06.pdf

70 . . . . , , . 1-2 2007 Tech. Chron. Sci. J. TCG, I, No 1-2 . . . . , , . 1-2 2007 Tech. Chron. Sci. J. TCG, I, No 1-2 71

. -, , . , , - (impedance functions) (stiffness indices) - (soil-structure interaction).

1.

[1]. , - , . - [2]. , , - (, .., ). - , , , ... , [3]. , Radon [4], - , [5]. - , . - [6]

[7].

, - () [8] - /, .., Nastran [9]. - - (), [10] , [11]. - [12], [13], [14] [15], - [16], ....

, - [17]. - - [18], - . - , - , [19], . , - , [20]. - , , . , , -

. . . . . .

... . . ., ... . . ., ...

: 19.1.2005 : 15.6.2007


[21]. , - .

2.

-,

x,

y ,

xz,

yz

,w

,x,w

,y ,

D, K(r) ,

0 ,

s ,

s - , .

3.

, , . - h, ( 1), :

: 3 21 )( rehrh (1)

: 3 2

24

1)(

rhrh

(2)

h (r)2

1h (r)

r

P

r

r

h

P

P

r

r

r

1: . Figure 1: Various types of plates.

() -: Winkler ( 2), ( 3).

[1] - .

Z, w

r

k

2: Winkler.Figure 2: Plate on Winkler foundation.

3: (b=4a) - k .

Figure 3: Plate on elastic half-space (b=4a) with spring constants k at the boundaries.

Es [kN/m2] -

0 [kN/m2]

200

00

21

)1(

EEs (3)

G0 [kN/m2] :

)1(2 000 GE (4), -

: Winkler,


, -

, -

.

[5], / Nastran [9].

4.

, h (h/r < 0.1), - . -, . , Kirchhoff [2].

x

x

w

zP

u=-z

w,x, u

x

4: .Figure 4: Deformation field in the plate.

x-y, ( 4). -, - . :

0z

wzz

0

y

w

zyz , 0

x

w

z

uxz (5)

x

ww xzxx

, ,y

ww yzyy

,

tan

(5)

:

2

2

2/32

22

)(])/(1[

/1

x

w

dx

dw

xxw

xw

rx

,

2

21

y

w

ry

(6)

D [kNm] :

)1(12 2

3

hED (7)

Laplace - :

dr

d

rdr

d 12

22 (8)

D

rp

dr

dw

rdr

wd

rdr

wd

rdr

wdw

)(11232

2

23

3

4

44 (9)

)()())(( 22 rprwrK (10) -

. K(r) - , . , Radon. - - : :

K(r) = D ( = 0), [12] :

)1(ln16

)( 2

rrD

Prw

(11)

P - . Winkler:

, [22] :

)(kei)2

()(2

rD

lPrw

(12)

kei Kelvin kD4 - . :

, Boussinesq [2], -

0()=(1-

0)2/( .

0.),

0

0

Poisson, , -. 0

30 kD


)]1(2[ 2000 Ek , :

0 300

20

12)(

drJ

D

Prw

(13)

J0 Bessel

0 . :

, [5] :

dnVVJ

yVn

yns

1,21

12

~~

)(4

1),(

(14)

sss

dd

s

HV lnln

)())((~1

ds

HV

))](sin()([1~ 1

211

2

))((sin))(si())((cos))(ci(~

11112 ssssV

dtt

tdt

t

t

zz

sin-si(z),

cos-ci(z) -

,

qp 1 sp .

, - [8] . , - , 5.

5: Figure 5: Plate modeling with surface finite elements.

, , ( 6).

, , .

6: . Figure 6: Finite element modelling of the non-homogeneous plate.

E=2,8107 kPa Poisson = 0,25. r = 2,5m h = 0,20m, - (h/r < 0,1). V

0=2,520,20 = 3,93m3,

:

32

0

5,2

0

3 22

0

5,2

0140,1)2,0()( mdrderdrdrhrV r

, - ( 0,36m 0,21m , ) :

32

0

5,2

21,0 3 2

22 99,1)

4

2,0(36,021,0 mdrd

r

rV

- P = 1000kN .

, - , Winkler . Hertz [22], , , - ( 7). p~ w :

7: Winkler.

Figure 7: Spring elements for modeling a Winkler foundation.


),(),(~ yxwkyxp (15)

( 7), , Winkler. ( 1) :

)1(2 20

00 Ek (16)

, , -, (welded contact) - . - Boussinesq [2]:

sEsK

0

20

0

)1()(

(17)

0

0 -

. 1

, Bowles [1]. k k

0 ,

, . 1:

.

Table 1: Ground categories and numerical values for their

properties.

0

[kN/m2]

0

k k0

[kN/m3] & [kN/m2] 200000 0,35 114000 100000 0,35 56980 50000 0,35 28490 30000 0,35 17090 20000 0,35 11400

, ( 8). - b=4a=20m (a=5m ) 20m. , [9]. , - -

[23]. - , , Givoli (1992). ( 8) - .

8: - ( ).

Figure 8: Solid elements for FEM modeling of the elastic half-space (the support conditions are shown at the edges).

5.

- ( 9), , . , [5], .

Winkler -

-0.003

-0.002

-0.001

0.000

0.001

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m

]


Winkler - B

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m

]

Winkler -

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m

]

Winkler -

-0.008

-0.007

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m

]

Winkler -

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m

]

-

-0.004

-0.003

-0.002

-0.001

0.000

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

-

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

-

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

-

-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]


-

-0.015

-0.012

-0.009

-0.006

-0.003

0.000

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

"" -

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

0.005

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

"" -

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

"" -

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

"" -

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

"" -

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

"" . -

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m

]

"" . -

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]


"" . -

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

"" . -

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

"" . -

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m]

9: - .

Figure 9: Displacements versus radial distance for various cases of plates resting on ground.

, , - . , - , - . , - . ,

, ( r/h > 10 r=2,5m) 8a4a ( a=2r). [23], .

6.

- , - . -, : , .

-. P ( 10) w

rig

.

rigw

h

P

10: .Figure 10: Deflections of rigid plate under a point load.

- P

~ ,

Hooke :

rigrig wKPP ~ (18)

Krig

[kN/m] - , - - ( 11).


P

wrig

rigKP~

11: - .

Figure 11: Modeling of the ground with a spring element for the rigid plate on elastic half-space.

- [21] , r ( 2), :

0

0

1

4

rGKrig (19)

wrig

( 11), k

[kN/m3],

, :

/rigKk (20)

2 K

rig k

.

.

2: .

Table 2: Impedance functions for rigid plate on elastic half-space.

Krig,z

[kN/m]k

[kN/m3] 710100 36160 355000 18080 177500 9040 106500 5424 71000 3616

. - . - w

fl ( 12).

flw

P

h

12: .Figure 12: Deflections of the flexible plate under a point load.

:

2

0 0)()(

~ rflfl drdrwkdAwkP (21)

- q :

qPP ~ (22)

, , PP ~

. k

( 2), .

,

[m2],

( 13).

-7-

11:

.

Figure 11: Modeling of the ground with a spring element for the

rigid plate on elastic half-space.

[21]

, r

( 2), :

(19)

wrig ( 11),

k [kN/m3],

, :

(20)

2

Krig k

.

.

2:

.

Table 2: Impedance functions for rigid plate on elastic half-space.

Krig,z[kN/m]

k[kN/m3]

710100 36160 355000 18080 177500 9040 106500 5424 71000 3616

.

.

wfl ( 12).

12: .

Figure 12: Deflections of the flexible plate under a point load.

:

(21)

q

:

(22)

, , PP ~ .

k

( 2),

.

,

[m2],

( 13).

- O

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.0 0.5 1.0 1.5 2.0 2.5

r [m]

w [m

]

13: .

Figure 13: Definition of the active surface of a plate on ground.

,

. ,

,

. ,

,

.

A

13: .Figure 13: Definition of the active surface of a plate on ground.

, - - . , - , . , , .


(21) 18a -

, -

. (21)

, :

fl

flwk

dAwkP ~

~

(23)

flw~

, -. Hooke :

flwKP~~ (24)

[kN/m]

Ak (25)

, :

PP~

(26)

(26) (24), :

flwk

P ~

(27)

(18), flw~

Kfl [kN/m]

, :

kK fl (28)

fl

rig -

. (18) (27), :

fl

rig

rig

ffl

w

w

K

K~ (29)

, (20) (28), :

1A

A

K

K

rig

fl (30)

. , , - . , K

fl K

rig

/ 1/, - . ,

fl

- ( 14).

q~

P

wfl

flKP~

14: .

Figure 14: Modeling of the groundfoundation system with a spring element for the rigid plate on elastic half-space.

, . - ( 15), ( wwwfl ~ ).

qP~

P

KK

15: .Figure 15: Equivalent spring model for the plate on ground.

(25),

D

= . D. ,

, 0)(lim DD . , , 0 D , .

:

-12-

,

:

PP~ (26)

(26) (24), :

flwk

P ~

(27)

(18), flw~

Kfl [kN/m]

, :

kK fl (28)

fl rig

. (18) (27),

:

fl

rig

rig

ffl

ww

KK

~ (29)

, (20) (28),

:

1A

AKK

rig

fl (30)

. ,

,

. ,

Kfl Krig

/ 1/,

.

, fl

( 14).

q~

P

wfl

flKP~

14:

.

Figure 14: Modeling of the groundfoundation system with a

spring element for the rigid plate on elastic half-space.

,

.

-

( 15),

( wwwfl ~ ).

qP~

P

KK

15: .

Figure 15: Equivalent spring model for the plate on ground.

(25),

D

D . ,

,

0)(lim DD . ,

,

0 D ,

.

:

ww fl ~

wq

wP ~ flK (31)

PqP ~

(=c),

(=x) .

( , k ),

,

(31)

(=c),

(=x) .

(

, k

), -


,

.

xc

xK

P

P

~

. ,

0)()(

1)

~(

22

xc

c

xc

x

xcP

P , c > 0.

, , -, , .

(

)

(=c)

(=x) .

cx

xK

P

q

,

,

, 0)(

)(2

cx

xc

P

q

(x'='


7.

, . - :

, , .

, - .

. - . , -, .

.

, .

, . [15], - .. ( ) . -, . - telos, , [17]. - , - . , , , , . , , [20]. , , -

. , , - [18].

1. J.E. Bowles, Foundation Analysis and Design, 5th Edition, McGraw-Hill, New York, 2001.

2. S.P. Timoshenko, S. WoinowskyKrieger, Theory of Plates and Shells, McGrawHill, New York, 1959.

3. R. Szilard, Theory and Analysis of Plates: Classical and Numerical Methods, Prentice-Hall, Englewood Cliffs, 1974.

4. A.I. Zayed., Handbook of Function and Generalized Function Transformations, CRC Press, Boca Raton, 1996.

5. G.D. Manolis, T.V. Rangelov, R.P. Shaw, The Non-Homogeneous Biharmonic Plate Equation: Fundamental Solutions, International Journal of Solids and Structures, 40, 5753-5767, 2003.

6. J.B. Brown, R.V. Churchill, Complex Variables and Applications, 7th Edition, McGraw-Hill, New York, 2003.

7. R.P. Shaw, G.D. Manolis, A Generalized Helmholtz Equation Fundamental Solution using Conformal Mapping and Dependent Variable Transformation, Engineering Analysis with Boundary Elements, 24, 177-188, 2000.

8. R.D. Cook, D.S. Malkus, M.E. Plesha, Concepts and Applications of Finite Element Analysis, J. Wiley, New York, 1989.

9. MSC/NASTRAN for Windows, Version 4.6, MacNeal-Schwendler Corp., Los Angeles, 2000.

10. M.D. Greenberg, Application of Greens Functions in Science and Engineering, Prentice Hall, Englewood Cliffs, 1970.

11. G. Bezine, Boundary Integral Formulation for Plate Flexure with Arbitrary Boundary Conditions, Mechanics Research Communications, 5, 197-206, 1978.

12. M. Stern, A General Boundary Integral Formulation for the Numerical Solution of Plate Bending Problems, International Journal of Solids & Structures, 15, 769 - 782, 1979.

13. J.T. Katsikadelis, E.J. Sapountzakis, A BEM Solution to Dynamic Analysis of Plates with Variable Thickness, Computational Mechanics, 7, 369-379, 1991.

14. J.T. Katsikadelis, A.E. Armenakas, Plates on Elastic Foundation by the BIE Method, ASCE Journal of Engineering Mechanics, 110, 1086-1105, 1984.

15. M.O. Faruque, M. Zaman, A Mixed Variational Approach for the Analysis of Circular Plate-Elastic Half-space Interaction, Computational Methods in Applied Mechanics & Engineering, 92, 75-86, 1991.

16. J.B. de Paiva , R. Butterfield, Boundary Element Analysis of Plate-Soil Interaction, Computers & Structures, 64, 319-328, 1997.

17. J.J. Johnson, Soil-Structure-Interaction: The Status of Current Analysis Methods and Research, Report No. NUREG/CR-1780, Lawrence Livermore Laboratories Publication, Pasadena, 1981.

18. J. Dominguez, Twenty-five years of Boundary Elements for Dynamic Soil-Structure-Interaction, Chap. 1 in W.S. Hall and G. Oliveto (eds.), Boundary Element Methods for Soil-Structure-Interaction, Kluwer Publishers, Dordrecth, 2003.

19. E. Kausel, R.V. Whitman, J.P. Moray, F. Elsabee, The Spring Method for Embedded Foundations, Nuclear Engineering & Design., 48, 377392, 1978.

20. J.P. Wolf, Foundation Vibration Analysis using Simple Physical Models, Prentice-Hall, Englewood Cliffs, 1994.

21. .. , -, , ...., , ..., , 2003.

22. H. Hertz, Gesammelte Werke, Vol. 1, Ernst und Sohn, Berlin, 1895.


23. D. Givoli, Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.

, . . ., , , , , 54124 , : 2310-995663, : 2310-995769, e-mail: [email protected] , . . ., , , , , 54124 , : 2310-995707 ,. . ., . . ., , , , 54124 , : 2310-995707


AbstractThe present work focuses on fundamental solutions for flexible inhomogeneous plates resting on an elastic foundation. More specifically, the solutions are for the displacement field generated in thin, circular, elastic plates with variable thickness under a vertical point load at the center. Subsequently, these solutions are integrated over the surface of the plate so as to yield impedance functions (or stiffness coefficients) that can be used within the context of dynamic soil-structure interaction (SSI) analyses involving discrete parameter representations of the structure-foundation-soil system. Similar results, shown for the rigid plate case and for the flexible homogeneous plate case, serve as points of reference.

1. INTRODUCTION

The inhomogeneous circular plates examined herein have variable thickness, which is gradually reduced away from their center. The reduced thickness may follow a power law or an exponential law, although other variations are possible as well. It is believed that variable thickness plates are an improvement over plates with constant thickness, especially when point loads are involved, as would be the case for columns reaching ground level and resting on spread footings. The fundamental solutions for such plates come from recent work reported in the literature, whereby techniques such as conformal mapping in conjunction with the Radon transformation have been used. Both free-standing as well as elastically supported inhomogeneous plate solutions are recovered. In addition, classical solutions for plates on a Winkler foundation (i.e. the floating plate case first solved by Hertz) and plates on the elastic half-space (Boussinesqs solution) are considered as the means for gaging the effect of inhomogeneity on the displacement field.

Once these solutions have been appropriately processed so as to yield the impedance functions that are the main thrust of this paper, the next step would be to conduct soil-structure interaction studies for vertical vibrations. The impedance

functions derived here represent the soil-foundation system and a finite element model is needed for the superstructure. The baseline solution for comparison purposes is that of the fixed-base structure. Parameters that influence the structural response and are represented in the impedance functions derived here are rigid versus flexible foundation, type of supporting soil, and degree of plate inhomogeneity when the foundation is flexible. Finally, these impedance functions can be augmented to include the remaining five possible degrees of freedom (two horizontal plus three rotational) for a fully 3D model and can be further supplemented by appropriate mass and damping coefficients.

2. ANALYTICAL SOLUTIONS

The computation of fundamental solutions (or Greens functions) for non-homogeneous plates resting on an elastic half-space and under a point load are the key step in developing impedance functions for the soil-foundation system. To that end, we recovered analytical results from our earlier work and used them to compute the displacement field that develops under the plate. Specifically, we employed a methodology based on conformal mapping in conjunction with the Radon transform. The specific type of plate inhomogeneity is dependent on the type of conformal mapping prescribed. For instance, an exponential mapping yields a plate modulus that also varies exponentially with distance from the point load, a quadratic mapping yields a similar quadratic function for the plate thickness, etc. We note here that conformal mapping methods for obtaining fundamental solutions have been introduced as an alternative to integral transforms (Fourier, Laplace, etc.), which despite their generality require an inverse transformation in the form of a contour integral over the complex plane. The second step in the solution procedure involves use of the Radon transform for computing the solution of the conformally mapped biharmonic equation. With the Radon transform, a

Extended Summary

Stiffness Coefficients for Problems in Soil-Structure Interaction

G.D. MANOLIS E. PARASKEVOPOULOS K. PLATSOUKAS

Professor, A.U.T.H. Civil Engineer, Ph.D. Civil Engineer, M.Sci.

Submitted: Jan. 19. 2005 Accepted: June 15, 2007


function of two variables is reconstructed from its integrals over all straight lines in the 2D plane or from contour integrals over smooth curves in 3D. This allows for compact expressions to be obtained that include the sub-grade effect. All results for the circular plate were checked by running parallel computations with the finite element method (FEM). Finally, appropriate integration of the displacement field over the plate-soil contact area, along with the use of basic principles of mechanics, yields stiffness coefficients for vertical vibrations.

3. IMPEDANCE FUNCTIONS

Once the stiffness coefficients Kfl,i

, for non-homogeneous plates on flexible soil are derived, they are tabulated for use in SSI problems. Specifically, we distinguish the following cases: (a) Index i = is the homogeneous (i.e. constant thickness) plate on a Winkler foundation, i = 0 is the homogeneous plate on the elastic half-space, i = 1 is the exponentially inhomogeneous plate on sub-grade and i = 2 is the quadratic inhomogeneous plate on sub-grade. Computation of the key integral involving plate displacement over the contact area with the soil, wdr

, is performed with

piece-wise linear interpolation over the circular area using a rather fine mesh. The analytical results previously derived were used in this process.

From an examination of the vertical impedance functions that were thus obtained, the following observations arise: (a) The Winkler foundation case yields stiffer coefficients when compared with the elastic half-space model, since the latter case includes mechanical action in the transverse direction; (b) The homogeneous plate case, where no uplifting occurs, serves to verify the elementary mechanical dictum for the spring, namely increasing soil stiffness yields higher soil reactions that translate into larger values for the impedance functions (in terms of dimensionless coefficient ); (c) Finally, a hierarchical classification of the plates, starting from stiff and moving to flexible (constant thickness-exponentially inhomogeneous-quadratically inhomogeneous), verifies a conclusion drawn based on the spring analogue, namely, reduction in the plate stiffness increases the value of the plate reaction and decreases the soil reaction (manifested by smaller values for coefficient ).

4. CONCLUSIONS

The basic conclusions reached have to do with (a) the plate displacement field, evaluated analytically as well as by the FEM; and (b) the stiffness factors for vertical movement. Specifically, we have:

The displacement distribution in the foundation plate becomes less smooth in soft soils, as expected in view of the fact that differential movement is more pronounced in flexible materials, with all other factors (e.g. load, geometry) remaining fixed.

The displacements that develop in homogeneous plates show excellent convergence, irrespective of the method of computation (analytical versus numerical) and regardless of the type of sub-grade.

The displacements that develop in the inhomogeneous plates show some divergence that depends on the method of computation. This has to do with the fact that the FEM employs polynomial functions to model the kinematic field variation in an element, while careful analysis reveals that this is not so. More specifically, it depends on the particular type of plate examined, and we have Bessel functions or sine and cosine integrals as the mathematical expressions that control kinematics. An additional source of divergence is the truncation of a semi-infinite mesh representing the half-space by the FEM, but that can be minimized by use of appropriate springs at the boundaries.

Any increase in soil stiffness causes an increase in the reaction to the load, which filters into the impedance function.

As the foundation plate becomes less flexible, the magnitude of the reaction it develops increases with a parallel drop in the soil reaction.

In terms of future developments, it should be mentioned that one key step in the present development of plate-soil impedance functions has to do with integration of the displacement field over the contact surface. For inhomogeneous plates under a point load, the analytical solution predicts uplift at a certain distance from the load, which in reality would cause loss of contact. Thus, the problem becomes geometrically non-linear and the results presented herein do not include this phenomenon. This could be the subject of future work, so that the present degree of approximation can be established.

George D. Manolis,Dr. Civil Engng., professor, Division of Structures, Civil Engineering Department, Aristotle University, Thessaloniki, GR-54124, Greece; Tel: +30-2310-995663; Fax: +30-2310-995769; E-mail: [email protected] Paraskevopoulos, Dipl. Civil Engng., Ph.D., Division of Structures, Civil Engineering Department, Aristotle University, Thessaloniki, GR-54124, Greece; Tel: +30-2310-995707 Konstantine Platsoukas, Dipl. Civil Engng., M.Sci., Division of Structures, Civil Engineering Department, Aristotle University, Thessaloniki, GR-54124, Greece; Tel: +30-2310-995707

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