3D stress

3DstressMostlyfromRaganandSchultz,unpublishedNotesignconventionchangeTractionsfromstressTractionsfromstressT=S*N(MATLAB)Tractionnormalandshearcomponentssigma=dot(T,N)(MATLAB)TractionnormalandshearcomponentsS=[-50-2010;-20-30-15;10-15-120][l,m,n]=plunge_trend_to_dir_cosines(30,290);N=[l;m;n];T=S*N;%equation13.11T_mag=sqrt(sum(T.^2));%normalizecomponentsofTtogetitsdirectioncosineslt=T(1)./T_mag;mt=T(2)./T_mag;nt=T(3)./T_mag;%plottractionvector[plunge,trend]=dir_cosines_to_plunge_trend2(lt,mt,nt);%weknowtheorientationofthenormaltraction,butwhatisitsmagnitude?sigma=dot(T,N);%equation13.13%Nowforthesheartraction;usetheMcKenzieconstructionB=cross(T,N);%vectornormaltotheplanecontainingTandNB_mag=sqrt(B(1)^2+B(2)^2+B(3)^2);lb=B(1)./B_mag;mb=B(2)./B_mag;nb=B(3)./B_mag;Ts=cross(N,B);%sheartractiondirectionTs_mag=sqrt(Ts(1)^2+Ts(2)^2+Ts(3)^2);Ts(1)=Ts(1)./Ts_mag;Ts(2)=Ts(2)./Ts_mag;Ts(3)=Ts(3)./Ts_mag;%let'scheckthatthenormalandsheararecomponentsofthetractiontestmag=sqrt(sum(sigma.^2+Ts_mag.^2));tractionvectorcomponentsare6.466010.9900-44.8311tractionmagnitude46.6091tractionvectordirectioncosines0.13870.2358-0.9619tractionplunge=74.1trend=239.5normaltractionmag-29.44sheartractionmag36.13checkthatcomponentsmakesamelengthastraction:46.6091=?=46.6091N,sigmatauTBTractionfromprincipalstressesTractionfromprincipalstressesalpha1=45;alpha2=60;S=[12000;0800;0020]l=cosd(alpha1);m=cosd(alpha2);n=sqrt(1-l.^2-m.^2);N=[l;m;n];[plunge,trend]=dir_cosines_to_plunge_trend(l,m,n);%poletoplaneld=-l;md=-m;nd=cosd(asind(n));[dip,dipdir]=dir_cosines_to_plunge_trend(ld,md,nd);%planeitselfT=S*N;%equation13.11T_mag=sqrt(sum(T.^2));%normalizecomponentsofTtogetitsdirectioncosineslt=T(1)./T_mag;mt=T(2)./T_mag;nt=T(3)./T_mag;%plottractionvector[plunge,trend]=dir_cosines_to_plunge_trend2(lt,mt,nt);plotdiamond(plunge,trend);%weknowtheorientationofthenormaltraction,butwhatisitsmagnitude?sigma=dot(T,N);%equation13.13%Nowforthesheartractionuse;theMcKenzieconstructionB=cross(T,N);%vectornormaltotheplanecontainingTandNB_mag=sqrt(B(1)^2+B(2)^2+B(3)^2);lb=B(1)./B_mag;mb=B(2)./B_mag;nb=B(3)./B_mag;Ts=cross(N,B);%sheartractiondirectionTs_mag=sqrt(Ts(1)^2+Ts(2)^2+Ts(3)^2);Ts(1)=Ts(1)./Ts_mag;Ts(2)=Ts(2)./Ts_mag;Ts(3)=Ts(3)./Ts_mag;S=1200008000020Pl=0.7071m=0.5000n=0.5000tractionvectorcomponentsare84.852840.000010.0000tractionmagnitude94.3398tractionvectordirectioncosines0.89940.42400.1060tractionplunge=6.1trend=25.2normaltractionmag85.00sheartractionmag40.93checkthatcomponentsmakesamelengthastraction:94.3398=?=94.3398N,sigmatauTB-O.ZielkeFullstresstensorrotationexampleGivenS=-100000-50000-10Withs1oriented020,s2oriented110ands3verticalS=s1000s2000s3xprimel=0.9397m=0.3420n=0.0000yprimel=-0.3420m=0.9397n=0.0000zprimel=-0.0000m=-0.0000n=1.0000checksfororthogonality:xy0.0000xz0.0000yz0.0000R=0.9397-0.342000.34200.93970001.0000rotatedS=-94.151116.0697016.0697-55.8489000-10.0000%MATLABmainscript%buildrotationmatrix2(xprimetrend,xprimeplunge,yprimetrend,yprimeplunge,zprimetrend,zprimeplunge,talkandplot)R=buildrotationmatrix2(20,0,110,0,0,90,1)S=[-10000;0-500;00-10]rotatedS=R'*S*Rfunction[R]=buildrotationmatrix2(xprimetrend,xprimeplunge,yprimetrend,yprimeplunge,zprimetrend,zprimeplunge,talkandplot)%Thistakestheorientationoftheoriginalcoordinatesystemastrendsand%plungesand%computesthedirectioncosinesofthefinalcoordinatesystemandthen%providestherotationmatrixoftheform:%ll'l''%R=mm'm''%nn'n%assumestheanglesareindegrees%assumethatyouwanttorotatetonorthsouthX=[100];Y=[010];Z=[001];[xprime(1),xprime(2)xprime(3)]=plunge_trend_to_dir_cosines(xprimeplunge,xprimetrend);[yprime(1),yprime(2)yprime(3)]=plunge_trend_to_dir_cosines(yprimeplunge,yprimetrend);[zprime(1),zprime(2)zprime(3)]=plunge_trend_to_dir_cosines(zprimeplunge,zprimetrend);%checktomakesureyouputtheminrightandtheyareorthogonal:checkxy=dot(xprime,yprime);checkxz=dot(xprime,zprime);checkyz=dot(yprime,zprime);iftalkandplot==1fprintf(1,'xprimel=%3.4fm=%3.4fn=%3.4f\n',xprime(1),xprime(2),xprime(3))fprintf(1,'yprimel=%3.4fm=%3.4fn=%3.4f\n',yprime(1),yprime(2),yprime(3))fprintf(1,'zprimel=%3.4fm=%3.4fn=%3.4f\n',zprime(1),zprime(2),zprime(3))fprintf(1,'checksfororthogonality:xy%3.4fxz%3.4fyz%3.4f\n',checkxy,checkxz,checkyz)primitive1;text(0,1,'N','HorizontalAlignment','center','VerticalAlignment','bottom')plotdiamond(xprimeplunge,xprimetrend);plotdiamond(yprimeplunge,yprimetrend);plotdiamond(zprimeplunge,zprimetrend);%plotfinalcoordinatesystem[plunge,trend]=dir_cosines_to_plunge_trend2(X(1),X(2),X(3));plotpoint(plunge,trend);[plunge,trend]=dir_cosines_to_plunge_trend2(Y(1),Y(2),Y(3));plotpoint(plunge,trend);[plunge,trend]=dir_cosines_to_plunge_trend2(Z(1),Z(2),Z(3));plotpoint(plunge,trend);endR(1,1)=X*xprime';R(1,2)=X*yprime';R(1,3)=X*zprime';R(2,1)=Y*xprime';R(2,2)=Y*yprime';R(2,3)=Y*zprime';R(3,1)=Z*xprime';R(3,2)=Z*yprime';R(3,3)=Z*zprime';ExampleapplicationtoSouthMountainsfaults%SetupReceiverFault%WeareinterestedinaplanewithP(75,225)poleplunge=45;poletrend=225;[l,m,n]=plunge_trend_to_dir_cosines(poleplunge,poletrend);ld