# Seismic Input Theory

Ef­fec­tive seis­mic in­put

We con­sid­er two pos­si­ble states of the soil do­main dur­ing an earth­quake: one with the struc­ture, and one with­out.

Con­sid­er first the struc­ture in an earth­quake, for which the equa­tions of mo­tion of the free body are:

where  is the to­tal mo­tion of the sys­tem, and  is the re­ac­tion force on the struc­ture from the soil.

For the as­so­ci­at­ed soil do­main, the free-body equa­tions of mo­tion are:

where  and  is the earth­quake force.

The same soil do­main in the ab­sence of the struc­ture will be gov­erned by the fol­low­ing equa­tion:

where  is the free-field ground mo­tion.

The scat­tered mo­tion in the soil do­main is ob­tained by tak­ing the dif­fer­ence be­tween the two, on all nodes oth­er than those on the in­ter­face with the struc­ture:

When put to­geth­er with the equa­tion of mo­tion for the struc­ture, the equa­tions for the whole sys­tem be­come:

where­in the right-hand side give the ef­fec­tive earth­quake forces that are equiv­a­lent to  —- these de­pend on­ly on the free-field ground mo­tion at the in­ter­face,  and be­cause of the spar­si­ty of the mass and stiff­ness ma­tri­ces, are con­fined to one lay­er of el­e­ments around the soil-struc­ture in­ter­face.

In oth­er words, us­ing the scat­tered mo­tion in the soil do­main cre­ates a dis­con­ti­nu­ity at the in­ter­face with the struc­ture, where the to­tal mo­tion is used, and this  dis­con­ti­nu­ity cre­ates ef­fec­tive forces at the in­ter­face. The dis­con­ti­nu­ity is ex­act­ly the free-field ground mo­tion at the in­ter­face, and thus ef­fec­tive forces de­pend sole­ly on that free-field ground mo­tion.

This is the ef­fec­tive seis­mic in­put method de­vel­oped by Bielak and co-work­ers - it di­rect­ly us­es the free-field earth­quake ground mo­tions at the soil-struc­ture  and does not re­quire their de­con­vo­lu­tion down to depth.

Non-lin­ear analy­sis of the struc­ture

Since the goal of tran­sient soil-struc­ture in­ter­ac­tion analy­sis is to pre­dict the non-lin­ear be­hav­iour of the struc­ture, the tran­sient analy­sis needs to start from a sta­t­ic state of the struc­ture. Fur­ther­more, the soil it­self may be­have non-lin­ear­ly, and this needs to be ac­count­ed for in the analy­sis. How­ev­er, the soil do­main it­self is

1. lin­ear by as­sump­tion, in or­der to al­low cal­cu­lat­ing the scat­tered mo­tion by sub­trac­tion, and
2. in­ca­pable of car­ry­ing any sta­t­ic load, be­cause (i) the sta­t­ic state is elim­i­nat­ed in cal­cu­lat­ing the scat­tered mo­tion, and (ii) the PML is meant to ab­sorb on­ly wave mo­tion and can­not sup­port sta­t­ic loads.

This con­flict may be re­solved as fol­lows:

1. As­sume that all the non-lin­ear­i­ty in the soil is lim­it­ed to a re­gion near the struc­ture, and de­fine the gen­er­al­ized struc­ture to be the phys­i­cal struc­ture it­self along with this non-lin­ear part of the soil. The rest of the soil do­main is then lin­ear and can be tak­en to be the soil do­main for the pur­pose of the in­ter­ac­tion analy­sis.

2. For the analy­sis, first cal­cu­late the sta­t­ic re­ac­tions at the base of the gen­er­al­ized struc­ture by a sta­t­ic analy­sis, and ap­ply those re­ac­tions at the base dur­ing the tran­sient analy­sis to sup­port the weight of the struc­ture and non-lin­ear soil.