makeup problem solution

Phong Nguyen Structural Reliability_Makeup Problem 1

Part I: My own Problem

Question 1:

a. Euler Bernoulli Beam Theory

The limit state function:

No. Iteration Variables Variable values gradients Direction

cosinesReliability

index

1X1 150000000 0.0003522 0.0865

3.35268X2 70000000 0.0004537 0.1115X3 12 -0.0040297 -0.9900

2X1 147824249.912 0.0006654 0.1569

3.32944X2 68691956.555 0.0008603 0.2029X3 21.9574 -0.0040988 -0.9666

3X1 146081907.58 0.0006735 0.1566

3.32944X2 67635876.107 0.0008736 0.2031X3 21.654 -0.0041562 -0.9665

4X1 146088824.86 0.0006735 0.1566

3.32944X2 67632683.595 0.0008736 0.2032X3 21.654 -0.0041562 -0.9665

Probability of failure (AFOSM): pf=Φ (−3.32944 )=4.351 x 10−4


Monte Carlo Simulation: pf=4.432 x10−4 ;COV=11%

b. Timoshenko beam theory



cosinesReliability

index

1

X1 150000000 0.000355 0.0866

3.29522X2 70000000 0.000457 0.1114X3 12 -0.004062 -0.9900X4 0.3 -1.66e-006 -0.0004

2

X1 147859795.967 0.000666 0.1559

3.27315X2 68715182.459 0.00086 0.2012X3 21.7868 -0.00413 -0.9671X4 0.3 -3.06e-006 -0.0007

3

X1 146173844.533 0.000674 0.1556

3.27315X2 67694908.156 0.000872 0.2015X3 21.496 -0.004187 -0.9671X4 0.3 -3.06e-006 -0.0007

4

X1 146180296.405 0.000674 0.1556

3.27315X2 67691930.201 0.000872 0.2015X3 21.496 -0.004187 -0.9671X4 0.3 -3.06e-006 -0.0007

Probability of failure: pf=Φ (−3.27315 )=5.318 x10−4

Monte Carlo Simulation: pf=5.408 x10−4 ;COV=10.61%


Question 2:



consines

Limit state

function

1

X1 150000000 -0.11217 -0.3843

0.911917X2 70000000 -0.14503 -0.4969X3 3400 0.01983 0.0680X4 2702 0.03161 0.1083X5 16 0.224 0.7675

2

X1 159005711.73 -0.10947 -0.3797

0.00569X2 75434182.099 -0.13955 -0.4840X3 3392.781 0.020606 0.0715X4 2692.857 0.032834 0.1139X5 15.233 0.224 0.7769

3

X1 158951699.254 -0.109475 -0.3796

2.2444e-6X2 75325284.135 -0.139688 -0.4844X3 3392.361 0.020601 0.0714X4 2692.327 0.032827 0.1138X5 15.218 0.224 0.7767

4

X1 158949909.436 -0.109477 -0.3796

3.2901e-9X2 75329300.95 -0.139683 -0.4843X3 3392.365 0.020601 0.0714X4 2692.331 0.032827 0.1138X5 15.218 0.224 0.7767

At design point:


Probability of failure: pf=Φ (−3.14369 )=8.341 x 10−4

Monte Carlo Simulation: pf=8.07 x10−4 ;COV=10.9%

Question 3:

a. Euler Bernoulli Beam Theory


At design point:


Omission sensitivity factor: b. Timoshenko beam theory


At design point:



Omission sensitivity factor:

Code Matlab:

Q.1

Bernoulli beam:

%% Use AFOSM to calculate probability of failure.% Limit state function: g=L/360-5*A*W*L^4/(384*(AD-B^2));syms betav x1 x2 x3 x4syms z lamdaformat long%% Inputb=0.2; h=0.5; L=12;p=2;muy=[150e6; 70e6; 12]; %Mean values of Ec,Em and Wcov=[0.05; 0.05; 0.25]; % coefficients of variantion of Ec,Em and W.xichma=cov.*muy; % Standard deviation.%%J1=int((z/h+0.5)^p,z,-h/2,h/2);J2=int(z*(z/h+0.5)^p,z,-h/2,h/2);J3=int(z^2*(z/h+0.5)^p,z,-h/2,h/2);AA=b*(x1-x2)*J1+x2*b*h;BB=b*(x1-x2)*J2;DD=b*(x1-x2)*J3+x2*b*h^3/12; gg=L/400-5/384*AA*x3*L^4/(AA*DD-BB^2);g1=diff(gg,x1);g2=diff(gg,x2);g3=diff(gg,x3); beta_trial=3; % guessed value of raliability index;tolerance=10^-7; % interation error.itermax=50; % the maximum number of iterationx0=muy; % Guessed valueser=1;i=1;


while er > tolerance beta(i)=beta_trial; x(:,i)=x0; g(1,i)=subs(g1,{x1,x2,x3},{x(1,i),x(2,i),x(3,i)}); g(2,i)=subs(g2,{x1,x2,x3},{x(1,i),x(2,i),x(3,i)}); g(3,i)=subs(g3,{x1,x2,x3},{x(1,i),x(2,i),x(3,i)}); grad(:,i)=g(:,i).*xichma; %Gradients in new coordinate system anpha(:,i)=grad(:,i)/sqrt(sum(grad(:,i).^2)); xnew(:,i)=muy-betav*anpha(:,i).*xichma; A=b*(xnew(1,i)-xnew(2,i))*J1+xnew(2,i)*b*h; B=b*(xnew(1,i)-xnew(2,i))*J2; D=b*(xnew(1,i)-xnew(2,i))*J3+xnew(2,i)*b*h^3/12; gbeta=L/400-5/384*A*xnew(3,i)*L^4/(A*D-B^2); betar=(double(solve(gbeta,betav))); %solve reliability index x0=muy-min(betar(betar>0))*anpha(:,i).*xichma; er=abs((min(betar(betar>0))-beta(i))); % error of reliability index beta_trial=min(betar(betar>0)); betaf(i)=min(betar(betar>0)); i=i+1; if i>itermax, break, endendpf=normcdf(-betaf);

Timoshenko Beam:

%% Use AFOSM to calculate probability of failure.% Limit state function: g=L/400-[(AD-B^2)pi^2/L^2+AH]W/[H(AD-B^2)pi^4/L^4];syms betav x1 x2 x3 x4syms zformat long%% Inputb=0.2; h=0.5; L=12;p=2;muy=[150e6; 70e6; 12; 0.3]; %Mean values of Ec,Em and W, poison ratiocov=[0.05; 0.05; 0.25; 0.1]; % coefficients of variantion of Ec, Em, W, poison ratio.xichma=cov.*muy; % Standard deviation.%%J1=double(int((z/h+0.5)^p,z,-h/2,h/2));J2=double(int(z*(z/h+0.5)^p,z,-h/2,h/2));J3=double(int(z^2*(z/h+0.5)^p,z,-h/2,h/2));AA=b*(x1-x2)*J1+x2*b*h;BB=b*(x1-x2)*J2;DD=b*(x1-x2)*J3+x2*b*h^3/12;HH=(5/12)*AA/(1+x4); gg=L/400-1.273*((AA*DD-BB^2)*pi^2/L^2+AA*HH)*x3/(HH*(AA*DD-BB^2)*pi^4/L^4);g1=diff(gg,x1);g2=diff(gg,x2);g3=diff(gg,x3);g4=diff(gg,x4); beta_trial=3; % guessed value of raliability index;


tolerance=10^-7; % interation error.itermax=10; % the maximum number of iterationx0=muy; % Guessed valueser=1;i=1;iternum=5;for i=1:iternum beta(i)=beta_trial; x(:,i)=x0; g(1,i)=subs(g1,{x1,x2,x3,x4},{x(1,i),x(2,i),x(3,i),x(4,i)}); g(2,i)=subs(g2,{x1,x2,x3,x4},{x(1,i),x(2,i),x(3,i),x(4,i)}); g(3,i)=subs(g3,{x1,x2,x3,x4},{x(1,i),x(2,i),x(3,i),x(4,i)}); g(4,i)=subs(g4,{x1,x2,x3,x4},{x(1,i),x(2,i),x(3,i),x(4,i)}); grad(:,i)=g(:,i).*xichma; %Gradients in new coordinate system anpha(:,i)=(grad(:,i)/sqrt(sum(grad(:,i).^2))); xnew(:,i)=muy-betav*anpha(:,i).*xichma; A=b*(xnew(1,i)-xnew(2,i))*J1+xnew(2,i)*b*h; B=b*(xnew(1,i)-xnew(2,i))*J2; D=b*(xnew(1,i)-xnew(2,i))*J3+xnew(2,i)*b*h^3/12; H=(5/12)*A/(1+xnew(4,i)); gbeta=L/400-1.273*((A*D-B^2)*pi^2/L^2+A*H)*xnew(3,i)/(H*(A*D- B^2)*pi^4/L^4); %limit state function in term of beta betar=(double(solve(gbeta,betav))); %solve reliability index betaf(i)=min(betar(betar>0)); x0=muy-min(betar(betar>0))*anpha(:,i).*xichma; er=abs((min(betar(betar>0))-beta(i))); % error of reliability index beta_trial=min(betar(betar>0)); endpf=normcdf(-betaf)

Q2.

%% Use New FOSM (Rackwitz-Fiessler algorithm) to calculate probability of failure.% Limit state function: g=L/360-5*A*W*L^4/(384*(AD-B^2));syms x1 x2 x3 x4 x5syms z lamdaformat long%% Inputb=0.2; h=0.5; L=5;p=2;muy=[150e6; 70e6; 3400; 2702; 16]; %Mean values of Ec,Em and pro_c, pro_m, omega_pcov=[0.05; 0.05; 0.01; 0.01; 0.02]; xichma=cov.*muy; % Standard deviation.%%

omega=1/14073748835532800*(-318954283212249450785993520220890*x3*x1-...636365057765537158911562306801735*x4*x1-636365057765537159064790073939380*x3*x2-...1269643098213150832808730680598670*x4*x2-1228285163894519192601457852416*x1*pi*x4-...1228285163894519192601457852416*x2*pi*x3+1228285163894519192601457852416*x2*pi*x4+...1228285163894519192601457852416*x1*pi*x3+...(-775950152915851682223433388725080334688732732710789356705546240*x3*x1*x2*pi*x4-...16897265771045921223758890041058086220520378114626105114624000*x1*pi^2*x4*x2*x3+...


7241685330448251953039524303310608380223019191982616477696000*x2^2*pi^2*x4^2+...8448632885522960611879445020529043110260189057313052557312000*x1*pi^2*x4^2*x2-...4827790220298834635359682868873738920148679461321744318464000*x1^2*pi^2*x4*x3-...14483370660896503906079048606621216760446038383965232955392000*x2^2*pi^2*x3*x4+...8448632885522960611879445020529043110260189057313052557312000*x2*pi^2*x3^2*x1+...7241685330448251953039524303310608380223019191982616477696000*x2^2*pi^2*x3^2-...779741890488142549074641931811401211393656294625276146525143040*x3*x1^2*pi*x4-...779741890488142549451056717955108233004107098497169948860743680*x3^2*x1*x2*pi+...1555692043403994231674490106680188567692839831207959305566289920*x4^2*x1*x2*pi+...1555692043403994231298075320536481546082389027336065503230689280*x3*x2^2*pi*x4+...100269203199468442259909210601902412590826076443469454441841148324*x3^2*x1^2+...399124254938931626098556067158630038575772652424994550311433368497*x4^2*x1^2+...395697153405116644263726859407719954223026314952595292554505684624*x3^2*x2^2+...1575030795402264382586435519595857903740975375801713688692324159172*x4^2*x2^2+...400099342658912552630490858778316650734268010539283991101585179884*x3*x1^2*x4+...387416055042219638950286615584330611099178246227970242231882679952*x3^2*x1*x2+...1541987404196006217152221930754029771503291352424995231110601756356*x4^2*x1*x2+...1578904939673404900066249094594156085375458443565438657703918244784*x3*x2^2*x4+...1545821047344481806722261812526463333008920294427829328267811034032*x3*x1*x4*x2+...2413895110149417317679841434436869460074339730660872159232000*x1^2*pi^2*x3^2-...783533628060433416302265261041429109709030660411656738680340480*x3^2*x1^2*pi+...1563275518548575965376907192852830321102686955036932885205483520*x4^2*x1^2*pi+...1563275518548575965753321978996537342713137758908826687541084160*x3^2*x2^2*pi-...3118967561952570197051397299533018888795526786244892190771773440*x4^2*x2^2*pi+...2413895110149417317679841434436869460074339730660872159232000*x1^2*pi^2*x4^2)^(1/2))/...(-113701209351838406*x3^2-454249228073565621*x3*x4-453693618739777618*x4^2+70368744177664*pi^2*x3^2-...140737488355328*pi^2*x3*x4+70368744177664*pi^2*x4^2); ggg=0.7*x5-omega^0.5;g1=diff(ggg,x1);g2=diff(ggg,x2);g3=diff(ggg,x3);g4=diff(ggg,x4);g5=diff(ggg,x5);beta_trial=3; % guessed value of raliability index;tolerance=10^-9; % interation error.itermax=5; % the maximum number of iterationx0=muy; % Guessed valueser=1;i=1;while er > tolerance x(:,i)=x0; u(:,i)=(x(:,i)-muy)./xichma; gg=double(subs(ggg,{x1,x2,x3,x4,x5},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i)}));g(1,i)=double(subs(g1,{x1,x2,x3,x4,x5},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i)})); g(2,i)=double(subs(g2,{x1,x2,x3,x4,x5},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i)})); g(3,i)=double(subs(g3,{x1,x2,x3,x4,x5},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i)})); g(4,i)=double(subs(g4,{x1,x2,x3,x4,x5},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i)})); g(5,i)=double(subs(g5,{x1,x2,x3,x4,x5},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i)})); grad(:,i)=g(:,i).*xichma; %Gradients in new coordinate system anpha(:,i)=double(grad(:,i)/sqrt(sum(grad(:,i).^2))); u(:,i+1)=((grad(:,i)'*u(:,i)-gg)/(grad(:,i)'*grad(:,i)))*grad(:,i);


x0=u(:,i+1).*xichma + muy; %neu design point in original coordinate er=abs(double((subs(ggg,{x1,x2,x3,x4,x5},{x0(1),x0(2),x0(3),x0(4),x0(5)})))); beta=sqrt(u(:,i+1)'*u(:,i+1)); betaf(i)=beta; i=i+1; if i>itermax, break, endendnormcdf(-betaf);

Part II: Lim’s Problem

Question 1:


At design point:

; ;

Probability of failure: pf=Φ (−1.6111 )=0.05358

Question 2:


Order of importance of random variables:

1 2 4 7 7 6 5 3

Question 3: probability of failure for different values of

100 150 200 250 300pf 0.0005215 0.00824 0.05358 0.17649 0.367475

We can see that The probability of failure increases rapidly when fs increases.

Figure 1: probability of failure for different values of fs

Code Matlab:

%% Use New FOSM (Rackwitz-Fiessler algorithm) to calculate probability of failure.% Limit state function: g=1.57*wc*sqrt(db1)/(k*beta*sqrt(s1*s2*dc))-c*fs;% db1,c are lognormal, others are normal.syms x1 x2 x3 x4 x5 x6 x7 x8 % wc,k,beta,s1,s2,dc,db1,cformat long%% Inputmuy=[0.35; 2.1e-5; 1.25; 80; 80; 25; 40; 1]; %Mean valuescov=[0.2; 0.15; 0.1; 0.05; 0.05; 0.06; 0.1; 0.11]; % coefficients of variantion.xichma=cov.*muy; % Standard deviation.fs=200;%% LogNormal Parameter Transformation for db1 & cshape7 = log( (muy(7)^2) / sqrt(muy(7)^2 + xichma(7)^2));


scale7 = sqrt(log((cov(7))^2 + 1));shape8 = log( (muy(8)^2) / sqrt(muy(8)^2 + xichma(8)^2));scale8 = sqrt(log((cov(8))^2 + 1)); %% Limit state function and its derivativesggg=1.57*x1*sqrt(x7)/(x2*x3*sqrt(x4*x5*x6))-x8*fs;g1=diff(ggg,x1);g2=diff(ggg,x2);g3=diff(ggg,x3);g4=diff(ggg,x4);g5=diff(ggg,x5);g6=diff(ggg,x6);g7=diff(ggg,x7);g8=diff(ggg,x8); beta_trial=5; % guessed value of raliability index;tolerance=10^-9; % interation error.itermax=20; % the maximum number of iterationx0=muy; % Guessed valueser=1;i=1; while er > tolerance x(:,i)=x0; % values in original coordinate system %% Transform from Lognormal to Equivalent Normal for x7 & x8 xichma7=x(7,i)*scale7; muy7=x(7,i)*(1-log(x(7,i))+shape7); xichma8=x(8,i)*scale8; muy8=x(8,i)*(1-log(x(8,i))+shape8); muy(7)=muy7; xichma(7)=xichma7; muy(8)=muy8; xichma(8)=xichma8; u(:,i)=(x(:,i)-muy)./xichma; % values in new coordinate system %% check limit state function and its derivatives gg=double(1.57*x(1,i)*sqrt(x(7,i))/(x(2,i)*x(3,i)*sqrt(x(4,i)*x(5,i)*x(6,i)))-x(8,i)*fs); g(1,i)=double(subs(g1,{x1,x2,x3,x4,x5,x6,x7,x8},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i),x(7,i),x(8,i)})); %Gradients in original coordinate system g(2,i)=double(subs(g2,{x1,x2,x3,x4,x5,x6,x7,x8},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i),x(7,i),x(8,i)})); g(3,i)=double(subs(g3,{x1,x2,x3,x4,x5,x6,x7,x8},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i),x(7,i),x(8,i)})); g(4,i)=double(subs(g4,{x1,x2,x3,x4,x5,x6,x7,x8},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i),x(7,i),x(8,i)})); g(5,i)=double(subs(g5,{x1,x2,x3,x4,x5,x6,x7,x8},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i),x(7,i),x(8,i)})); g(6,i)=double(subs(g6,{x1,x2,x3,x4,x5,x6,x7,x8},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i),x(7,i),x(8,i)})); g(7,i)=double(subs(g7,{x1,x2,x3,x4,x5,x6,x7,x8},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i),x(7,i),x(8,i)})); g(8,i)=double(subs(g8,{x1,x2,x3,x4,x5,x6,x7,x8},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i),x(7,i),x(8,i)}));


grad(:,i)=g(:,i).*xichma; anpha(:,i)=grad(:,i)/sqrt(sum(grad(:,i).^2)); u(:,i+1)=((grad(:,i)'*u(:,i)-gg)/(grad(:,i)'*grad(:,i)))*grad(:,i); x0=u(:,i+1).*xichma + muy; er(i)=abs(1.57*x0(1)*sqrt(x0(7))/(x0(2)*x0(3)*sqrt(x0(4)*x0(5)*x0(6)))-x0(8)*fs); beta=sqrt(u(:,i+1)'*u(:,i+1)); betaf(i)=beta; i=i+1; if er<tolerance, break,endendpf=normcdf(-betaf);Part III: Morsy’s Problem


Probability of failure for 2 elevations z

Z=2 ft Z=6 ftMethods MCS FORM SORM MCS FORM SORM

pf 0.00382 0.00441 0.00391 0.179095 0.194765 0.18619

The results obtained from second-order reliability method (SORM) are closer to Monte Carlo simulation than FORM.

At the design point (z=2 ft):

; ;


Order of importance of random variables:

5 6 3 1 4 2

Code:

%% Use New FORM (Rackwitz-Fiessler algorithm) and SORM (second-order) to calculate probability of failure.% Limit state function: g=(Tult/RF)-tan(45-phi/2)*(K/Ka)*[gama*z+qrb+qt+(qb+qll)/pi*(anpha1+sin(anpha1)*cos(anpha1+2*beta1))]*sv;% qll is exponential; qt,phi are lognormal; others are normalsyms x1 x2 x3 x4 x5 x6 % qt,qll,phi,K/Ka,Tult,RFformat long%% Inputmuy=[400; 2400; pi/6; 1; 2800; 5]; %Mean valuesvarr=[40; 240; pi^2/60/180; 0.1; 280; 0.5]; % coefficients of variantion.xichma=sqrt(varr) ; % Standard deviation.cov=xichma./muy;sv=8/12; b=2; ab=2; gama=110; qrb=300; qb=1600;z=6;beta1=atan(ab/z); anpha1=atan((b+ab)/z)-beta1;%% LogNormal Parameter Transformation for qt, qll & phishape1 = log( (muy(1)^2) / sqrt(muy(1)^2 + xichma(1)^2));scale1 = sqrt(log((cov(1))^2 + 1));shape2 = log( (muy(2)^2) / sqrt(muy(2)^2 + xichma(2)^2));scale2 = sqrt(log((cov(2))^2 + 1));shape3 = log( (muy(3)^2) / sqrt(muy(3)^2 + xichma(3)^2));scale3 = sqrt(log((cov(3))^2 + 1)); %% Limit state function and first derivativesggg=x5/x6-(tan(45*pi/180-x3/2))^2*x4*(gama*z+qrb+x1+(qb+x2)/pi*(anpha1+sin(anpha1)*cos(anpha1+2*beta1)))*sv;g1=diff(ggg,x1);g2=diff(ggg,x2);g3=diff(ggg,x3);g4=diff(ggg,x4);g5=diff(ggg,x5);g6=diff(ggg,x6); %% second derivativesg11=diff(g1,x1); g12=diff(g1,x2);g13=diff(g1,x3);g14=diff(g1,x4);g15=diff(g1,x5);g16=diff(g1,x6);g22=diff(g2,x2); g23=diff(g2,x3);g24=diff(g2,x4);g25=diff(g2,x5);g26=diff(g2,x6);g33=diff(g3,x3); g34=diff(g3,x4); g35=diff(g3,x5); g36=diff(g3,x6);g44=diff(g4,x4); g45=diff(g4,x5); g46=diff(g4,x6);g55=diff(g5,x5); g56=diff(g5,x6);g66=diff(g6,x6); %% FORMbeta_trial=5; % guessed value of raliability index;tolerance=10^-7; % interation error.


itermax=20; % the maximum number of iterationx0=muy; % Guessed valueser=1;i=1; while er > tolerance x(:,i)=x0; % values in original coordinate system %% Transform from Lognormal (x1, x2 & x3) to Equivalent Normal xichma1=x(1,i)*scale1; muy1=x(1,i)*(1-log(x(1,i))+shape1); xichma2=x(2,i)*scale2; muy2=x(2,i)*(1-log(x(2,i))+shape2); xichma3=x(3,i)*scale3; muy3=x(3,i)*(1-log(x(3,i))+shape3); muy(1)=muy1; xichma(1)=xichma1; muy(2)=muy2; xichma(2)=xichma2; muy(3)=muy3; xichma(3)=xichma3; u(:,i)=(x(:,i)-muy)./xichma; % values in new coordinate system %% check limit state function and its derivatives gg=double(subs(ggg,{x1,x2,x3,x4,x5,x6},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i)})); g(1,i)=double(subs(g1,{x1,x2,x3,x4,x5,x6},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i)})); g(2,i)=double(subs(g2,{x1,x2,x3,x4,x5,x6},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i)})); g(3,i)=double(subs(g3,{x1,x2,x3,x4,x5,x6},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i)})); g(4,i)=double(subs(g4,{x1,x2,x3,x4,x5,x6},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i)})); g(5,i)=double(subs(g5,{x1,x2,x3,x4,x5,x6},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i)})); g(6,i)=double(subs(g6,{x1,x2,x3,x4,x5,x6},{x(1,i),x(2,i),x(3,i),x(4,i),x(5,i),x(6,i)})); grad(:,i)=g(:,i).*xichma; %Gradients in new coordinate system anpha(:,i)=grad(:,i)/sqrt(sum(grad(:,i).^2)); anpha2=grad(:,i)/sqrt(sum(grad(:,i).^2)); % use form SORM u(:,i+1)=((grad(:,i)'*u(:,i)-gg)/(grad(:,i)'*grad(:,i)))*grad(:,i); x0=u(:,i+1).*xichma + muy; %neu design point in original coordinate er(i)=abs(x0(5)/x0(6)-(tan(45*pi/180-x0(3)/2))^2*x0(4)*(gama*z+qrb+x0(1)+(qb+x0(2))/pi*(anpha1+sin(anpha1)*cos(anpha1+2*beta1)))*sv); beta=sqrt(u(:,i+1)'*u(:,i+1)); gradd=grad(:,i); %% use for SORM betaf(i)=beta; i=i+1; if er<tolerance, break,endend %% Second-order reliability method (SORM);%% formulate matrix Rr0=[1 0 0 0 0 0;0 1 0 0 0 0;0 0 1 0 0 0;0 0 0 1 0 0;0 0 0 0 1 0;anpha2'];


%% Gram-Schmidt Orthogonalizationfor ii=5:-1:1 rr=zeros(1,6); for jj=ii+1:6 rr=rr+(r0(jj,:)*r0(ii,:)')/(r0(jj,:)*r0(jj,:)')*r0(jj,:); end r0(ii,:)=r0(ii,:)-rr;endr=r0;for j=1:6 r(j,:)=r(j,:)./sqrt(sum(r(j,:).^2));end%% Formulate matrix D%% Second derivativesd11=double(subs(g11,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(1)*xichma(1);d12=double(subs(g12,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(1)*xichma(2);d13=double(subs(g13,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(1)*xichma(3);d14=double(subs(g14,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(1)*xichma(4);d15=double(subs(g15,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(1)*xichma(5);d16=double(subs(g16,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(1)*xichma(6); d22=double(subs(g22,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(2)*xichma(2);d23=double(subs(g23,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(2)*xichma(3);d24=double(subs(g24,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(2)*xichma(4);d25=double(subs(g25,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(2)*xichma(5);d26=double(subs(g26,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(2)*xichma(6); d33=double(subs(g33,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(3)*xichma(3);d34=double(subs(g34,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(3)*xichma(4);d35=double(subs(g35,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(3)*xichma(5);d36=double(subs(g36,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(3)*xichma(6); d44=double(subs(g44,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(4)*xichma(4);d45=double(subs(g45,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(4)*xichma(5);d46=double(subs(g46,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(4)*xichma(6); d55=double(subs(g55,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(5)*xichma(5);


d56=double(subs(g56,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(5)*xichma(6);d66=double(subs(g66,{x1,x2,x3,x4,x5,x6},{x0(1),x0(2),x0(3),x0(4),x0(5),x0(6)}))*xichma(6)*xichma(6); D=[ d11 d12 d13 d14 d15 d16; d12 d22 d23 d24 d25 d26; d13 d23 d33 d34 d35 d36; d14 d24 d34 d44 d45 d46; d15 d25 d35 d45 d55 d56; d16 d26 d36 d46 d56 d66]; A=1/(sqrt(sum(gradd.^2)))*(r*D*r');A(6,:)=[];A(:,6)=[];k=eig(A);kk=1;for iii=1:5 kkk=(1+beta*k(iii))^-0.5; kk=kk*kkk;end pf1=normcdf(-beta);pf2=normcdf(-beta)*kk;


Part IV: Nick Kratkiewicz’s problem


g1=F yCSR−PL4; g2=Dmax−

P L3

24 ESH;

Dmax=1∈;R=1.05 ; L=20 ft ; H=13∈; E=29000 ksi ;

F y=X1 N (52,3 );C=X 2 N (0.75,0.0025 ); S=X3 N (4,0.1 ); P=X4 U (2.4,2.6 );

For Monte Carlo Simulation: pf=P (g1≤0∪ g2≤0 )=0.2065 ;

For AFORM: pf 1=P (g1≤0 )=0.1319 ; p f 1=P (g2≤0 )=0.1109 ;

In case of uni-modal: 0.1319≤ p f ≤1−(1−0.1319 ) (1−0.1109 )=0.2282

For bi-modal:

β1=1.11716 ; α1,1−4=[0.7641;0.4265 ;0.3176 ;−0.3652 ]

β2=1.22157 ;α 2,3−4=[0.7506;−0.6607 ]

So, ρ=0.3176∗0.7506+ (−0.3652 )∗(−0.6607 )=0.4797 ;

P (A )=p f 1∅ (−0.78145 )=0.02866 ;P (B )=p f 2∅ (−0.78145 )=0.02409;

So, 0.19005=0.1319+0.1109−0.05275≤ p f ≤0.1319+0.1109−0.02866=0.21414

Code Matlab:

Monte Cartlo Simulation:

syms x1 x2 x3 x4 x5 x6 % Fy, C, S, Pformat long%% Inputmuy=[52; 0.75; 4]; %Mean valuesxichma=[3; 0.025; 0.1]; % Standard deviation.. r=1.05; L=20*12; D=1; H=13;E=29000;%% Monte Carlo Simulation


Ns=1000;Nset=25;for i=1:Nset pf=0; for j=1:Ns % uniform random number u = rand(1,3); x1 = muy(1) + xichma(1)*norminv(u(:,1),0,1); x2 = muy(2) + xichma(2)*norminv(u(:,2),0,1); x3 = muy(3) + xichma(3)*norminv(u(:,3),0,1); aa=5; x4=2.35+0.05*ceil(aa.*rand(1,1)); % choose randomly x4 from [ 2.4; 2.45; 2.5; 2.55; 2.6] g1=double(x1*x2*x3*r-x4*L/4); g2=double(D-x4*L^3/24/E/x3/H); if g1>0 && g2>0 pff=0; else pff=1/Ns; end pf=pf+pff; end Pf(i)=pf;endpf=mean(Pf);cov=std(Pf)/pf;

AFORM for function g1

syms x1 x2 x3 x4 betav % Fy, C, S, Pformat long%% Inputmuy=[52; 0.75; 4; 2.5]; %Mean valuesxichma=[3; 0.025; 0.1; 0.0577]; % Standard deviation.. r=1.05; L=20*12; D=1; H=13;E=29000;%% Limit state function and its derivativesgg1=x1*x2*x3*r-x4*L/4;g1=diff(gg1,x1);g2=diff(gg1,x2);g3=diff(gg1,x3);g4=diff(gg1,x4); beta_trial=5; % guessed value of raliability index;tolerance=10^-8; % interation error.itermax=20; % the maximum number of iterationx0=muy; % Guessed valueser=1;i=1; while er > tolerance beta(i)=beta_trial; x(:,i)=x0; %% Transform from Uniform to Equivalent Normal for x4 xichma4=normpdf(norminv((x(4,i)/0.2-2.4/0.2)))/5; muy4=x(4,i)-norminv((x(4,i)/0.2-2.4/0.2))*xichma4;


xichma(4)=xichma4; muy(4)=muy4; g(1,i)=subs(g1,{x1,x2,x3,x4},{x(1,i),x(2,i),x(3,i),x(4,i)}); g(2,i)=subs(g2,{x1,x2,x3,x4},{x(1,i),x(2,i),x(3,i),x(4,i)}); g(3,i)=subs(g3,{x1,x2,x3,x4},{x(1,i),x(2,i),x(3,i),x(4,i)}); g(4,i)=subs(g4,{x1,x2,x3,x4},{x(1,i),x(2,i),x(3,i),x(4,i)}); grad(:,i)=g(:,i).*xichma; %Gradients in new coordinate system anpha(:,i)=double(grad(:,i)/sqrt(sum(grad(:,i).^2))); xnew(:,i)=muy-betav*anpha(:,i).*xichma; gbeta=xnew(1,i)*xnew(2,i)*xnew(3,i)*r-xnew(4,i)*L/4; betar=(double(solve(gbeta,betav))); %solve reliability index x0=double(muy-min(betar(betar>0))*anpha(:,i).*xichma); er=abs((min(betar(betar>0))-beta(i))); % error of reliability index beta_trial=min(betar(betar>0)); betaf(i)=min(betar(betar>0)); i=i+1; if i>itermax, break, endendpf=normcdf(-betaf);

makeup problem solution

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