Linear Seismic Analysis and Structural Optimization of Reinforced Concrete Frames Using OpenSeesPy

Abstract

Seismic design of reinforced concrete buildings in highly active seismic regions is challenging, as structural members are often oversized due to conservative design practices, leading to inefficient use of materials. This study proposes an optimization methodology based on the Peruvian seismic code E.030, implemented with the OpenSeesPy library for modeling and numerical analysis. The methodology automates the linear analysis of frame structures through the parametrization of member dimensions, span lengths, and material properties. Optimization is carried out using the Hill Climbing algorithm, which iteratively explores design alternatives and verifies compliance with code requirements for interstory drift and base shear. Results show material savings of up to 20% in beams and columns. Although interstory drifts increased by 60–85% compared to the initial configuration, they remained within code limits. The methodology establishes a framework for integrating optimization techniques into the seismic design of reinforced concrete frame buildings.

1. Introduction

Reinforced concrete moment-resisting frames constitute a predominant structural system in regions of high seismicity. However, these systems often face issues related to the oversizing of structural elements, primarily due to conservative design practices and the lack of coordination between architectural and structural disciplines during the early design stage [1]. Such practices result in excessive material consumption and higher construction costs, ultimately reducing structural efficiency. To address these limitations, recent years have witnessed increasing reliance on digital tools and optimization methodologies within the architecture, engineering, and construction (AEC) industry. Several studies have demonstrated the potential of integrating metaheuristic algorithms, artificial intelligence, and building information modeling (BIM) to improve both efficiency and sustainability in structural design. For instance, ref. [2] emphasized the evolution of generative design towards models incorporating artificial intelligence, genetic algorithms, and topology optimization, which facilitate the development of efficient structural solutions that also ensure architectural adaptability. Along this line, ref. [3] proposed a conceptual structural design framework based on genetic algorithms that evaluates alternatives in terms of cost, sustainability, and technical feasibility. Their approach enabled the simultaneous minimization of carbon emissions and construction costs, thereby supporting decision-making during the early design stage. Similarly, ref. [4] presented an automated approach for reinforced concrete slab design that, through BIM integration and open-source programming tools, achieved significant reductions in design time while improving accuracy. Additional studies, such as those of [5], further demonstrated the effectiveness of metaheuristic optimization methods for large-scale steel design problems, underscoring the importance of structural optimization during preliminary stages. Recent reviews on seismic design improvements of steel frames highlight the need for innovative strategies that combine rehabilitation and optimization techniques [6]. In parallel, machine-learning-based surrogate modeling has gained attention, with approaches such as recurrent conditional GANs (rcGAN) enabling accurate prediction of nonlinear seismic responses and facilitating structural optimization [7]. A complementary line of research has explored hybrid methodologies that integrate artificial intelligence with multi-objective optimization algorithms to enhance seismic performance and cost efficiency in reinforced concrete and steel frame systems [8]. More recently, optimization-based approaches have been applied to the development of hybrid shell/frame systems for long-span buildings, where geometric form finding and shell thickness distribution are optimized to improve seismic efficiency and minimize material usage. Nonlinear dynamic analyses of these systems demonstrated favorable interaction effects, with suspended steel frames acting as tuned mass dampers that significantly reduce stresses and displacements in the shell, highlighting the potential of optimization-driven methodologies for advancing sustainable seismic design [9]. Parallel to these developments, the adoption of open-source simulation platforms has gained relevance. In particular, OpenSeesPy has emerged as a powerful Python 3.9.13 interface for structural analysis, overcoming limitations of the legacy TCL environment. Recent contributions [10] highlighted the advantages of the opseestools library, which streamlines modeling, analysis, and post-processing within Python. Likewise, ref. [11] developed opstool, a complementary library that improves the efficiency of nonlinear simulations by automating workflows and integrating scientific libraries. Additional advances, such as OPS-ITO [12], combined isogeometric analysis and topology optimization within OpenSees, enabling the modeling of complex geometries and advancing sustainable design practices. Other applications have validated the capabilities of OpenSees for simulating reinforced concrete shear walls [13] and unreinforced masonry structures [14], thereby confirming its reliability for both conventional and innovative structural systems. Despite these advances, the automation of seismic analysis combined with optimization strategies for reinforced concrete frame structures remains limited, particularly under regional seismic design codes such as the Peruvian Seismic Code E.030 [15]. Most contributions in the literature have focused either on nonlinear modeling or on conceptual optimization frameworks, leaving a gap in practical methodologies that integrate automated linear analysis and optimization techniques tailored to real building configurations. In light of this background, the present study develops a computational model for the automated analysis and optimization of reinforced concrete frame structures using the OpenSeesPy library in combination with a Hill Climbing algorithm. The proposed approach, implemented in Python and applied to case studies representative of Lima, aims to improve the efficiency of preliminary structural design by reducing oversizing, lowering material consumption, and ensuring compliance with seismic safety requirements. In doing so, this research provides a practical foundation for advancing towards more sophisticated methodologies that incorporate nonlinear effects, seismic risk assessment, and long-term deterioration mechanisms.

2. Methodology

Currently, OpenSeesPy is a Python library that enables structural analysis of buildings through the finite element method. This tool allows the evaluation of structural performance under both static and dynamic loads, making it particularly useful for the development of advanced applications in structural engineering. The parametric model developed automates the generation and analysis of multiple frame system configurations, varying in structural geometry, member dimensions, and material properties. In this study, the evaluation is restricted to linear analysis, since the Peruvian Seismic Code E.030 establishes compliance criteria primarily based on elastic response parameters such as interstory drift and base shear. Complementarily, the Peruvian Load Code E.020 [16] specifies the gravitational actions considered in the model. Although the present implementation relies on the provisions of the Peruvian seismic code E.030 and load code E.020, the procedure is not limited to this regulatory framework. The response spectra were represented as mathematical functions defined by code parameters, which allows the methodology to be adapted to other seismic design codes by modifying these definitions accordingly. Regarding the optimization procedure, although more robust optimization methods exist, they often involve higher computational cost or complex sensitivity requirements. In this work, the Hill Climbing algorithm was adopted for its accessibility and efficiency, enabling rapid exploration of structural alternatives under the linear framework imposed by the Peruvian code. The model must comply with the seismic-resistant criteria of the Peruvian Seismic Code E.030, after which the Hill Climbing algorithm is applied to optimize the structural geometry. As illustrated in Figure 1, a logical sequence of steps is followed to obtain the results.

2.1. Geometric Definition and Structural Parameters

A regular square-plan reinforced concrete space frame system was considered, representative of mid-rise buildings in Lima, Peru. The building consists of four stories, with three bays in the X direction and four bays in the Y direction.

Table 1. Geometric parameters of the modeled building.

Parameter	Value
Number of stories	4
Story height	3.0 m
Total building height	12.0 m
Number of bays (X direction)	3
Number of bays (Y direction)	4
Span length (X direction)	3.3 m
Span length (Y direction)	2.9 m
Beam section	30 × 40 cm
Column section	40 × 45 cm

summarizes the adopted geometric parameters.

The model was automatically generated in OpenSeesPy, defining the nodes and elements according to the adopted configuration. Figure 2 shows the three-dimensional frame model with node and element numbering, providing a clear view of the structural grid and connectivity. For better interpretation, Figure 3 presents an extruded view, highlighting the volumetric representation of beams and columns.

2.2. Material Properties

The materials were modeled as linear-elastic, consistent with the scope of this study.

Table 2. Material properties of concrete.

Property	Value
Compressive strength $f^{\prime }c$	$280 \mathrm{k}\mathrm{g}/{\mathrm{c}\mathrm{m}}^2$
Elastic modulus $E$	$1500\sqrt{f^{\prime }c}$
Poisson’s ratio ν	0.20
Unit weight γ	$2400 \mathrm{k}\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$

lists the adopted properties for concrete.

2.3. Load Definition

The loads applied to the structural model follow the Peruvian code E.020 for gravity loads and E.030 for seismic actions.

Table 3. Adopted gravity loads.

Load Type	Value
Live load	$200 \mathrm{k}\mathrm{g}/{\mathrm{c}\mathrm{m}}^2$
Slab self-weight	$300 \mathrm{k}\mathrm{g}/{\mathrm{c}\mathrm{m}}^2$
Finishing load	$100 \mathrm{k}\mathrm{g}/{\mathrm{c}\mathrm{m}}^2$
Partition walls load	$150 \mathrm{k}\mathrm{g}/{\mathrm{c}\mathrm{m}}^2$

summarizes the adopted values.

The self-weight of beams and columns is automatically computed by OpenSeesPy using the unit weight of concrete. Loads are distributed using a tributary area function, which assigns to each node the corresponding portion of the applied loads based on geometry and position.

2.4. Seismic Parameters

For seismic analysis, the elastic response spectrum defined in the Peruvian seismic code E.030 was employed.

Table 4. Seismic design parameters according to E.030.

Parameters	Value
Seismic zone factor $Z$	$0.45$
$\mathrm{Use} \mathrm{factor} U$	$1.0$
Short-period spectral period $T_p$	$0.4 \mathrm{s}$
Long-period spectral period $T_p$	$2.5 \mathrm{s}$
Response modification factor $R$	$8$

summarizes the adopted site parameters.

2.5. Modeling and Boundary Conditions

The structural model was generated automatically in OpenSeesPy, with beams and columns modeled as frame elements. Rigid diaphragms were assigned at each story level by linking floor nodes to the corresponding center of mass. The base of the structure was modeled with fixed supports, assuming no soil–structure interaction.

2.6. Execution of Structural Analysis and Result Extraction

For each generated configuration, the following analyses are carried out in OpenSeesPy:

• Modal analysis: determination of the natural vibration periods and modal mass participation ratios.
• Linear static analysis: evaluation of displacements, internal forces, and the static base shear.
• Response spectrum analysis: calculation of the dynamic base shear and lateral displacements in accordance with the Peruvian Seismic Code E.030.
• Interstory drift verification: checking compliance with the maximum allowable drift specified by the code.
• Force diagrams: extraction of bending moment (BMD), shear force (SFD), and axial force (AFD) diagrams to assess structural behavior in detail.

2.7. Verification According to the Peruvian Seismic Code

The results obtained from the structural analyses are verified according to the requirements of the Peruvian Seismic Code E.030, with an emphasis on

• Interstory drift: verification against the maximum allowable limit of 7‰ of the story height.
• Modal mass participation: ensuring that the cumulative participation of the modes considered reaches at least 90%.
• Base shear scaling: checking whether the dynamic base shear requires scaling in order to satisfy the minimum base shear provision established by the code.

This verification ensures that all analyzed configurations comply with the seismic safety and performance standards mandated by the Peruvian code.

2.8. Parametric Optimization

The Hill Climbing algorithm is a local search optimization method that iteratively explores neighboring solutions and retains those that improve the objective function [17]. In this study, the algorithm was adapted to simultaneously vary the dimensions of beams and columns, as well as the value of $f^{\prime }c$, with the objective of reducing the total sectional area and, consequently, the stiffness of the structure. The optimization was constrained to satisfy the maximum drift limits specified by the Peruvian seismic code E.030 and to comply with basic design principles such as the “strong column–weak beam” criterion and the minimum section dimension of 0.25 m for structural elements. The objective function, or score, combines the total cross-sectional area with a penalty factor for the use of higher-strength concrete. Additional constraints invalidate a solution if the maximum interstory drift exceeds 7‰, if the beam inertia surpasses that of the column, or if the geometric ratio between beam depth and span is not admissible. Mathematically, the score function is defined as

$$score=\left\{\begin{array}{cc}\hfill \infty ,& if any constraints are no satisfied\hfill \\ \hfill \left(a^2+b \ast h\right) \ast {factor}_{f^{\prime }c},& otherwise\hfill \end{array}\right.$$

(1)

where

• $\mathrm{a}=$ column side;
• $b=$ beam width (m);
• $h$ = beam depth (m).
• The factor $f^{\prime }c$ takes the value of 1.0 for $f^{\prime }c$ = 210 kg/cm² and 1.15 for $f^{\prime }c$ = 280 kg/cm².
• A solution is deemed infeasible (score = ∞) if any of the following constraints are not satisfied:

  ○

  Maximum interstory drift exceeds 7 ‰ (Peruvian Seismic Code E.030).

  ○

  Beam inertia ($I_c$ =${ bh}^3/12)$ is greater than column inertia ($I_c$ =$a^4/12)$ (strong column–weak beam criterion).

  ○

  Beam depth-to-span ratio is outside admissible bounds.

  ○

  Minimum cross-sectional dimension of 0.25 m is not met.

Figure 4 illustrates the internal operation of the Hill Climbing algorithm, showing the iterative search process toward an optimal feasible solution.

3. Results

3.1. Modal Analysis and Mass Participation According to E.030

A modal analysis was conducted in compliance with the Peruvian Seismic Design Code E.030, which requires that the considered vibration modes account for at least 90% of the total mass participation in both principal directions. The analysis was performed on the initial structural configuration prior to optimization.

Table 5. Fundamental vibration periods for the first three relevant modes.

Mode No.	Period (s)
Mode 1	0.371021
Mode 2	0.348497
Mode 3	0.292926

summarizes the fundamental vibration periods for the first three relevant modes: translational motion in the X direction, translational motion in the Y direction, and torsional motion about the Z axis. These modes represent the dominant dynamic characteristics of the structure.

The computed vibration periods correspond to an elastic model without considering the effect of cracking, in accordance with the provisions of the Peruvian Seismic Code E.030. For validation, the approximate expression established in E.030 ($T\cong H_n/C_T$) where $H_n$ is the total building height and ($C_T=35$) for reinforced concrete moment-resisting frames. With ($H_n=12 m$), the estimated period is ($T\cong 0.343 s$) s, which is in close agreement with the numerical result presented in Table 5. The corresponding modal shapes are illustrated in Figure 5a, translation in the X direction; Figure 5b, translation in the Y direction; and Figure 5c, torsion about the Z axis.

3.2. Modal Mass Participation

The modal analysis confirms that the cumulative modal mass participation exceeds 90% in both the X and Y directions, in compliance with the Peruvian Seismic Code E.030. Detailed numerical values for the twelve modes considered are presented in Appendix A.

3.3. Static Analysis: Base Shear and Displacements

Using the response spectrum function defined in the Peruvian seismic design code (E.030), a static equivalent lateral force analysis was performed with OpenseesPy to obtain the base shear forces and maximum displacements in both principal directions.

3.3.1. Static Analysis in the X Direction

Table 6. Base shear and displacements obtained from the static analysis in the X direction using OpenseesPy.

Story	Vx (kN)	UxMax (cm)	UyMax (cm)	DriftX (‰)	DriftY (‰)
Story 1	380.0910	0.9660	0.0730	3.2199	0.2435
Story 2	342.0819	2.5662	0.1893	5.3342	0.3875
Story 3	266.0637	3.9779	0.2891	4.7054	0.3327
Story 4	152.0364	4.9218	0.3536	3.1465	0.2150

presents the results obtained for the equivalent static analysis in the X direction.

3.3.2. Static Analysis in the Y-Direction

Table 7. Base shear forces and displacements obtained using the OpenSeesPy library.

Story	Vy (kN)	UxMax (cm)	UyMax (cm)	DriftX (‰)	DriftY (‰)
Story 1	380.0910	1.0395	0.9391	3.4649	3.1305
Story 2	342.0819	2.7567	2.4481	5.7242	5.0299
Story 3	266.0637	4.2687	3.7556	5.0398	4.3583
Story 4	152.0364	5.2774	4.6087	3.3623	2.8437

presents the results obtained from the static analysis in the Y-direction.

Furthermore, as shown in Figure 6, the distribution of shear forces at each story can be visually assessed. The interstory displacements obtained from the static linear analysis are shown in Figure 7a for the X-direction and in Figure 7b for the Y-direction.

3.4. Dynamic Analysis: Ultimate Shear and Maximum Drifts

The final stage of the evaluation corresponds to the linear dynamic analysis, in which the results are obtained by considering whether scaling is required in both directions, as specified by the E.030 seismic code.

Table 8. Dynamic base shear forces and displacements obtained using the OpenSeesPy library.

Story	Vx (kN)	Vy (kN)	UxMax (cm)	UyMax (cm)	DriftX (‰)	DriftY (‰)
Story 1	322.5558	324.4534	0.7418	0.6808	2.4726	2.2693
Story 2	289.5358	290.5077	1.9618	1.7628	4.0692	3.6090
Story 3	228.7108	229.1734	3.0082	2.6705	3.6414	3.1673
Story 4	136.2960	135.6897	3.7353	3.2853	2.5037	2.1182

summarizes the dynamic base shear forces and interstory drifts. It is observed that the drift ratios comply with the limits prescribed by E.030; however, the relatively low values indicate that the structure exhibits excessive stiffness. Figure 8 illustrates the distribution of interstory drifts, where the maximum drift is concentrated at the second story.

It can be verified that all drift values comply with the maximum allowable drift ratio of 7‰, as prescribed by the E.030 code.

3.5. Extraction of Internal Force and Moment Diagrams

The OpenSeesPy library enables the generation of internal force diagrams, where Figure 9a shows shear forces, Figure 9b illustrates bending moments, and Figure 9c presents axial forces, providing a clear assessment of the structural response.

3.6. Optimization Using the Hill Climbing Algorithm

To validate the optimization performance of the Hill Climbing algorithm, the initial design parameters were compared with the optimized ones. As shown in

Table 9. Optimized dimensions obtained using the Hill Climbing algorithm.

Parameters	Beam (m)	Column (m)	Score
Initial	0.30 × 0.40	0.45 × 0.45	0.3708
Optimized	0.25 × 0.35	0.40 × 0.40	0.2415

, the algorithm achieved a reduction in both the structural dimensions and the specified compressive strength f′ c. Consequently, this optimization justifies a reduction in material consumption and associated construction costs.

3.6.1. Dynamic Base Shear Forces

Table 10. Dynamic base shear forces in the x-axis.

Story	Initial Shear Force (kN)	Optimized Shear Force (kN)	Variation (‰)
Story 01	322.5558	240.3829	25.48
Story 02	289.5358	212.0372	26.77
Story 03	228.7108	168.7928	26.20
Story 04	136.2960	103.218	24.27

presents the shear force variations for the X direction, while

Table 11. Dynamic base shear forces in the y-axis.

Story	Initial Shear Force (kN)	Optimized Shear Force (kN)	Variation (‰)
Story 01	324.4534	280.5651	21.25
Story 02	290.5077	225.4505	22.39
Story 03	229.1734	178.9891	21.90
Story 04	135.6897	107.8655	20.51

shows the corresponding values for the Y direction. These changes result from increased structural flexibility, which reduces seismic demand and enhances performance. As illustrated in Figure 10a, the largest variation in the X direction occurs at the second story and the smallest at the top, while Figure 10b confirms a similar distribution in the Y direction.

3.6.2. Story Drifts

As observed in

Table 12. Story drifts in the x-axis.

Story	Initial Drift (‰)	Optimized Drift (‰)	Variation (%)
Story 01	2.4726	4.3001	73.91
Story 02	4.0692	6.6134	62.52
Story 03	3.6414	5.7628	58.26
Story 04	2.5037	3.8524	53.87

and

Table 13. Story drifts in the y-axis.

Story	Initial Drift (‰)	Optimized Drift (‰)	Variation (%)
Story 01	2.2693	4.1947	84.85
Story 02	3.609	6.2329	72.70
Story 03	3.1673	5.3374	68.52
Story 04	2.1182	3.4725	63.94

, the optimized structure shows increased drifts along both axes due to the reduction in global stiffness from the optimization process. Nevertheless, all values remain within the Peruvian Seismic Code E.030 limit (<7‰), ensuring performance is not compromised. This confirms that the Hill Climbing algorithm balances safety and material efficiency. Figure 11a highlights the largest increase at the first story in the X-axis, while Figure 11b shows the same trend in the Y-axis, with the smallest variation at the fourth story in both cases.

4. Discussion of Results

4.1. Research Gaps and Contributions

The Hill Climbing algorithm produced consistent improvements in terms of material efficiency and seismic response. The value of the score function decreased by 34.87%, from 0.3708 to 0.2415. Dynamic base shear forces were reduced between 20.51% and 26.77%, depending on the story and direction. Interstory drifts increased between 53.87% and 84.85% due to the reduction in global stiffness; however, all values remained below the limit established by the Peruvian Seismic Code E.030 (≤7‰). The main contribution of this study is to show that a local search algorithm with low computational cost can yield measurable structural benefits. While previous studies [18] reported reductions in expected annual losses and retrofitting costs using evolutionary algorithms, the present results indicate that simpler methods can also provide material savings and satisfactory performance within code limits. This approach is particularly relevant for preliminary design, where fast evaluations are required. The findings also align with the development of recent frameworks such as OpenPyStruct [19] and OpenSeesPyView [20], which promote integrated analysis and optimization environments.

4.2. Study Limitations

This work has a methodological scope. The analysis was restricted to a regular mid-rise reinforced concrete frame without irregularities in plan or elevation. Only elastic response spectrum analyses were conducted, and nonlinear effects were not included. The optimization focused on section dimensions and concrete strength without addressing reinforcement detailing or constructability. Long-term effects such as corrosion, creep, or shrinkage were not considered. These simplifications limit the direct applicability of the results but provide a controlled framework for testing the proposed methodology.

4.3. Future Research

Future studies should apply the methodology to case studies of real buildings to validate its practical utility. Extensions may include other structural systems, such as shear walls, dual systems, and confined masonry walls, which are common in Peru. Nonlinear static and dynamic analyses, such as pushover and incremental dynamic analysis, should be incorporated to capture inelastic behavior. A probabilistic approach considering uncertainties in materials, geometry, and seismic demand could enhance the robustness of the results. The inclusion of deterioration mechanisms, such as reinforcement corrosion, would allow a more complete evaluation of long-term performance and resilience.

5. Conclusions

The application of the Hill Climbing algorithm resulted in significant changes in structural response. Interstory drifts increased on average by 62.14% in the X direction and 72.50% in the Y direction, but remained below the 7‰ limit prescribed by the Peruvian Seismic Code E.030. Base shear forces decreased on average by 25.68% in the X direction and 21.51% in the Y direction, primarily due to the reduction in global mass and stiffness from optimized cross sections. These results confirm that the structural safety and code compliance were preserved. From a practical perspective, the methodology demonstrates potential for use in early design stages, where rapid evaluation of section sizes and material choices is required. The reduction in the score function indicates benefits in terms of material consumption, cost, and sustainability. The study is limited to elastic analyses of a regular frame system and does not consider reinforcement detailing or degradation mechanisms, which reduces its applicability to final design. Nevertheless, it establishes a methodological basis that can be expanded to more complex scenarios. In summary, the integration of OpenSeesPy with optimization algorithms such as Hill Climbing provides an efficient tool for structural parameterization and optimization under seismic demands. Future studies should broaden the scope to other structural typologies, nonlinear analysis, probabilistic frameworks, and durability considerations, with the aim of supporting resilient and sustainable structural design.