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Journal of American Science, 2012;8(3) http://www.americanscience.org

http://www.americanscience.org [email protected] 565

Mathematical Modeling of Tall Buildings and its Foundation under Randomly Fluctuating Wind and Earthquake Ground Motions

Aly El-Kafrawy

Production Engineering and Mechanical Design Dept., Faculty of Engineering,

Port-Said University, Port-Said, Egypt [email protected]

Abstract: In the present paper, a non-dimensional mathematical model for high tower buildings and its foundation under randomly fluctuating wind loads and earthquake ground motions excitations is developed as a nonlinear model to study the system more extensively. The system main equations could be derived using two different derivation methods and linearized in minimal symbolic forms; which facilitate a subsequent numerical simulation in order to investigate the vibration characteristics of whole system. The analysis enables designers to have more insight into the characteristics of high tower buildings of similar configuration but with different geometry and material. The complexity of wind loading with its variations in space and time has been considered. A comprehensive mathematical model of six degrees of freedom is presented and solved for free and forced vibrations. [Aly El-Kafrawy. Mathematical Modeling of Tall Buildings and its Foundation under Randomly Fluctuating Wind and Earthquake Ground Motions. Journal of American Science 2012; 8(3):565-588]. (ISSN: 1545-1003). http://www.americanscience.org. 76 Keywords tall building vibrations, modal analysis, foundation vibrations, power spectral density, random wind excitation, earthquake ground motions Introduction and literature review Large investments have recently been made for the construction of new medium- and high-rise buildings in the world. In many cases performance-based designs have been the preferred method for these buildings. A main consideration in performance-based seismic design is the estimation of the likely development of structural and nonstructural damage limit-states given a hazard level. For this type of buildings efficient modeling techniques are required able to compute the response at different performance states. Certain structures are less vulnerable against vibration impacts whereas certain others are more vulnerable. As we all know that vibration effects are now cannot be neglected, as our day to day life is affected by them. Study of vibration responses of structures has always been a principal concern for design engineers. Therefore, we do put an eye on the vibrations of buildings and its foundations. Uncontrolled vibration causes devastation. Occurrences of Tsunami, earthquake, collapse of structures are few such most common devastating effects of vibration. Thus the study of vibration responses in advance is of immense importance for sustainable and positive effects of vibrations for the well being of humans. Nowadays, the new and emerging concept of seismic structural design, the so-called performance-based design, requires careful consideration of all aspects involved in structural analysis. One of the most important aspects of structural analysis is Soil-Structure Interaction (SSI). Such interaction may alter

the dynamic characteristics of structures and consequently may be beneficial or detrimental to the performance of structures. Soil conditions at a given site may amplify the response of a structure on a soil deposit. Not taking into account these structural response amplifications may lead to an under-designed structure resulting in a premature collapse during an earthquake. Analytical methods of SSI concentrate mainly on single degree of freedom systems and analysis/design of long and important structures such as large bridges and nuclear power plants, and rarely on regular type buildings. Studies which include SSI effects will help to better predict the performance of structures during future ground motions. State of the art knowledge and analytical approaches require, that, the structure-foundation system to be represented by mathematical models that include the influence of the sub-foundation media. A research work of Panagiotou (2008) was conducted at University of California San Diego (UCSD) on the seismic design, experimental response, and computational modeling of medium- and high-rise reinforced concrete wall buildings. Kim (2008) presented an investigation of the effect of vertical ground motion on reinforced concrete structures studied through a combined analytical-experimental research approach. Krier (2009) analyzed several soil-structure interaction problems. Buildings on elastic foundations were studied and comparisons were made between analytical results and the solutions obtained from a Tera Dysac finite element analysis. Gouasmia et al. (2009) studied the



seismic response of an idealized small city composed of five equally spaced, five storey reinforced concrete buildings anchored in a soft soil layer overlaid by a rock half space. These results show response amplification of the buildings in the near field in accordance with the results observed in similar cases. Antonyuk, Timokhin (2007) outlined a mathematical model describing the vibrations of buildings and engineering structures with general-type passive shock-absorbers, rigid bodies, and ideal constraints. Auersch (2008) predicted a practice-oriented environmental building vibrations. A Green’s functions method for layered soils is used to build the dynamic stiffness matrix of the soil area that is covered by the foundation. A simple building model is proposed by adding a building mass to the dynamic stiffness of the soil. Belakroum et al. (2008) studied the numerical prediction of the aerodynamic behaviour of rectangular buildings. Simulations were made for rectangles of different side coefficients and different angles of attack. The finite element method is used to simulate fluid flow considered Newtonian and incompressible. Davoodi, et al. (2008) used the ambient vibration tests to rely on natural excitations, consequently, it was recommended to perform impulsive test for identifying the hidden dynamic characteristics of the building. Kuźniar and Waszczyszyn (2006) applied neural networks for computation of fundamental natural periods of buildings with load-bearing walls. The analysis is based on long-term tests performed on actual buildings. The identification problem was formulated as the relation between structural and soil basement parameters, and the fundamental period of building. Uzdin, et al. (2009) derived equations for the vibrations of a building on the foundations under consideration. Impossibility of use of traditional methods of the linear-spectral theory for analysis of their earthquake resistance is demonstrated. It is established that the systems under consideration do not possess a natural vibration period, and may have ambiguous solutions for forced vibrations. The influence of city traffic-induced vibration on Vilnius Arch-Cathedral Belfry was investigated (Kliukas et al. 2008). Two sources of dynamic excitation were studied. Conventional city traffic was considered to be a natural source of excitation while excitation imposed artificially by moving a heavily loaded truck was considered to be the source of increased risk excitation. Configuration of equipment on springs is simplified for numerical analysis. A simplified approach and associated equations of motion can be developed to evaluate the response of the equipment with vertical and horizontal forcing functions (Turner 2004). Gong (2010) developed a free

vibration analysis method for space mega frames of super tall buildings. The physical model of a mega frame was idealized as a three-dimensional assemblage of stiffened close-thin-walled tubes with continuously distributed mass and stiffness. Yang et al. (2008) analyzed the wave propagation problems caused by the underground moving trains by the 2.5-dimensional finite/infinite element approach. The near field of the half-space, including the tunnel and parts of the soil, was simulated by finite elements, and the far field extending to infinity by infinite elements. Ground-borne vibrations due to subway trains have sometimes reached the level that cannot be tolerated by residents living in adjacent buildings (Shyu et. al. 2002). Also, approaches for predicting vibrations caused by metro trains moving through the tunnel were developed (Gupta et al. 2007), e.g., a semi-analytical pipe-in-pipe model (Forrest and Hunt 2006a,b) and a coupled periodic finite-element–boundary-element model (Clouteau et al. 2005; Degrande et al. 2006b). Clearly, ground-borne vibrations have become an issue of great concern, which will continuously attract the attention of researchers and engineers worldwide. Many research projects on ground-borne vibrations due to subway trains were conducted by field measurement (Vadillo et al. 1996; Degrande et al. 2006a) and empirical or semiempirical prediction models (Kurzweil 1979; Trochides 1991; Melke 1998). These studies provide practical references for solving related problems. However, most of these studies were performed for a specific condition, thereby suffering from the lack of generality. On the other hand, concerning the techniques of simulation, most previous works have been based on the two-dimensional (2D) models (Balendra et al. 1991; Yun et al. 2000; Metrikine and Vrouwenvelder 2000). Prowell (2011) presented an experimental and numerical investigation into the seismic response of modern wind turbines simultaneously subjected to wind, earthquake, and operational excitation. Ulusoy (2011) described a certain class of system identification algorithms with particular emphasis on civil engineering applications. The algorithms originated from system realization theory enabled one to identify finite dimensional, linear, time-invariant models of systems in the state space representation from observed data. Wieser (2011) used OpenSees finite element framework to develop full three dimensional models of four steel moment frame buildings. The incremental dynamic analysis method is employed to evaluate the floor response of inelastic steel moment frame buildings subjected to all three components of a suite of 21 ground motions. Ghafari Oskoei (2011) dealt with the dynamic behavior of tall guyed masts under seismic loads. Zhong (2011)



utilized a ground motion acceleration time-history as an input to an analytic model of a structure and solved the structural response at each time step of the ground motion record. Weng (2010) proposed a forward substructuring approach, the eigenproperties of the partitioned substructures were assembled to recover the eigensolutions and eigensensitivities of the global structure, which were tuned to reproduce the experimental measurements through an optimization process. Sonmez (2010) developed semi- active controllers, which were based on real-time estimation of instantaneous (dominant) frequency and the evolutionary power spectral density by time-frequency analysis of either the excitation or the response of the structure. Time-frequency analyses were performed by either short-time Fourier transform or wavelet transform. Soudkhah (2010) examined the dynamic response of surface foundations on sandy soils under both forced and ground motion disturbance. Yao (2010) used the direct method for modeling the soil and a tall building together and studied energy transferring from soils to buildings during earthquakes, which is critical for the design of earthquake resistant structures and for upgrading existing structures. Ahearn (2010) studied the dynamic effects of wind-induced vibrations on high-mast structures and proposed several retrofits that increase the aerodynamic damping, thereby reducing vibrations. The ground vibration induced by earthquake ground motions is a complicated dynamic problem due to the involvement of a number of factors along the paths of wave propagation, including the load generation mechanism, the geometry and location of tunnel structures, the irregularity of soil layers, etc. Previously, many research projects on ground-borne vibrations due to earthquakes were conducted by field measurement and empirical or semi-empirical prediction models. These studies provide practical reference for solving related problems. However, most of these studies were performed for a specific condition, thereby suffering from the lack of generality. Assumptions 1. The high tower building-foundation equivalent

system moves only in the y*- z* plane. 2. The wind effect is identified as randomly

fluctuating wind loads in horizontal direction. 3. Uy(t), UZ(t) are random ground motions of

earthquake in horizontal and vertical directions y and z.

4. The high tower building and its foundation are assumed as rigid bodies.

5. The soil kind under the foundation is assumed as a sandy clay.

6. The angular velocities ),t( ),t( *1

*o and )t(*

2 are

very small (<<1). 7. The equivalent spring stiffness k k EHH ,, and

vk are linear.

8. The equivalent damping coefficients r r EHH ,, and

vr are linear.

9. The density of building 2 is taken as 0.1 that of

the foundation. 10.The air friction was not considered.

11.The place pressure factor pC can be replaced

through the average load factor of total building.

12.The spectral power density )(S11UU is

independent on the Cartesian Coordinates z, y. 13.The wind velocity distribution along the height of

the building is )H(U)H

z()z(U .

14.The cross spectral power density )(S21UU can

be represented through the coherence spectrum of

the wind velocity )t,z(U 1' and )t,z(U 2

' :

)](S . )(S[)(S)(22112121 UUUU

2

UU2

UU

Derivation of system equations using D’alembert’s principle The model of the problem to be considered is schematically shown in Fig. 1. This model describing the vibrations of high-tower building and its foundation with general-type equivalent passive springs and dampers, rigid bodies, and some ideal constraints like linear springs and dampers under the effect of randomly fluctuating wind loads and the excitation of earthquake ground motions. In setting up the equations of motion of the equivalent system in Fig. 1, it should be born in mind that the geometric, elastic, and kinetic relations of both high tower building and its foundation must be derived. Moreover the external excitation of wind loads should be prepared. Foundation differential equations of motion Figure 2 shows the free body diagram of foundation with its accompanied vibrating soil.



Geometric relations of tall building and its foundation For the linearization of derived equations, let and 21o , << 1. Geometric relations of building’s foundation are

)(..)()( *** tb50tztz ooC , )(..)()( *** tb50tztz ooD , )(..)()( *** tb50tztz 11E , )t(.b5.0)t(z)t(z *1

*1

*F ,

)())(cos.(.)()( **** tzt1c50tztz o2o2 , )()( ** tt o2 , )()( ** tyty oC ,

)(..)()(sin..)()( ***** tc50tytc50tyty 2o2o2 , )()( ** tyty oD , )()( ** tyty 1E , and )()( ** tyty 1F

Rearranging the previous geometric relations leads to the following form

)t(.b5.0)t(z)t(z *2

*2

*C , )t(.b5.0)t(z)t(z *

2*2

*D , )t(.b5.0)t(z)t(z *

1*1

*E , )t(.b5.0)t(z)t(z *

1*1

*F

)t(.c5.0)t(y)t(y *2

*2

*C , )t(.c5.0)t(y)t(y *

2*2

*D , )t(y)t(y *

1*E , and )t(y)t(y *

1*F } (1)

Elastic relations of building’s foundation Elastic relations of building’s foundation have the form

)]t(z)t(z.[r)]t(z)t(z.[kF *C

*EV

*C

*EVV1 , )]t(y)t(y.[r)]t(y)t(y.[kF *

C*EH

*C

*EHH1 ,

)]t(z)t(z.[r)]t(z)t(z.[kF *D

*FV

*D

*FVV2 , )]t(y)t(y.[r)]t(y)t(y.[kF *

D*FH

*D

*FHH2

,

)]t(U)t(y.[r)]t(U)t(y.[kF y*1EHy

*1EHEH

, )]t(U)t(z.[r)]t(U)t(z.[kF z*1EVz

*1EVEV

} (2)

)t(.r)t(.kT *1EK

*1EKEK

Kinetic relations of building’s foundation Applying Newton’s second law for the forces in z- and y-directions and the moments about s1 results in

EVV2V1*11z FFF)t(z.mF ,

EHH2H1*11y FFF)t(y.mF

)t(cos.b5.0.F)t(.JM *1V1

*111s

)t(T)t(cos.b5.0.F EK*1V2

(3) )t(Tb5.0).FF( EKV1V2

Differential equations of motion of high tower building Figure 3 shows the free body diagram of high tower building with its forces and moments affecting on it. Aeroelastic relations of wind excitation

Nowadays, the study of the behavior of a structure subjected to hydro or aerodynamic loadings forms an integral part of tasks allocated to engineers. The effect of wind must be taken into consideration during the design phase of tall buildings. The mechanism of wind loads acting on a building is very complex. Substantial works have dealt with this problem. In civil engineering and construction of tall buildings, the assessment of wind loads is required to check the resistance of components of the construction and coating. In recent years, the methods proposed by scientists in this field are constantly being updated. The institutions of global standardization are thus forced each time to review the standards that are in force. Under the effect of wind, a building oscillates according to both directions parallel and perpendicular to the flow and in a torsional mode. Notwithstanding its enormous

fascination, wind loading is in fact a parasitic effect, and mostly an obstacle in the way of designing structures for their primary intended use. Without wind, structures – particularly large ones – would probably be a lot easier to design and cheaper.

Dynamic wind pressures acting on buildings are complicated functions of both time and space. The wind load per unit area has the form

)t,z(q.C)t,z(W p and )t,z( U2

1)t,z(q 2

)]t,z(U)t,z(U).z(U2)z(U[ .2

.C

)]t,z(U)z(U.[2

.C)t,z(U.2

.C)t,z(W

2''2p

2'p

2p



Fig. 1 Equivalent system of tall building and its foundation

Fig. 2 Free body diagram of foundation with its

accompanied vibrated soil

),()(),().(.)(..),( '' tzW zWtzUzU.C zU 2

CtzW p2

p

The total turbulent wind force in y*-direction as a function of time is

c

0

c

0p dz tzUzU.C dz tzWtW ),().(.),()( '' (4)

The total turbulent wind moment as a function of time is

c

0

'*2

*2W dz )t,z(W))].t(sin

2

b)t(cos

2

c(z[)t(M

c

0

' dz )t,z(W).2

cz(

c

0

'p dz )t,z(U).z(U..C ).

2

cz( (5)

Fig. 3 Free body diagram of the high tower building Elastic relations of high tower building

Elastic relations of high tower building have the form

)]t(z)t(z.[r)]t(z)t(z.[kF *E

*CV

*E

*CVV1 , )]t(y)t(y.[r)]t(y)t(y.[kF *

E*CH

*E

*CHH1

)]t(z)t(z.[r)]t(z)t(z.[kF *F

*DV

*F

*DVV2 , )]t(y)t(y.[r)]t(y)t(y.[kF *

F*DH

*F

*DHH2 (6)

Kinetic relations of high tower building Applying Newton’s second law for the forces in z and y-directions and also the moments about s2 results in

V2V1*22z FF)t(z.mF , )t(WFF)t(y.mF H2H1

*22y

)](cos.)(sin..[)(. *** t2

bt

2

cFtJM 22V1222s )](sin.

2)(cos.

2.[ *

2*21 t

bt

cF H

)]t(cos.2

b)t(sin.

2

c.[F *

2*2V2 )t(M)]t(sin.

2

b)t(cos.

2

c.[F W

*2

*2H2 } (7)

The previous equation can be linearized in the following form

]2

c)t(.

2

b.[F)]t(.

2

c

2

b.[FM *

2H1*2V12s )t(M]

2

c)t(.

2

b.[F)]t(.

2

c

2

b.[F W

*2H2

*2V2

Deriving the system’s differential equations of motion



Application of the geometric relations of the foundation Substitute from Eqs. 1 in Eqs. 2 of the elastic relations of foundation free body diagram

)](.5.0)()(.5.0)(.[)](.5.0)()(.5.0)(.[ *2

*2

*1

*1

*2

*2

*1

*11 tbtztbtzrtbtztbtzkF VVV

)]t(c5.0)t(y)t(y.[r)]t(c5.0)t(y)t(y.[kF *2

*2

*1H

*2

*2

*1HH1

)](.5.0)()(.5.0)(.[)](.5.0)()(.5.0)(.[ *2

*2

*1

*1

*2

*2

*1

*12 tbtztbtzrtbtztbtzkF VVV } (8)

)](5.0)()(.[)](5.0)()(.[ *2

*2

*1

*2

*2

*12 tctytyrtctytykF HHH

Application of the elastic relations of the foundation

Substitute from Eqs. 2 of foundation’s elastic relations in Eqs. 3 of its kinetic relations results in

)()(.)(.2)().2()(2)().2()(. *2

*1

*2

*1

*11 tUrtUktzrtzrrtzktzkktzm zEVzEVVVEVVVEV

)().2()(.)(.2)().2()(. *1

*2

*2

*1

*11 tyrrtcktyktykktym HEHHHHEH

} (9)

)(.)(.)(.)(.2 *2

*2 tUrtUktcrtyr yEHyEHHH

)(..5.0)(]..5.0[)(..5.0)(]..5.0[)(. *2

2*1

2*2

2*1

2*11 trbtrbrtkbtkbktJ VVEKVVEK

Application of the geometric relations of the building Substitute from Eqs. 1 of geometric relations in Eqs. 6 of elastic relations of the building

)](.5.0)()(.5.0)(.[)](.5.0)()(.5.0)(.[ *1

*1

*2

*2

*1

*1

*2


)](5.0)()(.[)](5.0)()(.[ *2

*2

*1

*2

*2

*11 tctytyrtctytykF HHH } (10)

)](.5.0)()(.5.0)(.[)](.5.0)()(.5.0)(.[ *1

*1

*2

*2

*1

*1

*2


)]t(c5.0)t(y)t(y.[r)]t(c5.0)t(y)t(y.[kF *2

*2

*1H

*2

*2

*1HH2

Application of the elastic relations of the building Substitute Eqs. 10 of building’s elastic relations in Eqs. 7 of its kinetic relations lead to the following differential equations

)(.)(.)(.)(.)(. ***** tzr2tzr2tzk2tzk2tzm 2V1V2V1V22

)()](.)(.2)(.2)(.)(.2)(.2)(. *2

*2

*1

*2

*2

*1

*22 tWtcrtyrtyrtcktyktyktym HHHHHH } (11)

)()..5.0.5.0()(.)(..5.0)(.)(. *2

22*2

*1

2*1

*22 tkbkctycktkbtycktJ VHHVH

)t(M)t().r.b5.0r.c5.0()t(y.cr)t(.r.b5.0)t(y.cr W*2V

2H

2*2H

*1V

2*1H

Arranging the differential equations of motion The differential equations of motion of both tall building and its foundation can be summarized in the form

)()(.)(2)().2()(.2)().2()(. *2

*1

*2

*1

*11 tUrtUktzktzkktzrtzrrtzm zEVzEVVVEVVVEV

)(.)(.2)().2()(.)(.2)().2()(. *2

*2

*1

*2

*2

*1

*11 tcktyktykktcrtyrtyrrtym HHHEHHHHEH

)t(U.r)t(Uk yEHyEH

0)(..5.0)()..5.0()(..5.0)(]..5.0[)(. *2

2*1

2*2

2*1

2*11 tkbtkbktrbtrbrtJ VVEKVVEK

0tzk2tzk2tzr2tzr2tzm 2V1V2V1V22 )(.)(.)(.)(.)(. *****

)()(.)(.2)(.2)(.)(.2)(.2)(. *2

*2

*1

*2

*2

*1

*22 tWtcktyktyktcrtyrtyrtym HHHHHH

)()..5.0.5.0()(.)(..5.0)(.)(. *2

22*2

*1

2*1

*22 trbrctycrtrbtycrtJ VHHVH

)()()..5.0.5.0()(.)(..5.0)(. *2

22*2

*1

2*1 tMtkbkctycktkbtyck WVHHVH



)(

)(

)(

)(

)(

)(

)5.05.0(05.00

20020

002002

5.0005.000

20020

002002

)(

)(

)(

)(

)(

)(

00000

00000

00000

00000

00000

00000

*2

*2

*2

*1

*1

*1

222

22

*2

*2

*2

*1

*1

*1

2

2

2

1

1

1

t

ty

tz

t

ty

tz

rbrccrrbcr

crrr

rr

rbrbr

crrrr

rrr

t

ty

tz

t

ty

tz

J

m

m

J

m

m

VHHVH

HHH

VV

VVEK

HHHEH

VVEV

+

)(

)(

)(

)(

)(

)(

)..(.

..

*

*

*

*

*

*

t

ty

tz

t

ty

tz

kb50kc50ck0kb50ck0

ckk200k20

00k200k2

kb5000kb50k00

ckk200k2k0

00k200k2k

2

2

2

1

1

1

V2

H2

HV2

H

HHH

VV

V2

V2

EK

HHHEH

VVEV

)(

)(

)(

)(

)(

)(

tM

tW

t

t

t

t

100000

010000

000000

000000

00rk00

0000rk

W

EHEH

EVEV

(12)

Derivation of system equations using Lagrange’s method

The previous obtained system differential equations 12 of motion can be verified using another derivation method, like Lagrange’s method using the following Lagrangian Differential Equation

i K

iiK

KKK qFQ

qq

L

q

L

dt

d

, L = E – U ,

n

2nnr

2

1 (13)

QK : General forces, Fi : External forces, and i : Velocity

Lagrangian function (a) Kinetic energy of the total equivalent system

)t(J2

1)t(ym

2

1)t(zm

2

1)t(J

2

1)t(ym

2

1)t(zm

2

1E

222222 *22

*22

*22

*11

*11

*11 (14)

(b) Elastic potential energy of the total equivalent system

2*C

*EV

*1EK

2y

*1EH

2z

*1EV )]t(z)t(z[k

2

1)t(k

2

1)]t(U)t(y[k

2

1)]t(U)t(z[k

2

1U

2

2*D

*FH

2*D

*FV

2*C

*EH )]t(y)t(y[k

2

1)]t(z)t(z[k

2

1)]t(y)t(y[k

2

1 (15)

(c) Lagrangian function Using Eqs. 1 and 14-15 to obtain the following Lagrangian function

)t(J2

1)t(ym

2

1)t(zm

2

1)t(J

2

1)t(ym

2

1)t(zm

2

1L

222222 *22

*22

*22

*11

*11

*11

)t(k2

1)]t(U)t(y[k

2

1)]t(U)t(z[k

2

1 2*1EK

2y

*1EH

2z

*1EV

2*2

*2

*1

2*2

*2

*1

*1 )](.5.0)()([

2

1)](.5.0)()(.5.0)([

2

1tctytyktbtztbtzk HV



2*2

*2

*1

2*2

*2

*1

*1 )](.5.0)()([

2

1)](.5.0)()(.5.0)([

2

1tctytyktbtztbtzk HV (16)

Rayleigh’s dissipation function

The Rayleigh’s dissipation function can be derived as

)t(r2

1)]t(U)t(y[r

2

1)]t(U)t(z[r

2

1r

2

1 2*1EK

2y

*1EH

2z

*1EV

61n

266

2*2

*2

*1H

2*2

*2

*1

*1V )]t(.c5.0)t(y)t(y[r

2

1)]t(.b5.0)t(z)t(.b5.0)t(z[r

2

1

2*2

*2

*1

2*2

*2

*1

*1 )](.5.0)()([

2

1)](.5.0)()(.5.0)([

2

1tctytyrtbtztbtzr HV (17)

General external forces

i K

ii

i K

iiK

q

rF

qFQ

, Where **

2221W21 and , y , MF , WF

Deriving the differential equations of motion

(a) Case of )(* tzq 11

)(.)(

**

tzmtz

L

dt

d11

1

, )(.)(.2)()2(

)(*2

*1*

1

tUktzktzkktz

LzEVVVEV

)(.)(.)().()(

***

tUrtzr2tzr2rtz

zEV2V1VEV1

, and 0Q *

1z

Substitute from the equations of case (a) in Eq. 13, the first differential equation of motion can be obtained

)()(.)(2)().2()(.2)().2()(. *2

*1

*2

*1

*11 tUrtUktzktzkktzrtzrrtzm zEVzEVVVEVVVEV

(18)

(b) Case of )(* tyq 12

)t(y.m)t(y

L

dt

d *11*

1

, )t(Uk)t(.ck)t(y.k2)t(y).kk2(

)t(y

LyEH

*2H

*2H

*1EHH*

1

)t(Ur)t(.cr)t(y.r2)t(y).rr2()t(y

yEH*2H

*2H

*1EHH*

1

, and 0Q *

1y

Substitute from the equations of case (b) in Eq. 13, the second differential equation of motion can be obtained

)(.)(.2)().2()(.)(.2)().2()(. *2

*2

*1

*2

*2

*1

*11 tcktyktykktcrtyrtyrrtym HHHEHHHHEH

)t(U.r)t(Uk yEHyEH (19)

(c) Case of )(* tq 13

)t(.J)t(

L

dt

d *11*

1

, )t(.k

2

b)t().k

2

bk(

)t(

L *2V

2*1V

2

EK*1

)t(.r2

b)t().r

2

br(

)t(

*2V

2*1V

2

EK*1

, and 0Q *

1

Substitute from the equations of case (c) in Eq. 13, the third differential equation of motion can be obtained

0)(..5.0)()..5.0()(..5.0)(]..5.0[)(. *2

2*1

2*2

2*1

2*11 tkbtkbktrbtrbrtJ VVEKVVEK (20)

(d) Case of )t(zq *24

)t(z.m)t(z

L

dt

d *22*

2

, )t(z.k2)t(z.k2

)t(z

L *2V

*1V*

2



)t(z.r2)t(z.r2z

*2V

*1V*

2

, and 0Q *

2z

Similarly, the fourth differential equation of motion can be obtained

0)(.2)(.2)(.2)(.2)(. *2

*1

*2

*1

*22 tzktzktzrtzrtzm VVVV

(21)

(e) Case of )t(yq *25

)t(y.m)t(y

L

dt

d *22*

2

, )t(.ck)t(y.k2)t(y.k2

)t(y

L *2H

*2H

*1H*

2

)t(.cr)t(y.r2)t(y.r2)t(y

*2H

*2H

*1H*

2

, and )t(WQ *

2y

Similarly, the fifth differential equation of motion can be obtained

)(.)(.2)(.2.)(.2)(.2)(. *2

*2

*1

*2

*2

*1

*22 tWcktyktykcrtyrtyrtym HHHHHH (22)

(f) Case of *26q

)t(.J)t(

L

dt

d *22*

2

, )t().k

2

bk

2

c()t(y.ck)t(.k

2

b)t(y.ck

)t(

L *2V

2

H

2*2H

*1V

2*1H*

2

)t().r2

cr

2

b()t(y.cr)t(.r

2

b)t(y.cr

)t(

*2H

2

V

2*2H

*1V

2*1H*

1

, and )t(MQ W*

2

Similarly, the sixth differential equation of motion can be obtained

)t().r.b5.0r.c5.0()t(y.cr)t(.r.b5.0)t(y.cr)t(.J *2V

2H

2*2H

*1V

2*1H

*22

)t(M)t().k.b5.0k.c5.0()t(y.ck)t(.k.b5.0)t(y.ck W*2V

2H

2*2H

*1V

2*1H (23)

Equations 18-23 can be written in the following matrix form

*

*

*

*

*

*

*

*

*

*

*

*

)(

)(

2

2

2

1

1

1

2

V2

H2

2

H

2

V2

2

H

2

H

2

H

2

H

2

V

2

V

1

V2

1

V2

EK

1

H

1

H

1

HEH

1

V

1

VEV

2

2

2

1

1

1

y

z

y

z

J2

rbrc

J

cr0

J2

rb

J

cr0

m

cr

m

r200

m

r20

00m

r200

m

r2J2

rb00

J2

rbr200

m

cr

m

r200

m

r2r0

00m

r200

m

r2r

y

z

y

z

100000

010000

001000

000100

000010

000001

+

*

*

*

*

*

*

2

2

2

1

1

1

2

V2

H2

2

H

2

V2

2

H

2

H

2

H

2

H

2

V

2

V

1

V2

1

V2

EK

1

H

1

H

1

HEH

1

V

1

VEV

y

z

y

z

J2

kbkc

J

ck0

J2

kb

J

ck0

m

ck

m

k200

m

k20

00m

k200

m

k2J2

kb00

J2

kbk200

m

ck

m

k200

m

k2k0

00m

k200

m

k2k



)(

)(

)(

)(

)(

)(

tM

tW

t

t

t

t

J

100000

0m

10000

000000

000000

00m

r

m

k00

0000m

r

m

k

W

2

2

1

EH

1

EH

1

EV

1

EV

(24)

Normalization of the system differential equations of motion The system differential equations of motion of the high tower building with its foundation can be presented in a dimensionless form using the following quantities

ooooo

ooo

tt

tt

ty

tyty

z

tztz

tt

y

tyty

z

tztz

***2

2

*2

2

*2

2

*1

1

*1

1

*1

1

)(,)(

)(,)(

)(,)(

)(,)(

)(

,)(

)(,)(

)(,)(

)(

where rad 1 and cm 1 cm, 1 y z ooooo .

Applying the time normalization through the following transformations to , dtd o , where rad/s 1 o

and

d

dz

dt

dzo ,

2

22o2

2

d

zd

dt

zd

, tt 1

o

11

Therefore the differential equations of motion will be written in the following dimensionless form

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

'

'

'

'

'

'

"

"

"

"

"

"

2

2

2

1

1

1

o2

V2

H2

o2

H

o2

V2

o2

H

o2

H

o2

H

o2

H

o2

V

o2

V

o1

V2

o1

V2

EK

o1

H

o1

H

o1

HEH

o1

V

o1

VEV

2

2

2

1

1

1

y

z

y

z

J2

rbrc

J

cr0

J2

rb

J

cr0

m

cr

m

r200

m

r20

00m

r200

m

r2J2

rb00

J2

rbr200

m

cr

m

r200

m

r2r0

00m

r200

m

r2r

y

z

y

z

100000

010000

001000

000100

000010

000001



)(

)(

)(

)(

)(

)(

2

2

2

1

1

1

2o2

V2

H2

o2o2

oH2o2

V2

o2o2

oH

o2o2

oH2o2

H2o2

H

2o2

V2o2

V

2o1

V2

2o1

V2

EK

o2o1

oH2o1

H2o1

HEH

2o1

V2o1

VEV

y

z

y

z

J2

kbkc

J

yck0

J2

kb

J

yck0

ym

ck

m

k200

m

k20

00m

k200

m

k2J2

kb00

J2

kbk200

ym

ck

m

k200

m

k2k0

00m

k200

m

k2k

)(

)(

)(

)(

)(

)(

W

o2o2

o2o2

oo1

EH

o2o1

EH

oo1

EV

o2o1

EV

M

W

J

100000

0ym

10000

000000

000000

00ym

r

ym

k00

0000zm

r

zm

k

(25)

Analytical solutions using the general modal analysis method Eigen value problem Homogeneous differential equations without damping

0 (t)xK (t)xM****

(26)

Assume that the exponential solutions of Eqs. 26 have the form tie x )t(*x

(27)

Applying the solutions of Eqs. 27 in Eqs. 26 leads to the general eigen value problem

0 e x )K M ( ti**2

or 0 x )I A( 2

Where the matrix A form thehas

J2

kbkc

J

ck0

J2

kb

J

ck0

m

ck

m

k200

m

k20

00m

k200

m

k2J2

kb00

J2

kbk200

m

ck

m

k200

m

k2k0

00m

k200

m

k2k

K.M A

2

V2

H2

2

H

2

V2

2

H

2

H

2

H

2

H

2

V

2

V

1

V2

1

V2

EK

1

H

1

H

1

HEH

1

V

1

VEV

1

**

(28)

Using equation 3 one can obtain 12 eigen values ),,,,,( 654321 and 6 eigen vectors

),,,,,( 654321 x x x x x x

.



Modal matrix The modal matrix has the form

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

654321 x x x x x x

,,,,, ,

665646362616

655545352515

645444342414

635343332313

625242322212

615141312111

T6

T5

T4

T3

T2

T1

T

x

x

x

x

x

x

Decoupling of the system differential equations The transformation of coordinates can be carried out using the equation

q . *x

and the system of the vibration differential equations will has the form

UB q K q R q M**T*T*T*T

Where I M*T

, ]D [2 diag. R*T , ][ diag. K 2*T , and Q] [ diag. (t)F 2*T

When the damping forces of the equivalent system are smaller than its elastic restoring forces, then the coupled terms of the transformed damping matrix can be neglected without any great error. The decoupled differential equations of the system will have the form

Q ][ diag. q ][ diag. q] [2D diag. q I 22

(t)Q (t)q (t)q 2D (t)q n2nn

2nnnnn , n = 1, 2, …, 6

tq

tq

tq

tq

tq

tq

2D00000

02D0000

002D000

0002D00

00002D0

000002D

tq

tq

tq

tq

tq

tq

100000

010000

001000

000100

000010

000001

6

5

4

3

2

1

66

55

44

33

22

11

6

5

4

3

2

1

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

tQ

tQ

tQ

tQ

tQ

tQ

00000

00000

00000

00000

00000

00000

tq

tq

tq

tq

tq

tq

00000

00000

00000

00000

00000

00000

6

5

4

3

2

1

26

25

24

23

22

2

6

5

4

3

2

1

26

25

24

23

22

211

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

(29)

The general external excitations of the system are

)t(UB .][ diag. (t)Q**T12

)(

)(

)(

)(

)(

)(

tM

tW

t

t

t

t

100000

010000

000000

000000

00rk00

0000rk

1/00000

01/0000

001/000

0001/00

00001/0

000001/

W

EHEH

EVEV

665646362616

655545352515

645444342414

635343332313

625242322212

615141312111

26

25

24

23

22

21

)t(F . . ]1

[ diag. (t)Q*T

2

and )t(F . .

1 (t)Q n

Tn2

n

n

, n = 1, 2, …, 6



)(

)(

)(

)(

)(

)(

))))))

))))))

))))))

))))))

))))))

))))))

tM

tW

t

t

t

t

100000

010000

000000

000000

00rk00

0000rk

(1/(1/(1/(1/(1/(1/

(1/(1/(1/(1/(1/(1/

(1/(1/(1/(1/(1/(1/

(1/(1/(1/(1/(1/(1/

(1/(1/(1/(1/(1/(1/

(1/(1/(1/(1/(1/(1/

W

EHEH

EVEV

662656

2646

2636

2626

2616

26

652555

2545

2535

2525

2515

25

642454

2444

2434

2424

2414

24

632353

2343

2333

2323

2313

23

622252

2242

2232

2222

2212

22

612151

2141

2131

2121

2111

21

)(

)(

)(

)(

)(

)(

.).).).).).)

.).).).).).)

.).).).).).)

.).).).).).)

.).).).).).)

.).).).).).)

tM

tW

t

t

t

t

1(1/1(1/r(1/k(1/r(1/k(1/

1(1/1(1/r(1/k(1/r(1/k(1/

1(1/1(1/r(1/k(1/r(1/k(1/

1(1/1(1/r(1/k(1/r(1/k(1/

1(1/1(1/r(1/k(1/r(1/k(1/

1(1/1(1/r(1/k(1/r(1/k(1/

(t)Q

W662656

26EH26

26EH26

26EV16

26EV16

26

652555

25EH25

25EH25

25EV15

25EV15

25

642454

24EH24

24EH24

24EV14

24EV14

24

632353

23EH23

23EH23

23EV13

23EV13

23

622252

22EH22

22EH22

22EV12

22EV12

22

612151

21EH21

21EH21

21EV11

21EV11

21

n

(30)

)]t(MB )t(WB )t(B )t(B )t(B )t([B 1

(t)Q Wn6n5n4n3n2n12n

n

Applying the total turbulent wind forces W(t) in y-direction and the total wind moments MW(t) on the previous equations.

dA )t,z(U).z(U..CB )t(B )t(B )t(B )t([B 1

(t)QA

'pn5n4n3n2n12

nn

]Ad )t,z(U).z(U..C ).2

cz(B

A'

pn6 (31)

The decoupled system of differential equations can be presented in the following form

)t(f)t(f)t(f)t(f)t(f)t(f)t(qk)t(qr)t(qm 661551441331221111111111


)t(f)t(f)t(f)t(f)t(f)t(f)t(qk)t(qr)t(qm 663553443333223113333333 } (32)




)t(f)m

(])t(r )t([k)m

(])t(r )t([k)m

()t(q)m

k()t(q)

m

r()t(q 3

i

i3EHEH

i

i2EVEV

i

i1i

i

ii

i

ii

)t(f)m

()t(f)m

()t(f)m

( 6i

i65

i

i54

i

i4 (33)

From the previous equations, one can obtain the following imaginary transformation functions

)])([)])([)()()(

ii

2

i

EVEVi

1i

ii

2

i

EVEV2ii

1i

ii2i

2

EVEVi

1i

1

D2( j1

]r j[kk

D2( j1

]r j[k1

.m

D2 j

]r j[km

H

)])([

)(

ii

2

i

EHEHi

2i

2

D2( j1

]r j[kk

H

,

)])([

)(

ii

2

i

i

3i

3

D2( j1

1 .k

H

,

)])([

)(

ii

2

i

i

4i

4

D2( j1

1 .k

H

} (34)



)])([

)(

ii

2

i

i

5i

5

D2( j1

1 .k

H

,

)])([

)(

ii

2

i

i

6i

6

D2( j1

1 .k

H

, where iii

i D2m

r and 2

ii

i

m

k

A dynamical system with known properties responds to a dynamical loading in a known manner, provided the time-description of the loading is available a priori. Such description is however not possible in case of the excitations due to earthquake ground motions or fluctuating wind loads. Therefore, the safety of a structural system has to be ensured by stochastic modeling of these motions for perceived seismic hazard at the site of the system and by predicting the structural response in probabilistic sense with the help of well-known concepts of random vibration theory. This theory estimates the statistical variations in the peak structural response due to possible variations in the time-description of the excitation (there may be several `different looking' time-histories corresponding to a given characterization of the excitation). The classical random vibration theory makes use of the frequency

distribution of input energy as obtained from the Fourier Transform of the excitation. However, since Fourier Transform gives only an `average' energy distribution in an excitation with time-evolving structure, this theory is insufficient for those cases where the non-stationary processes cannot be modeled as stationary or quasi-stationary. As a natural extension to double Fourier Transform for such processes is not considered to be practical, a large amount of effort has been devoted to modeling a (slowly-varying) non-stationary process through modulating function-based power spectral density function (PSDF). The auto power spectral density function of the response as a result of random wind and earthquake excitations with respect to general coordinates has the form

6

1s

ffs*r

6

1r

qq )( S)(H )(H )(Ssrii

)( S)(H )(H......)( S)(H )(H)( S)(H )(H )(S6ii f6

*12

*11

*1qq

......)( S)(H )(H......)( S)(H )(H)( S)(H )(H6f6

*22

*21

*2

)( S)(H )(H......)( S)(H )(H)( S)(H )(H6666 ff6

*6f2

*6f1

*6 (35)

The cross correlation function of excitation functions with respect to general coordinates is

2/T

2/T

6j62j21j16i62i21i1T

dt )]t(f...)t(f)t(f)][t(f...)t(f)t(f[T

1lim

)(R)(R)(R...)(R)(R 2212612111 ffj2i2ffj1i2ffj6i1ffj2i1ffj1i1

)(R...)(R)(R...)(R... 66261662 ffj6i6ffj2i6ffj1i6ffj6i2 (36)

The cross and auto power spectral density functions of excitation functions are

N

1l

ffljki

N

1k

QQ lkjiS S )()( , )()( lkl

*kff ).S().HA(HA S

lk, )()( ).S().HA(HA S l

*1ff l1

(37)

)t().2(u)t().1(u)t(f1 ,

)().2(u i)().1(u)(f1 ]i/)()].[2(u i)1(u[)()].2(u i)1(u[

The excitation functions can be represented as

6

1n

nnii tf tQ )()( ,

6

1i

iirr tf tQ )()( ,

6

1j

jjss tf tQ )()( ,

6

1j

jjsiir

6

1i

sr tf . tf tQ tQ )()()()( (38)

2/T

2/T

jiT

QQ dt )t(Q )t(QT

1lim )(R

ji



)()(

)()(

)()()()(

)()()()(

.

),().(.).(

),().(.

)()(

)()(

.

)(

)(

)(

)(

)(

)(

.)()(

'

')()(

tw6u

tv5u

0

0

t4ut3u

t2ut1u

1

dA tzUzU.C 2

cz

dA tzUzU.C

0

0

trtk

trtk

1

tf

tf

tf

tf

tf

tf

1tQ

T

121

A

p

A

p

EHEH

EVEV

T

121

6

5

4

3

2

1

T

121

1

)(..)()(

tf1

tQT

222

2

, )(..)()(

tf1

tQT

323

3

, )(..)()(

tf1

tQT

424

4

, )(..)()(

tf1

tQT

525

5

, )(..)()(

tf1

tQT

626

6

(39)

The cross correlation function of excitations is

2/T

2/T

srT

QQ dt )t(Q )t(QT

1lim )(R

sr

2/T

2/T

T

)s(2s

T

)r(2r

Tdt )]t(f

1)].[t(f

1[

T

1lim

2/T

2/T

N

1i

N

1j

jijsir2s

2r

Tdt )t(f)t(f

11

T

1lim

2/T

2/T

jiT

N

1i

N

1jjsir2

s2r

dt )t(f)t(fT

1lim

11

2/T

2/T

jiT

N

1i

N

1jjsir2

s2r

dt )t(f)t(fT

1lim

11

)(R.

11

ji ff

N

1i

N

1jjsir2

s2r

)](R.)(R.)(R.)(R.)(R..[11

6656655511 ffs6r6ffs5r6ffs6r5ffs5r5ffs1r12

s2r

)](.)(..)(..)(.[.{ RrRkrRrkRk11

2EVEVEVEVEV

2EVs1r12

s2r

)]}(.)(.)(.)(.[66566555 66566555 ffsrffsrffsrffsr RRRR

The cross power spectral density function of excitation functions has the form

)](.)(..)(..)(.[.{11

)( 221122

SrSkrSrkSkS EVEVEVEVEVEVsr

sr

QQ sr

]})(X.)(X.)(X.)(X.)[(S).z(U..C2

22s6r6

2

21s5r6

2

12s6r5

2

11s5r5uu222

f

)(S.11

)(Sjisr ff

N

1i

N

1jjsir2

s2r

QQ

(40)

The differential equations of motion can be written in the form

)t(Q.)t(Q.m

1)t(q

m

k)t(q

m

R)t(q i

2i

'i

ii

i

ii

i

ii

)t(Q.)t(Q.k

1)t(Q.

k

1)t(q.)t(q.D2)t(q i

2i

'i

i

2i

'i

2i

ii

2iiiii with )t(Q.

m

11)t(Q '

ii

2i

i

(41)

The cross power spectral density function of the vibration response with respect to general coordinates is

)().S().H(H )(Ssrsr QQs

*rqq (42)

The cross power spectral density function of the vibration response with respect to original coordinates is

)(.S )(Sjisr qq

n

1j

sjri

n

1i

XX

(43)

Substitute from Eq. 42 in Eq. 46 results in



)().S().H(.H )(Sjisr QQj

*i

n

1j

sjri

n

1i

XX

(44)

Substitute from Eq. 40 in Eq. 44, one can obtain the cross power spectral density function of the response with respect to original coordinates of the form

n

1l

ffljki

n

1k2jj

2ii

j*i

n

1j

sjri

n

1i

XX )(S.).1

.m

1().

1.

m

1).(().H(.H )(S

lksr

(45)

and the auto power spectral density function of the response with respect to original coordinates of the form

6

1s

ffsjri

6

1rjij

*i

6

1j

njni

6

1i

XX )(S.k

1.

k

1).().H(.H )(S

srnn (46)

The power spectral density function of the excitations Correlation function of the excitations The correlation function of the excitations with respect to general coordinates is

2/T

2/T

srT

QQ dt )t(Q )t(QT

1lim )(R

sr (47)

Substitute from Eq. 32 in Eq. 47 results in

2/T

2/T

A'

pr5r4r3r2r12r

TQQ dA )t,z(U).z(U..CB )t(B )t(B )t(B )t([B

1

T

1lim )(R

sr

)t(B )t(B )t([B 1

].Ad )t,z(U).z(U..C ).2

cz(B s3s2s12

s

A'

pr6

dt ]Ad )t,z(U).z(U..C ).2

cz(BdA )t,z(U).z(U..CB )t(B

A'

ps6

A'

ps5s4

2/T

2/T

s4r1s3r1s2r1s1r12s

2r

TQQ )t()t(BB )t()t(BB )t()t(BB )t()t(BB

11

T

1lim )(R

sr

Ad )t,z(U).z(U..C ).2

cz()t(BB dA )t,z(U).z(U..C)t(BB

A'

ps6r1

A'

ps5r1

)t()t(BB )t()t(BB )t()t(BB )t()t(BB s4r2s3r2s2r2s1r2

Ad )t,z(U).z(U..C ).2


A'

ps6r2

A'

ps5r2


Ad )t,z(U).z(U..C ).2


A'

ps6r3

A'

ps5r3


Ad )t,z(U).z(U..C ).2


A'

ps6r4

A'

ps5r4

)t(].dA )t,z(U).z(U..C[BB )t(].dA )t,z(U).z(U..C[BBA

'ps2r5

A'

ps1r5



)t(].dA )t,z(U).z(U..C[BB)t(].dA )t,z(U).z(U..C[BBA

'ps4r5

A'

ps3r5

]dA )t,z(U).z(U..C].[dA )t,z(U).z(U..C[BBA

'p

A'

ps5r5

]Ad )t,z(U).z(U..C ).2

cz(].[dA )t,z(U).z(U..C[BB

A'

p

A'

ps6r5

)t(].Ad )t,z(U).z(U..C ).2

cz([BB )t(].Ad )t,z(U).z(U..C ).

2

cz([BB

A'

ps2r6

A'

ps1r6

tAd tzUzU.C 2

czBB tAd tzUzU.C

2

czBB

A

ps4r6

A

ps3r6 )(].),().(.).([)(].),().(.).([ ''

]dA )t,z(U).z(U..C].[Ad )t,z(U).z(U..C ).2

cz([BB

A'

p

A'

ps5r6

dt ]Ad )t,z(U).z(U..C ).2

cz(].[Ad )t,z(U).z(U..C ).

2

cz([BB

A'

p

A'

ps6r6

)(RBB)(RBB )(RBB )(RBB 11

)(R s4r1s3r1s2r1s1r12s

2r

QQ sr

Ad )(R).z(U..C ).2

cz(BB dA )(R).z(U..CBB

A

Ups6r1

A

Ups5r1 ''

)(RBB )(RBB )(RBB )(RBB s4r2s3r2s2r2s1r2

Ad )(R).z(U..C ).2


A

Ups6r2

A

Ups5r2 ''

)(R.BB)(RBB )(R.BB )(R.BB s4r3s3r3s2r3s1r3

Ad )(R).z(U..C ).2

cz(BB )dA(R ).z(U..CBB

A

Ups6r3

A

Ups5r3 ''


Ad )(R).z(U..C ).2


A

Ups6r4

A

Ups5r4 ''

A

Ups3r5

A

Ups2r5

A

Ups1r5 dA )(R).z(U..CBB dA )(R).z(U..CBB dA )(R).z(U..CBB '''

A

21UU2122

p

A

s5r5

A

Ups4r5 dA.).dA(R).z(U).z(U..CBBdA )(R).z(U..CBB '2

'1

'

Ad )(R).z(U..C ).2

cz(BB dA.).dA(R)

2

cz).(z(U).z(U..CBB

A

Ups1r6

A

21UU22122

p

A

s6r5 '

2

'2

'1

1



Ad )(R).z(U..C ).2

cz(BBAd )(R).z(U..C ).

2

cz(BB

A

Ups3r6

A

Ups2r6 ''

2

'1

'2

1

'

A

21UU12122

p

A

s5r6

A

Ups4r6 dA.).dA(R).2

cz).(z(U).z(U..CBB Ad )(R).z(U..C ).

2

cz([BB

2

'2

'1

1 A

21UU2122

p21

A

s6r6 dA.).dA(R).z(U).z(U..C)2

cz)(

2

cz(BB (48)

Since the wind velocity U(z,t) and the underground excitations (t) ,t )( are uncorrelated, the following correlation

functions must have the values of zero.

0)(R)(R)(R)(R '''' UUUU

and 0)(R)(R)(R)(R '''' UUUU

(49)

The power spectral density function of the excitations The cross power spectral density function of the excitations with respect to general coordinates is

de )(R)}(RF{ )(S i-QQQQQQ srsrsr

(50)

Substitute from Eqs. 48 and 49 in Eq. 50 results in

)(RBB )(RBB )(RBB)(RBB )(RBB )(RBB 11

)(S s2r2s1r2s4r1s3r1s2r1s1r12s

2r

QQ sr

)(R.BB)(RBB )(R.BB )(R.BB )(RBB )(RBB s4r3s3r3s2r3s1r3s4r2s3r2


A

21UU2122

p

A

s5r5 dA.).dA(R).z(U)z(U..CBB '2

'1

2

'2

'1

1 A

21UU22122

p

A

s6r5 dA.).dA(R)2

cz).(z(U)z(U..CBB

2

'2

'1

1 A

21UU2122

p1

A

s5r6 dA.).dA(R).z(U).z(U..C )2

cz(BB

de dA.).dA(R).z(U).z(U..C)2

cz)(

2

cz(BB i

A

21UU2122

p21

A

s6r6

2

'2

'1

1

SBB SBB SBBSBB SBB SBB 11

S s2r2s1r2s4r1s3r1s2r1s1r12s

2r

QQ sr)()()()()()()(

)(S.BB)(SBB )(S.BB )(S.BB )(SBB )(SBB s4r3s3r3s2r3s1r3s4r2s3r2

)(SBB )(SBB )(SBB )(SBB s4r4s3r4s2r4s1r4

A

21UU2122

p

A

s5r5 dA.).dA(S)z(U)z(U..CBB21

2

21

1 A

21UU22122

p

A

s6r5 dA.).dA(S)2

cz).(z(U)z(U..CBB



2

21

1 A

21UU2122

p1

A

s5r6 dA.).dA(S).z(U).z(U..C )2

cz(BB

2

21

1 A

21UU2122

p21

A

s6r6 dA.).dA(S).z(U).z(U..C)2

cz)(

2

cz(BB (51)

The wind velocity )(zU depends on the height of the building, according to the following equation

)H(U )H

z( )z(U (52)

Using Eq. 52 in Eq. 51

)(SBB )(SBB)(SBB )(SBB )(SBB 11

)(S s1r2s4r1s3r1s2r1s1r12s

2r

QQ sr

)(SBB )(S.BB )(S.BB )(SBB )(SBB )(SBB s3r3s2r3s1r3s4r2s3r2s2r2

)(SBB )(SBB )(SBB )(SBB )(S.BB s4r4s3r4s2r4s1r4s4r3

2

21

1 A

21UU21

A

U222

fs5r5 dA..dA)()H

z()

H

z()(S).H(U..C.BB

2

21

1 A

21UU221

A

U222

fs6r5 dA..dA)()2

cz()

H

z()

H

z()(S).H(U..C.BB

2

21

1 A

21UU2

11

A

U222

fs5r6 dA..dA)()H

z)(

2

cz()

H

z()(S).H(U..C.BB

2

21

1 A

21UU22

11

A

U222

fs6r6 dA..dA)()2

cz()

H

z)(

2

cz()

H

z()(S).H(U..C.BB

(53)

These double integrals can be described as Aerodynamic Amplification Functions (Transformation Functions) are

2

21

1 A

21UU21

A2

11 dA..dA)()H

z()

H

z()(X

,

2

21

1 A

21UU221

A2

12 dA..dA)()2

cz()

H

z()

H

z()(X

2

21

1 A

21UU2

11

A2

21 dA..dA)()H

z)(

2

cz()

H

z()(X

, } (54)

2

21

1 A

21UU22

11

A2

22 dA..dA)()2

cz()

H

z)(

2

cz()

H

z()(X

)(SBB)(SBB )(SBB )(SBB 11

)(S s4r1s3r1s2r1s1r12s

2r

QQ sr

)(S.BB )(S.BB )(SBB )(SBB )(SBB )(SBB s2r3s1r3s4r2s3r2s2r2s1r2

)(SBB )(SBB )(SBB )(SBB )(S.BB)(SBB s4r4s3r4s2r4s1r4s4r3s3r3

])(X.BB)(X.BB)(X.BB)(X.B[B )(S).H(U..C2

22s6r6

2

21s5r6

2

12s6r5

2

11s5r5U222

f

Auto power spectral density function of the excitation with respect to the first general coordinates



)(SBB )(SBB )(SBB 11

)(S 13111211111121

21

QQ 11

)(SBB )(SBB )(SBB 121211121411

)(SBB )(S.BB )(S.BB )(SBB )(SBB 13131213111314121312

)(SBB )(SBB )(SBB )(SBB )(S.BB 14141314121411141413

2

121615

2

111515U222

f )(X.BB)(X.B[B )(S).H(U..C

])(X.BB)(X.BB2

221616

2

211516 (55)

Complex transformation matrix with respect to general coordinates Fourier transformation of the vibration response and excitation has the following form

)(Q . ] 2D i [- ).(q n2n

2nnn

2n

Where )(Q . )(H )(q nnn with ...,6 2, 1,n ,

)](2D[ i ])(-[1

1 )(H

nn

2

n

n

(56)

and its absolute value is ...,6 2, 1,n ,

)](D 2[ ])(-[1

1 )(AHF

2

nn

22

n

Response power spectral density function with respect to general coordinates Cross correlation functions of the response with respect to general coordinates have the form

d e )( S)(H )(H2

1 dt )t(q )t(q

T

1lim )(R i

QQs*r

2/T

2/T

srT

qq srsr (57)

The response power spectral density function with respect to general coordinates is

Cross: )}(RF{ )(Ssrsr qqqq and Auto: )(S .)(H )(S

nn Q

2

nq

Auto power spectral density function for n-eigen form with respect to general coordinates is

)(SBB )(SBB)(SBB )(SBB )(SBB 1

.)(H )(S n1n2n4n1n3n1n2n1n1n12n

2

nqq nn

)(S.BB )(S.BB )(SBB )(SBB )(SBB n2n3n1n3n4n2n3n2n2n2

)(SBB )(SBB )(SBB )(SBB )(S.BB)(SBB n4n4n3n4n2n4n1n4n4n3n3n3


22n6n6

2

21n5n6

2

12n6n5

2

11n5n5U222

f

} (58) Where the mechanical amplification functions (Transformation Functions) are

...,6 2, 1,n ,

)](2D[ i ])(-[1

1 )(H

2

nn

22

n

2

n

(59)

and the Aerodynamic Amplification Functions (Transformation Functions) are shown in Eqs. 54 Response power spectral density function with respect to original coordinates Cross correlation functions of the response with respect to original coordinates have the form

2/T

2/T

srT

XX dt )t(X )t(XT

1lim )(R

sr

2/T

2/T

ji

6

1j

sjri

6

1iT

dt )t(q )t(qT

1lim



)(R dt )t(q )t(qT

1lim

jiqq

6

1j

sjri

6

1i

2/T

2/T

jiT

6

1j

sjri

6

1i

(60)

The response power spectral density function with respect to original coordinates is

Cross: )(S )}(RF{ )(Sjisrsr qq

6

1j

sjri

6

1i

XXXX

and Auto: )(S. )(Snn q

6

1j

njni

6

1i

X

(61)

)(SBB)(SBB )(SBB )(SBB 1

.)(H. )(S n4n1n3n1n2n1n1n14n

2

n

6

1j

njni

6

1i

XX nn

)(S.BB )(S.BB )(SBB )(SBB )(SBB)(SBB n2n3n1n3n4n2n3n2n2n2n1n2

)(SBB )(SBB )(SBB )(SBB )(S.BB)(SBB n4n4n3n4n2n4n1n4n4n3n3n3


22n6n6

2

21n5n6

2

12n6n5

2

11n5n5U222

f

(62)

Mean square value response with respect to original coordinates Mean square value of the random vibration response with respect to original coordinates can be written as

d )(S2

1)0(R

nnnn XXX2X

(63) Conclusions This paper outlines a mathematical model describing the vibrations of high-tower buildings and its foundations with general-type equivalent passive springs and dampers, rigid bodies, and some ideal constraints under the effect of randomly fluctuating wind loads and the excitation of earthquake ground motions. Two derivation methods of the equivalent system’s differential equations have been considered, namely D’alembert’s principle and Lagrange’s method, which verified the acceptability of the developed equations of motion. Following conclusions can be withdrawn :

The mathematical model with 6 degrees of freedom presented in the present paper can be used to investigate the effect of both wind and earthquakes loading.

Analytical solution of the free vibrations of tall building and its foundation using the general modal analysis method has been performed.

Analytical solution of forced vibrations of tall building and its foundation has been developed, through the correlation function (time domain) and the power spectral density function (frequency domain) of system response with respect to general and also original coordinates.

Without wind and earthquakes, structures – particularly large ones – would probably be a lot easier to design and cheaper.

Random vibrations of building’s foundation subjected to seismic excitations of earthquake ground motions and also randomly fluctuating wind pressure fields acting on a building surface are analyzed. Nomenclature Cp Aerodynamic pressure factor (-) E Kinetic energy of the system (J) Ed Soil dynamic modulus of elasticity (kp/m3) F1H, F1V Spring and damping forces at C or E in horizontal and vertical direction (kp) F2H, F2V Spring and damping forces at D or F in horizontal and vertical direction (kp) FEH, FEV Spring and damping forces at s1 in horizontal and vertical direction due to earthquake effect (kp) H(iΩ) imaginary transformation function (-) J1, J2 Mass moment of Inertia of foundation with its accompanied vibrated soil and tall building (kg.s2.m) JF, JS Mass moment of Inertia of foundation and accompanied vibrated soil with it (kg.s2.m) kEH, kEV Linear horizontal and vertical equivalent spring stiffness of earth (kp/m) kEK Rotational equivalent spring stiffness of earth (kp.m/rad) kH, kV Linear horizontal and vertical equivalent spring stiffness of building-foundation connection (kp/m) L Lagrangian function (-) m1, m2 Total mass of foundation with its accompanied vibrated soil (mF+ mS) and tall building (kg) mF, mS Foundation and Vibrating soil mass (kg) MW(t) Total turbulent wind moment as a function of time (kp.m)



Kq General coordinates

and ,y ,z , ,y ,z *2

*2

*1

*1

*1

*2 (m, m, rad, m, m, rad)

)(Rji QQ Cross correlation function of

the excitations (m2)

)(R),(Rsrsr XXqq Cross correlation function of

response with respect to general and original coordinates (m2) Rayleigh’s dissipation function (kp.m/s) rb Vertical embedding damping constant : the damping constant of radiation (kp.s/m3) rEK Rotational equivalent damping coefficient of earth (kp.m.s/rad) rEH, rEV Linear horizontal and vertical equivalent damping coefficient of earth (kp.s/m) rH, rV Linear horizontal, vertical equivalent damping coefficient of building-foundation connection (kp.s/m) rS Damping coefficient of the elastic soil bed (kp.s/m3) s1, s2 Centre of gravity of the foundation and tall building (-)

)(S),(Ssrii qqqq Auto and cross power

spectral density function of response w.r.t. general coordinates (m2.s/rad)

)(S),(Ssrii QQQQ Auto and cross power

spectral density function of excitations (m2.s/rad)

)(S),(Ssrnn XXXX Auto and cross power

spectral density function of response w.r.t. original coordinates (m2.s/rad) t Time (s) TEK Spring and damping torques about s1 in rotational direction (kp.m) U Potential energy of the system (J)

)H(U Average wind velocity along the

building height H (m/s)

)t(U),t(U zy Random displacement excitation

of earthquake in horizontal and vertical direction (m)

)t,z(U Wind speed as a function of space and

time (m/s)

)z(U Constant part of wind speed as a

function of space (m/s)

)t,z(U' Turbulent part of wind speed as a

function of space and time (m/s)

SV Vertical wave velocity (m/s)

)t(W Total turbulent wind force in y*-

direction as a function of time (kp)

)t,z(W Wind load as a function of space and time

(kp)

)z(W Constant part of wind load as a function

of space (kp)

)t,z(W' Turbulent part of wind load as a

function of space and time (kp) x

Amplitude of exponential solution of

motion differential equations (m)

)t(z ),t(y *o

*o Displacement of point O in the

direction of axisz and y *o

*o (m)

)( ),(z ),(y ),( ),(z ),(y 222111

non- dimensional Displacements (-)

)( ),(z ),(y ),( ),(z ),(y '2

'2

'2

'1

'1

'1

non- dimensional velocities (-)

)( ),(z ),(y ),( ),(z ),(y "2

"2

"2

"1

"1

"1

non- dimensional accelerations (-)

)t(z),t(y *1

*1 Displacement of gravity centre s1 of

foundation in *1

*1 z and y - axis (m)

)t(z),t(y *2

*2 Displacement of gravity centre s2 of

high tower building in *2

*2 z and y - axis (m)

)t(z ),t(y *C

*C Displacement of point C in the

direction of axisz and y *C

*C (m)

)t(z ),t(y *C

*C Velocity of point C in the direction

of axisz and y *C

*C (m/s)

)t(z ),t(y *C

*C Acceleration of point C in the

direction of axisz and y *C

*C (m/s2)

)t(z ),t(y *D

*D

Displacement of point D in the

direction of axisz and y *D

*D (m)

)t(z ),t(y *D

*D Velocity of point D in the direction

of axisz and y *D

*D (m/s)

)t(z ),t(y *D

*D Acceleration of point D in the

direction of axisz and y *D

*D (m/s2)

)t(z ),t(y *E

*E Displacement of point E in

the direction of axisz and y *E

*E (m)

)t(z ),t(y *E

*E Velocity of point E in the direction

of axisz and y EE ** (m/s)

)t(z ),t(y *E

*E Acceleration of point E in the direction

of axisz and y *E

*E (m/s2)



)t(z ),t(y *F

*F Displacement of point F in the

direction of axisz and y *F

*F (m)

)t(z ),t(y *F

*F Velocity of point F in the direction of

axisz and y *F

*F (m/s)

)(),( ** tz ty FF Acceleration of point F in the

direction of axisz and y *F

*F (m/s2)

Profile constant (-)

B Specific weight of the high tower

building (kp/m3)

21 and,, Density of air , foundation , and

high tower building respectively (kg/m3) non-dimensional time [-]

)t(*o , )t(*

1 , )t(*2 Angular displacements

about axisx and,x,x *2

*1

*o [rad]

)t(o , )(t1 , )t(2 non - dimensional

angular displacement about

axisx and,x,x *2

*1

*o [-]

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APPENDIX

FFF V.m , Foundation weight gmW FF . ,

Lorenz, H. (1955) calculated the weight of the accompanied vibrating soil with the foundation using the equation

ton,ba8350AfW 3434FS

)/()/( ].].[.[. gWm SS /

kg.

SF1 mmm , 2SSS

2fFF1 lmJlmJJ .. ,

)]/([

]/)][(/[)(.

FS

FSF

F

SS

F

SF

mm1

2hdmml

2

hd

m

ml

m

m l

Fig. 4 Foundation with its accompanied vibrating

toned sand

SF

S

rA

W h

. , ]/)[( FS l2hdl ,

]12/)ed.[(mJ 22FF , ]/).[( 12ehmJ 22

SS

Vertical embedding damping constant: the damping

constant of radiation is Sdb VEr /

Mass of the high tower: the density of high tower can be assumed as 1/10 of that of the foundation, i.e.

10/12

g.m W, V.m 22222 , ]/).[( 12cbmJ 2222

2/25/12

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