RELIABILITY ASSESSMENT OF STEEL PORTAL FRAMES USING GAN FOR GENERATING SYNTHETIC DATA SAMPLE

This paper addresses the challenge of reliability estimation in structural engineering, where determining failure probabilities is often uncertain due to insufficient characterization of randomness and scale of design problems. Current approaches, like EN 1990, lack sufficient data for research and simulations, particularly for low failure probabilities (around 10‐4), making Monte Carlo simulations less accurate. The paper introduces Generative Adversarial Networks (GANs) as a solution to generate synthetic data to supplement existing examples. The study applies GANs to assess the reliability of steel portal‐framed industrial buildings and evaluate the safety of this structural solution according to Eurocodes.


Introduction
Structural design codes are designed to ensure that principles and rules applied to design result in suitable reliability levels.The safety margin directly signifies the level of risk society is willing to tolerate while maximizing an economical and sustainable design approach.The family of Structural Eurocodes, encompassing EN 1990 to EN 1999, is rooted in the limit state design philosophy and the utilization of partial factors.Since the inception of the Eurocodes, extensive research has been conducted both at the European and global scales, aiming to enhance design rules by providing more precise estimations of real-world behavior.
While EN 1990 specifies target reliability levels, the reliability standards across various Eurocodes and even within specific rules may vary.The ongoing revision of the Eurocodes has prompted scientific discussions regarding the partial factors associated with loads and resistances.This raises questions about the reliability of both existing and new constructions [1].
The classical problem of reliability, which involves estimating the probability of failure, often remains undefined due to its scale, primarily the size of the design problems, and the insufficient characterization of randomness in terms of statistical analysis of key random variables such as material properties and geometrical features [2].A common challenge in current approaches is the scarcity of data examples capable of supporting additional research, simulations, and experiments.In contemporary times, various intelligent data management strategies have emerged to enhance existing data by generating synthetic examples for subsequent use in

XIV Conference on Steel and Composite Construction
Portugal predicting specific behaviors.These approaches aim to maximize the probability of concealing whether a data point originates from the original distribution or not.
This paper focuses on establishing a practical approach to assess the safety of Eurocodes by concurrently addressing both the loading and resistance aspects of the reliability issue.To achieve this, a straightforward structural example, a steel pitched-roof portal frame, is chosen.This selection is deliberate because it encompasses all the attributes of more intricate structural systems while maintaining a manageable structural complexity.Consequently, it becomes feasible to generate a substantial array of structural configurations and load scenarios in an automated manner, facilitating a direct reliability evaluation employing the Monte Carlo method.Furthermore, utilizing this dataset, the paper investigates the viability of expanding the dataset artificially through the application of Generative Adversarial Networks (GANs).

Methods to assess reliability
EN 1990 [3] serves as the Eurocode laying down fundamental principles and prerequisites for the safety evaluation of structures.It forms the foundation of design and guides structural reliability.This code mandates specific reliability factors based on consequence classes, which consider the variations in loads, materials, geometrical properties, and the precision of design models.
Ideally, the reliability problem should be addressed by considering both loads and resistances together [4].Nevertheless, for practical purposes, EN 1990 permits the separation of variability attributed to loading and resistance.In contrast, certain codes like AISC [5] calibrate the reliability index beta based on a live-to-dead load ratio of 3, resulting in values of approximately 2.6 for members and 4.0 for connections.This approach imposes load-dependent reliability assessment in such codes.
The simplification introduced by EN 1990 plays a pivotal role in determining partial factors for resistance without simultaneous consideration of the action side.This approach has been embraced by the Eurocodes.More recently, it was applied in the European project SAFEBRIC-TILE [6], following Annex D of EN 1990, to systematically assess the design rules in EN 1993-1-1 [7] (EC3), encompassing stability, ductile, and brittle failure modes.The SAFEBRICTILE project's findings generally indicated that the rules within EC3 demonstrate adequate reliability.These results prompted recent modifications to reduce the variability in real-world reliability among different failure modes.These conclusions were derived from previous assessments [8] and probabilistic evaluations in PROQUA [9], which indicated that the current partial factors for resistance generally provide conservative estimates.Notably, these findings were recently extended to High Strength Steel (up to S700) within the European project STROBE.
Ongoing discussions within the Eurocodes revision [10] have contemplated altering the reference period for the reliability index from 50 years to 1 year, which would impact the division between loading and resistance and, consequently, the partial factors.Furthermore, some studies have suggested changes to the partial factors for permanent and variable loads [11,12], though these recommendations primarily pertain to simple failure modes, often focusing on cross-section resistance alone.
Regarding the resistance partial factors, [A.3] demonstrated that sufficient reliability is generally achieved.Additionally, a recent reliability assessment [13,14] examined various materials using different National Determined Parameters (NDPs) and concluded that the target reliability is met in most cases.However, disparities between different materials and load combinations were observed.Nonetheless, given the study's limitations, it underscores the necessity for a comprehensive evaluation of this issue.
Reliability assessment poses significant challenges primarily due to the sheer scale of the problem.It encompasses numerous random variables, each defined by statistical distributions that are frequently insufficiently characterized.Furthermore, the intricate nature of these phenomena gives rise to highly non-linear solutions for failure probabilities, often rendering them exceedingly difficult, if not impossible, to obtain.
The studies mentioned earlier employ FORM [15] as an approximation method, but its accuracy diminishes when dealing with non-linear failure surfaces.A more effective approach is Monte Carlo simulation (MCS); however, its precision hinges on the number of simulations conducted.Given that the failure probability specified in Eurocode is approximately 10 -4 , millions of simulations are necessary.There's potential for enhancement through variance reduction techniques [16], which can reduce the required number of simulations, or by expanding artificial datasets.Additionally, novel methods like surrogate models [17] are emerging to alleviate the computational burden of MCS by using approximate models to replicate finite element simulations.

Generative Adversarial Networks (GAN)
This section offers an introduction to Generative Adversarial Networks (GANs) and their utilization in this study.Figure 1 illustrates the standard architecture of a GAN.
GANs, introduced by Goodfellow et al. in [18], are a class of deep learning models renowned for their capacity to produce authentic synthetic data.As depicted in Fig. 1, GANs consist of two neural networks: a generator and a discriminator.The generator's role is to generate synthetic data that closely resembles the training data, while the discriminator's task is to differentiate between genuine and synthetic data.The training of GANs encompasses a dynamic interplay between the generator and discriminator networks.Initially, the generator generates synthetic data, which the discriminator assesses.The discriminator offers feedback to the generator regarding the fidelity of the synthetic data compared to real data.This feedback serves to refine the generator, enabling it to produce increasingly realistic synthetic data.Throughout the training, the generator improves its ability to create synthetic data that convincingly deceives the discriminator into perceiving it as authentic.
A notable application of GANs is dataset augmentation, where the synthetic data produced by the GAN is incorporated into the original training dataset to expand its size and enhance its diversity.This proves beneficial in scenarios where the training data is scarce or when it's necessary to introduce new samples to enhance the performance of a model.

Design of steel portal frames
In this section, the essential geometric parameters of a standard single-bay pitched-roof portal frame are first identified, based on which a set of portal frames is created and designed, following the guidelines of the structural Eurocodes.Since the goal of this study is to assess the reliability of such a structural solution, the frames are designed to push for a high design efficiency.This is achieved by implementing a Genetic Algorithm in which a novel index of efficiency is introduced and set as a target function.Finally, the results of the EC3-based optimized frames are compared with outcomes derived from a reality-simulating Finite Element Method (FEM) model, whereby geometric and material nonlinearities alongside imperfection are incorporated.

Frame geometry
The portal frame under examination is a symmetrical single-bay structure, with Fig. 2 depicting one-half of this structure.Key dimensions of interest include the system span (L), column height (h), and slope (P or α) in this study.The column typically adopts an H-section, while the rafter employs an I-section, featuring an eave haunch but no apex haunch.It's important to note that both hot-rolled and welded sections are considered for this work.
The primary geometric properties of the frame adhere to standard guidelines [19] and are illustrated in Fig. 2. As initial approximate dimensions, the beam depth is approximately L/70, the column depth is about L/80, and the haunch depth is roughly L/35.These dimensions will be further validated through analysis and verification.Furthermore, the slope P is set at 10% (equivalent to α = 5.71º), and the spacing between purlins and façade rail beams is adjusted to fall within common ranges, typically between 1.5-2.5m.Meanwhile, the following geometric parameters undergo variations: -Span: 25m £ L £ 45m (step 5m).
In total, this paper explores 110 distinct frame geometries, resulting in the optimization of 110 solutions using rolled profiles, as well as an additional 110 optimized solutions employing built-up welded sections.

Load actions and combinations
Concerning the loading conditions, the building in question serves an industrial purpose and features a lightweight roof.Located in Portugal, the following load factors are considered: -G0: self-weight of the frame.

Design efficiency index
The design of the portal frame adheres to all pertinent Ultimate Limit State (ULS) load combinations, calculated in accordance with EN 1990.Typically, the structural safety verification is condensed into a singular value for the entire structure, denoted as the utilization factor for resistance (UR).This value is determined as the maximum among all load combinations and is defined by Eq. (1) as: ( where the utilization factor for resistance (UR) is defined as the ratio of the design value of the effect of applied actions (Ed) to the design value of resistance (Rd) for the specific load combination (c).UR typically falls between 0 (inefficient) and 1 (optimal), serving as an indicator of structural safety.However, it can be overly conservative when assessing design efficiency.To address this, an index called the design efficiency index (IDE) is introduced.IDE offers a more nuanced representation of structural efficiency.To calculate IDE, the structure must be discretized into elements (e) of sufficiently small size, ensuring approximately constant properties and internal forces throughout their length.Considering all load combinations, a unique index is defined by Eq. (2) as: (2) where UR,c,e,max is the maximum value of the utilization factor for each element (e) considering all load combinations (c), and W is the total weight of the structure obtained by Eq. (3) as: (3) IDE is a non-dimensional weighted average of the utilization factor in the structure, with a value between 0 and 1.Higher values of IDE indicate a more efficient use of the material.More details regarding the design efficiency index can be found in [20].

Design efficiency index
In this study, a conventional real-coded genetic algorithm is implemented for the optimization of the steel portal frames [21].For steel portal frames, studied in this paper, the optimization problem is formulated by the objective function that refers to optimization searching for a maximum design efficiency index IDE, as shown in Eq. ( 4): subject to two main constraint,  !"# (̅ ) and  #"# (̅ ), where ULS refers to the ultimate limit state check and SLS to serviceability limit state check, as shown in Eq. ( 5), (5) where UR is defined by Eq. ( 1) and UD is the displacement utilization, defined by Eq. ( 6): (6) where dp,c is the displacement at the control point p on the direction of interest and for load combination c, and dp,max its maximum acceptable value.In this equation, C is the total number of combinations, and P is total number of control points.
In this study, the variable vector ̄ (genotype) comprises three variables for hot-rolled solu- tions -beam IPE profile Xb, column HEB profile Xc, and haunch length ratio Xh, whereas the number of variables for the built-up solutions is somewhat enlarged, i.e. 4 variables defining the column cross-section ( Xbf,c, Xtf,c, Xh,c, Xtw,c), 4 variables defining beam cross-section (Xbf,b, Xtf,b, Xh,b, Xtw,b), and another 2 variables defining haunch thickness and length ratio (Xt,H, XL,H).Each variable has boundaries coming from physical restrictions.
In Fig. 3, the index IDE is plotted for all studied geometries of portal frames, 110 optimized hot-rolled solutions (x-symbol), and 110 optimized welded solutions (dotted).The results are compared in Table 2.  Due to their flexibility in geometry, welded sections allow for a higher level of optimization and hence increased structural efficiency (40% on average, with c.o.v of 11%).Moreover, based on the presented results, it is indicative that IDE is a reasonable candidate to be used as a normalized index that reflects the efficiency of the structural design.

Design by finite element method (FEM)
In this study, Finite Element (FE) analysis is conducted for three portal frames constructed using hot-rolled sections (refer to Table 2), selected from the pool of 110 case studies outlined in section 4.1.1.These three frames have the same column height = 7.5m) and frame spacing (S = 5.0m).However, they differ in terms of span, with three different spans considered (L = 25m, 35m, and 45m).The frames are modeled in Abaqus with a 3D approach (as depicted in Fig. 4).

Fig. 4: model of the portal frame
The discretization of these models is accomplished using linear four-node shell elements with reduced integration (S4R).The chosen material, S355, is modeled as elastic-plastic without strain hardening.While the purlins and façade rail beams are not explicitly incorporated into the models, their presence is simulated by applying lateral restraints to the outer (exterior) flanges of the rafters and columns Two types of analyses are carried out: i) Linear elastic bifurcation analysis (LBA) to generate eigenmodes, which serve as the initial shapes of geometric imperfections, and ii) Geometrically and materially nonlinear analysis with imperfections included (GMNIA), wherein the ultimate resistance of the frame is determined.To achieve this, the Riks method from the software's library is employed, resulting in a non-linear static equilibrium solution.
For all three cases, the critical load combination for Ultimate Limit State (ULS) verification is the one characterized by a predominant vertical load (1.35G + 1.5Q + 0.75S).The most critical cross-section for assessment is situated at the junction between the rafter and column (as shown in Fig. 5).This critical cross-section aligns with the results obtained from the European Code for Steel Structures (EC3) verifications.Table 2 presents a comparison between the maximum loads of the frame, as calculated following Eurocodes and through advanced Finite Element (FE) analysis.In cases with shorter spans, the frame capacity according to Eurocode aligns closely with the capacity determined via FE analysis, differing by less than 5%.However, for larger spans, the Eurocode standards tend to be more conservative, resulting in a resistance difference of 24%, which is notably significant.

GAN applied to steel portal frames
Consider X as the collection of actual building frame designs, where each design is represented as a vector of design parameters (L, H, S, P, sp, sf, ̄).Correspondingly, let Y denote the set of IDE index values corresponding to these designs.Now, introduce G(z;θg) as the generator network, which accepts a noise vector z as input and generates a synthetic building frame design g, optimized to maximize the IDE index.Similarly, D(;d) represents the discriminator network, taking a building frame design x as input and producing a binary classification that indicates whether x is a genuine or synthetic design.The GAN training objective can then be defined as a minimax game between the generator and discriminator networks, following the formulation outlined in [19].
In this context, preal(x) represents the probability distribution associated with real building frame designs, while pz(z) corresponds to the probability distribution of the noise vectors z.
The objective is to determine the best-fitting parameters θg and d that minimize the objec- tive function described above.By achieving this, the generator network can produce synthetic building frame designs closely resembling those found in the real dataset X, while the discriminator network can effectively differentiate between real and synthetic designs.Upon successful GAN training, it becomes possible to generate additional synthetic building frame designs.This is accomplished by sampling noise vectors from pz(z) and passing them through the generator network.These newly generated designs can then be assessed using the IDE index.Designs that demonstrate favorable performance can be integrated into the original dataset X, thereby enhancing the overall performance of the optimization algorithm.
The quantity of examples necessary to effectively train a Generative Adversarial Network (GAN) can fluctuate depending on various factors, including problem complexity, the dimensions of input data, and the GAN's architecture.However, as a general guideline, GANs often require a substantial number of training examples to grasp the underlying distribution of real data.Given our current dataset of 110 generated examples, it may not suffice to train a robust GAN effectively.To overcome this limitation, we intend to augment our dataset.This can be achieved by either generating more designs using the existing optimization algorithm or by procuring additional data from external sources if feasible.The expansion of our dataset aims to enhance the quality and diversity of the generated designs, facilitating more effective learning by our GAN.This, in turn, will enable us to attain more accurate and reliable outcomes for our structural design task.q f + -

= E E
Recent research [21] has unveiled the possibility of achieving competitive outcomes with a few-shot adaptation of GANs, even when using as few as 5-100 examples.Their approach entails fine-tuning the generator network of a pre-trained GAN for a new task, which, in our case, would be structure design, with a small number of examples as training data.They also introduce a technique termed adaptive instance normalization (AdaIN), enabling the generator network to dynamically adjust the mean and variance of feature maps to better align with the style of input data.This method has shown promising results in tasks like image generation and could potentially be applicable to the domain of building structure design.

Conclusions
This paper serves as a pilot study, showcasing the viability of conducting a reliability assessment for the safety standards outlined in the Eurocodes.It does so by concurrently examining both the loading and resistance facets of the reliability issue.To maintain brevity, the paper did not treat the fundamental variables as random variables.Nonetheless, the methodology presented here possesses a broader applicability, enabling the direct evaluation of design codes against "advanced numerical experiments."Additionally, the paper illustrates that it's feasible to employ a deep learning model, such as GAN, to expand the dataset while achieving substantial computational savings:

Fig. 3 :
Fig. 3: Index IDE for studied set of portal frames: Hot-rolled vs Built-up solution

Fig. 5 :
Fig. 5: Failure mode under the critical load case (vertical load)

Table 1 :
Index IDE: Hot-rolled vs built-up solution

Table 2 :
EC3 vs FEM -maximum load capacity in kN/m