The main aim of FORM is finding the most probable point (MPP) as the nearest point on the limit state surface to the origin in the normal standard space. HL-RF as the traditional iterative FORM produces unstable and chaotic results [

21,

25,

26,

45]. On the other hand, the modified version of FORM requires a line search rule to compute a step size based on the Armijo [

38,

46], Wolfe condition [

25], or merit function [

37]. Thus, these improved FORM versions are more robust compared to HL-RF, however, they require additional iterations for evaluating a suitable step size, especially for highly nonlinear problems. FSL [

22] and CC [

21,

45] methods are computationally inefficient as they require a smaller step size than that in HL-RF (i.e., less than 1) for stable results. CHL-RF algorithm is based on a conjugate search direction using Fletcher and Reeves (FR) conjugate scalar factor, providing stable results for highly nonlinear problems more inefficiently than FSL and HL-RF [

47]. The CHL-RF is improved based on a limited FR (LFR) and dynamical chaotic finite-step size in chaotic conjugate control (CCC) to enhance its efficiency. However, the basic formulation of the conjugate search direction is extended by the FR scalar factor and therefore the effects of the previous and new grained vectors are not considered in computing the conjugate search direction [

47]. An iterative FORM algorithm based on conjugate search direction is developed in this section for the improvement of the efficiency and robustness of FORM without utilizing line search rules. This algorithm is validated by three nonlinear limit state functions and the converged results from the proposed CFORM are compared with those of HL-RF, CC, FSL, directional stability transformation method (DSTM), CHL-RF, CCC, and LFR to illustrate its performance.

#### 3.1. Conjugate Iterative Formula of FORM

The failure probability (

P_{f}) in the reliability analysis is estimated by Equation (1) [

25].

in which

g(

X) is the limit state function, separating the domain of design into failure (

g(

X) < 0) and safe (

g(

X) > 0) regions with respect to various uncertainties using the random variables of

X = (x_{1},x_{2},…,x_{n})

^{T}.

f_{X} is the joint probability density function of random variables, Φ is the standard normal cumulative distribution function, and

β is the reliability index. The iterative CFORM formula to search MPP is given as follows [

26,

38]:

in which

$U$ is the vector of normal standard random variables and

${\alpha}_{k+1}^{C}$ is the vector of conjugate unit normal at design point of

$U$_{k}. For reducing the parallel risk of the unit normal vector (

$\alpha $_{k}_{+1}) with the search direction vector,

$\alpha $_{k}_{+1} is proposed based on the conjugate search direction by

${\alpha}_{k+1}^{C}$, which is computed as:

where

d(

$U$_{k}) states the point-based conjugate search direction and is defined by Equation (4).

where

d_{k} is the vector of conjugate search direction defined as:

where

$\nabla g\left(U\right)={[\partial g/\partial {u}_{1},\partial g/\partial {u}_{2},\dots ,\partial g/\partial {u}_{n}]}^{T}$ is the gradient vector of limit state function at point

$U$. The iterative formula given in Equation (3) is used to compute the unit normal vector at

$U$_{k} based on the conjugate search direction.

Figure 1 shows schematically a cycle of the conjugate search direction vector in 2D normal space. It is illustrated that

${\alpha}_{k+1}^{C}$ is not parallel to

$\alpha $_{k}_{+1}, meaning that the new point using CFORM formula is not placed on the previous points. On the other hand, the new point is tended on the previous point

$U$_{k}. Therefore, the CFORM may converge rapidly in comparison with FORM-based steepest descent search direction. In addition, the vector

$\alpha $_{k}_{+1} is not parallel to the direction of

d(

$U$_{k}) point. Therefore, stable results with no oscillations can be provided through this formulation for highly nonlinear limit state functions while the

$\alpha $_{k}_{+1} and

$\alpha $_{k}_{−1} may locate on a same direction in HL-RF and provide

$U$_{k}_{+1} =

$U$_{k}_{−1}. This means that the HL-RF, the FSL with very large finite-step size, and CC having a large chaos control factor tended to 1 may provide unstable results for highly nonlinear problems. However, the proposed method provides stable results. Because the iterative FORM formula in Equation (2) is simply developed without any step size, the reliability index is directly computed without merit function [

37], Wolfe conditions [

25], sufficient descent condition [

25,

46], or Armijo rule [

38,

45]. Therefore, this method is simpler than the other modified versions of FORM formula.

#### 3.3. Validation of the Conjugate Reliability Analysis

In this section, the performance of the developed CFORM is compared with other algorithms including HL-RF, CC, FSL, CHL-RF, DSTM, conjugate FORM-based LFR, and CCC through the use of three nonlinear limit state functions. β and iterations are utilized to present the efficiency and robustness of the proposed CFORM. Parameters of different reliability algorithms are set as: finite-step length (λ) = 50 and adjusted factor (c_{1}) = 0.8 for FSL, λ = 0.1 and involutory matrix (C) = I for two algorithms of CC and DSTM, initial finite step length (λ_{0}) = 50 and c_{1} = 0.8 for CHL-RF, λ_{0} = 50 and limited conjugate factor (δ) = 1 for LFR, and logistic function with parameters of initial value of 0.375 and chaos scalar factor (a) = 4 for CCC. Three reliability examples are considered as follows:

Example 1: A highly nonlinear reliability problem with non-normal limit state function of Equation (7) is considered [

26].

where

x_{1} presents the Lognormal distributed variable with the mean (

μ) of 5 and standard deviation (

σ) of l, and

x_{2} presents the Gumbel distribution with

μ of 10 and

σ of 10. The converged

β was robustly obtained by CFORM after 11 iterations as

β = 3.259, while

β obtained by MCS with 1.2×10

^{6} samples was 3.501. Therefore, CFORM was more robust and efficient in solving Equation (7) compared to MCS.

Figure 2 shows the iterative histories of

β obtained from HL-RF, CC, FSL, CHL-RF, and the proposed CFORM. It is shown in the figure that HL-RF method yields to 4-periodic solutions as

β = (2.512, 1.855, 2.386, 1.597). CC and FSL robustly converged to the same

β of 3.259, but they are less efficient than the DSTM, LFR, CCC, CHL-RF, and CFORM. Unlike the CC, DSTM, and FSL, the formulation of HL-RF is iterated without chaos feedback control factor. Therefore, as can be seen in

Figure 2, the HL-RF provides periodic solutions but the improved versions of steepest descent algorithms as CC, DSTM, and FSL are robustly converged. On the other hand, the conjugate search direction in CHL-RF, CFORM, CCC, and LFR is provided by conjugate normal vector, which is not parallel to previous search directions. Therefore, as can be seen in the figure, stable results are obtained by these algorithms. The results also show the proposed method is more robust compared to HL-RF and is significantly more efficient than the other reliability methods. Notably, the CFORM converged about 5-times quicker compared with the CHL-RF method and twice quicker than CCC, DSTM, and LFR. It can also be seen in

Figure 2 that the iterations of the conjugate methods of LFR, CFORM, and CCC provide same results in the initial iteration, which means that their search directions are provided using similar normalized conjugate vector. However, the convergence of CFORM is faster than the algorithms of LFR and CCC, which is because of the applied adjusting factor of

$-\frac{0.1{\nabla}^{T}g\left({U}_{k}\right)\nabla g\left({U}_{k-1}\right)}{{\left|\right|\nabla g\left({U}_{k-1}\right)\left|\right|}^{2}}$ in the conjugate scalar factor. This factor increases the efficiency of FORM formula in the CFORM.

Example 2: A composite roof truss with compression members made of reinforced concrete, and tension bars made of steel illustrated in

Figure 3 is considered with the following serviceability limit sate function:

where

$g$ is the distributed load, and

Ac,

As,

Ec,

Es, and

l are cross-section area of reinforced concrete, cross-section area of steel bar, elastic modulus of concrete, elastic modulus of steel bar, and length of the truss member, respectively. This example included six normal independent random variables with statistical properties shown in

Table 1.

β obtained by MCS with 1.06×10

^{6} samples was 2.350. The iterative histories of

β obtained from different FORM algorithms are illustrated in

Figure 4. As shown, HL-RF, DSTM, and CC algorithms have unstable results, but the CHL-RF, FSL, CCC, LFR, and proposed CFORM converged robustly to the same

β of 2.422 after 88 (5.23 s), 44 (3.36 s), 37 (3.17 s), 38 (3.18 s), and 27 (2.61 s) iterations, respectively. The nonlinearity map of DSTM, HL-RF, and CC algorithms provides the chaotic search direction at final iterations due to their formulations while the FSL method with small finite-step length improves this drawback of the FORM-based HL-RF, CC, and DSTM. It is worth noting that, once again, the CFORM converged faster than the other converged algorithms while conjugate algorithms of CCC and LFR are shown the similar efficiency for this problem. The CFORM with nonlinear descript conjugate map provided stable results like the other conjugate approaches of CCC, LFR, and CHL-RF for this problem.

Example 3: A dam truss structure presented in

Figure 5 is considered using the following limit state function [

49]:

where

${\Delta}_{}^{z}$ is the maximum displacement at z-direction. This problem involves 32 random variables as

P_{1}–P_{7} loads,

E as Young’s modulus, and

A_{i} with

i=1, 2, 3, …, 24 as cross-section of 1–24 bars components with statistical properties shown in

Table 2.

The converged results of

β for different FORM algorithms are presented in

Figure 6. The obtained

β using MCS is 1.76 after 10

^{6} samples with CPU-run time of 26,484 s while the proposed FORM-based CFORM and FSL algorithms are converged after 21 (49.4 s) and 88 (161.6 s) iterations to

β of 1.735 and 1.944, respectively. As shown in

Figure 6, the HL-RF, DSTM, and CC show unstable chaotic results while the conjugate methods using formulation of CCC, LFR, and CHL-RF algorithms provide stable results for reliability index of 1.657 after 91 (163.4 s), 61 (113.1 s), and 98 (166.8 s), respectively. It can be conducted that the conjugate scalar factor combined with the previous conjugate vector may improve the accuracy of the results of this problem in comparison with the other FORM-based conjugate search direction as CCC, LFR, and CHL-RF while proposed CFORM closely agrees with the results of MCS compared to the FSL. Therefore, the CFORM is more robust in comparison with the FORM-based HL-RF, CC, and DSTM and it is significantly more accurate and efficient in comparison with the CCC, LFR, and CHL-RF.

The results of the three examples indicate that HL-RF, CC, and DSTM algorithms provide unstable solutions, whereas the CFORM, CCC, LFR, and FSL robustly converge. The CFORM converges quicker compared to the other reliability algorithms-based conjugate search direction of LFR, CCC, and CHL-RF. These observations indicate that the proposed CFORM provided superior results compared to existing reliability methods in terms of efficiency and robustness; hence it is selected in this study for the reliability analysis of FRP-confined concrete.