Western Ontario, London, Ontario N6A 5B9, ABSTRACT · 2014-05-19 · Response of long gravity dams to incoherent seismic ground motions 0. Ramadan, M. Novak Department of Civil Engineering,

Response of long gravity dams to

incoherent seismic ground motions

0. Ramadan, M. Novak

Department of Civil Engineering, University of

Western Ontario, London, Ontario N6A 5B9,

Canada

ABSTRACT

Spatially incoherent seismic ground motions are represented in bothfrequency and time domains. The frequency domain representation isemployed to obtain closed form solutions for axial and bending stressesthat develop in line structures on rigid foundations due to ground motionincoherence. Axial and lateral response of long gravity dams is theninvestigated including dam-reservoir-foundation interaction. The resultsindicate that ground motion incoherence can produce significantadditional stresses on long dams and other extensive structures.

INTRODUCTION

Seismic design of engineering structures is usually conducted with inputground motion specified in terms of a response spectrum, a powerspectral density function or a single accelerogram. Such representationsconsider the ground motion variation with time but ignore its variationin space. This is sufficient for structures with small bases, e.g. slendertowers and residential buildings, but may not be adequate for extensivestructures such as large dams, long tunnels, bridges and pipelines.Extensive structures suffer not only from the ground motion variationwith time but also from its variation in space.

Spatial variabilities of seismic ground motion are clearly indicated indata recorded at large scale seismographic arrays [2 and others]. Theyresult from random mechanical factors along the wave paths (spatialcoherency effects) as well as time lags due to non-vertical travellingwaves (wave passage effects). Previous studies have shown that the

Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509

462 Soil Dynamics and Earthquake Engineering

ground motion spatial variabilities (or incoherence) may significantlymodify the seismic response of extensive structures [4,5,7 and others].Spatial incoherence of the ground motions reduces the seismic responseof rigid structures and simply-supported beams but may produceconsiderable stresses in multi-supported systems.

This paper describes a comprehensive approach for the inclusion ofground motion incoherence in the seismic response of structures. First,spatially incoherent ground motion is mathematically described in bothfrequency and time domains. With soil-structure interaction neglected,variances of the axial and bending strains in line structures due to groundmotion incoherence are then established in closed forms. This yields theupper limits for the strains. Finally, the effects of ground motionincoherence on the axial and lateral response of concrete gravity damsare investigated in detail, including dam-reservoir-foundation interaction.

MODELLING INCOHERENT GROUND MOTIONS

Frequency Domain RepresentationTo fully describe the space-time random field of the ground motion,cross-spectra between every two points in the random field are needed.For statistically homogeneous random fields such as that of the groundmotion, the cross-spectrum between any two points can be described as

(1)

In Equation 1, S,(co) is the local, station-invariant, auto-spectrum of therandom field, a) is circular frequency, and R(F,w) is the spatial coherencyfunction which reflects ground motion incoherence due to all randomfactors at the source and along the wave paths; finally, F is the separationvector, and v, is the velocity vector of the predominant travelling wave.The study of the data recorded at large scale seismographic arrays hasshown that for the design of structures with plan dimensions muchsmaller than the epicentral distance, the coherency functions can beassumed direction independent, i.e. R(?,&>) = R(r,w) where r is the trueseparation [7].

A few mathematical models have been suggested for the coherencyfunction R(r,w) [2 and others]. This function is assumed here in thesimple, versatile exponential form, i.e.


Soil Dynamics and Earthquake Engineering 463

in which c,V,y are an exponential decay parameter, a velocity parameterand a power parameter, respectively. (Note that Equation 2 involves twoindependent parameters only.) The coherency function by Equation 2jointly describes the variations with both frequency and separation interms of the dimensionless frequency (wr/V) and is, hence, called jointcoherency. This coherency model is in agreement with simplifiedtheoretical studies on wave scattering in random media and comparesquite well with the data recorded during recent earthquakes [3].

The degree of spatial correlation of a random field over a certaindistance L* of interest, e.g. the length of the structure, can be assessed bycalculating the correlation coefficient. The latter is obtained bysuperimposing the effects of individual frequency components, i.e.

where S,,(w) is the local, normalized power spectrum of ground displace-ment. The numerical value of p ranges from 0 for no correlation to 1 forfull correlation. The value of p approaching unity indicates that thespatial incoherence effects are not significant for that particular structure.

Time Domain RepresentationTo include the ground motion spatial incoherence in time domainanalysis, time histories of the ground motions at all nodes along thestructure base are needed. Due to a paucity of data at large scaleseismographic arrays, artificially generated time histories are usually used.In this study, spatially variable stationary ground motions are generatedto fit the pre-selected auto-spectrum and coherency functions. For theline structures considered in this paper, a single spatial coordinate issufficient. The ground motion at time t is generated by

M

*-o i-1 (4)

In Equation 4, s is spatial coordinate and r, is the time lag due totravelling waves; <p\^ <p- ' are two statistically independent random phases



uniformly distributed between 0 and ZTT; and a%, L, are, respectively,amplitude modifiers and characteristic lengths that depend on thecoherency function. Equation 4, while simple and computationallyadvantageous, adequately satisfies the target functions. Details on andverification of this simulation technique are given in [6]. The techniquewas also extended to allow for two- and three-spatial coordinates, nearand far-field simulations, and anisotropy of the spatial coherency function[8].

The auto-spectrum employed here is the modified Kanai-Tajimi form[1]; the parameters of two filters are assumed to be dependent, i.e.

STRAINS IN LINE STRUCTURES NEGLECTING SOIL-STRUCTURE INTERACTION

Consider a line structure which is continuously supported on a foundationmuch more rigid than the structure itself. In this case, the structureexactly follows the motion of the ground. The strains and stresses whichdevelop under this condition represent the upper limits since soil-structure interaction usually reduces the structural response. For thiscase, closed form solutions for both axial and bending strains can bewritten and are given below.

Denote the axial and lateral components of the ground motion atnode i by w, and u*, respectively, and consider the three equidistant nodesshown in Figure 1. Then, the axial stress at node i can be evaluated by

a = E lim L l (5%)a id-o a

and the bending stress a distance y^ from the neutral axis by

c. = £>> lim ———

In Equations 5 a and 5b, E is Young's modulus of the structure material,and d is the distance between the nodes.

The auto-spectra of o, and a^ are found to be



S («) = 2£2 $ («) ||m JLi (6a),*

and

S» lim + , w - , w (6b)

where S (o>) and Su(o>) are the ground motion power spectra in the axialand lateral directions, respectively. Substitute Equation 2 into Equation6 and evaluate the limits. For y=2, Equations 6a and 6b reduce to

S (7a)

and

Note that the motion power spectra change to those of velocity andacceleration, respectively. Integrating Equations 7 and establishing thepeak values from the variances obtained, it follows that the peak axialstress is proportional to the peak ground velocity along the structurewhile the peak bending stress is proportional to the peak groundacceleration perpendicular to that axis, i.e.

E\j2c . fgala = —-— w \""/V

and

°b =

Equations 8 agree with what has long been recognized to hold fordeterministic harmonic waves. For random motions, the above relationssuggest that, when soil-structure interaction is neglected, the internalstresses monotonically increase as the degree of ground motionincoherence increases (i.e. as c increases or V decreases). This does nothold true, however, when soil-structure interaction is included as shownlater herein.



Most continuously supported structures are analyzed using a finiteelement discretization. From the point of view of the coherency effect,the length of the finite elements used should be small enough to capturethe spatial variability of the ground motion with adequate accuracy. Auseful limit may be obtained by evaluating the stress variances usingEquations 6a and 6b (disregard the limit operator) for different valuesof the distance, d. Then, an optimum element length may be selectedaccording to the convergence of the stress variances with d.

Figure 2 shows the convergence of both the axial strain (= oJE} andthe curvature (= cr Ey )) with separation for different values of thedecay parameter c. For the data shown, the auto-spectrum parametersare So=0.01, a) =5n and £g=0.6, while the coherency parameters are y=2and V=1000 m/s. For the case of no soil-structure interaction, Figure 2indicates that the axial stresses can be reasonably estimated with anelement length of as large as 100 m while a shorter length is needed toevaluate the bending stresses. The limiting values obtained for very smallseparations are identical with the exact values evaluated using Equations7a and 7b.

For structures on flexible foundations, reasonable results can beobtained with element lengths significantly larger than suggested earlierbecause soil-structure interaction de-amplifies the high frequencycomponents which require short lengths [4,5,7],

For 0<y<2, the limits in Equations 6a and 6b do not exist. Thisseems to indicate that coherency models which are exponential in r^ withy<2 are not adequate for separations approaching zero since they resultin unbounded strains and stresses.

DAM LATERAL VIBRATION

The seismic response of gravity dams is usually evaluated using a two-dimensional finite element model of a single slice or monolith. Thismodel overlooks the variations of the seismic input along the damlongitudinal axis which may be important for long dams. To account forthese variations, a simple mathematical model, complementary to theconventional 2D model, was formulated for long dams including dam-reservoir-foundation interaction and employed in an extensive parametricstudy [7]. Examples of the results are presented below.

The dam length/width ratio is assumed to be large enough so that itsresponse in the horizontal plane can be evaluated using the beam theory(Figure 3a). This analysis allows for dam bending in the horizontal planeand its twisting (rocking in the vertical plane). It yields the additional



stresses which develop due to ground motion spatial incoherence. Theresponse in the plane of the dam cross-section is evaluated using the 2DF.E. model as usual (Figure 3b). The final dam stresses should beevaluated based on the results from the two analyses.

For the parametric study, an 853 m long, 68.5 m wide and 103 m highconcrete gravity dam is considered. The dam is discretized into 10elements and the ground motions are generated at all 11 nodes usingEquation 4. The spectrum parameters are So=0.01, a) 5n, £g=0.6 whilethe coherency parameters are y=l, c=l, V=1000 m/s and no travellingwaves are considered. Thus, the correlation coefficient defined byEquation 3 corresponding to L<>=853 assumes the values of 0.578, 0.136and 0.004 for the ground displacement, velocity and acceleration,respectively. Time histories of the lateral ground displacements at nodes1 to 4 are displayed in Figure 4. The peak ground acceleration is 0.2 g.A time envelope function in the form of a half sine wave was superim-posed on the generated stationary motions to simulate nonstationarycharacteristics of real earthquakes. The dam response to these motionswas evaluated using the complex response analysis procedure.

Figure 5a shows the dam response in the horizontal plane, while theresulting bending moments are displayed in Figure 5b. The data areshown at five discrete time instances and for both spatially correlated(SC) and, for comparison, fully correlated (FC) ground motions. Whilethe dam vibrates almost as a rigid body under the FC motions, it bendsand twists significantly under the SC motions. The peak "additional"normal and shear stresses resulting from the ground motion spatialincoherence are 8.76 and 1.46 MPa, respectively. These stresses arequite high and should be considered together with those evaluated usingthe conventional 2D analysis.

Figure 6 compares the dam response and internal forces under theSC motions with those produced by the fully correlated, but otherwiseidentical, travelling waves. The figure shows that the dam response andinternal forces produced by the ground motion spatial incoherence arequalitatively different from those due to travelling waves. The peaknormal and shear stresses due to the more critical travelling waves(v,=2.133 km/s) are 1.83 and 0.83 MPa and thus are significantly smallerthan those due to the SC motions. More details on dam lateralvibrations, including the effects of dam-reservoir-rock interaction, damvertical joints and free field ground motion characteristics, can be foundelsewhere [7].



DAM AXIAL VIBRATION

Axial stresses in long concrete dams produced by ground motion spatialincoherence are obtained by evaluating the axial response of a beam onelastic foundation. The same dam as described above is assumed in theexample. This dam has a cross-sectional area of 3595 m*. The dam axialresponse and forces due to both SC and FC ground motions are shownin Figures 7a and 7b. The parameters of the coherency function and theauto-spectrum of the ground motion component parallel to the damlongitudinal axis are assumed to be c=l, y=l, V=1000, So=0.01, Wg=5#and £g=0.6. Again, the dam moves as a rigid body under the FC motionsbut is alternatively compressed and stretched by the SC motions. Themaximum axial stresses caused by the SC motions is 4.56 MPa which isnot insignificant. This vibration mode is important only for continuousdams such as the Old Aswan Dam in Egypt (2142 m long) and theWillow Creek Dam in Oregon, U.S.A. (543 m long).

Dam-rock interaction may be demonstrated by displaying the Fourieramplitudes of the dam motions normalized by those of the groundmotions. These data are presented in Figure 8 for both the FC and SCground motions; through-rock coupling of foundation tributary areas isconsidered for the data in Figure 8a but ignored for those in Figure 8b.Many high vibration modes are excited by the SC ground motions.Ignoring the through-rock coupling of foundation elements overestimatesthe stiffness parameters but underestimates the damping parameters.The overall effect of the through-rock coupling on the dam response isfound to be insignificant, however. For instance, the maximum stresswith coupling neglected is 4.42 MPa and 4.56 MPa when the coupling isincluded.

Finally, Figure 9 displays the dam axial stresses for a wide range ofthe coherency decay parameter c and variable foundation shear wavevelocity V,. The stresses increase as c increases and as the foundationstiffness increases. Previous studies [5] have shown that axial andbending stresses due to ground motion incoherence increase with thedecay parameter c up to a certain limit and then fall down withincreasing c. However, this peak occurs at a value of c much larger thancan be realistically expected for seismic ground motions.

CONCLUSIONS

A comprehensive approach for mathematical modelling of spatiallyincoherent seismic ground motions in both frequency and time domainsis described; this includes a novel simulation technique.



For continuously supported line structures, closed form solutions forthe spectra of both axial and bending stresses are derived for the case ofno soil-structure interaction.

Finally, axial and lateral vibrations of long concrete gravity dams areinvestigated, including dam-reservoir-rock interaction. It is found thatground motion incoherence results in significant stresses which remaintotally undetected when only the conventional 2D analysis is conducted.

When the space-time variations of the axial and lateral groundmotion components are assumed identical, the peak stress due to axialvibration of the gravity dam considered is about 50% of that producedin lateral vibration.

REFERENCES

1. Clough R.W. and Penzien, J. Dynamics of Structures. New York:McGraw-Hill, 1975.

2. Harichandran, R.S. and Vanmarcke, E.H. Stochastic Variation ofEarthquake Ground Motion in Space and Time. J. Eng. Mech.,ASCE, Feb., 154-174, 1986.

3. Novak, M. Discussion. J. Eng. Mech., ASCE, 113(8), 1267-1270,1987.

4. Novak, M. and Hindy, A. Seismic Response of Buried Pipelines.3rd Canadian Conf. on Earthq. Engrg., Montreal, 1,177-203,1979.

5. Novak, M. and Suen, E. Dam-Foundation Interaction UnderSpatially Correlated Random Ground Motion. In AS. Cakmak(ed.), Developments in Geotechnical Engineering: Soil-StructureInteraction, Elsevier, 25-33, 1987.

6. Ramadan, O. and Novak, M. Synthesizing Spatially IncoherentRandom Ground Motions for Earthquake Response Analysis.Report No. GEOT-14-91, Fac. Eng. Sc., U.W.O., London,Canada, 1991.

7. Ramadan, O. and Novak, M. Dam Response to SpatiallyVariable Seismic Ground Motions Parts I & II. Report No.GEOT-14-92, Fac. Eng. Sc., U.W.O., London, Canada, 1992.

8. Ramadan, O. and Novak, M. Simulation of Multi-Dimensional,Anisotropic Ground Motions. Report No. GEOT-15-92, Fac. Eng.Sc., U.W.O., London, Canada, 1992.



Ui-i

WM

Ui

Wi

UiH

Win

Figure 1 Ground motions along and across line structures

7= 2; V=1000 m/s; c=l( o ), 2( * ), 5( * ), B( - ), 10(

10"

'3 io_4_Jcn

'.-

(a)

•r—o o—o—n

C—O O—O—O O-OQ^

-j-10 v * ? *

10"'OJL,

<y

J3 10-^a

5—O O—o —r

10'10"

10"

Element Length d (m) Element Length d (m)

Figure 2 Variances of axial strains and curvatures in continuouslysupported line structures based on different finite elementdiscretizations (rigid foundation)



(a)

Soil Tributary Areas

% "9i

L-1_L— iy__i/c_j ' i

Nodal Points Dam Elements

(b)

Reservoir

" . » « , * '..; /• ' ."g .. .VFoundation /*-/*.' • /

Figure 3 Two-stage analysis of long gravity dams: (a) response inbending and torsion; (b) 2D response in the plane of damcross-section

£ o.io

^ o.os

"2 § o.oo2 Co 5 -o.os*"* O^ ^2 -o.io

S* -0.15

^ -0.20

At Nodes: (1) ; (2) ---; (3) ; (4)

\

2 4 6 8 10Time ( sec )

12

Figure 4 Stationary ground motions simulated at dam nodes 1 to 4



t = 2.0 sec

t = 2.0 sec SC; - - • FCSC; FC

t = 4.0 sec iO.lm t = 4.0 sec

10 MNm

t = 6.0 sect = 6.0 sec

V ^ \

t = 8.0 sec

t = 10.0 sec

t = 8.0 sec

(a)

t = 10.0 sec__

(b)

Figure 5 Dam horizontal response and bending moments


Spatially Correluted c/V= 0.001ng C v_« 2.133 Km/a

( v^= 4.205 Km/a

10 20 30 40

Frequency ( rad/ sec )

10

10

10d•«-«o

0

%nOX

cdPJo

oE-

(b)

30

g 200o

10

10 20 30 40 8'cc

g

m

a

10 20 30 40

Frequency ( rad/ sec )

Figure 6 Dam midpoint response to spatially correlated motions andtravelling waves



initial node locations; • response to FC motions; 10cm

t= 2 s

t= 4 |s I | | | | |

t= 6|s | | | | | 1

t= a is I 1 ! 1 ! !

t=10Js ! j 1 1 ! !

•=> r

_

\ p

i

t = 2.0 s

t - 4.0 s

t = 6.0 s

t = 8.0 s

t = 10.0

^mmmmmmwmmJ

12100 MN

SC

- - FC/- "* |

s

Figure 7 Dam axial response and forces


c.2 , (•)H-J «JOs

o

o

o 1• rHO

6 ocd 0Q

Soil Dynamics and Earthquake Engineering

(b)

475

Spatially CorrelatedFully Correlated

20 40 60

Frequency ( rad/ s )

o0 20 40 60

Frequency ( rad/ s )

Figure 8 Fourier amplitudes of dam motions normalized by those ofthe ground motions: (a) through-rock coupling included;(b) through-rock coupling neglected

cdCL,

60

40

w(U

OT 20

15•rH

010

(a) 7= 1; V=1000 m/s

/ov V,=800 m/s /• V^ = 3000 m/s /o very rigid / ,'

10° 10'

Decay Parameter c

60

40

20

0 *&

(b) 7= 2; V=1000 m/s

? V^ = 800 m/s• V, = JOOO m/s /o very rigid /

OX

,/""

10-2 10" 10° 10'

Decay Parameter c

Figure 9 Effect of coherency decay parameter on dam axial stressesfor different foundation stiffnesses and power factors


Documents

Western Ontario, London, Ontario N6A 5B9, ABSTRACT · 2014-05-19 · Response of long gravity dams to incoherent seismic ground motions 0. Ramadan, M. Novak Department of Civil Engineering,