Reliability-based design optimization for seismic structures considering randomness associated with ground motions

Modeling of stochastic ground motion

In Equation (1), the random input term $$F(\Theta, t)$$ is modeled using the physically motivated stochastic ground motion model, as described in ^[26]. This model incorporates the physical aspects of seismic wave propagation and combines the Fourier transfer representation with seismic source, propagation path, and site models, as outlined in ^[26], resulting in the following formulation:

where,

Here, the random parameter vector $$ \Theta = \{ {A_0}, \tau , {\zeta _g}, {\omega _g}\} $$ captures the randomness related to the seismic wave propagation process from the seismic source to the local site. This vector encompasses random parameters, including those governing the Fourier amplitude spectrum $$ {A_{\overline R }}\left( {\Theta , \omega } \right) $$ and the Fourier phase spectrum $$ {\Phi _{\bar R}}\left( {\Theta , \omega } \right) $$. It also includes parameters related to the seismic source, such as the amplitude parameter $$ {A_0} $$ and the Brune source model parameter $$ \tau $$. Furthermore, it accounts for site-specific characteristics, incorporating the equivalent damping ratio $$ {\zeta _g} $$ and the predominant circular frequency $$ {\omega _g} $$. K is the parameter accounting for propagation path attenuation, and the epicentral distance is denoted by $$ \bar R $$. In Equation (4), the empirical parameters connected to the propagation path are denoted as a, b, c, and d, completing the characterization of the seismic system.

The model outlined above addresses the stochastic nature of acceleration at the bedrock and the specific characteristics of the site's soil. To derive the input at the bedrock, a comprehensive integration of seismic source effects and wave propagation is conducted, guided by seismic hazard analysis. In this approach, the soil layer is effectively represented as an equivalent linear single-degree-of-freedom system, as shown in Figure 1. Subsequently, seismic excitation is applied at the base, and the absolute response of this system serves as an accurate representation of the seismic ground motion process. As a result, ground motions can be reliably simulated using a simplified theoretical formula that captures the physical relationship between surface ground motions and input motions at the bedrock while considering essential parameters such as the predominant circular frequency and equivalent damping ratio as depicted below^[7,27,39]:

Figure 1. Soil layer and equivalent single-degree-of-freedom-model of local site

where $$ {\ddot X_g}\left( {\Theta , \omega } \right) $$ represents the seismic acceleration at the surface of the local site in the frequency domain, while $$ {\ddot U_b}\left( {{\Theta _{\rm{b}}}, \omega } \right) $$ corresponds to the seismic acceleration at the bedrock, also in the frequency domain. The random vector $$ \Theta = \left\{ {{\rm{ }}\Theta {{\rm{ }}_{{\omega _g}}}, {\rm{ }}\Theta {{\rm{ }}_{{\zeta _g}}}, {\Theta _b}} \right\} $$ encompasses random parameters that account for the inherent randomness in ground motions at the surface of the local site. Furthermore, $$ {{\rm{\Theta }}_{\overline {{\omega _g}} }} $$ and $$ {{\rm{\Theta }}_{\overline \zeta }} $$ denote the random variables associated with the equivalent circular frequency and the equivalent damping ratio of the local site, respectively. Lastly, $$ {\Theta _{\rm{b}}} = \left\{ {{{\rm{\Theta }}_{b, i}}} \right\}_{i = 1}^{{s_{\rm{b}}}} $$ constitutes a vector of stochastic parameters that describe the randomness in ground motion at the bedrock introduced by the seismic wave propagation from the seismic source to the bedrock, with $$ {s_{\rm{b}}} $$ denoting the number of random variables involved.

The frequency domain equation of ground motion data presented in Equation (5) can now be obtained in terms of time domain by performing inverse Fourier transfer of this equation ^[39].

The time history of ground acceleration obtained from Equation (6) can be further enhanced by adding non-stationary characteristics. This is done using a uniform modulation function as follows ^[10]:

$$ T $$ in the above equation represents the total time duration of the ground motion; $$ t_a $$ and $$ t_b $$ represent the starting and ending time of the strong motion phase, respectively.

Probability density evolution method

In this study, a structural reliability analysis employs the PDEM introduced by Li and Chen ^[23]. Based on the probability conservation principle, this method is proficient at capturing the probability density function (PDF) of structural responses in the presence of random excitations and associated uncertainties at any given time instant ^[7,8,40].

In the context of a dynamic system, the likelihood of surpassing a predefined limit state can be characterized as the first passage probability. This helps us understand the probability of the system crossing a specific point within a certain time, providing insight into the system reliability^[10,41]. The dynamic structure reliability associated with the first-passage problem under stochastic ground motions is defined as:

where $$ \Omega_s $$ signifies the safety domain, while $$ Pr.\{\} $$ represents the probability operator applied to the random event $$ Z(\Theta, t) $$. This event pertains to relevant physical quantities such as structural drift, displacement, and so on.

The above definition of reliability holds when there is involvement of a single limit state function only, i.e., when a single mode of failure is considered or when only one member or element of the structure fails. However, when assessing the reliability of a structure, we typically need to consider multiple modes of failure or multiple failures of elements, as represented by the following equation ^[7,10,37]:

where $$ m $$ denotes the number of random events such as the number of modes of failure or failed elements; $$ \left\{ {{z_{Bi}}} \right\}_{i = 1}^m $$ denote the safe thresholds of the random events.

The above equation (Equation (9)) can be solved using commonly employed reliability theory based on the level-crossing process. However, this theory often struggles to ensure accuracy, and its procedure is not straightforward or convenient due to the requirement of the joint PDF of the response and its velocity for reliability assessment, along with the need to assume properties of the level-crossing events. Consequently, in this study, a more efficient method, devoid of the mentioned issues, is employed to assess dynamic reliability. This method transforms the problem into an EEV scenario, as detailed in ^[24,25]. By calculating the probability of such an EEV event, it becomes possible to evaluate the probability of compound random events involving combinations of more than one inequality.

The EEV event can be constructed in terms of random variables as:

Now, Equation (9) can be defined alternatively in terms of EEV as:

It can also be written in an integral form as:

where $$ {p_{{Z_{eq}}}}(z, T) $$ denotes the PDF of $$ {Z_{eq}}(\Theta , T) $$.

The equation above illustrates how system reliability, which involves a high-dimensional integral, can be simplified to a one-dimensional integral through the establishment of the EEV event.

The extreme value distribution or the PDF of this EEV event can be characterized by constructing a virtual stochastic process as follows:

The virtual stochastic process is to ensure that its extreme value coincides with the value of the virtual stochastic process at a specific time point, denoted as $$ ({\tau _c}) $$. This constructed random process adheres to the following conditions:

The variables used in the equation are defined as follows: $$ \tau $$ represents the generalized time, while $$ \tau_0 $$ and $$ \tau_c $$ represent the initial and final time instances of the pseudo-random process, respectively.

In this context, the constructed pseudo-random process $$ \nu (\tau) $$ adheres to the conditions outlined in Equation (14). The specific form of $$ \nu (\tau) $$ adopted in this study is $$ \nu (\tau) = {Z_{eq}}(\Theta , T)\sin (\omega \tau) $$, where $$ \omega = 5\pi /2 $$ and $$ {\tau_c} = 1 $$ are considered ^[8].

Here, both the pseudo-random process $$ \nu (\tau) $$ and the EEV event $$ Z_{eq}(\Theta , T) $$ share the same randomness parameter, $$ \Theta $$. Moreover, the final time instance of $$ \nu (\tau) $$ encapsulates the complete probabilistic information of $$ Z_{eq}(\Theta , T) $$. Consequently, the pair $$ (\nu (\tau), \Theta) $$ can be conceptualized as a probability-conserved system governed by the Generalized Probability Density Evolution Equation (GDEE) ^[10] expressed as follows:

The initial condition for the above partial differentiation equation is given by:

where $$ \delta ( \cdot ) $$ denotes the Dirac delta function, and $$ \theta $$ denotes a sample of stochastic vector $$ \Theta $$.

To derive the solution of $$ {p_{\nu \Theta }}(\upsilon , \theta , \tau ) $$, the initial value problem of a partial differential equation is solved, which results in:

where $$ {\Omega _\Theta } $$ denotes the sample domain for the vector of random parameters $$ \Theta $$.

Now, to determine the PDF of the EEV event, $$ {Z_{eq}}(\Theta , T) $$, we establish a connection with the virtual random process at the time instant $$ \tau_c $$ in the following manner:

Numerical procedure for solving the first-passage problem using PDEM

The theoretical derivation mentioned above can be explicitly expressed in terms of a combination of steps as mentioned below. The following algorithm first evaluates the first passage probability using PDEM, ultimately resulting in the structural reliability.

Step 1: Selection of representative points: Partition the probability space of the random vector $$ \Theta $$, which embodies the random parameters governing stochastic ground motion. Create a set of representative points denoted as $$ \{ {\theta _q}\} _{q = 1}^{{n_{{\rm{res}}}}} $$, where $$ n_{res} $$ signifies the set's size. Calculate $$ {{\rm P}_q} = \{ p_{assign}^1, p_{assign}^2, ....., p_{assign}^{{n_{res}}}\} $$ as the corresponding probabilities. The tangent sphere method is employed for selecting these representative points, where each representative point represents a time history of seismic ground acceleration.

Step 2: Deterministic structural analyses: For each representative point $$ \theta_q $$, perform deterministic structural analyses to attain the structural response of concern. From the results, construct the EEV event $$ {Z_{eq}}({{\theta }_q}, T) $$ and the pseudo-random process $$ \nu ({\theta _q}, \tau ) $$.

Step 3: Solve GDEE: Compute the velocity term of the pseudo-random process by taking its derivative and incorporating it into the GDEE presented in Equation (15). For each $$ \theta_q $$, solve the GDEE to obtain the solution for the joint PDF denoted as $$ {p_{\nu \Theta }}(\upsilon , {\theta _q}, \tau ) $$. To solve the GDEE, employ the finite difference method with a Total Variation Diminishing (TVD) scheme.

Step 4: Reliability evaluation: Derive the PDF $$ {p_\nu }(\upsilon , \tau ) $$ by integrating the joint PDF, as shown in Equation (17). The reliability is finally obtained using Equation (18).

Design variables

In a structural system with a fixed base, the pivotal role of distributing loads and maintaining stability falls upon components such as beams, columns, and slabs. The source of uncertainty in this study is the external excitation and the structural geometry uncertainties are not considered; therefore, given a pre-determined topology and layout of the superstructure, the dimensions of these structural members can be regarded as deterministic design variables within the framework of the superstructure's design. However, sometimes epistemic uncertainties do affect the RC structures, which should be considered in the analysis, as explained in ^[42]. When dealing with rectangular members, their cross-sectional property, such as the moment of inertia ($$ I_Z $$), can be formulated by means of the fundamental design parameters, namely, the width ($$ B $$) and depth ($$ D $$). Consequently, these two fundamental dimensions—width ($$ B $$) and depth ($$ D $$) emerge as the principal design variables in the context of the optimization procedures undertaken in this particular study.

Design Objective

The fundamental aim of this study is the minimization of construction costs pertaining to the superstructure. In scenarios involving a structure composed of $$ i $$ rectangular components, the pivotal factors governing their sizes are ($$ B_i, D_i $$). The assessment of the cost incurred by reinforcing rebars is excluded from this analysis as the variations in steel reinforcement exhibit negligible sensitivity to lateral elastic displacement ^[33]. Furthermore, the flexural members, such as beams and columns, have their flexural stiffness directly proportional to the cube of their depth that varies linearly with the width of the member. Thus, while enhancing the size of a frame member, it is more cost-effective to increase the depth ($$ D_i $$) of the member rather than enlarging its width ^[43].

While formulating the design objective, it is essential to properly consider the uncertainties associated with the objective function within the RBDO procedure. However, due to the lack of reliable cost data, it is generally acceptable to treat the cost function as deterministic, and this study also adopts this approach. The design objective is, thus, formulated as:

Here, $$ TC $$ represents the total construction cost, i.e., total concrete cost, and $$ w_i $$ is the unit weight of concrete; $$ L_i $$ is the length of the $$ i^{th} $$ member.

Design constraints

In this study, the primary design constraint for evaluating system reliability revolves around the global reliability of interstory drift ratios. Specifically, this constraint does not solely focus on the inter-story drift ratio between particular floors, such as drift ratios between the first and second floors, but instead encompasses a broader scope called "global reliability", necessitating that all other inter-story drift ratios remain below their corresponding thresholds ensuring the reliability of the specified parameter to remain above a designated threshold. This threshold might be established through a probabilistic model code or chosen based on specific performance criteria. The assessment of the reliability of this parameter is conducted through the methodology outlined in Section 2.2 and Section 2.3.

Here, $$ R_{(ISD)} $$ is the global reliability of the interstory drift ratio, i.e., a combination of multiple limit states pertaining to interstory drift of all the floors, and $$ X $$ represents the probabilistic target value.

RBDO problem formulation

Once the reliability threshold is established and the reliability is computed through the process outlined in Section 2.3, the RBDO scheme is formulated as a mixed continuous-discrete optimization problem, expressed as:

In the above equation, $$ R_{(ISD)} $$ is the global reliability of the interstory drift; $$ B_{i}^{L} $$ and $$ B_{i}^{U} $$ are the lower and upper bound sizing constraints for the width $$ B_i $$; $$ D_{i}^{L} $$ and $$ D_{i}^{U} $$ are the lower and upper bound sizing constraints for the width $$ D_i $$, respectively.

This study employs the GA within the pymoo module ^[44] in Python for optimization. The GA iteratively evaluates design solutions by creating an initial population, calculating failure probabilities, and analyzing reliability based on the constraints. Solution fitness is determined by their reliability, and reproduction is driven by selection, crossover, and mutation. Offspring also undergo reliability analysis, and this iterative process continues. The algorithm aims to find solutions that balance cost and reliability, aiming for designs to meet performance and safety requirements while fulfilling other criteria.

Reliability-based design optimization procedure

Step 1: Set initial values for the deterministic design variables and establish their corresponding bounds. Additionally, first, set the allowable limit and determine the targeted reliability for the specific parameter of interest, namely, the interstory drift ratio.

Step 2: Conduct structural analyses using the stochastic ground motions generated based on the principles detailed in Section 2.1 for the design variables. Calculate the drift ratio for each floor and record the maximum drift ratio among all floors at each time interval for further analysis.

Step 3: Perform reliability analysis using PDEM, as detailed in Section 2.3, to evaluate the global structural reliability against interstory drift ratios.

Step 4: Based on the results of reliability analysis, formulate the RBDO problem, as mentioned in Equation (22).

Step 5: Adopt the GA to perform the optimization procedure. Within each population, the algorithm adjusts the width $$ (B_i) $$ and depth $$ (D_i) $$ of the structural members to minimize the total cost of the superstructure while adhering to the imposed reliability constraints.

Step 6: End the design optimization procedure when the designated termination criterion for a solution is met. If not, proceed to Step 2 for the subsequent design iteration.

The procedure described above is depicted in a flowchart shown in Figure 2.

Figure 2. Flowchart of GA-based optimization

Deterministic Optimization

To initiate deterministic optimization, lower-bound depths and widths were adopted from Zou's prior study ^[34]. For beams and columns, the depths are set at a minimum of $$ 300 mm $$, and the widths at $$ 250 mm $$ and $$ 300 mm $$, respectively, in accordance with common buildability standards. Employing site class $$ III $$, seismic intensity $$ 8 $$, frequently occurring earthquake and damping ratio $$ (\zeta=5\%) $$, a response spectrum curve is generated according to Chinese seismic design code (GB50010-2010), as shown in Figure 3B.

A response spectrum modal analysis is conducted to determine the peak modal displacements. The Square Root of the Sum of Squares (SRSS) rule, known for its accuracy in two-dimensional problems, is employed to combine the first five peak modal responses ^[46]. Subsequently, the inter-story drift ratio is computed as the ratio between the difference in lateral displacement of $$ j^{th} $$ and $$ (j-1)^{th} $$ floor to the height of the story.

The optimal design problem is defined in a manner consistent with the RBDO formulation presented in Equation (22). The only distinction is that the reliability constraint is replaced by a deterministic constraint, which is expressed as follows:

where $$ |\delta u|_{max} $$ is the maximum interstory drift ratio among all the floors; X is the allowable inter-story drift ratio limit, which is taken as $$ 1/800 $$; $$ B_{i}^{L} $$ and $$ B_{i}^{U} $$ are the lower and upper bound sizing constraints for the width $$ B_i $$; $$ D_{i}^{L} $$ and $$ D_{i}^{U} $$ are the lower and upper bound sizing constraints for the depth $$ D_i $$, respectively. Upper bounds for all the members are arbitrarily taken to be 1.5m in this study.

The design optimization process aims to minimize the cost function, as described in Equation (20), ensuring that the maximum inter-story drift among all floors remains below the threshold of $$ 1/800 $$. As mentioned in the previous section, the pymoo module in Python is utilized for GA in the optimization procedure. The population size is assigned to be 100 with 15 variables (depths) to be optimized, and the optimization terminates when the number of generations reaches 100. The optimal objective function fulfilling the constraints within these generations is then obtained as the globally optimal solution.

Results of Deterministic Optimization

The initial structure, adopted with minimal member sizes to ensure basic constructability requirements ^[34], failed to meet multiple inter-story drift ratio criteria, leading to excessive flexibility, longer vibration periods, lateral displacement, and drift violations, as shown in Figure 4 and Table 2. After optimization, the member depths of the structure increased, making it stiffer, as shown in Table 1. This reduced lateral displacement across all floors and reduced the period of the first mode of vibration from 3.86 seconds to a suitable 1.5 seconds (as shown in Table 2). The increased member size also raised the structural cost (seismic weight) from the initial 46, 782.5 kg to 74, 813.60 kg for the optimal configuration. However, this cost increase is justified by the significant reduction in lateral displacement and drift ratios, which are now within acceptable limits, as shown in Figure 4.

Figure 4. Responses of deterministic optimization. A: Initial and final inter-story drift ratios; B: Initial and final lateral displacement profiles

Although Figure 4A illustrates that all floor drift ratios are well within acceptable limits post-optimization, it is essential to note that this optimization process does not consider uncertainties in external excitation or the system. Therefore, the reliability of this optimal solution remains unverified. Relying solely on deterministic optimization would necessitate an exceptionally high safety factor, leading to a rapid escalation in costs without guaranteeing reliability. Therefore, to address this concern and thoroughly assess the reliability of the optimized structure, a comprehensive reliability assessment is conducted in the subsequent section.

Reliability Assessment of optimal results from deterministic optimization

The optimal structure obtained from Section 4.1 is subjected to random ground motions derived using the physically motivated stochastic ground motion model detailed in Section 2.1 to incorporate the randomness within the external excitation. Subsequently, the global reliability of the structure with respect to the inter-story drift ratio is assessed through the utilization of PDEM.

Generation of stochastic ground motion

As detailed in Section 2.1, the two principle sources of randomness are defined as randomness in site soil and randomness in acceleration at bedrock. To evaluate the former, an engineering site is considered, characterized as site class $$ III $$, with a seismic fortification intensity of $$ 8 $$ according to the Chinese seismic design code (GB50010-2010). The random variables, $$ \omega_g $$ and $$ \zeta_g $$, have been ascertained to follow a log-normal distribution through the fitting of seismic records associated with the aforementioned site class. Consequently, a mean and a coefficient of variation values are obtained for $$ \omega_g $$ and $$ \zeta_g $$, resulting in 12 rad/s and 0.42 for $$ \omega_g $$, and 0.1 and 0.35 for $$ \zeta_g $$, respectively. Furthermore, to characterize the variability in bedrock acceleration, seismic hazard parameters are defined based on a frequently occurring earthquake with a 50-year return period, peak ground acceleration of 0.1g, and Fourier amplitude of 0.2 $$ m/s^2 $$. Additionally, it is assumed that the initial phase angle used to simulate stochastic ground motion follows a normal distribution, with a mean of $$ \pi $$ and a coefficient of variation of 1.2 ^[7].

In this study, PDEM, as described in Section 2.2, is utilized to perform reliability analyses on structures exposed to random ground motions. The initial step involves choosing representative points within the stochastic parameter space. Subsequently, the physical stochastic model outlined in Section 2.1 referred from ^[7] is employed to generate ground motion acceleration time histories. The tangent spheres method is then used for partitioning the probability space, resulting in the selection of 221 representative points. The generated ground motions have a sampling frequency of 50 Hz and a total duration of 20.48 s. To capture the non-stationary characteristics of these ground motions, Equation (7) is applied, with values of $$ t_a $$ and $$ t_b $$ set at 2s and 16s, respectively.

Now, the seismic load utilized in both deterministic optimization and stochastic analysis corresponds to the same site conditions and seismic intensity. To confirm their consistency, statistical properties of the response spectrum of these stochastic ground motions are computed. Figure 5A depicts the mean and the mean plus one times the standard deviation plotted against the design spectrum. The mean response spectrum is scaled to match the design spectrum, and it is evident that the calculated mean plus one times the standard deviation encompasses the entire design spectrum, affirming their validity. Additionally, the model effectively captures the non-stationary characteristics of the time histories and the sample-to-sample variation, with one representative time history illustrated in Figure 5B.

Figure 5. Stochastic ground motions from the physical model. A: Mean and mean plus standard deviation of stochastic ground motions; B: Representative time history of stochastic ground motion

Results of Reliability Assessment

The PDF and Cumulative Density Function (CDF) of the EEV interstory drift ratio resulting from the structure's exposure to stochastic ground motions are depicted in Figure 6. In contrast to the work of Zou, where reliability indices were employed, PDEM quantifies target reliability in terms of percentages. For this study, the target reliability in PDEM is set at 0.886, which corresponds to a reliability index of 1.2 in his study. This determination is made using a first-order approximation of the failure probability, which is calculated as the probability content of the half-space in the standard normal space, as described in the following equation ^[16,47]:

(24) $$ p_f \approxeq p_{f1}=\Phi(-\beta) $$

Figure 6. PDFs and CDFs of deterministic and reliability-based optimized structures subjected to stochastic ground motions. A: probability density functions; B: cumulative density functions

where $$ p_{f1} $$ represents the first-order approximation of the failure probability; $$ \Phi(.) $$ represents the standard Normal CDF, and $$ \beta $$ represents the reliability index.

The red curve in Figure 6B illustrates the CDF of the structure optimized deterministically. It is evident that with a drift ratio of 1/800, the reliability is only 0.41, which falls significantly below the target reliability. Examining the PDF curve depicted in Figure 6A, it becomes apparent that the probability of exceeding the drift limit is approximately 50%, a value considerably higher than the targeted exceedance probability. These results underscore the inadequacy of the deterministic optimization technique. Consequently, the subsequent section focuses on optimization while enforcing the constraint of drift ratio reliability.

Reliability-based design optimization

Single-objective optimization using GA

To initiate the RBDO procedure, the initial and final bounds, material properties, and structural configuration remain consistent with those in deterministic optimization. However, a fundamental distinction arises from the incorporation of random ground motions, resulting in a stochastic process. The central concept is to design an optimal structure in which the maximum interstory drift ratio remains within the permissible failure probability, ensuring global reliability where all limit states fall within safe margins. To achieve this, the concept of global reliability is integrated using the EEV event approach. In this context, the limit state of each floor must be considered. Consequently, the maximum drift ratio among all floors is evaluated for each representative point, as formulated in Equation (9). PDEM is then readily applied to obtain the PDF and CDF of the inter-story drift ratio. RBDO is a two-step optimization procedure; the inner loop is dedicated to reliability analysis, and the outer loop is focused on structural optimization. This process is mathematically defined in Equation (22), with a primary constraint stating $$ R_{(ISD)}\geq 0.886 $$. The GA, thus, selects offspring that primarily satisfy this critical constraint and subsequently minimizes the cost function, ultimately leading to a globally reliable optimal structure. In this case, a population of 100 for 100 generations is used for single-objective optimization using GA.

The root-mean-square time histories of the inter-story drift ratios of several floors for the optimal results obtained through deterministic and reliability-based optimization are depicted in Figure 7. Notably, the optimal outcome of RBDO showcases a substantial reduction in drift ratios compared to the deterministic optimization results, all while maintaining superior global reliability for the structure and incurring only a slight increase in cost. It is evident that the drift ratio of the structure comfortably meets the design requirements, as its root-mean-square time history remains well within the limit of $$ 1/800 \approx 0.00125 $$. To further assess the performance of the reliability-based optimal design, the PDF and CDF are presented in Figure 6. Figure 6B reveals that the reliability of this optimal design stands at approximately 89%, which is more than double the reliability achieved by the deterministically optimized structure. However, as detailed in Table 1, the seismic weight/cost of this optimal design is approximately 31% higher than that of the deterministically optimized structure and nearly double the initial structural cost. Despite this increase in cost, it is justified by the substantial enhancement in reliability from 0.413 of the deterministically optimized structure to 0.895 of the reliability-based optimal structure. This result underscores how a relatively modest cost increase of around 31% can yield nearly double the reliability. Furthermore, this enhanced reliability encompasses the entire structure, emphasizing its global reliability. Additionally, the first mode of vibration for this optimal structure exhibits a fundamental period of approximately 1.064 seconds, as shown in Table 2, aligning with the fundamental rule of thumb for structural dynamics and confirming the effectiveness of this design procedure.

Figure 7. Root mean square interstory drifts deterministic and reliability-based optimal structures. A: $$ 1^{st} $$ Interstory drift ratio; B: $$ 3^{rd} $$ Interstory drift ratio; C: $$ 5^{th} $$ Interstory drift ratio; D: $$ 10^{th} $$ Interstory drift ratio

Multi-objective optimization using NSGA-II

In this section, our aim is to introduce greater flexibility in design optimization, offering a spectrum of solutions for achieving a balanced trade-off between cost and reliability without degrading any of them. Utilizing NSGA-II, which stands for non-dominated sorting GA for multi-objective optimization, allows us to expand beyond the single global value generated by GA, introducing multiple solutions that serve as a Pareto set. The concept of Pareto dominance involves an exhaustive comparison of each solution with every other solution in the population. If no other solution dominates, it is considered non-dominated and is selected by NSGA-II to be part of the Pareto front set. While GA focuses on scalar optimization by providing a single outcome, multi-objective optimization provides a vector, enabling a holistic approach to objectives. Here, we emphasize minimizing structural cost while maximizing reliability. Originally, reliability was utilized only as a constraint in single-objective optimization; however, in this section, we have integrated an additional objective for maximizing reliability. The constraint remains unchanged at $$ R_{(ISD)}\geq 0.886 $$, leading to a suite of solutions that comply with the constraints while addressing both objectives.

(25) $$ \begin{array}{l} Minimize:\\ \quad TC(L_i, B_i, D_i)=\sum\limits_{i=1}^{N_i}w_i L_i B_i D_i\\[1ex] Maximize:\\ \quad R_{(ISD)}\\[1ex] subjected \quad to:\\ \quad R_{(ISD)}\geq 0.886\\[1ex] \quad B_{i}^{L}\leq B_i\geq B_{i}^{U}\\[1ex] \quad D_{i}^{L}\leq D_i\geq D_{i}^{U} \quad (i=1, 2, ...N_i) \end{array} $$

The optimization process involved a population of 100 candidates across 100 generations, yielding the results depicted in Figure 8. Adhering to the constraint $$ R_{(ISD)}\geq 0.886 $$, the solutions range from a cost of 95550 with a reliability of 0.8864 to a cost of 103400 with a notably higher reliability of 0.964. This broad spectrum of solutions offers the flexibility to choose as needed within this range.

Figure 8. Pareto solution obtained from multi-objective optimization using NSGA-II

The computational times for different optimization methods are tabulated in Table 3. The computational system utilized an Intel(R) Xenon (R) Gold 5218R CPU @2.10 GHz with 30 logical processors and 127 GB RAM. A marginal increase in time occurred due to the utilization of Python to run the MATLAB code using a MATLAB engine API within Python for reliability analysis. For the deterministic analysis, only response spectrum static analysis was employed. However, for RBDO, 221 deterministic time history analyses were conducted for each fitness check, and their outcomes were then processed in MATLAB for reliability analysis, which extended the time. The multi-objective optimization took approximately 119705.03 seconds, around 96% longer than the single-objective optimization, which consumed 60949.07 seconds. The deterministic analysis required only about 156.9 seconds due to a single static analysis per fitness check. However, it is worth noting that the computational time for reliability analysis was significantly reduced through the adoption of PDEM over classical reliability methods, substantially reducing the number of samples needed for each reliability analysis.

Table 3

Comparison of computational time for different optimization procedures

	Deterministic optimization	Single-objective optimization using GA	Multi-objective optimization using NSGA-II
Computational time (seconds)	156.9	60949.07	119705.03

Acknowledgments

The support of the Committee of Science and Technology of Shanghai, China (Grant No. 21ZR1425500) and the Ministry of Science and Technology, China (Grant No. SLDRCE19-B-26) is highly appreciated.

Authors' contributions

Software, investigation, implementation, numerical analysis, and formal analysis: Shrestha S

Critical input, supervision, visualization, validation, writing - review & editing, and resources: Peng Y

Availability of data and materials

Some or all data and materials that support the findings of this study are available upon reasonable request.

Financial support and sponsorship

Committee of Science and Technology of Shanghai, China (Grant No. 21ZR1425500) and the Ministry of Science and Technology, China (Grant No. SLDRCE19-B-26)

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Copyright

© The Author(s) 2023.