Unconfined Concrete Strength Research Articles

Finite element modeling and regression analysis of concrete filled – Glass fibre reinforced polymer tubes

This exploratory research narrates the axial compressive behavior of concrete filled circular GFRP (Glass Fibre Reinforced Polymer) tubes under concentric compression using FEA software ANSYS15.0, followed by optimizing the confinement models using Number Cruncher Statistical System (NCSS11). In this study about 40 specimens having (H/D ratio confined to 2) 150 mm diameter and 300 mm height were considered with tube thicknesses being 3, 4, 5 and 6 mm. The unconfined concrete strength was varied between 30 MPa and 120 MPa. Newer simplified linear confinement models were arrived for the short unreinforced CFFT columns which encourages optimality followed by acceptable correlation with the available codal equations of ACI 440.2R08.

Long-Term Performance of Seawater Sea-sand Concrete (SSC) Columns Wrapped with CFRP Strips Under Simulated Marine Environment

This article presents a durability study of carbon fiber-reinforced polymer (CFRP) partially wrapped seawater sea-sand concrete (SSC) columns exposed to natural seawater to explore the effect of exposure duration on the long-term performance of the specimens. Thirty-two cylinders were wrapped with CFRP jackets and exposed to different times of wet-dry cycles (up to 360 days) in an outdoor simulated marine environment. Test results indicate that exposure has no obvious influence on the failure process and ultimate strains of specimens, but the compressive strengths of confined columns (fcc) increase with the increment of exposure time, especially for the partially confined concrete specimens. Moreover, due to the significant variation of unconfined concrete strength fco*), the retentions of fcc and fcc/fco * exhibit an opposite trend. Therefore, the increase of fco* should be considered when using the parameter of the confined-to-unconfined ratio of strength to evaluate the long-term performance of the specimens.

Artificial Neural Network (ANN) and Finite Element (FEM) Models for GFRP-Reinforced Concrete Columns under Axial Compression

In reinforced concrete structures, the fiber-reinforced polymer (FRP) as reinforcing rebars have been widely used. The use of GFRP (glass fiber-reinforced polymer) bars to solve the steel reinforcement corrosion problem in various concrete structures is now well documented in many research studies. Hollow concrete-core columns (HCCs) are used to make a lightweight structure and reduce its cost. However, the use of FRP bars in HCCs has not yet gained an adequate level of confidence due to the lack of laboratory tests and standard design guidelines. Therefore, the present paper numerically and empirically explores the axial compressive behavior of GFRP-reinforced hollow concrete-core columns (HCCs). A total of 60 HCCs were simulated in the current version of Finite Element Analysis (FEA) ABAQUS. The reference finite element model (FEM) was built for a wide range of test variables of HCCs based on 17 specimens experimentally tested by the same group of researchers. All columns of 250 mm outer diameter, 0, 40, 45, 65, 90, 120 mm circular inner-hole diameter, and a height of 1000 mm were built and simulated. The effects of other parameters cover unconfined concrete strength from 21.2 to 44 MPa, the internal confinement (center to center spiral spacing = 50, 100, and 150 mm), and the amount of longitudinal GFRP bars (ρv = 1.78–4.02%). The complex column response was defined by the concrete damaged plastic model (CDPM) and the behavior of the GFRP reinforcement was modeled as a linear-elastic behavior up to failure. The proposed FEM showed an excellent agreement with the tested load-strain responses. Based on the database obtained from the ABAQUS and the laboratory test, different empirical formulas and artificial neural network (ANN) models were further proposed for predicting the softening and hardening behavior of GFRP-RC HCCs.

Confinement model for concrete wrapped with fiber reinforced cementitious mortar

Fiber reinforced cementitious mortar (FRCM) improves the performance of fiber-reinforce polymer (FRP) by replacing the organic matrix (e.g., epoxy) with inorganic one, resulting in a more sustainable alternative with a much improved fire resistance. While numerous models have been developed for FRP-confined concrete, FRCM-confined concrete have not received nearly as much attention. This study introduces a design-oriented model for FRCM-confined concrete. The model is the first to be derived from a very large database comprising 139 experimental specimens with various geometric and material properties, complemented with additional 144 numerical specimens obtained from finite element (FE) analysis. A nonlinear three-dimensional FE model was developed and verified with experimental results. It was used to examine the effects of mortar compressive strength (fm) and thickness (tm), type of FRP fabric, number of FRCM layers (n), compressive strength of unconfined concrete (fco), and height-to-diameter ratio (H/D). Results showed that the confined concrete strength (fcc) increases linearly as fm and tm increase, for all examined n, fco, and (H/D) values. fcc also varies almost linearly with n for all fabric types. A statistical analysis was performed on the combined database and a semi-empirical confinement model was developed to predict the peak strength and corresponding strain. It provided much better statistical performance and correlation with results than existing models and included a special coefficient for the effects of mortar properties.

Modeling for complete stress-strain curve of circular concrete columns confined with steel spiral and FRP

The results of 153 specimens from an experimental study on fiber-reinforced polymer (FRP) and steel spiral-confined (FRP-SS-confined) circular concrete columns were collected from accessible literature to establish a database. The existing models of FRP-SS-confined circular concrete columns were included and evaluated. A new stress-strain model with a simple, continuous, general expression was put forward for FRP-SS-confined circular concrete columns by taking into account various confining conditions during the loading process. Calculation models of the ultimate stress, ultimate strain, peak stress and peak strain were proposed by considering the contributions from both steel spiral confinement and FRP confinement. The comparison showed that the elastic stage, nonlinear transition stage, linear hardening stage and residual stage of the stress-strain curves were well captured by the proposed model for FRP-SS-confined circular concrete columns. Compared with other existing models, the proposed model showed better flexibility and versatility for practical application. In particular, various declining trends in the residual stages corresponding to the different confining conditions were well predicted by the proposed model. Furthermore, the effects of different parameters (steel tube thickness, FRP type, FRP layer and unconfined concrete strength) on the behavior of FRP-SS-confined circular concrete were discussed.

Axial Compressive Strength Models of Eccentrically-Loaded Rectangular Reinforced Concrete Columns Confined with FRP.

The majority of experimental and analytical studies on fiber-reinforced polymer (FRP) confined concrete has largely concentrated on plain (unreinforced) small-scale concrete columns, on which the efficiency of strengthening is much higher compared with large-scale columns. Although reinforced concrete (RC) columns subjected to combined axial compression and flexural loads (i.e., eccentric compression) are the most common structural elements used in practice, research on eccentrically-loaded FRP-confined rectangular RC columns has been much more limited. More specifically, the limited research has generally been concerned with small-scale RC columns, and hence, the proposed eccentric-loading stress-strain models were mainly based on the existing concentric-loading models of FRP-confined concrete columns of small scale. In the light of such demand to date, this paper is aimed at developing a mathematical model to better predict the strength of FRP-confined rectangular RC columns. The strain distribution of FRP around the circumference of the rectangular sections was investigated to propose equations for the actual rupture strain of FRP wrapped in the horizontal and vertical directions. The model was accomplished using 230 results of 155 tested specimens compiled from 19 studies available in the technical literature. The test database covers an unconfined concrete strength ranging between 9.9 and 73.1 MPa, and section’s dimension ranging from 100–300 mm and 125–435 mm for the short and long sides, respectively. Other test parameters, such as aspect ratio, corner radius, internal hoop steel reinforcement, FRP wrapping layout, and number of FRP wraps were all considered in the model. The performance of the model shows a very good correlation with the test results.

Experimental Study on the Confinement of Concrete Cylinders with Large Rupture-Strain FRP Composites

Large rupture strain (LRS) fiber-reinforced polymer (FRP) composites, typically formed from polyethylene naphthalate (PEN) and polyethylene terephthalate (PET) fibers, generally exhibit ultimate rupture strains >5%. Such fibers are particularly suited to the confinement of concrete columns on account of their LRS and sufficient elastic modulus. There are currently a limited number of studies on LRS FRP-confined concrete, particularly with high- and ultrahigh-strength concrete, so their behavior across a range of variables is still unknown. To improve this understanding, this paper systematically investigates the influence of fiber type, fiber thickness, and concrete strength on the behavior of FRP-confined concrete. To achieve this objective, the current investigation presents the results of 66 circular FRP-confined cylinders that are loaded concentrically. Three main parameters are investigated, namely, fiber type (i.e., PEN, PET, carbon, glass, and aramid), concrete strength (i.e., normal, high, and ultrahigh strength), and fiber thickness. The results show that regardless of fiber type, the stress–strain response is bilinear when the concrete is sufficiently confined. However, when there is insufficient confinement provided to the concrete core, the stress–strain response becomes trilinear. This trilinear response is more pronounced for LRS FRP-confined specimens because the confinement stiffness of the LRS FRP jacket is lower than that of a traditional FRP-confined specimen with an equivalent confinement ratio. Increasing the confining hoop stiffness (i.e., increasing FRP layers) reduces the magnitude of strength reduction after initial concrete cracking. It is also evident that as the unconfined concrete strength increases, the minimum confinement stiffness ratio necessary to prevent strength reduction after initial concrete cracking increases.

Axial compressive behavior of ultra-high-strength steel fiber-reinforced concrete-filled fiber reinforced polymer (FRP) tube columns

This study presents the results of the first experimental research on the axial compressive behavior of ultra-high-strength steel (UHSS) fiber reinforced concrete-filled FRP tubes (UHSSFR-CFFT). 24 circular UHSSFR-CFFT specimens were prepared and tested under axial compression to study the influence of unconfined concrete strength and fiber type. CFFTs were manufactured with glass, carbon and basalt reinforced polymer (GFRP, CFRP and BFRP) tubes with concrete compressive strengths (f′co) ranging from 35 to 105 MPa. Axial and lateral behavior were examined closely, including variation of lateral behavior along height of specimen. The results showed that the influence of concrete compressive strength on axial compressive behavior of UHSSFR-CFFTs is dependent on fiber type. For a similar normalized lateral confining pressure of FRP jacket at ultimate (flu/f′co), axial stress enhancement ratios (k1) were observed to decrease with an increase in f′co for all three fiber types. On the other hand, the obtained axial strain enhancement ratios (k2) indicated that k2 is influenced more by flu/f′co than f′co. Comparison of the recorded values of k1 and k2 for UHSSFR-CFFT with FRP-confined plain concrete showed similar trends on the influence of (f′co) on k1, but an opposite trend was observed for k2 of BFRP-confined specimens of the current study, in which k2 increased with increasing f′co. The results showed that the shape of the lateral strain-to-axial strain curves for UHSSFR CFRP- and BFRP-confined concrete specimens were similar to those of FRP-confined plain concrete, and this relationship was not influenced noticeably by f′co. Conversely, the obtained shape of the lateral strain-to-axial strain curves for UHSSFR GFRP-confined specimens was noticeably different when compared to FRP-confined plain concrete, with this behavior influenced by f′co. Finally, the results indicated that the addition of UHSS fibers in CFRP-confined specimens did not alter the influence of f′co on strain reduction factor (kɛ) where a decrease was observed by increasing f′co. However, the converse outcome was observed for GFRP- and BFRP-confined specimens where an increase in kɛ was observed when increasing f′co.

Three-dimensional finite element modeling and theoretical analysis of concrete confined with FRP rings

The confinement mechanism of concrete fully confined with fiber-reinforced polymer (FRP) jacket (FRP jacketed concrete) is different from that of concrete partially confined with FRP (i.e., FRP ring-confined concrete and FRP tie-confined concrete) in that the confinement in the latter is non-uniform along the longitudinal direction. In order to build a bridge between FRP jacketed concrete and FRP ring/tie-confined concrete, the current design of concrete confined with FRP rings/ties relies on the “arching action” assumption, which is not necessarily accurate as it was proposed for concrete confined with steel stirrups. Moreover, the arching action assumption usually adopts a hypothesis that the arching action angle equals to 45°, which has not been verified by any theoretical or experimental evidence. To this end, a revised analysis model has been implemented in an advanced finite element (FE) approach to study the axial stress distributions in concrete confined with FRP rings. The stress distribution at the center level of two adjacent FRP rings/ties is obtained, and the relationship between the arching action angle and controlling parameters (i.e., unconfined concrete strength, FRP width, FRP thickness and clear spacing of FRP rings) is established based on a proposed theoretical model of arching action angle. A new confinement effectiveness coefficient is then proposed, leading to a much more reliable prediction of the FRP ring-confined concrete in circular columns. The results presented in the current study can be easily extended to the concrete columns internally reinforced with FRP ties/spiral.

Numerical investigation of FRP-confined short square RC columns

Finite element (FE) simulation of confined RC columns is a complex task, as it involves the precise representation of a concrete material model that could express the volumetric change of concrete under a triaxial state of stress. The problem of confining RC columns using fiber-reinforced polymer (FRP) is more complex. This is due to the passive nature of FRP containment. Recently, the concrete damage plasticity (CDP) model available in an FE software product (ABAQUS) has been widely used to represent RC columns subjected to axial loading. The present paper aims to calibrate the concrete dilatancy angle and viscoplastic regularization parameters in the CDP model for the applications of square RC columns confined with FRP sheets. Ten experimental FRP-confined RC column specimens are incorporated in the calibration. In addition, a parametric study is investigated, including 96 cases, to obtain the influence of various design variables on the ratio between the confined to unconfined concrete compressive strength (fcc/fc). Four design variables are considered (i.e., the corner radius, concrete compressive strength, FRP reinforcement ratio, and cross-sectional size of columns). The study also focuses on assessing the reliability of four design codes: ACI-440, ECP-208, CSA-S806 and FIB-Bulletin-14. The numerical results show that a viscoplastic regularization with a fixed value equal to 0.0005 could provide accurate behavior. Nevertheless, the dilation angle cannot be considered a fixed value and should be related to the confinement degree coefficient Cd, which depends on the cross-sectional geometrical and FRP properties. Among the four considered design codes, ACI-440 and ECP-208 provide more accurate behavior than CSA-S806 and FIB-Bulletin-14.

Behaviour of preloaded concrete cylinders confined by fibre-reinforced polymer composites

The stress–strain behaviour of concrete confined by fibre-reinforced polymer (FRP) is an important part of studying the seismic rehabilitation of concrete structures. Most previous studies have investigated the stress–strain behaviour of concrete prisms confined by polymer composites without preloading, but concrete columns are under various loading conditions before being retrofitted with polymer confinement. Therefore, in this study, FRP-confined concrete cylinders with different preloading ratios were prepared to examine the effect of preloading on the behaviour of retrofitted concrete columns. The results indicated that preloading affects important points of stress–strain curve. Preloading up to 75% of the unconfined concrete strength had a positive effect on the stress–strain behaviour, but 90% preloading reduced the strength of the FRP-confined concrete. A modification for the stress–strain models of preloaded FRP-confined concrete is thus suggested.

Predicting ultimate strength of FRP and lateral steel confined circular concrete columns using Artificial Neural Networks

In this paper, a new model is proposed to predict the ultimate strength of FRP and conventional reinforced concrete confined by lateral steel bars. Accordingly, circular specimens models are developed and investigated using an Artificial Neural Network (ANN) technique. To study the effect of the various parameters, e.g., the strength of unconfined concrete, the diameter of concrete samples, the height of the samples, tensile strength of FRP and conventional lateral steel, the thickness of FRP layers, the diameter of lateral steel stirrups, the strength of lateral steel stirrups, and modulus of elasticity of FRP and lateral steel stirrups, were selected. The proposed model is trained, validated, and tested based on 49 samples collected from the literature. Moreover, based on the developed ANN model, a set of design-based equations and charts is recommended. In addition to the samples utilized in developing the proposed model, further new samples are selected to evaluate the accuracy of the proposed model. The results from both ANN and user-friendly equations and charts show a good agreement with the test results. Moreover, parametric study is conducted to compare the performance of the proposed model against the experiments and numerical models in the literature. The parametric study shows that the ANN models possess better accuracy and performance to predict the ultimate strength of FRP and conventional steel-reinforced concrete confined specimens, for which the influence of important parameters is considered.

Numerical modelling of the in-plane behaviour of concrete-filled circular steel tubular arches

Numerical modelling approaches are favoured in studying the instability failures of concrete-filled steel tube arches (CFSTAs) due to the complexity and high cost of experimental testing of such structural members. While some of the existing numerical models are able to capture the constitutive behaviour of the materials in CFSTAs accurately, they are computationally expensive to be used for studying the effect of key parameters affecting the CFSTA behaviour. This paper presents a simple yet accurate numerical modelling approach to predict the behaviour of CFSTAs considering in-plane instability failures. Numerical modelling was carried out using open-source finite element (FE) software OpenSees. CFSTAs were modelled using fibre section beam-column elements available in OpenSees. A modified stress–strain model was used to capture the constitutive behaviour of both confined normal and high strength concrete. Bi-axial stress state of steel tube was considered for failure modelling of steel. Proposed FE model for CFSTA was verified using existing experimental results of CFSTAs. Consideration of the bi-axial stress state of the steel tube was shown to increase the accuracy of the predictions. FE results showed that when a CFSTA is failing due to instability, geometric imperfection will significantly influence CFSTA behaviour. Both percentages of steel content and unconfined concrete strength was found to affect the ultimate load of the CFSTAs, but the effect was found to be dependent on the loading conditions. Rise-to-span ratio as well as the shape of the arch was also found to significantly influence the behaviour of CFSTAs, with the parabolic arch shape being found to be the best.

The axial and lateral behavior of low strength concrete confined by GFRP wraps: An experimental investigation

The purpose of this investigation is to investigate the axial and lateral behavior of low strength concrete confined by glass fiber reinforced polymer (GFRP) wraps. A total of 48 cylindrical samples having 150 mm diameter and 300 mm height were strengthened and examined. The mix designs were provided with four different target strength (5, 10, 15, and 20 MPa). The axial and lateral strain were studied to evaluate the axial strain relation and the lateral strain for axial compression. The results of this investigation present a comparison between the nine models previously reported for normal strength concrete and the lesser-known context of low strength concrete confined by fiber reinforced polymer (FRP) composites. The results obtained from this study show that the majority of models reported for normal strength concrete, can accurately predict the stress improvement ratio for low strength concrete (LSC), however, one of the models exactly predicts the strain improvement ratio for all ranges of LSC. The results also show that an increase in compressive strength of concrete causes a decrease in both stress improvement ratio and strain improvement ratio. The analysis of variance (ANOVA) was applied to study the percentage effect of the key criteria to the stress improvement ratio, strain improvement ratio, and the strain factor of LSC, where the results reveal that the compressive strength is a more effective criterion on the stress and strain improvement ratios and the strain factor than the confinement level (number of GFRP layers). Finally, lateral strain-axial strain behavior of GFRP and actively confined concrete is compared and it is understood that lateral strain- axial strain behavior of confined specimens is affected by lateral pressure and unconfined concrete strength.

BEHAVIOUR OF HIGH-STRENGTH CONCRETE CIRCULAR COLUMNS CONFINED BY HIGH-STRENGTH SPIRALS UNDER CONCENTRIC COMPRESSION

This paper tested the behaviour of 32 high-strength concrete columns confined by high-strength spirals under concentric compression. The test parameters included unconfined concrete compressive strength, spiral yield strength, volumetric ratio, and spiral spacing. The results showed that bulging and shear sliding were the two characteristic types of failure patterns of the thirty-two confined columns, depending on spiral spacing and concrete strength. Moreover, the spiral in most specimens did not yield at the confined concrete compressive strength. An analytical confinement model for high-strength concrete columns confined by high-strength spirals was proposed. In this proposed model, the calculated value of the spiral stress at the confined concrete compressive strength was used to calculate the feature points of the stressstrain curve. The proposed model showed good correlations with available experimental results of 64 columns.

Reliability analysis of proposed capacity equation for predicting the behavior of steel-tube concrete columns confined with CFRP sheets

Due to higher stiffness to weight, higher corrosion resistance, higher strength to weight ratios and good durability, concrete composite structures provide many advantages as compared with conventional materials. Thus, they have wide applications in the field of concrete construction. This research focuses on the structural behavior of steel-tube CFRP confined concrete (STCCC) columns under axial concentric loading. A nonlinear finite element analysis (NLFEA) model of STCCC columns was simulated using ABAQUS which was then, calibrated for different material and geometric models of concrete, steel tube and CFRP material using the experimental results from the literature. The comparative study of the NLFEA predictions and the experimental results indicated that the proposed constitutive NLFEA model can accurately predict the structural performance of STCCC columns. After the calibration of NLFEA model, an extensive parametric study was performed to examine the effects of different critical parameters of composite columns such as; (i) unconfined concrete strength, (ii) number of CFRP layers, (iii) thickness of steel tube and (iv) concrete core diameter, on the axial load capacity. Furthermore, a large database of axial strength of 700 confined concrete compression members was developed from the previous researches togive an analytical model that predicts the ultimate axial strength of composite columns accurately. The comparison of the predictions of the proposed analytical model was done with the predictions of 216 NLFEA models from the parametric study. A close agreement was represented by the predictions of the proposed constitutive NLFEA model and the analytical model.

Reliability analysis of strength models for CFRP-confined concrete cylinders

The present work describes a reliability analysis of strength models for predicting the confinement effect of concrete cylinders with Carbon Fibre Reinforced Polymers (CFRPs) through the Artificial Neural Network (ANN). For the reliability analysis, not only current available equations are used but present work also proposes the new empirical equation (Eq-Q) based on the previously conducted experimental work. For which, the database of 708 samples is developed with the detailed information of critical parameters, i.e.; diameter of cylinder (d), height of cylinder (h), unconfined concrete strength (fco'), thickness (nt), elastic modulus of CFRP (Ef), and the ultimate strength of confined concrete (fcc'). The aim of this investigation is achieved in two phases: In first phase, the comparison of current available equations with the proposed ANN model and proposed Eq-Q. In second phase, for the validation purposes of proposed ANN model and Eq-Q, new experimental scheme is developed for the concrete cylinders confined with CFRP and its nonlinear finite element analysis (NLFEA). The NLFEA model after calibration with experimental results is used for further parametric studies. The Comparative studies show that the proposed ANN model, proposed Eq-Q and NLFEA results are in close agreement with the experimental results and give better results as compared to their counterparts.

Structural state of stress analysis of confined concrete based on the normalized generalized strain energy density

This paper investigates the behavior of 32 high-strength concrete square columns confined by high-strength transverse reinforcement under concentric compression, by applying structural state of stress theory. First, the sum of generalized strain energy density values (Eij) of columns at every load value (Fj) is normalized as Ej,norm to describe the structural state of stress. Then, the Ej,norm-Fj curves reveal the characteristics of the state of stress and failure mechanism of confined concrete columns under load-bearing condition. The Ej,norm-Fj curves also reveal an updated failure load definition and failure criterion. Furthermore, the effects of unconfined concrete compressive strength, volumetric ratio, and transverse reinforcement yield strength on the state of stress are studied. Finally, a formula is proposed to predict the updated failure loads of confined concrete columns. In summary, a novel method to identify some working characteristics of structures is explored.

A Proposed Soft Computing Model for Ultimate Strength Estimation of FRP-Confined Concrete Cylinders

In this paper, the feed-forward backpropagation neural network (FFBPNN) is used to propose a new formulation for predicting the compressive strength of fiber-reinforced polymer (FRP)-confined concrete cylinders. A set of experimental data has been considered in the analysis. The data include information about the dimensions of the concrete cylinders (diameter, length) and the total thickness of FRP layers, unconfined ultimate concrete strength, ultimate confinement pressure, ultimate tensile strength of the FRP laminates and the ultimate concrete strength of the concrete cylinders. The confined ultimate concrete strength is considered as the output data, while other parameters are considered as the input data. These parameters are mostly used in existing FRP-confined concrete models. Soft computing techniques are used to estimate the compressive strength of FRP-confined concrete cylinders. Finally, a new formulation is proposed. The results of the proposed formula are compared to the existing methods. To verify the proposed method, results are compared with other methods. The results show that the described method can forecast the compressive strength of FRP-confined concrete cylinders with high precision in comparison with the existing formulas. Moreover, the mean percentage of error for the proposed method is very low (3.49%). Furthermore, the proposed formula can estimate the ultimate compressive capacity of FRP-confined concrete cylinders with a different type of FRP and arbitrary thickness in the initial design of practical projects.

Confinement Coefficient Predictive Modeling of FRP-Confined RC Columns

Abstract This article proposes an approach to determine the confinement coefficient of concrete columns with square and rectangular cross sections wrapped with all types of fiber-reinforced polymer (FRP) based on soft computing. Five inputs, including the length or longer side of the column, radius of the corner, elastic modulus of FRP, total thickness of FRP, and the ratio of the ultimate tensile strength of FRP to the compressive strength of unconfined concrete, are used as predictors to estimate the confinement coefficient of concrete columns wrapped with FRP. The model was trained and validated based on a set of experimental data. The equations based on the proposed approach were determined and the steps to calculate the confinement coefficient were presented. The results indicated that the extracted equations of the proposed approach have a high degree of precision in comparison with existing models.