Risk−Based Cost−Benefit Optimization Design for Steel Frame Structures to Resist Progressive Collapse

Abstract

The design of structures to resist progressive collapse primarily focuses on enhancing structural safety and robustness. However, given the low probability of accidental events, such designs often lead to a negative cost–benefit. To address this problem, this paper uses risk analysis to optimize the progressive collapse resistance design of steel frame structures. The elements’ cross-section design for the progressive collapse resistance of steel frame structures is optimized using genetic algorithms and SAP2000 23, which identify the structural model with the minimum robustness index while ensuring safety. The results show that the risk-based robustness index can effectively assess the cost of progressive collapse design. More importantly, the optimization model can rapidly identify the most cost-effective structural design solution that complies with progressive collapse resistance guidelines, enhancing the simplicity and usability of the structure design optimization process. Additionally, the integration of the SAP2000 API with Python 3.8 automation streamlines the parameterization process, minimizes manual errors, and enhances the precision and efficiency of the structural design optimization. Finally, the model’s effectiveness is validated through a case study, where the refined single-frame structure shows a reduction in initial construction and collapse-related costs by 2.4% and 9.1%, respectively. Meanwhile, the three-dimensional frame shows a 2.9% rise in initial costs but a 13.5% decrease in total collapse-resistant design costs, illustrating the model’s ability to balance safety with cost-effectiveness.

1. Introduction

Buildings may be subjected to accidental loads such as fire, earthquakes, and explosion impacts during their long service life, which often leads to local damage to structures. Notable incidents include Ronan Point (UK, 1968), Skyline Plaza (USA, 1973), Oklahoma City (1995), the World Trade Center (New York, 9/11, 2001), and the Changsha self-built houses (China, 4/29, 2022). These events have raised awareness about robust design against progressive collapse. In robust design, uncertainties such as material parameters, geometric parameters, and loads play a significant role. Under multiple uncertainties and hazards, the structure’s ability to resist damage can be assessed through structural reliability analysis and probability assessment [1]. Additionally, Bassam et al. [2] introduced a probabilistic risk assessment framework for multi-story steel buildings under extreme load conditions, which has been effectively utilized in further studies [2,3]. The probability of progressive collapse under variable uncertainties and hazards is calculated as follows [2,3]:

$$p_C=P\left[C\right]=\sum _H\sum _{\mathit{LD}}P\left[C\right|\mathit{LD},H\left]P\right[\mathit{LD}\left|H\right]P\left[H\right]$$

(1)

where $P\left[H\right]$ is the probability of abnormal event occurrence; $P\left[\mathit{LD}\right|H]$ is the conditional probability of local damage given $H$; and $P\left[C\right|\mathit{LD}, H]$ is the conditional probability of progressive collapse given H and LD. Typically, $P\left[H\right]$’s range is between ${10}^{-6}$ and ${10}^{-5}$ annually [4]. To mitigate these risks, non-structural measures are recommended, including the installation of barriers outside buildings to prevent impacts from vehicles. However, due to the large uncertainty regarding accidental loads and events [5], Stewart [6] proposed that protecting all US public buildings under conservative discretionary column removal scenarios is not economical. A similar study also shows [7] that applying control measures to iconic bridges to prevent terrorist attacks is only economical when the threat probability exceeds ${10}^{-4}$. Consequently, it is not economical to design structural components to prevent the occurrence of $H$. For $P\left[\mathit{LD}\right|H]$, the primary approach is the key structural element, which is used to enhance the structure’s resistance to collapse, but the large uncertainty makes this method easily uneconomical. For instance, Shi and Stewart [8] found that the failure probability of reinforced concrete columns significantly varies with changes in the load magnitude and impact distance; similar results have been confirmed for steel columns [9]. Regarding $P\left[C\right|\mathit{LD}, H]$, current anti-collapse design codes [10,11,12] allow for local damage under abnormal loads and events while requiring structures to have sufficient capacity to resist stress redistribution, and this is the basic framework of the alternate load path (ALP) method [10,12]. According to the studies above, this paper focuses on $P\left[C\right|\mathit{LD}, H]$ and employs the ALP method to quantify the structure’s collapse probability.

Currently, the capacity of structures to resist progressive collapse is primarily indicated by their robustness. EN 1991-1-7-2010 [13] and GSA (2013) [10] define structural robustness as “the capacity of the structure to withstand abnormal events, such as fire or explosion, without disproportionate collapse relative to the initial damage”. Recently, extensive research has been conducted on structural robustness, focusing on deterministic and stochastic robustness. Deterministic robustness is defined as the ratio of certain metrics between undamaged and damaged structures based on structural performance, such as deformation, energy absorption, and load-bearing capacity [14,15,16]. The deterministic robustness index is easily and conveniently calculated but does not account for the randomness of loads and the unique characteristics of the structure, particularly given the low-probability–high-consequence (LPHC) nature of accidental events [17]. To address these uncertainties, researchers have proposed corresponding robustness indexes [6,18,19,20,21]. Among these, risk-based robustness considers both the causes of structural collapse and the consequences. It includes several factors such as the type of accidental event, its probability of occurrence, the local damage caused by the event, and the subsequent direct damage, as well as indirect consequences (casualties, social and environmental impacts, economic losses, etc.) [22].

It should be noted that it is feasible to design buildings with sufficient robustness to withstand structural damage caused by abnormal loads. However, due to the LPHC of accidental events, in addition to considering the structural safety redundancy, the economic feasibility should also be taken into account. The occurrence of abnormal loads is highly uncertain and may not happen during the entire life cycle of the structure, leading to a conflict between the initial construction cost and the future expected losses [23,24,25,26]. Therefore, there has been an increasing number of studies focused on optimizing structural design for progressive collapse [27,28].

In a pioneering study, Beck et al. [29,30,31] proposed a cost–benefit analysis of progressive collapse design. They considered $P\left[C\right|\mathit{LD}, H]$ as an independent parameter and divided the collapse cost of the structure into two categories: initial construction cost and collapse-related cost. The construction costs were nondimensionalized based on the structure’s load-bearing capacity, while the collapse-related costs included expenses associated with building closure (such as revenue loss), debris removal, and reconstruction, as well as impacts on the surrounding environment and human casualties. Beck also highlighted that the probability of column failure is crucial in optimization design and calculated a threshold for this probability. It was proposed that designing for collapse resistance is economical and effective only when the failure probability exceeds this threshold.

It is observed that the majority of research on progressive collapse design primarily focuses on structural safety, with insufficient consideration for economic efficiency. Due to the low probability of accidental events, there is a need to integrate both collapse probability and initial construction costs, yet current studies lack a comprehensive robustness index that considers both structural collapse risk costs and initial construction costs. Additionally, while the main goal of structural optimization design focuses on minimizing structural mass and material usage, the process becomes complex due to the need to consider uncertain factors such as accidental loads. Consequently, few studies effectively integrate reliability analysis, robustness indices, and the structural optimization design process. Beyond structural safety, economic considerations are essential for achieving the highest overall benefits in progressive collapse design. Moreover, while most secondary development efforts with SAP2000 have been concentrated on parameterizing modeling or analysis for specific models, there has been no secondary development aimed at optimizing design for continuous collapse resistance. Considering a large number of structural scenarios necessitates significant manual modeling and calculation by engineers, which is time-consuming, labor-intensive, and prone to human error.

Consequently, this paper contributes to three main areas. First, it introduces a new risk-based robustness index that not only accounts for structural collapse risk costs but also integrates initial construction costs, providing a novel and more economically viable method for progressive collapse design. Second, by combining genetic algorithms with the SAP2000 API for structural optimization design, this study not only enhances structural safety but also improves economic efficiency. We demonstrate significant reductions in initial construction and collapse-related costs while maintaining structural safety. Finally, through the secondary development of SAP2000, we have automated the extraction of parameters to model analysis, significantly improving computational efficiency and reducing manual errors, which is particularly crucial for handling numerous structural scenarios.This research adopts the analytical framework developed by [2,3,4], incorporating the SAP2000 API and Python to optimize the design for progressive collapse in steel structures. Section 2 employs Monte Carlo and Latin Hypercube Sampling (LHS) for structural reliability analysis, considering the risk-based robustness index in the design process to evaluate both economic and structural robustness. In Section 3, Python and the SAP2000 API are used to parameterize variables such as loads, materials, section properties, and outputs, applying these to a case study that assesses the reliability of a steel frame structure. Section 4 uses a parametric design approach and genetic algorithm to optimize a steel structure. The cross-sectional area of structural components is targeted as the primary optimization variable, with the objective of minimizing the robustness index while adhering to the structure’s limit state function as a constraint. This approach can effectively select the optimal structural system that fulfills the design intentions, significantly enhancing the efficiency of the structural design process.

2. Structure Reliability Analysis

2.1. Parametric Reliability Analysis Steps for Steel Frame Structures

Due to the complexity of the progressive collapse process, this paper will use the Monte Carlo method to calculate the reliability indices for the collapse resistance of steel structures. Additionally, the optimization process involves significant computations and data handling, necessitating the use of LHS to reduce the number of samples required, which will ensure precise simulation results with minimal iterations.

Latin Hypercube Sampling utilizes stratified sampling to enhance sampling efficiency with smaller sample sizes, thus addressing the issues of large sample requirements and low efficiency associated with Monte Carlo methods.

Suppose the function has N basic variables and M sampling instances, denoted as $x =(x_1,x_2,\dots x_m)$. The cumulative distribution function for the input variables is given as below:

$$Y_i=F_i(X_i), i=1,2,\cdots \mathrm{M}$$

(2)

As shown in Figure 1, the cumulative distribution function $F_i\left(X_i\right)$ is divided equally into N parts, and uniform sampling from 0 to 1 is conducted within each interval. The sampled value $Y_i$, used as $L_n$, is then transformed through the inverse function of $F_i\left(X_i\right)$ to obtain $X_i$, as specified in Equation (3).

$$x_{\mathit{ni}}=F_i^{-1}(L_n), n \in \mathrm{N}, i \in \mathrm{M}$$

(3)

By repeatedly sampling all random variables in the limit state function and arranging each sample value as a column vector in matrix A, an “N × M” sampling matrix A is obtained. By randomly permuting the column vectors of matrix A, the final sampling matrix B is produced.

Furthermore, Python is integrated with SAP2000 to automate both parameter input and output result extraction. The Python programming is organized into five distinct modules: LHS sampling, load parameterization, material parameter input, section parameterization, and results extraction. The LHS sampling module is used to sample the predefined random variables, which will be integrated into various parameterization modules (section, material, and load). Through nonlinear static analysis and the results extraction module, the required load coefficients are extracted, and their statistical moments are calculated. Finally, the failure probability and reliability indices of the steel frame structures are determined. This process is illustrated in Figure 2.

2.2. Limit State Function

According to the requirements for nonlinear static analysis in GSA 2013, when applying loads, the dynamic amplification factor ${\Omega }_N$ must be considered. The calculation method for ${\Omega }_N$ is explicitly given in

Table 1. Dynamic amplification factor ${\Omega }_N$.

Material	Structure	Dynamic Amplification Factor
Steel	Frame	$1.08+0.76/({\theta }_{\mathit{pra}}/{\theta }_y+0.83)$

[10].

Here, ${\theta }_{\mathit{pra}}$ is the plastic hinge rotation, and it is determined according to Table 9-7.1 (

Table 2. Modeling parameters and acceptance criteria for nonlinear procedures—flexural actions.

Parameters		Acceptance Criteria
Plastic Rotation Angle a\b (Radians) Residual Strength Ratio c		Plastic Rotation Angle (Radians) Performance Level
		IO	LS	CP
1. ${\displaystyle \frac{b_f}{2t_f}}\le 0.30\sqrt{{\displaystyle \frac{E}{F_{\mathrm{ye}}}}}$ and ${\displaystyle \frac{h}{t_w}}\le 2.45\sqrt{{\displaystyle \frac{E}{F_{\mathrm{ye}}}}}$	$\mathrm{a}={9\mathsf{\theta }}_{\mathrm{y}}$ $\mathrm{b}=11{\mathsf{\theta }}_{\mathrm{y}}$ $\mathrm{c}=0.6$	${0.25}^{ \mathrm{a}}$	a	b
2. ${\displaystyle \frac{{\mathrm{b}}_{\mathrm{f}}}{2{\mathrm{t}}_{\mathrm{f}}}}\ge 0.38\sqrt{{\displaystyle \frac{\mathrm{E}}{{\mathrm{F}}_{\mathrm{ye}}}}} \mathrm{o}\mathrm{r} {\displaystyle \frac{\mathrm{h}}{{\mathrm{t}}_{\mathrm{w}}}}\ge 3.76\sqrt{{\displaystyle \frac{\mathrm{E}}{{\mathrm{F}}_{\mathrm{ye}}}}}$	$\mathrm{a}={4\mathsf{\theta }}_{\mathrm{y}}$ $\mathrm{b}=6{\mathsf{\theta }}_{\mathrm{y}}$ $\mathrm{c}=0.2$	${0.25 }^{\mathrm{a}}$	${0.75}^{ \mathrm{a} }$	a

Other: linear interpolation between the values on lines 1 and 2 for both flange slenderness (first term) and web slenderness (second term) shall be performed, and the lower resulting value shall be used.

) in ASCE 41 [11]. The steel frame structures discussed in this study adhere to the strong column–weak beam design principle, which primarily predisposes the beams to failure. Consequently, the failure criteria for these structures are predominantly determined by the limits of plastic hinge rotation specified for beams.

${\theta }_y$ is the yield rotation, which is calculated according to Equation (4) in ASCE 41:

$${\mathsf{\theta }}_{\mathrm{y}}={\displaystyle \frac{\mathrm{Z}{\mathrm{F}}_{\mathrm{y}}{\mathrm{l}}_{\mathrm{b}}}{6\mathrm{EI}}}$$

(4)

where Z is the plastic section modulus, ${\mathrm{F}}_{\mathrm{y}}$ is the yield strength of steel, ${\mathrm{l}}_{\mathrm{b}}$ is the length of the beam, E is the modulus of elasticity for the steel section, and I is the beam’s moment of inertia.

In this section, SAP2000 software is utilized to perform a pushdown analysis on the steel frame structure. The ultimate load factor ${\mathsf{\alpha }}_{\mathrm{u}}$ is defined as the maximum load-bearing capacity of the structure during progressive collapse analysis. Therefore, the limit state function of the structure is as follows:

$$\mathrm{Z} ={\mathsf{\alpha }}_{\mathrm{u}}-{\mathsf{\Omega }}_{\mathrm{N}}$$

(5)

As the beam and column cross-sections are continuously changing during the optimization design process, their sectional dimensions also vary with the changes in the sections. Therefore, during the analysis process, it is necessary to dynamically change these values using the SAP2000 API.

2.3. Risk-Based Robustness Index

Risk-based robustness assessments for structures involve analyzing the causes and probability of collapses, offering crucial insights for decision-makers. The collapse cost analysis includes the selection of types of accidental events, the probability of occurrence, the direct consequences of local damage to the building, and the indirect consequences such as casualties and social impacts that result from the direct consequences. Pinto et al. [18] calculated the risk cost of structural failure SR by multiplying the probability of structural failure ${\mathrm{P}}_{\mathrm{f}}$ by its consequences “C”. Equation (6) illustrates this calculation, with “C” determined by either the ultimate limit state or the serviceability limit states.

$$\mathrm{SR}={\mathrm{P}}_{\mathrm{f}} \times \mathrm{C}$$

(6)

Beck et al. [31] nondimensionalized the collapse cost associated with ultimate limit states, only considering the alternate load paths in scenarios of column removal in steel frame structures. Other design measures, such as enhancing structural constraints, increasing component ductility, the compressive arching effect, and catenary actions were not considered. The GSA guidelines [10] specify the load combinations for extreme events as follows:

$$\mathsf{\varphi }{\mathrm{R}}_{\mathrm{m}} \ge 1.2{\mathrm{D}}_{\mathrm{n}}+0.5{\mathrm{L}}_{\mathrm{n}}$$

(7)

where R is the resistance; D is the dead load; L is the live load; subscript (n) is the nominal values; subscript (m) is the mean values; and $\mathsf{\varphi }$ is the strength reduction factor, typically set to 1.

Based on the load combination coefficient in Equation (7), the optimization coefficient ${\mathsf{\lambda }}_{\mathrm{PC}}$ is also introduced and is used as a design variable. Ultimately, by adjusting ${\mathsf{\lambda }}_{\mathrm{PC}}$, the optimization of the structural design is achieved, which is shown in Equation (8):

$$\mathsf{\varphi }{\mathrm{R}}_{\mathrm{m} }\ge { \mathsf{\lambda }}_{\mathrm{PC}}(1.2{\mathrm{D}}_{\mathrm{n}}+0.5{\mathrm{L}}_{\mathrm{n}})$$

(8)

The initial construction cost is considered to be directly proportional to ${\mathrm{\lambda }}_{\mathrm{P}\mathrm{C}}$. In this paper, the construction cost ${\mathrm{C}}_{\mathrm{construction}}$ is derived from the volume ratio of the structure:

$${\mathrm{C}}_{\mathrm{construction}}\left({\mathsf{\lambda }}_{\mathrm{PC}}\right)={\displaystyle \frac{\mathrm{V}1}{\mathrm{V}0}}={ \mathsf{\lambda }}_{\mathrm{PC}}$$

(9)

Here, V0 is the volume of the initial structure, and V1 is the volume of the modified structure. Following a progressive collapse, the associated cost of losses is determined using a nondimensional cost multiplier cost multiplier, k, which is set at 10 according to JCSS guidelines [1].

$${\mathrm{C}}_{\mathrm{collapse}}=\mathrm{k}{\displaystyle \frac{{\mathrm{R}}_{\mathrm{m}}\left({\mathsf{\lambda }}_{\mathrm{PC}}=1\right)}{{\mathrm{R}}_{\mathrm{m}}\left({\mathsf{\lambda }}_{\mathrm{PC}}=1\right)}}=\mathrm{k}$$

(10)

Therefore, the expected loss cost due to collapse can be calculated by multiplying the collapse cost by the probability of collapse, that is, $\mathrm{k}{\mathrm{P}}_{\mathrm{f}}$. Using Equations (9) and (10), the total expected loss cost ${\mathrm{C}}_{\mathrm{T}}$ can be determined:

$${\mathrm{C}}_{\mathrm{T}}\left({\mathsf{\lambda }}_{\mathrm{PC}}\right)={\mathsf{\lambda }}_{\mathrm{PC}}+\mathrm{k}{\mathrm{P}}_{\mathrm{f}1}+\mathrm{k}{\mathrm{P}}_{\mathrm{f}2}$$

(11)

where ${\mathrm{P}}_{\mathrm{f}1}$ is the probability of the failure of the internal columns, and ${\mathrm{P}}_{\mathrm{f}2}$ is the probability of the failure of the perimeter columns.

This paper combines the risk-based structural robustness indices proposed by Pinto and Beck, analyzing the robustness of structures using the anti-collapse design cost ${\mathrm{C}}_{\mathrm{T}}\left({\mathsf{\lambda }}_{\mathrm{PC}}\right)$ as the robustness index.

2.4. Model Validation

Ma Yadong [32] developed a scaled model of a two-story three-dimensional steel frame structure based on an actual building. The model features spans of 2 m in both directions and a story height of 1 m for each frame. After an internal middle column was removed, the pushdown simulation method was used to analyze the structural integrity, applying vertical static loads at the failure point until the structure collapsed. For ease of load application, a 400 mm × 400 mm hole was incorporated into the center of the second-floor section. The first- and second-floor plans of the experimental steel frame structure are depicted in Figure 3.

A pushdown analysis was used on the steel frame model, with load application controlled by displacement. The load–displacement curve of the failed column was generated, which was subsequently compared to the actual load–displacement curve obtained experimentally, as shown in Figure 4. In the experimental curve, when the vertical displacement of the failed column reaches 420 mm and the load reaches 380 kN, the bottom flange of the node connected to the weak axis of the main beam and the failed column is torn, followed by a rapid decrease in the load; the structure fails. The graph shows that the load–displacement curve of the failed column derived from finite element simulation closely matches the experimental curve, thus validating the effectiveness of the settings in this paper for hinge attributes and locations, material selection, I-beam placement angles, and the rigid connection style of beams and columns.

2.5. Practical Implementation

The main types of distributions for random variables are shown in

Table 3. Functional random variables for the steel structure.

Variable	Distribution	$\frac{\mathbf{M}\mathbf{e}\mathbf{a}\mathbf{n}}{\mathbf{N}\mathbf{o}\mathbf{m}\mathbf{i}\mathbf{n}\mathbf{a}\mathbf{l}}$	Coefficient of Variation (COV)	Ref.
Dead load (DL)	Normal	1.05	0.1	[33,34]
Live load (LL)	Lognormal	0.24	0.6	[33,34]
Expected yield strength of the material ($f_{ye}$)	Lognormal	1.0	0.07	[33,34]
Expected ultimate strength ($f_{ue}$)	Lognormal	1.0	0.07	[33,34]
Beam section height (hZ)	Normal	1.0	0.05	[33,34]
Beam section width (bZ)	Normal	1.0	0.05	[33,34]
Beam flange thickness (tZ1)	Normal	1.0	0.05	[33,34]
Beam web thickness (tZ2)	Normal	1.0	0.05	[33,34]
Column section height (hC)	Normal	1.0	0.05	[33,34]
Column section width (bC1)	Normal	1.0	0.05	[33,34]
Column flange thickness (tC1)	Normal	1.0	0.05	[33,34]
Column web thickness (tC2)	Normal	1.0	0.05	[33,34]

.

This section analyzes the original model derived from a nine-story 6 × 3-span steel frame structure [35]; one of the frames from this model was selected for study, which is shown in Figure 5. The specific parameters are shown in

Table 4. Section parameters.

Section	Section Dimensions
Main beams	H-535 × 165 × 16 × 10
Column (floors 1–5)	H-380 × 395 × 30 × 19
Column (floors 6–9)	H-356 × 369 × 18 × 11

.

The reliability index of steel frame structures is analyzed in two scenarios: removal of a middle column and a perimeter column. By changing the section dimensions of the structure, the impact of different section sizes on the probability of structural failure is also analyzed. A Monte Carlo simulation involving 1000 random variables is used to produce outputs as precise solutions. Additionally, the load coefficient ${\alpha }_u$ is fitted to a normal distribution to verify the distribution type of the structure’s limit state function. Figure 6 and Figure 7 show the results for the removal of the middle column (scenario 1) and perimeter column (scenario 2), respectively. From the probability density distribution graphs, ${\alpha }_u$ is approximately normally distributed. The actual cumulative probability of ${\alpha }_u$ compared with the expected cumulative probability on a P-P plot shows that the data points closely align with the theoretical line (i.e., the diagonal), indicating that the structure’s limit state function Z satisfies a normal distribution.

Figure 8 illustrates the comparison of the load coefficient cumulative distribution functions, fitted from 1000 samples obtained through Monte Carlo simulation against 100 samples from LHS. It is observed that the simulation results from the LHS method are in agreement with those from the Monte Carlo method, which indicates that LHS can cover a broader and more effective range of sample points with fewer samples. For this research, LHS will be employed with 100 samples.

This paper conducts a finite element analysis on the progressive collapse resistance of a nine-story steel frame structure. The impact of different main beam section sizes on the collapse probability $P_f$ is explored, with the specific section sizes detailed in

Table 5. Beam section size.

Beam Number	Cross Section Size 2
1	H-560 × 191 × 21 × 15
2	H-555 × 186 × 20 × 14
3	H-550 × 181 × 19 × 13
4	H-545 × 176 × 18 × 12
5	H-540 × 171 × 17 × 11
6	H-535 × 165 × 16 × 10
7	H-530 × 161 × 15 × 9
8	H-525 × 156 × 14 × 8

.

Figure 9 and Figure 10 illustrate the probability density functions (PDFs) and cumulative distribution functions (CDFs) for the ultimate load coefficient ${\mathsf{\alpha }}_{\mathrm{u}}$ under scenarios 1 and 2. Figure 9a and Figure 10a reveal that the mean and standard deviation of ${\mathsf{\alpha }}_{\mathrm{u}}$ fluctuate in response to variations in beam cross-sectional sizes. Furthermore, Figure 9b and Figure 10b present the CDFs for ${\mathsf{\alpha }}_{\mathrm{u}}$ across different structural cross-sections, demonstrating how structural volume changes distinctly impact different failure scenarios. For scenario 1, enlarging the main beam’s cross-sectional area generally increases the mean and standard deviation of ${\mathsf{\alpha }}_{\mathrm{u}}$. Conversely, for scenario 2, enlarging the beam’s cross-sectional area results in a decrease in these metrics. The reason for this is that a larger beam cross-section increases the structure’s initial stiffness and load-bearing capability during a progressive collapse, but at the same time, it reduces the ductility of the structure. Therefore, increasing the beam cross-sectional area improves the collapse resistance without affecting the structure’s ductility.

Table 6. Failure probability.

Section Number	Failure Probability Pf1 (Scenario 1)	Failure Probability Pf1 (Scenario 2)
1	3.4 × 10−2	5 × 10−2
2	4.1 × 10−2	5.9 × 10−2
3	5.1 × 10−2	8.2 × 10−2
4	4 × 10−2	1.1 × 10−1
5	4.7 × 10−2	1.9 × 10−1
6	9 × 10−3	6.8 × 10−2
7	4.2 × 10−2	9.2 × 10−2
8	8.2 × 10−2	1.3 × 10−1

lists the calculated results for the failure probabilities Pf1 and Pf2 of a nine-story steel frame structure under scenario 1 and scenario 2 with different section sizes. It can be observed that the risk of structural failure is higher when perimeter columns are removed. This increased risk is attributed to fewer load transmission paths in the frame when the perimeter columns, which provide essential horizontal support and axial load-bearing capabilities, are absent. As a result, the beams linked to these removed columns cannot engage in effective arching actions to support loads, leading primarily to a reliance on plastic deformation mechanisms. When the position of the removed column moves from the perimeter to the middle, the increased lateral constraint forces, the compressive arch, and the catenary mechanisms enhance the structure’s collapse resistance.

After increasing the cross-sectional area of the structure, the failure probability does not necessarily show a decreasing trend; instead, it may fluctuate. This implies that merely increasing the beam’s cross-sectional area does not necessarily enhance the structure’s collapse resistance and might even increase the probability of failure.

Table 7. Volume ratio of different section types.

Section Number	Structural Volume (m3)	Volume Ratio
1	13,540.43	1.218
2	13,031.67	1.173
3	12,534.48	1.128
4	12,048.89	1.084
5	11,574.88	1.042
6	11,336.54	1.000
7	10,661.61	0.959
8	10,097.60	0.908

lists the volumes of structures with different section types, with the steel frame structure of Section 6 serving as the reference model to calculate the volume ratios of the structures.

Figure 11 shows the relationship between the volume ratios of different steel frame structures and their failure probabilities. The graph illustrates that, initially, as the cross-sectional area increases, the failure probability of the structures generally tends to decrease. However, as the structural volume continues to increase, the failure probability begins to rise, followed by fluctuations up and down. As a result, simply increasing or decreasing the main beam’s cross-sectional area does not necessarily enhance or reduce the structure’s collapse resistance. Effective collapse-resistant design, therefore, requires extensive testing of diverse section sizes to determine the most effective structure.

2.6. Robustness Analysis

By setting the construction cost of the Section 6 structure as a baseline of 1, the construction costs and anti-collapse design costs for eight different types of structures are shown in

Table 8. Cost analysis of different structures.

Section Number	Total Volume (m3)	Construction Cost (${\mathsf{\lambda }}_{\mathit{pc}}$)	Collapse Cost (Scenario 1)	Collapse Cost (Scenario 2)	Total Cost (${\mathit{C}}_{\mathit{TE}}$)
1	13,540.43	1.218	0.345	0.508	2.072
2	13,031.67	1.173	0.414	0.594	2.181
3	12,534.48	1.128	0.519	0.821	2.469
4	12,048.89	1.084	0.406	1.116	3.732
5	11,574.88	1.042	0.473	1.928	3.782
6	11,112.45	1.000	0.091	0.679	1.77
7	10,661.61	0.959	0.422	0.918	2.301
8	10,097.60	0.908	0.824	1.361	3.094

. It can be observed that as the structural volume changes, the construction and collapse costs also vary. In scenarios 1, 2, and 6, the initial construction costs account for more than 50% of the total costs, whereas in other scenarios, the proportion is less than 50%. This indicates that the cost–benefit analysis of structures should not only consider the initial construction costs but also the costs incurred after the collapse. Furthermore, it is evident that the collapse costs in scenarios involving the removal of perimeter columns are significantly higher than those in scenarios involving the removal of middle columns, which corresponds with the failure probabilities determined in Section 2.5. Due to the lack of horizontal constraints, the collapse probability in scenarios with perimeter columns removed is higher than that in those with middle columns removed; thus, the corresponding collapse costs are also higher.

Figure 12 shows the relationship between construction costs and total costs. It can be observed that as construction costs (beam cross-sectional area) increase, the collapse and anti-collapse design costs fluctuate and do not necessarily correlate positively or negatively. Additionally, it can be observed that construction costs have a smaller impact on total costs when the structure’s failure probability is high; however, in scenarios with a low failure probability, such as at point 5, the construction costs form a larger fraction of the total costs. Therefore, finding the structure type that minimizes the expected total costs necessitates conducting comprehensive calculations across various structural models.

3. Structural Collapse Resistance Optimization Design

3.1. Optimization Design Workflow

The workflow for optimizing structural designs is illustrated in Figure 13, which encompasses four key components: building the model, defining design variables, setting optimization goals, and establishing constraints. This research utilizes the SAP2000 API alongside Python to develop a model capable of evaluating the collapse resistance of steel frame structures. The analysis of this model is conducted using SAP2000, while Python facilitates the integration of genetic algorithms to optimize the structural design. The design parameters include the dimensions of the beams’ and columns’ sections, specifically the widths and thicknesses of I-beam flanges and webs, chosen within predefined limits. The robustness index, described in Section 3 as the structural collapse-resistant design cost function $C_T\left({\lambda }_{PC}\right)$, is adopted as the objective function. The optimization strategy aims to minimize the collapse-resistant design costs by adjusting the steel sections’ dimensions, thereby enabling fully automated computations and outputs for the optimized design of steel frame structures.

Using the limit state function proposed in Section 3 as a constraint, as shown in Equation (12), when Z = 0, the structure is at its limit state of load-bearing capacity.

$$\mathrm{Z}={\mathsf{\alpha }}_{\mathrm{u}}-{\mathsf{\Omega }}_{\mathrm{N}}=0$$

(12)

The transformation from Equation (12) leads to Equation (13), where F is the factor of safety. When F ≥ 1, the structure has not collapsed; when F < 1, the structure has collapsed.

$$\mathrm{F}={\mathsf{\alpha }}_{\mathrm{U}}/{\mathsf{\Omega }}_{\mathrm{N}}$$

(13)

3.2. Steel Frame Structural Optimization Example

3.2.1. Development of the Optimization Model

In this study, the model’s efficiency is demonstrated through the optimization of the nine-story steel frame structure discussed in Section 3. This optimization adheres to the Chinese architectural standards for anti-collapse design (CECS 392-2021), which include guidelines for the methodical analysis of structural components designated for removal, such as perimeter columns and those centrally located along the longer sides of buildings. This paper selects the two scenarios most likely to lead to structural failure—removing columns along the long side and perimeter columns—as depicted in Figure 14 and Figure 15, to formulate the cost calculations for potential collapse losses, and the optimization of the structure is conducted using two different scenarios, namely, the single-frame and three-dimensional structures.

3.2.2. Optimization Design of a Single-Frame Structure

The initial parameters for the frame structure, including the construction, collapse, and anti-collapse design costs, are detailed in

Table 9. Cost analysis results of a single-frame structure.

Section Size	Construction Cost	Collapse Costs	Total Costs
Main beams (535 × 166 × 10 × 16)	1	0.77	1.77
Columns from floors 1–5 (380 × 395 × 19 × 30)
Columns from floors 6–9 (356 × 369 × 11 × 18)

. The optimization variables selected for this design focus on the section sizes of the main beams and the columns from floors 1–5 and 6–9, with specifics on the range, steps, and initial dimensions provided in

Table 10. Structural optimization design variables.

Section Size	Design Parameters	Optimization Range (mm)	Step (mm)	Initial Size (mm)
Main beams (ZL)	Height (hZ)	485–585	5	535
	Section width (bZ)	116–216	5	166
	Flange thickness (tZ1)	11–21	1	16
	Web thickness (tZ2)	5–15	1	10
Columns from floors 1–5 (C1–5)	Height (hC1)	330–430	5	380
	Section width (bC1)	345–445	5	395
	Flange thickness (tC1)	25–35	1	30
	Web thickness (tC2)	14–24	1	19
Columns from floors 6–9 (C6–9)	Height (hC2)	306–406	5	356
	Section width (bC2)	319–419	5	369
	Flange thickness (tC3)	13–23	1	18
	Web thickness (tC4)	6–16	1	11

.

Table 11. Parameter settings for genetic algorithm in steel frame structure calculation.

Population Size	Maximum Evolutionary Generations	Mutation Rate	Crossover Rate	Objective Function	Constraints
8	20	0.2	0.8	$C_{\mathrm{TE}}$	F1, F2

outlines the settings for the genetic algorithm used in this optimization, encompassing aspects such as population size, evolutionary generations, mutation and crossover probabilities, constraints, and the objective function. The optimal parameters achieved post optimization are recorded in

Table 12. Optimization results of single-frame structure.

Section Size	Design Parameters	Optimized Dimensions (mm)
Main beams (ZL)	Height (hZ)	495
	Section width (bZ)	181
	Flange thickness (tZ1)	20
	Web thickness (tZ2)	6
Columns from floors 1–5 (C1–5)	Height (hC1)	410
	Section width (bC1)	365
	Flange thickness (tC1)	26
	Web thickness (tC2)	23
Columns from floors 6–9 (C6–9)	Height (hC2)	346
	Section width (bC2)	364
	Flange thickness (tC3)	20
	Web thickness (tC4)	15

.

To ensure the safety of these optimized sections, their dimensions were tested in SAP2000 via pushdown analysis to ascertain the ultimate load coefficients ${\alpha }_u$ and safety factors F1 and F2 for scenarios involving the removal of middle and perimeter columns. The dynamic amplification factor ${\Omega }_N$, calculated from Table 1 and Equation (3), was found to be 1.08. The pushdown analysis curves, depicted in Figure 16, demonstrate that the ultimate load coefficient for the scenario removing the middle column is 1.443 with a safety factor F1 of 1.336, and for the perimeter column removal scenario, it is 1.218 with a safety factor F2 of 1.128. These results confirm that the optimized structure complies with the GSA’s anti-collapse standards.

The optimization curves for the steel frame structure using the genetic algorithm and the computational results are shown in Figure 17 and

Table 13. Cost analysis results of a single-frame structure.

Section Size (mm)	Construction Cost	Collapse Costs	Total Costs
Main beams (535 × 166 × 10 × 16)	1	3.54	4.54
Columns from floors 1–5 (380 × 395 × 19 × 30)
Columns from floors 6–9 (356 × 369 × 11 × 18)

The optimized structures meet all the constraint conditions. As can be seen from Figure 17, after nine iterations, the objective function value C_TE tends to converge and reaches its minimum value of C_TE = 1.609.

The original structure’s volume was 111,124.53 m³, reduced to 108,430.02 m³ in the optimized design. This redesign resulted in a total cost reduction of 0.161, with construction costs down by 0.024 and collapse costs down by 0.137. Specifically, the optimized structure’s construction expenses are 2.4% lower than those of the original, and the overall costs associated with enhancing the collapse resistance have dropped by 9.1%. The approach ensures that the minimized total cost of the collapse-resistant design does not compromise the structure’s safety and efficacy.

3.2.3. Structural Optimization Design of the Three-Dimensional Frame

The initial parameters for the frame structure, including the construction, collapse, and anti-collapse design costs, are detailed in Table 13. The structural design optimization variables are the same as the genetic algorithm parameter settings and the single-frame settings, and the optimized dimensions of the structure are shown in

Table 14. Optimization results of the three-dimensional frame structure.

Section Size	Design Parameters	Optimized Dimensions (mm)
Main beams (ZL)	Height (hZ)	515
	Section width (bZ)	201
	Flange thickness (tZ1)	20
	Web thickness (tZ2)	7
Columns from floors 1–5 (C1–5)	Height (hC1)	420
	Section width (bC1)	400
	Flange thickness (tC1)	31
	Web thickness (tC2)	14
Columns from floors 6–9 (C6–9)	Height (hC2)	401
	Section width (bC2)	334
	Flange thickness (tC3)	16
	Web thickness (tC4)	6

.

To ensure the safety of these optimized sections, their dimensions were tested in SAP2000 via pushdown analysis to ascertain the ultimate load coefficients ${\alpha }_u$ and safety factors F1 and F2 for scenarios involving the removal of middle and perimeter columns. The dynamic amplification factor ${\Omega }_N$ is 1.08. The pushdown analysis curves, depicted in Figure 18, demonstrate that the ultimate load coefficient for the scenario removing the middle column is 1.163 with a safety factor F1 of 1.076, and for the perimeter column removal scenario, it is 1.082 with a safety factor F2 of 1.002. These results confirm that the optimized structure complies with the GSA’s anti-collapse standards.

The optimization curves for the steel frame structure using the genetic algorithm and the computational results are shown in Figure 19 and Table 14. The optimized structures meet all the constraint conditions. As can be seen from Figure 19, after 13 iterations, the objective function value C_TE tends to converge and reaches its minimum value of C_TE = 3.928. The optimized structure’s volume is recorded at 1.029. When compared to the original design, the revised structure demonstrates a comprehensive cost reduction of 0.612, while the construction expenses show a slight increase of 0.029. Notably, the cost associated with potential collapse scenarios has been reduced by 0.583. Specifically, the construction costs of the optimized framework are 2.9% higher than those of the initial design, yet the overall expenses related to enhancing the collapse resistance have been successfully reduced by 13.5% relative to the original structure.

4. Conclusions

This study concentrates on steel frame structures, assessing their reliability and robustness. It employs genetic algorithms to enhance their design against collapse, complemented by bespoke optimization software. The main findings and conclusions are as follows:

1. This paper introduces robustness indices that effectively evaluate the costs associated with the entire life cycle of designs aimed at preventing collapse. These indices show fluctuations with changes in cross-sectional areas, suggesting that increasing size alone does not align with enhanced structural safety or cost efficiency. Detailed simulations are necessary to find the model with the smallest robustness index.

2. Through a detailed analysis that includes reliability, robustness, and safety factors, this study applies genetic algorithms to refine the dimensions of beams and columns in steel frames. The refined single-frame structure shows a reduction in the initial construction and collapse-related costs by 2.4% and 9.1%, respectively. Meanwhile, the three-dimensional frame shows a 2.9% rise in initial costs but a 13.5% decrease in total collapse-resistant design costs, illustrating the model’s ability to balance safety with cost-effectiveness.

3. By combining the SAP2000 API and python for the overall development of the structural optimization design, the automatic extraction of parameters such as the plastic section modulus, yield strength, and steel section modulus, the automatic input of random variables such as the load and material parameters, and the automatic extraction and analysis of the output results such as the plastic angle, yield angle, load coefficient, and amplification factor have been realized, which effectively reduces the number of parameters that are required by manual labor and the number of parameters that are required by the structural optimization design. This effectively reduces the accidental errors caused by manual operation and greatly improves the calculation efficiency and result accuracy.

4. In the structural optimization design process, an increased number of design parameters can be included. In practical engineering contexts, the factors that need to be considered are significantly more complex than those in theoretical models, and constraints related to failure must be comprehensively addressed. In future designs, a more thorough and integrated selection of design parameters, constraints, and failure mechanisms should be undertaken, along with an analysis of the importance of these parameters, to enhance the rationality of the optimization model.