75504 2712-8172 19 4 144 2026

RAR RUS 14401-14401 0009-0009-5382-8876 Donbas National Academy of Civil Engineering and Architecture - branch of NRU MGSU Levchenko Victor v.n.levchenko@donnasa.ru

Makiyivka, Russian Federation

0000-0002-6035-3532 Donbas National Academy of Civil Engineering and Architecture - branch of NRU MGSU Dmitrenko Evgeniy d.e.a_2008@mail.ru

Makiyivka, Russian Federation

57190980263 0000-0001-8391-3099 Donbas National Academy of Civil Engineering and Architecture (DonNACEA) – Branch of National Research Moscow State University of Civil Engineering (MGSU) Nedorezov Andrei a.v.nedorezov@donnasa.ru

Makiyivka, Russian Federation

0009-0009-5967-2008 Donbas National Academy of Civil Engineering and Architecture - branch of NRU MGSU Sevostyanov Nikita n.a.sevostyanov@donnasa.ru

Makiyivka, Russian Federation

Influence of cross-section shape on the stress-strain state of centrally compressed reinforced concrete columns The study focuses on centrally compressed reinforced concrete columns with various cross-sectional shapes, including square, rectangular, cruciform, T-shaped, and L-shaped sections. In structural engineering, the load-bearing capacity of columns is typically enhanced through transverse reinforcement, external jackets, or composite confinements that create a triaxial compression effect. However, the influence of cross-sectional geometry on the formation of a 3D stress state has not been sufficiently investigated. A numerical analysis of centrally compressed reinforced concrete columns with different cross-sectional configurations was carried out using the finite element method in the LIRA-SAPR software package, accounting for the nonlinear behavior of concrete. The results indicate that the shape of the cross section has only a minor effect on strength characteristics (differences below 3 %), but significantly affects the distribution of volumetric stress zones, which are pronounced in T- and L-shaped columns when dilation effects are considered. It was found that volumetric confinement primarily develops in the support regions and has little influence on the overall structural performance of the columns. Future research will focus on eccentrically loaded columns and on the effect of external confinement systems, including concrete-filled steel tubes, steel jackets, and indirect reinforcement, on the formation and evolution of the 3D stress-strain state of reinforced concrete elements. The proposed research direction confirms the practical relevance of the study and the potential for further investigation in this field. 1.Introduction Reinforced concrete columns with various cross-sectional shapes are widely used in building and civil engineering, particularly when the layout configuration or architectural requirements call for efficient use of interior space. In addition to conventional rectangular and square columns, cross-shaped, T-shaped, and L-shaped columns are used in structural systems because they enable structural members to be integrated into walls and beams and allow multiple functions to be combined within a single element. The complex geometry of the cross-section has a significant impact on the structural stress-strain state. Local stress concentrations in massive members lead to the formation of a 3D stress-strain state. Considering this 3D stress-strain state in such zones greatly influences the overall strength, crack resistance, and durability of the structures. A number of studies have shown that a 3D stress state in reinforced concrete columns can develop under the influence of various factors, including the level and eccentricity of loading, end-restraint conditions, the presence of longitudinal and transverse reinforcement, as well as the geometry of the cross-section [1–17]. For rectangular and square columns, the stress state is generally close to plane stress conditions, whereas in members with complex shapes, such as cross-shaped, L-shaped, and T-shaped sections, local zones of spatial stress interaction may arise due to the mutual influence of the flanges and webs. These characteristics lead to non-uniform deformation, asymmetry in crack development, and differences in failure modes, all of which should be taken into account when analyzing the load-bearing capacity and stability of such elements. Under uniaxial compression, concrete exhibits relatively brittle failure, whereas under biaxial or triaxial confinement its load-carrying capacity and deformability increase significantly [18, 19]. In nonstandard cross-sections (such as cross-shaped, T-shaped, and L-shaped), the distribution of lateral stresses is nonuniform, which leads to the formation of zones with different levels of confinement and alters the concrete stress-strain relationship within a single section. In [20, 21], reinforced concrete columns with equivalent cross-sectional areas and different cross-sectional shapes were considered, including square, cross-shaped, L-shaped, and T-shaped sections. It was found that, for a slenderness ratio of less than 40, the load-bearing capacity of equivalent columns differs only slightly: an increase of approximately 4.5 % was observed for the cross-shaped section, whereas reductions of about 1.4–6.9 % were reported for the T- and L-shaped sections, which may be considered insignificant. For a slenderness ratio greater than 40, the influence of the cross-sectional shape becomes significant, and cross-shaped, L-shaped, and T-shaped columns provide an increase in load-bearing capacity of up to 12 % compared with square columns. In [22], theoretical investigations were carried out on short reinforced concrete columns with equivalent cross-sectional areas and different cross-sectional shapes, namely circular, square, rectangular, octagonal, cross-shaped, L-shaped, and T-shaped sections. It was established that rectangular, square, and cross-shaped columns differ only slightly in terms of load-bearing capacity: the increase for the cross-shaped section was approximately 1.4 %, while the difference between square and rectangular columns was practically negligible. For L- and T-shaped columns, the reduction in ultimate load compared with the rectangular section was 6.9 % and 9.7 %, respectively. Thus, the results reported in [20–22] are comparable: the advantage of the cross-shaped section is relatively small, while the reduction in load-bearing capacity for L- and T-shaped sections remains within the range of a moderate calculated difference. In [1, 2], the behavior of cross-shaped concrete-filled steel tubular (CFST) columns and their steel counterparts was investigated. The composite columns, consisting of concrete confined within a steel tube, had a load-bearing capacity 75–80 % higher than that of the steel columns. This increase is attributed to the composite action of the materials: the steel shell restrains lateral expansion and prevents cracking of the concrete, creating a confinement effect, while the concrete core, in turn, enhances the local stability of the steel walls and delays buckling. In [3, 4], cross-shaped and L-shaped steel-reinforced concrete (SRC) columns subjected to seismic loading were examined. Experimental results and finite element simulations showed that encasing the steel profile in concrete provides improved structural stability. In [5, 6], the focus is on enhancing the confinement effect in T-shaped CFST columns by introducing stiffeners and tie elements, as well as on finite element modelling (FEM) of such members considering the strengthening of the concrete core. One of the aspects examined in [7] was the influence of cross-section geometry on the load-bearing capacity of special-shaped CFST columns. A comparison of L- and T-shaped specimens showed that the cross-section configuration has a noticeable, although not decisive, effect on the behavior of the columns under axial compression. T-shaped columns exhibited slightly higher load-bearing capacity and stiffness, which can be attributed to a more uniform stress distribution and the effective composite resistance provided by the flange and the web. L-shaped members showed greater longitudinal deformability and stress localization in the corner regions, resulting in a less symmetric failure pattern. The influence of the cross-section geometry of CFST columns was also addressed in [8], which presents an analysis of various special-shaped configurations (L-shaped, T-shaped, cross-shaped, and other combined forms), along with corresponding beam-to-column connection details. The authors note that the cross-section shape significantly affects the stress distribution within both the steel tube and the concrete core, which in turn influences the overall load-bearing capacity and stiffness of the members. In [9], the behavior of L-shaped CFST columns with internal stiffening plates was simulated. The analysis showed that the asymmetry of the cross-section leads to a non-uniform stress distribution between the flanges and the interior corner regions, as well as to the formation of local zones with a 3D stress-strain state. The introduction of internal plates reduced the effects of geometric asymmetry and enhanced the composite action between the concrete and the steel tube. Similar conclusions were reported in [10], which showed that the configuration of an L-shaped cross-section, including the flange length ratio, has a noticeable influence on the stiffness, failure pattern, and stability of the columns. In [11], FEM in ABAQUS showed that the angle between the branches of L-shaped CFST columns has a significant effect on the stress distribution: as the angle decreases, the edge effect intensifies and stress concentrations increase in the outer corner regions, whereas increasing the angle to 135° leads to a reduction of stresses in the inner zones and a redistribution of forces along the flanges of the section. In rectangular columns, a 3D stress-strain state can be achieved by installing confinement jackets made of various materials [12–14] or by applying indirect reinforcement [15–17]. In [13], the axial compressive behavior of square and rectangular reinforced concrete columns strengthened with a composite fiber-reinforced polymer (FRP) jacket was investigated experimentally. It was shown that the effectiveness of confinement decreases as the aspect ratio of the cross-section increases; however, even at h/b = 2, a significant strength enhancement can still be achieved with an increased number of FRP layers. The results of calculations for short columns showed an increase in load-bearing capacity of up to 30–50 % compared with unconfined specimens [15]. Experimental studies [16] demonstrate that reducing the spacing of closed stirrups and using welded or zigzag reinforcement meshes can increase the ultimate load by approximately 7–22 %, depending on the reinforcement configuration. This effect is pronounced when the stirrup spacing is small. Replacing traditional stirrups with reinforcement meshes increases the concrete strength by 15–28 % and reduces lateral deformations, providing a more ductile failure mode [17]. In the numerical analysis of reinforced concrete structures, numerous finite element method-based approaches are employed that allow both the physical and geometric nonlinear behavior of concrete and reinforcement to be taken into account. A significant contribution to the advancement of numerical modelling has been made by studies conducted by domestic and international researchers using various software packages, including LIRA-SAPR, ANSYS, SCAD Office, ABAQUS, and ATENA. In [15], the nonlinear behavior of concrete was modeled in LIRA-SAPR using 3D materially nonlinear universal 8-node isoparametric solid finite elements (FE type 236), while the nonlinear behavior of reinforcement was represented by materially nonlinear truss-bar finite elements (FE type 204). The failure criterion was defined as the attainment of ultimate concrete stresses considering multiaxial stress conditions, based on the strength criteria of Klovanich–Bezushko [23] and Willam–Warnke [24], as well as the attainment of the yield stress in the reinforcement. The difference in strains obtained from comparison between the experimental results and the numerical simulations based on these models did not exceed 5 %. In [25], a method for modelling reinforced concrete structures in LIRA-SAPR was presented, followed by comparison of the numerical results with experimental data. Concrete and longitudinal reinforcement were modelled using 3D materially nonlinear finite elements of types 231 and 233, while transverse reinforcement was represented by bar elements of type 210. The interaction between concrete and reinforcement was simulated through link elements based on bar-type elements 295 and 296. The calculated results for cracking, strength, and stiffness showed a deviation from the experimental data of no more than 10 %. A similar approach was implemented in [26], where comprehensive nonlinear modelling of reinforced concrete structures was performed using the ESPRI and LIRA-SAPR software packages. The analysis was carried out in a materially nonlinear formulation, accounting for concrete creep, temperature effects, and time-dependent stress redistribution. The deformation behavior was described using the 25th nonlinear constitutive law for concrete implemented in LIRA-SAPR. In [27], an analytical-numerical approach for determining the stress-strain state of CFST columns under eccentric compression was presented, based on the deformation theory of concrete plasticity. The method allows for consideration of the triaxial stress state of the concrete core by analytically describing the relationship between the principal stresses and strains. In [28], an algorithm for modelling the behavior of reinforced concrete in LIRA was developed, based on the principles of fracture mechanics, which takes into account crack opening, concrete discontinuity, and reinforcement response. This approach provides a reliable representation of stress redistribution processes during cracking, allows refinement of element stiffness characteristics, and improves calculation accuracy compared with traditional linear-elastic models. In [29, 30], the development of a 3D finite element model of reinforced concrete columns implemented in the ATENA software environment was presented. In [29], the modelling was carried out using 3D solid elements and the nonlinear Kent–Park concrete material model, which accounts for the material response under compression, tension, and cyclic loading. In contrast, in [30], ATENA’s built-in concrete model was employed, based on automatic parameter identification from cube strength and an exponential tensile fracture law. This approach made it possible to reproduce various failure modes of the columns and to validate the numerical results against experimental data. In structural systems, the load-bearing capacity of columns is commonly enhanced through the use of transverse reinforcement [15–18], external jackets [1–11, 14], or composite wraps [12, 13], which provide a confinement effect. An analysis of [20–22] shows that, for reinforced concrete columns with complex cross-sectional shapes and a slenderness ratio of less than 40, where failure is governed by the material load-bearing capacity rather than by loss of stability, the influence of the cross-sectional shape on the ultimate load is generally limited. However, the published studies do not establish whether the cross-sectional geometry itself, without additional confinement measures, can lead to the formation of a 3D stress state within the concrete volume. This issue is particularly relevant for columns with complex configurations, such as cross-shaped, L-shaped, and T-shaped sections, which are used in frame and wall structural systems, where the interaction between flanges and webs may potentially cause stress redistribution within the concrete volume. In this context, there is a need for a comprehensive assessment of how the cross-sectional geometry influences the stress-strain state parameters of concentrically compressed reinforced concrete columns under 3D stress conditions. The aim of the present study is to evaluate the influence of cross-sectional geometry on the development of zones with a 3D stress-strain state in concentrically compressed reinforced concrete columns and to determine the feasibility of accounting for this effect in the strength design of such elements. 2.Methods For the analysis of the stress-strain state of reinforced concrete columns, five cross-section types with equivalent area and a height of 3300 mm were considered: square; cross-shaped; T-shaped; L-shaped; rectangular. Longitudinal reinforcement consisted of 12 bars with a diameter of 10 mm, while transverse reinforcement was represented by 6 mm stirrups spaced at 150 mm along the entire column height. Concrete of class B25 and reinforcement of classes A500 (longitudinal) and A240 (transverse) were used. These material grades were selected due to their wide application in conventional construction practice, ensuring the comparability of the analysis results with real structures and the practical relevance of the conclusions. The overall dimensions of the cross-section and the height of the element were selected based on the applicability condition of a bar-type calculation scheme for a centrally compressed reinforced concrete member. The geometric characteristics are shown in Fig. 1. a b c d e Figure 1. Geometric characteristics of the square (a), cross-shaped (b), T-shaped (c), L-shaped (d), and rectangular (e) cross-sections. Table 1. Strength and deformation characteristics of reinforcing steel No. Rebar grade Diameter, mm Yield strength fy, MPa Ultimate strength fu, MPa Ultimate strain components Compression, εsc2·103 Tension, εs2·103 1 A500 10 500 600 –25 25 2 A240 6 240 380 –25 25   Table 2. Mechanical and deformation properties of concrete No. Concrete grade Prism strength fc,pr, MPa Initial elastic modulus E0, MPa Ultimate strain components Poisson’s ratio, µ Compression, εc2·103 Tension, εt2·105 1 В25 19.4 26000 –2.6 15 0.2   The modelling of reinforced concrete columns was performed using the licensed version of the LIRA-SAPR 2022 R1.1 software package. The software package has a verification certificate issued by the Russian Academy of Architecture and Construction Sciences (RAACS)[1]. The computational model consists of solid finite elements (type 236) with dimensions of 50×50 mm used to model the concrete, and bar elements (type 204) used to model the reinforcing bars. Material nonlinearity was taken into account through a piecewise-linear stress-strain law. The concrete stress-strain diagrams were adopted in accordance with the experimental data. The behavior of reinforcement was described using a bilinear stress-strain diagram, which is widely used in FEM of reinforced concrete members [3, 7, 15, 25]. The mechanical properties of concrete and reinforcing steel used in the analysis are summarized in Table 1 and Table 2, respectively. The columns were loaded incrementally, in steps of 10 % of the ultimate load. Based on the condition that the concrete and reinforcement simultaneously reach their respective ultimate stress levels, the ultimate load was determined as:                                                            (1) where  is the prism compressive strength of concrete,  is the yield strength of the reinforcing steel,  and  are the areas of concrete and longitudinal reinforcement, respectively. A subsequent analysis was performed to determine the theoretical ultimate load for each cross-sectional geometry, based on the value obtained from equation (1). The final theoretical ultimate load was taken as the magnitude of the external force at which the stresses in the concrete 3D finite elements reached the prism compressive strength, or the stresses in the bar elements of the reinforcement reached the yield strength. The theoretical ultimate load was defined as the value of the applied load at which the stresses in the concrete solid elements reached the prism compressive strength (in regions of biaxial and triaxial compression, according to the failure criterion of K.J. Willam and E.P. Warnke [18, 24]), while in the reinforcement bar elements the stresses reached the yield strength of steel. Once the specified limit states were attained in any finite element, that element was excluded from the structural resistance by artificially reducing its elastic modulus by a factor of 1000. If, at this reduced stiffness, the stresses in the adjacent elements did not exceed their ultimate values after the redistribution of internal forces, the failure was considered local and not significantly affecting the overall load-bearing capacity of the member. In the computational model, the bottom nodes of the column were fully restrained in the X, Y, and Z directions, providing a fixed support, while the top nodes were restrained only in the X and Y directions, allowing free vertical displacement. 3.Results and Discussion As a result of the analysis of the numerical simulations of centrally compressed columns with different cross-section geometries, theoretical curves based on the nonlinear stress-strain laws of concrete were generated, as shown in Fig. 2.   |s3| ei×103     square section*   square section**   cross-shaped section*   cross-shaped section**   T-shaped section*   T-shaped section**   – L-shaped section*   – L-shaped section**   – rectangular section*   – rectangular section** * theoretical concrete stress–strain curves obtained from the linear analysis; ** theoretical concrete stress–strain curves obtained from the nonlinear analysis. Figure 2. Theoretical curves based on the nonlinear stress-strain laws. Based on Fig. 2, it can be concluded that the cross-sectional shape of a centrally compressed column does not have a significant effect on its strength or deformation characteristics. The difference between the curves obtained using the nonlinear stress-strain laws of concrete is less than 3 %. The obtained results are comparable with those reported in [20–22]: the increase in load-bearing capacity for the cross-shaped section was approximately 1 %, whereas for the T-shaped, L-shaped, and rectangular sections, the reduction was 2 % compared with the square column. This comparison confirms that the influence of the cross-sectional shape on the load-bearing capacity of columns is insignificant when the slenderness ratio does not exceed 40. It has been established that, at a slenderness ratio greater than 40, the influence of the cross-sectional shape on the load-bearing capacity becomes significant, and the increase in strength may reach 12 % [20, 21]. However, failure of such columns is generally associated with loss of stability, whereas, in the case of failure governed by the material load-bearing capacity, the influence of the cross-sectional shape on the ultimate load remains insignificant. Based on this distinction, further research should include a comparative analysis of eccentrically compressed columns with different cross-sectional shapes. A comparison of the results (Fig. 2) shows that up to 30 % of the ultimate load, the concrete deforms linearly; therefore, the linear and nonlinear analyses yield nearly identical values, which is consistent with the conventional understanding of the initial elastic stage of material behavior. However, beyond this threshold, the curves begin to diverge: the nonlinear model captures the gradual reduction in stiffness, whereas the linear model preserves a constant modulus of elasticity. As a result, in the linear analysis the prism strength is reached at approximately 80 % of the ultimate load, while in the nonlinear analysis the concrete reaches its prism strength at about 95–100 % of the ultimate load calculated according to equation (1). In the nonlinear analysis, the reinforcement reaches a stress level of 87–90 %, whereas in the linear analysis it reaches only 27–30 %. This comparison confirms that the nonlinear formulation more accurately reflects the behavior of a reinforced concrete element under high load levels. The values of the ratio  were also calculated for each solid finite element according to the Willam–Warnke [24] failure criterion. Based on these results, contour plots of the  ratio were generated at the mid-depth of each column cross-section (Fig. 3), as well as contour plots of the principal stress ratios  at elevation +1.650 m, corresponding to the mid-height of the columns (Fig. 4). The main analyses were performed with a Poisson’s ratio of  = 0.2. To assess the possibility of developing lateral confinement due to the increased transverse deformability of concrete in the pre-failure stage, an additional analysis was performed with  = 0.4. The adoption of this value of Poisson's ratio makes it possible to parametrically account for the dilatancy effects characteristic of concrete at high levels of compressive stress [31, 32]. a b c d e Figure 3. I – segment of the analyzed specimen; II – contour plot of the s3/fc,pr ratio for μ = 0.2; III – contour plot of the s3/fc,pr ratio for μ = 0.4 (square column (a); cross-shaped section column (b); T-shaped column (c); L-shaped column (d); rectangular column (e)). a b c d e Figure 4. Contour plot of the σ₃:σ₂:σ₁ ratios in the cross-section at elevation +1.650 (rectangular column (a); cross-shaped section column (b); T-shaped column (c); L-shaped column (d); rectangular column (e)).             An analysis of the results obtained for  = 0.2 reveals the following patterns. Within the support zones, regardless of the cross-section geometry, regions of triaxial compression are observed, where the  ratio exceeds 3 over a height of 50–100 mm. Further upward, within the range of 100–200 mm, the  ratio decreases to 1 (uniaxial compression) for the cross-shaped section. In the remaining sections, local zones of compression accompanied by transverse tension occur between 500 and 1150 mm, where the  ratio falls within the range from 0.99 to slightly below 0.91. In the square and cross-shaped sections, localized regions of triaxial stress are observed along the entire column height, with  values ranging from 1.09 to slightly below 1.11. Accounting for dilatational effects significantly increases the horizontal stresses in the concrete. The region of triaxial compression, where the  ratio exceeds 3, expands to 100–150 mm. Further along the height, a layered variation of the  ratios is observed, ranging from compression-compression, with  = 1.5, to compression-tension, with  dropping below 0.91. The compression-compression zones are governed by the confinement effect provided by the transverse reinforcement. The increase in horizontal stresses when concrete dilatancy is taken into account is in qualitative agreement with the results reported in [15, 33], where the increase in transverse deformations of concrete is considered one of the factors contributing to the development of lateral confinement stresses in the presence of elements restraining transverse expansion. At the same time, the obtained results show that the cross-sectional shape alone does not provide stable lateral confinement of concrete comparable to that produced by external jackets or indirect reinforcement. Increasing Poisson’s ratio to  = 0.4 mainly led to the formation of a compression-tension-tension stress state over a significant part of the cross-section, whereas the compression-compression-compression zones remained local. Thus, the cross-sectional shape affects the redistribution and localization of principal stresses but does not create a stable lateral confinement effect characteristic of special confinement systems. It can also be observed that in the T- and L-shaped cross-sections, the triaxial stress zones in the support regions are more pronounced. This phenomenon is attributed to the more massive flanges in these geometries, which enhance the restraining effect on lateral deformations in the central part of the cross-section. 4.Conclusions The analysis of existing experimental and numerical studies has shown that the confinement effect in concrete is achieved by providing external restraining reinforcement, such as concrete-filled tubular shells, steel jackets, or indirect reinforcement systems. The use of these methods increases the load-bearing capacity of compressed members, reduces the risk of brittle failure, and thereby improves the performance of concrete under a complex stress state. Investigation of the influence of the 3D stress-strain state in concentrically and eccentrically compressed members, including the effect of external restraining reinforcement, such as concrete-filled tubular shells, steel jackets, and indirect reinforcement systems, is therefore of significant scientific and practical interest. The study of the 3D stress state of concentrically compressed members carried out by the authors shows that the cross-sectional shape of a concentrically compressed column has almost no effect on the strength and deformation properties of the structure, since the differences in the values of the investigated parameters did not exceed 3 %. At the same time, the cross-sectional shape has a significant influence on the nature and size of the zone of the 3D stress state in the support regions. With an increase in the length of the web and flanges in T- and L-shaped sections, the restraining effect on horizontal deformations becomes more pronounced, which leads to an expansion of the 3D confinement zone and an increase in the level of triaxial compression in the concrete. Accounting for dilatancy effects when determining the parameters of the 3D stress state of concentrically compressed members showed a significant increase in horizontal stresses and an increase in the height of the 3D stress-state zone up to 100–150 mm. In addition, a layer-by-layer alternation of zones characterized by compression-compression and compression-tension stress states was observed, indicating a complex spatial stress distribution. By contrast, the use of the transverse strain coefficient  = 0.2, including at load levels close to the ultimate load, results in the 3D confinement effect appearing only in the support regions of the columns. In the remaining part of the section, zones of uniaxial compression and local tensile regions predominate, which indicates a relatively uniform distribution of compressive stresses in the middle part of the member height. Extending the scope of research on the influence of geometric parameters on the formation and development patterns of the 3D stress-strain state from concentrically compressed members to eccentrically compressed members is of significant scientific and practical interest.   [1] Rossiyskaya akademiya arkhitektury i stroitelnykh nauk. Svidetelstvo o verifikatsii programmnogo sredstva LIRA-SAPR No. 06/LIRA-SAPR/2015 [Verification certificate for the LIRA-SAPR software No. 06/LIRA-SAPR/2015] [Online]. URL: https://rflira.ru/files/verify/Verif_LIRA-SAPR_Sign.pdf?ysclid=mu440t1v4u582410961 (accessed 16.09.2026) 10.34910/MCE.144.1 624.012.45 reinforced concrete column cross-section shape centrally compressed column finite element method nonlinear concrete behavior https://engstroy.spbstu.ru/article/2026.144.1/

RAR RUS 14402-14402 Gusev Kirill gusev_kirill@mail.ru 56091980300 0000-0003-3850-424X Peter the Great St. Petersburg Polytechnic University Lalin Vladimir vllalin@yandex.ru

29 Politechnicheskaya St., St. Petersburg, 195251, Russia

57194431559 0000-0003-4992-2084 Peter the Great Saint Petersburg Polytechnic University Galyamichev Alexander galyamichev@yandex.ru

Polytechnicheskay, 29

0000-0003-4901-4967 Istanbul Okan University Dogru Selcuk seltrue@hotmail.com

Istanbul, Turkey

Experimental and numerical investigation into the properties of silicone bonded connections in façade structures The object of the study is a structural silicone adhesive sealant. The aim of the work is to investigate the stress-strain state (SSS) of a bonded connection in hinged façade systems (HFS) using numerical analysis (finite element method, FEM) based on the results of experimental research. Method. During the study, experimental tests of a silicone adhesive sealant in combination with glass and aluminum were carried out. Based on the results of the experimental investigation, the favorable FE mesh refinement of the bonded connection into 3D FEs was selected for subsequent numerical calculations of HFS with adhesive-type cladding fixation. In addition, the suitable strength theory for evaluating the bonded connection according to the first group of limit states was identified based on the results of numerical modeling. Results. To determine displacements in HFS structures with bonded connections, the favorable FE mesh refinement in terms of the obtained results and the complexity of the computational model is an 8 × 8 FE mesh (FE size across the adhesive width not exceeding 2.75 mm; FE size through the adhesive thickness not exceeding 1 mm). With an 8 × 8 FE mesh refinement of the bonded connection, the stresses according to the first strength theory are significantly higher than the engineering stresses obtained from experimental studies. From the standpoint of stress evaluation in the bonded connection with an 8 × 8 FE mesh refinement, the fourth strength theory is appropriate. 1.Introduction Bonded connections are widely used in various industrial sectors, including mechanical engineering, aeronautics, and aerospace engineering [1–3]. In these fields, they are characterized by high structural efficiency and a broad range of applications. In civil engineering, bonded connections are also increasingly employed, and research aimed at developing approaches to their application is actively ongoing [4]. It should be noted that bonded connections often function as load-bearing elements; therefore, accurate determination of adhesive material properties and the development of reliable calculation methods are required. A significant number of studies reported in the scientific literature are devoted to the evaluation of the mechanical properties of bonded connections and the standardization of methods for their determination [5–8]. In addition, numerous investigations have examined the behavior of bonded connections used in timber and metallic structural systems [9–11]. In addition to the strength criterion, adhesion plays an important role in the bonding process. Papers [12, 13] present the results of tests of various adhesive joints using different types of adhesives. The study [14] notes that an MS polymer-based adhesive sealant exhibits high adhesion to a wide range of construction materials. Furthermore, the quality of adhesion is significantly influenced by the use of surface pretreatment agents applied prior to bonding [15]. Study [16] demonstrates that, in order to form a durable adhesive joint, mechanical surface treatment in combination with chemical treatment is required. Analytical evaluation of the stress-strain state (SSS) of bonded connections often encounters complexities due to the absence of exact analytical solutions [17, 18]. In this case, it is advisable to use numerical calculation methods. In references [19–21], numerical simulations of bonded connections were performed using the finite element method (FEM) implemented in software suites. In [22], the authors employed numerical modeling of adhesive joints and obtained results that are in good agreement with experimental data. It should be emphasized that the widespread implementation of bonded connections is constrained by a scarcity of comprehensive data regarding factors influencing their strength and long-term durability. Publications [23–26] explore the impact of hygrothermal environmental conditions and other exogenous factors on the mechanical performance of bonded connections. Issues related to the study of bonded connections are also actively addressed by researchers in other countries. In most of these works, the authors primarily focus on experimental investigations of bonded connections [27, 28]. In [29], experimental tests of bonded connections under shear loading showed that the type and thickness of the adhesive significantly affect the joint strength: increasing the plasticity and thickness of the adhesive layer increases the strength but reduces the stiffness of the joint. In [30], based on experimental tests, it was found that the overlap length, the thickness of the bonded material, and the overlap geometry (for example, wavy patterns or stepped configurations) significantly influence the fatigue performance of the joint, while some other parameters, such as fillets, have a negligible effect. It is also important to note that the results of experimental studies confirm the significant influence of cyclic loading on the durability of bonded connections [31]. In this study, the object of investigation is a structural silicone adhesive sealant. Applications of such bonded connections in hinged façade systems (HFS) have been previously investigated by domestic researchers. Reference [32] evaluates a calculation method for load-bearing bonded joints based on international regulatory frameworks [33]. In [34], a numerical study of silicone adhesive sealant, which is used in façade structures, was carried out. The implementation of bonded connections in HFS offers a superior alternative to traditional mechanical fastening, as it preserves the integrity of metallic components by eliminating the need for perforations or welding. Based on the literature review, it is evident that a substantial knowledge base exists regarding the numerical modeling and calculation of bonded connections across multiple industrial fields. However, the specific behavior of bonded connections employing structural adhesive sealants within HFS has not been comprehensively investigated. Despite the similarity of the principles of calculating adhesive joints, the structural adhesive sealant used in façade structures has its own characteristics that must be taken into account in analytical and numerical calculation methods. The present research aimed to investigate SSS of bonded connections in façade structures using FEM based on experimental results. To achieve this aim, the following tasks were defined: Conducting experimental tensile tests of a silicone adhesive sealant on specimens of two types. Determining a rational discretization scheme of the bonded connection into 3D FEs in a software package implementing FEM, based on the results of the experimental study (Type 1 specimen). Substantiating the selection of an appropriate strength theory for evaluating the bonded connection used as a fastening element of the cladding to the façade substructure according to the first group of limit states. Investigating the influence of considering geometric nonlinearity in determining the SSS of the bonded connection. Validating the developed numerical computational model by comparison with the results of experimental tests (Type 2 specimen). The object of the study is a structural adhesive sealant. The subject of the research is the SSS of bonded connections in HFS. 2.Materials and Methods Initially, the investigation of the bonded connection based on structural silicone was conducted through experimental testing. The specimens shown in Figs. 1 and 2 were subjected to testing. A glass element measuring 92 × 50 mm with a thickness of 8 mm was bonded to two aluminum tubes welded together (50 × 50 × 50 mm, wall thickness 2 mm) using a silicone adhesive sealant. For Type 1 specimens (5 units), the adhesive joint was located at the center of the glass plate and the aluminum tube; the cross-sectional dimensions of the adhesive layer were 22 × 7 mm. For Type 2 specimens (5 units), bonding was performed at the edge of the aluminum tube; the cross-sectional dimensions of the adhesive layer were 21 × 8 mm. Due to the limited number of specimens, the obtained results should be considered preliminary and are mainly intended for comparison with the numerical analysis. Figure 1. Specimen for experimental testing, Type 1. Figure 2. Specimen for experimental testing, Type 2. All test specimens were subjected to initial conditioning for at least 28 days after fabrication at a temperature of 23 ± 2 °C and a relative humidity of 50 ± 5 %. The experimental study of the bonded connection was carried out using Zwick/Roell Z100 universal testing machine (Fig. 3) with maximum load capacity of 100 kN, manufactured by Zwick Roell AG (Ulm, Germany). During testing, the tensile loading rate of the specimens was maintained at 5 mm/min. Figure 3. Zwick/Roell Z100 universal testing machine. The subsequent stage of the study was devoted to the numerical modeling of the test specimens, including the determination of the appropriate FE mesh size for the silicone adhesive sealant and the selection of a suitable strength theory for evaluating the material according to the first group of limit states within the HFS. To verify the static analysis of the bonded connection within a HFS, numerical models of the test specimens were developed using the SCAD Office software[1]. The bonded connection was modeled using volumetric FEs (FE Type 36). The connection between the glass, silicone adhesive sealant, and the aluminum tube in the computational model was implemented through shared FE nodes. Examples of the numerical model in SCAD Office with an 8 × 8 FE mesh partition for the bonded connection for test specimens of Type 1 and Type 2 are demonstrated in Fig. 4. Figure 4. Numerical models of the test specimens in SCAD Office: Type 1 specimen (left); Type 2 specimen (right). The load application schemes using metal fixtures in SCAD Office are shown in Fig. 5. Figure 5. Schemes of applying the experimental load in SCAD Office: Type 1 specimen (left); Type 2 specimen (right). In the numerical models, support constraints in the X, Y, and Z directions were assigned at the locations where the test specimens were fixed in the universal testing machine (Fig. 6). Figure 6. Assignment of support constraints in SCAD Office: Type 1 specimen (left); Type 2 specimen (right). The bonding system consists of a two-component, high-modulus, neutral-cure structural silicone. The technical characteristics of the silicone bonded connection are as follows: Design tensile strength: Modulus of elasticity in tension or compression: Shear modulus: It is important to note that, according to regulatory documentation [33], the adhesive sealant used must exhibit a cohesive failure mode under both tensile and shear loading. In the case of adhesive failure, the use of this adhesive sealant as a fastening element for cladding in façade structures is not permitted. The numerical models of the Type 1 specimen and the static analyses were performed with a gradual refinement of the FE mesh in the bonded connection region. After determining appropriate modeling conditions based on the experimental tests of the Type 1 specimen, the obtained parameters were validated using the numerical model of the Type 2 test specimen. First, a series of static analyses of the Type 1 test specimen was performed with successive refinement of the FE mesh along the thickness and width of the bonded connection by a factor of two (Fig. 7). Six static analyses were carried out: 1 × 1 FE (22 × 7 mm); 2 × 2 FE (11 × 3.5 mm); 4 × 4 FE (5.5 × 1.75 mm); 8 × 8 FE (2.75 × 0.875 mm); 16 × 16 FE (1.375 × 0.438 mm); 32 × 32 FE (0.688 × 0.219 mm). The values in parentheses indicate the dimensions of a single solid FE along the width and thickness of the bonded connection, respectively. 1 × 1 FE 2 × 2 FE 4 × 4 FE 8 × 8 FE 16 × 16 FE 32 × 32 FE Figure 7. FE mesh for the bonded connection. For the purpose of improving the numerical models of the HFS involving bonded connection modeling, reduce the total number of FEs, and consequently decrease the computational time for static analyses, an additional series of numerical calculations was carried out. In these cases, the FE mesh of the bonded connection was refined only through the material thickness, while the number of elements along the width remained constant. An example of mesh refinement through the thickness with a 4-element subdivision along the width is shown in Fig. 8.                   4 × 1 FE 4 × 2 FE 4 × 4 FE 4 × 8 FE Figure 8. FE mesh refinement through the thickness of the bonded connection. Similarly, numerical models with various FE subdivisions of the bonded connection were developed and analyzed. Ultimately, static calculations were performed for the following partitions: 2 FE: 2 × 1 FE (11 × 7 mm); 2 × 2 FE (11 × 3.5 mm); 2 × 4 FE (11 × 1.75 mm); 2 × 8 FE (11 × 0.875 mm); 2 × 16 FE (11 × 0.438 mm); 2 × 32 FE (11 × 0.219 mm); 4 FE: 4 × 1 FE (5.5 × 7 mm); 4 × 2 FE (5.5 × 3.5 mm); 4 × 4 FE (5.5 × 1.75 mm); 4 × 8 FE (5.5 × 0.875 mm); 4 × 16 FE (5.5 × 0.438 mm); 4 × 32 FE (5.5 × 0.219 mm); 8 FE: 8 × 1 FE (2.75 × 7 mm); 8 × 2 FE (2.75 × 3.5 mm); 8 × 4 FE (2.75 × 1.75 mm); 8 × 8 FE (2.75 × 0.875 mm); 8 × 16 FE (2.75 × 0.438 mm); 8 × 32 FE (2.75 × 0.219 mm). The values in parentheses indicate the dimensions of a single solid FE along the width and thickness of the bonded connection, respectively. In order to study the influence of geometric nonlinearity on the SSS of the bonded connection, static nonlinear analyses were performed in SCAD Office with the following FE discretizations of the bonded connection: 4 × 8 FE (5.5 × 0.875 mm); 8 × 8 FE (2.75 × 0.875 mm); 32 × 32 FE (0.688 × 0.219 mm). The values in parentheses indicate the dimensions of a single solid FE along the width and thickness of the bonded connection, respectively. The bonded connection incorporating geometric nonlinearity was modeled using volumetric FEs (FE Type 336). The nonlinear analysis was performed using a simple incremental method until the load reached 300 N with a load step of 10 N (a total of 30 increments). 3.Results and Discussion Experimental Results During the experimental study, the cross-sectional area of the bonded connection, the breaking load, and the test duration were determined (Tables 1 and 2).     Table 1. Test results (Type 1 specimen). No. S0 (mm2) Fmax (MPa) t (sec) 1 1124.98 0.970 309.62 2 1088.82 1.170 285.47 3 1080.14 0.764 255.39 4 1278.24 0.911 327.21 5 1145.24 1.100 223.53   Table 2. Test results (Type 2 specimen). No. S0 (mm2) Fmax (MPa) t (sec) 1 1008.10 1.169 152.24 2 1150.82 1.147 238.7 3 1117.70 1.074 326.49 4 1069.88 1.168 311.79 5 1029.62 1.158 305.8   As a result of the conducted tests, diagrams showing the dependence of displacements and conditional stresses on the applied load were also obtained for the Type 1 test specimen (Figs. 9 and 10). Figure 9. Dependence of displacements on applied load (Type 1 specimen). Figure 10. Dependence of engineering stresses on applied load (Type 1 specimen). Similar dependencies were obtained for the Type 2 test specimen (Figs. 11 and 12). Figure 11. Dependence of displacements on applied load (Type 2 specimen). Figure 12. Dependence of engineering stresses on applied load (Type 2 specimen). Engineering stresses  in the bonded connection, presented in the diagrams, were determined according to the following formula:                                                                         (1) where  is the tensile load acting on the test specimen;  is the cross-sectional area of the bonded connection, oriented orthogonally to the load application vector. According to ETAG 002, the stresses occurring in the center of the longest bonded joint  can be determined by the following formula:                                                                     (2) where  is the peak wind pressure acting on the cladding element;  is the length of the short side of the cladding element;  is the width of the bonded joint. Based on (2), the stresses in the longest bonded joint can be determined using the following formula:                                                                     (3) where  is the tributary width of wind load. The total tensile force acting on the bonded connection can be determined by the following formula:                                                                    (4) where  is the length of the bonded joint. Thus, by expressing the peak wind pressure from (4) and substituting it into (3), the following expression is obtained:                                                             (5) Therefore, the determination of engineering stresses in the bonded connection using (1) is consistent with the stress determination method according to ETAG 002 using (2). The deformations of the bonded connection under the action of the tensile load in the testing machine are shown in Fig. 13. Figure 13. Deformations of the bonded connection (Type 1 specimen). Tensile testing of the specimens was conducted until failure of the bonded connection. Cohesive failure occurred in all tests. The failed specimen is shown in Fig. 14. Figure 14. Failed specimen (Type 1 specimen). Results of Numerical Investigation with FE Mesh Refinement through the Thickness and Width of the Bonded Connection An analysis of the relationships presented in Figs. 10 and 12 shows that the conditional stresses arising in the bonded connection reach the design resistance value at a test load level of up to 200 N. Therefore, the assessment of the displacements of the test specimen and the stresses arising in the bonded connection, based on the calculation results obtained using the SCAD Office software package, was performed within the load range from 0 to 200 N. During the numerical analysis, the displacements of the test specimen and the stresses according to the 1st, 3rd, and 4th strength theories were evaluated. The results of the numerical calculations performed in SCAD Office with a sequential refinement of the FE mesh along the thickness and width of the bonded connection are presented in Table 3. Since the analysis in SCAD Office was carried out within a linear formulation, a single calculation point is sufficient to construct the dependency diagram. Table 3. Results of calculations in SCAD Office. FE mesh Load (N) Displacements (mm) Stresses 1st strength theory (MPa) Stresses 3rd strength theory (MPa) Stresses 4th strength theory (MPa) 1 ´ 1 0 0 0 0 0 200 0.13 0.175 0.007 0.007 2 ´ 2 0 0 0 0 0 200 0.23 0.197 0.042 0.036 4 ´ 4 0 0 0 0 0 200 0.34 0.237 0.12 0.104 8 ´ 8 0 0 0 0 0 200 0.39 0.258 0.198 0.172 16 ´ 16 0 0 0 0 0 200 0.41 0.307 0.261 0.226 32 ´ 32 0 0 0 0 0 200 0.42 0.407 0.322 0.279   A comparison of the results of numerical modeling and experimental studies is presented in the diagrams shown in Figs. 15–18. The results of the experimental study were averaged and presented as a single generalized graph. Figure 15. Dependence of displacements on applied load. Figure 16. Dependence of maximum principal stresses on the applied load.     Figure 17. Dependence of equivalent stresses (3rd strength theory) on the applied load. Figure 18. Dependence of equivalent stresses (4th strength theory) on the applied load. According to the results presented in Table 3, the displacement values in the computational model with mesh divisions of 16 × 16 and 32 × 32 FEs differ by: Thus, further refinement of the FE mesh was not performed, since the increase in displacements in the computational models did not exceed 5 %. At the same time, further reduction in the size of FEs could increase the probability of numerical errors in stress determination associated with the application of the FEM. The following conclusions can be drawn from the obtained diagrams: For determining displacements within the considered load range, an appropriate mesh, considering both the obtained results and the computational model complexity, is the 8 × 8 FE discretization (the difference in displacements does not exceed 7 %). With an 8 × 8 FE discretization of the bonded connection, the stresses determined according to the 1st strength theory are significantly higher (by 47 %) than the nominal stresses obtained from the experimental study. The stress results obtained according to the 3rd and 4th strength theories with an 8 × 8 FE discretization show the closest agreement with the nominal stresses obtained experimentally. The difference is 13 % according to the 3rd strength theory and 2 % according to the 4th strength theory. From the standpoint of evaluating the stress state of the bonded connection with an 8 × 8 FE discretization, the 4th strength theory is appropriate. Another study [34] also notes that the 4th strength theory allows the most accurate assessment of sealant adhesive failure.     Numerical Study Results for FE Mesh Refinement Across the Thickness of the Bonded Connection The results of numerical calculations in SCAD Office software with sequential FE mesh refinement across the thickness of the bonded connection and their comparison with the experimental study results are presented in the diagrams. The results of the experimental study were averaged and presented as a single graph. Partitioning of the bonded connection into 2 FEs across the width and FE mesh refinement across the thickness (Figs. 19–22). Figure 19. Dependence of displacements on applied load. Figure 20. Dependence of maximum principal stresses on the applied load. Figure 21. Dependence of equivalent stresses (3rd strength theory) on the applied load.     Figure 22. Dependence of equivalent stresses (4th strength theory) on the applied load. Partitioning of the bonded connection into 4 FEs across the width and FE mesh refinement across the thickness (Figs. 23–26). Figure 23. Dependence of displacements on applied load. Figure 24. Dependence of maximum principal stresses on the applied load.     Figure 25. Dependence of equivalent stresses (3rd strength theory) on the applied load. Figure 26. Dependence of equivalent stresses (4th strength theory) on the applied load. Partitioning of the bonded connection into 8 FEs across the width and FE mesh refinement across the thickness (Figs. 27–30). Figure 27. Dependence of displacements on applied load.     Figure 28. Dependence of maximum principal stresses on the applied load. Figure 29. Dependence of equivalent stresses (3rd strength theory) on the applied load. Figure 30. Dependence of equivalent stresses (4th strength theory) on the applied load. The presented diagrams clearly demonstrate the results, on the basis of which the following conclusions can be formulated: To refine the static analysis of HFS with bonded connections, FE mesh refinement across the thickness of the bonded connection is of primary importance. To simplify the numerical model of the HFS when modeling the bonded connection and determining displacements, FE meshes of 2 × 8 and 4 × 8 may be used. The displacement values obtained using these meshes differ from the experimental data by no more than 5 %. However, the application of the 4th strength theory for evaluating stresses in the bonded connection with these mesh configurations is inappropriate, since the calculated stresses are 24–83 % lower than the values obtained from the experimental study. Results of Numerical Investigation Considering Geometric Nonlinearity The results of nonlinear calculations in the SCAD Office software suite, considering the geometric nonlinearity of the bonded connection’s volumetric FEs, are summarized in the diagrams (Figs. 31 and 32). Figure 31. Dependence of displacements on applied load. Figure 32. Dependence of equivalent stresses (4th strength theory) on the applied load. The following conclusions can be drawn from the obtained diagrams: Taking geometric nonlinearity into account, the displacement results obtained from experimental studies and numerical simulations show good agreement provided that the bonded connection is discretized with a mesh no finer than 8 × 8 FEs and the applied load does not exceed 50 N. With further mesh refinement (up to 32 × 32 FEs), the displacement values remain consistent with the experimental data up to a load of 105 N. At higher load levels, the numerical model with the bonded connection and geometric nonlinearity demonstrates a significant increase in stiffness, which does not correspond to the experimental results. Stress analysis according to the 4th strength theory for a bonded connection modeled with geometrically nonlinear FEs leads to significant discrepancies (from 24 to 60 %) compared with experimentally obtained stress values. A mesh of 8 × 8 FEs provides a conservative stress assessment with a safety margin of approximately 33 %. These factors indicate the need to refine the numerical model by accounting for the physical nonlinearity of the material. In modeling the HFS with geometric nonlinearity taken into account, the bonded connection should be represented using geometrically linear 3D FEs. Otherwise, the stiffness of the HFS numerical model will be significantly overestimated. Validation of the Obtained Modeling Conditions for the Silicone Adhesive Sealant In order to verify the validity of the previously established modeling conditions for the silicone adhesive sealant, a validation study was performed based on the results of tests of Type 2 specimen, which is characterized by a different bonding configuration and different cross-sectional dimensions of the bonded connection. The validation was carried out through the analysis of load–displacement diagrams obtained during the experimental study (Figs. 11 and 12). The silicone adhesive sealant in the numerical model of Type 2 specimen was simulated using 3D FEs (FE Type 36) with the bonded connection discretized by an 8 × 8 FE mesh. The FE size along the width of the bonded connection was 2.625 mm, while the element size through the thickness was 1 mm. A comparison of the displacement values obtained from the numerical simulation and the experimental study is presented in Figs. 33 and 34. Figure 33. Dependence of displacements on applied load (Type 2 specimen). Figure 34. Dependence of equivalent stresses (4th strength theory) on the applied load (Type 2 specimen). Based on the interpretation of the obtained diagrams, the following conclusions can be drawn: Discretization of the adhesive joint using an 8 × 8 mesh of linear 3D FEs in numerical modeling provides displacement values that differ from the results of experimental studies by no more than 7 %. In the evaluation of the stress state of the silicone adhesive sealant according to the 4th strength theory using an 8 × 8 FE discretization, it was found that the obtained stress values differ from the results of experimental tests by approximately 4 %. 4.Conclusions The main conclusions and results of the conducted study are as follows: The experimental study was limited to two types of specimens and a small number of tests; therefore, the obtained results are preliminary. Further experimental studies with a larger number of specimens are required to obtain statistically reliable characteristics of the bonded connection. It is proposed to determine displacements in HFS with bonded connections using discretization of the adhesive layer into an 8 × 8 FE mesh (the FE size along the adhesive width should not exceed 2.75 mm; through the adhesive thickness, not more than 1 mm). The displacement values obtained using this discretization differ from the experimental results by no more than 7 %. It was established that when the bonded connection is discretized using an 8 × 8 FE mesh, the stresses determined according to the 1st strength theory are significantly higher (up to 47 %) than the nominal stresses obtained from experimental studies. This result indicates that stress analysis based on principal stresses leads to an excessive safety margin for the bonded connection. It is proposed that, when using the 8 × 8 FE discretization of the bonded connection, stresses should be evaluated according to the 4th strength theory. In this case, the deviation from the experimental results ranges from 2 to 4 %. It has been shown that, to improve the accuracy of static analyses of HFS with bonded connections, a significant factor is the refinement of the FE mesh of the adhesive layer through its thickness. To simplify the computational model of HFS with bonded connections and to determine displacements, discretization schemes of 2 × 8 FEs (FE size through the adhesive thickness not exceeding 1 mm and along the width not exceeding 11 mm) and 4 × 8 FEs (FE size through the adhesive thickness not exceeding 1 mm and along the width not exceeding 5.5 mm) may be used. These discretization schemes provide good agreement with experimentally obtained displacement values (up to 5 %). However, this modeling approach is not appropriate for evaluating the bonded connection according to the first group of limit states, since the calculated stresses differ from the experimental values by 24–83 %. Therefore, these discretization schemes are recommended only for preliminary static analyses of HFS with bonded connections. In analyses that take geometric nonlinearity into account, the results of experimental studies and numerical modeling are consistent only at the initial stage of loading (up to 105 N). In this case, the stress assessment in the bonded connection provides a safety margin of up to 33 %. As the load increases, the computational model with geometric nonlinearity demonstrates a significant increase in stiffness, which does not correspond to the experimental results. Thus, it has been established that in modeling HFS with consideration of geometric nonlinearity, the bonded connection should be represented by geometrically linear 3D FEs. Otherwise, the stiffness of the computational model of the HFS will be significantly overestimated. 10.34910/MCE.144.2 69.04 numerical modelling stress-strain state façade structures structural silicone adhesive-sealant tensile strength theory finite element method nonlinear static calculations bonded connections https://engstroy.spbstu.ru/article/2026.144.2/

RAR RUS 14403-14403 56315295500 0000-0001-5546-0950 Babol Noshirvani University of Technology Hasanzadeh Ali a_hasanzade64@yahoo.com

Babol, Mazandaran, Iran

0000-0002-0090-5745 ANO SAFAS Hematibahar Mohammad eng.m.hematibahar1994@gmail.com

Moscow, Russian Federation

M-6585-2013 6508103761 0000-0002-1196-8004 Peter the Great Saint Petersburg Polytechnic University Vatin Nikolai vatin@mail.ru

Polytechnicheskay, 29

Moscow State University of Civil Engineering Kharun Makhmud miharun@mail.ru

Moscow, Russian Federation

Effects of three types of concretes on the dynamic performance of helical shell structure through finite element method In this research, the dynamic performance of helical shell structure under earthquake loading is investigated using the ANSYS Workbench Finite Element Software. Three different types of concretes including high-performance concrete (HPC), basalt fiber reinforced high-performance concrete (HPC-12) and ordinary concrete (C50) were selected and the dynamic behavior of helical shell structures made by these concretes were analyzed through Response Spectrum (RSM) and Time-Domain History methods (THM), as two valuable approaches for dynamic analysis. The displacements of internal and external edges and also the displacement of the whole structure were determined by ANSYS. The results of RSM showed that C50 depicted the greatest internal and external displacements. The THM results indicated that C50 had the maximum displacement of external edge. However, the maximum displacement of internal edge in the structure was obtained for HPC-12. Furthermore, both types of analyses (RSM and THM) depicted that for all three types of concretes, the maximum displacement of external edge of helical shell structure has been more than the internal one. It was concluded that the mechanical properties of materials considerably affect the displacement of helical shell structure. 10.34910/MCE.144.3 691.32 helical shell structure dynamic performance finite element method time-domain history method response spectrum method https://engstroy.spbstu.ru/article/2026.144.3/

RAR RUS 14404-14404 16441812200 0000-0003-0560-5933 Kaliningrad State Technical University Naumov Vladimir van-old@mail.ru

Kaliningrad, Russian Federation

Optimization of the pipeline diameter of the water supply system This study addresses the main pipeline of a water supply system. The paper reviews existing methods for determining the optimal pipeline diameter with the aim of minimizing both capital investment and operating costs. The aim of the study is to improve the method for calculating the optimal diameter by incorporating the dependence of electric motor power consumption on pump performance. The input data comprise the performance, energy and cost characteristics of centrifugal pumps; the physical and cost characteristics of high-pressure polyethylene (HPPE) pipes; and electricity tariffs. As in the conventional approach, the dependence of the cost and hydraulic losses of HPPE pipes on internal diameter is established. The proposed method introduces three substantive improvements. First, the performance and energy characteristics of pumps are approximated by second-order polynomials. Second, the pipeline inner diameter is determined such that the required flow rate is achieved at the operating point for each pump installed. Third, the hypothesis of constant efficiency is no longer required: power consumption is calculated directly from the corresponding dependence provided in the technical documentation. It is shown that efficiency changes significantly as the design flow rate or pipeline length increases. Under the conditions considered, efficiency decreases from 0.793 to 0.648 as the water flow rate rises from 60 to 80 m3/h. The influence of various factors on the optimal diameter of the main pipeline is examined. The optimal diameter decreases from 0.175 to 0.166 m as the pipeline length increases from 1000 to 9000 m. It increases with a longer project lifecycle and decreases with a higher interest rate. 1.Introduction Water is essential for meeting both present and future human needs. However, its uneven distribution in time and space remains the principal barrier to reliable access [1, 2]. The importance of water supply systems increases as infrastructure becomes more complex. Therefore, a substantial body of research has addressed the challenges of improving these systems. Given the breadth of this literature, no single article can cover it in full, and specialized reviews have consequently emerged in particular subfields. A review of publications on the issues of public water supply and water scarcity is presented in [3]. It is indicated that fresh water availability in certain regions is minimal – in some cases insufficient even to meet basic population needs. The need for sustainable and reliable water supply is emphasized. To do this, a revision of water management system is proposed, particularly in water-scarce regions. General recommendations and possible solutions are discussed, such as advanced technological interventions that enhance water-use efficiency, the use of unconventional water resources, and others. Water supply networks are essential for delivering high-quality water in sufficient quantities to consumers. A water supply system comprises interconnected hydraulic components that ensure the supply and distribution of water in accordance with demand. Optimization of water distribution networks is carried out in order to minimize resource use and costs or to enhance system efficiency and realize greater performance gains. Genetic algorithms (GA) are widely used to solve problems of nonlinear optimization in water supply systems. In [5], a review of research on the optimization of water supply systems using GA methods is provided. These studies address the design of new water supply systems and the expansion, strengthening, and rehabilitation of existing infrastructure. A comprehensive analysis of all GA-based optimization studies spanning the period from the 1990s to 2021 is presented. Determining pipe diameters is the simplest application of GA. A set of pipes is selected from commercially available standard sizes so as to minimize construction costs for a given network. Within the GA framework, the network structure is defined as a “chromosome”, a binary or integer sequence of numbers. The first GA simulation program analyzes each investment project in terms of nodal load. The second GA simulation program analyzes the cost of construction. A reasonable choice of the objective (penalty) function enables GA to converge on an optimal solution. GA and related methods are based on the classical problem of determining the optimal economical diameter of a simple (unbranched) pipeline [6–9]. This problem was first solved in [10]. An increase in the internal pipeline diameter  on the one hand, raises construction cost through greater material consumption. On the other hand, a larger  reduces the hydraulic pressure loss in the pipeline, thereby lowering the operating cost of the pump motor. The capital cost of water supply is determined by the unit cost per pipe length:                                                               (1) where    are the empirical constants, the correction  takes into account valve cost, fittings and pipeline installation. The operating costs of  are defined as the product of the power consumed by the pumping unit during operation and the cost of 1 kWh of electricity. The power required for pumping is the product of the volumetric flow rate (pump supply) and the pressure drop, divided by the efficiency of the pumping unit. In [10], the coefficient of hydraulic losses along the pipe length was adopted as for new steel pipes under turbulent flow (the resistance region of hydraulically smooth pipes). Total costs reduced to one year are given by:                                                             (2) where  is a coefficient that takes depreciation into account. Equating the derivative of the  function to zero yields the diameter  at which this function reaches a minimum. The approach proposed in [10] proved highly fruitful: a large number of subsequent publications used and developed it. Modern research considers various special cases or accounts for factors that were previously overlooked. Many applications of the method [10] to the calculation of specific water supply systems have been published. In [11], for instance, two special cases of economic optimization of pipe diameter under turbulent flow are considered: hydraulically smooth and rough pipes. Different optimal pipe diameters were obtained for flow rate of 20 l/s: 155 mm for rough pipes and 147 mm for smooth pipes. The calculation of hydraulic losses in particular cases made it possible to simplify the Genereaux equation. Hydraulic losses in local resistances have been taken into account, but static pressure drop was not. The approach considered in [11] is therefore unsuitable when water must be raised to a certain height. Capital recovery factor is used to calculate the amount of each level payment to be made at the end of each  period, to recover the present amount at the end of the period, at a predetermined interest rate of  [12]. The factor is computed as follows:                                                      (3) A significant part of [12] focuses on the analysis of the dependence of pipe cost on diameter and on determining the empirical constants  and  in (1). Costs for pipes of various materials (two types of reinforced cement concrete, galvanized iron, and three types of polyvinyl chloride) with a diameter from 40 to 160 mm are curved. The index of  degree ranged from 0.61 to 1.53. In some of the graphs shown, the number of points is small and their scatter is large (e.g., PVC Grey Plain). This scatter is likely due to the fact that the author used price lists from various retail stores. Therefore, the representativeness of the sample is questionable. The solution of the set mathematical problem is postponed. Subsequently, the application of the Generaux method in combination with (3) to determine the economic diameter of the pipeline became known as Life-Cycle Cost Analysis (LCCA). In [13], twenty different sections of a water pipeline in four regions of Brazil were considered, with length ranging from 533 to 45000 m and flow rate from 0.558 to 800 l/s. The material of the pipes is high-pressure polyethylene (HPPE). The calculated optimal diameters of  ranged from 50 to 1000 mm across various sections. The authors compare three methods for calculating  In fact, only one LCCA method was used (with accepted values of  = 0.12 and  = 20), but with two models for pipe cost: the linear dependence of pipe cost on its weight; the linear dependence of pipe cost on inner diameter. The results obtained made it possible to determine the values of the empirical coefficient  in the approximate Bresse formula:                                                                    (4) Values of  ranging from 0.6 to 1.7 were selected for different sections of the water pipeline. Clearly, this approach is untenable, since the value of  depends on a large number of factors. In [14], an optimized selection of pipe diameter for a farm at Hamelmalo Agricultural College (HAC) in the Anseba Region of Eritrea was presented. The LCCA method was used to formulate the problem, and the head loss gradient method was employed to solve it. Another attempt to apply an approximate formula was made to determine the optimal pipeline diameter. In particular, Smith’s empirical formula was used:                                                                   (5) where  is the pipe diameter, mm;  is the system discharge, m3/h. The value of  is selected from a dedicated table. It can vary from 25 to 34 depending on the operating time of the pumping unit  (h/y) and friction losses. The value of  affects the annual cost of electricity. The annual pumping time was assumed to be 2920 hours, and  was determined to be 27.92 based on the analysis of hydraulic losses. It is clear that (5) is suitable only for preliminary, approximate estimates. In [15], the results of calculated values of  under different flow regimes are compared: laminar, transient, and turbulent. The calculations were implemented using media of different viscosities. The volumetric flow rates  were 0.111, 0.167 and 1.667 m3/s, respectively. The pipeline length in all cases was the same  = 4000 m. Depreciation charges were 18 rubles per meter of pipeline length and per meter of diameter per year; the electricity tariff was 3 rubles per kWh. The calculated pipe diameters were 0.311, 0.392, 0.419 m, respectively. Therefore, with high viscosity of the pumped liquid, energy costs increase substantially. The LCCA method was used in [16] to calculate the optimal diameter of the pipeline network on another farm located in the Anseba Region of Eritrea. The results for  were compared with those obtained using 11 approximate formulas from different authors, including (4) and (5). This comparison was based on calculated total pressure losses and an economic analysis of the pipe diameters of the network in question. The authors claim that the Bresse formula proved the most adequate compared with the LCCA model. The number of publications using specialized software in the field is increasing [17–19]. The WaterGEMS program is used in [17] to optimize the water supply system of Wukro Town in Ethiopia. Neural networks are used in [18] to determine the inner diameter of a drinking water supply system. These programs rely on the same LCCA method. However, specialized programs are not always used correctly. In [19], the optimal pipe diameter for transporting treated water in Karbala in Iraq using WaterCAD V8i software is determined. WaterCAD is used for designing and analyzing the operation of a water supply network. Options with four pipe materials (ductile iron, GRIP, concrete, and plastic) with diameters from 0.8 to 2.0 m are compared. Operating costs for pipes of different diameters and materials were given. GRP pipes had the highest operating costs. They are more than twice higher than those of concrete pipes. This is clearly an error. Operating costs are calculated based on electricity costs. The cost of electricity depends on the hydraulic losses along the pipeline. But GRP pipes have noticeably lower hydraulic losses than concrete pipes. The opposite was found in [19]. These works do not take into account that a decrease in diameter and an increase in hydraulic losses will require a more powerful pump. This means that the cost of the pump depends on the diameter. Moreover, this dependence is inverse. In [15], it was noted that the required pump power depends on the diameter. In particular, for fully developed turbulent flow in the pipeline, the required pump power depends inversely on  However, this fact is not taken into account in the costs. It was proposed to take into account not only the dependence of pipeline cost on the diameter, but also the cost of pumping station on the required capacity in the investment costs [20]:                                                                  (6) where  is the cost of one installed kW pump at its actual capacity  It was proposed to calculate  as hydraulic power divided by efficiency (  = 0.77). In [20], the speed of water movement through pipes (0.8 m/s) is recorded instead of flow rate. The examples given have not been brought to a decision. The analysis of (1) has been performed. The dependence of  on  is not defined in (6). Optimal diameters have not been found. A similar approach was used in [21]. It was noticed that the cost of a polyethylene pipeline is directly proportional to its weight. The mass was calculated using the formula:                                (7) where  is the ratio of the inner diameter of the pipe  to its wall thickness    is the density of polyethylene;  is the cost of 1 kg of HPPE. It is important that plastic pipes are selected from a line with the same ability to withstand a given pressure  which is determined by the  value. In [21], in order to evaluate investment projects of water supply systems, in addition to discounting (interest rate) according to (2), it was proposed to take inflation into account. But the method of accounting for inflation expectations is not yet generally accepted, unlike (2). The hypothesis about the connection between the pipeline diameter and the capacity of the pumping station has been put forward:                                                                    (8) where    are the empirical constants of the regression dependence. They should be selected when processing the results of multivariate hydraulic modeling for each water supply network. There are no specific examples and no solution to the problem in [21]. The approach proposed in [20] was brought to its logical conclusion in [22, 23]. It was found that the dependence of pump cost on pump power is close to a linear. In [22], the power in this function was additionally multiplied by the ratio  (the ratio of suction diameter to discharge diameter). The cost per pipe unit length, directly proportional to its volume  was taken from price lists. It is known that the volume of a cylindrical pipe can be found by the formula:                                                        (9) where  is the wall thickness of the pipe. But in [22], a formula like (9) does not contain a multiplier of 2 before the second term. Therefore, the calculation results cannot be considered reliable. The approach proposed in [23] formally differs from the one in [20, 22], it is called “a new concept for the economic pipeline design”. In fact, it was a transition from the independent variable  to the “economic design velocity” variable  according to the formula:                                                            (10) Using the example of a simple pipeline, a mathematical model was presented that allows relating various types of annual costs to  The relationship between the design velocity and various pipeline parameters was presented. The influence of the pipeline length, static pressure, discount coefficient, and flow rate on the value of  was shown. Three different markets were considered in order to estimate the annual cost of the various pipeline components: those of Lebanon, the USA and India. Two different types of pipe materials were considered: ductile iron pipes and low-pressure polyethylene. The economic velocities for low-pressure polyethylene were 1.75 m/s, 1.00 m/s and 1.95 m/s for Lebanon, the USA and India, respectively. This approach is unlikely to hold much promise for design purposes. Apparently recognizing this, the authors of [23] calculated the values of  using  at the end of their article. It is known that the power consumed by an electric pump unit  can be found by dividing the hydraulic power  by the efficiency                                          (11) where  is the density of water, kg/m3;  is the acceleration of gravity, m/s2;  is the pump head, m;  is the efficiency of the pump;  is the efficiency of the electric motor. All the articles listed above use (11). A common disadvantage of these articles is the assumption that efficiency can be considered constant (  = const). In fact, efficiency depends on the design and power of the pump. Efficiency is a function of flow rate for a particular pump. A method for calculating the optimal diameter of the pipeline using the pump performance characteristics was proposed in [24]. This made it possible to take into account the change in efficiency when switching to a different pipeline diameter. In addition, it was shown that the mass and cost of HPPE pipes should be calculated not according to (7), but according to the following formula:                                      (12)                    (13) The use of the dependence of the consumed power  of the CNS-38 sectional pumps on the flow rate required additional recalculation to account for changes in the efficiency of the electric motor. Currently, many pump manufacturers include in the technical documentation not only the dependence of  (often denoted as ) on  but also the dependence of the consumed power of the electric motor    This makes it possible to improve the calculation method proposed in [24]. The aim of this article is to improve the method of calculating the optimal diameter  of the pipeline of a water supply system using the energy characteristics of an electric pump unit (accounting for changes in efficiency). The tasks to be solved are as follows: development of an improved method for calculating the optimal diameter verification of the significance of the change in efficiency at the operating point of the pumping unit when calculating investigation of the influence of various factors on 2.Materials and Methods In addition to the water flow rate  the pipeline length  the static head  and the pipeline material HPPE, the cost of a pipe unit length, the cost and performance characteristics of pumps must be determined. The investment and operating costs of the pipeline of the water supply system are presented in [22–24]:                                 (14) where  is the investment cost with discount;  is a part of investment cost that does not depend on the pipeline diameter;  is annual operating cost;  is a part of operating cost that does not depend on the pipeline diameter. To find the optimal pipeline diameter, we equate the derivative (14) to zero:                                                         (15) In order to find  the operating costs, depending on the pipeline diameter, are calculated using the following formula:                                                         (16) where  is the cost of electricity, $/kWh;  is the work of the pump’s electric motor per year, kWh;  is the pump’s operating time per day, h;  is the power consumed by the electric motor, kW;  is the number of days per year. The consumed power of the electric motor was calculated using (11) in previous publications. One of the important improvements of the calculation method is that the value of  in this article is calculated based on the energy characteristics of the pumps used. Here we will use the performance characteristics of a line of vertical multistage pumps CR-64[1]. An example of the characteristics is shown in Fig. 1. The dependence of the head  and the electric motor consumed power  on the flow rate  can be approximated by second-order polynomials:                             (17) where            are the empirical coefficients that are determined by the least squares method based on experimental data for each pump. For example, their values for CR-64-4 are shown in Fig. 1:  = 118.3 m,  = 0.1663 m/(dm3/s),  = –0.0918 m/(dm3/s)2,  = 12.04 kW,  = = 0.6904 kW/(dm3/s),  = –0.00827 kW/(dm3/s)2. Figure. 1. Dependence of the CR-64-4 head and the electric motor consumed power on the flow rate. Points are experimental data[2]; lines are calculations using (17). The characteristics of the pipeline will be calculated as in [22–24]:             (18) where  is the static head, m;  is the length of the pipeline, m;  is the coefficient of hydraulic friction losses along the length of the pipeline;  is the total coefficient of head losses in local resistances;  is the velocity of water in the pipeline, m/s. The coefficient of hydraulic friction losses will be calculated using Altschul formula:                            (19) where  is the Reynolds number;  is the coefficient of kinematic viscosity of water, m2/s;  is the ratio of the absolute roughness of the pipe wall to the inner diameter. The analysis of experimental data in [24, 25] allowed for obtaining the dependence of this ratio on the diameter. For pipes of small and medium diameter (up to 340 mm) of Russian production, the  relationship is well approximated by a linear relationship:  = 8.09·10–5·  This means that in the specified range, the relative roughness of small and medium HPPE can be considered constant  ≈ 8.09·10–5. An accurate calculation is not possible in extended pipeline. Instead, we increase  by 10 %, as in previous works. Then substituting (19) into (18), we get the function:                  (20) Next, for each pump of the CR-64 line, we will find the diameter of the pipeline  at which the flow rate will be equal to the set value  (at the operating point of the pumping unit). These values are determined from the equation:                                                        (21) The pump head of the CR-64  line and the electric motor expended power  at the operating point can be calculated by substituting  into (17). Let us next find an approximation of the dependence of the consumed power on the values found:                                                             (22) where      are the empirical constants determined by the least squares method for the pipeline and pump line. Let us substitute (22) into (16) and differentiate:                                             (23) The cost of electricity  does not remain constant over time. It may vary markedly in different countries and regions. Here we will use the tariff values of Rusenergosbyt LLC for Russian regions in January 2024 for consumers of the first price category (power consumption up to 670 kW, voltage up to 0.4 kV)[3].  = 0.04776 $/(kW·h) in the Irkutsk region,  = 0.07794 $/(kW·h) in the Kostroma region,  = = 0.1158 $/(kW·h) in the Leningrad region. Let us find the first term in (15). The discount coefficient will be calculated using (3). We will find the cost of polyethylene pipes using (11). For certainty, we will take the cost of a unit of HPPE mass from the price list[4] (from the 13th of July 2024)  = 2.1 $.kg. CR-64 pumps with different number of sections at maximum efficiency have the same nominal capacity (64 m3/h), but different nominal head  It is necessary to install a pump with a high  value while reducing the diameter of the pipeline. This, however, leads to an increase in investment costs. The cost of  pumps of the CR-64 line was taken from the price list[5]. Some of the source data are presented in Table 1. The number of impellers is indicated in the pump model designation after the number 64. If another digit follows, this indicates the number of impellers of reduced size. Note that in some cases reducing the impellers does not affect the cost of the pump: for example, CR-64-4 and CR-64-4-1 cost the same. Table 1. Price list4 of CR-64 pumps with calculated pipeline diameters, head and power consumption (L = 3000 m, HC = 40 m). No. Model CP, $ d1, m HWP, m NWP, kW 1 CR-64-2 7772.9 0.264 43.4 11.6 2 CR-64-3-2 8796.2 0.213 45.5 14.0 3 CR-64-3-1 8796.2 0.170 53.5 15.7 4 CR-64-3 10110.7 0.152 63.4 17.4 5 CR-64-4-2 10589.4 0.147 67.2 19.8 6 CR-64-4-1 11758.0 0.139 75.9 21.5 7 CR-64-4 11758.0 0.133 84.9 23.2 8 CR-64-5 12544.0 0.122 106.3 29.0 9 CR-64-6 15339.3 0.115 127.9 34.8     An approximation of the dependence of the pump cost  on the electric motor consumed power  (at the operating point) is necessary for calculating  This dependence turned out to be linear, as in [22, 23]:  Then the first term in (15) can be expressed as follows:                                      (24) Let us substitute (23), (24) into (15) and obtain the equations for determining the optimal diameter:                             (25) The algebraic equation (25) was solved numerically for different parameter values. The efficiency at the working point of the pumping unit will be calculated using the following formula:                                                        (26) 3.Results and Discussion The power used and the pump pressure at the operating point depend on the pipeline parameters. Table 1 (column 4) shows the results of calculating such a diameter so the flow rate at the operating point is equal to the specified  The calculation was performed using the following parameters:  = 72 m3/h (20 dm3/s),  = 40 m,  = 3000 m. Moreover,  includes necessary free head from the consumer. The pump head and the expended electric motor power at the operating point of the corresponding pumping unit are placed in the last two columns of Table 1. A graphical interpretation of the examples of determining the operating point of three pumping units from Table 1 with the specified pipeline parameters is shown in Fig. 2. According to it, in order to ensure the flow rate of  at d = 0.122 m, it is necessary to install the CR-64-5 pump (the upper point in Fig. 2). At  = 0.152 m, it is enough to install CR-64-3 pump (the lower point). The latter is noticeably cheaper. Figure 2. Examples of determining the operating point of pumping units at Q0 = 72 m3/h, L = 3000 m, HС = 40 m: 1) CR-64-3, 2) CR-64-4, 3) CR-64-5; 4) characteristic of the pipeline with d1 = 0.152 m, 5) d1 = 0.133 m, 6) d1 = 0.122 m. The data of their price list[6] and the approximation of the dependence of the cost CR-64 pumps on the electric motor consumed power at the operating point are shown in Fig. 3. The linear function is in good agreement with the points. The determination index, adjusted due to the small number of points, is 0.95. In this case,  =  = 309.21 $/kW. Figure 3. Relationship between the cost of CR-64 pumps and the electric motor consumed power at the operating point. Fig. 4 shows that the calculation results according to (22) are in good agreement with the experimental points. The empirical constants in this case are  = 11.1,  = 0.00274,  = 4.239. The adjusted index of determination is 0.986. Figure 4. Dependence of the electric motor consumed power (NWP) at the operating point and the derivative (NWP′) of CR-64 units on the pipeline diameter. Points are experimental data; lines are calculations using (22). The results of (25) solved by numerical method for various parameter values are presented in Tables 2–4 and in Figs. 5 and 6:  = 30,  = 0.12, discount coefficient  = 0.124,  = 0.07794 $/(kWh). The efficiency decreases from 0.793 to 0.648 with an increase in water consumption  from 60 to 80 m3/h according to Table 2. Efficiency  = 0.692 at  = 1000 m,  = 0.753 when increasing the length of the pipeline to 9000 m according to Table 3. Efficiency  = 0.637 at  = 10 m,  = 0.753 with an increase in static head to 40 m according to Table 4. The efficiency changes very significantly. The change in efficiency is less significant (Figs. 5 and 6). Therefore,  = 0.754 at  = 4 h,  = 0.720 at  = 24 h (Fig. 5, line 1,  = 21),  = 0.742 at  = 4 h,  = 0.737 at  = 24 h (Fig. 5, line 3,  = 13.6). In Fig. 6, the efficiency increases from 0.731 to 0.755 with an increase in  from 0.05 to 0.3 (line 2, the project lifecycle is 15 years). At a constant value of  = 0.1, if  = 15 years, then  = 0.740; if  = 30 years, then  = 0.734. Table 2. Effect of Q0 on the optimal diameter and efficiency, L = 3000 m, HC = 30 m, T0 = 12 h. Q0, m3/h d0 Hwp, m Nwp, kW η 60 0.147 47.1 11.6 0.793 64 0.151 48.2 12.5 0.758 68 0.154 49.6 13.4 0.725 72 0.157 51.3 14.4 0.695 76 0.160 53.5 15.7 0.669 80 0.162 56.1 16.9 0.648   Table 3. Effect of L on the optimal diameter and efficiency, Q0 = 68 m3/h, HC = 30 m, T0 = 24 h. L, m d0 Hwp, m Nwp, kW η 1000 0.175 31.9 9.0 0.692 2000 0.172 32.2 10.2 0.694 3000 0.170 41.2 11.4 0.707 6000 0.167 57.2 15.2 0.739 9000 0.166 71.9 18.7 0.753   Table 4. Effect of Hc on the optimal diameter and efficiency, Q0 = 68 m3/h, L = 3000 m, T0 = 24 h. Hc, m d0 Hwp, m Nwp, kW η 10 0.155 26.5 8.1 0.637 20 0.154 37.8 10.8 0.688 30 0.154 49.6 13.4 0.725 40 0.153 62.8 16.2 0.757   Tables 2–4 and Figs. 5 and 6 allow for analyzing the influence of various factors on the value of the optimal pipeline d 10.34910/MCE.144.4 621.67+628.25 water supply system pipeline diameter centrifugal pump performance characteristics energy efficiency numerical methods costs optimization https://engstroy.spbstu.ru/article/2026.144.4/

RAR RUS 14405-14405 Kalinga Institute of Industrial Technology (KIIT) Deemed to be University Ghimire Deekshant ghimiredg55@gmail.com

Bhubaneswar, Odisha, India

Kalinga Institute of Industrial Technology (KIIT) Deemed to be University Rout Prasanna Kumar prasanna.rout@kiit.ac.in

Bhubaneswar, Odisha, India

Kalinga Institute of Industrial Technology (KIIT) Deemed to be University Ahmed Zaki zaki111094@gmail.com

Bhubaneswar, Odisha, India

0000-0002-8353-7799 Kalinga Institute of Industrial Technology (KIIT) Deemed to be University Acharya Prasanna Kumar pkacharya64@yahoo.co.in

Bhubaneswar, Odisha, India

International PranaGraf Mintech Research Centre (IGMRC) Dash Tapan tapanphy@gmail.com

Bhubaneswar, Odisha, India

International PranaGraf Mintech Research Centre (IGMRC) Biswal Surendra Kumar skbiswal61@gmail.com

Bhubaneswar, Odisha, India

Effect of graphene oxide on resistance to action of aggressive environments, tortuosity and economic index of concrete Cement concrete though has emerged as one of the best construction materials concerning versatility and strength, suffers from durability when subjected to aggressive environments and action of water because of its internal cracks and voids. Further, because of its huge cement demand, it is blamed for concerning carbon emissions. As such when there is one need to strengthen its microstructure by modifying internal defects, there is another need to reduce the consumption of cement by enhancing the efficiency of concrete. To address these issues, this paper explores the impact of incorporating different percentages (0.01–0.1 % of binder content) of graphene oxide (GO) into concrete made with Portland slag cement and its subsequent effects on strength and durability. Apart from compressive strength, durability parameters such as water absorption, sorptivity, and resistance against attack of aggressive environments caused due to chloride (NaCl), sulphate (MgSO4), and acid (H2SO4) have been examined. The results of this study emphasize the beneficial effects of adding GO to concrete regarding compressive strength enhancement 1.35–2.25 times, similarly increased durability against chemical exposure, and improved resistance to transport properties. The performance-related economic analysis in terms of the economic index revealed that the use of GO up to 0.03 % is beneficial from both a technical and an economic point of view. Use of GO more than 0.04 % may be beneficial where strength and time are more important than cost. The findings may contribute to the advancement of construction materials, which can potentially lead to the development of more durable and sustainable infrastructure. 1.Introduction Cement concrete has emerged as one of the best construction materials because of its strength and versatility. On the other hand, its durability is challenged due to its poor microstructure containing inherent cracks, voids and weak transition zones. Huge research works have been documented on the use of various kinds of fibers for arresting the micro-cracks and strengthening the microstructure, but the achievement is still far from the intended one. Past studies also reveal the use of finely ground materials as filler materials used for filling the microvoids. In recent years, researchers have explored the use of graphene to enhance the performance of concrete. Experiments and studies have shown that the addition of graphene improves the strength of concrete, making it more resistant to loads. In addition, the incorporation of graphene has led to significant findings indicating improved waterproofing properties. Concrete is also blamed because of the embodied carbon and energy associated with cement production. As such development of the efficiency of concrete with a reduction of cement consumption is a vital issue. Graphene and its related compounds are chosen as the most promising candidates for reinforcing material concerning their outstanding elastic modulus strength [1]. Graphene-reinforced materials have been reported in the past, and among these materials, cement-based materials, like concrete, stand out as the most widely consumed human-made material. The reinforcement of concrete and other cementitious materials is essential because they exhibit inherent weaknesses such as susceptibility to crack formation, low tensile strength, and limited strain capabilities [2]. According to Gong et al., the addition of 0.03 wt% graphene oxide (GO) to the cement paste (CPS) resulted in a significant increase, developing more than 40 % of compressive and tensile strengths [3]. In a different study, Pan et al. reported that incorporating 0.05 wt% of GO into the cement mortar led to notable improvements. Specifically, they observed enhancements up to 33 % in compressive and up to 59 % in tensile strength [4]. Lv et al. conducted studies that confirmed the positive impact of incorporating GO into CPS and concrete, leading to improved mechanical properties [5]. Despite numerous studies on the mechanical properties of GO in cement-based materials, information is scarce regarding its impact on durability [6]. Zeng et al. conducted experiments and found that the addition of 0.03 % of GO significantly improved the structural life of nearly 2.3 times cement mortars concerning external sulphate attack. The study informed that the addition of 0.03 % improved compressive strength by nearly 69 % and corrosion resistance by nearly 41 % [7]. Lu et al. investigated the dispersion and stability of GO in neutral water with and without silica fume (SF), as well as in CPS with and without SF. Results indicate that SF contributes to reducing the size of GO in water, enhancing its dispersion [8]. De Souza et al. discussed how graphene-based nanomaterials (GN) improve the properties of cementitious materials like concrete, making them more resistant to various stresses and extending their service life. However, challenges remain in accurately predicting their performance and conducting comprehensive assessments to determine economic and environmental benefits, highlighting the need for further research [9]. Yu et al. reported the impact of GO on the mechanical properties and microstructure of ultra-high-performance concrete. The addition of 0.04 % GO reduces total porosity, especially capillary porosity, enhancing hydration product formation. However, excessive GO content above 0.04 % leads to increased porosity and a slight decrease in mechanical properties due to poor dispersion and fluidity loss [10]. Muthu et al. explained that the characterization of GO synthesized via Hummers' method revealed a highly oxygenated and hydrophilic material, with functional groups including epoxy, hydroxyl, carbonyl, and carboxyl bonded to the graphene sheets. These findings underscore the role of GO content in influencing the hydration kinetics and rheological properties of CPS, with optimal dosage balancing improvements in mechanical strength against potential issues like workability and hydration retardation [11]. Zhao et al. investigated the effects of incorporating modified carbonaceous mesoporous materials (mCMK) and GO on cement mortar properties. Results showed that mCMK and GO improved both flexural and compressive strengths due to pozzolanic reactions, refined pore structure, and enhanced bonding between the matrix and aggregates. Furthermore, mCMK exhibited better dispersion, leading to superior performance compared to CMK, as evidenced by reduced porosity and increased hydration products, ultimately enhancing the mechanical properties of the cement mortar [12]. Dung et al. found that incorporating graphene, whether in pristine (PG) or aggregated (AG) form, enhanced the compressive strength of Portland cement (PC) mortar due to increased hydration and the formation of a denser microstructure. Moreover, the compressive strength improvement was more pronounced at earlier ages and decreased over time, with AG demonstrating better stability and enabling higher hydration levels compared to PG, ultimately resulting in superior compressive strength of AG samples [13]. Du and Dai Pang explained that the introduction of graphene nanoplatelets (GNP) into cement mortar results in a refinement of the pore structure, reducing the critical pore diameter and increasing tortuosity, leading to improved resistance against water penetration and chloride diffusion. However, at higher GNP concentrations, agglomeration becomes more pronounced, compromising the effectiveness of GNP in enhancing barrier properties as reflected in the decrease in water penetration resistance and chloride migration coefficient [14]. According to Yu et al., by incorporating a graphene Nano fluid additive (GNA) into carbon fiber-reinforced cement mortar (CFRCM), a synergistic effect was achieved, resulting in significant enhancements in flexural and compressive strengths by up to 101 and 126 %, respectively, surpassing control specimens. Moreover, the electrical and thermal conductivities of the modified material exhibited remarkable increases of 68 and 36 %, respectively, when compared to controls, indicating its potential for high-efficiency cathodic protection systems with a service life extension of over 200 % [15]. Sui et al. explained the microstructure of GO revealed amorphous multi-layers and wrinkles, with a high purity and oxidation rate as indicated by SEM and EDS analysis. The addition of GO enhanced compactness and mechanical strength while accelerating cement hydration, counteracting the hydration-retarding effect of waste ceramic powder (WCP), thereby enhancing the characteristics of the cement-WCP composite [16]. Jaramillo and Kalfat discuss the impact of different mixing methods on the performance of concrete, focusing on workability, density, compressive strength, and flexural strength. It highlights that the addition of non-sonicated and sonicated GNP dispersions slightly decreased workability but increased compressive strength, with sonicated dispersions showing a more significant improvement. Moreover, increasing mixing time enhanced the compressive strength, indicating better distribution of GNPs within the concrete matrix [17]. Zeng et al. explore the impact of GO nanosheets on sulphate resistance in cement-based materials, revealing significant improvements in transport performance and durability. Through numerical simulations and experimental validation, the proposed chemo-mechanical model demonstrates that adding 0.03 wt. % GO extends structural life expectancy by 2.3 times, offering a cost-effective and environmentally friendly approach to enhance sulphate resistance without contributing to additional carbon dioxide emissions [7]. Salami et al. explained that the incorporation of graphene-based materials in cement alters the pore structure, promoting the formation of compact hydrated crystals, which enhances mechanical strength by minimizing large pores and cracks. Optimal quantities of graphene ensure the formation of regular-shaped crystals, improving microstructure densification and minimizing voids in the cement-graphene composite [18]. Sheikh et al. characterize three types of GO as lowly oxidized GO (LOGO), moderately oxidized GO (MOGO), and highly oxidized GO (HOGO) and investigate their hydronium activity and influence on cement performance. It suggests that the presence of a dense hydronium layer, particularly in HOGO, accelerates cement hydration, leading to denser microstructures and improved mechanical properties in GO-incorporated concretes [19]. Cui and Chang explained that the addition of multi-layer graphene (MLG) to ultra-high-performance concrete (UHPC) accelerated the hydration process, evidenced by earlier exothermic peaks, and increased ion concentrations, particularly calcium ions, promoting hydration product formation. Moreover, MLG refinement of the pore structure, evidenced by decreased pore volume and improved pore size distribution, contributed to enhanced mechanical properties, such as increased flexural and compressive strengths, indicating improved performance of UHPC [20]. Du et al. explained that the addition of GNPs to concrete refined the pore structure, reducing pore sizes and enhancing resistance to water and chloride ion penetration. However, beyond a certain threshold (1.5 % GNP), further addition led to diminished dispersion efficiency, resulting in clustered GNPs that negated the benefits of pore refinement [21]. Fonseka et al. reported that the relation between the compressive strength and abrasion coefficient of concrete containing GO is strongly correlated as indicated by the high coefficient of determination values. Essentially, as the compressive strength of the GO-incorporated concrete increases, the abrasion coefficient tends to decrease. This relationship suggests that enhancing the compressive strength of concrete through GO incorporation can lead to improved resistance against surface abrasion, thereby enhancing the durability of the concrete structure [22]. Suh et al. reported that the incorporation of well-dispersed Carbon Nanotubes (CNTs) in CPS generates interconnected and dense structures, enhancing sample durability by reducing pore scattering and forming three-dimensional networks. This highly connected structure leads to increased durability compared to GO-incorporated samples and ordinary CPS, highlighting the critical role of connectivity in determining material strength [23]. Wang and Zhong reported the characterization of synthesized GO revealed a thickness of about 2 nm and identified chemical functional groups through Atomic Force Microscope (AFM) and Fourier Transformed Infrared Spectra (FTIR) analysis. Three dispersion states of GO in Tricalcium Silicate (C3S) pore solution were observed: well-dispersed (WDS), self-agglomerated (SAS) and grain-adsorbed (GAS), each affecting hydration differently. GO in GAS delayed hydration peak and promoted poison effects due to its conformal coating on C3S grains, while GO in SAS exhibited delaying effects possibly due to its absorption capability for Ca2+ ions, as evidenced by isothermal calorimetry and ion absorption experiments [24]. Ghazizadeh et al. reported the role of GO in PC paste was investigated, revealing that GO aggregates in the pore solution rather than interacting with the surface of hydrating PC particles. While low concentrations of GO slightly increased compressive strength, the extent of improvement was minor and correlated with changes in hydration heat, suggesting that GO primarily enhances strength by increasing hydration degree rather than reinforcing the paste microstructure. Additionally, the presence of polycarboxylate ether (PCE) copolymer did not prevent GO aggregation in PC pastes, indicating that reported strength improvements in PC-GO-PCE systems are likely not due to well-dispersed GO but rather to changes in hydration degree [25]. Lu et al. introduced a novel method for enhancing the mechanical properties of strain-hardening cementitious composites (SHCC) by coating polyethylene (PE) fibers with GO, which improves chemical reactivity and surface wettability. The resulting GO/PE fibers significantly strengthen the bond between fibers and the matrix, leading to a notable increase in strain capacity and tensile strength of SHCC without compromising compressive strength [26]. The literature survey indicated that a good number of papers are documented concerning the use of various forms of graphene in concrete. Most of the papers have focused on mechanical properties. Though some papers are available concerning durability describing a few aspects, there's still a gap in understanding the performance of GO-modified cementitious materials under real-world conditions, like aggressive environments of chemical exposure and tortuosity under the action of water. Since graphene is a very costly material in comparison to the ingredients of concrete, there exists a need to understand the economic benefits of GO-modified concrete. Further, it is understood that most of the available literature is based on the use of ordinary Portland cement (OPC). Concerning the high embodied energy and carbon of OPC in comparison to slag-based cement like Portland slag cement (PSC) and fly ash-based cement like Portland pozzolana cement (PPC), the use and production of OPC is discouraged. As such, there is a need for knowledge of GO-modified concrete using PSC and PPC to encourage the production and utilization of waste-based cement. The present work may help bridge the above gaps. By incorporating oxide into the concrete mix, this study aims to enhance the material's mechanical properties, such as compressive strength, and improve its resistance to chemical attack. This advancement could lead to the construction of more durable and sustainable infrastructure, resulting in longer-lasting buildings and structures that require less frequent maintenance and repair. Additionally, the reduced water absorption and improved waterproofing properties of graphene-modified concrete could prevent moisture-related issues, extending the service life of constructions in various environments. The performance-related economic analysis made in terms of the economic index may be a deciding factor in issues related to time, cost and quality in construction management. Ultimately, this research has the potential to contribute to the development of innovative and resilient construction materials, advancing the field of civil engineering and enhancing the sustainability of infrastructure worldwide. 2.Methods 2.1.Materials In this research, PSC was utilized, adhering to the specifications outlined in IS: 455-2015 (Reaffirmed 2020) [27]. Table 1 displays the various properties of the cement. As for the aggregates, natural granite stone and natural riverbed sand are employed, meeting the requirements of IS: 383-2016 [28] for coarse and fine aggregates. Table 2 provides a presentation of their properties. For the incorporation of GO, it was procured from International PranaGraf Mintech Research Centre (IGMRC), Bhubaneswar, Odisha, India. Table 1. Properties of PSC. Property Results Requirements Setting time-initial (min) 89 ≥30 Setting time-final (min) 198 ≤600 Standard Consistency (%) 32 – Soundness (mm) 1 ≤10 Sulphur trioxide (%) 2.49 ≤3 Chloride content (%) 0.01 ≤0.1 Magnesium oxide (%) 4.37 ≤6 Sulphide sulphur (%) 0.16 ≤1.5 Insoluble residue (%) 2.28 – Loss in ignition (%) 1.51 ≤5     Table 2. Properties of the aggregates. Property Coarse Aggregate Fine Aggregate Flakiness index (%) 17.5 – Elongation Index (%) 12.5 – Specific Gravity 2.83 2.7 Impact Value (%) 14.7 – Abrasion Resistance 18.4 – Water Absorption (%) 0.3 0.76 Crushing Value (%) 19.7 –   2.1.1. Characterizations of graphene oxide The GO used in this work has a purity of about 99.99 % carbon and the Brunauer–Emmett–Teller (BET) specific surface area value of 384.180 m2/g. The micro Raman spectroscopy of GO is shown in Fig. 1. It shows peaks of G (corresponds to the lattice of graphite), D (corresponds to 1st order-disorder peak in the graphite lattice structure), and 2D (2nd order-disorder peak in the graphite lattice structure). The D, G, and 2D peaks were observed at 1344, 1573, and 2686 cm−1, respectively. The intensity ratio of peaks of 2D to G (I2D/IG) is 0.4, indicating a few layers (4–5 layers) of graphene sheets present in the GO. The band gap energy value of GO was determined by ultraviolet–visible diffuse reflectance spectroscopy (UV–Vis DRS) using the Kubelka–Munk function vs. the energy of light. The band gap energy was found to be 3.59 eV (Fig. 2). Figure 1. Micro Raman spectrum of graphene oxide. Figure 2. Determination of band gap energy (Eg) of graphene oxide from the UV-Vis DRS spectrum. 2.2.Methodology 2.2.1. Mix design and ingredients Following the guidelines of IS: 10262-2019 [29], a mix design of M20 grade concrete was prepared using PSC, 12–20 mm size black hard granite aggregates, river sand, portable tap water, and a carboxylate-based superplasticizer. The mix design suggested a proportion of 1:2:4, wherein the fine aggregate quantity was 2 times and that of coarse aggregate was 4 times of cement used. A water-cement ratio of 0.45 and a superplasticizer dosage of 0.5 % of cement were used. As such, for making 1 cum of concrete, 330 kg of PSC, 660 kg of fine aggregates, 1320 kg of coarse aggregates, 149 kg of water, and 1.65 kg of superplasticizer were used. The GO was added to such concrete in 0, 0.01, 0.02, 0.03, 0.04, 0.05, and 0.1 %. These mixes were named as PSC-0, PSC-1, PSC-2, PSC-3, PSC-4, PSC-5, and PSC-10 respectively. 2.2.2. Preparation of concrete and samples For the preparation of concrete, the GO was mixed with cement thoroughly in dry conditions in a laboratory-type pan mixture for 5 minutes. This process was carried out under controlled laboratory conditions to avoid agglomeration and achieve better dispersion. Then the dry mixture of GO and PSC was further mixed with fine aggregate for 5 minutes and then with coarse aggregate for 5 minutes in two different stages, one after another. The required amount of superplasticizer was added to water and mixed with the dry mixture of all other materials. Mixing continued for nearly five minutes to get a uniform homogeneous concrete mix. The mixing time adopted was more than twice of normal concrete for good dispersion of GO. After measuring the slump of concrete, the required number of cubes of 100 mm size were prepared for the test of compressive strength, acid resistance, sulphate resistance, chloride resistance, water absorption and sorptivity. The prepared samples covered with PE were allowed to be cured at room temperature for 24 hours, after which they were put under water curing till attaining the required test age. 2.2.3. Test ages When the strength test was conducted at the ages of 7, 28, and 91 days, the durability parameters concerning acid resistance, sulphate resistance, chloride resistance, water absorption, and sorptivity were examined at the ages of 28 and 91 days. The average results of three samples were considered as the result of a particular mix. 2.2.4. Test procedure The workability of concrete mixes was examined as per the guidelines of IS: 1199-1959 (Reaffirmed 2004) [30]. For the compressive strength test, IS: 516-2021 [31], for the test of acid and sulphate resistance, the procedure, as reported in past publications [32], was followed. For this purpose, 28 days of water-cured cubes were further cured for 28 days in 1 % acid and sulphate solutions. The results of acid and sulfate-cured samples were studied, considering strength and weight loss and compared with the results of water-cured samples of the same age. The test of sorptivity, the procedure published by Acharya and Patro [33] was followed, wherein the samples were wax-painted from all sides. The unpainted bottom surface was kept over the stands in such a position that only 5mm of the bottom was immersed in the water. The ingress of water was noted at the end of 9, 49, 100, 169, 256, and 361 minutes by taking the weight before and after water immersion. The water absorption test was conducted as per IS 1124-1974 (Reaffirmed 2003) [34]. The chloride resistance test was conducted by immersing the samples in a solution containing 20 % sodium chloride. The effect of sodium chloride was checked in terms of strength change and mass change. The results of GO-modified concrete were compared with normal concrete without GO. 3.Results and Discussions Workability The workability of all concrete mixtures was checked after 15 minutes of adding water. The slump of normal concrete (PSC-0) was measured as 75 mm. The same for mixes (PSC-1, PSC-2, PSC-3, PSC-4, PSC-5, and PSC-10) having different dosages of GO were recorded as 68, 67, 65, 64, 63, and 60 mm. Thus, it can be said that there is not much influence of GO on workability. With the highest dosage application of GO (0.1 %), the slump value reduction is only 20 %. Compressive Strength The first set of data concerning (PSC-0) shows the compressive strength values of the material without adding any graphene. At 7 days, the compressive strength is 15.96 MPa, which increases to 20.93 MPa at 28 days, and further increases to 26.94 MPa at 91 days. These values represent the baseline compressive strength of the material, which is used for comparison with the values obtained after adding different amounts of graphene. The results are presented in Fig. 3. The subsequent data shows the compressive strength values after adding different amounts of graphene to the material concerning mixes PSC-1, PSC-2, PSC-3, PSC-4, PSC-5, and PSC-10 that contained GO 0.01, 0.02, 0.03, 0.04, 0.05, and 0.10 %. Adding 0.01 % graphene results in a significant increase in compressive strength by nearly 27 % at 7 days, 35 % at 28 days, and nearly 38 % at 91 days. As the amount of graphene added increases, so does the compressive strength of the material. Adding 0.02 % graphene increases the compressive strength by nearly 62 % at 7 days, 70 % at 28 days, and a similar amount at 91 days. A dosage of 0.03 % graphene results in compressive strength development of 95 % at 7 days, 1.04 times at 28 days, and 1.05 times at 91 days in comparison to PSC-0, which was made without GO. Adding 0.04 %, the strength development is nearly 1.05, 1.16, and 1.08 times of normal concrete at 7, 28, and 91 days. The same inclusion of 0.05 % grapheme oxide content was found to be 1.08, 1.21, and 1.21 times. The results on a higher dosage of grapheme oxide amounting to 0.1 % were found to be 1.08–1.24 times of normal concrete made of PSC. On reviewing the percentage of strength increase it is seen that at the end of 7 days, the strength development varied 0.27–1.08 times of normal concrete on application of 0.01–0.1 % of GO. The same at 28 days was 0.35–1.25 times and at 91 days it was 0.38–1.24 times. The result indicated very good strength development at an early age, which may bring huge technical and economic advantages due to less requirement of stripping time. Concerning the 28-day results, it is seen that the normal concrete (PSC-0) attained a strength of 20.93 MPa. At a dosage of 0.01 % GO, the achieved strength is related to the strength of M25-grade concrete. Similarly, with the addition of 0.02 % of GO, the achieved strength relates to the M35 grade of concrete. The trend of results on the application of 0.03 and 0.04 % is related to the strength of M40 and M45 grades. Further increases in the dosage of GO, like 0.05 and 0.1 % enhanced the strength but not significantly, as was noticed in the case of the application of GO up to 0.04 %. The results may be very useful in structural applications by enhancing the efficiency of concrete by applying small dosages of GO based on the requirement for achieving different grades of concrete. Visiting the results of 91 days, it is seen that there is a continuous development of strength, which is very encouraging. The results showed that after 13 weeks of water curing, M20 grade concrete is converted to M35–M60 grades of concrete on application of 0.01–0.1 % of GO. The results revealed that upon the application of GO, 0.03 % more than 55 MPa strength was achieved. After that, the strength development slows down and with the application of GO 0.1 %, 60 MPa is achieved. This data may be very useful in construction activities where curing can be continued for 13 weeks. The reason behind the development of strength is attributed to the physical properties of GO. It affects concrete health as a filler as well as a reinforcing material. As a filler, at an early age, it modified the transition zone by increasing bonding between the interface of coarse aggregate and paste. The addition of graphene improves the dispersion and bonding within the concrete matrix. This enhances load transfer and increases compressive strength. Findings are found in line with the views of the authors of the past [36]. At a later age, it stitched the micro-cracks and microvoids. It also helped in arresting the development and propagation of internal cracks. The GO also contributed to the refinement of the pore structure and reduced the pore size. Graphene acts as a nucleation site for hydration products, accelerating the hydration process and promoting the formation of a denser microstructure. GO fine particles help in promoting the formation of regularly shaped compact crystals of hydration products that enhance the internal structure of GO-modified concrete. The studies of the past by Bai et al. and Wang et al. validate the results of the present study [35, 36]. Comparing the results of all ages, the rate of strength development was best up to 0.04 % application of GO. Further application of GO, though it enhanced the strength, the rate of strength development was slower, which may be due to improper dispersion and loss of fluid. The observed strength enhancement can be attributed to the optimized dispersion of GO, achieved through thorough dry blending with cement, followed by separate blending with fine and coarse aggregates for five minutes each. This stepwise mixing process ensured effective utilization of GO’s high specific surface area and abundant functional groups. Such uniform dispersion likely facilitated accelerated hydration kinetics and the development of a refined, denser matrix, thereby contributing to improved mechanical performance. Figure 3. Compressive strength of concrete with and without GO. Chloride Resistance The chloride resistance of concrete samples with and without GO was examined through the calculation of weight change and strength change after 28 days of chloride solution curing. The mass and strength change were compared with the 28-day results of the water-cured samples. The results of mass change and strength change are reported in Figs. 4 and 5. After 28 days of chloride curing, there is a minor decrease in weight up to 1 % in concrete samples without GO (PSC-0). On the other hand, there is an increase in weight up to 1 % in samples with GO (PSC-1 to PSC-10). A much smaller amount of loss in weight of normal concrete is due to its slag cement, made of ground granulated blast furnace slag. The elemental composition of slag cement makes the cement composite resistant to aggressive environments. The increase in weight of the GO-modified samples is due to the robustness of their internal structure and surface hardness. From visual inspection, it was seen that the surface of the GO-modified samples was very hard, which repelled the ingress of chloride and external moisture thereof. Due to the presence of moisture outside the sample, concerning loss of moisture from the internal part of the sample. As such, the hydration continued with the help of internal moisture. Because of the hydrated products, the weight of the samples increased. In the case of normal concrete, it was seen that the surface was a little affected, for which there was a little mass loss. Figure 4. Change in mass of concrete with and without GO under chloride curing. After 28 days of chloride curing, the samples with GO and without GO were subjected to a compressive load test to estimate the effect of chloride on concrete health. It is seen that chloride could not affect negatively. When the concrete made of PSC suffered a minor loss of strength up to less than 1 %, there was an improvement in the strength of 4–19 % in GO-modified concrete. PSC, which is made of blast furnace slag, is said to have better resistance to aggressive environments. In the case of GO-modified concrete, there was no moisture movement due to the dense internal structure on modification of transition zones, filling of microvoids and arresting the microcracks. In the absence of moisture movement, there was also no loss of internal moisture. The bound water and capillary water helped in continuing the hydration process, for which the strength improvement is noticed. Further to state, GO has a high surface area and can chemically adsorb chloride ions onto its surface, reducing their concentration in the concrete pore solution and mitigating the chemical degradation of concrete due to chloride attack. Functionalized graphene nanosheets can undergo ion exchange reactions with chloride ions, leading to the formation of less soluble compounds or complexes that are less harmful to concrete. The results and the findings thereof are validated by the previous studies [37, 38]. Figure 5. Change in strength of concrete with and without GO under chloride curing. Sulphate Resistance The resistance of GO-modified samples against sulphate attack was determined in terms of weight and strength variation after immersion of samples in sulphate solution for 28 days. The results were compared with concrete samples made without GO. For this purpose, the weight and strength of samples were measured before and after immersion and the variation was computed. Such results are presented in Figs. 6 and 7. The effect of magnesium sulphate on normal concrete was noticed, wherein the weight loss was more than 1 %. However, this loss is much less because of the positive effect of the slag, from which the cement was made. The action of sulphate on GO-modified concrete was defended by the hard surface structure and dense core structure, for which there was no negative impact. Rather, a little growth in weight was noticed. This growth may be due to the production of hydrated products. From visual inspection, it was seen that the surface of the GO-modified samples was very hard, which repelled the ingress of sulphate and moisture thereof. In the case of normal concrete, it was seen that the surface was a little affected, for which there was a little mass loss. Figure 6. Change in mass of concrete with and without GO under sulphate curing. On a compressive strength test of normal concrete (PSC-0), it was estimated that there was a loss of strength of 3 %. There was no detrimental effect of magnesium sulphate on GO-modified concrete. There was a growth of strength of 5–13 % after 28 days of sulphate curing. Here, the sulphate solution did not contribute to the health and hydration of concrete but at least helped in arresting the release of internal moisture. The sulphate solution could not enter due to the hard external structure. In the absence of moisture movement from the sample or into the sample, the hydration continued with the internal moisture available in gel pores and capillaries, for which strength development was noticed. GO's high surface area and reactivity can facilitate chemical interactions with sulphate ions, potentially forming barrier layers or stable compounds that inhibit sulphate attack [7]. Through its ability to adsorb and immobilize ions, graphene may reduce the availability of sulphate ions for detrimental reactions, thereby preserving the integrity of the concrete matrix. By incorporating graphene into concrete, both physical and chemical actions during sulphate attack can be influenced positively, resulting in improved resistance and durability. These advancements underscore graphene's potential as a multifunctional additive in concrete materials. The findings of the present study are somewhat in line with past studies [7, 39]. Figure 7. Change in strength of concrete with and without GO under sulphate curing. Acid Resistance The response of GO-modified samples against the action of sulphuric acid was determined by calculating the change in weight and strength after immersion of 28-day-old water-cured samples in the sulphuric acid solution for 28 days. The results were compared with concrete samples made with PSC only. For this purpose, the weight and strength of samples were measured before and after immersion and the variation was computed. The results are presented in Figs. 8 and 9. The change in weight in normal concrete was negatively calculated as 0.74 %. On the other hand, the weight change was positive in the case of GO-modified concrete varied by 0.39–0.75 %. Thus, it can be said that the effect of acid was not efficient on the Go-modified concrete sample. Graphene, when incorporated into concrete, forms a physical barrier against acid ingress due to its impermeable nature, hindering the penetration of acid ions into the concrete matrix. In the interior of the sample, the hydration of cement continued and because of the hydrated products, the weight increased. The increase in weight during an acid attack can be explained by the absorption of acid into the material matrix. This absorption occurs due to the porous nature of the material, allowing the acid molecules to infiltrate its structure. As a result, the material gains weight. Figure 8. Change in mass of concrete with and without GO under acid curing. The negative effect of sulphuric acid on the strength of PSC-made concrete was 2.05 %. It is seen that the effect of acid on samples containing GO was not active because of the hard surface structure and dense microstructure. When the inner part of the samples remained unaffected, the hydration process continued. The cohesive nature of graphene enhances the interfacial bonding between cementitious materials and aggregates, resulting in higher compressive strength, for which the strength developed 7.51–16.02 %. The reason behind the development of the strength of GO-modified concrete may be described as the two-dimensional structure of GO provides an impermeable barrier to acid molecules, effectively limiting their penetration into the material. This barrier effect prevents direct contact between the acid and the material's reactive components, reducing the rate of chemical degradation. Further, GO is chemically inert and resistant to corrosion, making it highly stable in acidic environments. Unlike traditional materials, graphene does not undergo chemical reactions with acids, thereby preserving its structural integrity and mechanical properties. This inert behavior ensures the long-term durability and performance of graphene-enhanced materials under acid exposure. As such, the remarks of past authors [40, 41] validate the results and findings of the present study. Figure 9. Change in strength of concrete with and without GO under acid curing. Water Absorption Water absorption characteristics of concrete affect its durability and strength. In this study, the water absorption percentage was measured for concrete cubes containing different percentages of graphene. The results show that as the percentage of graphene increases, the water absorption percentage decreases. Concrete cubes with no graphene had a water absorption percentage of 2.8 %. This value decreased to 2.29 % when 0.01 % graphene was added to the mix. The reduction in water absorption was more significant as the percentage of graphene increased. Concrete cubes with 0.02 % graphene had a water absorption percentage of 2.15 %, while those with 0.03 and 0.04 % graphene had values of 1.94 and 1.89 % respectively. The water absorption percentage for concrete cubes with 0.05 % graphene was 1.62 %. Finally, the addition of 0.1 % graphene resulted in a water absorption percentage of 1.6 %, indicating the highest reduction in water absorption among all percentages tested. Thus, due to the addition of GO (0.01–0.1 %), the water absorption was reduced by 18–43 % in comparison to normal concrete. GO, being a two-dimensional nanomaterial, can physically block water ingress into concrete pores due to its impermeable nature. This barrier effect reduces the absorption of water by the concrete. The results suggest that the addition of graphene to concrete can significantly reduce its water absorption percentage. This can lead to better durability and strength of the concrete, as lower water absorption means less moisture in the concrete, reducing the likelihood of damage due to freeze-thaw cycles or other forms of weathering. The reduction in water absorption may also result in less corrosion of the steel reinforcement in the concrete, further increasing the durability of the structure. Overall, the addition of graphene to concrete can lead to a more sustainable and long-lasting construction material. Sorptivity The sorptivity (S) is a function of water ingress due to the capillary action of water. This transport property of concrete samples with GO and without GO was examined using oven-dried samples, considering weight gain due to sorption of water. The samples were placed in water, with 5 mm immersed from the bottom face. The water ingress was calculated at 9, 16, 25,36, 49, 64, 91, 100, 121, 144, 169, 196, 225, 289, 324, and 361 minutes by finding the difference in weight after and before immersion. The interval of time (t) for measurement of weight gain was so selected that the value of √t would be a whole number for ease of plotting. The S was calculated using a linear regression of i versus √t, where i is cumulative water absorption per unit area inflow in g/mm2. The results of the S test are shown in Fig. 10. When the S index of normal concrete was found as 0.9035, it was reduced to 0.7592 on application of 0.01 % of GO, which shows a reduction of sorption due to capillary action of nearly 16 %. Further application of GO reduced the sorption value by up to 23 %. GO acts as a barrier to water penetration due to its impermeable nature. As water molecules attempt to pass through the concrete matrix, the presence of graphene can hinder their movement, reducing S. Incorporation of GO increases the tortuosity of the pore structure within concrete. This means that water molecules have a more convoluted path to travel through the material, which slows down their movement and reduces S. The reports of the past [42, 43] validate the findings of the present study. Figure 10. Sorptivity of concrete with and without GO. Economic Analysis The economic analysis of all concrete mixes with GO and without GO was analyzed in terms of an economic index (EI), considering both cost and performance at the end of 28 and 91 days. The EI was computed as a ratio of compressive strength to the cost of the mix. The cost of mixes per cubic meter has been derived based on the market available price of all materials those are mentioned in Table 3. Since the cost of machinery, labour, and tools is equal for all concrete mixes, the same has not been taken into the calculation of cost. Table 3 presents the EI of all concrete mixes. The results show that though GO is very costly in comparison to other ingredients of concrete, still the concrete mixtures containing 0.01 % (PSC1), 0.02 % (PSC2), and 0.03 % (PSC3) have economic viability with EI values more than the mix without GO (PSC0) at the end of 28 and 91 days. On application of 0.01–0.03 % of GO to concrete, having 0 % GO has changed the concrete strength M20 grade to M40 grade at 28 days and to M55 at 91 days. At the same time, it is also estimated that the EI has increased 1.38–2.54 % at 28 days and 2.75–3.43 % at 91 days. Such concrete mixes (PSC1, PSC2, and PSC3) are found suitable for all types of structural works. On application of further increased doses of GO, like 0.04–0.1 %, the strength has increased up to 47 MPa at 28 days, but the EI decreased 7.01–47.94 %. Similarly, the mixes containing 0.04–1 % of GO enhanced the strength up to 60 MPa but brought the EI value 10.77–48.18 %. Such types of concrete mixes (PSC4, PSC5, and PSC10) may be suitable for restoration and rehabilitation works in case of natural calamities and war situations where cost is not a factor in comparison to time and strength requirements.           Table 3. Economic indices of various concrete mixes per cum. Mix CS-28d CS-91d Concrete ingredients Cost EI-28d EI-91d PSC GO CA FA W SP Rate (INR)     7 50000 1.25 0.45 0.2 420       PSC0 20.93 26.94 330 0 1320 660 149 1.65 4979.8 0.004203 0.005410 PSC1 28.25 37.10 330 0.033 1320 660 149 1.65 6629.8 0.004261 0.005596 PSC2 35.50 45.70 330 0.066 1320 660 149 1.65 8279.8 0.004288 0.005519 PSC3 42.80 55.20 330 0.099 1320 660 149 1.65 9929.8 0.004310 0.005559 PSC4 45.25 55.90 330 0.132 1320 660 149 1.65 11579.8 0.003908 0.004827 PSC5 46.35 59.65 330 0.165 1320 660 149 1.65 13229.8 0.003503 0.004509 PSC10 47.00 60.20 330 0.33 1320 660 149 1.65 21479.8 0.002188 0.002803   (CS: Compressive strength, PSC: Portland slag cement, GO: Graphene oxide, CA: Coarse aggregate, FA: Fine aggregate, W: Water, SP: Superplasticizer and EI: Economic index) 4.Conclusions The experimental work on graphene-reinforced concrete has provided valuable insights into the potential of GO as a reinforcement material for concrete. By varying the dosage of graphene from 0.01–0.1 % and conducting various tests on strength and durability, the present work has yielded the following findings: On application of 0.01 % of GO, a concrete of M20 yields strength more than that of M25 and M35 at the end of 28 and 91 days. The strength of M20-grade concrete achieved the strength of M35 and M45 at 28 and 91 days on application of 0.02 % of GO. At the same period, a dosage of 0.03 % of GO converted M20 grade concrete to M40 and M55 grade. At 28 days, the strength further increased to M45 grade on application of 0.04 % GO. Further application of GO at the rate of 0.05 % and 0.1 % enhanced the strength up to 47 MPa and 60 MPa at 28 and 91 days, but the rate of growth in strength was lower than the GO dosages 0.01–0.04 %. Facing an adverse environment of chloride, when PSC-made normal concrete suffered a loss of weight and strength up to 1 %, GO-modified concretes gained in weight and strength up to 1 % and 19 % respectively. Fighting with an aggressive environment of magnesium sulphate, when concrete without GO lost weight by up to 1 % and strength by up to 3 %, the concrete mixed with GO gained a little weight and strength by 5–13 %. On an attack of sulphuric acid, when normal concrete lost a mass of 0.74 %, GO-modified concretes gained a mass of 0.75 %. Under the same condition, strength loss was 2 % in normal concrete and strength gain was 8–16 % in GO-modified concrete. The water absorption is reduced by 18–43 % on the application of 0.01–0.1 % of GO. The sorption due to capillary action is reduced by 16–23 % when 0.01–0.1 % of GO was added to concrete. The EI concerning to strength and cost of concrete mixes containing 0.01–0.03 % of GO is found to be better than normal concrete. Beyond 0.03 % of the application of GO, the strength increases, but the EI decreases. However, such types of concrete mixes containing 0.04–0.1 % of GO may be suitable for restoration and rehabilitation works in case of natural calamities and wars, where cost is not a factor in comparison to time and strength requirements. The findings of this study may support the transition to a more sustainable and resilient built environment. 10.34910/MCE.144.5 624 chloride resistance sulphate resistance acid resistance sortivity water absorption https://engstroy.spbstu.ru/article/2026.144.5/

RAR RUS 14407-14407 6602934825 0000-0003-0632-6784 V.G. Shukhov Belgorod State Technological University Logachev Konstantin kilogachev@mail.ru

Belgorod, Russian Federation

57202250955 0000-0001-7335-7797 Kazan State University of Architecture and Engineering Ziganshin Arslan ziganshin.arslan@gmail.com

Kazan, Russian Federation

57201773316 0009-0005-1651-9696 Belgorod State Technological University named after V.G. Shukhov Popov Evgenii evg-popov@yandex.ru

Belgorod, Russian Federation

0009-0001-8496-9104 Xi’an University of Architecture and Technology Huang Yanqiu huangyanqiu@xauat.ed

Xi’an, China

23984180700 0000-0002-4227-8663 Belgorod State Technological University named after V.G. Shukhov Averkova Olga olga_19572004@mail.ru

Belgorod, Russian Federation

0000-0003-1989-0632 Belgorod State Technological University named after V.G. Shukhov Elistratova Yuliya tgv.info@mail.ru

Belgorod, Russian Federation

57216545677 0000-0002-9497-349X Kazan State University of Architecture and Engineering Kareeva Julia jkareeva2503@gmail.com

Kazan, Russia

57807235700 0009-0000-2825-0536 Belgorod State Technological University named after V.G. Shukhov Tiron Oleg tiron1997@gmail.com

Belgorod, Russian Federation

0009-0003-8818-9791 Moscow Aviation Institute (National Research University) Stepanova Angelina angellogacheva@mail.ru

Moscow, Russian Federation

Peculiarities of separated flows in ventilation elbows and their nodes studied numerically This study examines vortex zones (VZs) that form in single ventilation elbows and their U-shaped and Z-shaped nodes. Reducing local drag in these elements is important for reducing energy consumption in ventilation systems, which can account for up to 80 % of the total network drag. A mathematical model based on the discrete vortex method (DVM) was developed to calculate the VZ boundary in a steady state. The model places attached vortices along the flow boundary and free vortices detached from sharp edges, iteratively determining their circulation and position. Convergence is achieved with a sampling step of  = = 0.00125 m (3522 boundary elements), yielding an average vertical velocity in the control section that differs by only 0.18 % from the solution with  = 0.0025 m, and a difference in the VZ width of 0.47 %. For a single 90° elbow, with an increase in the relative channel width from  = 0.5 to 2.0, the maximum VZ width increases from approximately 0.27  to 1.20  The DVM results are in good agreement with the experimental outlines of VZs at a distance of up to two calibers from the sharp edge. Two-dimensional calculations of the Reynolds-averaged Navier-Stokes equations were performed in the Ansys Fluent computational fluid dynamics (CFD) software package using the standard  model with extended wall modeling. The determined local drag coefficients (LDCs) for a single elbow differ by less than 1 % from the previous numerical data and have the same change trend as the experimental results, with a difference of about 20 %. For U-shaped and Z-shaped nodes with distances between individual elbows equal to  = 2 and 4, the DVM shows the merging of VZs at a shorter distance, while the CFD method shows separate VZs, demonstrating the influence of the downstream ventilation elbow. The developed approach using DVM and CFD allows for the precise determination of VZ outlines in nodes consisting of ventilation elbows and opens up prospects for the design of shaped energy-efficient ventilation system components. 1.Introduction Mechanical ventilation and air conditioning systems together with pollutant capture or localization systems are used in both civil and industrial buildings to ensure the necessary microclimate parameters for sanitary, hygienic, or technological requirements [1, 2]. According to statistics, the operation of heating, ventilation, and air conditioning systems accounts for approximately 40–60 % [3] of the total electricity consumption of buildings. Overcoming the aerodynamic drag of ventilation networks, imposed by duct fitting elements (“local resistances”), accounts for up to 70–80 % of their total resistance [4]. The main consumers of electrical energy are draft equipment (fans, smoke exhausters, pumps), the power of which is directly proportional to the flow rate of medium passing through them and the network resistance (pressure losses in the elements of the networks). Therefore, one of the areas for energy conservation in ventilation and air conditioning systems is the reduction of energy losses in the ductwork elements, which will reduce the carbon footprint of electricity generation and is an important task of environmental engineering [5–7], especially in conditions of increasing loads on systems under changing climatic parameters [8]. The scientific works for reducing pressure losses by overcoming local resistances by shaping the element walls or by installing variously curved guide vanes are being well developed. Various types of fitting elements have been studied. For elbows and bends, a resistance reduction of 25–31 % was obtained [9, 10]. The reduction in local resistance coefficients for tees was found to be from 6 to 99 % for the forward flow direction and from 2 to 98 % for a side branch, depending on the ratio of the transverse dimensions and the flow rates of air flowing through the branches [11–14]. For a tee connecting kitchen exhaust hoods to a flue, a design and installation location in the main shaft of a cycloidal guide vane was developed. This design reduces the resistance to flow in the forward direction by up to 44–197 % and increases the resistance to flow in the opposite direction by up to 4–171 % [15]. When installing a guide vane in a side branch, the flow is separated, separation processes are reduced, and resistance decreases from 8 to 74 % [16]. By selecting the appropriate position and curvature shape of the guide vane installed in the forward direction, the flow resistance can be reduced by up to 44 % [17]. The improved designs of air and check dampers, including optimized shutter position and profiling of the pressure bars [18, 19] result in a decrease in resistance in the open state by up to 87 %, which is especially important for “normally open” fire dampers. Optimization of the curved shape of the fairing of the supply ventilation diffuser is used to increase the quality index of the air distribution device [20], which also includes the aerodynamic resistance. Flows in static pressure chambers [21] and in other fitting elements in case of constrictions [22] and expansions are also the subject of research and design optimization. In these cases, a significant reduction in aerodynamic resistance is also achieved. The considered studies used both numerical and natural experiments, and a wide range of methods for finding the outlines for shaping: biomimicry and machine learning [9]; biomimicry of branched plant structure [12]; methods based on the principle of field synergy and viscous dissipation; methods using the principle of field synergy and entropy increase [10, 20]; the method of shape optimization by gradient descent [11]; the use of guide vanes to reduce fluid deformation and convert mechanical energy into internal energy and turbulent energy dissipation [13, 14, 17, 18]; methods based on the calculus of variations [15]; a combined method of optimization of geometric parameters and the installation of deflectors, using the principle of energy dissipation and vortex force [16]; an improved method of topology optimization for the design of low-drag elements and methods of machine learning and numerical modeling [19]; a method of structural-parametric optimization of static pressure chambers [21]. The shape of the Boltzmann curve was obtained and applied for a shaped element in the form of a confuser [22]. For an elbow, a development of the method of topological shape optimization using the Forchheimer model was proposed [23]. In this case, solved were the problems of reducing the resistance of elements of not only air ducts of ventilation, air conditioning and smoke removal systems but also of water pipes, pipelines of heating systems [24, 25], in a heat exchanger [26] cooling and water supply [9, 16, 19], and distribution static pressure chambers [21]. Studies of flow characteristics in engineering systems are also developing in the direction of identifying peculiarities in a specific operating condition. For example, the work [27] summarizes the obtained data on the influence of various parameters on local resistance losses, including atmospheric pressure, inlet velocity, humidity, oxygen concentration, pipe diameter, wall roughness, velocity ratio, and area ratio between branches and the main duct. In addition, when considering methods for improving the energy efficiency of duct networks, an important issue is the effective balancing of all branches, which can be taken into account by, for example, the criterion of minimum life cycle cost (LCC) of the system [28]. The paper provides a detailed review of the existing methods for designing duct networks and demonstrates the advantages of the developed balancing technique based on minimum LCC. In this case, one should perform an accurate calculation of the aerodynamic resistance of all shaped elements, including their mutual influence on each other. The work [29] considers the problems associated with balancing airflow distribution in duct networks, taking into account the undeveloped turbulent flow regime and the complex dependence of the resistance of fitting elements on their velocity. The methods developed based on machine learning and computational fluid dynamics (CFD) make it possible to regulate such complex systems with an accuracy of up to 5 %. Special stabilizers are also being developed to perform balancing and automatic regulation of airflows [30], and issues of regulating the exhaust air flow for natural ventilation systems of multi-story buildings are considered. When studying fitting elements in a three-dimensional formulation, it becomes important to find a method for determining the vortex zone (VZ) outlines [31, 32]. The approach being developed by the authors (a scientific school) is devoted to the study of VZ outlines and shaping based on them [4, 33]. CFD and the discrete vortex method (DVM) were used for the study. The DVM was used to determine the VZ outlines at the inlet of the exhaust ducts. It is of interest to consider the application of the DVM to the study of other fitting elements of mechanical ventilation systems, in particular, to consider the possibility of its application to describing the VZ when the flow turns in a duct fitting element in the form of an elbow and nodes composed of consecutively arranged elbows. A similar study is conducted using the CFD method to compare the results with each other and with the existing data from other authors. The goal of this work is to develop a discrete mathematical model, a computational algorithm, and a computer program for calculating the VZ boundary for flows inside air ducts – in elbows and their nodes using the DVM in a steady-state formulation. To achieve this goal, it is necessary to solve the following problems: to construct numerical CFD and DVM models; to configure and validate the models, to compare the calculation results with each other and with the calculated and experimental data known from literature. 2.Methods During numerical modeling of elbows for channels with a depth (dimension in the direction normal to the plane of view) significantly (more than 7–10 times) exceeding the channel width, they can be considered as two-dimensional, which is confirmed by the known experimental data on the resistance of elbows [34]. The numerical modeling of the VZ outlines in channels with elbows confirmed the three-dimensional shape of the separation zone [35–36]. However, the existing deviation from the VZ outlines obtained in the two-dimensional formulation does not lead to a significant increase in efficiency, when they are subsequently used for profiling shaped elements (an increase in efficiency is about 17 %). Moreover, from a technological standpoint, manufacturing an insert with a three-dimensional shape or profiling the channel wall along such a surface poses a highly challenging task, and therefore, it is not justifiable for practical application. Manufacturing of a shaped element profiled along the outlines (a two-dimensional curve) merely involves bending the plane of the channel wall and can be accomplished without any changes to the manufacturing technology for standard shaped elements on the existing production lines equipped with sheet-metal plasma cutting machines. For this reason, determining the two-dimensional VZ outlines is relevant. When constructing a numerical DVM model, the non-stationary nature of the vortex formation process allows for the use of a non-stationary formulation. However, to determine the VZ outlines for the purpose of using them for profiling, it would be necessary to further average them over time, which is time-consuming and results in the same shape of the separation zone as when using a more rapidly converging solution in a stationary formulation. 2.1.Construction of a Discrete Mathematical Model A discrete mathematical model (Fig. 1b) of flow separation from a sharp edge  during a turn had been constructed as follows. The boundary of the region had been constructed symmetrically with respect to the  axis, that had ensured the absence of air flow across the  axis, since each attached or free vortex located in the upper half-plane (Fig. 1b) corresponded to a vortex in the lower half-plane, symmetrical with respect to the  axis. The circulations of these vortices had been set equal in magnitude but opposite in sign:  where    was the number of vortices in the upper half-plane.  attached vortices had been uniformly distributed along the flow boundary. The distance between them had been set equal to the discretization step  Flow separation occurred at point  and point  symmetrical with respect to the axis, at which a free vortex layer of  vortices had been formed.           a) b) Figure 1. Formulation of the problem: a) flow boundary; b) discrete model. The influence of all vortices on a point  along the direction  had been determined by the sum:            (1) where  was the velocity at a point  along the direction  caused by the action of a vortex of single circulation located at a point    was the circulation of free vortices at a point    was the location point of the -th attached vortex;   , was the location point of free vortices that have come off the sharp edge    was their amount. If the distance from the point  to the point  had exceeded  then  In case of their coincidence The free vortex locations had been determined iteratively. At the first iteration, the coordinates of the control point  had been substituted instead of  in formula (1). Varying  from 1 to  had yielded a system of  linear algebraic equations with  unknown vortex circulations. After solving this system, a streamline was constructed from point  along which free vortices were uniformly distributed with a step of  When the streamline had approached the domain boundary at a distance smaller than the discretization step, its construction was stopped. At this point, the first iteration had been completed. The second iteration also consisted of solving the system of equations, taking into account the new location of the free vortices. Iterations had been performed until the difference in the absolute value of the free vortex circulations at the subsequent and previous iterations was greater than the accuracy of 10–8. 2.2.Construction of a Numerical CFD Model The Ansys Fluent (Release 18.2) computational software was used for the problem solution. The geometries of the computational domains for all three problems: a single elbow, and nodes consisting of two consecutive elbows in U-shaped and Z-shaped form, are shown in Fig. 2. Figure 2. The geometry of the computational domains: a) single elbow at 90°, b) U-shaped elbow, c) Z- shaped elbow. At the boundaries  (single elbow) and  (Z- and U-shaped elbows), the boundary condition (BC) of constant and uniform velocity  = 10 m/s (Re ~ 5.9·104) had been set; at the boundaries  (single elbow) and  (Z- and U-shaped elbows), the condition of constant excess pressure equal to zero (release to atmosphere) had been set. The remaining boundaries had been set as impermeable walls with a no-slip condition. The channel widths had been set as follows:  = 0.1 m,  = 0.05 (0.5 ); 0.1 (1.0 ); 0.15 (1.5 ); 0.2 (2.0 ). The channel length before the elbow  had been chosen equal to 40  to be sufficient to equalize the flow after the inlet. Similarly, for a single elbow, the length after it had been chosen  = 40  Previously, numerous comparative studies have shown (e.g., [37]) that such a length is sufficient to ensure a fully developed profile after the inlet BC in the form of a uniform profile, as well as to equalize the flow after shaped elements, for the correct determination of the local resistance coefficient. For nodes in the form of Z- and U-shaped elbows, the flow had been modeled and studied for the following distances between two consecutive elbows:  = 2  and 4  The length of the outlet channel after the nodes  = 40  had been also chosen to be sufficiently large to equalize the flow parameters in the channel cross-section. Since flows in shaped elements can be quite complex, many researchers note the lack of universal models for all ventilation elements and the need to validate numerical models for the elements being studied specifically [38]. In the cited work, the limited scope of application of  models in the study of flows in three-dimensional bends is noted, and the acceptable accuracy is observed for Re < ~1·105. In this study, the use of the “standard”  model with “enhanced wall treatments” (two-layer concept with automatic switching (crosslinking) based on the dimensionless distance value ) for two-dimensional nodes from elbows is based on detailed preliminary studies and model validation [37, 39] and, in addition, fits into the previously specified range (Re ~ 5.9·104) [38]. As previously, here the first-order upwind discretization was used for validation, and convergence was assessed by achieving absolute residuals for all equations of less than 1·10–10 and maintaining the control parameter constant over more than 100 iterations. The control parameter chosen was the mass-averaged total pressure at the channel outlet, which characterizes the overall resistance and its constancy. During the verification process, a “grid dependence” study had been conducted for each problem by sequentially refining the computational grid. At the first stage, refinement had been performed over the entire computational domain, until a dimensionless number  reached ~ 30, and further the layers along all solid boundaries decreased until  reached ~ 1 [40]. The grid solution had been accepted as final if the difference in the local drag coefficient (LDC) values between the last and penultimate grids had not exceeded 1 %. The parameters of the initial and final grids for several problems are given in Table 1. Table 1. Computational grid parameters. Problem Initial grid parameters Final grid parameters Number of cells, thousand pcs. Max. cell size, mm Max. cell size, mm Number of cells, thousand pcs. Max. cell size, mm Max. cell size, mm Single elbow (b2/b1 = 1.5) ~8 12.5 12.5 ~3029 1.56 4.9·10–2 U-shaped elbow (b2/b1 = 2.0) ~5.2 12.5 12.5 ~2413 1.56 4.9·10–2 Z- shaped elbow (b2/b1 = 2.0) ~5.4 12.5 12.5 ~2472 1.56 4.9·10–2                   The LDC values had been determined using the methods described in previous works [33, 39]. The VZ outlines had been determined by the outermost current line separating the main flow from the circulation flow inside the VZ. 3.Results and Discussion DVM Study Figs. 3–6 present the calculation results for  = 0.1 m;  = 0.1 m;  = 1 m/s;  = 0.4 m;  = = 0.8 m. Figs. 4 and 5 show the relationship between the maximum width of the separation (vortex) zone  (Fig. 3) and average velocity in the section 1–1 (Fig. 1a) on the discretization step  Analysis of these graphs show that the step  = 0.00125 m is sufficient. For the discretization steps of  = 0.0125 m;  = = 0.0025 m,  = 0.00125 m,  = 0.000625 m, the separation lines having the free vortices are practically overlapped. The value of the average vertical component of velocity in section 1–1 is equal to  = = 0.99678 and differs by 0.18 % from the value obtained with the step  = 0.0025 m. The  value in a similar comparison differs by 0.47 %. Figure 3. Separation streamlines. Figure 4. Change in average velocity in section 1-1 of the duct: 1 – h = 0.05 m, 90 boundary elements (BE); 2 – h = 0.025 m, 178 BE; 3 – h = 0.0125 m, 354 BE; 4 – h = 0.0025 m, 1762 BE; 5 – h = 0.00125 m, 3522 BE; 6 – h = 0.000625 m, 7042 BE. Figure 5. Change in the VZ width. Fig. 6 shows the VZ outline formed from the sharp edge  (Fig. 1a). The experimental data are taken from [41], which, although they were obtained long ago, have not lost their relevance, since there are no newer ones. Moreover, these data confirm their validity, since many newly obtained independent results from various analytical and numerical methods agree with each other and with the cited experimental data. Calculation using the conformal mapping method (CMM) had been performed using the Zhukovsky–Roshko flow model [42] and using smoothed particle hydrodynamics (SPH) method [43] Note that the results obtained using the DVM8 method (the separation streamline had been calculated up to a cross-section distance of  =  from the sharp edge ) and the CMM and SPH method are almost overlapped. Up to two calibers (caliber is ), the DVM calculations agree with the experiment. The VZ outline calculated using CFD methods [39] is significantly smaller. It should be noted that [42] suggests that the VZ outline constructed using CFD more closely matches the physical representation, while the ideal incompressible fluid flow model does not adequately describe the process. However, the previous visualization of similar flows [4] shows that a clear VZ outline is visible only at the initial stage. Further, the VZ boundary becomes blurred due to the high turbulence of the flow, and the point of flow attachment to the solid boundary is not visible. Figure 6. Separation line (VZ outline) constructed using different methods for b2 = b1. Fig. 7 shows streamlines for the case when the separation streamline (free vortices) is constructed before the exhaust section (DVM8) for different channel width ratios. Also shown here (Fig. 7a) is a dotted line of possible profiling during a turn, obtained by parallel horizontal shift of the VZ boundary by  The maximum VZ width increases as the width  increases relative to Note that when limiting the calculation length of the separation streamline (free vortices) to a section of  from the sharp edge  a flow pattern more similar to the experimental one is obtained (Figs. 6 and 8). In this case, shaping does not require increasing the dimensions of the fitting element, and a shaping insert can be inserted along the determined VZ boundaries. Note that the plots of  versus channel width ratio in Figs. 7b and 8b coincide qualitatively and quantitatively, so, the maximum VZ width almost does not depend on the calculation length. Figure 7. Streamlines for different width ratios b2/b1 = 0.5; 1; 1.5; 2.0 (a) and the relationship between the maximum VZ width and the channel width ratio (b). Figure 8. DVM calculation of streamlines for various distances to the section of separation streamline up to 2.8 b2/b1 (b2/b1 = 0.5; 1; 1.5; 2.0) (a) and the relationship between the maximum VZ width and the channel width ratio (b). During a turn, VZ arises as a result of flow separation not only from the sharp edge but also from the smooth surface near angle  (Fig. 1a). To simulate such separation, the discrete model shown in Fig. 8 should be used without placing vortices symmetrical with respect to the -axis. Here, at some point  where the attached vortex is located, a free vortex appears at a distance  from  A free vortex sheet is constructed along the streamline, starting from this free vortex. The final arrangement of the free vortices is determined iteratively. Accordingly, the resulting free vortex sheet determines the VZ outline. The calculation ratios in this case have the following form: The influence of all vortices on the point  along the direction  is determined by the sum:                            (2) where  is the location of free vortices descending from point    is their number (fig.9). The location of point  had been determined by a computational experiment. The point  had been moved horizontally from angle  until the shape of the resulting VZ ceased to change significantly. Point  had been located at a distance  from the attached vortex. The experiments had shown that this point could not be placed within the vortex itself, since, due to errors and the discrete nature of the model, the free streamline could extend beyond the flow boundary into the outer region. The circulation  is equal to the circulation of the vortex located on the boundary, above which point  is located. Figure 9. The discrete elbow model without using symmetric vortices. The flow pattern for the considered case is shown in Fig. 10. The figure also shows VZ boundaries in the bend constructed using CFD methods [44]. It can be seen that the vortex flow region constructed using DVM significantly exceeds its outline. Note that the VZ outline determined from sharp edge  is almost identical to the case of the absence of VZ forming from point VZ outline for multiple elbows can be determined using DVM. For example, Figs. 11 and 12 show the VZ outlines for a Z-shaped and U-shaped elbows. These figures show that VZs can merge at a small distance between bends. Figure 10. An elbow with two vortex zones. Figure 11. A node with a Z-shaped elbow with the following length of channel between single elbows: a) d2 = 4b1; b) d2 = 2b1.     Figure 12. A node with a U-shaped elbow with the following length of channel between single elbows: a) d2 = 4b1; b) d2 = 2b1. CFD Calculations CFD methods make it possible to construct not only velocity field and streamlines but also the fields of other hydrodynamic parameters – pressure, turbulence, etc. The calculated pressure fields enable determining the LDC values. For the considered nodes and their dimensions, the LDC values are known from the previous experimental [34] and numerical [33, 39] studies. The available data made it possible to validate the results obtained in this numerical experiment. Fig. 13 shows (“CFD”) the relationship between LDC values (calculated by CFD) as a function of the relative channel width in the elbow  after a turn. Figure 13. LDC values ζ as a function of the relative channel width in the elbow b2/b1 after a turn for a current “CFD”, previous “CFD prev.” numerical and experimental “Exp.” studies. The Fig. 13 also shows the relationship between  and  obtained experimentally (“Exp.”) [34] and numerically by the authors in the previous work (“CFD prev.”) [39]. In this study, a similar dependence was determined for partially different values of  It can be seen that the results are in good coincidence with each other, the LCD values obtained in this work have the same trend and differ from the previously obtained data by no more than 1 %. The difference from the experimental data is about 20 % (as before). Given that when comparing the results of studies conducted by different methods, quite large deviations (over 40 %) are often observed, many researchers [45, 46] consider a deviation level of 15 % to be a good approximation. Therefore, in our case, the level of agreement with the experimental data [34] is not good but satisfactory, and it demonstrates the validity of the model used. Fig. 14 shows VZ outlines obtained by numerical calculation using CFD, compared with data obtained using discrete vertices method DVM2.8. Figure 14. VZ outlines. It can be seen that for  ≥ 1.0, VZ outlines, constructed using two methods, are relatively equal and have the same type of dependence on  up to the maximum width coordinate. Further, the nature of the VZ width  dependence changes. VZ width  calculated by CFD method, shows a more smooth and gradual change than that calculated by DVM. Figs. 15 and 16 show VZ outlines in Z- and U- shaped elbows. VZs, constructed by DVM and VZ for a single elbow, are also shown here for comparison. Figure 15. VZ outlines in a Z-shaped elbow. Figure 16. VZ outlines in a U-shaped elbow. At the nodes, as previously in the single elbow, similarity in outlines is observed up to the point of maximum width, after which a divergence begins. Furthermore, for the short variant (  = 2) of the U-shaped elbow node, a significant difference in the VZ character is observed. DVM calculation shows that VZ1, breaking away from the sharp edge of the first elbow, does not have time to close onto the channel wall and merges with the second VZ2 into a single VZ. At the same time, the VZ outlines found by the CFD method are separate. Analyzing the behavior of the VZ outlines as the distance increases to  = 4, it can be noted that in the DVM calculation, the VZ1 and VZ2 outlines and for the single elbow practically merge, indicating the absence of mutual influence between the elbows at this distance. In the CFD calculations, the VZ1 outlines and for the single elbow merge, while the VZ2 outline differs significantly. This indicates no upstream influence but a significant downstream influence from one elbow to the other. 4.Conclusions For the first time, a mathematical flow model has been developed using the DVM for air duct elements in the form of elbows, nodes, and their combinations. The VZ outlines for duct elbows calculated using DVM demonstrate sufficient accuracy and allows one to obtain the VZ outlines of much faster than, for example, using CFD methods. Convergence of the developed mathematical model with increasing number of attached vortices was established. The calculated and experimental VZ outlines agree satisfactorily. There are some differences for small distance nodes. The developed mathematical model and software make it possible to study VZ outlines in duct elbow-shaped elements, taking into account VZ merging and flow separation from a smooth surface. The relationship between maximum VZ width and the duct channel width ratio were obtained. CFD methods were used to construct the VZ outlines and their dependence on the distance between single elbows in a node, as well as their convergence with the DVM results in the first part of the VZ outlines. Future research could include DVM modeling to establish the relationship between the VZ size in an elbow and the elbow angle; VZ outlines construction for a sudden flow expansion, in tees and other fitting components; calculation of LDC as a function of VZ dimensions. It is important to further explore methods for studying VZ outlines, including experimental ones, in order to determine VZ outlines along its entire length and to find models that most adequately describe them. It is of interest to determine the LDC of shaped fitting components based on the determined outlines and to compare the obtained values for the corresponding shaped components based on VZ modeling results obtained using other methods. 10.34910/MCE.144.7 697.9 vortex zones numerical study discrete vortex method ventilation node local drag coefficient https://engstroy.spbstu.ru/article/2026.144.6/

RAR RUS 14408-14408 Peter the Great Saint Petersburg Polytechnic University Sabri Mohanad Muaya mohanad.m.sabri@gmail.com

Polytechnicheskay, 29

0000-0002-9850-6565 Department of Civil Engineering, Semnan Branch, Islamic Azad University Akhavan Salmassi Mehran m.akhavan.s@stu.semnaniau.ac.ir

Semnan, Iran

Seismic influence of end shear walls in tall buildings with linker bridge systems Contemporary architectural demands significantly influence structural analysis studies. In this regard, some tall buildings are linked by linker bridge to satisfy the functional demand. However, lateral loads significantly influence the seismic performance of such interconnected systems. This seismic load is a key consideration for structural designers working on tall buildings subjected to earthquake-induced forces. End shear walls in high-rise structures help alleviate stress concentrations caused by increased pressure at the extremities of the walls. In this study, nonlinear time history analysis was employed to compare the seismic effects of end shear walls in two 30-story buildings, one with and one without end shear walls connected by a steel linker bridge. The results showed that the average maximum absolute displacement decreased by 14 % under far-field ground motion records at the floor level where the linker bridge is located, specifically in the corner region adjacent to the bridge, in structures incorporating end-positioned shear walls. Additionally, the skewness coefficient of the acceleration record decreased by 65 % and 66 % due to the presence of end shear walls at the same floor level. In general, the two tall buildings equipped with connecting bridges and end shear walls showed notably better seismic effectiveness under earthquake loading. Previous investigations on linked tall buildings have studied parameters such as bridge location, stiffness, and end conditions, but the specific role of end shear walls within such interconnected systems remains unexplored. This study addresses this gap by investigating the seismic impact of end shear walls in two 30-story buildings connected by a steel linker bridge. 10.34910/MCE.144.8 624.04 tall buildings nonlinear time history analysis linker bridge reinforced concrete structures end shear walls. https://engstroy.spbstu.ru/article/2026.144.7/

RAR RUS 14409-14409 57194112309 0000-0001-6184-2365 Vyatka State University Tyukalov Yury yutvgu@mail.ru Finite element approach for reinforced concrete beam cracking The study aimed to construct a finite element model of bending reinforced concrete beams taking into account crack formation. An incremental orthotropic model was used for modeling the elastic-plastic deformation of concrete. The finite element stiffness matrix was formulated based on a complementary energy functional. By employing Lagrange multipliers, the functional was augmented with nodal equilibrium equations derived via the principle of virtual displacements. Crack formation was incorporated by introducing additional unknown nodal displacements and using independent stress approximation functions within the element subdomains separated by the crack. The stiffness matrix of the rectangular finite element was derived by approximating the stress fields within the element domain using linear functions. To solve the non-linear problem, an incremental-iterative algorithm was proposed to track the progressive sequence of crack initiation within the finite elements. The Euclidean force norm was used to evaluate the solution residual. The number of required iterations is determined chiefly by the total number of initiated cracks. Once no new crack initiation occurs during a load step, the residual norm approaches zero within just two iterations. To validate the proposed model, a numerical analysis of a reinforced concrete beam subjected to a uniformly distributed load was performed. The numerical results derived via the proposed method were validated against the LIRA-SAPR simulation data. 1.Introduction As is well known, cracks can occur in tension zones of reinforced concrete structures even under relatively light loads. These cracks, when further loaded, significantly affect the stress distribution and strength of the structure. The crack width affects the rate of reinforcement corrosion and the durability of reinforced concrete structures. Numerous scientific studies have been devoted to modeling the process of crack formation and opening in the analysis of reinforced concrete structures. In [1], a continuous-discontinuous method was used to model cracking deformation and rock contact, taking into account the internal cracking of quadrangular elements. This method divides the computational domain into quadrilateral elements with high accuracy. It combines the advantages of the continuous and discontinuous methods. Before crack formation, this method does not require the introduction of normal and tangential contact stiffness. After formation, cracks can propagate along the element boundaries and diagonal lines of the quadrilateral elements. In [2], a crack model was proposed that focuses on calculating the crack spacing based on the reinforcement strain profile. For small beams, a linear crack width profile was assumed. For primary cracks in large beams, a bilinear width profile was used. The crack spacing and width predictions obtained using the proposed model are in good agreement with test results. One of the most widely adopted approaches for modeling crack formation is Extended Finite Element Method (XFEM) [3–6]. Within this framework, additional degrees of freedom along with corresponding enrichment functions are introduced at the nodes of the cracked element, utilizing the Heaviside step function to represent the displacement jump across the crack faces. The strain fields within the finite element are subsequently determined through the derivatives of these displacement approximation functions. In [5], this methodology was successfully employed to analyze massive structures while accounting for crack propagation. Furthermore, [7] integrated XFEM with the substructure technique for crack modeling. By partitioning the structure into multiple substructures, XFEM could be efficiently applied to capture localized cracking within each subdomain. Papers [8, 9] focus on experimental investigations of crack propagation in reinforced concrete beams, evaluating crack widths and spacing in comparison with standardized design values. In [10], a novel family of finite elements based on deformation analysis was introduced to capture material softening in planar frame structures. Within this framework, the softening zone is represented by a discrete crack treated as an ‘excluded’ point of the finite element. Consequently, crack strains are computed separately from the bulk element strains, being linked to the element solely via kinematic variables that define the crack opening. To further advance structural modeling, a rectangular finite element formulated through a complementary energy functional was proposed in, where nodal equilibrium equations were integrated using the Lagrange multiplier method and derived via the principle of virtual displacements. Additionally, a new approach for modeling frictional contact within XFEM was developed in. This work introduced an innovative nodal scheme incorporating a coupled Coulomb friction model. The primary novelty of this technique lies in its unprecedented ability to account for stress redistribution within localized regions of the crack surface, offering a distinct advantage in mitigating the blocking problem during numerical simulations of frictional contact. [14] introduces a finite element formulation for the analysis of reinforced concrete slabs, accounting for both concrete cracking and reinforcement yielding. Developed within the framework of localized damage mechanics, this formulation assumes that all inelastic effects are concentrated along discrete lines, directly linking all numerical parameters to the mechanical properties of the slab. Furthermore, this finite element approach has been numerically validated to yield mesh-independent results. Alternatively, an approach for modeling discrete cracks in three-dimensional space using a particle method was described in. Under this framework, cracks can be arbitrarily oriented, and their propagation is discretely represented by activating crack surfaces on individual particles, thereby eliminating the need for explicit crack topology tracking. In [16], an adaptive crack propagation algorithm utilizing the multifunctional super singular element  method was proposed. This approach yields highly accurate numerical solutions for singular stress fields and stress intensity factors, thereby enabling precise trajectory predictions. The implementation of adaptive mesh refinement technology allows for the automatic generation of background meshes for complex geometries based on three distinct intersection strategies applied within the update zone, significantly reducing the complexity of mesh subdivision. Similarly, the primary objective of study [17] was to develop a local multiscale mesh refinement strategy. To capture singularities accurately during crack growth, a localized refinement scheme around the crack tip region was implemented. The formulation integrates elements with a variable number of nodes and their corresponding shape functions to effectively resolve the mismatch problem arising from adjacent elements of different sizes. Various plastic flow theories are frequently employed to model the elastic-plastic behavior of concrete under loading [18–20]. In these theories, the relationship between stress increments and plastic strains is governed by specifying yield and loading surfaces. This approach enables the development of effective incremental-iterative algorithms for analyzing reinforced concrete components under diverse loading scenarios. Under active loading, however, a simpler orthotropic concrete model can be utilized, such as the one outlined in the CEB-FIP Model Code 1990. In [21], an innovative approach to describe crack evolution in quasi-brittle materials, modeled in the framework of isotropic damage mechanics, has been presented. Based on innovative Finite Element Method – Virtual Element Method coupled discretization of the body, this technique gives satisfactory results. They are comparable to the results obtained by well established approaches in the literature, where crack propagation is correctly caught by usually employing remeshing procedures or adding degrees of freedom. The aim of the work [22] was to reduce the computing efforts in the framework of the Global Cracking Elements Method, which involves finite elements with nonlinear interpolation of the displacement field. This aim was achieved with the help of hybrid linear and nonlinear interpolation. The numerical examples have shown that the proposed approach saves a lot of computing time as compared to the original global cracking elements. Besides, with this approach, curved boundaries can be more accurately modeled by adding edge nodes. In [23], to solve the mesh dependency problems of the cracking elements method with a 6-node triangular element (CEM-T6), the numerical formulation and procedures of CEM-T6 were investigated. The characteristic length used in the original CEM-T6 was found to be inaccurate and inconsistent. Hence, a new form of the characteristic length that depends only on the size and shape of the T6 element was presented. With the new form, the mesh dependency of CEM-T6 was greatly mitigated. The study aimed to construct a finite element model of bending reinforced concrete beams taking into account crack formation. An incremental orthotropic model was used for modeling the elastic-plastic deformation of concrete. A stepwise iterative algorithm was used to solve the nonlinear problem. 2.Methods To determine the location and opening width of cracks in a bending reinforced concrete beam, the plane elasticity problem was formulated under physically nonlinear conditions. An orthotropic model was applied for the stress-strain relationship, following the CEB Model Code 1990 guidelines. Under this model, the concrete stresses under uniaxial compression (Fig. 1a) were calculated:                                             (1) a) b)   Figure 1. The stress-strain diagrams for concrete: a) under compression; b) under tension.  is the initial modulus of elasticity of concrete;  = –0.0022;    is the compressive strength of concrete,  is the tensile strength of concrete,  is the ultimate deformation under compression.                                      (2) Figure 2. Biaxial strength of concrete (fcm = 37 MPa). Under biaxial stress condition, the concrete strength was evaluated using Equations (3)–(5), which represent failure criterion curves (Fig. 2). In case of biaxial compression or tension-compression at                                                        (3) under biaxial tension                                                                    (4) under biaxial tension-compression at                                                               (5)      are the greatest, mean and least principal stresses. In the coordinate system aligned with the principal stress axes, the relationship between strain and stress increments was determined as follows:                                            (6) The tangent deformation moduli    expressed as functions of the principal stresses, were determined using the concept of equivalent uniaxial strains                                                                 (7)                                                                      (8)  is the number of stress increments. Tangent modulus of elasticity:                             (9) Equation (7) can be expressed in matrix form as follows:                                                      (10) To determine the tangent moduli  based on equivalent uniaxial deformations  it is necessary to replace  in Equation (1) with the corresponding value of the ultimate concrete strength By substituting  for  in Equation (3), the concrete strength value was obtained.                                                (11) Similarly, by substituting  for  in Equation (3), the following relation was obtained:                                               (12) This approach implies that at each iteration step, the concrete strength in the principal direction under consideration is determined based on the current principal stress in the orthogonal direction. For the biaxial tension-compression state  the following relations were derived in accordance with Equation (5):                                                            (13)                                                             (14) The strain value  must be replaced by the equivalent strain  accompanied by the corresponding Poisson’s ratio  In the case of biaxial compression or tension-compression at  the following relation was obtained:                         (15) Under biaxial tension                                                    (16) Under biaxial tension-compression at  the following relation was obtained:      (17)                                    (18) The shear modulus was determined using Equation (19).                                                           (19) In Equation (6), the relationship between the shear stress and shear strain increments was incorporated as follows:                            (20) The following notations were introduced for the vectors and the concrete stiffness matrix:        (21) Equation (20) can be expressed in matrix form as follows:                                                                 (22) The prime symbol indicates that the equation is formulated in the principal axes The principal stresses and the inclination angle of the crack normal are evaluated as follows:                                                 (23)                                                             (24) A crack forms in a finite element when the principal stress satisfies the following condition: The vector of stress increments  in the global coordinate system  was expressed via the matrix                         (25) Using Equations (22) and (25), the strain vector  in the global coordinate system can be expressed as follows:                                                (26) Within the finite elements where a crack has formed, the concrete deforms elastically, and the matrix  is given by:                              (27) The stress-strain relationship for the reinforcement is represented by a bilinear diagram as follows:                                           (28) where  is the yield strength of the reinforcement,  is the modulus of elasticity of the reinforcement steel, and  is the hardening modulus. 3.Results and Discussion Figure 3 Finite element model.   The stiffness matrix of the rectangular finite element was derived by approximating the stress fields within the element domain using linear functions [12, 24, 25]:                                             (29) The local coordinate system was centered at the finite element origin. The stress functions (29) satisfy homogeneous differential equilibrium equations in the finite element domain for plane elasticity theory. In [24], such approximations were used to obtain the stiffness matrix of a rectangular finite element with a linear stress-strain diagram of the material. Using Equation (29), the tangent stiffness matrix of the finite element is derived, accounting for physical nonlinearity:                            (30) After a crack had formed in the finite element, additional unknown displacements were introduced into its nodes (Fig. 4b), and the stresses on each side of the crack were approximated independently, yielding:                     (31)                                               (32) The superscript in Equation (31) denotes that the corresponding parameters are associated with the finite element subdomains  or  (Fig. 4a). a) b) Figure 4. a) The variants of the crack cd location; b) the unknowns’ number in a node, depending on the crack location [24].   Increments of finite element complementary strain energy:                                      (33) where  is the element thickness. The matrix  (26) for a finite element without a crack is calculated at its geometric center point; its elements depend solely on the parameters      and are assumed to be constant. The tangent matrix  is calculated numerically using the 3-point Gauss quadrature rule:                               (34)    are the weighting factors. The expressions for the matrix components of a cracked finite element were provided in [24]. To this end, the equilibrium equations derived via the principle of virtual displacements were incorporated into the complementary energy increment (Fig. 4) following the approach in [24]:                                           (35) where  is the element nodal displacement increment vector,  is the element nodal reaction vector. Expressions for the matrix  elements, which depends only on the geometric parameters of the finite element, are given in [24]. Each node has two primary and two additional displacements. Based on these nodal displacements, the crack opening width within the finite element was evaluated (Fig. 5). The expressions for evaluating the crack opening width were provided in [24]. a) b) c) d) Figure 5. The basic δu and additional δucr possible displacements of nodes after the crack cd forming: а) δu4,x(y); b) δucr4,x(y); c) δu3,x(y); d) δucr3,x(y). The figure is given in [24].   For the iterative scheme, the iteration index  was incorporated into Equation (35):                                    (36) Expression (36) was transformed as follows:                          (37) where  is the vector of unknowns from the previous iteration. Equating the derivatives of  to zero, yields the following equations:                                 (38) Solving system of Equation (36) yields:                                              (39)                                                          (40)                                                              (41) where  is the finite element tangent stiffness matrix. Figure 6. Reinforcement modeling within the finite element.   If the reinforcement passes through the finite element (Fig. 6), its stiffness matrix must be incorporated into the element stiffness matrix defined by Equation (41). The bond slip of the reinforcement relative to the concrete was neglected, therefore, the end displacements of the reinforcement were expressed in terms of the nodal displacements of the rectangular finite element. Thus, the following relation was obtained:                                                                 (42)                                    (43) The reinforcement stresses are constant along the element length and  is the only unknown parameter for the reinforcement.  is the tangent modulus of elasticity of the reinforcement. Accounting for the reinforcement, equation (40) is expressed as follows:                                                (44) The global system of linear equations for the entire domain is formulated as follows:                                    (45) where  is the global tangent stiffness matrix for the entire region,  is the global nodal displacement increment vector,  is global external load vector,  is the total number of finite elements,  is the number of reinforcement-yielding elements. At each step of the iterative process, the finite element  with the maximum principal stress  was determined. If  a crack initiates within the element. Subsequently, the vector  was updated using the following formula:                                                                   (46) Reducing the step size is permissible because the tangent stiffness matrix is constant. Thus, at each iteration, crack initiation is restricted to a single finite element exhibiting the maximum principal stress. This algorithm allows one to track the sequence of crack formation. Upon crack initiation, additional unknown nodal displacements are introduced into the finite element structure [24], and a new iteration is executed. If no cracks initiation occurs within any finite element, the iteration termination condition is checked. The Euclidean norm was used to calculate the solution residual, and the iteration termination criterion was the defined by the following inequality:                                                                  (47) where  is an acceptable computational tolerance. If the vector  approaches zero during the iteration, the nodal equilibrium conditions of the finite element mesh nodes are satisfied, and the iterative process terminates. Fig. 6 shows the step-by-step algorithm of iterations. 1. , , , , . 2. For  = 1 to number of load step do 3.    . 4.    . Crack=false. 5.     For  = 1 to number of iterations do 6.       Compute for all finite elements:             (9),            .            If (j > 1) .            If (j > 1) and (Crack=false) and ( )              then                 , .                  Terminate the current load step. 7.       Compute: . 8.       Solve the system: . 9.      ,           Compute for all finite elements:           , ,          , ,           principal stresses (23),           10.     Find out , . 11.     If               then                  Crack=true.                   Add the new unknowns to the nodes of finite element .                   Compute for all finite elements:                   , ,                   , ,                    ,                   , ,                    principal stresses  (23).                                 else                    Crack=false.              Compute the finite element’s equivalent deformations           .  12.   End of iterations loop.         Form the load step results.  13. End of load step loop. Figure 7. Step-by-step algorithm of iterations.   To validate the proposed model, a numerical analysis of a reinforced concrete beam subjected to a uniformly distributed load was performed. The beam parameters are listed in Table 1. The reinforcement consists of four 24-mm diameter bars Fig. 3. Table 1. The beam parameters. Length, m h, m b, m fcm, MPa fctm, MPa Eci, MPa Es, MPa Rs, MPa As, m2 ys, m from the bottom 6 0.6 0.3 22 1.805 32500 200000 235 0.00181 0.03     Due to symmetry, the numerical analysis was performed for only half of the beam. To model the support, vertical displacements were restricted at the nodes of the leftmost cross-section. The beam was analyzed under a uniformly distributed vertical load  = 10, 20, 30, 34 kN/m. Fig. 8 shows the evolution of the solution residual (47) during the iterations. The loading step size was set to  = 2 kN/m. The spikes in the residual norm correspond to the initiation of new cracks during the loading process. Once no new crack initiation occurs during a load step, the residual norm approaches zero within just two iterations (as indicated by the nearly vertical lines in Fig. 8). Thus, the number of required iterations is determined chiefly by the total number of initiated cracks. Figure 8. Convergence history of the solution residual. qz = 24 kN/m, grid 20 by 80.   Figs. 9–12 show the numerical results for the beam analysis under various loads. The scale of the crack width visualization is 50 times larger than that of the finite element mesh. The maximum crack width is 0.128 mm under a load of  = 34 kN/m. Because the rightmost crack (numbered 1 in Fig. 12) lies directly on the symmetry plane, its calculated width was doubled to reflect the actual physical crack opening. Figure 9. qz = 10 kN/m. Grid 20 by 80. Figure 10. qz = 20 kN/m. Grid 20 by 80. Figure 11. qz = 30 kN/m. Grid 20 by 80. Figure 12. qz = 34 kN/m. Grid 20 by 80.   The concrete stresses in the X-direction are presented in Fig. 13. Fig. 14 presents the mid-span load-deflection curves obtained using the proposed method in comparison with the LIRA-SAPR simulation results. In LIRA-SAPR, an exponential stress-strain diagram for class B40 concrete was adopted. According to LIRA-SAPR calculations, the maximum tensile stress in the reinforcement is 107182 kN/m2, and the peak compressive stress in the concrete along the X-direction is –10529 kN/m2. These findings are in good agreement with the numerical predictions obtained via the proposed method. Maximum vertical displacements obtained using LIRA-SAPR calculations are fifteen percent smaller than those obtained using the proposed method (Fig. 14). Figure 13. The stresses along the X-axis. LIRA-SAPR. Figure 14. The blue line represents the results of the proposed method; the red line shows the LIRA-SAPR results.   It should be noted that the proposed method does not take into account possible slippage of the reinforcement relative to the concrete, which can lead to increased crack width and displacements. To account for reinforcement bond slip, independent degrees of freedom must be defined for the rebar nodes. The reinforcement-to-concrete interface can be captured via shear spring elements, governed by a prescribed bond stress-slip diagram. 4.Conclusion A finite element model for the non-linear analysis of reinforced concrete beams incorporating crack initiation and propagation has been presented. Plane stress rectangular finite elements were employed to discretize the concrete beam. The finite element stiffness matrix was formulated based on a complementary energy functional. By employing Lagrange multipliers, the functional was augmented with nodal equilibrium equations derived via the principle of virtual displacements. An incremental orthotropic model was applied for the stress-strain relationship of concrete. Crack formation was incorporated by introducing additional unknown nodal displacements and using independent stress approximation functions within the element subdomains separated by the crack. The stiffness matrix of the rectangular finite element was derived by approximating the stress fields within the element domain using linear functions. To solve the non-linear problem, an incremental-iterative algorithm was proposed to track the progressive sequence of crack initiation within the finite elements. The Euclidean force norm was used to evaluate the solution residual. The number of required iterations is determined chiefly by the total number of initiated cracks. Once no new crack initiation occurs during a load step, the residual norm approaches zero within just two iterations. To validate the proposed model, a numerical analysis of a reinforced concrete beam subjected to a uniformly distributed load was performed. The numerical results derived via the proposed method were validated against the LIRA-SAPR simulation data. 10.34910/MCE.144.9 624 crack formation reinforced concrete orthotropic model equilibrium equations additional possible displacements https://engstroy.spbstu.ru/article/2026.144.8/

RAR RUS 14410-14410 University of Baghdad Al-Saadi Semaa semaazeyad1@gmail.com

Baghdad, Iraq

0000-0003-1304-0303 University of Baghdad Karkush Mahdi mahdi_karkush@coeng.uobaghdad.edu.iq

Baghdad, Iraq

Numerical modeling of unconnected piled raft system In recent decades, increasing attention has been directed toward the behavior of unconnected piled raft foundations (UCPRFs) under various soil conditions. In this foundation system, a cushion layer is introduced between the pile heads and the raft, preventing direct structural contact between these components. The cushion provides an intermediate load-transfer medium and facilitates the redistribution of the applied load between the raft and the piles. Understanding this interaction is essential for evaluating the overall response of the foundation and the contribution of its individual components. This study numerically investigated the behavior of UCPRFs using the finite element method implemented in PLAXIS. The analysis considered bearing pressure, foundation settlement, axial stress distribution along the piles, and bending moment response. A parametric investigation was conducted to evaluate the effects of pile length, pile diameter, and raft stiffness on settlement reduction and foundation performance. The results showed that axial stresses in the piles increased with increasing pile length and diameter, demonstrating the influence of pile geometry on load transfer within the foundation system. Furthermore, the maximum axial load occurred at a certain depth below the pile head, indicating that load transfer continued along the upper portion of the pile rather than being concentrated at its head. Within the investigated range, changes in the raft’s modulus of elasticity had no significant effect on settlement reduction. These findings improve the understanding of load-transfer mechanisms in UCPRFs and clarify the relative influence of pile geometry and raft stiffness on their numerical response. 1.Introduction The main concern associated with a piled raft foundation system is the connection between the piles and the raft. Although the full geotechnical capacity of the piles may not be mobilized, significant shear forces and bending moments can be transferred from the raft to the pile heads, potentially resulting in excessive structural stresses or failure [1]. To address these problems, several researchers have proposed the use of unconnected piled raft foundations (UCPRFs), in which the piles are separated from the raft by an intermediate cushion layer [1–5]. In this system, a compacted granular cushion is placed between the raft and the pile heads, allowing the piles to function primarily as subsoil reinforcement rather than as structural components directly connected to the raft [3, 6]. The cushion layer facilitates load redistribution between the raft and the piles and modifies the load-transfer mechanism within the foundation system [2, 7]. The use of a structural fill cushion can also reduce stress concentrations at the pile heads and protect both the pile heads and the underlying soil from localized structural damage [6–9]. A structural fill cushion can be used to maintain the separation between the raft and the unconnected piles [12, 13]. Typically, the cushion consists of a sand–gravel mixture compacted in layers between the underside of the raft and the pile heads. This cushion plays an important role in mobilizing the bearing capacity of the underlying soil and modifying the load-transfer mechanism between the raft, piles, and surrounding soil. The granular cushion redistributes the applied stresses beneath the raft and reduces the stress concentration transferred directly to the pile heads. Tradigo et al. [14] investigated the variation of skin friction along the pile and the location of the maximum axial force,  in unconnected piled raft foundations. The settlement of the raft center, pile, and soil at the pile–soil interface was represented by    and  respectively. For the connected piled raft configuration,  and  were approximately equal because the raft and piles were directly connected, whereas  was lower than  and  Under these conditions, positive skin friction developed along the pile, resulting in the maximum axial force occurring near the pile head. In contrast, different settlement patterns were observed in the unconnected piled raft configuration. Because the pile and raft were separated by the cushion layer, relative displacement between the pile and the surrounding soil varied along the pile length, resulting in a transition between negative and positive skin friction. According to Tradigo et al. [14],  in the lower portion of the pile, resulting in positive skin friction extending toward the pile base. The transition between negative and positive skin friction defines a neutral plane, where the relative displacement between the pile and surrounding soil is approximately zero and the axial force reaches its maximum value,  Therefore, the axial force distribution along the unconnected pile is governed by the combined effects of negative and positive skin friction, as illustrated in Fig. 1. The researchers concluded that the negative skin friction observed by Cao et al. [2] is attributed to the existence of  at the upper part of the pile shaft. While, along the lower portion of the pile shaft;  so that positive skin friction develops up to the pile base. Figure 1. Settlement of the CPR and UPR [14]. The main advantages of unconnected piled raft (UPR) foundations can be summarized as follows: A flexible cushion layer is placed beneath the relatively rigid raft to redistribute the applied external loads and modify the stress distribution between the piles and the surrounding soil [15]. Because the piles are not rigidly connected to the raft and cannot effectively transfer significant bending moments from the raft, they primarily function as subsoil reinforcement elements rather than conventional structural components [16]. The UPR system can provide an attractive alternative to conventional ground-improvement techniques, such as preloading, installation of vertical drains, electro-osmosis, grouting, and lightweight fills. The system can be implemented relatively quickly and generally does not require extensive soil replacement [17]. The UPR system uses vertical piles to improve the shear resistance and stiffness of the subsoil and reduce the potential for total and differential settlement. The granular cushion layer also limits the transfer of excessive shear forces and bending moments between the raft and the piles [18]. The cushion layer reduces stress concentrations at the pile heads and limits the direct transfer of excessive shear forces and bending moments between the raft and the piles, thereby improving the overall performance of the foundation system [19]. Despite these advantages, several limitations should be considered when using UPR foundations: The UPR system may provide limited resistance to horizontal and lateral loads because the piles are not rigidly connected to the raft. Therefore, its application to structures subjected to significant lateral loads, such as structures located in areas with strong wind or seismic loading, requires careful assessment of lateral stability and deformation, as illustrated in Fig. 2 [2]. The stiffness and strength of the cushion layer should be carefully controlled to ensure adequate load transfer between the raft, piles, and surrounding soil. An improperly designed or poorly compacted cushion may result in excessive deformation and an inefficient distribution of load to the piles [15]. Figure 2. Transfer mechanisms for horizontal loads, (a) piles structurally connected to raft; (b) piles structurally unconnected from raft [2]. 2.Methods Using PLAXIS 3D [20], a finite element analysis was conducted to investigate the behavior of an unconnected piled raft foundation (UPRF) resting on sand. The Mohr–Coulomb (MC) constitutive model was employed to simulate the mechanical behavior of the soil and cushion layer. The soil domain was modeled with dimensions of 700 × 700 × 700 mm, with a single borehole located at the center of the model. The soil volume was discretized using 10-node tetrahedral elements. The UPRF was subsequently analyzed under static loading. The overall numerical model and its boundary conditions are illustrated in Figs. 3–5. The standard Mohr–Coulomb constitutive model available in the PLAXIS 3D material library was used to represent both the sand and cushion sand. Standard fixities were applied to the model boundaries; the lateral boundaries were restrained against horizontal movement, while the bottom boundary was fixed in all directions. The material properties of the soil were defined in the PLAXIS 3D material data sets based on the results of laboratory tests. The raft and piles were modeled as linear-elastic structural elements. The Mohr–Coulomb elastic–perfectly plastic constitutive model was used to simulate the soil layers and cushion material. Table 1 presents the experimentally determined soil properties, whereas Table 2 summarizes the material properties of the raft and piles. A relatively coarse mesh consisting of 57,960 nodes and 36,429 soil elements was initially generated. The mesh size was selected based on mesh-convergence considerations, while local mesh refinement was applied around the foundation components to obtain a more accurate estimation of the foundation response. The unconnected piled raft foundation (UPRF) consists of four main components: the raft, piles, surrounding soil, and cushion layer. The raft was modeled as a solid volume with dimensions of 11.5 × × 11.5 cm and a thickness of 0.03 cm. The pile group consisted of four piles, each with a diameter of 1.6 cm and a length of 50 cm. Each pile was represented as a solid volume element. The interaction between the piles and the surrounding soil was governed by both shaft and base resistance. Special interface elements were introduced to simulate the pile–soil interaction and to account for the shear behavior at the pile–soil contact. The cushion layer, which represents the fourth component of the UPRF system, had dimensions of 23 × 23 × 5.8 cm. Interface elements were also employed to simulate the interaction between the cushion and the surrounding soil. These interface elements allow the thin zone of highly sheared material that develops at the contact between different materials to be represented in the finite element model. Table 2 presents the material properties of the raft, pile cap, pile shafts, foundation soil, and cushion layer used in the PLAXIS 3D finite element analysis.   (a)     (b)   Figure 3. a): Boundary conditions of model and b) Finite element mesh of the model. Figure 4. Finite element mesh of the unconnected piled raft foundation. Figure 5. 3D modeling of unconnected pile raft model, H/B=0.5. Table 1. Soil properties considered in MC model analysis. Parameter Unit Soil bed Cushion Type of drainage – drain drain Dry unit weight, γd kN/m3 14.8 17 Angle of internal friction, ϕ degree 30o 40o Dilatancy angle, ψ degree 0 10.2 Cohesion, c kPa 0.2 0.2 Young’s modulus, Es MPa 10 60 Poisson’s ratio, μ – 0.3 0.35     Table 2. Material properties of (raft and pile cap, and pile shaft, bed soil, cushion) used in PLAXIS 3D. Model Parameters Unit Steel square plate Aluminum pile shaft Model type Linear Elastic Linear Elastic Linear Elastic Diameter, D m ----- 0.016 Wall thickness, t m 0.003 0.001 Unit weight, γp kN/m3 70 27 Young’s modulus MPa 7.5×104 60×103 Poisson ratio, μ – 0.29 0.30   3.Results and Discussion 4.1. Effect of Pile Length To evaluate the effect of pile length on the performance of the unconnected piled raft foundation (UPRF), five different pile lengths 10, 20, 30, 40, and 50 cm were considered in the numerical analysis. According to Eq. (1), the piles with lengths of 10, 20, 30, and 40 cm were classified as rigid piles, whereas the 50-cm-long piles were classified as flexible piles, based on the criterion proposed by Fleming et al. [21], as expressed in Eq. (1).                                                             (1) where  and  are modulus of elasticity of pile and shear modulus of bearing soil layer. Based on the results presented in Fig. 6, increasing the pile length resulted in a reduction in foundation settlement, which is consistent with the findings reported by [17]. The maximum settlement decreased, while the bearing capacity of the unconnected piled raft foundation (UPRF) increased as the pile length increased. The beneficial effect of increasing pile length on reducing foundation settlement was particularly evident in the results reported by [12]. The longer piles transferred a greater proportion of the applied load to the pile group, thereby reducing the settlement and bending moments of the model raft. This observation is also consistent with the findings of Cao et al. [2]. The variation in axial stress distribution along the piles is presented in Fig. 7. The results indicate that the axial stress generally increased with increasing pile length. However, the axial stress near the lower end of the pile decreased as the pile shaft became longer. In addition, the location of the neutral plane shifted to a greater depth as the pile length increased, which agrees with the observations reported in [12]. Furthermore, in the unconnected piled raft system, increasing the pile length caused the location of the maximum axial stress to shift downward from the pile head to a specific point along the pile shaft, consistent with the findings reported in [22,23]. Fig. 8 presents the bending moment distribution along the centerline of the model raft under loading for the different pile lengths. The results indicate that increasing the pile length did not produce a significant reduction in the bending moment within the raft foundation. A similar observation was reported by Cao et al. [2]. The total displacement distributions obtained from the finite element analyses for all investigated pile lengths are presented in Fig. 9. Figure 6. Variations of bearing pressures with relative settlements for different pile length of unconnected piled raft.   Figure 7. Distribution of axial stress along the center pile for unconnected pile raft for different pile length. Figure 8. Bending moment diagrams along center line of the raft for unconnected pile raft for different pile length. (a) (b) (c) (d) (e)   Figure 9. The total displacement finite element (scale up tp 5 times) of a) L = 10 cm, b) L = 20 cm, c) L = 30 cm, d) L = 40 cm, and d) L = 50 cm. 4.2. Effect of Pile Diameter To investigate the effect of pile diameter on the behavior of the unconnected piled raft foundation (UPRF), numerical analyses were conducted using pile diameters of 12, 16, 20, and 24 mm, while maintaining a constant pile length of 50 cm. The corresponding slenderness ratios (length-to-diameter ratio, L/D) were 41.67, 31.25, 25, and 20.83, respectively. According to the criterion given by Eq. (1), the piles were classified as rigid or flexible based on their slenderness ratios. Fig. 10 presents the relationship between the relative settlement (S/B) (%), and bearing pressure (q) for the different pile diameters. The results indicate that increasing the pile diameter while maintaining a constant pile length resulted in a reduction in the overall foundation settlement. This behavior can be attributed to the increase in pile stiffness associated with the larger diameter, which improves the load-transfer capacity of the pile group. In addition, increasing the pile diameter reduces the clear spacing between the piles for a constant raft configuration, thereby influencing the interaction between the piles and the surrounding soil [24, 25]. Fig. 11 presents the variation of axial stress along the pile shafts with depth for the different pile diameters. The results show that the axial stress distribution along the pile shafts generally increased with increasing pile diameter, indicating enhanced load transfer through the larger-diameter piles. Fig. 12 illustrates the bending moment distribution along the centerline of the model raft for the different pile diameters. The results indicate that increasing the pile diameter had no significant effect on reducing the bending moment in the raft; however, a slight variation in the bending moment was observed as the pile diameter increased. Finally, the total displacement distributions obtained from the finite element analyses for all investigated pile diameters are presented in Fig. 13. Figure 10. Variations of bearing pressures with relative settlements for different pile diameter of UPR. Figure 11. Distribution of axial stress along the center pile for UPR for different pile diameter. Figure 12. Bending moment diagrams along center line of the raft for UPR for different pile diameter. (a) (b) (c) (d)   Figure 13. The total displacement finite element (scale up to 5 times) for a) d=12, b) d=16, c) d=20, and d) d=24. 4.3. Effect of Modulus of Elasticity of Raft To investigate the effect of raft rigidity on the behavior of the unconnected piled raft foundation (UPRF), particularly the relationship between relative settlement (S/B) and bearing pressure, five different values of the raft elastic modulus  were considered: 5, 7.5, 10, 60, 75, and 180 GPa. The raft–soil relative stiffness,  was used to quantify the relative rigidity of the raft with respect to the supporting soil. According to [26], an expression for determining the raft–subsoil stiffness ratio  was proposed, as given in Eq. (2).                                                        (2) where  – width of the raft,  – thickness of the raft,  – length of the raft,  – Poisson’s ratio of the soil. According to the above equation, the model raft with a thickness of 3 mm can be classified as a flexible raft when the raft–soil stiffness ratio,  is relatively low, with values of 0.486, 0.727, and 0.970. In contrast, the raft can be considered rigid when  is sufficiently high, with values of 5.82, 8.10, and 17.46. Fig. 14 illustrates the relationship between bearing pressure and relative settlement (S/B) for the different values of raft elastic modulus. The results indicate that increasing the elastic modulus of the raft had no significant effect on the overall settlement of the unconnected piled raft foundation. The results clearly indicate that raft rigidity had a negligible effect on the overall foundation behavior, and no significant improvement in bearing capacity was observed with increasing raft stiffness. Furthermore, as shown in Fig. 15, increasing the elastic modulus of the raft did not result in a significant increase in the axial stress distribution along the pile depth. These results suggest that, within the range of raft stiffnesses investigated, increasing raft rigidity has a limited influence on load transfer to the piles and the overall response of the unconnected piled raft foundation. Fig. 16 illustrates the bending moment distribution along the centerline of the model raft for the different modulus of elasticity of pile. The results indicate that increasing the modulus of elasticity of pile foundation had significant effect on reducing the bending moment in the raft. The total displacement distributions obtained from the finite element analyses for all investigated pile diameters are presented in Fig. 17. Figure 14. Variations of bearing pressures with relative settlements for different the modulus of elasticity of the raft of UPR.     Figure 15. Distribution of axial stress along the center pile for different the modulus of elasticity of the raft of UPR. Figure 16. Bending moment diagrams along center line of the raft for different the modulus of elasticity of the raft of UPR. (a) (b) (c) (d) (e)   Figure 17. The total displacement finite element (scale up to 5 times) of a) E = 5 GPa, b) E = 7.5 GPa, c) E = 10 GPa, d) E = 60 GPa, and e) E = 180 GPa. 4.Conclusions The maximum settlement is reduced and the bearing capacity of an unconnected pile raft foundation is increased as the pile length is increased, As the pile length increases, the settlement reduces. The pile length increases in the unconnected system, the position of the maximum axial stress shifts lower to a specific location below the pile head. An increased pile length did not lead to a significant reduction in the bending moment in center line of raft foundation. The axial stress along pile shaft was increases as the pile diameter increased. The effectiveness of increase the diameter of piles for bending moment reduction along the center line of the model raft is not significantly changed. It is clear that the raft elastic modulus has no effect on the behavior of unconnected piled raft, the bearing capacity has been not improved. The bending moment reduction along the center line of the model raft is significantly changed, increase with increase raft modulus of elasticity. 10.34910/MCE.144.10 624 unconnected piled raft cushion numerical analysis Plaxis sandy soil https://engstroy.spbstu.ru/article/2026.144.9/