Probabilistic Topology Optimization Framework for Geometrically Nonlinear Structures Considering Load Position Uncertainty and Imperfections

Abstract

In this manuscript, a novel approach to topology optimization is proposed which integrates considerations of uncertain load positions, thereby enhancing the reliability-based design within the context of structural engineering. Extending the conventional framework to encompass imperfect geometrically nonlinear analyses, this research discovers the intricate interplay between nonlinearity and uncertainty, shedding light on their combined effects on probabilistic analysis. A key innovation lies in treating load position as a stochastic variable, augmenting the existing parameters, such as volume fraction, material properties, and geometric imperfections, to capture the full spectrum of variability inherent in real-world conditions. To address these uncertainties, normal distributions are adopted for all relevant parameters, leveraging their computational efficacy, simplicity, and ease of implementation, which are particularly crucial in the context of complex optimization algorithms and extensive analyses. The proposed methodology undergoes rigorous validation against benchmark problems, ensuring its efficacy and reliability. Through a series of structural examples, including U-shaped plates, 3D L-shaped beams, and steel I-beams, the implications of considering imperfect geometrically nonlinear analyses within the framework of reliability-based topology optimization are explored, with a specific focus on the probabilistic aspect of load position uncertainty. The findings highlight the significant influence of probabilistic design methodologies on topology optimization, with the defined constraints serving as crucial conditions that govern the optimal topologies and their corresponding stress distributions.

1. Introduction

Topology optimization (TO) plays a vital role in contemporary structural engineering, using mathematical models to distribute materials efficiently within given constraints to improve performance in specific areas. Over the years, TO has made substantial progress, expanding its effectiveness and applications across various fields, like civil and mechanical engineering. These advancements have provided designers with new tools to boost creativity in their project [1,2,3,4,5].

Among the evolving techniques, the bidirectional evolutionary structural optimization (BESO) method has attracted attention as it enables the removal and insertion of material in each iteration, based on sensitivity values [6]. Many studies have explored and enhanced the BESO approach. For instance, research has shown BESO’s improved effectiveness in certain areas, like constrained layer damping and contact stress distribution [7,8,9,10,11,12]. One study by Habashneh et al. [13] proposed a modified BESO technique to optimize steel beams exposed to high temperatures, while Zhao [14] introduced a nodal variable approach, employing a material density field and Shepard interpolation, to optimize the topology of continuum structures. Xu et al. [15] proposed a TO algorithm for double-sided, curved reinforced panels, focusing on the optimization of design variables and structural stiffness. Moreover, using additive manufacturing technologies, Tang et al. [16] proposed a design methodology for the fabrication of periodic lattice structures through the use of BESO. The BESO approach was used in the work of Kazakis and Lagaros [17] to create a MATLAB code for multiscale optimization problems.

In parallel with these advancements, structural stability has emerged as a critical consideration in addition to conventional characteristics, like strength and rigidity. This shift in perspective has prompted extensive research into stability challenges, particularly in complex structures [18,19,20,21]. Notable contributions include optimization approaches addressing buckling constraints, large-scale binary programming problems, and conflicts between structural rigidity and stability requirements [21,22,23,24]. Zhang et al. [25] investigated formulations for topology optimization incorporating linear buckling constraints, employing a mean value function. Additionally, Movahedi et al. [26] developed a new BESO method that accounts for imperfect structural analysis. The growing focus on integrating uncertainty into structural optimization is evident, with researchers exploring how to manage uncertainty through various methods, from reliability-based design to optimizing complex systems in uncertain conditions [27,28,29,30,31].

Reliability-based topology optimization (RBTO) has become essential in ensuring that structures can endure real-world variables, which means designers must carefully consider uncertainties in materials, loads, and other factors. For example, Maute and Frangopol [32] presented a methodology combining first-order reliability analysis with material-based topology optimization. Similarly, Meng et al. [33] worked on a reliability-based approach using fuzzy and probabilistic theories to address uncertainties in both material properties and loads. There has also been a focus on load uncertainties, with tailored methods improving robust topology optimization [34,35,36,37]. Guo et al. [38] looked into multiscale frameworks for robust optimization under bounded load uncertainties. In consideration of external load uncertainties, Jung and Cho [36] devised a method for minimizing the volume of a structure in accordance with reliability design.

Building upon recent advancements, this research introduces an innovative refinement to the BESO method, integrating probabilistic considerations of load position into topology optimization. The proposed approach prioritizes reliability design to address imperfections in geometrically nonlinear analyses, extending beyond load position uncertainty to encompass additional random variables, such as geometric imperfections and material properties. In this work, the reliability index is determined based on the desired volume of the structure at the end of the optimization process. This index serves as a governing limit that controls the final optimized layouts. The numerical examples provided herein illustrate the impact of uncertainty regarding load position on the optimization of structural layouts, highlighting the efficacy of incorporating randomness. Modeled random variables with mean values and standard deviations conform to a normal distribution, further substantiating the effectiveness of the proposed methodology.

The following divisions of this paper are organized as follows: Section 2 offers a comprehensive examination of the theoretical foundation that underpins the expanded BESO method and the proposed algorithm. Section 3 pertains to the algorithm chart that has been devised. The effectiveness of our methodology is demonstrated through numerical examples in Section 4. Ultimately, Section 5 includes conclusions, observations, and the future direction of our research.

2. Problem Statement

This section provides an extensive synopsis of the problem’s context, establishing the groundwork for the topology optimization problem. To accommodate nonlinearity in materials and geometry, this article examines imperfect and perfect analyses employing shell bending theory. The proposed work addresses structural imperfections through the utilization of nonlinear finite element analysis (FEA), with a specific emphasis on stability-oriented design. To mitigate the impact of imperfections, ABAQUS [39] software is used to perform FEA. The proposed methodology of probabilistic elastoplastic imperfect topology optimization considering load position as a random variable is proposed in the subsequent subsections providing specific details on the novel enhancements. Subsequently, the algorithm presented for execution via the BESO method is detailed, offering a comprehensive guide for developing designs resilient to uncertainties.

The manuscript highlights several advancements in the BESO method, particularly integrating load position uncertainty into the topology optimization process. The following enhancements are added: introduction of stochastic variables for load position variability using a Gaussian distribution, incorporation of Monte Carlo simulations for computing reliability indices, and refinement of material addition/removal strategies. Detailed explanations of these modifications are provided in the following subsections.

2.1. Exploring Finite Element Analysis

The optimal layouts of the considered structures are achieved based on analyses: perfect and imperfect. The perfect analysis incorporates shell bending theory to model nonlinearities, while the imperfect analysis introduces initial geometric imperfections, crucial for stability-oriented design. Through nonlinear finite element analysis, imperfections are efficiently addressed, ensuring reliable structural representations that reflect real-world scenarios under loading conditions.

Notably, the emergence of geometric nonlinearity is more pronounced in structures that are subjected to significant deformations, such as those which include deformable or flexible materials. Linear assumptions are rendered invalid in such instances, as structural behavior significantly diverges from linear models. Consequently, the objective of the proposed research is to examine the response of imperfect structures to topology optimization in probabilistic scenarios.

The implementation of an imperfect nonlinear FEA in ABAQUS requires a structured methodology. We first define the geometric and material properties. Through nodal displacements, we introduce initial geometric imperfections that simulate real-world deviations. It is absolutely essential to conduct a linear buckling analysis (LBA) to evaluate how the global buckling modes affect the structure. This fundamental phase provides the groundwork for following nonlinear analyses, providing valuable insights into the mechanisms of failure.

In this study, ABAQUS is employed for finite element analysis (FEA) to perform both linear and nonlinear simulations on imperfect and perfect structures. ABAQUS’s role is limited to performing FEA, which includes modeling material and geometric nonlinearities, as well as introducing initial geometric imperfections. To integrate probabilistic aspects and perform the topology optimization, a custom code is developed to work in conjunction with ABAQUS. Specifically, following the execution of the simulation in ABAQUS, the resultant data, contained inside an output file, are analyzed using MATLAB R2020a. Custom MATLAB scripts were created to retrieve key response variables, including displacements, stresses, node and element coordinates, and element volume from the ABAQUS output file. The data are then reintegrated into the optimization cycle to modify the variables to use in subsequent simulations.

2.2. Deterministic Approach

The BESO method has been extensively discussed in the literature [40,41], with thorough explanations of its fundamental principles. However, in this analysis of plastic ultimate limit behavior, the focus is on understanding how elastoplastic structures respond to gradually increasing loads. A scaling factor, $m_i$, is applied to the initial load ${\mathit{F}}_0$, resulting in a progressively larger force ${\mathit{F}}_i$. As $m_i$ increases, plastic zones begin to expand throughout the structure, eventually reaching a point of full plastic deformation under higher loads. The plastic limit state is reached when the load ${\mathit{F}}_0$ is multiplied by a plastic multiplier $m_p$, yielding the ultimate plastic load ${\mathit{F}}_p$. At this critical stage, the stress field ${\mathit{\sigma }}_{\mathit{i}\mathit{j}}$ is analyzed under quasi-static conditions for the plastic limit load ${\mathit{F}}_{\mathrm{i}}=m_p{\mathit{F}}_0$, while ensuring that the admissible stress field (${\mathit{\sigma }}_{ij}^s$) and force ${\mathit{F}}_{is}=m_{\mathrm{i}}{\mathit{F}}_0$ meet the yield criteria. By employing the principle of virtual velocities, deformable structures under such loading scenarios can be assessed according to the following Equation (1):

$$f\left({\mathit{\sigma }}_{ij}^s,k\right)\le 0$$

(1)

where k represents the plastic properties of the material. By considering a deformable body with volume V and loading surface $S_q$, the concept of virtual velocities can be applied to the stress and force fields. This is achieved by utilizing the strain rate ${\dot{\mathit{\epsilon }}}_{ij}$ and velocities ${\mathit{v}}_i$ that are kinematically permissible, as in Equations (2) and (3):

$$\underset{V}{{\displaystyle \int }}{\mathit{\sigma }}_{ij}{\dot{\mathit{\epsilon }}}_{ij}dV=m_p\underset{S_q}{{\displaystyle \int }}{\mathit{F}}_0{\mathit{>v}}_{i}dS$$

(2)

$$\underset{V}{{\displaystyle \int }}{\mathit{\sigma }}_{ij}^s{\dot{\mathit{\epsilon }}}_{ij}dV=m_s\underset{S_q}{{\displaystyle \int }}{\mathit{F}}_0{\mathit{>v}}_{i}dS$$

(3)

Subtracting these two equations yields the following Equation (4):

$$\underset{V}{{\displaystyle \int }}({\mathit{\sigma }}_{ij}-{\mathit{\sigma }}_{ij}^s){\dot{\mathit{\epsilon }}}_{ij}dV=\left(m_p-m_s\right)\underset{S_q}{{\displaystyle \int }}{\mathit{F}}_0{\mathit{>v}}_{i}dS$$

(4)

To satisfy the yield surface convexity, each surface point must be normalized to ensure that it is contained within the normal distribution. Equation (5) is as follows:

$$({\mathit{\sigma }}_{ij}-{\mathit{\sigma }}_{ij}^s){\dot{\mathit{\epsilon }}}_{ij}\ge 0$$

(5)

Therefore, Equation (4) yields the following Equation (6):

$$\left(m_p-m_s\right)\underset{S_q}{{\displaystyle \int }}{\mathit{F}}_0{\mathit{>v}}_{i}dS\ge 0$$

(6)

The integral in this formulation represents the work when the body’s actual velocities are influenced by external forces. As the plastic limit state is unable to have a negative work, $m_s-m_p\le 0$ must be positive.

The formulation for topology optimization for continuum structures based on elastoplastic limit analysis can be succinctly expressed as follows:

$$\mathrm{Minimize}\text{:} C={\displaystyle \frac{1}{2}}{\mathit{f}}^{\mathrm{T}}\mathit{u}={\mathit{u}}^{\mathrm{T}}\mathit{K}\mathit{u}.$$

(7)

$$Subject to\text{:} V^{*}-{\displaystyle \sum }_{i=1}^N{V}_{\mathrm{i}}{x}_{\mathrm{i}}=0.$$

(8)

$$x_{\mathrm{i}}\in \left\{0,1\right\}.$$

(9)

$${\lambda }_{\mathrm{j}}\ge \underset{\_}{\lambda }>0.$$

(10)

$$m_{\mathrm{i}}-m_{\mathrm{p}}\le 0.$$

(11)

In this context, the objective function is to minimize the structural compliance where the displacement vector is denoted by “$\mathit{u}$” the vector of force is represented by “$\mathit{f}$”, and the global matrix is denoted by “$\mathit{K}$”, and the expression “${\mathit{u}}^{\mathrm{T}}\mathit{K}\mathit{u}$” corresponds to the strain energy of the structure under the applied loads. The binary constraint on $x_{\mathrm{i}}$ ensures that each design variable $x_{\mathrm{i}}$ is either 0 or 1, indicating the absence or presence of material in the corresponding element. Constraints regarding the volume of elements, and the total volume of the structure, respectively, are denoted by $V_{\mathrm{i}}$, and V*. Structural stability is ensured by Equation (10), where ${\lambda }_{\mathrm{j}}$ denotes the buckling load factor associated with the $j-th$ load case and $\underset{\_}{\lambda }$ signifies the minimal value of buckling factor. Equation (11) represents the constraint related to the ultimate plastic load multiplier.

2.3. Introducing Probabilistic Approaches to Topology Optimization

In the midst of uncertainties, fluctuations in load positions are the subject of particular attention within the complex domain of probabilistic topology optimization. Under conditions of uncertainty, the way for robust optimization is paved by estimating failure probabilities and reliability indices.

Expanding upon the foundation laid by prior studies carried out by the authors in uncertain topology optimization [26,42], the current work ventures into a specific scenario exemplified in Figure 1, where the variability in load position emerges as a pivotal factor. The schematic diagram depicts how the external force $\left(\mathit{F}\right)$ experiences stochastic fluctuations within a Gaussian distribution, which is defined by a mean and a standard deviation. Throughout this process, the force remains within the confines of the boundary of the structural domain $\left(\mathsf{\Gamma }\right)$.

To evaluate the structural performance under load position variability, Monte Carlo simulations are conducted with a sample size of $Z={10}^6 \mathrm{to} {10}^9$ for each example. This sample size is chosen to ensure that the Gaussian distribution accurately reflects load position uncertainty, and significantly influences the optimization results.

The main goal of the proposed optimization algorithm is to create designs that are reliable and can handle changes in load positions, effectively transferring forces to the boundary constraints. This particular optimization challenge, different from the typical fixed-load scenario, requires a more resilient approach to finding a solution.

By employing a Monte Carlo simulation, the reliability index (β) is calculated based on the probability of failure ($P_{\mathrm{f}}$). By utilizing random sampling from the probability function $f_X\left(x\right)$ of the random vector $X$, this approach calculates the probability of failure through an examination of the ratio of points within the failure domain to the total number of samples.

Crucial engineering parameters, such as the positioning of loads, material attributes, structural imperfections, and $V_f$, play pivotal roles in the analysis and design processes. Therefore, they are all incorporated into the proposed probabilistic methodology as random variables with means and standard deviations. The Gaussian distribution is used in this investigation because of its simplicity.

Through the establishment of the reliability index, reliability constraints pertaining to $V_f$ are enforced. This constraint, captured by the optimization problem, guarantees the structural integrity even when faced with uncertain loading conditions, and it is constructed as follows:

$${\beta }_{\mathrm{target}}-{\beta }_{\mathrm{calc}}\le 0.$$

(12)

The reliability index (β) is defined as the inverse of the standard normal cumulative distribution function corresponding to the probability of failure. Therefore, to calculate ${\beta }_{\mathrm{target}}$ and ${\beta }_{\mathrm{calc}}$, the following equations are used:

$${\beta }_{\mathrm{target}}=-{\Phi }^{-1}\left(P_{\mathrm{f},\mathrm{target}}\right).$$

(13)

$${\beta }_{\mathrm{calc}}=-{\Phi }^{-1}\left(P_{\mathrm{f},\mathrm{calc}}\right).$$

(14)

In this framework, $P_{\mathrm{f},\mathrm{target}}$ refers to the target probability of failure that the design aims to achieve, reflecting the acceptable level of risk for the structure. $P_{\mathrm{f},\mathrm{calc}}$ is the calculated probability of failure determined from the Monte Carlo simulations.

The subsequent Equations (15)–(19) show the mathematical formulation of the proposed optimization problem:

$$\mathrm{Minimize}\text{:} C={\displaystyle \frac{1}{2}}{\mathit{f}}^{\mathrm{T}}\mathit{u}={\mathit{u}}^{\mathrm{T}}\mathit{K}\mathit{u}.$$

(15)

$$Subject to\text{:} V^{*}-{\displaystyle \sum }_{i=1}^{\mathrm{N}}{V}_{\mathrm{i}}{x}_{\mathrm{i}}=0$$

(16)

$$x_{\mathrm{i}}\in \left\{0,1\right\}$$

(17)

$$P({\lambda }_j-\underset{\_}{\lambda }\ge 0)\ge \mathrm{stab};$$

(18)

$$m_{\mathrm{i}}-m_{\mathrm{p}}\le 0$$

(19)

where ${\beta }_{\mathrm{target}}$ and ${\beta }_{\mathrm{calc}}$ represent the target and calculated reliability indices, respectively, enforced by the cumulative distribution function (CDF). These constraints ensure structural reliability of the resulting layouts.

3. The Developed Algorithm Based on BESO

The proposed algorithm leverages the efficacy of the BESO method, which is known for its robustness in optimizing structural layouts by gradually removing inefficient material while maintaining the overall performance. An algorithmic framework for topology optimization, as depicted in Figure 2, is developed by adopting a systematic process that includes parameter establishment, model specification, and iterative refinement. The algorithm follows a step-by-step procedure designed to improve the structural design’s resilience against probabilistic uncertainties.

The key steps of the proposed algorithm are as follows:

• Material properties, boundary conditions, and loading scenarios are defined using ABAQUS 2021 software for initial simulations.
• A custom MATLAB code utilizes the BESO method to iteratively refine the material distribution by removing inefficient elements based on sensitivity analysis. Each iteration evaluates the objective function subject to constraints, continuing until the predefined constraints and convergence criteria are met.

4. Results and Discussion

This study examines three numerical examples: U-shaped plates, 3D L-shaped beams, and steel I-beams. These scenarios illustrate the efficacy of the proposed RBTO algorithm. To capture the probabilistic essence of the problem, material properties, geometrical imperfections, volume fraction, and the position of the applied load are regarded as random variables that adhere to a normal distribution. Moreover, the results of the first two examples are compared to those presented by Movahedi et al. [43]. Additionally, the results of the third example are compared to those obtained by Habashneh et al. [24], where in this particular example the effect of initial geometrical imperfection is considered.

The foundation of the developed code is based on performing finite element analysis in the iterative scheme by using ABAQUS 2021 software, which makes use of (S4) elements for mesh discretization. These elements are ideal for structural engineering applications due to their adeptness at accurately simulating bending and membrane effects. Furthermore, the impact of mesh size fluctuations on optimization results was thoroughly examined, considering its critical significance in establishing the accuracy, convergence, and reliability of the simulation. As a result, an adequate mesh size was chosen for each model investigated in our research.

4.1. Example #1: U-Shaped Plate

The numerical demonstration focuses initially on a U-shaped plate in order to illustrate the proposed topology optimization framework. As illustrated in Figure 3, the left end of this plate is constrained by fixed boundary conditions that forbid such rotation or displacement. For deterministic and probabilistic designs, the proposed optimization strategy combines the elastoplastic approach with the BESO technique. In order to further evaluate the performance of the proposed algorithm and validate our findings against the benchmark problem developed by Movahedi et al. [43], an additional random variable that represents the uncertainty associated with the position of the applied load is incorporated.

The plate has a thickness of 10 mm. The BESO parameters are defined as follows: τ is set to 1%, the minimum radius $r_{min }$ is 6 mm, and the evolutionary ratio (ER) is 1%. The coefficient of variation (COV) for Young’s modulus is specified at 5%, with a mean value of 160 GPa, and the Poisson’s ratio is set at 0.3. Additionally, the random variable $V_f$ is characterized by a mean of 50% and a standard deviation of 5%. To calculate ${\beta }_{\mathrm{target}}$, a Monte Carlo simulation is conducted using a sample size of $Z=3.0\times {10}^6$. Furthermore, it is important to highlight that the locations of the applied loads, as shown in Figure 3, are treated as random variables with a standard deviation of 5%

The yield stress of the plate is denoted by ${\mathit{\sigma }}_y=135 \mathrm{MPa}$, whereas the initially predefined load is $F_0=1.0 \mathrm{kN}$. Consider the entire design domain with a plastic limit load multiplier of $m_p=3.25$. Therefore, the subsequent four scenarios are presented to elucidate the operation of the plastic ultimate load multiplier: The values are $F_1=0.9 F_0$, $F_2=1.5 F_0$, $F_3=2.5 F_0,$ and $F_4=3 F_0$.

The outcomes of integrating reliability-based topology optimization with the assumption that ${\mathsf{\beta }}_{\mathrm{target}}=3.79$ are presented in

Table 1. The obtained layouts according to various load multiplied in the case of the U-shaped plate.

Applied External Force	Movahedi et al. [43]	Proposed Algorithm
$F_1=0.5 F_0$
$F_2=2 F_0$
$F_3=2.8 F_0$
$F_4=3 F_0$

. It is crucial to emphasize that this comparison compares the results reported by Movahedi et al. [43] with those of the algorithm under consideration, which incorporates uncertain load positions. The observed differences in structural layouts between the proposed algorithm and the findings of Movahedi et al. [43] are attributed to variations in optimization methodologies and underlying assumptions. The distinct layouts generated by the proposed algorithm may signify the exploration of novel design paradigms, potentially offering insights into alternative structural solutions with improved performance characteristics.

A noticeable reduction in stress intensity within our generated layouts compared to those reported by Movahedi et al. [43] is noted, underscoring the efficacy of the considered optimization strategy in enhancing structural robustness, since the lower stress levels are indicative of a more resilient design, capable of withstanding applied loads while minimizing the risk of structural failure.

The results presented in Table 1 reveal that the U-shaped plate shows a variation in final optimized shape compared to the benchmark study by Movahedi et al. [43], attributable to the stochastic nature of the load position. This variation in design demonstrates how sensitivity to load position can influence the final layout and, subsequently, the structure’s overall robustness. It should be noted that in Table 1, the white area depicted at the loading point in the final configuration (on the right side) represents the region within which the load is applied. The position of the load is assumed to follow a normal distribution along this defined region, rather than being applied at a fixed point.

Moreover, a direct correlation between the load multiplier and stress intensity is observed, affirming the fundamental principles of structural mechanics. As the applied load multiplier increases, the resultant stress within the structure is proportionally escalated, reflecting the sensitivity of the system to external loading conditions.

The incorporation of uncertain load positions represents a significant advancement in structural optimization, addressing the inherent variability present in real-world loading conditions. This consideration not only enhances the reliability of optimization outcomes but also reinforces the applicability of the findings in practical engineering scenarios, where uncertainties are inherent and must be accounted for in design processes.

4.2. Example #2: 3D L-Shaped Beam

This paper expands the investigation of the reliability-based topology optimization approach by incorporating a 3D L-shaped beam problem, which signifies an alternative structural configuration. Probabilistic design considerations are the focal point of the analysis methodology, which also accounts for uncertain load positions and geometrically nonlinear analysis. To comprehensively assess the efficacy of our proposed algorithm and validate our findings against the established benchmark problem by Movahedi et al. [43], an additional random variable is introduced. This variable encapsulates the uncertainty linked to the position of the applied load, further enhancing the robustness and realism of our analysis.

In the context of this specific problem, a downward force denoted as $\mathit{F}$ is exerted on the unrestricted end of the beam. It is important to highlight that in Figure 4, the positioning of the applied loads is considered random in both horizontal and vertical directions, adhering to a normal distribution with a standard deviation of 5%. Similar to the previously considered problem, a Monte Carlo simulation is conducted with a sample size of $3.0\times {10}^8$ to calculate ${\beta }_{target}$.

An elastoplastic model is implemented, characterized by a Young’s modulus of $200 \mathrm{GPa}$ and a Poisson’s ratio of $0.25$. A minimum filter radius ($r_{min }$) of $18 \mathrm{mm}$, an element removal ratio ($ER$) of $1\%$, and an allowable error ($\tau$) of $0.1\%$ are the BESO parameters that are employed during the optimization procedure. A yield stress (${\mathit{\sigma }}_y$) of $93 \mathrm{MPa}$ is assumed, with an initial load (${\mathit{F}}_0$) of $10 \mathrm{kN}$. The design domain of the three-dimensional L-shaped beam and the normal distribution under consideration for the uncertain load position are depicted in Figure 4.

For the entirety of the design domain, $m_p$ is set to be $3.48$. In tandem with this parameter, the analysis encompasses four distinct load multiplier cases, denoted as $F_1=0.348 F_0$, $F_2=2.262 F_0$, $F_3=2.784 F_0,$ and $F_4=3.3 F_0$. The resulting layouts, which are presented in

Table 2. Obtained layouts according to various load multiplied in the case of L-shaped beam.

Applied External Force	Movahedi et al. [43]	Proposed Algorithm
$F_1=0.348 F_0$
$F_2=2.262 F_0$
$F_3=2.784 F_0$
$F_4=3.3 F_0$

, show a comparative analysis between those obtained by the proposed algorithm and those achieved by Movahedi et al. [43].

The proposed research work of reliability-based topology optimization highlights how critical it is to consider load position uncertainty.

The findings reveal a noticeable difference in optimized designs when load position uncertainty is accounted for, compared to those that ignore it. This reduction in stress suggests that designs produced under stochastic load conditions can better distribute stress, enhancing structural resilience. This contrasts with earlier studies, such as the work by Movahedi et al. [43], where neglecting load position uncertainty resulted in higher stress intensities.

Moreover, the results also indicate a clear relationship between the load multiplier and stress intensity. As the load multiplier increases, the stress within the structure rises accordingly, which underscores the system’s sensitivity to loading conditions.

4.3. Example #3: Steel I-Beam

This research encompasses the impact of uncertainty regarding load position in the context of RBTO of steel beam considering geometrical imperfection. To comprehensively assess the efficacy of our proposed algorithm and validate our findings against the established benchmark problem by Habashneh et al. [24], an additional random variable is introduced. This variable encapsulates the uncertainty linked to the position of the applied load, further enhancing the robustness of the analysis.

Given its extensive use in construction applications, ISMB 100 cross-sections were selected for analysis. Figure 5 offers a detailed depiction of the geometry and cross-section of the selected beam. The beams that are being examined have a flange width of 55 mm and a total height of 170 mm. Each of the webs and flanges have a nominal thickness of 4.7 mm and 5 mm, respectively, adhering to standard specifications for steel beams of this nature. Additionally, transverse stiffeners, crucial for bolstering structural stability, are incorporated into the design. These stiffeners take the form of flat plates with a nominal thickness of 5 mm and a width of 5 mm. The volume fraction $V_f$ and ultimate load-bearing capacity are both set to $25\%$ and $30.80 \mathrm{kN}$, respectively. Moreover, the proposed methodology incorporates variability by assuming that the load position is randomly distributed, with a standard deviation of 5%.

In this analysis, material properties and initial imperfections are key factors influencing the geometrically nonlinear behavior. The model assumes steel with a Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3. To account for real-world deviations, an initial imperfection ratio of L/1000 is introduced, while the yield point is considered at 355 MPa. These parameters are essential for capturing the nonlinear response of the structure. As stated previously in this article, the BESO method was implemented to resolve the topology optimization issue. To achieve an optimal structural design, specific parameters associated with the BESO technique were carefully selected. $ER$ = 1%, $r_{min}=6 \mathrm{mm}$, $A{R}_{max}=1\%$, and $\tau =1\%$. In addition to the position of the applied force, the initial imperfection value and $V_f$ were considered random variables. The value of $3.0\times {10}^8$ is considered as Z to calculate ${\beta }_{target}$ in the Monte Carlo simulation. The specified value for “stab” is $99\%$.

Table 3. Random variables consideration.

Variable	Mean Value	Standard Deviation
$V_f$	$0.25$	$5\%$
$L/1000$	$1.15$

also provides the mean values and standard deviations of the initial imperfections and random variables $V_f$ for the beam under consideration.

As the proposed methodology incorporates a plastic-limit ultimate load multiplier,

Table 4. ${\mathit{F}}_0$ and $m_i$ values.

${\mathit{F}}_0$ (KN)	${\mathit{m}}_1$	${\mathit{m}}_2$	${\mathit{m}}_3$	${\mathit{m}}_4$	${\mathit{m}}_{\mathit{p}}$
$8.00$	$0.20$	$1.90$	$2.50$	$3.20$	$3.85$

represents the value of the considered initial predefined load as well as the various load multipliers.

The results derived from the proposed framework are presented in

Table 5. Resulting layouts when ${\beta }_{target}=4.00$.

Applied Load	Habashneh et al. [24]	Proposed Algorithm
${\mathit{F}}_1$
${\mathit{F}}_2$
${\mathit{F}}_3$
${\mathit{F}}_4$

and

Table 6. Resulting layouts when ${\beta }_{target}=3.00$.

Applied Load	Habashneh et al. [24]	Proposed Algorithm
${\mathit{F}}_1$
${\mathit{F}}_2$
${\mathit{F}}_3$
${\mathit{F}}_4$

, providing profound insights into the intricate interplay between load position uncertainty, geometric imperfections, and structural performance within the context of topology optimization. A pivotal observation arises from this analysis: the optimized layouts derived under the consideration of load position uncertainty markedly differ from those obtained in scenarios devoid of such considerations. This difference further emphasizes how crucial it is to consider uncertain loading conditions when aiming for robust and resilient structural designs. A comparison with previous work by Habashneh et al. [24] highlights the impact of geometric imperfections. Our optimized layouts showed higher stress intensities, largely due to the introduction of these imperfections. The increased stress can be explained by the fact that steel structures are inherently sensitive to imperfections, which lead to localized stress concentrations.

The influence of the load multiplier on stress intensity is also unmistakable in our results. As the load multiplier increases, the corresponding rise in stress intensity highlights the importance of selecting appropriate load parameters when optimizing structural designs.

Additionally, the proposed methodology reveals subtle but meaningful changes in optimized layouts corresponding to different load multipliers. These variations show how sensitive the optimization process is to the chosen reliability targets. These findings point to the complexity of reliability-based topology optimization, providing valuable guidance for engineers and researchers working to create more robust and resilient designs.

5. Conclusions

In conclusion, this study advances reliability-based topology optimization by refining the BESO method to incorporate probabilistic load position uncertainty. Through detailed numerical analysis, the proposed research demonstrates the significant impact of uncertain loading conditions on optimized designs. A key finding is the noticeable difference between optimized layouts generated with and without load position uncertainty, underscoring the effect of introducing these uncertainties on the optimal structural layouts.

The results also confirm the relationship between the load multiplier and stress intensity, reinforcing the fundamental principles of structural mechanics. As the load multiplier increases, so does the stress intensity, emphasizing the importance of careful load parameter selection. The interplay between load position uncertainty, load multiplier, and the resulting designs offers deep insights that could help engineers and researchers enhance structural resilience when facing uncertain loading conditions.

Looking forward, future research in reliability-based topology optimization could explore integrating advanced machine learning techniques to improve predictive capabilities. Leveraging vast datasets and sophisticated algorithms can enhance our understanding of complex design–performance relationships. Additionally, investigating optimization strategies for novel materials and advanced manufacturing methods, like additive manufacturing, holds promise for achieving higher levels of structural reliability and performance. Embracing these emerging technologies and methodologies will drive innovation in structural engineering, pushing the boundaries of what is possible in optimized structural design.