Effective Seismic Input

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When in­cor­po­rat­ing earth­quake ex­ci­ta­tion in­to the soil-struc­ture mod­el, we not on­ly have to ap­ply ef­fec­tive forces that are equiv­a­lent to the in­com­ing earth­quake, but we al­so have to ac­count for the non-lin­ear­i­ty of the struc­ture, or in oth­er words, start the tran­sient earth­quake analy­sis from a sta­t­ic state of the struc­ture.

The ef­fec­tive forces to be ap­plied to the mod­el are de­rived from a scattering analysis framework as de­scribed below, where­in we con­sid­er the dif­fer­ence  be­tween the ground mo­tion with the struc­ture and with­out, and this scat­tered mo­tion – en­tire­ly out­go­ing from the struc­ture – is ab­sorbed by a PML bound­ary.

scattered motion in the soil domain is absorbed by the boundary

The non-lin­ear analy­sis of the struc­ture can al­so be in­cor­po­rat­ed in the same con­text, as de­scribed lat­er.

Scattering analysis framework

A scat­ter­ing analy­sis frame­work, de­vel­oped by Bielak and co-work­ers as part of the ef­fec­tive seis­mic in­put method, is adopt­ed as the ap­proach for soil-struc­ture in­ter­ac­tion analy­sis in LS-DY­NA. This ap­proach con­sid­ers soil-struc­ture in­ter­ac­tion to be caused by the scat­ter­ing of the free-field ground mo­tion by the pres­ence of the struc­ture, and the ap­pro­pri­ate earth­quake forces and the use of an ab­sorb­ing bound­ary fol­low ra­tio­nal­ly from this view­point.

Con­sid­er the fol­low­ing two in­de­pen­dent states as part of a thought ex­per­i­ment:

earthquake excites free-field

(a) soil is ex­cit­ed by an earth­quake in the ab­sence of the struc­ture, and

structure scatters free-field motion

 (b)  the struc­ture dis­turbs and scat­ters the in­com­ing earth­quake wave.

 Note that both these states can­not oc­cur in re­al­i­ty: it can be ei­ther one or the oth­er.

If we take the dif­fer­ence of the foun­da­tion mo­tion in the two states, we are left with on­ly the scat­tered mo­tion, hav­ing elim­i­nat­ed both the earth­quake source and the in­com­ing wave. The scat­tered mo­tion — be­cause it is gen­er­at­ed sole­ly by the struc­ture — prop­a­gates en­tire­ly out­ward from the struc­ture.

structure causes scattered motion

Now the un­bound­ed do­main can be re­placed by a trun­cat­ed bound­ed do­main, but with­out on­ly sim­ple bound­ary con­di­tions, the out­er bound­ary will re­flect spu­ri­ous waves back to the struc­ture. This may be avoid­ed by us­ing an ab­sorb­ing bound­ary to re­duce the wave re­flec­tion and ap­pro­pri­ate­ly mod­el the un­bound­ed do­main be­yond.

absorbing boundary models the unbounded domain

We shall use per­fect­ly matched lay­ers as the ab­sorb­ing bound­ary, and the scat­tered-wave for­mu­la­tion will pro­vide the equiv­a­lent earth­quake forces, termed as the ef­fec­tive seis­mic in­put.

Theory

structure and soil domainFree-field soil domain

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:

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

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

soil structure
where ptf and eq force is the earth­quake force.

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

soil free-field
where u0 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:

scattered motion
scattered motion

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

full equation

where­in the right-hand side give the ef­fec­tive earth­quake forces that are equiv­a­lent to eq force —- 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.

ESI discontinuity

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

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.

non-linear 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.

Implementation

Ef­fec­tive seis­mic in­put has been im­ple­ment­ed in LS-DY­NA, with IN­TER­FACE_­SSI cards used to iden­ti­fy the soil-struc­ture in­ter­face, and LOAD_­SEIS­MIC_­SSI used to spec­i­fy the ground mo­tion on such an in­ter­face. Typ­i­cal­ly, on­ly ground ac­cel­er­a­tion his­to­ries are re­quired to spec­i­fy the ground mo­tion, but if ground ve­loc­i­ty and dis­place­ment curves are al­so avail­able from sig­nal pro­cess­ing of the ac­celero­grams, then the ground mo­tion may be spec­i­fied us­ing DE­FINE_­GROUND_­MO­TION.

The vari­a­tions on the IN­TER­FACE_­SSI cards (_­AUX, _­AUX_­EM­BED­DED and _­STA­T­IC) are meant for dif­fer­ent stages in the analy­sis. All the IN­TER­FACE_­SSI cards (ex­cept for _­AUX) cre­ate a tied-con­tact in­ter­face be­tween two spec­i­fied seg­ment sets, the mas­ter sur­face be­ing on the soil side and the slave on the struc­ture side.

Soil-struc­ture in­ter­ac­tion analy­sis un­der earth­quake ex­ci­ta­tion may then be car­ried out in LS-DY­NA us­ing these cards as fol­lows:

1. Car­ry out a sta­t­ic analy­sis of the soil-struc­ture sys­tem (e.g. us­ing dy­nam­ic re­lax­ation; see *CON­TROL_­DY­NAM­IC_­RE­LAX­ATION), with the soil-struc­ture in­ter­face iden­ti­fied us­ing *IN­TER­FACE_­SSI_­STA­T­IC_­ID.

Op­tion­al­ly, car­ry out a free-field analy­sis to record free-field mo­tions on the fu­ture soil-struc­ture in­ter­face, us­ing ei­ther *IN­TER­FACE_­SSI_­AUX or *IN­TER­FACE_­SSI_­AUX_­EM­BED­DED, for sur­face-sup­port­ed or em­bed­ded struc­tures re­spec­tive­ly.

2. Car­ry out the tran­sient analy­sis as a full-deck restart job (see *RESTART), with on­ly the struc­ture ini­tial­ized to its sta­t­ic stress state (see *STRESS_­INI­TIAL­IZA­TION), and the same soil-struc­ture in­ter­face iden­ti­fied us­ing *IN­TER­FACE_­SSI_­ID with the same ID as in sta­t­ic analy­sis:

  1. The struc­ture mesh must be iden­ti­cal to the one used for sta­t­ic analy­sis.
  2. The soil mesh is ex­pect­ed to be dif­fer­ent from the one used for sta­t­ic analy­sis, es­pe­cial­ly be­cause non-re­flect­ing bound­ary mod­els may be used for tran­sient analy­sis.
  3. The mesh­es for the struc­ture and the soil need not match at the in­ter­face.
  4. On­ly the struc­ture must be sub­ject­ed to sta­t­ic loads, via *LOAD_­BODY_­PARTS.
  5. The earth­quake ground mo­tion is spec­i­fied us­ing *LOAD_­SEIS­MIC_­SSI:
      1. The NODE, SET, and POINT options allow specifying the ground motion at particular nodes, node sets, or coordinate points, repectively, on the soil-structure interface.
      2. The AUX option employs mo­tions record­ed from a pre­vi­ous analy­sis us­ing *IN­TER­FACE_­SSI_­AUX or *IN­TER­FACE_­SSI_­AUX_­EM­BED­DED.
      3. The DECONV option will take a motion specified at or below the top soil surface of the model and use the SHAKE algorithm to deconvolve it to all points on the interface.

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

Results

The im­ple­men­ta­tion of ef­fec­tive seis­mic in­put in LS-DY­NA was val­i­dat­ed by analysing a mod­el of the Mor­row Point Dam and com­par­ing the re­sults against the re­sponse com­put­ed from EACD, which us­ing the sub­struc­ture method for ground mo­tion in­put and a bound­ary-el­e­ment mod­el for the foun­da­tion rock, serves as a bench­mark.

Morrow Point Dam

The up­stream-down­stream dis­place­ment am­pli­tude at the cen­ter of the crest of the dam, as com­put­ed from LS-DY­NA and from EACD, are shown be­low. It is seen that the LS-DY­NA re­sults close­ly match the se­mi-an­a­lyt­i­cal re­sults from EACD.

displacement amplitude

Examples

Building on soil

 

Soil-structure interface

The fig­ure above shows a mod­el rep­re­sent­ing a build­ing (red) up­on soil — the blue part is the elas­tic soil do­main and the green part is the PML. The fig­ure on the right shows the soil-struc­ture in­ter­face.

An ex­am­ple of a pure­ly dy­nam­ic analy­sis of this sys­tem — with­out any grav­i­ty load — is giv­en in the in­put deck ssi-dy­nam­ic.k. Tran­sient analy­sis of the sys­tem fol­low­ing an ini­tial sta­t­ic analy­sis is demon­strat­ed by a pair of in­put decks: ssi-sta­t­ic.k and ssi-tran­sient.k, to be run one af­ter the oth­er. An example with deconvolved ground motion input is given in planessi.k. The ground mo­tions used in the analy­sis are giv­en in el­cen­tro-x.ath, el­cen­tro-y.ath, and el­cen­tro-z.ath.

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

References