A fast decoupled reliability-based design optimization of structures using B-spline interpolation curves

Abstract

This paper introduces a new method for reliability-based design optimization (RBDO) of structures. In RBDO of structural problems unlike the conventional two-level approaches, sometimes it is not necessary to carry out reliability analysis for each deterministic design. The proposed method may be categorized as a decoupled method for reliable optimum design; however, it is based on the safety factor (SF) concept. To briefly describe the proposed method, a deterministic design optimization (DDO) point is obtained based on an arbitrary SF. The corresponding failure probability (P _f) is then determined using Monte Carlo simulation (MCS). The P _f is then compared with the targeted P _f. If the relative distance error is greater than a desirable tolerance, the cubic B-spline interpolation concept is then employed as a result of which a modified SF is extracted. For the modified SF found, DDO procedure is carried out. The above procedure is iteratively repeated until convergence occurs and the reliable optimum point is found. Finally, the proposed method was applied to solving some structural problems. The obtained results were favourably in accordance with those recorded in the literature while only a fraction of P _f computations was necessary.

1 Introduction

Reliability-based design optimization (RBDO) implements structural optimization considering simultaneously the uncertainties in the structural materials properties and/or applied loading. The general structural RBDO problem with both deterministic and probabilistic design constraints can be expressed as:

$$ \begin{aligned} &{\text{Min/Max}}\;f\left( \varvec{d} \right) \hfill \\ & {\text{Subject to:}} \hfill \\ & P\left( {G_{i} \left( {\varvec{d},\varvec{ X}} \right) \le 0} \right) \le P_{\rm f}^{i} ,\quad i = 1, \ldots ,N_{\rm PC} \hfill \\ & {\text{and/or}}\;\sigma_{j} (\varvec{d}) \le \sigma_{\rm all.}, \quad j = 1, \ldots ,N_{m} \hfill \\ & {\text{and/or}}\;u_{k} (\varvec{d}) \le u_{\rm all.}, \quad k = 1, \ldots ,N_{\rm DOF} \hfill \\ & {\text{and/or}}\;\varvec{d}^{L} \le \varvec{d} \le \varvec{d}^{U} \hfill \\ \end{aligned} $$

(1)

where $\varvec{d} = [d_{1} ,d_{2} , \ldots ,d_{n} ]^{\text{T}}$ is a column vector of n deterministic design variables, $\varvec{X} = [x_{1} ,\varvec{ }x_{2} , \ldots ,\varvec{ }x_{m} ]^{\text{T}}$ is the m-dimensional vector of random variables, $f(\varvec{d})$ is the objective function, $P\left( {G_{i} \left( {\varvec{d},\varvec{ X}} \right) \le 0} \right)$ denotes the failure probability for the i-th limit state function $G_{i} \left( {\varvec{d},\varvec{ X}} \right)$. $P_{\rm f}^{i}$ is the target failure probability of i-th constraint and N _PC is the number of probabilistic constraints. In Eq. (1), $\sigma$ and $u$ are the stress of jth member and the nodal displacement of k-th degree of freedom, respectively. $\sigma_{{{\text{\rm all}} .}}$, $u_{\rm all.}$, $\varvec{d}^{\text{L}}$, $\varvec{d}^{\text{U}}$, N _m and N _DOF are respectively allowable member stress, allowable nodal displacement, lower and upper bounds of $\varvec{d}$, total number of members of the structure and total number of degrees-of-freedom. The target failure probability can be expressed in terms of the target reliability index, $P_{\rm f}^{i} = \varPhi ( - \beta_{ti} )$, as $\varPhi (.)$ is the standard normal cumulative distribution function.

The most common routines to solve Eq. 1 include: (1) A two-level or loop-nested approach that introduces a reliability analysis as a probabilistic constraint within a deterministic optimization loop. This approach is namely referred to as the reliability index approach (RIA) based on FORM, or as the performance measure approach (PMA) based on an inverse FORM [1]. (2) A mono-level approach that attempts to solve RBDO problems by avoiding the reliability analysis [2–6]. This method reformulates the reliability constraints and replaces them by utilizing the optimality conditions resulting in a mono DDO loop. This technique expresses inconstancy in some cases since it requires calculation of second-order derivatives [7]. (3) Decoupled approaches where the concept of reliability analysis is carried out whether before or after the DDO procedure [8–10]. Recently many studies have been conducted in this field to decouple the reliability and DDO loops. However, a recent benchmark study [7] shows that such approaches may suffer from non-unicity of the most probable failure point (MPFP) in cases where they rely on the assumption of MPFP.

One of the most popular approaches to reduce computational cost of reliability analysis is the response surface method (RSM) and its adaptive versions (see for example [11–17]). Numerous studies have also been done on the application of RSM to solve structural reliability analysis and structural RBDO problems (see [17–20]). RSM employs polynomial function to approximate the unknown implicit performance function. However, RSM becomes computationally impractical for problems involving a large number of nonlinear random variables, particularly when mixed or statistically dependent random variables are involved [21].

One could refer to the study by Qu and Haftka [22] who proposed the concept similar to safety factors to relate DDO and reliability analysis, namely referred to as the probabilistic sufficiency factor (PSF). In that research PSF is computed by Monte Carlo simulation (MCS) combined with response surface method (RSM) approximations.

In addition, Wu et al. [23] presented a method in 2001 based on safety factors by which the optimization procedure and reliability analysis are employed using a decoupling approach. In fact they substituted random variables with deterministic ones based on safety factors.

The current study attempts to implement a fast method to find reliable optimum solutions, based on the concept of safety factors embedded into the essential formulation of RBDO. The deterministic optimum points are computed in an inverse manner iteratively until the global optimum point is determined whose failure probability, estimated using MCS method, meets that of the target value. The safety factor corresponding to the global optimum point is called the reliable demand factor (RDF). It is worthwhile noting that the dynamic modification of the SFs throughout the proposed approach is performed using the cubic B-spline interpolation concept. It modifies the fitting curve accuracy and reduces the relative error distance with the aim of finding a RDF encountered with the targeted failure probability. The reasons for applying MCS is its versatility for all problems, easy to implement and its capability of computing the probability of failure with the desired precision [24, 25].

The proposed decoupled RBDO approach possesses the following unique advantages: (1) it incorporates safety factors in structural design, a methodology that engineers are familiar with, (2) it identifies solutions which are near the reliable optimum design with a few number of failure probability computations, (3) it avoids computation of optimization and reliability analyses simultaneously, (4) utilization of cubic B-spline interpolation concept is due to its advantageous characteristic to fit modified curves in a piecewise manner without increasing the degree of the polynomial. It finally determines an optimum response corresponding to the target failure probability. This leads to a fast convergence of an optimum SF via a few P _f calculations, and (5) the efficient decoupling approach employed here allows for any type of reliability method to be exploited along with the optimization procedure.

Finally, some structural problems will be attempted containing two and three-dimensional truss structures and the results will be performed.

2 The proposed methodology

2.1 Constraints handling

Considering the deterministic design constraints of Eq. (1), the squared normalized degree of constraints violation V for each design, related to the member stresses and nodal displacements, may be defined as:

$$ V\left( \varvec{d} \right) = \sum\nolimits_{i = 1}^{NM} {\left[ {{ \hbox{max} }\left( {0,\left| {\frac{{\sigma_{i} \left( \varvec{d} \right)}}{{\sigma_{{{\text{all}} .}} }}} \right| - 1} \right)} \right]^{2} } + \sum\nolimits_{j = 1}^{ND} {\left[ {{ \hbox{max} }\left( {0,\left| {\frac{{u_{j} \left( \varvec{d} \right)}}{{u_{{{\text{all}} .}} }}} \right| - 1} \right)} \right]^{2} } $$

(2)

By incorporating a set of safety factors into Eq. (2), it can be modified as a squared normalized safety factor-based degree of constraints violation V’ given by:

$$ V '\left( {\varvec{d},\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }} \right) = \sum\nolimits_{i = 1}^{NM} {\left[ {{ \hbox{max} }\left( {0,\left| {\frac{{\sigma_{i} \left( \varvec{d} \right)}}{{\gamma_{s} \sigma_{\rm all.} }}} \right| - 1} \right)} \right]^{2} } + \sum\nolimits_{j = 1}^{ND} {\left[ {{ \hbox{max} }\left( {0,\left| {\frac{{u_{j} \left( \varvec{d} \right)}}{{\gamma_{d} u_{\rm all.} }}} \right| - 1} \right)} \right]^{2} } $$

(3)

where, $\gamma_{s}$ and $\gamma_{d}$ are defined here as safety factors corresponding to stress and displacement, respectively and $\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }$ denotes the vector of mean values of random variables. Based on the definition of safety factors, $\gamma_{s}$ and $\gamma_{d}$ are allowed to vary between 1 and 0. The safety factor that satisfies the target level of reliability is called RDF.

2.2 RBDO formulation based on safety factors

In structural design, due to the stochastic nature of material properties and applied loads and thus design uncertainties, use of stochastic analysis is inevitable. As an alternative, use of safety factors in the process of structural design is a simple way familiar to all engineers. These coefficients are referred to as load or resistance factors. Conventionally in the processes of optimization based on reliability, the deterministic objective function, constrained with the structural failure probability is a repetitive procedure that requires thousands of analyses to be carried out for each design. In fact, the conventional RBDO method utilized in structural optimization, is a nested or a double-loop method which contains two loops; the outer loop that contains optimization procedure and the inner loop, which is related to the reliability analysis. The design vectors of deterministic variables are transferred to the inner loop from the outer one in order to compute the failure probability for each design. Therefore, a gigantic number of analyses should be carried out for the RBDO.

The purpose of the present work is to search for the optimum point on the different safety levels of V′ as in Fig. 1 to satisfy the targeted level of reliability. These surfaces are generated by utilizing a series of safety factors. In the proposed approach, use of SF in the objective function simply allows a combined reliability based objective function for optimization. The proposed formulation of decoupled RBDO using SFs is introduced as the following equation:

$$ \begin{aligned} & {\text{Find}} \gamma^{\varvec{*}} \hfill \\ & {\text{subject to}}: \hfill \\ & \left\{ {\begin{array}{l} {V^{\prime} = 0 } \\ {P_{\text{f}} \left( {\varvec{d}^{*} ,\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }} \right) = P_{\text{f}}^{\text{Target}} } \\ \end{array} } \right. \hfill \\ \end{aligned} $$

(4)

Fig. 1

Series of deterministic design optimizations based on a sequence variation of safety factors

The objective of Eq. (4) is to find an optimum safety factor $\gamma^{ *}$ utilized by the cubic B-splines. This will lead to the optimum solution vector d ^* under the deterministic safety factor-based constraint V′. After that, the instituted optimum solutions d ^* are controlled for the probabilistic constraint as to check the corresponding failure probability over the targeted probability of failure.

2.3 Proposed RBDO procedure using B-spline interpolation concept

The procedure for RBDO of structures using proposed inverse method may be performed according to the flowchart shown in Fig. 2.

Fig. 2

Flowchart of the proposed method for RBDO of structures

Figure 3 indicates the procedure by which the range of response for $\gamma^{*}$ is decided upon. The following steps details the whole procedure. In all steps the optimum points are computed for the mean values of the random variables $\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} } = [\mu_{{x_{1} }} , \mu_{{x_{2} }} , \ldots ,\mu_{{x_{m} }} ]^{\text{T}}$. $P_{\rm f}^{a}$ means the probability of failure corresponding to $\gamma = a$.

Fig. 3

Structural failure probability against the different values of safety factor

Step 1—Primary check of the design space First for the mean value of safety factor, $\gamma = 0.5$, according to Eq. (3), the corresponding optimum design d ^* will be determined. With respect to the type of constraints, $\gamma$ could represent either $\gamma_{s}$ and/or $\gamma_{d}$ in most structural problems. The $P_{\rm f}^{0.5}$ is then computed using MCS as described in Sect. 2.4. The reason for this selection is based on the fact that within the range 0 < γ < 1, a mid-point $(\gamma = 0.5)$ may be a suitable starting point. As an example, if one uses the medians of $\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }$, the median-design will have a reliability of approximately 0.5 because the MPP will have a minimum distance of zero resulting in Φ(0) = 0.5 [23].

Step 2—Movement decision for $\varvec{\gamma}$ towards $\gamma^{*}$ Compare $P_{\text{f}}^{\text{Target}}$, and determine the range of $\gamma^{ *}$ according to Eq. (5):

$$ \left\{ {\begin{array}{l} {\left( {\text{I}} \right) \quad {\text{if }} \ P_{\text{f}}^{0.5} > P_{\text{f}}^{\text{Target}} \quad {\text{search }} \ {\text{for}}\;\gamma^{ *} \quad 0 < \gamma < 0.5 } \hfill \\ {\left( {\text{II}} \right) \quad {\text{if }} \ P_{\text{f}}^{0.5} < P_{\text{f}}^{\text{Target}} \quad {\text{search }} \ {\text{for}}\;\gamma^{ *} \quad 0.5 < \gamma < 1} \hfill \\ \end{array} } \right. $$

(5)

In most structural problems, computing $P_{\text{f}}^{0.4}$ for region I and $P_{\text{f}}^{0.6}$ for region II could be very efficient.

Therefore, in case of (I), compute the values of P _f for the two auxiliary points $\gamma = 0$ and 0.4. In case of (II), compute the values of P _f for the two auxiliary points of two safety factors $\gamma = 0.6$ and 1.

Step 3—Curve fitting In this stage, cubic B-spline interpolation concept is used in order to more accurately fit $(P_{\text{f}} {\text{vs}}. \gamma )$ by having the coordinates of the points found. The authors recommend de Boor, [26] in case further studies about the advancements of B-splines are required.

Step 4—Extracting data from fitted curve Extract the approximate safety factor ($\bar{\gamma }$) corresponding to the value of $P_{\text{f}}^{\text{Target}}$ given, from the fitted curve of step 3.

Step 5—deterministic design optimization (DDO) Use DDO approach to find the optimum design point $\varvec{d}^{ *}$ subject to V′ based on the extracted $\bar{\gamma }$ using the mean values of random variables $\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }$.

Step 6—Reliability analysis Determine the $P_{\text{f}}^{\text{MCS}}$ for the deterministic optimum design $\varvec{d}^{ *}$ of step 5 using MCS.

Step 7—Error estimation Compute the relative distance ${\text{error}}^{{P_{\text{f}} }}$ between $P_{\text{f}}^{\text{MCS}}$ and $P_{\text{f}}^{\text{Target}}$ using Eq. (6)

$$ {\text{error}}^{{P_{\text{f}} }} = \frac{{P_{\text{f}}^{\text{MCS}} - P_{\text{f}}^{\text{Target}} }}{{P_{\text{f}}^{\text{Target}} }} $$

(6)

Step 8—Check convergence If relative distance ${\text{error}}^{{P_{\text{f}} }}$ is less than or equal to a tolerance $\varepsilon$, the $P_{\text{f}}^{\text{MCS}}$ will be assigned as the $P_{\text{f}}^{\text{Target}}$ and the reliability based optimum design and the RDF $\gamma^{*}$ is said to being found.

Step 9—Repeating the procedure until convergence If according to Eq. (6) the convergence has not occurred, the $P_{\text{f}}^{\text{MCS}}$ will be considered as another auxiliary point to more accurately fit the curve. Steps 3–9 will be repeated until convergence and the reliable optimum design point $\varvec{d}^{ *}$ will be recorded.

2.4 Monte Carlo simulation

Monte Carlo simulation, named after the casino games of Monte Carlo, Monaco, originates from the research work of Metropolis and Ulam in 1949 [27]. Monte Carlo simulation is known as a simple random sampling method or statistical trial method that makes realizations based on randomly generated sampling sets for uncertain variables. A basic advantage of sampling methods is their direct utilization of experiments to obtain mathematical solutions or probabilistic information concerning problems whose system equations cannot be solved easily by known procedures.

In general, a reliability problem is formulated using a limit state function, $G(\varvec{X})$, where $\varvec{X} = [X_{1} , X_{2} , \ldots ,X_{m} ]^{\text{T}}$ is a vector with m random variables. Violation of the limit state function or failure is defined by the condition $G(\varvec{X}) \le 0$ and the probability of failure, $P_{\text{f}}$, is expressed by the following expression (Eq. 7):

$$ P_{\text{f}} = P\left[ {G\left( \varvec{X} \right) \le 0} \right] = \mathop {\iint }\limits_{{G\left( \varvec{X} \right) \le 0}}^{{}} \ldots \mathop \smallint \nolimits J_{{X_{1} X_{2} \ldots X_{m} }} \left( \varvec{X} \right){\text{d}}x_{1} {\text{d}}x_{2} \ldots {\text{d}}x_{m} $$

(7)

where $J_{{X_{1} X_{2} \ldots X_{m} }} \left( \varvec{X} \right)$ is the joint probability density function. The Monte Carlo method allows the determination of an estimate of the probability of failure, given by equation:

$$ P_{\rm f} \cong \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} I(\varvec{X}_{i} ) $$

(8)

where N is the total number of samples and $I(\varvec{X})$ is a function defined by Eq. (9):

$$ I\left( \varvec{X} \right) = \left\{ {\begin{array}{*{20}c} {1\quad {\text{if }}\quad G\left( \varvec{X} \right) \le 0} \\ {0\quad {\text{if }}\quad G\left( \varvec{X} \right) > 0} \\ \end{array} } \right\} $$

(9)

Equation 9 states that the violation of the ultimate limit state function ($G\left( \varvec{X} \right) \le 0$) occurs when either the strength or serviceability capacity of structure is less than the loading effects. The basic computation procedure of MCS is as follows:

• Step 1 Generate N samples for each random variable based on its probability distribution type and construct N sampling set vectors $\varvec{X}$.

• Step 2 Compute the value of limit state function for each sample to obtain $G(\varvec{X})$.

• Step 3 Check Eq. (9) for the computed $G(\varvec{X})$).

• Step 4 Use Eq. (8) to estimate the $P_{\rm f}$.

To generate a sample value $x_{i}$ for each random variable, the following general procedure for any type of distribution can be formulated [24, 25]:

• Step 1 Generate the random value $u_{i}$ from a uniformly distributed random variable between 0 and 1.

• Step 2 Calculate a sample value $x_{i}$ from the following equation:

$$ x_{i} = F_{X}^{ - 1} (u_{i} ) $$

(10)

where $F_{X}^{ - 1}$ is the inverse of $F_{X}$ and $F_{X}$ is a cumulative distribution function of random variable x.

3 Examples

In this section three benchmark RBDO structural problems were studied: a 10-bar truss with 10 design variables, a 13-bar truss with 7 design variable and a 72-bar spatial truss problem with 16 design variables. The structural P _f computation was performed using MCS. As Nowak [24] suggested, the number of samples produced by MCS, with regard to the coefficient of variation for P _f as to be less than or equal to ν, is determined by the following equation:

$$ N^{\text{MCS}} = \frac{{1 - P_{\text{f}} }}{{\nu^{2} P_{\text{f}} }} $$

(11)

For each problem, ν was set as 0.1 and P _f should be equal to the targeted value for that problem.

In all examples, the sequential quadratic programming (SQP) algorithm was used to carry out the optimization procedure. For this purpose, “fmincon” function of MATLAB was utilized. Also, the starting point $\varvec{d}^{0}$ of DDO was considered as the middle of the allowable design bounds, shown as follows:

$$ d_{j}^{0} = \frac{{d_{j}^{U} + d_{j}^{L} }}{2} $$

(12)

where j denotes the design variable number and $\varvec{d}_{\varvec{j}}^{\text{U}}$ and $\varvec{d}_{\varvec{j}}^{\text{L}}$ are the upper and lower bounds of j-th variable, respectively. For all three examples, computational expenses will be compared with those from the literature by means of the number of estimations of the failure probability. Also, a package which contains finite element method codes written by the authors in MATLAB software was prepared as an analyser where the linear elastic behaviour of elements was considered.

3.1 A 10-bar truss problem

This benchmark problem is selected to verify the proposed methodology. It was addressed, for example, by Zhao and Qiu [20] as a RBDO problem. The geometry of this 10-bar truss is shown in Fig. 4. The deterministic design variables of the structure are the members cross-sectional areas and the stochastic variables include external loads P ₁, P ₂ and P ₃, module of elasticity E and the length of the horizontal and vertical members L. Their statistical properties are given in Table 1.

Fig. 4

A 10-bar planar truss

Fig. 5

Convergence histories of objective function, 10-bar truss. a For total sample points; and b magnified view around the reliable optimum solution with f = 62.068 cm²

Table 1 Statistical properties of random variables, 10-bar truss

The cross-section areas of all bars are design variables and their lower and upper bounds are 0.0001 and 0.002 m², respectively. The target reliability index is 2.5, i.e., the target failure probability is 6.21 × 10⁻³. The total area of bars is to be minimized. The limit state is defined as nodal maximum vertical displacement of node 3, which should be less than 0.004 m. The modified implicit limit state function G ^* is expressed as in Eq. (13) for P _f calculation:

$$ G^{*} = 0.004 - u_{3,y} (\varvec{d}^{*} ,\varvec{X}) $$

(13)

Considering Eq. (13), the following formulation is introduced as utilizing the displacement safety factor ($\gamma_{d}$) to obtain optimum deterministic points:

$$ V^{\prime }\left( {\varvec{d},\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }} \right) = \left[ {{ \hbox{max} }\left( {0,\frac{{u_{3,y} \left( {\varvec{d},\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }} \right)}}{{\gamma_{d} 0.004}} - 1} \right)} \right]^{2} $$

(14)

In Eq. (14), the aim is to find the optimum $\gamma_{d}^{*}$ that gives an optimum solution d ^*, such that P _f(d,^* X) is less than the $P_{\text{f}}^{\text{Target}}$ considering the given tolerance $\varepsilon$.

Having found $P_{\text{f}}^{0.5}$ and compared the value with $P_{\text{f}}^{\text{Target}}$, the range 0.5 < γ^* < 1 was nominated. To first fit the curve, the two initial points corresponding to $\gamma = 0.6$ and 1 were computed for P _f. The procedure then was repeated for three iterations until the B-spline interpolation curves were fitted for three auxiliary points (see Fig. 6a, b), as a result of which ${\text{error}}^{{P_{\text{f}} }}$ was found less than 1 %. Figure 5a, b illustrates the convergence leading to the optimum design corresponding to the $P_{\text{f}}^{\text{Target}}$. The results are as listed in Table 2. It is worthwhile noting that through the whole process of reliability-based optimization with the proposed technique, the $P_{\text{f}}$ was only computed six times, wherein the method proposed by Zhao and Qiu [20] requires a total of 1904 P _f calculations to find an optimum solution with the objective value of 63.649 m² with the corresponding P _f equal to 2.742 × 10⁻³. One could claim that the proposed RBDO approach requires less P _f calculations. This optimum solution also possesses a P _f less than the target one with an error less than 1 %.

Fig. 6

Fitted cubic B-spline curves for 3, 4 and 5 sample points, 10-bar truss. a For 0.5 < FS < 1; and b magnified view around the reliable optimum solution with P _f = 6.15 × 10⁻³

Table 2 RBDO results of 10-Bar truss

3.2 A 13-bar bridge truss problem

A 13-bar bridge truss shown in Fig. 7 is considered which has been studied by some authors as a RBDO problem such as Nakib and Frangopol [28] and Ghorbani and Ghasemi [29], previously. Table 3 lists the seven design variables’ groups as the cross-sectional areas of members. Their lower and upper bounds are between 6.4516 × 10⁻⁵ and 6.4516 × 10⁻³ m², respectively. The material density and modulus of elasticity were considered as 7850 kg/m³ and 206 GPa, respectively. A load P and material yield stress, F _y are considered as random variables associated with normal distribution with the corresponding parameters defined in Table 4. System failure is assumed to occur when the stress in each member $\sigma_{i}$ reaches the yield stress. The modified limit state function G ^* of this truss for P _f calculation is therefore defined as:

$$ G_{i}^{*} = F_{y} - |\sigma_{i} (\varvec{d}^{ *} ,\varvec{X})| $$

(15)

Fig. 7

A 13-bar bridge truss

Table 3 Member grouping details of a 13-bar bridge truss

Table 4 Statistical properties of random variables, 13-bar bridge truss

In this example, by embedding the stress safety factor, ($\gamma_{s}$) to Eq. (15), we can reformulate it as the following equation to define V′

$$ V '\left( {\varvec{d},\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }} \right) = \sum\nolimits_{i = 1}^{13} {\left[ {{ \hbox{max} }\left( {0,\frac{{\sigma_{i} \left( {\varvec{d},\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }} \right)}}{{\gamma_{s} F_{y} }} - 1} \right)} \right]^{2} } $$

(16)

Having found $P_{\text{f}}^{0.5}$ and compared the value with $P_{\text{f}}^{\text{Target}}$, the range 0 < γ^* < 0.5 was selected. Figure 8a, b shows the converged cubic B-spline-fitted curves of ($P_{\text{f}}$ vs $\gamma$) according to the six sample points. At the third iteration where $\bar{\gamma }$ was found equal to 0.42 with the corresponding truss mass of 353.0202 kg, the computed P _f reached the $P_{\text{f}}^{\text{Target}}$ with a discrepancy of 1 % in the safe region. Therefore, the authors may claim that the optimum solution is the most reliable since 99 % of the $P_{\text{f}}^{\text{Target}}$ was met in the safe region. The results of the proposed RBDO of this problem are listed in Table 5, where the reliable optimum design was slightly lighter than that in the literature.

Fig. 8

Fitted cubic B-spline curves for 3, 4 and 5 sample points, 13-bar truss. a For 0 < FS < 0.5; and b magnified view around the reliable optimum solution with P _f = 0.99 × 10⁻⁵

Table 5 RBDO results of 13-bar bridge truss cross-sectional area (m²) × 10⁻³

Figure 9a, b also illustrates the convergence of total truss mass leading to the optimum design corresponding to the $P_{\text{f}}^{\text{Target}}$.

Fig. 9

Convergence histories of objective function, 13-bar truss. a For total sample points; and b magnified view around the reliable optimum solution with f = 353.0202 kg

As seen from the results listed in Table 5, the efficiency of the proposed RBDO is the best in terms of the number of P _f evaluations. The proposed RBDO approach requires a total of 5 P _f calculations to find the global optimum solution with the objective value of 353.0202 kg, being 1.7 % less than the best value recorded in the literature.

3.3 A 72-bar spatial truss problem

The 72-bar space truss shown in Fig. 10 was first investigated by Shayanfar et al. [30] in 2014 as a reliability-based design optimization problem. The geometry of the truss with its nodes numbering and grouping of the members are indicated in Fig. 10. In the current study, the design variables were selected as discrete for the cross-sectional areas where the objective function was kept as to minimize the total truss weight. The density of material is 2767.99 kg/m³. To enforce symmetry, members of the truss were set into sixteen groups as in Table 7. Loading condition, consists of 22.2411 kN in x, y and −22.2411 kN in z directions at node 1. The lower and upper bounds of members C.S.A. were confined to 6.4516 × 10⁻⁵ m² and 2.903 × 10⁻³ m², respectively. Properties of the random variables are shown in Table 6. This problem has two reliability constraints to control the lateral displacement of node 1 in x and y directions. The allowable lateral displacement $\Delta _{{{\text{all}} .}}$ is considered as 7.62 × 10⁻³ m in both x and y directions. The modified limit state functions of this space truss for P _f calculation is therefore defined as:

$$ G_{i}^{*} =\Delta _{{{\text{all}} .}} - |\Delta _{i} (\varvec{d}^{ *} ,\varvec{X})| $$

(17)

where i is equal to x or y direction of node 1. By embedding the displacement safety factor ($\gamma_{d}$) to Eq. (17), we can reformulate it as the following equation to define the hypothetical constraint surfaces V′ as:

$$ V^{{\prime }} \left( {\varvec{d},\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }} \right) = \sum\nolimits_{i = 1}^{2} {\left[ {{ \hbox{max} }\left( {0,\frac{{\Delta _{i} \left( {\varvec{d},\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} }} \right)}}{{\gamma_{d}\Delta _{{{\text{all}} .}} }} - 1} \right)} \right]^{2} } $$

(18)

Fig. 10

The 72-bar space truss structure

Table 6 Statistical properties of random variables, 72-bar space truss example

As depicted in the results of Table 7, Shayanfar et al. [30] could produce an optimum weight of 243.030 kg for a reliability index of $\beta = 3$ corresponding to a P _f of 0.00135, slightly in the unsafe region. According to the proposed method here, a minimum weight of 243.2706 kg was determined for a targeted P _f of 0.00135.

Table 7 RBDO results of 72-bar space truss cross-sectional area (m²) × 10⁻⁵

As listed in Table 7, three iterations as $\bar{\gamma }_{d} = 0.58$, 0.59 and 0.5913 were done, while the third iteration where $\bar{\gamma }_{d}$ was found equal to 0.42, the computed P _f reached the $P_{\text{f}}^{\text{Target}}$ with a relative distance of less than 1 % in the safe region. Figure 11a, b illustrates the convergence of optimum truss weight leading to the optimum design corresponding to the $P_{\text{f}}^{\text{Target}}$. Figure 12a, b also shows the converged B-spline interpolation curves of ($P_{\text{f}}$ vs $\gamma$) according to the 3, 4 and 5 sample points. As seen from the results in Table 7, the proposed method yields high efficiency in terms of the number of P _f evaluations. This is where the proposed method found an optimum weight of 243.2706 kg after a total of 6 P _f evaluations that is less than the total 200 number of P _f evaluations reported by Shayanfar et al. [30].

Fig. 11

Convergence histories of objective function, 72-bar truss. a For total sample points; and b magnified view around the reliable optimum solution with f = 536.32 lb

Fig. 12

Fitted cubic B-spline curves for 3, 4 and 5 sample points, 72-bar truss. a For 0.5 < FS < 1; and b magnified view around the reliable optimum solution with P _f = 0.001345

4 Conclusions

In this article an efficient method based on safety factors and cubic B-spline interpolation concept was introduced to solve RBDO of structures. This technique may be regarded as one in the category of decoupling methods. The proposed method may be distinctive for its ability to highly reduce the number of reliability analyses required for structural reliability-based optimization compared to conventional methods. MCS was utilized for reliability analysis to show the efficiency of the proposed approach. The performance of the proposed technique was verified on three structural problems. The results obtained well state that this method possesses a sufficient speed and accuracy. The features of the proposed technique may be summarized as follows:

• The proposed RBDO framework contains an efficient decoupling approach that allows for a reasonably high speed of convergence.

• The proposed method may solely require a relatively simple understanding of RBDO together with a low effort of programming.

• This technique could be distinctive with a minimum number of P _f computations required since the P _f determination is carried out only after a global deterministic optimum design is found with respect to the mean values of random variables and the intended safety factors.

• B-spline interpolation curves were utilized as a tool to find an optimum safety factor that yields an optimum solution with the corresponding failure probability equal to the targeted P _f. These interpolation curves were modified each time with the newly founded optimum solution.

• One may realizes that the method contains a firm and straightforward framework since in all examples it leads to the global and reliable optimum design that satisfies the targeted P _f with a maximum dispute of 1 % in the safe region.

• In all examples studied, the reliable optimum design point was found only with less than a total of 10 P _f calculations.