Triangular Layered Finite Element Method for Reinforced Concrete Slabs

Full Text

1. Introduction From the 2010-2025 construction period, reinforced concrete (RC) slabs are essential structural components, serving as flooring systems while carrying vertical loads. Accurate performance analysis is critical to ensure both safety and cost-effectiveness in RC building designs [1; 2]. However, predicting the nonlinear response of RC slabs remains a significant challenge due to the complex behavior of reinforced concrete structures, making it an active research area [3; 4]. This complexity arises from material nonlinearity in concrete and steel, cracking, imperfect bond-slip behavior, and time-dependent effects such as creep and shrinkage [5]. Several material models have been developed to capture the layered nonlinear behavior of RC slabs. For reinforcing steel, a uniaxial elastic-plastic stress-strain relationship is typically employed, exhibiting symmetrical response under both tension and compression. Similarly, concrete behavior can be effectively represented using a bilinear stress-strain approximation that incorporates tensile capacity [5; 6]. To address the limitations of conventional 3D finite element models, researchers have developed innovative layered methods [7; 8]. Unlike simplified effective stiffness approaches, these layered FE models enable precise prediction of ultimate bending and shear capacity in RC slabs [9; 10]. The methods employ triangular plate elements composed of perfectly bonded, superimposed equivalent layers representing both concrete and steel reinforcement. This layered triangular element facilitates detailed tracking of concrete failure mechanisms (including cracking and crushing) and progressive steel yielding throughout the slab depth [11; 12]. Although numerous layer-based FE models exist for RC slab analysis, current implementations remain predominantly limited by Kirchhoff-Thin Plate Theory (KTPT) assumptions [13; 14]. Current research indicates a strong preference for displacement-based formulations in finite element modeling of RC slabs and plate structures [15; 16]. While these layered triangular elements derive stress components indirectly through numerical differentiation of displacement fields, the resulting post-processed stresses particularly out-of-plane components, often demonstrate reduced accuracy compared to their displacement counterparts. In contrast, advanced layered FE formulations for composite structures treat stresses and displacements as independent variables, thereby achieving superior stress prediction accuracy [17; 18]. Although layered finite element formulations have been widely adopted for laminated composite analysis, their application to nonlinear RC slab modeling remains relatively limited. Recent advances by Liguori et al. [19; 20] introduced a mixed finite element formulation for nonlinear material analysis of RC shell structures. These layered triangular elements utilize conventional Mindlin-Reissner plate theory to describe displacement fields in RC structural analysis. In contrast to their methods the present study implements an innovative global-local kinematic framework for displacement field representation in plate analysis. Notably, while Liguori et al. [20] treated both membrane/flexural stresses and displacement fields as primary unknowns in their formulation, the current approach adopts a distinct strategy for variable selection. Wang et al. proposed an efficient quasi-three-dimensional mixed finite element formulation based on a refined layered global-local plate theory for nonlinear analysis of RC slabs. In this approach, the cross-section is discretized into distinct concrete and steel layers, with each layer modeled as an independent plate element characterized by unique material properties [21]. This study introduces an innovative computational framework for nonlinear analysis of RC slabs, based on an advanced triangular-layered global-local plate theory formulation. The proposed triangular layered plate methods offer optimal computational advantages, combining superior geometric flexibility with adaptive finite element analysis capabilities for RC slab modeling. The framework employs a 3-node triangular composite plate element augmented with additional nodal degrees of freedom to explicitly represent out-of-plane stress components. While requiring additional field variables, this approach enables direct computation of through-thickness stress distributions during nonlinear solution procedures. The formulation is derived through a parameterized mixed variational principle, providing rigorous mathematical foundations for the methods The RC slab is modeled as an assembly of perfectly bonded concrete and steel layers, with material nonlinearities addressed through: (1) a smeared crack formulation for concrete behavior, and (2) elasto-plastic theory for steel reinforcement response. 2. Methods For reinforced concrete slab elements, the principles of membrane and plate bending theory exist, as will be demonstrated in the subsequent steps. 2.1. Membrane Element Analysis For the membrane component, a standard 3-node triangular element is defined by its node numbering and their (x, y) coordinates. (Figures 1 and 2) [6; 13]. (1) where (uᵢ, vᵢ) represent the horizontal and vertical displacements at node i, and Nᵢ denotes the corresponding shape function for that node. The shape functions for the 3-noded triangular element are derived as follows: (2) The system was solved for coefficients, and substituting these solutions back into Equation (2) produces: where is the element area and, The coefficients and are determined through cyclic permutation of the indices . A comparison of Equation (2) with Equation (1) yields the explicit expressions for the shape functions: (3) Figure 1. Discretization of a structure into 3-noded triangular elements S o u r c e: made by H. Werkle [6]. Figure 2. Shape functions for the 3-noded triangular element S o u r c e: made by E. Oñate [13]. 2.1.1. Membrane-Induced Strain The strain components () are computed via differentiation of the displacement fields u(x,y) and v(x,y) represented by their respective shape functions: , (4) , (5) where . 2.1.2. Membrane-Induced Stress The stresses in the element are calculated from the strains by applying Hooke’s law, as shown in [11; 12]: or , and with the strains defined in (eq. 5), as: , (6) where . These stresses are also referred to as membrane stresses [13]. 2.1.3. Membrane Stiffness Formulation Based on the stresses obtained from Equation (6), which were derived from displacement shape functions, the equivalent nodal forces are calculated using the principle of virtual displacements. The element stiffness submatrix , representing the interaction between nodes and within the element, is typically calculated as: (7) substituting Equation (5) and (6) into Equation (7), yields: . For a homogeneous material, the integrand in Equation (8) remains constant, leading to: . (8) 2.2. Component for Bending The Reissner - Mindlin plate theory is an advanced plate theory that incorporates shear deformation effects. This theory is commonly preferred for formulating finite plate elements. Plate deformations are described by the vertical displacement () and the rotational angles at each point on the plate. Consequently, every node in the plate element possesses three degrees of freedom: one translational displacement () and two rotational components . The corresponding nodal forces consist of a transverse force () and two bending moments (). Figures 3, а, b show that, this 3-node element possesses nine degrees of freedom in total. The shape functions of a 3-node triangular element are constructed through bilinear interpolation of the nodal variables [6; 12]. , (9) where u, v w a b 1 2 S o u r c e: made by E. Oñate [13]. S o u r c e: made by R.Wang [21]. Figure 3. 3-node triangular element: a - Triangular plate elements; b - Graphical illustration of the displacement field: 1 - in plane components, 2 - transverse component ; (10) N are denoted as shape function. 2.2.1. Bending Strain Components The strain state of a plate element is determined by its curvature components and transverse shear deformations [5; 6]: , and shear strain or ; (11) 2.2.2. Stiffness Matrix for Plate Bending The stiffness matrix establishes the proportional relationship between applied nodal forces/moments and resulting nodal displacements/rotations. The virtual displacement field utilizes identical interpolation functions as the real displacement field, satisfying: (12) The virtual curvatures () and virtual shear strains () derived from the virtual displacement field are: or ; or . The internal virtual work comprises two terms: the product of real bending moments and their corresponding virtual curvatures, plus the product of actual shear forces and associated virtual shear angles. (13) The virtual work done by external loads consists of force-displacement and moment-rotation products at all nodes: (14) The principle of virtual work is satisfied when the sum of internal virtual work equals external virtual work: (15) (Bending) (Shear) The plate element’s stiffness matrix [] comprises two distinct sub-matrices: a bending stiffness component and a shear stiffness component. As a conforming element, it maintains continuity across both displacement and rotation fields. 2.3. Mathematical Principle Among existing analytical approaches, the layered finite element method (LFEM) with triangular formulation demonstrates high efficacy in evaluating the flexural behavior of RC slabs. The methods employs a stratified representation of plate elements, discretizing the cross-section into distinct concrete and steel layers, as illustrated conceptually in Figure 4 [5; 22; 23]. This layered approach facilitates accurate modeling of stress distributions across the RC slab using an assembly of plane stress elements. Figure 4. Typical triangular plate element for reinforced concrete plate structures S o u r c e: made by D.A. Mawlood. In this modeling approach, the RC slab is idealized as a composite system of perfectly bonded, uncracked concrete layers and equivalent steel layers. The reinforcement is represented using a smeared-layer approximation, with horizontal steel layers positioned at the centroidal levels of the actual reinforcement bars. The computational model employs the same number of smeared layers as physical reinforcement layers in the cross-section. Each equivalent steel layer is assigned uniaxial material properties corresponding to the orientation of the actual rebars. The equivalent thickness () of each steel layer is determined from the rebar cross-sectional area () and spacing () according to the relationship: [21]. Reddy’s Third-Order Shear Deformation Theory (TSDT) overcomes the fundamental limitations of classical plate theories by eliminating both the normal hypothesis constraint and the requirement for planar cross-sections to remain plane after deformation [11]. The theory is founded on the following kinematic relations: (16) The kinematic functions , , , , , and represent undetermined parameters that characterize the cross-sectional warping deformation. Within the TSDT framework, these functions collectively introduce seven independent displacement variables. Specifically, the bending rotation components and describe the slope of the warped cross-section at the neutral plane , while the remaining variables account for higher-order deformation effects [11]. where . Figure 5. Unreformed and deformed plate segment S o u r c e: made by C. Mittelstedt [11]. The displacement field is mathematically represented by Equations (1) in the following form: (17) where . Consequently, Reddy’s Third-Order Shear Deformation Theory incorporates just three degrees of freedom, as depicted in Figure 5. 2.4. Interconnections Between Layers 2.4.1. Strain Distribution Field Based on the derived displacement-strain relationships, the strain components in the i-th layer of the RC slab can be determined [11], as illustrated in Figure 5. ; ; ; ; ; . These can be represented in vector form as: ; ; ; . The strain field can be expressed as follows: (18) 2.4.2. Stress Fields in the Out-of-Plane Direction The normal stress components are obtained from the strain field by applying Hooke’s law, as given below: [6; 13] ; . Using the strain definitions from Equation (18), the strains can be expressed as: . (19) The transverse shear stress components for each layer in the reinforced concrete slab are determined using the following expressions: (20) where represents the elastic modulus of the i-th layer in the reinforced concrete slab system, and denotes the Poisson’s ratio characteristic of the reinforced concrete layers. The layer-wise constitutive formulation accounts for material heterogeneity through distinct elastic modulus, and Poisson’s ratios for each layer . The governing equations employ standard stress resultants obtained via thickness integration of stress components [24]: . (21) Here’s a rigorous academic formulation of the additional force/moment resultants in TSDT: ; . (22) 2.4.3. Layer Interaction Mechanics in Reinforced Concrete Slabs 2.4.3.1. Layer of Steel Reinforcement The reinforcing steel layers exhibit elastoplastic behavior, characterized by an idealized bilinear response. For the i-th reinforcement layer in the pre-yield regime, the constitutive [11] relationship is expressed as: , where () is represented by Equation 19. The sectional properties for steel; . (23) In Equation (23), represents the elastic modulus of the i-th reinforcing steel layer. Upon yielding, the constitutive relationship transitions to a plastic regime, with the post-yield behavior described by the following incremental formulation: . The term refers to the plastic modulus of the steel layer after it has yielded [21]. 2.4.3.2. Concrete Layer Properties In its uncracked state, the concrete material exhibits isotropic, homogeneous linear elastic behavior. The constitutive relationship governing the i-th concrete layer’s pre-cracking response is expressed as: , (24) where is defined by Equation (18), and . The transverse shear stress components andin each concrete layer can be evaluated through application of the appropriate constitutive relationship: (25) where denotes the elastic modulus of the i-th concrete layer, and represents the Poisson’s ratio characterizing the concrete material’s transverse strain response. When the principal stress state exceeds the concrete’s tensile capacity (), the elastic constitutive relations (Eqs. 24-25) no longer apply. This investigation adopts a smeared crack formulation [24] to model post-cracking concrete behavior. The smeared crack approach necessitates a material symmetry transition from isotropic to orthotropic behavior in the local coordinate frame. Here, the -axis normal to the crack plane defines the material softening direction, while the h- and -axes plane (aligned with principal stresses) maintains elastic stiffness. In the post-cracking phase, the constitutive relationship for the i-th concrete layer transforms to an orthotropic formulation in the local crack-aligned coordinate system, expressed as: ; (26) . (27) Note that () represents the shear retention factor, which is utilized to model the effects of aggregate interlock. 2.5. Mixed Element Stiffness Formulation The element stiffness matrix components can be decomposed into membrane and bending contributions as follows: ; (28) ; . The stiffness contribution exhibits material-specific behavior: the steel reinforcement provides only membrane (in-plane) stiffness, while the concrete contributes to both membrane and bending (flexural) resistance. ; . (29) where and represent the membrane and bending stiffness matrices for the element, respectively. 2.6. Discretization of Finite Elements The reinforced concrete slab is modeled using a three-node, triangular layered plate element that accounts for thickness effects. The development of this composite finite element requires discrete approximations of three key field quantities: (1) geometric configuration, (2) displacement fields, and (3) stress distributions. The proposed layered plate formulation incorporates 15 degrees of freedom (DOFs) per element, with the complete nodal configuration detailed in Figure 6. Figure 6. Meshing process of the RC slab using layered plate elements S o u r c e: made by D.A. Mawlood. 2.7. Load Distribution Elements The element loads are converted into equivalent nodal loads that yield identical external virtual work under virtual displacements as the original loads. This relationship is defined by Equations (10) and (15): or . The reinforced concrete slab, subjected to element loads combining membrane and bending effects, can be expressed as: , where: dm, db…membrane (u1, u2, u3, v1, v2, v3) and bendingelement displacement; Pm, Pb ……membrane and bending element loads. The nonlinear algebraic system is solved using a mixed-step iterative method that combines incremental loading with Newton - Raphson equilibrium iterations. Figures 7 and 8 illustrate how this approach applies loads incrementally while performing iterative corrections at each step to satisfy equilibrium conditions. Although the Newton-Raphson method improves solution accuracy, it requires additional computational effort. The computational algorithm for the non-incremental Newton - Raphson method (Figure 8) executes the following sequence: 1. In the initialization phase, the structure is loaded with , followed by computation of the first displacement approximation according to: , (30) where the global stiffness matrix is computed using the initial elastic modulus E₀. 2. From the computed displacements, element stresses (or strain ) and updated moduli Eᵢ⁽ᴺ⁾ are determined. Equilibrium verification with the updated stiffness matrix yields residual forces: . (31) 3. The residual forces induce corresponding corrective displacements: . (32) 4. The iterative correction process computes successive deflections , where each step generates residual forces and displacement increments , until equilibrium is attained. The algorithm’s general form is given by: for (33) The iterative process terminates when the residual forces diminish to a negligible magnitude. The total plate deflection is subsequently obtained through superposition of all incremental displacement components [5]. . (34) The described numerical procedure is successively applied to all load increments (). Figure 7. Step iteration or mixed procedure S o u r c e: made by D.A. Mawlood. Figure 8. Iterative tangent stiffness procedure S o u r c e: made by D.A. Mawlood. 3. Results and Discussion This work presents a MATLAB implementation, based on the preceding theoretical framework, for nonlinear layer-wise finite element analysis specialized for triangular element formulations. To evaluate computational performance, the proposed method was applied to analyze three experimentally validated reinforced concrete slab specimens. All test cases had been previously characterized under controlled laboratory conditions, enabling direct comparison between numerical predictions and experimental results. 3.1. Analysis of One-Way Reinforced Concrete Slabs (S1) A semi-precast one-way RC slab with dimensions 75 mm × 600 mm × 1650 mm was analyzed using the proposed layered finite element method with triangular elements. The slab incorporated mesh reinforcement consisting of 12 mm diameter steel bars spaced at 200 mm center-to-center, with a 25 mm concrete cover. Figure 9 illustrates the slab’s geometric configuration, loading conditions, and boundary constraints. The material properties were defined as follows: concrete with a modulus of elasticity of 26.420 GPa, Poisson’s ratio of 0.15, and compressive strength of 31.6 MPa; steel reinforcement with an elastic modulus of 190 GPa, Poisson’s ratio of 0.3, and yield stress of 535 MPa. This configuration replicates the experimental setup by Mohamed et al. [25], employing a two-point loading system with 516.7 mm spacing at mid span. The loading was applied through a hydraulic jack on a spread steel beam to create a pure bending region, with continuous load monitoring via a calibrated load cell. Strain gauges and LVDTs provided comprehensive deformation measurements through a high-frequency data acquisition system. Taking advantage of symmetry, the finite element analysis modeled only half of the slab structure. Mesh convergence studies determined an optimal 3×4×2 grid of triangular laminated plate elements, with the thickness layered into six layers (five concrete layers and one equivalent steel layer). Figure 10 illustrates this configuration. Figure 9. Schematic of the one-way RC slab: geometrical parameters, loading, and boundary conditions S o u r c e: made by D.A. Mohamed et al. [25]. 1 2 Figure 10. FE mesh of semi precast of reinforced concrete slab with toping concrete 1 - plan discretization; 2 - cross section discretization S o u r c e: made by D.A. Mawlood. Figures 11 and 12 compare the predicted load-deflection response with experimental data from Mohamed et al. and 3D ABAQUS simulations [25]. The proposed model demonstrates excellent agreement, showing a maximum deflection prediction error of 11.2% while maintaining a 99% correlation coefficient for the load-deflection relationship. Displacement (mm)Load (kN) Figure 11. Load-deflection curve of the one-way RC slab at the mid-span S o u r c e: made by D.A. Mawlood. Figure 12. Deflection chat area S o u r c e: made by D.A. Mawlood. 3.2. Two-Way Reinforced Concrete Slab with Single-Layer Reinforcement (S2) The specimen had dimensions of 2.2 meters by 2.2 meters and a depth of 160 millimeters. All slabs had a loaded span of 2 meters in each direction, with a 0.1-meter overhang extending from the center of the supports on both sides. This study examines the two-way square reinforced concrete slab tested by Sara Nurmi et al. [26]. A point load is applied to the center of the slab, which is supported at all four corners. The tensile region of the reinforced concrete slab contains bidirectional steel reinforcement ( and directions), with equal reinforcement ratios of . The mechanical properties of the slab are presented in Table 1, while Figure 13 depicts the geometric configuration and reinforcement arrangement. A central load was gradually applied to the slab using a hydraulic jack during testing, as conducted by Sara Nurmi et al. Deflections were measured at multiple locations, including the slab center, using LVDT sensors in Sara Nurmi’s experimental setup [26]. Table 1 Material parameters of two-way slab with one layer of reinforcement Material Elastic modulus, MPa Poisson’s ratio Yield stress, MPa Compressive strength, MPa Concrete 28,200 0.15 36 Steel 190,000 0.3 450 - S o u r c e: made by D.A. Mawlood. Figure 13. Geometrical parameters and reinforcement details of two way reinforced concrete slab S o u r c e: made by Sara Nurmi’s [26]. Figure 14 shows that the RC slab cross-section was discretized into five concrete layers, complemented by two equivalent steel layers representing reinforcement in orthogonal directions (x and y). Figures 15 and 16 show the slab deflections computed using the proposed layered approach with triangular element discretization, along with the corresponding numerical results. For validation purposes, Figure 15 compares the model predictions with both experimental data from Nurmi et al. [26] and nonlinear finite element results obtained from Abaqus simulations. The comparative analysis demonstrates the effectiveness of the layered model in predicting deflection responses across the complete loading spectrum, from serviceability conditions to ultimate capacity. Figure 14. FE mesh of RC Slab: cross-section discretization S o u r c e: made by D.A. Mawlood. The benchmarking study reveals that the proposed layer method achieves superior accuracy in predicting two-way slab behavior compared to conventional Abaqus nonlinear solutions. The proposed model demonstrates excellent agreement, showing a maximum deflection prediction error of 2.59% while maintaining a 99.6% correlation coefficient for the load-deflection relationship. Load (kN) Displacement (mm)Figure 15. Load-deflection curve in the center of RC slab Figure 16. Deflections point S o u r c e: made by D.A. Mawlood. S o u r c e: made by D.A. Mawlood. 3.3. Two-Layer Reinforced Two-Way Concrete Slab (S3) The experimental investigation conducted by Yao Xiao et al. examined a two-way RC slab system featuring dual reinforcement layers and full peripheral restraint. The test specimen comprised a 1200×1200×100 mm slab subjected to center-point loading. Key experimental parameters including material characteristics, are summarized in Table 2. The slab’s reinforcement configuration consisted of orthogonal steel reinforcement distributed in both top and bottom layers, with detailed arrangement illustrated in Figure 17 [27]. Table 2 Material parameters of two-way reinforced concrete slab Material Elastic Modulus, MPa Poisson’s ratio Longitude reinforcement diameter, mm Yield stress, MPa Compressive strength, MPa Concrete 30,784 0.15 - - 42.9 Steel 190,000 0.3 10 576 - S o u r c e: made by D.A. Mawlood. Clear cover (15 mm) and all reinforcement used diameter Փ10mm Figure 17. Reinforced concrete slab details S o u r c e: made by Yao Xiao et al. [27]. The two-way slab was reinforced with 10 mm diameter deformed steel bars (Grade 500) in orthogonal arrangements for both top and bottom layers, maintaining a uniform 15 mm concrete cover throughout. Figure 17 details the cross-sectional reinforcement layout and corresponding finite element discretization scheme. Figure 18 shows the load-deflection response at the slab center, capturing the complete nonlinear behavior from initial loading to ultimate capacity. This study presents the experimental results obtained by Yao Xiao et al., along with numerical results from nonlinear ABAQUS simulations and predictions from the triangular-shaped layer method. The load-deflection relationship predicted by the proposed methods shows excellent agreement with experimental observations. Compared to the nonlinear ABAQUS simulations, the triangular layer method demonstrates comparable accuracy in predicting both the ultimate load capacity and maximum deflection of the RC slab. The proposed model demonstrates a 7.32% error in maximum deflection prediction while maintaining a strong correlation coefficient of 99.6% between the predicted and experimental curves (Figures 19-20). Figure 18. FE mesh of RC Slab: cross-section discretization S o u r c e: made by D.A. Mawlood. Load (kN) Deflection (mm) Displacement (mm) Deflection Point Figure 19. Load-deflection curve in the center of RC slab S o u r c e: made by D.A. Mawlood. Figure 20. Deflection point S o u r c e: made by D.A. Mawlood. Figures 21-28 present the ABAQUS-simulated deflection patterns and stress distributions throughout the RC slab. Figure 21. Stress in reinforced steel layer (S1) S o u r c e: made by D.A. Mawlood. Figure 22. Deflection of concrete slab (S1) S o u r c e: made by D.A. Mawlood. Figure 23. Effect of deflection on reinforced bars (S2) S o u r c e: made by D.A. Mawlood. Figure 24. Effect of deflection on concrete slab (S2) S o u r c e: made by D.A. Mawlood. Figure 25. Effect of stress distribution on concrete slab (S2) S o u r c e: made by D.A. Mawlood. Figure 26. Stress in top and bottom reinforcement bars (S3) S o u r c e: made by D.A. Mawlood. Figure 27. Stress in concrete slab (S3) S o u r c e: made by D.A. Mawlood. Figure 28. Deflection in concrete slab (S3) S o u r c e: made by D.A. Mawlood. 4. Conclusion 1. The proposed triangular-layered finite element model enables direct computation of out-of-plane stress components in RC slabs, eliminating both the need for through-depth integration of equilibrium equations and the requirement for shear correction coefficients. 2. The proposed triangular-layered formulation maintains a constant number of unknown parameters for displacement and stress fields, independent of the number of layers, while preserving its layer-based framework. 3. The static nonlinear behavior of RC slabs up to failure was analyzed using a partial mixed stress-displacement variational principle combined with a three-noded triangular plate element. 4. The effectiveness of the proposed triangular-layered finite element model in predicting nonlinear structural responses was validated through comprehensive analyses of RC slabs with diverse geometries, reinforcement configurations, and boundary conditions. 5. Numerical results demonstrate that the proposed formulation accurately predicts both the ultimate load-carrying capacity and failure deflection of RC slabs. 6. Benchmark validation studies confirm the method’s accuracy and computational efficiency, with ultimate deflection predictions exhibiting errors ranging from 2.59% (minimum) to 11.2% (maximum). 7. The proposed triangular-layered model accurately captures the complete load-deflection behavior of RC slabs while simultaneously predicting detailed structural responses, including deformation characteristics and stress component distributions. 8. The triangular-layered finite element model achieves an optimal balance between computational efficiency and predictive accuracy. Future work will extend this stress-displacement formulation to coupled material and geometric nonlinear analysis of reinforced concrete plate structures.

About the authors

Dara A. Mawlood

Siberian Federal University

Author for correspondence.
Email: dara.mawloud@univsul.edu.iq
ORCID iD: 0009-0003-2819-3107

Master student, Department of Building Structures and Controlled Systems, Institute of Civil Engineering

79 Svobodny аv., 660041, Krasnoyarsk, Russian Federation

Alexandr A. Koyankin

Siberian Federal University

Email: KoyankinAA@mail.ru
ORCID iD: 0000-0001-5271-9904
SPIN-code: 2779-8314

Candidate of Technical Sciences, Associate Professor of the Department of Building Structures and Controlled Systems, Institute of Civil Engineering

79 Svobodny аv., 660041, Krasnoyarsk, Russian Federation

References

Supplementary files