Cavitation Effects on Dam Response

and the tension cutoffs have little effect as well (figs. 3.9 and 3.10). As shown in fig. 3.11, the extent of cavitation is confined to the first and second columns of elements upstream of the dam. No isolated regions of cavitation occurred.

The high frequency oscillations noted in fig. 3.8 are similar to those of fig. 3.4 and are again judged to be spurious. Their removal was accomplished by increasing the stiffness-propodional damping in the water io 3 % of criLical in the fundamental pressure mode; results appear in figs. 3.12 to 3.16 which parallel figs. 3. 7 to 3.11. Again, the linear responses shown are for zero internal damping in the water. As with the 1 % damping case, effects of the cavitation on the dam response are small (figs. 3.14 and 3.15) and the extent of cavitation is similar (fig. 3.16). Indications are from the figures that the 3 % damping in the water is still low enough not to significantly affect the fundamental components of the system response.

Part of the success of the increased damping to eliminate the spurious os- cillations lies in its reduction of the expansive strains associated with cavitation (compare positive strains in figs. 3.7 and 3.12) which, in turn, may have reduced the subsequent compressive impact, so its use may be viewed as somewhat du- bious. However, because of the high cost of analyzing the finer meshes which are needed to reduce the damping requirement, the impact responses following cavitation closure seen in fig. 3.13 are the best available estimates for the "true"

behavior. Each impact involves a single pressure spike of moderate amplitude and short duration and is too small to noticeably affect the dam response. The tension cutoffs would appear to have greater potential in this regard, but their effect is small as well.

The domination of the hydrodynamic response by high frequency compo- nents seen in [7] and [3] when cavitation wa.s included is due, apparently, to their use of either no internal damping for the water or a much smaller amount than used here. Also, their more extensive spread of cavitation in the upstream di- rection can likely be attributed to the same feature. Incidentally, the measured pressure responses of [6] show impacts without any subsequent high frequency

oscillation and with only moderate increases, if any, in compressive pressures.

This is in qualitative agreement with the results achieved here with stiffness- proportional damping in the water on the order of 1

%

to 3

%

of critical in the fundamental pressure mode. The final conclusion with regard to the effect of cavitation on the dam response is that it was found to be small in [7], [3], and here.

0

-30 300

ele 2

LO it

0

T"""

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it it

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ele 3

E

:J

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0 -100

600 ele 4

-200

!,,---!:=---.,...._---!:=---~

0 2 4 6 8

time in seconds

Figure 3.1. Volume strain time histories along the dam face (Pine Flat Dam analysis). Solid line - cavitation included, 1 % critical damping in the water.

Dashed line - linear response, no added damping in the water.

600

ele 5

0 -200

500

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ti me in seconds

Figure 3. 7. Continued.

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Figure 3. 7. Continued.

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time in seconds

Figure 3.7. Continued.

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Figure 3.8. Detailed view of the compressive volume strains shown in fig. 3.7.

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time in seconds

Figure 3.8. Continued.

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horizontal

•

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,,

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⁰

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⁰

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Figure 3.9. Horizontal and vertical motions of the dam crest at node 1 (Pine Flat Dam analysis). Solid line - cavitation included, 1 % critical damping in the water. Dashed line - linear response, no added damping in the water.

·en

a.

C:

(/) (/) (])

...

I..

(/)

1500

ELE 81

0

-1500 1500

ELE 96 0

-1500 1500

,,

_{ELE 225}

II

0

-1500 OL---4.2~----41;;---~6~---~8 time in seconds

Figure 3.10. Vertical component of the normal stress in the dam (Pine Flat Dam analysis). Solid line - cavitation included, 1 % critical damping in the water. Dashed line - linear response, no added damping in the water.

~

....

) ~><

...

,

;x )

><:

)x

~ )<._

Figure 3.11. Envelope of cavitation for the Pine Flat Dam analysis (1 % critical damping in the water).

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30

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0

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600

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time in seconds

Figure 3.12. Volume strain time histories along the dam face (Pine Flat Dam analysis). Solid line - cavitation included, 3 % critical damping in the water.

Dashed line - linear response, no added damping in the water.

600

ele 5

0

-200

500

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time in seconds

Figure 3.12. Continued.

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time in seconds

Figure 3.12. Continued.

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-100

0

-100 0

-100

0

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ele 14

ele 15

ele 16

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0 2 4 6 8

time in seconds

Figure 3.12. Continued.

0

ele 1 -25

0

V ¹ V

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,j<

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v \v·~ ·v·V\r-

0

>

ele 3 -100

0

v \v~ v··V\r-

ele 4 -130

2 3 4 6

time in seconds

Figure 3.13. Detailed view of the compressive volume strains shown in fig. 3.12.

0

-120 0

L()

jC i<

0 -r-

jC

C:

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-75 2

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time in seconds

Figure 3.13. Continued.

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^{\ I I I I ~ I I}

~

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ele 7

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0.4

0

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0

-0.1 200

0

-200 80

0

horizontal

vertical

horizontal

vertical

-80!:,,,---,!,,---L::---~~---l,

0 2 4 6 8

time in seconds

Figure 3.14. Horizontal and vertical motions of the dam crest at node 1 (Pine Flat Dam analysis). Solid line - cavitation included, 3 % critical damping in the water. Dashed line linear response, no added damping in the water.

1500

ELE 81

0

-1500 1500 'ci)

ELE 96 a.

C:

en

0

en

Q)

I,,..

- ^en

-1500 1500

ELE 225

II

0

- 1 5 0 0 k - - - ~ - - - ; ; - - - ~ - - - - " " " 7

0 2 4 6 8

time in seconds

Figure 3.15. Vertical component of the normal stress in the dam (Pine Flat Dam analysis). Solid line - cavitation included, 3 % critical damping in the water. Dashed line - linear response, no added damping in the water.

~ "><

~ >< ...

>

X

, ;x..._

)Ix

)C

~

Figure 3.16. Envelope of cavitation for the Pine Flat Dam analysis (3 % critical damping in the water).

I 00 <:11

I

References

[ 1] Bleich, H. H. and Sandler, I. S., "Interaction between structures and bilinear fluids," International Journal for Solids and Structures, Vol. ^6, No. ^{5, 1970.}

[ 2] Clough, R. W. and Chu-Han, C., "Seismic cavitation effects on gravity dams," Numerical Methods in Coupled Systems, edited by R. W. Lewis, P.

Bettes and E. Hinton, John Wiley & Sons Ltd., 1984

[ 3] Fenves, G. and Vargas-Loli, L. M., "Nonlinear dynamic analysis of fluid- structure systems," Journal of Engineering Mechanics, Vol. 114, No. 2, February 1988.

[ 4] Hall, J. F., "Study of the earthquake response of Pine Flat Dam," Earth- quake Engineering and Structural Dynamics, Vol. 14, No. 2, March-April 1986.

[ 5] Newton, R. E., "Effects of cavitation on underwater shock loading - Part I," NPS-69-78-019, Naval Postgraduate School, Monterey, California, July 1978.

[ 6] Niwa, A. and Clough, R. W., "Shaking table research on concrete dam models," Report No. UCB/EERC-80/05, University of California, Berkeley, September 1980.

[ 7] Zienkiewicz, 0. C., Paul, D. K. and Hinton E., "Cavitation in fluid-structure response ( with particular reference to dams under earthquake loading),"

Earthquake Engineering and Structural Dynamics, Vol. 11, No. 4, July- August 1983.

Chapter Four

TENSILE CRACKS IN

CO'NCRETE GRAVITY DAMS

4.1 Introduction

Experience has shown that strong ground motions are capable of producing ten- sile stresses in concrete gravity dams that are well above the tensile strength of plain concrete. From a practical viewpoint, geometrical stress concentrations, joints, weak zones, and initial cracks constitute potential factors that may lead to cracking even under moderate shaking. Long term effects including gravity loadings, temperature gradients, and chemical activities ( alkali aggregate reac- tion) are also capable of producing cracks in absence of dynamic loadings.

There are two distinct approaches that are in common usage in finite el- ement studies 'of cracks in solid media: the smeared crack approach and the discrete crack approach. In the former, cracks are smeared in the finite elements by adjusting the element constitutive description for open and closed crack cases.

The mesh, in general, can accommodate random crack patterns without mod- ifications in geometry. The mechanics at the crack tip, however, are crudely represented and may depend on the mesh design. The discrete crack approach, on the other hand, models individual cracks which, therefore, are better de- fined allowing for more accurate representation of the crack tip field. However, continual remeshing is required as crack extension takes place.