Response Surface Method (RSM)

Table 5.13: Reliability indices and probabilities of failure (SFEM-FORM-Case II) Member β Pf 1 3.2598 0.00056 2 3.2177 0.00065 3 5.5177 1.72E-08 4 7.3255 1.19E-13 5 3.2675 0.00054 6 3.5368 0.00020 7 3.0731 0.00106 8 5.6005 1.07E-08 9 5.2553 7.39E-08 10 3.0980 0.00097

can be performed in two ways. One approach is to find a closed-form (explicit) relationship between the applied load and the internal member forces. It will lead to finding a relationship such as fi = mi× F for each one of the truss members. A regression method is used to find the relationship between the applied force (F ) and member internal forces (fi). This approach can be referred to as a quasi-response surface method since it only deals with the response of load effect part of the performance function (only considers the load effect random variables). Another approach is to get the response of the whole structure for the performance function. This way a response surface can be formed with respect to all of the random variables involved in the limit state function (F and fy). In such a way in each deterministic analysis the whole limit state function G needs to be assessed. Here, the first method was chosen since the resistances are already explicitly available with respect to the yield stress, and regression is only executed to approximate the response of the system due to the load effect random variable F .

A least-squares regression method is used to find a closed-form expression for the element internal forces with respect to the applied load. Five regression values were considered for the applied load F. These values are µ−2×σ, µ−σ, µ, µ+σ, µ+2×σ (factorial method). They are respectively 52.5, 113.1, 174, 234.9, 295.8. For each one of these values a deterministic finite element analysis is done, and each time the member internal forces are stored in a vector. Through all of the obtained values a regression is done and the response of each member to the applied load is evaluated. The obtained results from the finite element analysis for different members of the ten-bar truss are listed in Table 5.14.

Table 5.14: Internal forces due to five different applied loads

F f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 -52.2 -66.64 -26.13 6.92 5.19 66.47 -58.58 58.87 12.03 -8.65 43.55 -113.1 -144.39 -56.61 15.00 11.25 144.01 -126.92 127.56 26.06 -18.75 94.35 -174 -222.14 -87.09 23.08 17.31 221.56 -195.26 196.24 40.09 -28.85 145.15 -234.9 -299.89 -117.57 31.16 23.37 299.10 -263.60 264.92 54.12 -38.94 195.96 -295.8 -377.64 -148.06 39.23 29.42 376.65 -331.94 333.61 68.15 -49.04 246.76

The first Column of Table 5.14 shows the values of the applied loads and the columns 2 through 11 display the internal member forces in each of the components of the truss. The minus sign shows that the member is in compression, whereas the positive sign shows that the member is in

tension. The minus sign for the applied loads is only to show the direction of the applied load, which means that the load is applied downwards. Using these values, it is possible to perform a least squares regression (using built-in functions of Matlab) and find an explicit expression for the truss members. Table 5.15 shows the obtained closed-form expression for the load effect of each member together with the resistances.

Table 5.15: Closed-form expressions for load effect and resistance

Component Load Effect Resistance Member 1 E1= −1.2767F R1= 1.2293fy Member 2 E2= −0.5005F R2= 0.4778fy Member 3 E3= +0.1326F R3= 0.1885fy Member 4 E4= +0.0995F R4= 0.1885fy Member 5 E5= +1.2733F R5= 1.2260fy Member 6 E6= −1.1222F R6= 1.1352fy Member 7 E7= +1.1278F R7= 1.0490fy Member 8 E8= +0.2304F R8= 0.3310fy Member 9 E9= −0.1658F R9= 0.2257fy Member 10 E10= +0.8342F R10= 0.7783fy

Looking at the expressions in the table, it can be seen that the intercept for all of the internal force explicit expressions is zero and the internal force in each member is just a coefficient multiplied by the applied external force (m). For the resistance, in the case of compression members the coefficient that reduces the resistance (χ) was multiplied by the area and was divided by 1000 so that the resistance values are in kN. In case of tension members the coefficient is just the area divided by 1000 (refer to Table 5.6). As was mentioned earlier, the limit state function is explicitly available with respect to the resistances.

Once the closed-form expression for the member internal forces is established, it is possible to do a Monte Carlo analysis of the strength limit state function of each member. Random numbers between zero and one can be generated, and using the inverse cumulative limit state functions of the resistance and the load effect numerous realisations will be extracted and the number of failures will be counted. Dividing the number of failures for each of the performance functions by the total number of simulations will yield the probability of failure. Two types of Monte Carlo analysis were performed: direct Monte Carlo analysis and Latin Hypercube sampling. For the direct Monte Carlo analysis 10,000; 100,000; 1,000,000 and 3,000,000 simulations were performed and the simulation results for each member are presented in Tables 5.16 and 5.17 as shown below.

Table 5.16: Reliability indices and probabilities of failure RSM-DMCS (Case I) n 10000 100000 1000000 3000000 members Pf β Pf β Pf β Pf β 1 0.0056 2.5364 0.00739 2.4377 0.00657 2.4801 0.00670 2.4730 2 0.0060 2.5121 0.00787 2.4149 0.00704 2.4555 0.00716 2.4494 3 0.0001 3.7190 0.00014 3.6331 0.00014 3.6387 1.62E-04 3.5948 4 0 ∞ 0 ∞ 4.00E-06 4.4652 2.67E-06 4.5512 5 0.0056 2.5364 0.00739 2.4377 0.00657 2.4800 0.00670 2.4729 6 0.0032 2.7266 0.00490 2.5828 0.00443 2.6173 0.00449 2.6130 7 0.0073 2.4422 0.00931 2.3531 0.00859 2.3828 0.00873 2.3770 8 0.0001 3.7190 0.00010 3.7190 0.00012 3.6664 1.44E-04 3.6265 9 0.0001 3.7190 0.00023 3.5030 2.37E-04 3.4950 2.68E-04 3.4621 10 0.0072 2.4471 0.00916 2.3591 0.00839 2.3913 0.00854 2.3852

In Table 5.16, the result of the Direct Monte Carlo simulation are given when the load effect follows an extreme value distribution and the resistance a lognormal distribution. In this case since both of random variables are non-normal, the number of failures is counted to calculate the failure probability (Equations 3.35 and 3.36). Then by using Equation 2.12 the reliability index is calculated.

In Table 5.17, the calculation of the results of the Monte Carlo simulation are given where both of the variables are normally distributed. In this case since both of the random variables are normal, the limit state function (G) is also normally distributed. Therefore, it is easier to calculate the mean and standard deviation by using Equations 3.38 and 3.39, and compute the reliability index using Equation 2.17. This way, the results have a reasonable accuracy even with a smaller number of simulations as is evident in Table 5.17.

Table 5.17: Reliability indices and probabilities of failure RSM-DMCS (Case II)

n 1000 10000 100000

members µ σ β Pf µ σ β Pf µ σ β Pf

1 282.04 82.19 3.43 0.00030 278.51 85.79 3.25 0.00058 278.19 85.56 3.25 0.00057 2 108.89 32.17 3.38 0.00036 107.51 33.58 3.20 0.00068 107.38 33.49 3.21 0.00067 3 54.11 9.40 5.75 4.36E-09 53.68 9.78 5.49 2.03E-08 53.65 9.73 5.51 1.76E-08 4 59.80 7.89 7.58 1.79E-14 59.44 8.18 7.27 1.85E-13 59.41 8.12 7.31 1.30E-13 5 281.28 81.97 3.43 0.00030 277.76 85.56 3.25 0.00058 277.44 85.33 3.25 0.00058 6 270.21 72.92 3.71 0.00011 267.06 76.09 3.51 0.00022 266.77 75.87 3.52 0.00022 7 234.08 72.16 3.24 0.00059 231.00 75.34 3.07 0.00108 230.71 75.15 3.07 0.00107 8 95.43 16.40 5.82 2.93E-09 94.69 17.05 5.55 1.40E-08 94.63 16.96 5.58 1.21E-08 9 63.58 11.60 5.48 2.11E-08 63.06 12.07 5.22 8.75E-08 63.01 12.01 5.25 7.78E-08 10 174.11 53.40 3.26 0.00056 171.83 55.76 3.08 0.00103 171.62 55.62 3.09 0.00101

In Table 5.17 for different number of simulations the mean values and standard deviations are shown for each member. Mean and standard deviation values are rounded up to two digits after the floating point to fit the table. The convergence of the reliability index values and probabilities of failure can be observed in Table 5.17 where the number of required simulations is smaller compared to the case of non-normal random variables (Table 5.16).

The obtained limit state equations are also evaluated using Latin Hypercube Sampling Monte Carlo simulation (LHCSMC). The results are presented in Tables 5.18 and 5.19. For LHCSMC

also 1,000; 10,000; 100,000; and 1,000,000 simulations are performed. Again, for non-normal random variables, the failure probability was calculated by counting the number of failures.

Table 5.18: Reliability indices and probabilities of failure RSM-LHCSMC (Case I)

n 1000 10000 100000 1000000 members Pf β Pf β Pf β Pf β 1 0.006 2.5121 0.0065 2.484 0.00654 2.482 0.006556 2.481 2 0.007 2.4573 0.0067 2.473 0.00706 2.454 0.007024 2.456 3 0 ∞ 0.0002 3.540 0.00014 3.633 0.000151 3.614 4 0 ∞ 0 ∞ 0 ∞ 2.00E-06 4.611 5 0.006 2.5121 0.0065 2.484 0.00654 2.482 0.006559 2.481 6 0.005 2.5758 0.0044 2.620 0.00442 2.618 0.004407 2.619 7 0.009 2.3656 0.008 2.409 0.00863 2.381 0.008637 2.381 8 0 ∞ 0.0002 3.540 0.0001 3.719 0.000131 3.650 9 0 ∞ 0.0003 3.432 0.00024 3.492 0.000256 3.474 10 0.009 2.3656 0.008 2.409 0.00847 2.388 0.00843 2.390

Table 5.19: Reliability indices and probabilities of failure RSM-LHCSMC(Case II)

n 1000 10000 100000

members µ σ β Pf µ σ β Pf µ σ β Pf

1 278.26 86.74 3.21 0.000669 278.19 85.19 3.27 0.0005467 278.18 85.35 3.26 0.000559 2 107.41 33.95 3.16 0.00078 107.38 33.35 3.22 0.0006416 107.38 33.41 3.21 0.000655 3 53.65 9.90 5.42 2.99E-08 53.65 9.69 5.54 1.526E-08 53.65 9.71 5.53 1.64E-08 4 59.41 8.28 7.18 3.57E-13 59.41 8.09 7.35 1.018E-13 59.41 8.11 7.33 1.17E-13 5 277.51 86.51 3.21 0.000669 277.44 84.97 3.27 0.0005469 277.43 85.12 3.26 0.000559 6 266.83 76.95 3.47 0.000263 266.77 75.53 3.53 0.0002064 266.76 75.68 3.52 0.000212 7 230.78 76.17 3.03 0.001223 230.71 74.84 3.08 0.0010248 230.71 74.97 3.08 0.001044 8 94.64 17.26 5.48 2.09E-08 94.63 16.89 5.60 1.047E-08 94.63 16.93 5.59 1.13E-08 9 63.02 12.22 5.16 1.25E-07 63.01 11.96 5.27 6.824E-08 63.01 11.98 5.26 7.29E-08 10 171.67 56.37 3.05 0.001161 171.62 55.38 3.10 0.0009709 171.62 55.48 3.09 0.00099

In Table 5.19 the results for the case of normal random variables are shown. The mean and standard deviation of the limit state function of each member is calculated in order to get the reliability index and probability of failure. This is performed precisely the same way as for the direct Monte Carlo analysis.

Another Method to check the results for the case where both of the random variables are normal is by using Equation 3.2 (First Order Second Moment Method). Thanks to the response surface method, a closed-form linear expression for the limit state functions is available, and since case II is being considered, it can give exact results for the reliability index as far as the obtained limit state functions are concerned. Equation 3.2 is presented here again.

β = a0+ n P i=1 aiµi s n P i=1 (aiσi)2

3.2 each one of the member limit state functions are evaluated to get the reliability index. The results are presented in Table 5.20 below.

Table 5.20: Reliability indices and probabilities of failure RSM-FOSM(Case II)

Component β Pf 1 3.2620 0.00055 2 3.2164 0.00065 3 5.5312 1.59E-08 4 7.3359 1.10E-13 5 3.2618 0.00055 6 3.5280 0.00021 7 3.0799 0.00103 8 5.5967 1.09E-08 9 5.2631 7.08E-08 10 3.0959 0.00098