Theoretical study for predicting the fundamental period of concrete buildings including non-structural elements

Estudo teórico para a estimativa do período fundamental de edifícios de concreto incluindo-se a contribuição de elementos não-estruturais

Abstract

Abstract  This paper presents a theoretical study for estimating the fundamental period of multi-storey buildings with plan symmetry and without horizontal or vertical irregularities, considering a structural system composed of reinforced concrete frames. In this study, non-structural elements (masonry infill walls) are assumed to participate in the distribution of seismic loads. The fundamental period was estimated using simplified expressions obtained through regression analysis of results from modal analyses performed on 288 buildings with geometric and mechanical characteristics representative of Brazilian construction practice. The structural models consisted of frames with equivalent diagonal struts used to simulate the presence of masonry infill walls. The simplified expressions obtained from the regression analyses were compared with the approximate expression recommended by the Brazilian code for the design of seismic-resistant structures. In addition, the seismic base shear values calculated using the investigated expressions were evaluated through the Equivalent Lateral Load Method, revealing relevant differences among the predicted seismic demands.

Keywords:
fundamental period; seismic analysis; modal analysis; masonry-infilled frames

Resumo

Resumo  Apresenta-se neste trabalho um estudo teórico para a estimativa do período fundamental de edifícios de múltiplos andares, simétricos em planta e sem irregularidades em planta e na vertical, com sistema estrutural formado por pórticos de concreto. Neste estudo, considera-se que os elementos não estruturais (alvenarias de preenchimento) participam na distribuição das forças sísmicas. A estimativa do período fundamental é feita por meio de expressões simplificadas, as quais foram obtidas por meio de regressão de resultados oriundos de análise modal de 288 edifícios, com características geométricas e mecânicas usuais no Brasil. Nas análises empregou-se o modelo de pórtico com diagonais comprimidas equivalentes, sendo estas utilizadas para simular as alvenarias de preenchimento. As expressões simplificadas obtidas por regressão foram comparadas com a expressão aproximada recomendada na Norma Brasileira de Projeto de Estruturas Resistentes a Sismos (ABNT 15421). Também foram comparadas as forças horizontais totais na base da estrutura obtidas com as expressões investigadas, empregando-se o Método das Forças Horizontais Equivalentes, observando-se diferenças relevantes.

Palavras-chave:
período fundamental; análise sísmica; análise modal; pórticos preenchidos

1 INTRODUCTION

The fundamental natural period of a building constitutes a key parameter in the seismic design of structures, as it governs the determination of seismic loads through the Equivalent Lateral Load Method, as prescribed by several international codes and the Brazilian code ABNT NBR 15421:2023 [¹].

In structural design, masonry infill walls fixed to reinforced concrete frame systems are usually not considered as bracing elements to resist lateral loads. Therefore, the stiffness of these infill walls is disregarded, considering only the corresponding mass in static and dynamic analysis.

However, several studies found in the international literature, such as those by Amanat and Hoque [²], Kose [³], Ricci et al. [⁴], Perrone et al. [⁵], Asteris et al. [⁶], Al-Balhawi and Zhang [⁷] and Asteris et al. [⁸], indicate that the fundamental period and seismic structural behavior are significantly influenced by the presence of the masonry infill walls in reinforced concrete building structures. Although more recent than abroad, in Brazil numerical and experimental investigations have been carried out on concrete and steel frames with infill walls subjected to lateral loads, with greater emphasis on wind effects, such as those by Alvarenga [⁹], Santos [¹⁰], Silva [¹¹], Sousa [¹²], Montandon [¹³], Grandi [¹⁴] and Rigão [¹⁵]. More recently, Silva [¹⁶] and Alva et al. [¹⁷] investigated the influence of masonry infill walls on the fundamental period of concrete buildings, with emphasis on seismic analysis.

Regarding the approximate assessment of the fundamental period, several formulations are found in the literature, including design codes. Most formulations only take into account the total height of the building and the type of structural system. It is known that, in real behavior under a seismic loading, the main components of the frame structure interact with non-structural elements. Thus, Su et al. [¹⁸], Kim et al. [¹⁹] and Asteris et al. [⁶] indicate that the presence of masonry infill walls exerts a great influence on the fundamental period of vibration of buildings. The number and length of beam spans, the stiffness of the masonry and the presence of openings in the infill walls are also parameters that influence the fundamental period. In reality, encompassing all the parameters involved in the fundamental period of a building structure in a single analytical expression is a challenging task.

Although the fundamental period can be obtained via modal analysis, the use of analytical expressions for estimating this parameter is still useful, especially for application in simplified methods for calculating seismic forces – such as the Equivalent Lateral Load Method.

The main objective of this study is to investigate a simplified analytical formulation for obtaining the fundamental period of concrete buildings, symmetrical in plan and without vertical irregularities in mass and stiffness, considering the infill walls. To this end, numerical simulations were performed on 288 buildings with different characteristics relating to the main structure (number and length of beam spans) and the infill walls (compressive strength of the unit and percentage of openings), considering or not the effects of cracking.

The equivalent strut model was employed to consider the presence of infill walls in the structural system under lateral loads, with the fundamental period obtained via modal analysis. Simplified expressions for the fundamental period were determined using the results of the modal analysis, and these results were compared with those provided by the approximate expression in ABNT NBR 15421:2023 [¹]. The seismic base shear was also calculated and compared using the Equivalent Lateral Load Method, employing the fundamental periods obtained with the determined expressions.

2 INVESTIGATED FORMULA FOR THE PREDICTION OF THE FUNDAMENTAL PERIOD

The typical form of the expression used in several international design codes to estimate the fundamental period (T) of a structure is presented in Equation 1:

$T=\alpha H^{\beta }$ (1)

where H represents the building height.

For masonry infilled concrete frames, part of the seismic forces is assumed to be resisted by the frames, while the remaining portion is resisted by the infill walls. In this case, the classification provided in Item 9.2 of ABNT NBR 15421:2023 [¹] indicates “other structures,” with α = 0.0488 and β = 0.75. More recently, the experimental and numerical investigation conducted by Ditommaso et al. [²⁰] reports β = 0.75 for masonry infilled reinforced concrete frames. Therefore, the expression investigated in this study is that presented in Equation 1, adopting β = 0.75 and α varying according the damage state.

According to Amanat and Hoque [²] formulas provided by codes are based on observed periods of real buildings during ground motion. In these empirical formulas the fundamental period is generally expressed as a function of the building height and depends on the structural system, as shown in Equation 1. A compilation of formulas for predicting the fundamental period provided by several codes in the world can be found in Dias-Oliveira et al. [²¹].

3 BUILDING CHARACTERISTICS USED IN THE NUMERICAL SIMULATIONS

3.1 Structural arrangement

• Symmetrical rectangular (square and non-square) base, without horizontal and vertical irregularities (mass or stiffness).

• Concrete frames arranged in two orthogonal axes X-Y in plan.

• Constant column and beam cross-sections along the entire height of the building.

• Infill walls placed over all beams, with defined opening percentages.

• Ratio between the larger plan dimension (BX) and the smaller plan dimension (BY): 1.0 (buildings with a square base), 1.5, and 2.0 (buildings with a non-square base).

• Beam spans (in two orthogonal axes): L = 4.0 m; L = 5.0 m; L = 6.0 m.

• Number of beam spans: 2; 3 and 4.

• Floor-to-floor distance: 3.0 m.

• Number of stories: 3; 6; 9 and 12.

Figure 1 presents a simplified general plan layout of the simulated buildings, showing the arrangement of columns, beams, and slabs.

Figure 1
Simplified scheme: structural plan of the simulated buildings

3.2 Beam and column dimensions

• Columns: 30x30 (cm); 45x45 (cm); 55x55 (cm); 65x65 (cm) for buildings of 3, 6, 9 and 12 stories, respectively.

• Beams: 25x60 (cm) for L = 6.0 m; 25x50 (cm) for L = 5.0 m; 25x40 (cm) for L = 4.0 m.

The selection of the cross-sectional dimensions of the reinforced concrete elements was based on the investigation conducted by Silva [¹⁶], who used Ultimate Load Combinations (including seismic loads) to check the frame members and calculate the steel reinforcements. The seismic forces were calculated using the Equivalent Lateral Load Method. Silva’s simulations [¹⁶] considered exclusively L = 6.0 m, with beams having cross-sections of 25x60 (cm). In the present study, for the remaining spans, beam depth equal to L/10 was adopted.

3.3 Slab thickness

• 10 cm for L = 4.0 m; 11 cm for L = 5.0 m, 13 cm for L = 6.0 m.

3.4 Concrete strength class

• C25 for columns, beams and slabs.

3.5 Mechanical properties and geometrical characteristics of the infill walls

• Thickness: 19 cm.

• Units: Hollow clay bricks (horizontal and vertical holes).

• Characteristic compressive strength of the prism: f_pk = 0.75 MPa (weak infill) and f_pk = 2.0 MPa (strong infill).

• Modulus of elasticity (uncracked infills): 450 MPa (weak infill) and 1200 MPa (strong infill).

• Equation for the calculation of the equivalent strut width: Mainstone [²²].

• Opening percentages: 20% and 40%.

The compressive strength values were selected based on the minimum requirements established by ABNT NBR 15270-1:2023 [²³] for non-structural clay bricks: 1.5 MPa for hollow bricks with horizontal holes and 4.0 MPa for hollow bricks with vertical holes. It is worth noting that the adopted stiffness and strength values are consistent with the recent study by Pereira et al. [²⁴] on seismic risk assessment of Brazilian buildings. The prism-to-unit compressive strength ratio was assumed equal to 0.5, in accordance with the reference values recommended in Annex F of ABNT NBR 16868-1:2020 [²⁵].

Using the concept of the equivalent diagonal strut, the infill wall is replaced by a pin-jointed diagonal strut whose area is equal to the product of the wall thickness and the equivalent strut width. The expression proposed by Mainstone [²²] was chosen because it is the most widely known among researchers in infilled frame analysis. This expression is also the most conservative compared to other expressions in the literature, as it provides the lowest values of equivalent strut width.

Table 1 presents the values of equivalent strut widths calculated using Mainstone’s expression, considering uncracked members (concrete frame and masonry infills) and infill walls without openings.

Table 1
Equivalent strut width (in cm): uncracked structural members and infill walls without openings

H (m)	L=4.0m (weak infill)	L=4.0m (strong infill)	L=5.0m (weak infill)	L=5.0m (strong infill)	L=6.0m (weak infill)	L=6.0m (strong infill)
9.0	58.38	52.93	69.33	62.85	81.41	73.80
18.0	66.72	60.49	79.37	71.96	93.43	84.70
27.0	70.92	64.29	84.46	76.57	99.58	90.28
36.0	74.37	67.42	88.65	80.37	104.70	94.92

The effects of cracking members and the presence of openings in infill walls are explained in Sections 4.2 and 3.6, respectively.

3.6 Effects of openings on the lateral stiffness

To account for the reduction in lateral stiffness due to the presence of openings, the equivalent strut width was multiplied by a stiffness reduction factor (F_rr). Based on Finite Element Method analyses performed by Silva et al. [²⁶] and Silva [¹⁶], an intermediate value (average) between that obtained from the expression proposed by Al-Chaar et al. [²⁷] and that proposed by Asteris et al. [²⁸] was adopted for this factor. Accordingly, the stiffness reduction factors were 0.51 for 20% openings and 0.29 for 40% openings.

3.7 Total number of simulated buildings

For square base buildings (BX = BY), three variations in the number of spans, three variations in span length, four different numbers of storeys, two different masonry strength classes, and two distinct opening percentages were considered, resulting in 144 buildings. For non-square base buildings with BX/BY = 1.5, three variations in span length were considered, while maintaining two spans in the X direction, along with four different numbers of storeys, two different masonry strength classes, and two distinct opening percentages, totaling 48 buildings. For buildings with BX/BY = 2.0, three variations in span length, two variations in the number of spans, four different numbers of storeys, two different masonry strength classes, and two distinct opening percentages were considered, resulting in 96 buildings. Thus, the total number of simulated buildings was 288.

4 METHODOLOGY APPLIED IN NUMERICAL SIMULATIONS

4.1 Structural model

The structural reference model for the analyses was the frame model with equivalent diagonal struts (Figure 2), which were modeled using the LINK10 truss element from ANSYS program, with only axial compression activated.

Figure 2
Structural models used in numerical simulations

Due to the large number of simulated models, a Modified Shear Building Model (Figure 2) was employed, in which the lateral stiffness of each storey was estimated from the equivalent diagonal strut frame model. The estimation of storey lateral stiffness was based on the Lateral Force–Deformation Method proposed by Vijayanarayanan et al. [²⁹]. The simulation of some buildings in ANSYS indicated small differences between the fundamental periods obtained using the Modified Shear Building Model and those from the reference model (approximately 8%).

4.2 Cracking of the structural members

For reinforced concrete columns and beams, cracking was considered by means of flexural stiffness reduction factors of concrete members, as indicated in item 15.7.3 of ABNT NBR 6118: 2026 [³⁰]. For infill walls, cracking was considered by means of an axial stiffness reduction of the equivalent diagonal strut, according to Annex D (D.2.3.4.2) of ABNT NBR 16868-1:2020 [²⁵].

4.3 Rigid-end offsets and slab stiffness contribution

In the intersection region between beams and columns, rigid-end offsets were considered. For practical purposes, rigid offsets were introduced only along the beam axes, with lengths defined according to the recommendations of ABNT NBR 6118:2026 [³⁰].

The contribution of slab stiffness was incorporated into the geometric properties of the beams (area and bending moment of inertia) in the frame model. The effective slab width was determined in accordance with the recommendations of ABNT NBR 6118:2026 [³⁰].

4.4 Modulus of elasticity of concrete

In all analyses, the dynamic modulus of elasticity was adopted, which is related to the static modulus of elasticity according to the expression presented by Gidrão [³¹]. The static modulus of elasticity was estimated using the expression provided in ABNT NBR 6118:2026 [³⁰], assuming granite as the coarse aggregate. For greater accuracy, the increase in the modulus of elasticity of the structural members due to the presence of steel reinforcement bars was considered using the expression proposed by Su et al. [¹⁸].

4.5 Modulus of elasticity of masonry infill walls

The masonry modulus of elasticity was determined according to the expression presented in Table 1 of ABNT NBR 16868-1:2020 [²⁵].

5 RESULTS

In concrete buildings, structural elements may be assumed cracked, in accordance with Section 8.7.3 of ABNT NBR 15421:2023 [¹]. Accordingly, the analyses were performed considering both uncracked and cracked structural members. Based on the fundamental periods obtained through modal analyses of the 288 simulated buildings, two curves were obtained by regression analysis: (i) one in which the coefficients α and β were determined; and (ii) another in which β was fixed at 0.75 and α was determined.

5.1 Square base buildings

As shown in Figures 3 and 4, the curves obtained by fixing β = 0.75 provide a better fit to the numerical results from the modal analyses, confirming the findings of Ditommaso et al. [²⁰].

Figure 3
Curves obtained from NBR 15421:2023 [¹] and from regression analysis: uncracked structural members (BX=BY)

Figure 4
Curves obtained from NBR 15421:2023 [¹] and from regression analysis: cracked structural members (BX=BY)

Figures 3 and 4 also include the curve derived from the approximate expression of ABNT NBR 15421:2023 [¹], with α = C_T = 0.0488 (where C_T is the structural period coefficient). It can be observed that the curve obtained for the uncracked members condition is the one that most closely matches the expression provided by the Brazilian standard. In contrast, the curve corresponding to the cracked members condition resulted in fundamental periods approximately 49% higher than those obtained using the approximate expression of the Brazilian code.

Table 2 presents a summary of the investigated expressions. Figure 5 provides a graphical comparison of these expressions.

Table 2
Coefficients for calculating the fundamental periods (BX=BY)

Description	α	β
Approximated formula in NBR 15421: T = Ta	0.0488	0.75
Upper limit in NBR 15421: T = CupTa	0.0768	0.75
(Cup = 1.6 for Seismic Zone 3)
Simplified formula (regression): modal analysis – uncracked members	0.05298	0.75
Simplified formula (regression): modal analysis – cracked members	0.07275	0.75

Figure 5
Comparison of curves: NBR 15421:2023 [¹] and regression analysis (BX=BY)

In seismic analyses using the Equivalent Lateral Load Method, ABNT NBR 15421:2023 [¹] does not allow the fundamental period obtained through modal analysis to exceed C_up times the approximate fundamental period (T_a), where C_up is the upper limit coefficient. Thus, Figure 5 also includes the expression defining this upper bound on the fundamental period, assuming Seismic Zone 3 (C_up = 1.6). It can be observed that the curve which results in the highest fundamental periods (cracked members) did not exceed the upper limit prescribed by the code.

In order to illustrate how differences among the fundamental period expressions lead to variations in seismic forces, the seismic base shear was calculated for buildings with the following characteristics: L = 5.0 m (span length), NB = 3 (number of spans), and masonry infill walls with a modulus of elasticity of 1200 MPa (strong infill) and a wall opening ratio of 20%. Figure 6 presents a graphical comparison of the seismic base shear obtained using the investigated fundamental period expressions and applying the Equivalent Lateral Load Method.

Figure 6
Seismic base shear: NBR 15421:2023 [¹] vs modal analysis (uncracked and cracked members) (BX=BY)

From Figure 6, it can be observed that no differences exist among the expressions for low-rise buildings (in this case, three-storey buildings). For taller buildings (six-, nine-, and twelve-storey) there were differences among the expressions, with the expression provided by ABNT NBR 15421:2023 [¹] consistently leading to more conservative results. Compared to the expression derived for uncracked elements, the standard expression resulted in seismic base shear between 3% and 9% higher; in relation to the expression obtained considering cracked elements, these values ​​were between 41% and 49% higher.

5.2 Non-Square (Rectangular) base buildings

Following the same procedure adopted for square-plan buildings, Figures 7 - 8 present the results obtained for rectangular-plan buildings, which are summarized in Table 3.

Figure 7
Curves obtained from NBR 15421:2023 [¹] and from regression analysis: uncracked structural members (BX≠BY)

Figure 8
Curves obtained from NBR 15421:2023 [¹] and from regression analysis: cracked structural members (BX≠BY)

Table 3
Coefficients for calculating the fundamental periods (BX≠BY)

Description	α	β
Approximated formula in NBR 15421: T = Ta	0.0488	0.75
Upper limit in NBR 15421: T = CupTa	0.0768	0.75
(Cup = 1.6 for Seismic Zone 3)
Simplified formula (regression): modal analysis – uncracked members	0.05009	0.75
Simplified formula (regression): modal analysis – cracked members	0.06879	0.75

As observed for square-plan buildings, the curves with β = 0.75 provide a better fit to the numerical results from the modal analyses. Among them, the curve derived for the uncracked elements condition is the one that most closely matches the expression provided by the Brazilian code. The curve obtained from the cracked members condition resulted in fundamental periods approximately 41% higher than those obtained using the approximate expression of the Brazilian code.

Similarly to the comparisons performed for the buildings analyzed in Figure 6, no differences were observed among the expressions for low-rise buildings (three storeys) in terms of seismic forces. For taller buildings (six-, nine-, and twelve-storey), there were differences among the expressions, with the one provided by ABNT NBR 15421:2023 [¹] leading to the most conservative results. Regarding the expression derived for uncracked members, the standard expression resulted in seismic base shear values approximately 3% higher. When compared to the expression derived considering cracked members, the differences were 34% to 41% higher.

6 CONCLUSIONS

This study presented a theoretical investigation of a simplified formulation for estimating the fundamental period of multi-storey buildings, symmetric in plan (square and non-square base) and without vertical irregularities in mass and stiffness, with structural system composed of reinforced concrete frames. Non-structural elements (masonry infill walls) were considered in the distribution of seismic forces. Expressions following the format recommended by several international codes (Equation 1 of this study) were investigated.

Based on the numerical simulations performed, the following conclusions can be highlighted:

• The value of β = 0.75 proved to be appropriate for masonry infilled concrete frames, confirming the findings of Ditommaso et al. [²⁰];

• The approximate expression provided by ABNT NBR 15421:2023 [¹] yielded lower fundamental period values compared to the expressions obtained through regression of modal analysis results, showing closer agreement with the condition in which cracking of the structural elements is not considered;

• When applying the Equivalent Lateral Load Method, the approximate expression of ABNT NBR 15421:2023 [¹] led to more conservative values of seismic forces in the structure compared to the expressions obtained through regression of modal analysis results;

• A significant difference in seismic force results was observed between the curves obtained considering cracked and uncracked structural members, with the latter representing the most conservative condition (seismic forces between 34% and 49% higher, considering all simulated cases).

Finally, it should be emphasized that these conclusions are limited to buildings with geometric and mechanical characteristics similar to those considered in this study, particularly with regard to building height (up to 36 m, or 12 storeys) and plan configuration (square and non-square plans).

ACKNOWLEDGEMENTS

The authors acknowledge the National Council for Scientific and Technological Development (CNPq) for the financial support (grant n. 403378/2023-0).

The authors did not use generative AI tools or AI-assisted technologies in the preparation of this manuscript, its text, figures, or underlying data.

Financial support:

CNPq (grant n. 403378/2023-0).

Data Availability:

The data that support the findings of this study are available from the corresponding author, GMSA, upon reasonable request.

How to cite:

G. M. S. Alva, A. O. Rigão, M. A. S. Pinheiro, and J. Kaminski Junior, “Theoretical study for predicting the fundamental period of concrete buildings including non-structural elements,” Rev. IBRACON Estrut. Mater., vol. 19, no. 4, e19403, 2026, https://doi.org/10.1590/S1983-41952026000400003

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Correspondence

Corresponding author: Gerson Moacyr Sisniegas Alva. E-mail: [e-mail]

Conflict of interest:

Nothing to declare.

Editors:

Mauro de Vasconcellos Real, Leandro Trautwein.

Publication Dates

• Publication in this collection
  05 Oct 2026
• Date of issue
  2026

History

• Received
  09 Mar 2026
• Reviewed
  22 June 2026
• Accepted
  20 Aug 2026