Günther SCHMID STRUCTURES - stuba.sk

Y. KOLEKOVÁ, M. PETRONIJEVIĆ, G. SCHMID

SPECIAL DYNAMIC SOIL-STRUCTURE ANALYSIS PROCEDURES DEMONSTATED FOR TWO TOWER-LIKE STRUCTURES

KEY WORDS

• Soil-structure interaction,• frequency domain, • spectral elements

ABSTRACT

Many problems in Earthquake Engineering require the modeling of the structure as a dynamic system including the sub-grade. A structural engineer is usually familiar with the Finite Element Method but has a problem modeling the sub-grade when its infinite extension and wave propagation are the essential features. If the dynamic equation of a soil-structure system is written in a frequency domain and the variables of the system are total displacements, then the governing equations are given as in statics. The dynamic stiffness matrix of the system is obtained as the sum of the stiffnesses of the structure and sub-grade sub-structures. To illustrate the influence of the sub-grade on the dynamic behavior of the structure, the frequency response of two tower-like structures excited by a seismic harmonic wave field is shown. The sub-grade is modeled as an elastic homogeneous half-space. The structure is modeled as a finite beam element with lumped masses.

Yvona KOLEKOVÁ email: [email protected] field: Soil-structure Interaction, Stability problems, SeismicityAddress: Department of Structural Mechanics , Faculty of Civil Engineering, Slovak University of Technology, Radlinského 13, 813 68, Bratislava, Slovakia

Mira PETRONIJEVIĆ email: [email protected] field: Structural Analysis, Dynamic of Structures, Trafic induced vibrationsAddress: Građevinski fakultet, Bul. Kralja Aleksandra 73, 11000 Beograd

Günther SCHMID email: [email protected] field: Dynamics of Structures, Earthquake structural response, Dynamics of Sub-soilAddress: Građevinski fakultet, Bul. Kralja Aleksandra 73, 11000 Beograd

2010/2 PAGES 1 – 8 RECEIVED 21. 9. 2009 ACCEPTED 20. 1. 2010

1. INTRODUCTION

Structures are founded on a sub-grade. In a dynamic analysis the total system may therefore be understood as consisting of two sub-structures: the structure and the sub-grade. Depending upon the relations of the mass and stiffness between the sub-structures, their interaction effects may be neglected. If the sub-grade is much stiffer than the structure, the sub-grade can be replaced by the kinematic boundary conditions at the structure - sub-grade interface. If the structure is much stiffer than the sub-grade, the structure can be modeled as a rigid body resting on a deformable sub-grade. In other cases the kinematic and inertial interaction between the two sub-structures should be considered in a dynamic analysis of the system. Such a situation will be discussed in this presentation.

We will assume that the external forces are zero at the structure and along the structure - sub-grade interface and that an excitation of the system will arise from a seismic ground motion, which we can assume as known at the location of the (future) structure - sub-grade interface. These displacements are part of the seismic free field (a wave field not disturbed by the structure). If the mass of the structure is big, the inertial interaction will contribute considerably to the structure - sub-grade interaction. In this presentation we will further assume that the system behaves linearly with respect to loading. One can observe in figure 1, where the system is indicated, that the structure has a finite extension, whereas the sub-grade may extend to infinity. Therefore, the vibration energy introduced into the system is trapped by the finite boundaries of the structure but can radiate away from the system

2010 SLOVAK UNIVERSITY OF TECHNOLOGY 1

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in the sub-grade. This dissipation of energy results in the damping of the motion of the structure even if no material damping is assumed. In the following the system is represented in the frequency domain through its dynamic stiffness matrix (see section 2). It is obtained according to the direct stiffness method as the sum of the dynamic stiffness matrices (upper index S for the structure) and

(upper index F for the foundation). The system displacements are the discrete total displacements, including the inertial and the kinematic parts of the interaction as well as the rigid body motion of the structure. The tilde indicates a complex valued function.

2. EQUATION OF THE MOTION IN A FREQUENCY DOMAIN

For the derivation of the equation of motion, we further assume that at the source of the earthquake we know the soil-motion uG (t) from which we obtain its frequency content through the Fourier Transform. The displacements due to the source excitation in the system can be represented by the discrete values . A corresponding nodal force value belongs to each term . These values are related through the relations

or . (1)

In equation (1) and are the dynamic flexibility and stiffness terms, respecctively of the equation of motion in the frequency domain, which can be written as a partitioned matrix equation:

or . (2)

The terms in equation (2) have the same meaning as in statics. But in a frequency domain they are complex, representing harmonic motions defined by teirs amplitudes and phases, such as, for example

(3a)

which represents the harmonic motion

with and

(3b)

Alternatively, the response in the time domain can be obtained by an inverse Fourier Transform from equation (3a).

3. STRUCTURE-SOIL SYSTEM

Since the dynamic behaviour is different for the structure and the sub-grade, one should model each “sub-structure” of the system by the “appropriate” method.

StructureThe most well-established procedure for the structure is the Finite Element Method. The linear equation of motion for a structure in a time domain results in

(4)

where are the mass, damping and stiffness matrices of the structure, respectively. and are the time histories of the nodal displacements and nodal forces. The Fourier transformation of equation (4) leads to the equation of motion in the frequency domain:

(5a)

or with the partitioning shown in equation (2)

(5b)

or with . (5c)

is the dynamic stiffness of the structure. Note that equation (5) can alternatively be obtained by assuming the load and response Fig. 1 Structure-soil model, left; system partitioning, right

structure

sub-grade

base rock(wave source)

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as the time harmonic functions and inserting them in equation (4). In equation (5) the dynamic stiffness matrix is constructed from element matrices which are based on shape functions that approximate the static deformation within the element due to the unit deformations of its degrees of freedom (DOF). For beam elements the shape functions are the solutions of the static equilibrium equation of the beam, which satisfies the kinematic beam end conditions. For static analyses the nodal values of the beam structure are therefore exact. For dynamic analyses of beam structures the shape functions of one structural beam element cannot represent the vibration forms excited at high frequencies with its shape functions. For higher frequencies the beam therefore has to be subdivided into many elements to improve the approximation. If the solution of the wave equation of the beam are chosen as shape functions, then these shape functions are frequency dependent sin, cos, sinh and cosh functions defined by the beam’s kinematic end conditions. They are exact for any frequency. The corresponding elements, with dynamic element stiffness , are called spectral elements. With spectral elements the dynamic stiffness matrix of the structure is obtained through the usual FE-procedure

as , with E as the number of spectral elements. For

further detail see [1, 2].

FoundationWe assume at the moment that we have numerical methods available, which enable us to obtain the dynamic stiffness of the foundation, including the sub-grade defined for the degrees of freedom at the source G and the soil-structure interface I (see figure 1, right). The corresponding dynamic stiffness matrix is

. (6)

SystemThe kinematic interface conditions , together with the principle of virtual displacements (expressing the fact that the sum of

and results in zero external forces), give the system equation

(7)

In equation (7) it is assumed that seismic motion is prescribed at the source. are the reactions due to the prescribed motion. The first two equations in (6) allow us to calculate the system’s structural nodal displacements :

. (8)

Usually the motion at the source is not available, instead, the free field motion at the site, which is defined as the earthquake motion of the sub-grade without the structure, is available. In this case we can use equation (7) without the structure to express the effective earthquake forces , which were transferred from the sub-grade to the soil-structure interface, through the free field motion. The corresponding equation of motion is:

. (9)

From the first equation in (9) we have at the soil-structure interface the relation

. (10)

Equations (8) and (10) give the dynamic system equation

with . (11)

Equation (11) allows us to calculate the structural vibration at the soil-structure interface and at the remaining nodal degrees of freedom of the structure. These deformation are the total displacements (rigid body motion plus elastic deformations) created by the free field motion at the site. The system displacements contain the kinematic interaction and the inertial interaction.

4. DYNAMIC STIFFNESS MATRIX OF THE FOUNDATION

For general cases the dynamic stiffness matrix of the soil-structure interface (and if the source is included in the discrete model), has to be obtained numerically. There are powerful methods available to model the dynamic stiffness matrix. We mention here the Boundary Element Method (BEM) [3], the Thin Layer Method [4] and an effective approach for inhomogeneous media [5]. Each method has its advantages and disadvantages, which will not be addressed here. In many practical applications, from the vibration of a block foundation through the analysis of soil-structure interaction in nuclear power plants, the simplification of a rigid soil-structure interface may be introduced. Dynamic stiffness functions (also

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called impedance functions) of rigid interfaces, where the sub-grade is either a homogenous half-space or a homogenous layer, are found in the literature [6]. In the following example the dynamic stiffness of the foundation was obtained using the boundary element method. Therefore, a short summary of the boundary element method within the context of the derivation of the dynamic stiffness matrix of a rigid foundation is given.

5. EXAMPLE

5.1 Beam on a rigid foundation blockThe derivation of the dynamic stiffness matrix of a tower-like structure based on a homogenous elastic half-space is illustrated. For illustrative purposes, the structure is modeled through one axial beam element with constant properties along the beam, which is erected on a rigid foundation block. The mass of the beam is lumped to its end cross sections.

Vertical excitationFor vertical loading (see figure 2, left), the dynamic stiffness matrix of the structure results in

. (12)

For horizontal loading we correspondingly obtain (see figure 2, right):

(13)

5.2 Sub-grade modelThe boundary element method is applied here to model the wave propagation in the sub-grade. The procedure is based on the solution of the wave equation of the infinite space due to a unit load at position xi (source point) in space. In the frequency domain this unit load is a time harmonic function with a frequency ω and unit amplitude. This unit load creates body waves, which propagate from the source into the infinite space (the fundamental solution). At any other position, these waves have xj (generic point) and response Fij. If point xj represents the boundary point of our problem, we can specify either the displacement or the traction at this position. A numerical solution can be obtained by subdividing the boundary of the problem – in our case the surface of the half-space - in the E boundary elements, where (most simply) the displacement and the traction are constant. These values could be allocated to the node in the centre of the element. Either the displacement or the traction in element j is known through the boundary conditions. In the case of a half-space its boundary is the soil’s free surface discretized by the boundary elements where the tractions (corresponding to the displacements ) are zero. By applying a ‘virtual’ unit load in each element node at each degree of freedom, we obtain N=3E equations (N=2E in 2-D problems) in order to calculate the remaining unknowns, which in this case are the displacements at the surface of the half-space. One can demonstrate that the Principle of Virtual Work leads to a linear equation system which relates the displacement and traction variables at the boundary

i, j = 1, 2, ...N (14a)

or in the matrix notation

. (14b)

Equation (14) is a generalization of the unit load principle that a structural engineer uses to calculate the influence coefficients of the flexibility matrix in statics. From equation (14) a relation between the nodal displacements and nodal forces can be established. This relation defines the dynamic stiffness of the sub-grade:

(15)with

(16)

where A is a diagonal matrix which, element by element, transforms the nodal traction in the nodal forces.In the frequency domain one can obtain, without any loss of accuracy Fig. 2 Models for vertical (left) and horizontal (right) excitation

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through condensation, a reduced model, which only contains the nodes on the soil-structure interface I. The original model is therefore condensed according to equation (17) to the dynamic stiffness of the flexible foundation-soil interface. It can be considered to be the stiffness of a “finite element” representing the sub-grade. If the sub-grade is given as in equation (6) we obtain

. (17)

If the lower part of the structure which will come in contact with the soil is very stiff, this part moves as a rigid body, which is represented as degrees of freedom through its rigid body: 3 DFOs in 2D problems and 6 DOFs in 3D problems. The same rigid body motion is imposed onto the soil-structure interface. The point of reference is usually chosen as the centre O of the soil-structure interface. The condensation and kinematic constraints are illustrated in Figure 3.The constraint equation between the flexible soil-structure interface and its rigid body motion is expressed in 2D problems as

(18)

The dynamic flexibility matrix of the rigid foundation is then

. (19)

5.3 Numerical resultsAs an example of the computation we will consider a chimney and an antenna, both of which were modeled through a 1 beam element. The constant mass distribution is modelled as lumped at the cross

sections of the beam’s ends. The beam is based on a rigid square foundation block and founded on a homogenous half space. The system’s properties are given in Table 1. A harmonic Rayleigh wave is assumed as excitation. Since the vertical and horizontal motions of the structure are decoupled, one can analyze the vertical and horizontal motions separately. We chose a free field with amplitude

and in the vertical and horizontal directions, respectively at interface I.For an interpretation of the results, it is helpful to calculate the

eigenfrequency , of the fixed

structures (no structure – sub-grade-interaction) with and

also compare the static stiffnesses of the structure, ,

, , with the static stiffness of the sub-grade,

, , , at the

structure - sub-grade interface. The corresponding numerical values

discretized soil surface

soil-stucture interface

half-space

Vi, vVi, h

i

“static“ condensation:flexible interface

Vi, v

Vi, h

i

kinematic constraint:rigid interface

Vo, v

Vo, h

0

ϕ0

Fig. 3 Models with different levels of condensation and with kinematic constraints

Tab. 1 System properties

Structure height [m]

E [N/m2]

A [m2]

I [m4]

ρ [kg/ m3]

Chimney 180 28x109 22 517 3500Antenna 62 21x109 0.075 2.19 7850

Foundation Half edge-length a [m]

Thickness [m]

ρ [kg/ m3]

Foundation block

7.5 1.5 2400

Sub-grade G [N/m2] ν ρ [kg/ m3]Soil 24x106 0.4 1900

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are given in Table 2. With BEM we obtain the static stiffness terms of the sub-grade from the stiffness functions for a0=0 (see figure7).

ChimneyTable 2 shows that in the case of the chimney, the static stiffnesses of the structure for any vertical and rotational motions are about 3 and 4 times larger than the corresponding stiffnesses of the foundation, whereas in a horizontal motion, the stiffness of the foundation is about two orders of magnitude larger than that of the structure. Without SSI (soil-structure interaction) the vertically base-excited 1DOF system approaches an infinite amplitude when the excitation frequency approaches its eigenfrequency. With SSI the system shows a first resonance at , where the mass of the structure essentially vibrates with only a little in-phase elastic elongation. The second resonance at corresponds to a vibration where the top and base of the chimney vibrate in an opposite phase and equal amplitude (figure 4, left). The damping is due to the large wave radiation in the half-space. For horizontal free field excitation the structure with a rigid sub-grade (no SSI) again shows the singularity for the 1DOF system at its eigenfrequency. With the SSI on the considered frequency range, only one of the resonances of the 3DOF-system shows up (see figure 4, right). The motion is essentially a rigid body rotation of the chimney (see figures 6a, b). The radiation damping is small

due to the small foundation motion in a horizontal direction and due to the small radiation-damping effect in rocking at low frequencies. Figures 6a and 6b show snapshots of the deformation when the harmonic free field motion has its maximal value for the excitation frequencies β1=0.2 and β2=1.1 .

AntennaIn the case of the antenna the static stiffness of the foundation is larger than the stiffness of the structure in all three directions; in a horizontal direction the foundation´s stiffness can even be considered as rigid. The result is a very small resonance amplitude in the vertical vibration with the total mass of the antenna and a strong resonance at about the same frequency as the fixed structure. For horizontal free field excitation SSI somewhat reduces the resonance frequency with respect to the case with no SSI and introduces a small amount of damping, since the radiation energy is small for the relatively stiff half-space. Figures 6c and 6d show snap shots of the deformation when the harmonic free field motion has its maximal value for the excitation frequencies β1 and β2.

Tab. 2 Eigenfrequencies (no soil-structure interaction) and static stiffnesses of the system

ωe, v [rad/s]

ωe, h [rad/s]

Kv [N/m] Kh [N/m] [Kr [Nm]

Chimney 22.22 1.04 3.42x109 7.45x106 241x109Antenna 117.97 17.81 2.50 x107 5.75 x105 2.23x109Sub-grade - - 0.98.2x109 1.31 x109 57.41x109

Fig. 4 Frequency response of the vertical motion at the top of the chimney, a), Frequency response of the horizontal motion at the top of the chimney, b)

Fig. 5 Frequency response of the vertical motion at the top of the antenna, a), Frequency response of the horizontal motion at the top of the antenna, b)

Fig. 6 Maximal horizontal deformation of the harmonic motion: a) chimney, β=0.2, b) chimney, β=1.1; c) antenna, β=0.2, d) antenna, β=1.1

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SUMMARY

If the dynamic interaction between a structure and sub-grade is formulated in total displacements (they include the kinematic and inertial interaction effects), then the system response in the frequency domain is obtained using the usual Finite Element procedure. The given formulation can be easily extended and applied to general structures and layered soil. In the two examples the soil-structure interaction effects are shown for vertical and horizontal excitations. It should be noted that all the material damping in the given

examples is set to zero, so the damping effects solely arise from the outgoing waves. The results present the importance of the relation of the dynamic stiffness of a structure to the dynamic stiffness of a foundation and sub-grade. They also show that different vibration forms create different damping effects through the outgoing waves produced.

AknowledgementThe authors acknowledge support by the Slovak Scientific Grant Agency under Contract No. 1/0652/09.

Fig. 7 Non-dimensional dynamic stiffness functions for horizontal, vertical and rocking motions of a rectangular rigid soil-structure interface. The foundation is not embedded; sub-grade: homogenous halfspace with ν=0.4. The coupling term between the swaying and rocking is not shown.

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REFERENCES

[1] Petronijevic, M., G. Schmid, Y. Kolekova: “Dynamic soil-structure interaction of frame structures with spectral elements – Part I”, GNP2008, Žabljak 3-7 Mar, 2008

[2] Penava, D., N. Bajrami,G. Schmid, M. Petronijevic, G. Aleksovski: “Dynamic soil-structure interaction of frame structures with spectral elements – Part II”, GNP2008, Žabljak 3-7 Mar., 2008

[3] Dominguez, J. “Boundary elements in dynamics”, Computational Mechanics Publications, Southampton, Boston, 1993

[4] Kausel, E, “Thin-layer method”, International Journal for Numerical Methods in Engineering, Vol. 37, 1994, pp. 927-941

[5] Kalinchuk, V.V., T.I. Belyankova, A. Tosecky: “The effective approach to the inhomogeneous media dynamics modeling“, 5th Structural Dynamics Conference EURODYN 2005, Vol. 2, pp. 1309-1314, 2005

[6] Sieffert, J.-G., F. and Cevaer, “Handbook of Impedance Functions”, Ouest Edition, Nantes, 1992

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Günther SCHMID STRUCTURES - stuba.sk