Dam 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.

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.

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.