HARDFILL3

RESEARCH ARTICLE

Wei XIAO, Yunlong HE, Yanfeng ZHANG

Simplified analytical solution for free vibration characteristicsof Hardfill dam

E Higher Education Press and Springer-Verlag 2008

Abstract Formulas for computing natural frequencies

and modes of Hardfill dams are derived based on one-

dimensional shear wedge theory, in which the influences

of the upstream concrete face and hydrodynamic pressure

of water on the dams natural frequencies and modes are

discussed. Furthermore, the seismic responses of Hardfill

dams are calculated using response spectrum method. An

example is analyzed to compare the differences of natural

frequencies and modes between shear wedge method and

FEM. Then the applicability and accuracy of shear wedge

method to analyze free vibration characteristics of

Hardfill dams are proven.

Keywords shear wedge, natural frequency, mode, face,

hydrodynamic pressure

1 Introduction

Hardfill dam (called CSG dam in Japan) was put into

practice in recent years which is a new dam type between

traditional gravity dam and concrete face rockfill dam. It

is constructed to have lower cost and less impact on the

environment than other dams. The dam material is made

by mixing a certain percentage of cement and water with

material such as riverbed gravel or excavation muck

which can be obtained easily near the dam site. The dam

cross-section is trapezoidal, and its upstream side is cov-

ered with waterproof concrete face plate [1]. The surface

slopes of upstream and downstream are both about 1:0.7.

Because the height is so much smaller than the width of

the dam base, dam deformation under lateral (upstream -

downstream) earthquake is mainly shear deformation.

Compared with shear deformation, the bend deformation

is negligible. This conclusion has been demonstrated by

Japanese scholar Hirose [2], so calculating the traditional

shear wedge method used to analyze the seismic dynamic

responses of embankment dams is appropriate for deriv-

ing the lateral seismic dynamic response of Hardfill dams.

In recent years, shear wedge method has had significant

progress and improvement. 2D and 3D results of lateral,

longitudinal and vertical vibration have been published for

several idealized canyon shapes, such as rectangular, semi-

cylindrical, trapezoidal and triangular, and the inhomogen-

eous distribution of shear modulus is also taken into

account. Based on shear wedge method, wavemotion-shear

wedge method and a modified equivalent linear technique

for analysis of seismic response of embankments are pre-

sented. Whats more, the shear wedge method is not only

used to calculate free vibration characteristics for earth

dams but also for CFRDs. The influences of concrete face

are discussed by additional mass and stiffness method when

the formulas for computing natural frequencies and modes

of CFRDs are derived, and the conclusion is verified by the

results of shaking table tests [314].

However, because the slopes of earth dams are so gen-

tle, the effect of hydrodynamic pressure is always ignored.

The effect of hydrodynamic pressure should be considered

for CFRDs because of the upstream concrete face and the

dams steeper slopes. The slope of Hardfill dam is about

1:0.5 to 1:0.8, which is much steeper than earth dams and

CFRDs, and there is concrete face on the upstream. Thus,

the influences of hydrodynamic pressure to natural fre-

quencies and modes should be considered, which are lar-

ger than those for earth dams and CFRDs. An equivalent

system is applied in this paper to calculate the influences

to amend the results by traditional shear wedge method.

The closed-form solutions for the lateral response of

Hardfill dams under empty and full storage are developed

based on the traditional shear wedge method. Results are

present for natural frequencies, modes and participation

factors. Hydrodynamic pressure is considered by addi-

tional mass method when free vibration characteristics

under full storage are analyzed.

Through one example, the reasonability and accuracy of

results by shear wedge method are evaluated by comparing

Received December 4, 2007; accepted May 26, 2008

Wei XIAO, Yunlong HE (*), Yanfeng ZHANGState Key Laboratory of Water Resources and HydropowerEngineering Science, Wuhan University, Wuhan 430072, ChinaE-mail: [email protected]

Front. Archit. Civ. Eng. China 2008, 2(3): 219225DOI 10.1007/s11709-008-0037-3

with the finite element numerical results. At last, it is con-

cluded that shear wedge method is appropriate in calculat-

ing seismic responses of Hardfill dams.

2 Analysis method

2.1 Shear wedge method (SWM)

The cross section of Hardfill dam is simplified into triangu-

lar section(Fig. 1), and the triangular vertices are on the

crest of the dam. Then the slopes are slightly changed so

that the area is equal to a trapezoidal section. As to high

dam, results with this simplification are the beginning of

getting the results by trapezoidal section because the height

of the dam is far less than the width of the dam base.

Some simplifying assumptions are still introduced with

shear wedge method.

1) Three-dimensional geometry of the Hardfill dam is

neglected when the dam axis is infinite long, and a plane

strain approximation is used.

2) The dam is modeled by uniform elastic material,

resting on a rigid foundation.

3) Only lateral shear deformation is allowed and

assumed to be uniformly distributed across the lateral(upstream-downstream) direction of the dam under lat-

eral earthquake.

4) Hydrodynamic effect from the reservoir water is

taken into account.

If the hydrodynamic effect is ignored, considering the

dynamic equilibrium of an infinitesimal horizontal ele-

ment of the dam body and accounting for the stress-dis-

placement leads to the following equation

arCzbrH L2uLt2

~ aGCzbGH L2u

Lz2z bGCzmGH LuLz , 1

where a5 a+bz is the lateral width of the concrete face,

b~B

Hz~mz is the lateral width of the dam, u is the lateral

displacement relative to the boundaries. r,G and E are den-sity, shearmodulus and elasticmodulus, and subscriptC and

H respectively represent concrete and Hardfill material.

If the effect of concrete face is ignored, Eq. (1) may be

written asL2uLt2~

GH

rH

L2uLz2z

GH

zrH

LuLz

: 2

So when the effect of the concrete face plate is ignored,

the ith natural frequency vi and corresponding mode

shape Wi are

vi~biH

GH

rH

s, 3

Wi~J0 bi:z

H

, 4

in which J0 is the Bessel function of the first kind and

order zero, bi are the zeros of J0.

2.2 Effect of concrete face plate on free vibration

characteristics

Dvi,DWi respectively represent the influences of ith naturalfrequency and mode shape function caused by mass matrix

and stiffness matrix of upstream concrete face plate, the

results have been presented byKongXianjing &Liu Jun [12]

2

GH

rH

sbiH rHmzrCb HBizrCaAi Dvi

zXkn~1n=i

ain

10

{GHm{GCb Hb2nx2J0 bix J0 bnx dx

zGCa:bn

10

J0 bix J1 bnx dx

z

10

rHmzrCb GH

rHHb2i x

2J0 bix J0 bnx dx

~{rCGH

rHb2i aAizbBiH zGCbHb2i Bi{GCaCi, 5

2

GH

rH

srHmzrCb biH2

10

x2J0 bix J0 bjx

dxDvi

zXkn~1n=i

ain rHmzrCb GH

rHb2i H{ GHmzGCb b2nH

|

10

x2J0 bjx

J1 bnx dxzGCa10

J0 bjx

J1 bnx dx

{ GCb2j{rC

GH

rHb2i

aAjaij

~ GC{rCGH

rH

bb2i H

10

x2J0 bjx

J0 bjx

dx

{GCa:bi

10

J0 bjx

J1 bix dx, 6

Fig. 1 Cross-section of Hardfill dam

220 Wei XIAO, et al.

Dvi and aij can be obtained by solving the above sim-ultaneous Eqs. (5) and (6). Then DWi is expressed as

DWi~Xkj~1j=i

aijWj : 7

In which

Ai~J21 bi

2, Bi~

10

x2J20 bix dx,

Ci~0:5{b2i Ai, x~z=H:

8

2.3 Effect of hydrodynamic pressure on free vibration

characteristics

Eigenvalue li is defined as

li~v2i : 9

There exist

K Wi~li M Wi, 10

where [K], [M] is stiffness matrix and mass matrix of the

dam under empty storage, respectively.

Water is assumed to be incompressible without damp-

ing. Water vibration in the reservoir surface and vertical

displacement of the reservoir bottom are overlooked.

Then additional mass matrix caused by hydrodynamic

pressure can be obtained according to Westergaard solu-

tion.Dli

pand DWi respectively present the change of

natural frequency and mode shape by the additional mass

matrix [DM] under full storage. Thus,

K WizDWi ~ lizDli MzDM WizDWi : 11Assume that DWi is defined as

DWi~Xkj~1j=i

aijWj : 12

That is

Dli~{WTi DMWiWTi MWi

li, 13

aij~li

lj{li:W

Ti DMWiWTj MWj

: 14

As the results by shear wedge method are based on the

assumption that shear deformation is uniformly distribu-

ted in the lateral direction, so it is acceptable to represent

the dam as a one-dimensional particle system as Fig. 2

shows. The system is about K isometric points with the

spacing is l, mass is mn

l~H=k, 15

mn~nrHmH2k2: 16

The additional mass applied on the points based on

Westergaard solution is

Dmn~7rWHh.720:

n=k

p: 17

Then the mass matrix [M], additional mass matrix [DM]and mode shape matrix [Wi] of the equivalent system can

be given by

M~

rHmH2

1k2

2k2

P(k{1)

k2

kk2

2666666664

3777777775, 18

DM~

7rWhH

720

1=k

p2=k

p

P(k{1)=k

pk=k

p

26666666664

37777777775

, 19

Wi~J0 bi=k J0 2bi=k . . . J0 (k{1)bi=k J0 kbi=k T :

20

Substituting Eqs. (13)(15) in Eqs. (17)(19) yields

Dli~{

7rWhPkn~1

J20 nbi=k :n=k

p

720rHmHPkn~1

n:J20 nbi=k =k2li, 21

Fig. 2 Equivalent system of dam

Simplified analytical solution for free vibration characteristics of Hardfill dam 221

DWi~Xkj~1j=i

lilj{li

:7rWh

Pkn~1

J20 nbi=k :n=k

p

720rHmHPkn~1

n:J20 nbi=k =k2Wj: 22

Thus natural frequencies and mode shape functions under

full storage are given

v0i~vizDvzDli

p, W0i~WizDWizDWi: 23

2.4 Seismic responses

Once the solutions of natural frequencies and mode shape

functions are given, the maximum seismic responses of the

dam, which are the most important for engineering, can be

derived by response spectrum method.

Maximum displacement:

di, max~ Wi:gij jSd,i: 24Maximum velocity:

vi, max~ Wi:gij jSv,i: 25Maximum acceleration:

ai, max~ Wi:gij jSa,i: 26

In which gi is the participation factor defined by

gi~

H0W0izdzH

0W20izdz

: 27

Sd, Sv, Sa respectively present relative displacement res-

ponse spectrum, relative velocity response spectrum and

absolute acceleration response spectrum. In general, the

first three to four maximum responses of the modes are

combined by SRRS method to get the maximum res-

ponses of the dam. For example, the maximum seismic

displacement is given by

dmax~

X4i~1

d2i, max

vuut : 28

3 Calculation example

A Hardfill dam is 50 m in height and 6 m in width of the

dam crest, and the upstream and downstream slopes are

both 1:0.7. The Hardfill material is assumed to be elastic,

and its properties are: mass density rH5 2200 kg/m3,Youngs modulus EH5 2 GPa, Poissons ratio nH5 0.25and those of the concrete face plate are: rC5 2400 kg/m

3,EC5 20 GPa, nC5 0.167.The width of the concrete faceplate increase as the depth like this:

a~0:3z0:00235z: 29Three schemes are respectively calculated by SWM and

FEM to analyze the effects of upstream concrete face

plate and hydrodynamic pressure. Three schemes are

given in Table 1.

The first four natural frequencies and mode shapes of

the dam calculated by SWM and FEM are compared

under three schemes. Then the accuracy of SWM to ana-

lyze free vibration characteristics of Hardfill dams is illu-

strated (Table 2).

In general, the first four natural frequencies of the

three schemes are closer between SWM and FEM as

Table 2 shows. Results by SWM are a little higher than

FEM and the gap is almost within 10%.The effects ofconcrete face plate and hydrodynamic pressure on nat-

ural frequencies are almost the same by SWM and

FEM. From Scheme 1 and Scheme 2, when the effect

of the face is considered, natural frequencies have dif-

ferent degrees of increase within 0.5% by SWM, andthe increase is a little larger by FEM. From Scheme 2

and Scheme 3, natural frequencies under full storage

have different degrees of decreases by 10%, which iscorrect in analysis by SWM and FEM. In general, nat-

ural frequencies of the three schemes by SWM and

FEM have few discrepancies.

The first four modes by SWM and FEM are shown in

Fig. 3, and the effects of concrete face and hydrodynamic

pressure (Scheme 3) are considered. And the results by

FEM choose the mode displacement of the central axis

of the dam. From Fig. 3, it is observed that differences of

the first four modes by SWM and FEM are pretty small.

The differences of higher mode become larger between the

two methods.

The influence of concrete face onmodes is much smaller

than natural frequencies. Because the ratio of concrete

Table 1 Three calculating schemes

effect of concrete face plate effect of hydrodynamic pressure

Scheme 1 ignored ignored

Scheme 2 considered ignored

Scheme 3 considered considered

Table 2 Frequencies of three schemes by SWM and FEM/Hz

Scheme 1 Scheme 2 Scheme 3

SWM FEM SWM FEM SWM FEM

1st 4.62 4.07 4.64 4.17 4.28 3.93

2nd 10.60 9.60 10.66 9.90 10.12 9.31

3rd 16.61 15.32 16.71 15.78 16.04 14.51

4th 22.63 21.38 22.75 21.85 21.97 19.55


and Hardfill material modulus is about 10, and effect of

the face can be neglected when we calculate free vibration

characteristics of Hardfill dams by SWM, which is con-

sistent with what Ref. [15] shows.

By comparing the first four natural frequencies and

modes of the two methods, a conclusion is obtained: the

free vibration characteristics of Hardfill dam can be cal-

culated by shear wedge method as accurately as finite

element method.

Once the free vibration characteristics are given by

shear wedge method, seismic responses including seismic

displacement, velocity and acceleration can be calculated

by response spectrum method. Quasi-velocity response

spectrum and velocity response spectrum have little dif-

ference for most earthquake movement that they can be

approximately equal

viSd,i~Spv,i~1

viSa,i&Sv,i: 30

Standard response spectrum in Specifications for

seismic design of hydraulic structures(SL 20397) is

applied [16]. The maximum value of the spectrum,

characteristic period of the court and the largest lateral

acceleration are 2.0, 0.2 s and 1.96 m/s2. Acceleration

response spectrum of the dam can be obtained in

accordance with natural periods. Then velocity res-

ponse spectrum and displacement response spectrum

can be calculated by Eq. (30). The maximum lateral

seismic response is calculated by SRRS method in the

final.

The distributions of seismic displacement, velocity and

acceleration of the Hardfill dam by response spectrum

method are shown in Fig. 4. Responses increase along

the dam height and the maximum values all appear on

the top of the dam. Distribution of seismic velocity by

SWM is absolutely consistent with those by FEM. The

differences of seismic displacement and acceleration

between SWM and FEM become larger along the dam

height. Largest gaps between the two methods are

0.792 mm and 1.49 m/s2, which occur on the dam crest.

Below half of the dam height, distributions of seismic

acceleration by SWM and FEM are almost the same.

Generally speaking, seismic responses by SWM and

FEM are appropriate.

Through comparison of natural frequencies, modes and

seismic responses between shear wedge method and finite

element method, shear wedge method is appropriate for

use in seismic design of Hardfill dams.

Fig. 3 First four mode shapes of Scheme 3 by SWM and FEM


4 Conclusions

Nowadays, computer technology and finite element

method have had much development. However, there is

still superiority of shear wedge method. It is particularly

suitable for feasibility study at comparative stage because

its simpler to calculate acceleration distribution along the

height of the dam.

The closed-form natural frequencies and modes for-

mulas of Hardfill dams are derived in this paper based

on one-dimensional shear wedge theory. Then the effects

of concrete face and hydrodynamic pressure of water in

the reservoir on the free vibration characteristics of the

dam are studied: natural frequencies have some reduction

caused by water and have a little increase by concrete face.

The differences of natural frequencies between SWM

and FEM are within 10%. And the differences of modesare much smaller than natural frequencies between the

two methods. Furthermore, seismic responses of

Hardfill dams including seismic displacement, velocity

and acceleration are calculated using response spectrum

method by SWM and FEM. The distributions of these

seismic responses along the dam height by SWM are close

to results by FEM.

In general, shear wedge method is accurate and reliable

for calculating free vibration characteristics and seismic res-

ponses of Hardfill dams. The distribution laws of the seis-

mic responses are helpful for the design of Hardfill dams.

Acknowledgements This study was supported by the National NaturalScience Foundation of China (Grant No. 50679058).

References

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International Symposium on Roller Compacted ConcreteDams, Nov. 2003, 457464

2. Hirose T, Fujisawa T, Nagayama I, et al. Design Criteria forTrapezoid-Shaped CSGDams. In: ICOLD-69th Annual meet-ing, Dresden, 2001

3. Mete Oner. Shear vibration of inhomogeneous earth dams inrectangular canyons. Soil Dynamics and EarthquakeEngineering, 1984, 3(1):1926

4. Mejia L H, Seed H B, Lysmer J. Dynamic analysis of earthdams in three dimensions. J Geotech Eng Div ASCE, 1982,108(GT12): 13541376

5. Dakoulas P, Hsu C. Lateral response of dams in semi-ellipticalrigid canyons. Soil Dynamics and Earthquake Engineering,1993, 12: 497507

6. Ahmed M. Abdel-Ghaffar, Sik-Siong Koh. Longitudinalvibration of non-homogeneous earth dams. EarthquakeEngineering and Structural Dynamics, 1981, 9: 279305

7. George Gazetas. Vertical oscillation of earth and rockfilldams: analysis and field Obsercation. Soils and Foundations,1981, 21(4): 5668

8. Panos Dakoulas, George Gazetas. Seismic shear vibration ofembankment dams in semi-cylindrical alleys. EarthquakeEngineering and Structural Dynamics, 1986, 14: 1940

9. Shen Zhenzhong, Xu Zhiying. Approximate solution to ver-tical vibration of earth-rock dams in triangular canyons.International Journal Hydroelectric Energy, 1999, 17(4): 912 (in Chinese)

10. Shen Zhenzhong, Xu Zhiying. Simplified analysis of trans-verse vibration for inhomogeneous earth-rock dams in tri-angular canyons. Journal of Hydraulic Engineering, 2002,(3): 7479 (in Chinese)

11. Shen Zhenzhong, Xu Zhiying. Approximate solution to ver-tical vibration of earth-rock dams in triangular canyons.Journal of Hohai University, 2002, 30(2): 8589 (in Chinese)

12. Kong Xianjing, Liu Jun, Han Guocheng et al. Shear wedgebeam method of seismic response analysis for concrete facedrockfill dams (CFRDs). Journal of Hydraulic Engineering,2000, (7): 5560 (in Chinese)

13. Kong Xianjing, Han Guocheng. Wave motion-shear wedgebeam method of seismic response analysis for embankmentand soil deposit. Journal of Dalian University ofTechnology, 1994, 34(2): 173179 (in Chinese)

14. Luan Maotian, Li Zhan. Equivalent linearization techniquebased on discrete shear-slice model for nonlinear seismic

Fig. 4 Seismic responses by SWM and FEM


response of embankments. Chinese Journal of RockMechanics and Engineering, 2006, 25(1): 4046 (inChinese)

15. Kong xianjing, Han Guocheng, Li junjie, Lin Gao. Influenceof facing slab on free vibration characteristics of rockfill dam.

Journal of Dalian University of Technology, 1989, 29(5): 583588 (in Chinese)

16. Ministry of Water Resources P. R. China. Specifications forSeismic Design of Hydraulic Structures (SL 20397). Beijing:China Water Power Press, 1997


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