Нелинейная послойная формулировка: моделирование сборно-монолитных железобетонных плит

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Approaches [9-11]. Over the past four decades, advances in FE formulations have produced increasingly sophisticated models capable of capturing the complete nonlinear load-deflection response of reinforced concrete slabs [12-14]. Among these, the layered method was developed to offer a computationally efficient alternative to conventional 3D solid elements, thereby overcoming their inherent limitations [15-17]. This challenge of accurate shear modeling contributes to a broader issue: standardized numerical modeling protocols for semi-precast slabs are still lacking [18-20]. Existing studies and formulations largely concentrate on specialized configurations, such as pre-stressed variants, fiber-reinforced composites, and hybrid structural solutions [21-23]. This research presents a novel six-node, triangular-layered finite element formulation for the materially nonlinear analysis of semi-precast reinforced concrete slabs with cast-in-place concrete overlays. The work complements prior experimental investigations on prototype systems by introducing numerical models to analyze flexural capacity as a function of systematically varied interfacial surface textures, enabling a rigorous evaluation of composite performance [3]. The proposed formulation is based on an innovative global-local plate theory. Semi-precast RC slabs are modeled by discretizing the cross-section into perfectly bonded concrete and steel layers via a superposition approach, with material nonlinearities captured by distinct constitutive models: a smeared crack formulation for concrete and an elasto-plastic framework for steel reinforcement. A key innovation is the unique incorporation of transverse stress components as nodal degrees of freedom within the six-node triangular composite plate element, which significantly enhances computational capability and accuracy. Within this displacement-based framework, primary solution variables are displacement field parameters, while stress components are derived as secondary quantities through constitutive equations. The model’s robustness is validated against benchmark cases, demonstrating superior accuracy and computational efficiency compared to conventional approaches for predicting the nonlinear load-deflection response of semi-precast slab systems. 2. Materials and Methods The theoretical development of the proposed six-node triangular layered element is based on a plate formulation for combined membrane and bending action. It proceeds according to the following steps. A more versatile member of the triangular element family is the six-node linear strain triangle (LST) [24-26], depicted in Figure 1. Figure 1. Linear strain triangular element S o u r c e: made by A. Khennan [26]. The displacement field within the element is approximated by: u Nu 1 1 N u2 2 N u3 3 N u4 4 N u5 5 N u6 6, v Nv N v Nv N v N v N v 1 1 2 2 3 3 4 4 5 5 6 6 or U N a , (1) N1 ξ,η λ 1 2 λ N2 ξ,η 4ξλ N ξ,η ξ 1 2 ξ where 3 with λ 1 ξ η. N4 ξ,η 4ξη N5 ξ,η η 1 2 η N6 ξ,η 4ηλ The Jacobian matrix [J] of the geometric transformation is given by: x y 6 Ni 6 Ni ξ ξ i 1 ξ xi i 1 ξ yi J ; x y η η 6 Ni 6 Ni i 1 η xi i 1 η yi 1 1 4λ J 4 1 4λ with 4 λ ξ 1 4ξ 4η 0 4η 0 4ξ 1 4η x1 x 2 4η x3 4 λ η x4 x5 x6 y1 y2 y3 , y4 y5 y6 (2) λ 1 ξ η. (3) Then, the membrane strain can be determined: εm B am ; (4) To evaluate the [B] matrix, the partial derivatives with respect to global coordinates (x, y) must be transformed to the local (ξ, η) coordinates using the chain rule: Ni x y Ni Ni Ni ξ ξ ξ x x 1 ξ Ni x y Ni then Ni J Ni . (6) η η η y y η The membrane stiffness matrix for the element is given by using the Gauss quadrature (often called the Hammer formula for triangular domains) [26; 27]: 1 1 ξ T i 0 0 Bm ξ,η D Bm ξ,η det J ξ,η d dη ξ t i 1Wi Bm ξi ,ηi Di Bm ξi ,ηi det J ξi ,ηi , where nhp represents the number of Hammer points. (7) 1 μ 0 E Di 2 μ 1 0 . 1 μ 1 μ 0 0 2 (8) 2.1. Bending Element Formulation nhp T Modern finite element formulations for plates commonly adopt Mindlin - Reissner plate theory. Its primary advantage in FEM context is the use of independent rotational degrees of freedom, θx and θy, and transverse displacement, w(x, y) [24; 26; 27]. w 6 Ni ξ,η wi ,θx 6 Ni ξ,η θ xi,θy 6 Ni ξ,η θ yi. (9) i 1 0 N1 y N1 x L L L L L L L L L N1 [5] x With BB 0 0 N1 0 y N1 y Bs N1 x i 1 N6 0 x 0 0 N6 0 y N6 y N6 x i 1 0 N6 , y N6 0 N6 N6 . 0 x Therefore, the element stiffness matrix is composed of two distinct contributions: the bending stiffness matrix and the shear stiffness matrix. Ke B KB Ks Ae BB T D B dAi B Ae Bs T D B dAis s . (10) 2.2. Layer Kinematic Formulation Reddy’s third-order shear deformation theory (TSDT) relaxes the restrictive kinematic assumptions of classical plate theory by explicitly accounting for transverse shear strains and their associated deformation. The displacement field within this framework is expressed as [28]: ui x y z, , zψx x y, 4z3 / 3h2 ψx x y, wi0 x y, / x ; (11) v x y zi ( , , ) zψy (x y, ) 4z3 / 3h2 (ψy (x y, ) wi0 (x y, ) / y ; w x yi , w x y0i , , w0 w0 where ψx ,ψy . x y 2.2.1. Strain Field Components Using the kinematic strain-displacement relations, the strain state in the i-th layer of the reinforced concrete slab is formulated vector form as [19; 28]. ε xx1 1 1 εi εyy 1 ψx x ψy y ε xx3 3 ε 3 , εi yy γ 3 4 ψ w - 2 x 20 3h x x 4 ψy 2w0 - 2 2 3h y y , γ 2 xy ψyx ψxy xy 34h2 ψyx ψxy 2 x y2w0 γi γ xz0 ψx w0 , γi γ xz2 h42 ψx w0 . x x The strain field can be expressed as follows: (12) εi zεi 1 z3εi 3 ; γ γi i 0 z2γi 2 . (13) γ yz0 ψy wy0 2 2 42 ψy w0 0 γyz h y 2.2.2. Out-of-Plane Stress Formulation The element normal stresses are derived from the strains via Hooke’s law [24; 29] as follows: σi Diεi or σi Di zεi 1 z 3 εi 3 . (14) Transverse shear stresses, τxz and τyz, are calculated independently for each material layer within the reinforced concrete slab. Their expressions are derived as follows: τxz Ei 1 0 γxz i i 0 2 2 τyz 2 1 μ 0 1 γyz or τi Ds γi,τi Ds γ z γi , (15) where parameter Ei is the elastic modulus of the i-th layer, and μ is the Poisson’s ratio of the reinforced concrete. Within Reddy’s Third-Order Shear Deformation Theory, the plate’s constitutive law is: Mi0 Di i 0 εi(3)(1) QPii F s D0is γεii(0) . (16) Ri γi(3) Following the standard finite element procedure, the stiffness matrix is assembled as: F11 Fi F12 0 F12 0 Q11 Q12 0 zi F22 0 zi 1 Q12 Q22 0 z dz4 ; 0 F66 0 0 Q66 H11 H i H12 0 H12 0 Q11 Q12 0 zi 6 H22 0 zi 1 Q12 Q22 0 z dz; 0 H66 0 0 Q66 D11 Di D12 0 D12 0 Q11 Q12 0 zi 2 D22 0 zi 1 Q12 Q22 0 z dz; 0 D66 0 0 Q66 D Dis 44 0 and 0 zi C44 zi 1 0 D55 0 2 z dz C55 F Fis 44 0 0 zi C44 zi 1 0 F55 0 4 z dz; C55 A Ais 44 0 where 0 zi C44 zi 1 0 A55 0 2 z dz, C55 E μE E Q11 , Q12 Q21 2 , Q22 2 , 1 μ2 1 μ 1 μ 3. Methods Q66 G and C44 C55 G. The reinforced concrete slab geometry (Figure 2) is defined within a Cartesian coordinate system (x1, x2, z), with the mid-plane at z = 0 serving as the reference plane. The slab is idealized as a laminated composite structure comprising perfectly bonded layers of normal concrete (NC) and steel [8]. Computationally, a smeared reinforcement approach is adopted, representing steel rebars as equivalent uniaxial layers positioned at their centroids. The thickness of an equivalent steel layer, tens, is determined from the rebar cross-sectional area As and spacing S as tens = As/S. a b Figure 2. Configuration of the RC slab (a); Idealized laminated composite model (b) S o u r c e: made by D.A. Mawlood, A.A. Koyankin. 3.1. Steel Reinforcement Layer Steel reinforcement layers are modeled with an idealized elastoplastic constitutive law. For layer i, the elastic stress-strain relationship is given by [28]: σisteel Dsteeli ε . (17) Es The sectional properties Dsteel for steel: Dsteeli 0 0 0 0 0 0 Esp 0 , Dsteeli 0 0 0 0 0 0 0 0 . 0 i Es(i) represents the elastic modulus of the i-th steel reinforcement layer. The parameter Eps denotes the post-yield plastic modulus of the steel reinforcement layers [8]. 3.2. Concrete Layer Properties In its pre-cracked state, the i-th concrete layer exhibits isotropic, homogeneous, and linear elastic behavior, characterized by the constitutive relation: σiconcrete Dconcretei εi , τiconcrete Dsi γi . (18) When the tensile strength f is exceeded in any principal direction, the isotropic constitutive relations (Eqs. 18) become invalid. This study employs the smeared crack approach [29] to characterize post-cracking concrete behavior, necessitating a transformation from isotropic to orthotropic material representation in a local coordinate system ξ, η, ζ . The post-cracking constitutive relationship for layer i in this local coordinate system is expressed as: σξ Ei 1 ση 1 μ2 μ τξη 0 μ 1 0 0 εξ τξζ Ei 1 0 γεξηη , τηζ 2 1 μ 0 1 μ ρ 2 0 γξζ γηζ . ρ (19) Note that ρ (0, 1] represents the shear retention factor, which is utilized to model the effects of aggregate interlock. 3.3. Mixed Stiffness Matrix The element stiffness matrix, Kei, can be additively decomposed into its membrane and bending components as follows: Kei in 1 Ke im, Ke iB, , where n: number of layers. Ke im, Ae Bm T D Bm tdA , B T T Ke i, KB Ks Ae BB DB BB dA Ae Bs Ds B dAs . Within this formulation, steel reinforcement contributes solely to the membrane stiffness, while concrete contributes to both membrane and bending stiffness components. K mm m m m m m B 12,12 0 BB . K18,18 (20) Ke i, Kconcrete Ksteel K12,12 , then Ke i, Ke i, Ke i, 0 3.4. Application of Load The applied load can be calculated using: K mm 0 d mm Pmm K d P , then 12,12 BB BB BB , 0 K18,18 d P (21) where d mm , d BB are the membrane u1, u2, u3, u4, u5, u6, v1, v2, v3, v4, v5, v6 and bending w1, φx1, φy1, w2, φx2, φy2, w3, φx3, φy3, w4, φx4, φy4, w5, φx5, φy5, w6, φx6, φy6 element displacements; Pmm , PBB are the membrane and bending element loads. The nonlinear algebraic system is solved using an incremental-iterative strategy that combines load stepping with the Newton - Raphson method [30; 31]. The algorithm for the standard (non-incremental) Newton - Raphson method is summarized as follows: 1. Find initial displacements; d0 K0 E0 1 P1 , where K0 is found by equation (20). 2. Computing the residual (unbalanced) force vectorr0 : r0 K1 E1 d0 P1 . 3. A displacement correction, d0 , is computed by: d1 K1 E1 1 r0 . 4. The total displacement is updated: d1 d0 d . Thus, the iterative Newton - Raphson step can be expressed in its standard form [12]: 1 rj K j 1 E j 1 d j P, d j 1 K j 1 E j 1 rj for j 1,2,3 m 5. The final, total displacement vector is obtained by: d d 0 j 0 d j 1 . 4. Results and Discussion A MATLAB-based computational tool was developed to implement the proposed six-node triangular layered finite element method for nonlinear analysis. The program’s performance was validated by numerically analyzing two semi-precast reinforced concrete slabs with cast-in-place toppings. These test specimens identical in configuration to those experimentally studied by Mohamed et al. [3] were modeled using the proposed finite element approach, as illustrated in Figure 3. Two semi-precast slab specimens were modeled using the developed six-node triangular layered FEM. The slabs had a total thickness of 125 mm (75 mm precast slab with 50 mm top slab), are illustrated in Figure 3. Complete material and structural details are provided in Tables 1 and 2. Figure 3. Schematic four-point testing arrangement of the slab S o u r c e: made by Mohamed et al. [3]. Table 1. Description of the specimens and casted concrete strengths. No. Thickness, mm Description Semiprecast slab Overtopping - - - Compressive strength, MPa Splitting strength, MPa Compressive strength, MPa Splitting strength, MPa S2 125 Untreated full thick slab 29.4 4.3 24.9 3.4 S3 125 Wire brushed roughened at the interface slab 30.1 4.1 23.4 3 S o u r c e: made by D.A. Mawlood, A.A. Koyankin. Table 2. Input parameters for the numerical model Parameters Values Young’s moduli of concrete 17,000-24,000 MPa Poisson’s ratio of concrete 0.15 Dilation angle 31º Eccentricity 0.1 Biaxial strength ratio 1.16 K 0.667 Peak compressive strengths of concrete 23.4-33.2 MPa Peak tensile strengths of concrete 1.3-1.9 MPa Young’s modulus of steel 190,000 MPa Poisson’s ratio of steel 0.3 Yield tensile strength of steel 535 MPa S o u r c e: made by D.A. Mawlood, A.A. Koyankin. A 12×8 mesh of laminated triangular plate elements discretized the semi-precast slab, with the mesh size determined via a convergence study, illustrated in Figure 4. (1) plan discretization а b (2) cross section discretization Figure 4. FE mesh of semi precast reinforced concrete slab with concrete topping: a - neutral axes position before bonding failure; b - neutral axes position, after bonding failure S o u r c e: made by D.A. Mawlood, A.A. Koyankin. Specimen S2 - a one-way semi-precast RC slab with an untreated, full-depth precast section and a concrete topping - was analyzed using the six-node triangular layered finite element methodology. The evaluation of mid-span deflection established a constant moment region, enabling a detailed assessment of flexural performance. Predictions from the layered FE model were compared against experimental data from Mohamed et al. [3] and results from high-fidelity 3D ABAQUS simulations in a three-way validation. While the layered approach yielded a conservative estimate of the initial cracking load (Figures 5 and 6), excellent agreement with all reference data was observed beyond the 42 kN load threshold. The model demonstrated high fidelity in deflection prediction, with maximum values deviating by only 4.68% from experimental measurements and 7.40% from the ABAQUS simulations. The triangular layered FE model demonstrated excellent predictive capability for Specimen S3, which featured a wire-brushed interface, accurately capturing its ultimate load capacity. As shown in Figures 7 and 8, the numerical load-deflection response exhibited remarkable agreement with both the experimental data and the ABAQUS results (Table 3). The maximum deflection prediction deviated by only 0.50% from the experimental measurement and by 12.5% from the ABAQUS simulation. The deflection behavior and stress distributions resulting from the ABAQUS finite element analysis of the precast reinforced concrete slab are presented in Figures 9 through 14. Figure 5. Load-deflection curve for sample (S2) Figure 6. Deflection chart for sample (S2) S o u r c e: made by D.A. Mawlood, A.A. Koyankin. S o u r c e: made by D.A. Mawlood, A.A. Koyankin. defelection, mm 0 10 20 30 40 50 60 70 Figure 7. Load-deflection curve for sample (S3) S o u r c e: made by D.A. Mawlood, A.A. Koyankin. 80 1 2 3 4 5 6 7 8 9 10 Figure 8. Deflection chart for sample (S3) S o u r c e: made by D.A. Mawlood, A.A. Koyankin. 11 Table. 3. Experimental and theoretical data Sample Exp. Moment = PL/4, N. mm Theo. Moment, N. mm % diff. Exp. strain σ / E , mm/mm Theo. Strain, mm/mm % diff. S2 17.94·10^6 17.02·10^6 -5.13 0.00287 0.00270 -5.92 S3 25.98·10^6 24.02·10^6 -7.54 0.003158 0.002897 -8.26 S o u r c e: made by D.A. Mawlood, A.A. Koyankin. Figure 9. Plot of deflection for sample (S2) Figure 10. Plot of steel stress for sample (S2) S o u r c e: made by D.A. Mawlood, A.A. Koyankin. S o u r c e: made by D.A. Mawlood, A.A. Koyankin. Figure 11. Plot of stress for sample (S2) Figure 12. Plot of stress for sample (S3) S o u r c e: made by D.A. Mawlood, A.A. Koyankin. S o u r c e: made by D.A. Mawlood, A.A. Koyankin. Figure 13. Plot of deflection for sample (S3) Figure 14. Plot of steel stress for sample (S3). S o u r c e: made by D.A. Mawlood, A.A. Koyankin. S o u r c e: made by D.A. Mawlood, A.A. Koyankin. 5. Conclusion 1. This study presents a six-node, triangular layered plate finite element model for the nonlinear analysis of semi-precast reinforced concrete slabs. 2. The proposed formulation employs a layer-wise discretization to accurately capture complex throughthickness material behavior. 3. The slab is discretized into multiple concrete layers, with steel reinforcement modeled as equivalent smeared layers positioned at the rebar centroids under a perfect bond assumption. 4. The method directly computes out-of-plane stress components without requiring through-thickness integration and inherently satisfies transverse shear deformation requirements, thereby eliminating the need for shear correction factors. The formulation is derived from a partial mixed variational principle, incorporating both stress and displacement fields within a six-node triangular element. 5. Numerical results demonstrate that the model successfully predicts the ultimate load capacity and failure deflections of semi-precast slabs. 6. Benchmark tests confirm the method’s accuracy and computational efficiency, with maximum deflection prediction errors ranging from 0.5 to 4.68% compared to experimental and high-fidelity reference data. The model achieves an excellent balance between accuracy and computational cost. 7. Future research will extend this approach to the materially and geometrically nonlinear analysis of reinforced concrete shell structures within the presented layered framework.

Об авторах

Дара А. Мавлуд

Сибирский федеральный университет

Автор, ответственный за переписку.
Email: dara.mawloud@univsul.edu.iq
ORCID iD: 0009-0003-2819-3107

магистрант кафедры строительных конструкций и управляемых систем, Инженерно-строительный институт

Российская Федерация, 660041, г. Красноярск, пр. Свободный, д. 79

Александр Александрович Коянкин

Сибирский федеральный университет

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

кандидат технических наук, доцент кафедры строительных конструкций и управляемых систем, Инженерно-строительный институт

Российская Федерация, 660041, г. Красноярск, пр. Свободный, д. 79

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