evaluating foundation mass damping and stiffness

8/7/2019 Evaluating foundation mass damping and stiffness

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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2003; 32:14311442 (DOI: 10.1002/eqe.285)

Evaluating foundation mass, damping and stiness

by the least-squares method

S. H. Ju;

Department of Civil Engineering; National Cheng-Kung University; Tainan 70101; Taiwan

SUMMARY

This paper discusses how to use the three-dimensional (3D) time-domain nite-element method incor-porating the least-squares method to calculate the equivalent foundation mass, damping and stinessmatrices. Numerical simulations indicate that the accuracy of these equivalent matrices is acceptablewhen the applied harmonic force of 1+sine is used. Moreover, the accuracy of the least-squares methodusing the 1+sine force is not sensitive to the rst time step for inclusion of data. Since the nite-elementmethod can model problems exibly, the equivalent mass, damping and stiness matrices of very compli-cated soil proles and foundations can be established without diculty using this least-squares method.Copyright ? 2003 John Wiley & Sons, Ltd.

KEY WORDS: damping; nite-element method; foundation; least-squares method; mass; soil; stiness

INTRODUCTION

For the vibration analysis of foundations placed on soft ground, it is necessary to con-sider the dynamic eect produced from the soil. Such a soil dynamic eect can usuallybe modelled by using equivalent mass, damping and stiness matrices. A substantial amountof research on calculating these three soil parameters of embedded foundations is reportedin the literature. Wolf and Song [1] calculated the static stiness and mass matrices of abounded medium and the dynamic stiness and unit-impulse response matrices of an un-bounded medium by using the scaled boundary nite-element method. Bernal [2] presentedthe formulation of a frequency-domain substructure approach for the analysis of secondarysystems using a dynamic-stiness matrix in physical co-ordinates to characterize each oneof the substructures. Qian et al. [3] used the boundary-element procedure to analyse the dy-namic response of rigid surface footings on an elastic half-space. Mulliken and Karabalis [4]

Correspondence to: S. H. Ju, Department of Civil Engineering, National Cheng-Kung University, Tainan 70101,Taiwan.

E-mail: [email protected]

Contract=grant sponsor: National Science Council, Republic of China; contract=grant number: NSC-90-2211-E-006-063.

Received 19 November 2001Revised 14 August 2002 and 2 November 2002

Copyright ? 2003 John Wiley & Sons, Ltd. Accepted 2 November 2002


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EVALUATING FOUNDATION MASS, DAMPING AND STIFFNESS 1433

If the scheme of Equation (3) is used for the transformation of the mass, damping andstiness matrices and vectors, the components of the matrix [Qi] are

Qi(i1)=2+j; j =1 for ij; Qj;j(j1)=2+i =1 for ji and Qi; j = 0 for others (5)

Substituting Equation (4) into Equation (2), one obtains

[{kta}T{kty}

T{ktx}T]

[Qi] [0] [0][0] [Qi] [0]

[0] [0] [Qi]

{Mv}{Cv}{Kv}

= ktFi (6a)

or written in a simplied form {ktA}T[Ti]{B} =

ktFi (6b)

where

{ktA} =

{kta}{ktv}

{ktx}

; [Ti] =

[Qi] [0] [0][0] [Qi] [0]

[0] [0] [Qi]

and {B} =

{Mv}{Cv}

{Kv

}

(6c)

The sum of the squares of the error for a number of applied forces and time steps is

=

i

k

t

({ktA}T[Ti]{B}

ktFi)

2 (7)

To minimize the sum of the squares of the error (@=@{B} = 0), one obtains the followinglinear equation:

[S]{B} = {F} (8a)

where

[S] =

i

[Ti]T

k

t

{ktA}{ktA}

T

[Ti] (8b)

[F] =

i

[Ti]T

k

t

{ktA}ktFi

(8c)

When the applied force k is dened, the time-dependent nodal accelerations, velocities anddisplacements can be obtained using the nite-element method for a soilstructure interactiveproblem. Alternatively, the nodal accelerations and velocities can also be calculated from thenodal displacements using the central dierence method as shown in Equations (9a) and (9b);thus, the only input is the displacement eld. Then, Equation (8a) is obtained by substitutingthem into Equations (8b) and (8c). Finally, mass, damping and stiness matrices can be foundby solving Equation (8a).

{ktv} = ({kt+tx} {

kttx})=(2t) (9a)

{kta} = ({kt+tx} 2{

ktx} + {

kttx})=(t

2) (9b)

where t is the time-step length.

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:14311442


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1434 S. H. JU

SELECTING AN APPROPRIATE HARMONIC APPLIED FORCE {ktFi}

A simple way to nd the exible matrix of a structure is to put a unit force at a certain degreeof freedom, and then a column of the exible matrix can be obtained. In this study, a similar

way is used. A harmonic force is arranged at the degree of freedom whose mass, dampingand stiness needs to be calculated, and zero forces are arranged at other degrees of freedom.Thus, if the number of degrees of freedom whose dynamic parameters need to be calculated isN, there are N dynamic analyses required for the arrangement of Equation (8a). For example,for a foundation with six degrees of freedom, there are six nite-element analyses requiredto nd the dynamic parameters. The harmonic force is applied to each degree of freedom foreach nite-element analysis. For this arrangement of applied forces, the index i of Equations(8b) and (8c) is counted from 1 to N, and the index k is also counted from 1 to N.

A suitable harmonic function should have the features of continuity and dierentiability;moreover, the rst and second derivatives should have the same period as the original function,such as sine and cosine functions. However, using them as applied forces will make Equation(8a) nearly singular. This condition can be validated from a dynamic problem with a singledegree of freedom as follows:

ma + cv + kx = sin( !t) (10)

where m;c;k;a;v and x are mass, damping, stiness, acceleration, velocity and displacement,respectively. The solution of this equation is

x = xh + p sin( !t ); v = xh + p ! cos( !t ) and a = xh p !2 sin( !t ) (11)

where xh is the homogeneous solution of Equation (10), p is a factor and is a phase angle.Those two values are functions of m;c;k and !. When ! is larger than the natural frequencyof Equation (10) and the damping c is not zero, the eect of the homogeneous solutioncan be minor, especially for the time after a number of cyclic periods. Thus, neglecting thehomogeneous solution and substituting Equation (11) into Equation (8a), one can obtain

[S]=

t

[p sin( !t)]2

t

p2 ! sin( !t) cos( !t)

t

p2 !2 sin( !t) sin( !t)

t

p2 ! cos( !t) sin( !t)

t

[p !2 cos( !t)]2

t

p2 !2 cos( !t) sin( !t)

!2

t

[p sin( !t)]2 !2

t

p2 ! sin( !t)cos( !t) !2

t

p2 !2 sin( !t) sin( !t)

(12)

The above equation indicates that the components of the third column of matrix [S] are !2

times those of the rst column, which means that matrix [S] is singular. If the homogeneoussolution is not neglected in Equation (8a), matrix [S] will not be singular. However, when theeect of the homogeneous solution is minor, matrix [S] is nearly singular, and this conditioncauses the solution of Equation (8a) to be unstable. This condition will be demonstrated in thesection of numerical simulations. The functions of 1+sine and 1cosine will also be testedin numerical simulations as they do not have the above-mentioned drawback.



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Figure 1. Mass, damping and stiness errors vs Nt using Equation (13) for Example 1.

NUMERICAL VALIDATION

Example 1: The example with two degrees of freedom

In this numerical example, a problem with two degrees of freedom was rst solved. To realizethe accuracy of the analysis, an accuracy ratio R is dened in Equation (13). The dynamicmatrix equation of this two-degree-of-freedom problem is shown in Equation (14), where Hviis the actual ith component of the mass, damping or stiness vector illustrated in Equation(3), Hvi is that component calculated from Equation (8a) and NH is the total number ofcomponents of the vector.

R = NHi=1 |H

vi H

vi |

NHi=1 |H

v

i |

(13)

2 1

1 3

a1a2

+

10 1010 20

v1v2

+

2000 500500 1000

x1x2

= {f} (14)

First, the accuracy of the harmonic forces including sine, 1+sine and 1cosine with theperiod T of 2 s was investigated. The nite-element method was used to nd the solution ofEquation (14) for 500 time steps with the time-step length of T=100=0:02s. The summationsof t in Equations (8b) and (8c) were arranged from time step Nt to time step 500. Figure 1shows the error ratio of the calculated mass and stiness matrices changing with Nt. Thisgure indicates that the accuracy of the sine function is highly dependent on Nt, but theaccuracy of those under the other two functions is almost independent of Nt. For the sine

function, the calculation error increases when Nt increases. This is because the eect ofthe homogeneous solution is minor for a large Nt. At this time, the sine function causesEquation (8a) to be unstable. This condition has been explained in the previous section.

Then, the periods T of the harmonic forces were changed from 0.1 to 4 s in order toinvestigate the variation of the error ratio. The results are shown in Figure 2, which demon-strates that the calculated mass is inaccurate for the sine function when the period of the sinefunction increases. For the other two harmonic forces, the results are accurate enough.



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1436 S. H. JU

Figure 2. Mass, damping and stiness errors vs the period of applied forceusing Equation (13) for Example 1.

Example 2: A rigid circular footing resting on an elastic half-space

A rigid circular footing of radius a resting on an elastic half-space investigated by Luco andMita [9] was compared. The footing radius a, shear modulus G, Poissons ratio and mass

density of the homogeneous soil are 10 m, 8e4 kN=m2

, 13

and 2 t=m3

, respectively. Withinthe circular footing area, a masterslave node scheme [10] was used. The master node wasarranged at the area centre and other nodes within this area were set to slaved nodes, whosethree degrees of freedom are dominated by the master node which has vertical, horizontal,torsional and rocking degrees of freedom. Luco and Mita [9] obtained the complex stinessmatrix [KLuco] with the following relationship of real [M]; [C] and [K] in Equation (1).

[Ke] =1

Ga1Re([KLuco]) =

1

Ga1([K] !2[M]) (15)

[Ce]=Im([KLuco])=(a0 Ga1) = ![C]=(a0 Ga1) (16)

where [Ke] is the equivalent dimensionless stiness, [Ce] is the equivalent dimensionlessdamping, ! is the wave frequency, a1 equals the footing radius a for horizontal and verticaltranslation terms, equals a3 for the rocking term and equals a2 for the horizontal-rockingcoupled term and a0 is the dimensionless frequency as follows:

a0 = !vs=a (17)

where vs(=

G=) is the shear velocity.The nite-element mesh is shown in Figure 3, where the absorbing boundary conditions

[11; 12] were used along the ve boundaries, except the top surface, to simulate semi-innite

soil stratum. Three nite-element analyses were performed to nd the vertical, horizontal androcking terms in Equations (15) and (16) using a unit 1 + sine load with the frequencies of15, 22.5, 30, 45 and 60 rad=s, and the applied force was subjected to each degree of freedomat the master node in each analysis. The Newmark direct integration method was used inthe time-history nite-element analysis with Newmark parameters and of 0.25 and 0.5,respectively. Eight-node 3D isoparametric elements were used to simulate the soil prole.The solution method of the nite-element analysis was the conjugate gradient method with



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Figure 3. Finite-element mesh of Example 3 (model dimensions = 290 m

290 m

113 m,471240 8-node solid elements).

Table I. Horizontal, rocking, vertical and horizontalrocking stiness and damping values of Example 3.

a0 (Equation (17)) Horizontal Rocking Vertical Horizontalrocking

Ke Ce Ke Ce Ke Ce Ke Ce

0.775 4.943 3.567 4.106 0.517 5.899 5.976 0:538 0.0161.162 4.958 3.333 3.769 0.821 5.568 5.674 0:556 0.1101.549 4.705 3.317 3.412 1.020 4.779 5.679 0:552 0.1382.324 4.413 3.376 2.745 1.265 3.405 6.224 0:478 0.187

3.098 4.210 3.351 2.058 1.419 2.824 6.611

0:261 0.193

the SSOR scheme [13]. The time-step length was set to 1100

of the loading period and a totalof 500 time steps were used. Table I shows the horizontal, rocking, vertical and horizontal-rocking stiness and damping (Equations (15) and (16)) of the least-squares result. Theleast-squares and the Luco and Mita [9] results are shown in Figure 4, which indicates anacceptable agreement.

Example 3: Flexible foundation and soilstructure interactive analyses of buildings

A 14-storey reinforced concrete building with the storey height of 3:2 m and the columnspacing of 7:5 m was analysed. Each column is connected to a pile cap with four reinforcedconcrete piles as shown in Figure 5, in which the pile diameter is 1:2 m and the length is50:4 m. The top surface of piles is connected to a cap with thickness of 2:2 m. Within thearea of the 1:4 m diameter disk at the pile cap surface center, a masterslave node scheme[10] was used to model the connection eect of the column. The master node was arrangedat the area centre, and other nodes within this area were set to slaved nodes, whose three



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1438 S. H. JU

Figure 4. Comparison between least-squares and theoretical [9] results for Example 2.(a) Ke (Equation (15)) changing with a0 (Equation (17)), (b) C

e (Equation (16))changing with a0 (Equation (17)).

Figure 5. Foundation dimensions and nite-element mesh of the foundation of Example 3.

degrees of freedom are dominated by master node. The master node having six degreesof freedom, three translations and three rotations, represents the structural behaviour of thefoundation. First, the least-squares method was used to nd the 6 by 6 mass, damping andstiness matrices at the master node. The Youngs modulus, Poissons ratio and mass density

of the concrete are 2e7 kN=m2

, 0.15 and 2:4 t=m3

, respectively. The Youngs modulus of the

surface soil is 0:75 e5 kN=m2

and more than 50 m under the ground is 10e5 kN=m2

. Thelinear interpolation is applied to determine the Youngs modulus between these two depths.

The mass density and Poissons ratio of the soil are 2 t =m3

and 0.48, respectively. The twofactors of Rayleigh damping ([Damping] = [Mass] + [Stiness]) and equal 0:4=s and



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Figure 6. The plane view of the nite-element mesh for Example 3. (The foundation of Figure 5 islocated at the mesh centre, and along the ve mesh surfaces except the top surface, absorbing boundary

conditions were set. The Z-direction contains 32 element layers.)

Table II. Residual ratios of Example 3 at three dierent frequencies using Equation (18).

Frequency (rad=s) 15 30 60

Re, Residual ratio (Equation (18)) 0.0029 0.0025 0.0066

7:3 e4 s, respectively, which provides approximately 2% soil damping ratio at a frequency of

7Hz. The nite-element mesh is shown in Figure 6, where the absorbing boundary conditions[11; 12] were used along the boundaries shown in Figure 6 to simulate semi-innite soilstratum. Six nite-element analyses were performed to nd the three dynamic matrices for aunit 1 + sine load with the frequencies of 15, 30 and 60 rad=s (periods of 0.419, 0.209 and0:105 s), and the applied force was subjected to each degree of freedom in each analysis.The Newmark direct integration method was used in the time-history nite-element analysiswith Newmark parameters and of 0.25 and 0.5, respectively. Eight-node 3D isoparametricelements were used to simulate the soil prole. The solution method of the nite-elementanalysis was the conjugate gradient method with the SSOR scheme [13]. The time-step lengthwas set to 1

100of the loading period and a total of 500 time steps were used. Since there is

no exact solution of this example, Equation (7) was rearranged to obtain the residual ratio(Re) as shown in Equation (18). Table II shows the residual ratios of the 1+sine loads at

three dierent frequencies using Equation (18). Since the residues are small, the least-squaresmethod obtains accurate simulations. Figure 7 shows the wave propagation at 2:17 s afterapplying the vertical loading under the frequency of 30 rad=s, which shows the reasonablewave propagation without the error reection from nite-element boundaries.

Re =

i

k

t

|{ktA}T[Ti]{B}

ktF|

i

k

t

|ktF|

(18)



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1440 S. H. JU

Figure 7. Wave propagation at 2:17s after applying the vertical loading under the frequency of 30rad=sof Example 3 (magnifying factor = 4e7).

Then, the 14-storey building was analysed using the same numerical parameters mentionedabove. The square column size is 1m, the rectangular foundation beam size is 1m by 2:8m andthe other rectangular beam size is 0:5 m by 0:7 m. In the nite-element analysis, the buildingoors were assumed to be rigid and the numerical scheme was used from Ju and Lin [14].There are two types of nite-element analyses including the exible-foundation analysis andthe soilstructure interactive analysis. The exible-foundation analysis assumes the foundationis exible, and 6 by 6 mass, damping and stiness matrices are arranged at each connectionof the foundation beam and the column end. These matrices were averaged from the 30 and60 rad=s matrices. The soilstructure interactive analysis models the building, soils and pilefoundations using a whole nite-element mesh and absorbing boundary conditions. Figure 8shows the mesh of the building and pile foundations, in which the soil mesh with dimensionsof 216 m 216 m 92:2 m (length width depth) is not shown in this gure. The appliedforce (N) of sin(40t) is subjected to the mass centre of the building roof in the X and Ydirections, and a torsion (T-m) of 2 sin(40t) is also subjected to the roof mass centre, wheret is time (with unit = second). Figure 9 shows the wave propagation at 1:25 s after applyingthe 40rad=s sine loads, which indicates reasonable wave propagation results. Figure 10 showsthe displacements and rotation of the foundation cap centre (point A in Figure 8). Figure10 indicates that similar results are obtained from the exible-foundation analysis and thesoilstructure interactive analysis. A major dierence of the two methods is that the soilstructure interactive analysis includes the coupled eect of all the foundations but the exiblefoundation does not. This example indicates that the least-squares method can obtain accurateequivalent matrices of the foundation.

CONCLUSION

In this study, a least-squares method was developed to calculate the equivalent foundationmass, damping and stiness matrices using the time-domain nite-element results. Numericalsimulations indicate that the accuracy of these equivalent matrices is acceptable when the



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Figure 8. Finite-element mesh of the building and piles of Example 3 (the soil meshis not shown in this gure).

Figure 9. Wave propagation of Example 3 at 1:25 s after applying the 40 rad=s sineloading (magnifying factor = 3e8).

applied harmonic force of 1+sine is used. Moreover, the accuracy of the least-squares methodis not sensitive to the rst time step (Nt) for inclusion of data. Using the least-squaresformulation deduced in this paper is not complex. Only the nodal displacements calculatedfrom time-domain nite-element analyses are used to nd a linear matrix equation. Aftersolving this matrix equation, one obtains the equivalent mass, damping and stiness matrices.Since the nite-element method can model problems exibly, the equivalent mass, damping



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1442 S. H. JU

Figure 10. X displacement, Z displacement and X rotation at the foundation capcentre (point A in Figure 8) of Example 3.

and stiness matrices of very complicated soil proles and foundations can be established

without diculty using this least-squares method.

ACKNOWLEDGEMENT

This study was supported by the National Science Council, Republic of China, under the contractnumber: NSC-90-221l-E-006-063.

REFERENCES

1. Wolf JP, Song CM. The scaled boundary nite-element methoda fundamental solution-less boundary-elementmethod. Computer Methods in Applied Mechanics and Engineering 2001; 190:55515568.

2. Bernal D. A dynamic stiness formulation for the analysis of secondary systems. Earthquake Engineering andStructural Dynamics 1999; 28:12951308.

3. Qian J, Tham LG, Chung YK. Dynamic analysis of rigid surface footings by boundary element method. Journalof Sound and Vibration 1998; 214:747759.

4. Mulliken JS, Karabolis DL. Discrete model for dynamic through-the-soil coupling of 3-D foundations andstructures. Earthquake Engineering and Structural Dynamics 1998; 27:687710.

5. Wolf JP. Spring-dashpot-mass models for foundation vibrations. Earthquake Engineering and StructuralDynamics 1997; 26:931849.

6. Zhao JX, Carr AJ, Moss PJ. Calculating the dynamic stiness matrix of 2-D foundations by discrete wave numberindirect boundary element methods. Earthquake Engineering and Structural Dynamics 1997; 26:115133.

7. Qian J, Beskos DE. Dynamic interaction between 3-D rigid surface foundations and comparison with the ATC-3provisions. Earthquake Engineering and Structural Dynamics 1995; 24:419437.

8. Song C, Wolf JP. Dynamic stiness of unbounded medium based on damping-solvent extraction. EarthquakeEngineering and Structural Dynamics 1994; 23: 169 182.

9. Luco JE, Mita A. Response of a circular foundation on a uniform half-space to elastic waves. EarthquakeEngineering and Structural Dynamics 1987; 15:105118.

10. Ju SH. Investigating contact stresses on articular surfaces by 3-D rigid links. Journal of Engineering Mechanics(ASCE) 1997; 123:12531259.

11. Ju SH, Wang YM. Time-dependent absorbing boundary conditions for elastic wave propagation. InternationalJournal for Numerical Methods in Engineering 2001; 50:21592174.

12. Ju SH. Finite element analyses of wave propagations due to high-speed train across bridges. InternationalJournal for Numerical Methods in Engineering, 2002; 54:13911408.

13. Ju SH, Kung KS. Mass types, element orders and solving schemes for the Richards equation. Computers andGeosciences 1997; 23:175187.

14. Ju SH, Lin MC. Building analyses comparisons assuming rigid or exible oors. Journal of StructuralEngineering (ASCE) 1999; 125:2531.


evaluating foundation mass damping and stiffness

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