W e analyze the military bridge resp onse to a random loading/. /{/1/{ CBPF/-NF/-/0/1/7///0/1/1 In tro ductionThe random stresses in military bridges /(/[/1/]/, sp ecially under retreat /(a common militarypro cedure in south america armies/: remem be r F alkland Islands/!/)/)/, are alw a ys caused b ythe irregular comp osition and sp eed of random motions/. In the sto c hastic approac h/, thebridge is assumed to b e a mec hanical system of giv en elastic prop erties/. The main problemis/, th us/, ho w to determine the sto c hastic c haracteristics of the de/ ections and stress ofthe bridge under the random excitation /(a disordered retreat/!/) with the ob jectiv et odetermine the / rst/-passage sto c hastic bridge de/ ection/, the main parameter in the bridgedesign in order to prev en t the bridge colapses /(see App endix /1/) and the marines and thearmour/-cars on the bridge fall in the jungle/'s riv er /(igarap / e/) full of piranhas/, anacondasand gian t alligators /(jacar / es/-a/ c/ us/) when in retreat/.The di/ eren tial equation go v erning the bridge de/ ection U /( x/;; t /) will tak et o b e t h eDarb oux equation/@ /4U /( x/;; t /) /@ /4x /+ /@ /2U /( x/;; t /) /@ /2t /+ / /( t /+ a /) /@U /@t /( x/;; t /)/= / /( x/;; t /)U /( x/;; t/0 /)/= f /( x /) /(/1/)Ut /( x/;; t/0 /)/= g /( x /)The external load will b e denoted b y / /( x/;; t /) and it is sup osed of the form/ /( x/;; t /)/= / /( x /; vt /) h /( t /) /(/2/)where the concen trated force h /( t /) is assumed to satisfy a gaussian /(normal/) statistich h /( t/1 /) h /( t/2 /) i /= C /( t/1 /;;t/2 /) /(/3/)In what follo ws w e supp ose /(at least/!/) that the /\cro wd/" under retreat under enem y/ re will k eep a minim um of order b ym o ving themselv es with a c onstant sp e e d v /(thereis alw a ys a patriotic sargen t/-ma jor with a taurus/-pistol /4/5 trying to imp ose order on theheard stampid/!/)/.Let us presen tly in a resumed form the solution of eq/. /(/2/)/, /(/3/)/, /(/4/)/. Its solution readsU /( x/;; t /)/= / t /; / /2 /!/ /( x/;; t /) /(/4/) /{/2/{ CBPF/-NF/-/0/1/7///0/1/ k /( t /)/= /1 /2 / Z/+ /1/;/1 dx e ik x/ /( x/;; t /) /(/5/)/ k /( t /)/= k p t nA /( k /) J/( / /; /1 /2 /) /( k /2t /)/+ B /( k /) Y/( / /; /1 /2 /) /( k /2t /) o/(/6/)where A /( k /)a n d B /( k /) are adjusted b y the initial conditions for t /= t/0 /> /0/.The particular solution is easily written do wn in terms of the homogeneous solutionseq/. /(/6/) and dep ending linearly on the /\heard stampid/" sto c hastic loading eq/. /(/3/)/.Finally /, let us presen tt h e y /( t /)t w o/-p oin t out/-put correlation function asso ciated to theo v er/-damp ed case of a harmonic oscillator with mass m and damping parameter / and theimput mo deled b y a shot noise S /( t /) with h /( / /) /2 L /2/( R /) and f ti g are p oisson distributedrandom times with exp ected arriv al rate / /(a stupid parac h ute arm y brigade with theirairplane hitted and jumping without command on the civilians ro of /'s houses since thetransp ort is on / re and going to crash/!/)S /( t /)/= Xf i g h /( t /; ti /) /(/7/)In this case/, w eh a v e the exactly form ulae for the ro of /'s t w o/-p oin t resp onseE /[ y /( t/1 /) y /( t/2 /)/] /= h y /( t/1 /) y /( t/2 /) i /= //1 m /2w /2 /exp /( /; / /( t/1 /+ t/2 /)/) Zt/1/0 d//1 Zt/1/0 d//2 exp /( / /( //1 /+ //2 /)/) /[sin /( w /( t/1 /; //1 /)/)/]/[sin /( w /( t/2 /; //2 /)/)/] //f / Z/+ /1/;/1 d/ h /( / /) h /( / /+ j //2 /; //1 j /) g /(/8/)The ab o v e written ob ject giv e us directly the out/-put pro cesses sp ectral densit yi nterms of the con v olution of the function h /( / /) /(the air/-b orne soldier w eigh t/!/) with itself/(last reference in /[/3/]/)/.Ac kno wledgmen ts /{ Luiz C/.L/. Botelho is grateful to DCP//CBPF/, Rio de Janeirofor w arm hospitalit y /. /{/3/{ CBPF/-NF/-/0/1/7///0/1App endix A /{ On the Bridge Collapse P arameterIn this App endix w ep o i n t out ho w to obtain theoretically the parameter b asso ciatedto the lev el/-crossing of the bridge de/ ection collapse v /( x/;; / /)/= b in the simple case of atensioned string/.Let us / rstly p oin t out the form of the complete w a v e equation asso ciated to the stringwith a space/-time densit y / /( x/;; t /) and a space/-time in ternal tension T /( x/;; t /) /(/\Quan tum Ge/-ometry of Bosonic Strings/-Revisted /(CBPF/-NF/-/0/4/2///9/9/) Luiz C/.L/. Botelho and Raim undoC/.L/. Botelho/)// /( x/;; t /) T /( x/;; t /) //@ /2U /( x/;; t /) /@ /2t /= /@ /@x /ar csinh //@U /( x/;; t /) /@x / //(A/./1/)The parameter b is the threshold v alue of the string de/ ection U /( x/;; t /) whic h mak esthe non/-linear term of eq/. /(A/-/1/) /(the righ/-hand side/) to safely b e substituted b y the usualterm /@ /2 /@ /2x U /( x/;; t /)/, whic h/, b y its turn/, allo ws the existence of harmonic string excitations/(and decen t guitar m usic comp ositions/!/)/.It is w orth p oin t out that ther e is no harmonic solutions of e q/. /(A/-/1/) /( U /( x/;; t /)/=A /( x /) exp /( iw t /)/) in the simple c ase of / /( x/;; t /) /=T /( x/;; t /) /; /1 v /2 /= c te/, that ma y b e p ossibleexplanation for the non/-harmonic guitar m usic of the musicists/-physicists Jorj/~ ao/-Marc/~ aoof the band /\Planet Bo y/-Uol/(/ as/)/.com/.br/"/. Ma y b e they pla y their comp ositions guitarab o v e the linear parameter b /! /(a rev erb eran t noisy/?/) /(/[/2/]/)/. /{/4/{ CBPF/-NF/-/0/1/7///0/1