Evaluation and optimisation of regular RC frame structures of varying heights with graphene oxide

Abstract

Frame structures are commonly used in places with low seismic activity. Their structural behaviour is characterised by low rigidity and lateral resistance, due to the small sections of the supporting elements (columns), and for this reason, this structural typology requires the help of rigid elements such as filler, structural walls, or other bracing elements. The use of materials in the sections that can improve the resistant behaviour of these structures, such as graphene oxide (GO), offers an alternative solution to the massive use of walls. The results presented here show that introducing small amounts of GO into the composition of the concrete (4% with respect to the weight of the cement used in traditional 30 MPa concrete test specimens) in the sections (beams and columns) improves the mechanical and resistant behaviour of framed structures by between 3 and 15%, and that this effect increases as the structures increase in height. In this research, regular frame structures in plan and elevation of three, eight and 15 storeys are analysed, with proportions of steel equal to 2.5% in the columns and 4% in the beams. Optimisation of the structural sections of the frames using GO concrete, in which the ultimate strengths of the structures with traditional concrete are matched, gives reductions in the amount of concrete in the sections of between 10 and 13%, with the greatest reductions in the highest structures.

Introduction

Frame structures are typically used in environments that are not excessively demanding, for example where seismic activity is not particularly high (such as in Spain or Italy); they are popular due to their quick assembly, low cost^1,2 and the small structural sections (beams and columns) that compose them.

However, the structural vulnerability of these structures to seismic events is not very favourable^3,4. The structural behaviour of this type of structure towards the frames is characterised by low strength, rigidity and ductility. On the other hand, the transmission of moments from the beams to the columns is problematic, since part of the longitudinal reinforcements of the beams are located outside the width of the columns, which can generate torsion mechanisms due to the greater width of the beams in these structural types compared to the columns^3–6. In the transverse direction, the situation is even more unfavourable, since there are no anchoring elements between the structural elements, which produces moment reversals and brittle fractures^3,7,8. Hence, these structural typologies require the use of stiffening elements, such as braces, filler walls or structural walls^9–11.

The need for walls or stiffening elements to improve and solve structural behaviour problems means that the structural design and costs for this type of structure can be affected, especially in moderately seismic areas. Studies and research have been carried out in recent years to solve this problem by improving structural behaviour. One of the most frequently studied approaches involves improvements to the properties of the materials most commonly used in construction, such as concrete or steel, by introducing materials into their chemical composition that mitigate the most unfavourable properties^12–16.

In regard to the use of concrete, there are a variety of materials in the scientific literature that have been added to the mixture, with the aim of improving the most deficient properties of concrete (such as the ductility, flexural tensile strength, or thermal insulation), including plastics in the form of fibres^17–19 and more rigid materials such as glass^20,21, among others. The purpose of this research is to improve the mechanical and structural properties of frame structures. From among the many materials that have been added to concrete, graphene oxide (GO) is chosen here. In the scientific literature, it has been shown that small amounts of GO can significantly improve the mechanical properties of concrete^22–24 and other materials. Hence, the inclusion of small amounts of GO in the mixtures can reduce the size of the structural sections needed and therefore the construction costs, thereby improving the structural behaviour and reducing the vulnerability of the structures to seismic events of medium and low magnitude.

The choice of GO rather than graphene was made due to its availability and the cost of these materials²⁵, which make it unfeasible for graphene to be used on a massive scale in construction. Further reasons why this material was chosen include its physical and chemical properties²⁶, which are beneficial in the field of construction. The characteristics of GO perfectly complement the good properties of concrete in the field of construction. For example:

• It has a high density of between 1.12 g/cm³ (paste) and 0.35 g/cm³ (powder)²⁷;

• It can be easily dispersed in water by mechanical agitation or ultrasound;

• Depending on its chemical and structural properties, it ranges from being an insulator to a semiconductor. This characteristic is beneficial in regard to its thermal and electrical properties;

• It is antibacterial, and like graphene, it resists attacks by microbial agents;

• It is impermeable to all types of gas and liquid;

• It has a modulus of elasticity in the range of 23 to 42 GPa;

• Its use in cement mortars has shown improvements in compressive strength of 38.9% compared to ordinary mortars²⁸. It has been reported that the improvements produced with this material ranged between 13% and 48%²⁹;

• It has a flexural-tensile strength of 0.13 gigapascals. Improvements in materials such as concrete have been shown to range from 30 to 60%²⁹;

• It has a high tensile strength: improvements in materials such as concrete have been shown to range from 35 to 78%²⁹.

Compared to the existing scientific literature, the novelty of this research lies in an analysis of the effect of adding GO on construction (structures), which will determine its viability for real-world applications. Although GO has a lower cost than graphene, its price is still high³⁰.

The aim of this research is to demonstrate that the effect of adding GO to concrete for low, medium and high-rise frame structures is beneficial in their structural behaviour. Furthermore, it has been shown that the addition of GO to the mix produces economic and material savings compared to structures with traditional concrete. As a reference, concrete with a resistance of 30 MPa will be used, a value that is typically used in structures located in areas of medium and high seismicity^3,4. Structural models with three, eight and 15-storey frames will be considered, and a nonlinear static (push-over) analysis will be performed to assess the plastic and ultimate forces and displacements, and the stiffness and ductility, characteristics that determine and define good structural performance^3,4.

Methodology

Frame design

Structural designs based on reinforced concrete (RC) frames are some of the most widespread systems in Latin America and Europe. These are simple and economical, and are characterised by regular and simple geometries in elevation and plan^31–33 compared to other existing systems, such as those using walls. In this research, structures of different heights will be analysed to determine the impact of the solution in terms of improvement due to the use of a new concrete mix with GO. 3D regular frame structures of three, eight and 15 stories with a floor height of 3 m will be analysed, corresponding to structures with total heights of 9, 24 and 45 m, respectively. The regularity in plan and elevation of the models used in this research is due to compliance with and the advice considered in the earthquake-resistant design of buildings that appear in the different regulations existing in the world. The floor configuration of these structures is based on a rectangular scheme, composed of 4 × 2 spans of 6 m each measured to the column axes, giving floor dimensions of 24 × 12 m². The heights and spans of these structures were selected based on the configurations of real structures³⁴. Slabs or one-way slabs have been used as structural surface elements (Fig. 1), as they are widely used in this structural typology³⁵. Since these are frame structures, the direction of the frame or the resistant direction of the structures is in the Y direction, while in the X direction they are weaker. This is due to the presence of fragile joints (between the slab and the beam or column), which arise from the lack of continuity of reinforcement between the structural elements in the interior spans; the only continuity of reinforcement in this direction is in the exterior spans, where the enclosures and the structural mesh (#15Ø10) in the 5 cm compression layer in the slab or one-way slab (Fig. 1) of thickness 25 + 5 cm are located. The structural configurations of the structures are shown in plan and elevation view in (Figs. 2 and 3), respectively. The dimensions and the quantities of longitudinal reinforcement used in the columns and beams were designed following the requirements of EC-2 and ACI318, standards that are accepted by most European and American countries (North and South America), in order to ensure that our study covered a wide field of applications across the world.

Fig. 1.

Details of the unidirectional slab.

Fig. 2.

Structural plan for the models.

Fig. 3.

Structural frames for the models (Y-direction).

The combination of loads considered in the analysis was based on those established by the standards for reinforced concrete structures (CTE-AE, ACI318 and EC-2) and by the European earthquake-resistance EC-8 regulations. A value of 2 kN m^− 2 was used for live loads (use)³⁶, as this is associated with the category of residential, administrative and small commercial buildings (Type II). In addition, a load of 1 kN m^− 2 was considered for the upper floor (roof), as this is associated with maintenance loads³⁶. The dimensions of the structural sections are shown in (Table 1), where the sections of the square columns vary by 10 cm every four floors (with those on the upper floors being 30 × 30 cm²). The dimensions of the structural sections of the columns decrease with the height of the frame structure (better structural performance against seismic events^3,4), to comply with the minimum dimensions established by ACI318. In addition, all beams used in the models are rectangular in Sect. (30 × 40 cm²) in the three, eight and 15-story structures.

Table 1.

Sections of the structural elements of the frames (dimensions in cm).

Frame/height	Columns			Beams (cm2)
	3-story (cm2)	8-story (cm2)	15-story (cm2)
1–4	30 × 30	40 × 40	60 × 60	40 × 50
5–8	–	30 × 30	50 × 50
9–12	–	–	40 × 40
13–15	–	–	30 × 30

The amounts of longitudinal reinforcement used in the sections are 2.5–3.0% for the columns and 4.0% for the beams. The arrangement of the reinforcement is distributed equally along the four faces that make up the sections. In addition, the transverse reinforcement used in all sections (columns and beams) is Ø10 every 10 cm. The number of reinforcements for the sections was chosen based on a consideration of the execution of the sections, to ensure correct vibration of the concrete and a uniform distribution along the section. When calculating the amount of steel used in the sections, the action of the infill walls, corresponding to the divisions or enclosures of the real buildings, was not considered. Finally, the covering used in all sections had a thickness of 2.5 cm.

The longitudinal reinforcement steel used in each of the structural sections of the frames is shown in (Table 2).

Table 2.

Sections and longitudinal reinforcement used in the structural sections of the models.

Columns (cm2)	Longitudinal reinforcement steel	Beams (cm2)	Longitudinal reinforcement steel
60 × 60	20Ø24	30 × 40	18Ø19
50 × 50	20Ø20
40 × 40	12Ø20
30 × 30	12Ø16

The characteristic strength of the traditional concrete used in the models is taken as a comparative reference in the analyses carried out here, with a value of f_ck = 300 kg/cm² (H-30), and the elastic limit of the steel used for the reinforcement of the sections is f_yk = 5000 kg/cm² (AEH500). These values were chosen since they are typically used in RC structures in areas of medium and high seismicity.

Although the simple scheme used in the regular configuration of the models (Figs. 2 and 3) is widely extended in the configuration of real structures that use this structural system, it has been proven that it does not work properly under horizontal loads; this is mainly due to the slenderness of the columns, which gives low rigidities to the structural configurations^3,4. For this reason, in real buildings, these structures require stiffening elements such as braces, fillers or structural walls^37,38 among others, as solutions to improve their structural and seismic behavior. However, in this research, these stiffening elements will not be used, because the main basis of this research is the effect that GO has on traditional concrete, which is why the structural sections used in the models have not taken these stiffening elements into account in their design].

Structural analysis

The Seismostruct v.2025 finite element structural program from Seismosoft^®39 was used in the structural analysis carried out in this research. To enable a comparison of the results obtained for the structural performance of the different frames, the following parameters were evaluated: the basal shear force (F_u), the plastic and ultimate displacements (d_y,d_u) on the upper floors of the frames, the ductility (µ), and the elastic stiffness (K_y). All of the structural performance results were compared with those for frames made with traditional H-30 concrete.

The results obtained in this research have been validated by many existing works in the scientific literature^{22,25,40–50}. The simulation of the mechanical behavior of each material in the structural elements in this research has been carried out with the introduction of various data corresponding to the properties of the material from the laboratory tests obtained from the corresponding hydraulic machines. The values entered correspond to the experimental values of plastification (yielding) and fracturing (collapse) from the capacity curves of each material. For the unit deformations corresponding to the fracture processes of concrete and steel, the standard values of Seismostruct^38,39,51 were used. Expanding the veracity of what was done in this work, there are various investigations and studies that use this program, obtaining successful results^52–56.

When entering data into the software, the following aspects were considered:

1. The selection of structural elements (beam and columns) was based on inelastic linear frame elements with plastic hinges^51,57, with a formation in which the plastic hinges were grouped in the areas close to the joint, specifically at 10% of the length of each bar⁵⁸.

2. The nonlinear method of Mander et al.⁵⁹ for concrete is a nonlinear uniaxial constant confinement model that follows the constitutive relationship proposed by the same author and suggested by the cyclic laws described in⁶⁰. The confinement effects due to the stirrups were considered in terms of the laws proposed in⁵⁹, in which a constant confining pressure is assumed throughout the entire stress-strain range of each bar. The deformation and strength values were based on the capacity curves (stress-strain) for cylindrical samples obtained in the laboratory. The correction factor was defined as the ratio between the confined and unconfined f_c of the concrete, used to amplify the stress-strain relationship in the deformation range. To determine this factor, bibliographical data from^59,61 were used. Its value fluctuated between one and two for RC, and between 1.5 and four for elements made of concrete and steel.

3. The bilinear model of Ferrara et al.⁶² was used for the steel, with the default values ​​of the program assigned to a steel of 500 MPa. This is a uniaxial stress-strain model with kinematic hardening, in which the elastic range remains constant throughout the loading phases; in the hardening range, a linear function of the plastic strain increment is assumed.

4. Poisson’s ratios of 0.2 for concrete and 0.3 for steel were assumed⁶³.

5. The normalised structural stiffnesses were calculated based on the elastic properties of the main material of the section (in this case, the concrete used for an RC section).

6. The geometry of the structure was obtained from the dimensions of the section and the connectivity of the joints in space (X, Y, Z), thus establishing internal restrictions on the structure. These restrictions apply at the base of the columns (as these are elements embedded in the ground) and in the vertical deformations of the slabs or floors, restricting their degrees of freedom as rigid diaphragms with infinite rigidity in this direction⁶⁴.

7. Axial deformations were restricted⁶⁵.

8. (h) The rotations and stresses of the elements that made up the frames were verified using different standards^66,67.

9. Geometric nonlinearity was incorporated through a corotational formulation, as developed and implemented by^68–71, considering the effects of large displacements/rotations and large deformations of the structure frame (P-Delta effects⁶⁹).

10. The inelasticity of the materials was represented by simple concentrated plasticity models^68,72 at the hinges. The behavior of the cross sections was done through each of the fibers that make up the section, in relation to the uniaxial stress-strain behavior) The sectional state of the beam-column elements was obtained by integrating the nonlinear uniaxial stress-strain response of the fibres that make up each Sect. (300 in this case). This model has several advantages, for example:

• (i)There is no need to perform a moment-curvature analysis prior to defining the elements;
• (ii)There is no need to define any type of hysteretic response of the elements;
• (iii)The interaction between axial tension and flexural moment (strength and stiffness) can be modelled directly;
• (iv)The biaxial load can be represented directly;
• (v)The interaction of the frame elements (beams and columns) can be obtained using classical displacement-based finite element formulations^70,71.

• (k)The structural design of the models used is regular in plan and elevation, meeting the performance characteristics required by current earthquake-resistant regulations worldwide. This is intended to avoid torsion problems that could occur in buildings, jeopardizing their structural and earthquake-resistant performance. Taking this effect into account would significantly expand this research, analyzing a new case.
• (l)The algorithm used to establish the different vibration modes of the structure was developed using the Jacobi algorithm with a Ritz transformation⁷³. The structural results were obtained following the approaches established by Seismostruct through fibre modelling. These approaches consider the deformation of the material, which is often the best parameter to represent the performance of a structure.

The criteria used to obtain the deformations in the structures were as follows:

1. The values of the curvatures of the sections were taken from⁶⁴;

2. The columns and beams were represented as nonlinear bar finite elements^59,71,74, with the nonlinearities concentrated in the plastic hinges located in areas close to the nodes (10% of the total length of each structural element);

3. The connections between columns and beams were considered to be rigid, with the hysteretic behaviour of the connections represented by the stress distribution⁵⁹;

4. The deformations were represented by a fibre model (300 units) based on the material properties and the section of the structural element;

5. An infinite stiffness was assumed for the vertical displacements of the slabs or floors (diaphragms)⁷⁵;

6. The loads of the structural system were applied to the beams;

7. The tolerances for the displacement and the rotations were adjusted to 10 and five, respectively. The number of iterations was limited to 300. In each iteration, the numerical convergence seeks the instability of the structural element when increasing the applied load. The maximum number of iterations was obtained using the modified Newton-Raphson method⁷⁶. A value of 50 was taken to limit the number of divergence iterations, since lower values could generate nonlinear problems;

8. The Newmark methodology was applied, with values of β = 0.25 and γ = 0.5⁷⁷;

• (i)The Rayleigh damping model was used⁷⁸, with a damping coefficient of 4% in Mode 1 and 6% in Mode 2;

• (j)To determine the results for each structural element, several performance criteria were defined; concrete cracking (0.0001), concrete spalling (− 0.002)⁷⁹, concrete core crushing (− 0.002), steel yield strength (0.0025)⁸⁰ and steel fracture (0.06)⁸¹;
• (k)The static behaviour of the buildings was calculated using the results for the seismic forces obtained from Eurocode 8;
• (l)The distribution of incremental horizontal loads obtained from the push-over analyses followed a triangular load pattern in the calculations.

Characterisation of concrete with graphene oxide

To determine the characteristics of concrete with the inclusion of GO in the mix, laboratory tests were carried out. To do this, cylindrical concrete test tubes were generated in order to analyse the density, granulometry, and humidity, among other parameters of the compounds. Finally, compression and flexural traction tests were conducted using hydraulic machines in the laboratory of the Faculty of Engineering of Talca University.

The choice of GO as a material was made due to the characteristics that it offers to concrete, since these can improve the most deficient properties of the mix, as described in the Introduction.

The GO used in this research took the form of an aqueous solution; that is, it was dissolved in water in a proportion of 4 mg/ml. The solution (Fig. 4) had a dark colour. The amount of GO used in the test tubes (in this case 4%) was determined with respect to the amount of cement used in the dosage of traditional 30 MPa concrete.

Fig. 4.

Aqueous GO solution used in the tests.

The following regulations were considered in the process of generating the concrete:

• NCh 164 Of. 2009⁸², for the quartering of the material, in order to maintain the homogeneity of the sample;

• NCh 165 Of. 2009⁸³, for the granulometric study and the screening of the gravel used in the mixtures;

• NCh170 of 2016⁸⁴, for the preparation of the mixtures and the cylindrical test tubes of 30 cm height and 15 cm diameter (Fig. 5) and for the mechanical tests of the concretes;

• NCh 1017 Of. 2009⁸⁵, for setting, curing and stripping the mixture;

• NCh 1018 Of. 2009⁸⁶, for the concreting procedure;

• NCh 1019 Of. 2009⁸⁷, for the execution of the Abrams cone and for determining the type of concrete according to its workability, with the descent that the cone undergoes;

• NCh 1116 Of. 2008⁸⁸, for the calculation of the bulk density of the mixtures;

• NCh 1117 Of. 2010⁸⁹, for the washing and drying of the material. Drying was carried out in an oven at 110º C ± 5ºC, until a constant mass was achieved, over a period of 24 h. The water losses will be considered later for the dosage of the concrete. The losses that will be considered are 10%;

• NCh 1239 Of. 2009⁹⁰, to calculate the density of the sand;

• NCh 1172 Of. 2010⁹¹, for the facing of the test tubes (Fig. 5);

• NCh 1037 Of. 2009⁹², for concrete compression tests (Fig. 6a). The rate of incremental force used in the tests was 0.20 N/mm²/s.

• NCh 1038 Of. 2009⁹³, for the concrete flexural-tensile tests (Fig. 6b). The rate of incremental force used in the tests was 0.20 N/mm²/s.

Fig. 5.

Photograph of the facing and a cylindrical specimen.

Fig. 6.

Mechanical tests of concrete specimens.

Once the granulometry, density and water absorption data had been obtained, the dosage required to achieve the concrete strength needed for this research was determined. To do this, the Faury-Joisel (FJ) method⁹⁴ was used, in accordance with the ACI 211.1–91 Standard (ACI, 1991 (Rev. 2002)). This method uses granulometric data based on a graph, an ideal curve (L) and the proportion of aggregates corresponding to the desired dosage.

To determine the average dosage strength, a strength of 30 MPa was used, corresponding to the traditional concrete in this study. The formula Eq. (1) used for the average dosage strength is as follows:

1

Where, F_d is the average dosing strength (MPa), F_p is the specified strength of the project (MPa), t is a factor defined in NCh 170 of. 2016⁸⁴, a value that depends on the level of confidence that is desired to be achieved, and s is the estimated deviation (MPa) extracted from NCh 170 of. 1985/2016⁸⁴. A confidence level of 95% and “good” working conditions are assumed, corresponding to t = 1.645 and s = 5.0.

Next, the quantity of cement per m³ was determined, using the following expression Eq. (2):

2

where C is the quantity of cement, F_d is the average strength of the dosage, and E = 1.05, which is a correction factor defined by the type of concrete. This value is essential to determine the quantity of GO used in the mixtures.

The values for the water/cement ratio and the compactness of the mixture were taken from those published by the National Road Laboratory in 1993⁹⁵. In the first batch, workability and compactness tests of the concrete were carried out with an Abrams cone⁹⁶ with water losses of 10%. The final dosages with 10% water loss are shown in (Table 3).

Table 3.

Dosages used in the preparation of traditional H-30 concrete.

Sand (kg/m3)	706
Gravel (kg/m3)	1025
Water (l/m3)	204
Cement (kg/m3)	412

The average results obtained from the mechanical tests in the laboratory, which were used for the modelling of the structures, are shown in (Table 4). In this table, the density, the compressive strength and flexural-tensile strength values are obtained in the laboratory, the ultimate strain values (ε_u) are obtained from the deformation of the last point of the capacity curve corresponding to the maximum stress or force prior to the descent of the slope of the curve, the plastic strain (ε_p), is obtained from the area equity method used and explained in ATC-40, in which an intersection is made between the original curve of the capacity curve of the behavior of each mixture and a bilinear curve generated by the user and, finally, ductility, being a dimensionless value obtained from the quotient between the ultimate strain and the plastic deformation of the capacity curves. These data have been compared with the values considered by default in the Seismostruct program³⁹. On the other hand, the elastic compression modulus of elasticity (E_{y, comp}), is obtained through the quotient between the compressive plastic stress and the plastic strain of that same point. The value obtained from the elasticity calculation has been carried out using the laws of elasticity, where E_y = σ_y/ε, considering the mechanical laboratory results obtained from the compression tests, although there are other forms of calculation for this parameter, such as those established by ASTM C469, ACI 318 − 19 or ATC-40. For this purpose, the ACI-318 or ATC method has been established to make a comparison of results, with no excessive differences, as shown in (Table 4).

Table 4.

Mechanical results for traditional H-30 concrete and GO concrete.

	Traditional concrete H-30	Concrete with GO (4%)
Density (kg/m3)	2347	2290 (−3%)
Compressive strength (MPa)	29.98	35.50 (+ 18%)
Flexural-tensile strength (MPa)	3.98	6.90 (+ 73%)
Ey, comp (MPa) (Laws of elasticity /ACI318)	24,662 /25,734	28,430/28,003 (+ 15%/+ 9%)
εp (comp)	0.0011	0.0019
εu (comp)	0.0011	0.0020
µ (comp)	1.75	1.85

It should be noted that, from the average results corresponding to the tests carried out in the laboratory that appear in (Table 4), a total of 6 cubic test pieces (3 cubic ones of 15 × 15 × 15 cm³ for the compression tests, complying with Standards NCh1037 and NCh1017 and 3 cubic ones of 15 × 15 × 45 cm³, complying with Standard NCh1038 (h, h, 3 h) and NCh1017) and 3 cylindrical test pieces (Ø15 cm and 30 cm high for the compressive strength tests), have been made, using the same materials and the same dosage, being tested at the same time as mentioned in Standard NCh170 (1985 in cubic test pieces and 2016 in cylindrical test pieces), which are shown in (Figs. 5 and 6). These results were validated with the realization of 3 other RILEM test pieces (size 40 × 40 × 160 mm³) tested in compressive strength and, another 3 test pieces of 40 × 40 × 120 mm3 tested in flexural-tensile strength, using the same dosages as the test pieces initially made. In addition to this, the results obtained were compared with other investigations that used similar dosages, there were no discordant results between the test pieces, which justifies and validates the mechanical results obtained. Finalize the explanation that the mechanical tests to compressive strength and flexural-tensile strength followed the process explained in the Standards NCh 170, NCh1017, NCh1037 and NCh1038.

Push-over analysis

A push-over analysis was performed to estimate the maximum load-bearing capacity of the structures. This is also known as an “incremental static analysis”, where the incremental load P applied on each floor is proportional to the nominal load pattern (P°), and P = λ(P°). To carry out these calculations, SeismoStruct software³⁹ was used, with the parameters defined in Sect. 2.2. This program automatically increases the applied incremental load by a factor λ, until reaching the limit defined by the researcher or a numerical instability that causes the analysis to stop. The pattern of incremental loads used by is through a triangular load, proportional to each of the floors. The criterion used to analyze the load-bearing capacity of the structures used in this research is a control of deformations, not forces. This method consists of determining, as the deformation of the structural models increases, a series of plastic hinges appear in the different elements that make up the frame, until the structures become completely unstable. When this occurs, the program stops calculating.

Results and discussion

The capacity curves obtained from the push-over analysis show improved resistance for the concrete frames with GO, which increases with the height of the structures. The capacity curves for the different models are shown in (Fig. 7a–c). The legends in the images describe the behaviour of the frames in the X and Y directions, using traditional concrete and concrete with GO. The purpose of the regularity in plan and elevation existing in the models used in this research is to avoid the effects of torsion that could occur in the models, complying with the seismic-resistant structural design requirements mentioned in the vast majority of earthquake-resistant regulations existing in the world. X-direction_trad and Y-direction_trad represent the capacity curves in the X and Y directions, respectively, for the frames using the traditional 30 MPa concrete, whereas X-direction_GR and Y-direction_GR represent the capacity curves in the X and Y directions, respectively, for the frames using concrete with GO. We note again that the strong direction for the frames corresponds to the Y direction. Table 5 show the most characteristic values of the frames (F_y; yielding force, F_u; ultimate force, d_y; yielding deformation, d_u; ultimate displacement, K_y; elastic stiffness, and µ; ductility (d_u/d_y)) corresponding to the capacity curves in (Fig. 7a–c). As can be seen in the Figures, the improvement in the resistant capacities of the models with GO concretes occurs from the initial part of the behavior shown in the curves for each of the points analyzed, due to the greater rigidity existing in the models with GO. In addition, Fig. 7a–c show the yielding (F_y) and ultimate (collapse) (F_u) points, the performance points (IO, LS and CP) and the damage states (Sd₁, Sd₂, Sd₃ and Sd₄), corresponding to (Tables 5, 6 and 7) respectively, in order to show in a more visual way, the detailed structural behaviors of each case. The percentages shown in the Table 5 represent the existing relationships between the results for the concrete structures with GO and those for the structures with traditional concrete. As can be seen from the tables, an increase in height for the structures significantly increases the differences between these values. The greater demand in terms of resistance and ductility with height produce these differences in the results for these structures. In addition, the improvements produced in the frames with GO coincide with the mechanical improvements in compressive strength and flexural-tensile strength for the modified mixtures. Comparing the ultimate strengths between X-Y directions for the three- and eight-storey structures are approximately 10%, increasing to 25% for the 15- storey structures. The difference in the ultimate strengths between the cases in which traditional concrete is used and those in which GO is used increase progressively, with values of at least 3% in the three-storey structures, increasing to 11% in the 15-storey structures.

Fig. 7.

Capacity curves of the models.

Table 5.

Characteristic results of the capacity curves of the models.

3-story frame	X-direct_TRAD	Y-direct_TRAD	X-direct_GR	Y-direct_GR
Fy (kN)	1176	1297	1198 (+ 2%)	1302 (+ 1%)
Fu (kN)	1321	1455	1370 (+ 4%)	1503 (+ 3%)
Dy (m)	0.12	0.09	0.11	0.09
Du (m)	0.19	0.14	0.19	0.14
Ky (kN/m)	20,543	24,718	26,042 (+ 27%)	33,481 (+ 35%)
µ (duct)	1.69	1.47	1.81 (+ 7%)	1.59 (+ 8%)

8-story frame	X-direct_TRAD	Y-direct_TRAD	X-direct_GR	Y-direct_GR
Fy (kN)	1626	1808	1693 (+ 4%)	1830 (+ 1%)
Fu (kN)	1838	2018	1932 (+ 5%)	2107 (+ 4%)
Dy (m)	0.28	0.25	0.26	0.23
Du (m)	0.39	0.31	0.38	0.30
Ky (kN/m)	12,445	12,973	13,683 (+ 10%)	16,030 (+ 24%)
µ (duct)	1.39	1.24	1.44 (+ 4%)	1.31 (+ 6%)

15-story frame	X-direct_TRAD	Y-direct_TRAD	X-direct_GR	Y-direct_GR
Fy (kN)	2287	2912	2495 (+ 9%)	3021 (+ 4%)
Fu (kN)	2558	3190	2830 (+ 11%)	3423 (+ 7%)
Dy (m)	0.56	0.50	0.54	0.46
Du (m)	0.79	0.62	0.81	0.60
Ky (kN/m)	8613	10,372	10,707 (+ 25%)	14,487 (+ 40%)
µ (duct)	1.42	1.24	1.51 (+ 6%)	1.30 (+ 4%)

Table 7.

Damage states of the models.

The colors represents the damage states according to Lagomarsino and Penna, where the green color includes a slight damage state (Sd,1), the yellow color a moderate damage state (Sd,2), the orange color a extensive damage state (Sd,3) and the red color a complete damage state (Sd,4).

Table 8.

Comparative sections, volumes and weight of the optimized sections of concrete with GO and traditional concrete H-30.

Traditional concrete H-30
3-story frame	X-direction (m)	Y-direction (m)	M3	W (kg)
Columns	0.30	0.30	12.15	27,945
Beams	0.30	0.40	38.88	89,424
Total			51.03	117,369
Concrete with GO
Columns	0.30	0.30	12.15	27,945
Beams	0.30	0.35	34.02	78,246
Total			46.17 (−10%)	106,191 (−10%)

Traditional concrete H-30
8-story frame	X-direction (m)	Y-direction (m)	M3	W (kg)
Columns	0.30	0.30	16.20	37,260
	0.40	0.40	28.80	66,240
Beams	0.30	0.40	103.68	238,464
Total			148.68	341,964
Concrete with GO
Columns	0.30	0.30	16.20	37,260
	0.40	0.40	28.80	66,240
Beams	0.30	0.34	88.13	202,694
Total			133.13 (−10%)	306,194 (−10%)

Traditional concrete H-30
15-story frame	X-direction (m)	Y-direction (m)	M3	W (kg)
Columns	0.30	0.30	16.20	37,260
	0.40	0.40	28.80	66,240
	0.50	0.50	45.00	103,500
	0.60	0.60	48.60	111,780
Beams	0.30	0.40	194.40	447,120
Total			333.00	765,900
Concrete with GO
Columns	0.30	0.30	16.20	37,260
	0.35	0.35	22.05	50,715
	0.45	0.45	36.45	83,835
	0.55	0.55	40.82	93,925
Beams	0.30	0.36	174.96	402,408
Total			290.50 (−13%)	668,144 (−13%)

Table 6 show the displacements at the performance points (IO: immediate occupancy, LS: life safety; CP: prevention of collapse) for each model, based on ATC-40⁹⁷ and FEMA 1050⁹⁸. The colours used in the table correspond to the state of damage of the structure, following the study carried out by Lagomarsino and Pena (2003)⁹⁹, where green represents a state without damage, yellow indicates mild damage, orange represents moderate damage and red severe damage.

Table 6.

Performance points of the frame structures models.

The colors represents the damage states according to Lagomarsino and Penna, where the green color includes a slight damage state (Sd,1), the yellow color a moderate damage state (Sd,2), the orange color a extensive damage state (Sd,3) and the red color a complete damage state (Sd,4).

The damage thresholds were evaluated based on the idealised bilinear capacity spectrum according to Lagomarsino and Penna (2003), using the yielding displacement (d_y) and the ultimate displacement (d_u). The four damage thresholds are as follows:

Sd,1 = 0.7dy,

Sd,2 = dy,

Sd,3 = dy + 0.25(du-dy),

Sd,4 = du,

representing ‘slight’, ‘moderate’, ‘extensive’, and ‘complete’ damage states.

Table 7 below show the limiting damage states as established by Lagomarsino and Pena⁹⁹.

As shown by the performance points of the structures, there is a slight improvement in the behaviour of the structures in which GO was used in the concrete. The greatest damage occurs in the lower structures, which may be due to the greater rigidity requirements that these structures demand.

To complete this investigation, the following study was conducted, matching the maximum (ultimate) strengths of the capacity curves with GO concrete with the capacity curves of the structures made with traditional concrete. By matching the strengths of both concretes, the amount of GO concrete used in the structural sections was reduced. The comparative capacity curves for this case are shown in (Fig. 8a–c). The dimensions, volumes and weights of the optimised sections of the new structures using concrete with GO are shown in (Table 8), together with the percentages of concrete reduction and the weights of the beams and columns corresponding to the two proposed solutions. The largest reductions of up to 13% of concrete are found for the tallest structures (15 storeys).

Fig. 8.

Matched capacity curves of the models.

Conducting a quick economic study, considering an average price of G30 concrete at $150/m³, with the dosages considered in Table 3 and the quantities of concrete shown in (Table 8), the greatest profitability of using GO would be with the greatest quantity of concrete used (in this case with the greatest height, because the floor plan of all models is the same). With the total quantities used in each model, it is concluded that, in the framework of 3 heights, the use of GO would be profitable when its price was less than $90/kg; in the framework of 8 heights, when the price of GO was less than $100/kg and; in the framework of 15 heights, when the price of GO was less than $120/kg.

Among the improvements that new concretes offer in buildings compared to traditional concrete are the following:

1. Greater strength: the addition of graphene oxide significantly improves the compressive and flexural tensile strength of concrete.

2. Greater durability: the addition of graphene oxide increases the durability of concrete, making it more resistant to corrosion, moisture, and fire.

3. Impermeability: the quality of graphene oxide acts as an impermeable barrier, potentially protecting concrete from moisture and corrosion.

4. Thermal insulation: the quality of graphene oxide can improve the thermal insulation properties of concrete, which could reduce energy costs for heating and cooling buildings.

5. Improved structural seismic vulnerability of buildings: the reduction in seismic forces due to the reduction in the weight of buildings produced by the reduction of structural sections and the increase in ductility are determining factors for improving the seismic conditions of buildings.

6. (6) More significant economic improvement with increased building height: the greater amount of concrete used in relation to the small amount of GO in the mixes increases their profitability.

Conclusion

This study is supported by other research on graphene oxide^{22,25,40–50}, as are the calculation methods used in this study with the software used³⁹.

Numerous studies have investigated the effects of graphene and its derivatives, most of which have focused on improving cement paste, mortars, and concrete¹⁰⁰. These studies have shown that small doses (i.e., 0.01 to 0.03% by weight) of graphene oxide (GO) significantly improve the compressive and flexural tensile strength of concrete (i.e., an average of 20–60%)¹⁰¹. Furthermore, there are studies that justify the existing improvement in the use of GO, with the greater height of structures, as shown in the study conducted on the use of GO in the structural analysis performed, including the economic and pollution benefits (CO₂ emissions) of its use in construction⁵².

However, the benefits of GO may not be fully achieved unless it is uniformly dispersed in the matrix, which is why some studies vary some of the results obtained and that could lead to contradictions. Achieving a uniform dispersion of GO in concrete mixtures is crucial to extend its application in the concrete industry¹⁰², since its chemical inertness and hydrophobic nature can lead to the formation of tangled lumps, which would reduce the mechanical strength due to the concentration of stresses in some parts of the mixtures¹⁰³. The temporal dispersion effect, as demonstrated by Divya et al.¹⁰⁴, is crucial in this type of research. Furthermore, these authors found that, at an equal dose of GO, the compressive strength improved by 40% and the flexural tensile strength by 70% by increasing the temporal dispersion from 30 min to 60 min¹⁰⁴. This tendency limits the maximum amount of GO that can be added to approximately 0.1% by weight of the cement^100,105, as done in this investigation. For example, in M20 grade concrete, the maximum increase in strength is observed with a GO dosage of 0.03% by weight¹⁰⁶; however, above 1.0% by weight, the strength tends to decrease¹⁰⁰.

The structural results obtained in this research were analysed in order to compare models based on traditional H-30 concrete frames with those based on concrete with the inclusion of 4% GO rather than cement. The following conclusions can be drawn:

1. In general, regular frame structures using concrete with GO show significant improvements in structural performance. These improvements increase with the height of the structures.

2. Regarding the strengths between the structural directions of regular frames, the structures have greater strength in the Y direction, being 10% higher than in the X direction in the lowest buildings and reaching 25% higher in the tallest structures. This is due to the greater resistance of the structures in the direction of the frames (Y).

3. The ultimate strength values ​​of structures with GO concrete increase significantly with height. This increase is 4% for three- and eight-story structures and, rises to 11% for 15-story structures. These improvements are due to the greater mechanical resistance of the concretes.

4. The increase in the ductility of the structures using concrete containing GO is not very significant, although the improvement increases with the height of the structure. This is due to the greater rigidity requirements for structures with a lower height. These small differences are due to the lack of difference between the ductility of the concretes, since they are fragile and rigid materials, even with the addition of GO.

5. There are no major differences between the plastic and ultimate deformations of the structures with traditional concrete and the new mixtures with GO. This is because the behaviour of the concretes continues to be fragile and rigid, as reflected in the mechanical deformation of the concretes and the small ductility of the mixtures.

6. There are slight improvements in the damage states of the structures when the concrete contains GO, but these are not significant. An analysis of the performance points and damage states in the structures indicates that the most significant damage occurs in the structures with lower height (three storeys), due to the greater rigidity of these structures.

7. From an analysis of the optimisation of the main sections of the structures (columns and beams), we see that there are significant reductions with the use of concrete containing GO. These reductions increase significantly with the height of the structures, with values of 10% for three- and eight-story structures, and 13% for 15-story structures.

8. The reductions in the sections of the elements that make up the frames produce significant reductions in the weights of the structures. This will benefit the earthquake-resistant behaviour of the structures, since the seismic forces will be reduced.

9. Considering the quantities of concrete used in buildings, the profitability of GO in the mixtures increases with the greater quantity of concrete used (corresponding to the greater height of the buildings), which corroborates what was mentioned in this research.

On the other hand, like all research, there are barriers that limit the applicability of this research. The biggest drawback in this study is the profitability and feasibility of its widespread application in real-life situations, due to the current high price of graphene oxide (GO). Even so, this study has addressed this issue with optimized prices that graphene oxide should have to be profitable in each case, in Chap. 3 of this study. In addition, there are more detailed economic studies on the use of this product in concrete⁵², should the reader wish to delve deeper into the topic. To address this potential drawback regarding its applicability, other lines of research are offered that could help the researcher or reader continue advancing this topic of great interest to society.

This research could be expanded with future research and studies related to this topic, such as validating the results obtained in this research with dynamic analysis (Time-history), using different types of records from around the world. Other studies that would complement what has been done in this research include the use of other GO dosages in concrete mixes, the use of other construction typologies such as shear walls, the use of non-regular structures by analyzing the torsional effects existing in the structures, or the use of other heights in the structures. To conclude this broad field of research, other, more economical additives derived from graphene could be used, which would make the use of these new mixtures in real buildings even more profitable, although they would not be as effective in terms of structural, thermal, and waterproofing.

Author contributions

D. Dominguez-Santos: Conceptualization, Formal analysis, Project administration, Writing - Original DraftP. Muñoz: Conceptualization, Formal analysis, Writing - Original Draft.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.Any information needed should be consulted with the project authors to be provided.

Declarations

Competing interests

The authors declare no competing interests.