Generally, contact surface between tank and soil is modelled in two ways: complete cohesion or


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friction contact surface in which there is soil and tank slip and segregation between them. In this research, contact elements have been used for modelling contact surface of soil and tank. In Abaqus program, mechanical contacts between two objects are possible in two ways: Surface to surface contact and surface to node contact which are more accurate than the first state. In this research, two contacts between interfaces have been considered: friction contact and vertical contact. In friction contact, penalty formulation with friction coefficient of 0.7 has been used. In vertical contact, penalty formulation has been used. In this research, effect of segregation between tank and soil has been considered and segregation between tank and soil was found in time intervals of seismic analysis in upper and lower parts of the tank. It is possible to select Rayleigh damping in Abaqus program in specifications of materials. Critical damping coefficients have been assumed 5% for this research which are in line with studies of Stuart and Campetla (19) . Radiation damping play very important role in modelling of soil and dynamic issues especially issues of soil-structure interaction. In this research, infinite elements and dampers have been used for modelling the soil-tank interaction. Using infinite elements, one can be release from effect of undesirable phenomenon of packing in modelling. Cubic three-dimensional elements with 8 nodes which have three degrees of freedom in each node have been used for modelling soil and tank system. For modelling the tank, brick three-dimensional elements with 8 nodes have been used. In this research, structure and soil environment has been used altogether and directly with finite elements method. In order to analyse their interactional system, direct method has been used. In direct method which is widely applied for solving problems of soil-structure interaction, soil environment is considered as an infinite space. Considering that it is not possible to model a half-space in finite elements method, soil environment is considered with some elements. In this case, it is important to consider support conditions in soil boundaries for accurate answer of the problem. Degree of freedom of the lateral parts of soil is bound. In dynamic problems, one can use this method but these boundaries cause reflection of wave and create many errors by unreal intensification of the system response. This problem for small dimensions in soil environment causes more errors in the system’s response. In this regard, larger environment of soil should be modelled and more elements should be used which require high expense and long time for calculation. In order to solve this problem, idea of using the consistent boundaries was used which tried to make characteristics of soil environment close to characteristics of a half-space. These boundaries are applicable in thetime-frequencyzone which is used in nonlinear problems of time-zone boundaries such as energy absorbent boundaries. One of the most common and simplest boundaries is Lysmer viscous absorbent boundaries using two orthogonal dampers in each boundary node to absorb the contact waves in each direction (MirHosseini and Aref Poor, 2008) . Modelling In Abaqus Software For static and dynamic analysis of the storage tanks, it is necessary to model geometry of these tanks desirably. For modelling the tank and its surrounding soil, a mass of soil was considered as a cube with dimension of 50 m and the studied tank was put in centre of the soil environment and frequency analyses were repeated for different harmonic seismic loadings. Tank was made from concrete and its length and width were 10x10 m and depth was 6 m. Dimensions of the internal columns were considered for square tanks and equal to 0.5 x 0.5 m. Thickness of the floor, wall and ceiling of the tanks has been considered 0.3 m for all models. Three types of soil were classified as hard, middle and soft and considering time limitation of calculation, models were meshed such that time of calculation was not so long on the one hand and results were carefully presented on the other hand. For this purpose, coarse finite elements were selected on sides of the tank model and fine near the tank model. The elements used in all models were selected to be cubic models with 8 nodes and they are finite in middle of the model and infinite on sides of the model. In all analyses, segregation between tank and soil was considered and vertical and friction contact between tank and soil was introduced to the software by defining contact surfaces made from both materials. Soil behavioural law wasDrucker-Prager.

Solid element have been used in order to model wall of the tank and fluid and solid element has been used in

order to model soil. Combine element has been used in order to model absorbent boundaries. Damping coefficients in this element is obtained from the following relations. For elements vertical to surface of soil (compressive elements):

1p pC C A (56)

For elements tangential to soil surface (shear elements):

2s sC C A (57)

In the above relations,vp andvsare compressive wave velocity and shear wave velocity respectively,A is cross sectional surface, is density andC1 andC2 are damping coefficients. The important point in geometrical

modelling of soil is that dimensions of element are in suitable range. Based on research of Lysmer, the


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maximum length of element should be below 8 to 10 to make the answer close to reality where is

wavelength of the earthquake (23) . A section of finite elements geometry is shown in figure (8). As shown in the figure, density of meshing has increased in the sensitive regions of the model such as tank and support joint and the location in which pressure forces are applied (Naderi et al., 2011) .

Figure 8. Shows correct meshing.

In this study, damping ratio has been selected to be 5% considering soil material and by obtaining effective shear strain through primary dynamical analyses and specifications of concrete with body mass of 2400 kg/m

3 and elasticity modulus of 20 GPa and Poisson coefficient of 0.18. Soil is located over the bedrock

and record of earthquake is applied through lower boundaries to the system. Earthquake record is applied on the models as acceleration time history based on Elcentro accelerogram. Considering time-consuming analysis and massive models, intensive oscillation area of earthquake record has been used for calculation. For this purpose, Elcentro earthquake record with length of 2s from 0.4 to 2.4 s has been used (this zone has the highest energy of Tabas earthquake). In order to do transient dynamic analysis on the model, then, dynamic analysis has been done with 1.3 acceleration history. It is necessary to note that history of the desired earthquake in the lower part of model has been applied as displacement.

Figure 9. Longitudinal acceleration record of Elcentro earthquake

Figure 10. Modelling of tank and surrounding soil Figure 11. Tank modelling

Table1. characteristics of soil materials and resistance parameters of all kinds of soil used in the model

Soil type Kg/m

3 v E (dynamic) MPa E (static) MPa

Internal friction angle

CohesionC N/m

2

Dilation

angle

Hard soil (1) 2000 0.25 260 150 35 20000 10 Middle soil (2) 1900 0.3 90 50 30 10000 4 Soft soil (3) 1800 0.35 19 10 26 1000 0


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In this study, three types of soil have been used. Elastic parameters of all types of soil are described in table 1 where is density and v is Poisson Coefficient in soil.

Dimensions of elements are effective on the system response. Using large finite element grids causes high frequency components because their short wavelength cannot be muddled with node points with long distances. Correct interaction of soil and tank is obtained by meshing the adjacent points carefully. This requires careful and regular meshing of the structure and the fluid therein. Because cubic hexagonal elements have been used for meshing fluid elements, we should use regular cubic meshes for meshing crust of the tank. In this case, points of fluid and tank are superposed onto each other and coupling action is correctly performed. In figure 13, meshing method is shown. Three-dimensional models of finite elements are given in ABAQUS software for cubic tanks in figures 12 to 15.

Figure 12. meshing of tank and surrounding soil Figure 13. position of columns inside the tank

Figure 14. position of fluid inside the tank Figure 15. magnitude of A between soil and tanks in figure 14

Static Analyses Study As mentioned above, static analysis is regarded as the first step where force resulting from mass inertia of tank and liquid and soil causes structural response due to gravitational acceleration of the ground. It is very important to consider support conditions in soil boundaries for accurate determination of the problem’s response. In the studied model in static analysis, vertical and horizontal degrees of freedom of the lower part of soil and horizontal degree of freedom at the bottom of the soil have been bound. Diagrammatic comparison of the results is presented in different states. On this basis, the soils which have been specified in table (1) are used. Of the suitable outputs for studying behaviour of tanks and durability of tank are stress and deformation of wall and floor of the tank. It is necessary to note that fluid leakage in tanks is one of the damages which cause irreparable losses despite low destruction. On this basis and due to space shortage for representing all diagrams, the most critical state has been used in the tank for presenting the diagram. After static analysis of the tanks and presented models, some results as described in diagrams of figures 16 to 21 have been obtained. In these diagrams, attempt has been made to evaluate stress and deformation at specified points based on changes in height and floor of the tank to study changes of parameters such as soil type for stress and deformation of wall and floor of the tank. In the first step, results of research results and evaluated deformations and stress in wall of the soils types 1 to 3 as described in diagrams 16 and 17. As observed, changes in deformation of soft soil relative to height in wall of tanks are higher than that of hard soil. For stress values, diagrams of soft and hard soils are equal in form but they have different scales. In other words, when material of soil changes (reducing elasticity modulus, internal friction angle and cohesion of soil) or when soil is softened, stress and deformation in wall of tank considerably increase and the main reason is hardening of wall of tank relative to soil which causes wall of tank to tolerate more stress. It is necessary to note that the highest stress is applied on floor of the tank and in longitudinal direction. In floor of tank, the highest stress and deformation are observed in the third type soil, which are shown in figures 20 and 21. By comparing figures, we find that when material of soil changes (reducing elasticity modulus, internal friction angle and cohesion of soil),stress considerably increases.


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Figure 16. Diagram of defromation in wall of tank Figure 17. Diagram of stress in wall of tank

Figure 18. Diagram of deformation in wall of tank Figure 19. Diagram of stress in wall of tank

Figure 20. Diagram of deformation in floor of tank Figure 21. Diagram of stress in floor of tank

Dynamic Analyses The geometrical model prepared for dynamic analyses is the model prepared for static analysis with this difference that the previous model of combined damping elements has been added to ABAQUS program. These elements are located in all external boundaries which are related to earth. Dynamic properties of materials are different from static properties of materials. Considering focus of studies on soil and structure interaction,dynamic properties of materials were also considered elastic. Damping of all materials was considered 5%. In the next step, we study results of dynamical analysis in tanks as described in diagrams of figures 22 to 25 after discussing results of static analysis and the presented models. In these diagrams, attempt has been made to evaluate changes of deformation and stress in wall of type 1 to 3 soils as described in diagrams of 22 and 23. Values of stress and deformationshould be evaluated at specified points based on changes in height and floor of the tank to study changes of parameters such as soil type for stress and deformation of wall and floor of the tank. In dynamic analysis, soil pressure applied on wall of tank considerably changes when material of the soil changes. In these studies, when material of soil changes from soft to hard state (from clay to sand and gravel , elasticity modulus , increase of , decrease ofc, increase in density of

), the maximum pressure of soil on wall of tank is reduced. As shown in figures 22 and 23, the maximum stress increases in wall of tank when soil is softened (decrease of elasticity modulus, decrease of , increase ofc,

decrease in density of ) and tensile stress is maximised in the middle of wall of tank. The obtained results

show that changes in deformation of soft soil relative to height in wall of tanks are higher than that of hard soil. For stress values, diagrams of soft and hard soils are equal in form but they have different scales. In other

Hig

ht

(m)

U (m)

hard middle soft

Hig

ht

(m)

S (Mpa)

hard middle soft

Hig

ht

(m)

U (m)

soft middle hard

Hig

ht

(m)

S

hard middle soft

Hig

ht

(m)

U (m)

soft hard

Hig

ht

(m)

S

hard middle soft


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words, when material of soil changes (reducing elasticity modulus, decrease of internal friction angle and cohesion of soil) or when soil is softened, stress and deformation in wall of tank considerably increase and the main reason is hardening of wall of tank relative to soil which causes wall of tank to tolerate more stress. This is obtained from soil and structure interaction which has considerable effect on values of stress for the storage tanks in soft soil. In other words, one can say that when soil surrounding the storage tanks is softened, soil and structure interaction increases. Therefore, it is recommended not to build heavy structures in soft soils and if it is necessary to do so, dynamic soil and structure interaction should be considered based on nonlinear behaviour of soil. It is necessary to note that the changes in deformation of soft soil in floor of tanks and in longitudinal direction are higher than those of hard soil. The highest stress is applied on floor of the tank. In floor of tank, the highest stress is observed in the third type soil, which is shown in figure 25. By comparing figures, we find that when material of soil changes (reducing elasticity modulus, decrease of internal friction angle and increase in cohesion of soil), stress considerably increases in floor of the tank.

Figure 22. Diagram of defromation in wall of tank Figure 23. Diagram of stress in wall of tank



CONCLUSIONS

Stresses on the wall in tank are higher than those in floor of tank; therefore, it is more important to study probability of crack in wall than the floor. Using or not using the contact elements between soil and tank has considerable effect on dynamic pressure of the soil on external wall of the tank; however, this doesn’t cause considerable change in dynamic

Hig

ht

(m)

U (m)

hard middle soft

Hig

ht

(m)

S (Mpa)

hard middle soft

Hig

ht

(m)

U (m)

hard middle soft

Hig

ht

(m)

S (Mpa)

hard middle soft

Hig

ht

(m)

U (m)

hard middle soft

Hig

ht

(m)

S (Mpa)

hard middle soft


531

pressure of fluid on internal wall of the tank. Therefore, value of hydrodynamic forces in storage tanks can be regarded independent of contact modelling between soil and tank. Although use of contact elements between soil and external wall of the tank considerably reduces forces applied on external wall of the tank compared with the case where contact element has not been used while it mentions interaction between soil and structure more really. In static and dynamic analyses, when specifications of soil change, increase of elasticity modulus, increase of , decrease of c , increase in density of ), pressure applied on wall of tank considerably

increases. In the performed static analyses, the maximum tensile stress is found in the middle of wall of tank and the maximum compressive stress is found in 1 m at the bottom of wall of tank. Energy absorbing and transmitting boundaries prevent reflection of waves from boundaries to great extent. Priority of the transmitting boundaries is that static state modelling is easily possible in this method and general convergence of the problem is much rapider than the model with energy absorbent boundaries in all states. Soil type has considerable effect on soil and structure interaction. In other words, soft soil has more interaction with structure than hard soil has. This fact increases stresses of different points of structure in wall of the tank for soft soils. On the other hand, range of stress increases in wall of concrete tank also increases. When soil surrounding the tank is softened, horizontal displacements of wall of tank increases. On the other hand, when soil is more softened, the term during which motion of tank is damped also increases. Therefore, soil surrounding the tank is of special importance in deformation of the wall of tank and issue of soil and structure is very important. Considering the obtained results, it is recommended not to build heavy structures in soft soils and if it is necessary to do so, dynamic soil and structure interaction should be considered based on nonlinear behaviour of soil. Dynamic response of storage tanks in soil depends on specifications of the site and location of structure. Type of soil has considerable effect on behaviour of soil and structure interaction. Fine-grained or softer soils have higher interaction with structure. This interaction shows stresses of different parts of structure and soil and increase of displacements. When shear modulus of the surrounding soil decreases or the surrounding soil is softened, displacements and the term of damping motion of tank is elongated. Therefore, one should be careful that type of the surrounding soil is one of the important parameters in design and should be considered as important. With decrease of shear modulus of the surrounding soil, vertical stresses in different parts of tank also increase but because the tanks used in the practical projects have been designed uneconomically, stresses don’t increase to such extent that they crack wall of tank and consequently leakage of water from wall of the storage tank.

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