Dam Analysis

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Earth­quake analy­sis of dams pro­vide an added chal­lenge be­yond analy­sis of soil-struc­ture sys­tems be­cause here there are two un­bound­ed do­mains — the wa­ter and the foun­da­tion rock, in­ter­act­ing with each oth­er as well as the dam — as op­posed to on­ly one un­bound­ed do­main in soil-struc­ture in­ter­ac­tion.

pine flat dam

We show next how the the­o­ry of ef­fec­tive seis­mic in­put for soil-struc­ture in­ter­ac­tion is ex­tend­ed to the earth­quake analy­sis of dams, and how the ex­ist­ing im­ple­men­ta­tion in LS-DY­NA is re­pur­posed for such analy­sis.

Theory

As men­tioned ear­li­er, the ef­fec­tive seis­mic in­put method is de­rived by view­ing soil-struc­ture in­ter­ac­tion as a scat­ter­ing prob­lem, where­in the pres­ence of the dam caus­es the scat­ter­ing of the free-field ground mo­tion in the lin­ear foun­da­tion do­main. This is en­tire­ly anal­o­gous to acoustic scat­ter­ing, where­in acoustic waves in a lin­ear acoustic flu­id are made to scat­ter by the pres­ence of a sol­id body. In both cas­es, the waves in a lin­ear back­ground medi­um — the soil foun­da­tion or the acoustic flu­id — are scat­tered by a sol­id struc­ture placed in the free-field, and the so­lu­tion is for­mu­lat­ed by con­sid­er­ing the scat­tered mo­tion in the back­ground medi­um, which re­places a dis­tant ex­ci­ta­tion source with equiv­a­lent ef­fec­tive forces at the in­ter­face with the struc­ture.

scattering analysis

This shows that this scat­tered-mo­tion ap­proach to solv­ing a scat­ter­ing prob­lem de­pends on­ly on the lin­ear­i­ty of the back­ground medi­um, and not on the par­tic­u­lar phys­i­cal prop­er­ties of the medi­um. There­fore, for dam analy­sis, we can con­sid­er the wa­ter do­main and the foun­da­tion rock to­geth­er as one free-field back­ground do­main, and the dam as the struc­ture. How­ev­er, we know the free-field ground mo­tion in the foun­da­tion rock on­ly, not in the wa­ter, so to de­ter­mine that, we need an ini­tial aux­il­iary analy­sis where the wa­ter do­main plays the part of the struc­ture. This re­sults in a two-step analy­sis pro­ce­dure, where we first an­a­lyze the aux­il­iary wa­ter-foun­da­tion rock to com­pute the free-field mo­tion in the wa­ter, and then use this in the analy­sis of the whole dam-wa­ter-rock sys­tem.

dam analysis

Note that there is a phys­i­cal sig­nif­i­cance to the aux­il­iary sys­tem, even though it is not a phys­i­cal en­ti­ty by it­self. The earth­quake ground mo­tion af­fects the dam not on­ly di­rect­ly through its base, but al­so through pres­sure waves in the wa­ter, ex­cit­ed by the earth­quake at the reser­voir bot­tom. The aux­il­iary sys­tem brings in the ef­fect of the far-field pres­sure waves in the fi­nal analy­sis of the whole sys­tem with­out need­ing to mod­el a large length of the reser­voir.

pressure wave through water

Example

Dam analy­sis can be for­mu­lat­ed in LS-DY­NA ac­cord­ing to two-step ap­proach out­lined ear­li­er us­ing the IN­TER­FACE_­SSI and LOAD_­SEIS­MIC_­SSI cards im­ple­ment­ed for soil-struc­ture in­ter­ac­tion.

Two pre­lim­i­nary analy­ses are re­quired — a sta­t­ic analy­sis, and the aux­il­iary analy­sis to cap­ture the ef­fect of up­stream ground mo­tion.

Sta­t­ic analy­sis

In sta­t­ic analy­sis, the foun­da­tion rock and wa­ter are con­strained on the out­er bound­ary, and they are cho­sen to be suf­fi­cient­ly large to mod­el the sta­t­ic stress dis­tri­b­u­tion. This is not as oner­ous a con­straint as in dy­nam­ic analy­sis, where the out­ward prop­a­gat­ing waves would have re­flect­ed back from a fixed out­er bound­ary. On­ly the dam and the wa­ter are sub­ject­ed to grav­i­ty load­ing, the rea­son­ing be­ing that the cur­rent state of the foun­da­tion is al­ready its de­formed shape due to aeons of grav­i­ty act­ing up­on it. The fol­low­ing fig­ures show the mod­el and the “soil-struc­ture” in­ter­face, spec­i­fied us­ing IN­TER­FACE_­SSI_­STA­T­IC to record the sta­t­ic re­ac­tions. In a man­ner sim­i­lar to soil-struc­ture in­ter­ac­tion analy­sis — where the near­by non-lin­ear part of the soil was tak­en as part of the gen­er­al­ized struc­ture — here the near-field wa­ter, which may be­have non-lin­ear­ly, is tak­en to be part of the struc­ture, and the far-field wa­ter and the foun­da­tion rock are to­geth­er tak­en to be the “soil” so­main. Of course, any near­by non-lin­ear por­tion of the foun­da­tion rock may al­so be tak­en to be part of the struc­ture as nec­es­sary.

static mesh
static interface

Aux­il­iary analy­sis

Aux­il­iary analy­sis in­volves on­ly the far-field wa­ter and the foun­da­tion rock, with the wa­ter do­main act­ing as the struc­ture, and the un­bound­ed wa­ter and foun­da­tion rock trun­cat­ed us­ing PML. The earth­quake ex­ci­ta­tion is ap­plied us­ing LOAD_­SEIS­MIC_­SSI on a seg­ment set at the wa­ter-foun­da­tion in­ter­face away from the PML bound­ary, and the mo­tions at the fu­ture “soil-struc­ture” in­ter­face are record­ed us­ing IN­TER­FACE_­SSI_­AUX for use in the sub­se­quent tran­sient analy­sis.

static mesh
static interface
static interface

Tran­sient analy­sis

In the tran­sient analy­sis of the en­tire dam-wa­ter-rock sys­tem, the di­rect earth­quake ex­ci­ta­tion at the base of the dam is ap­plied us­ing LOAD_­SEIS­MIC_­SSI on a seg­ment set, and the ef­fect of the up­stream ground mo­tion, com­put­ed in the aux­il­iary analy­sis at the “soil-struc­ture” in­ter­face, is in­cor­po­rat­ed us­ing IN­TER­FACE_­SSI, which al­so reads in the sta­t­ic re­ac­tions com­put­ed in the sta­t­ic analy­sis.

static mesh
static interface

The cor­re­spond­ing ex­am­ple in­put files for these analy­ses are dam-sta­t­ic.k, dam-aux­il­iary.k and dam-tran­sient.k, us­ing the ground mo­tion files el­cen­tro-x.ath, el­cen­tro-y.ath, el­cen­tro-z.ath. Please al­so see the README file.

NOTE: Please ig­nore er­ror mes­sages from LS-Pre­post flag­ging the IN­TER­FACE_­SSI cards as in­valid.