Finite Difference Energy Method for nonlinear numerical analysis of reinforced concrete slab using simplified isotropic damage model

Abstracts

Neste trabalho é apresentado um modelo para previsão do comportamento à flexão de lajes de concreto armado, combinando o modelo de dano de Mazars, para simulação da perda de rigidez do concreto durante o processo de fissuração e a Teoria Clássica de Laminados, para reger a flexão do elemento estrutural. Uma formulação variacional com base no princípio dos trabalhos virtuais foi desenvolvida para o modelo, sendo em seguida tratada numericamente segundo o Método das Diferenças Finitas Energéticas (MDFE), tendo como resultado final um programa desenvolvido em Fortran. Para validar o modelo proposto, foram simulados com o programa dois casos de lajes sob flexão, encontrados na literatura. A avaliação dos resultados obtidos nas análises demonstrou a potencialidade do modelo, tendo em vista a boa capacidade de previsão do comportamento de lajes sob flexão, varrendo a trajetória de equilíbrio até a ruptura do elemento estrutural. Além da satisfatória previsão do comportamento observou-se, como aspectos positivos do modelo, a sua relativa simplicidade e o número reduzido de parâmetros experimentais necessários à modelagem.

laje de concrero armado; mecânica do dano; método das diferenças finitas energéticas

Finite Difference Energy Method for nonlinear numerical analysis of reinforced concrete slab using simplified isotropic damage model

M. V. A. Lima; J. M. F. Lima; P. R. L. Lima

Universidade Estadual de Feira de Santana, Programa de Pós-graduação em Engenharia Civil e Ambiental, Feira de Santana, Bahia, Brasil. [e-mail], [e-mail], [e-mail]

ABSTRACT

This work presents a model to predict the flexural behavior of reinforced concrete slabs, combining the Mazars damage model for simulation of the loss of stiffness of the concrete during the cracking process and the Classical Theory of Laminates, to govern the bending of the structural element. A variational formulation based on the principle of virtual work was developed for the model, and then treated numerically according to the Finite Difference Energy Method, with the end result a program developed in Fortran. To validate the model thus proposed have been simulated with the program, some cases of slabs in flexure in the literature. The evaluation of the results obtained in this study demonstrated the capability of the model, in view of the good predictability of the behavior of slabs in flexure, sweeping the path of equilibrium to the rupture of the structural element. Besides the satisfactory prediction of the behavior observed as positive aspects of the model to its relative simplicity and reduced number of experimental parameters necessary for modeling.

Keywords: reinforced concrete slabs, damage mechanics, finite difference energy method.

1. Introduction

The concrete structures are used throughout the world with many different applications, as bridges, buildings, roads, etc. However, the reduction of natural resources and the large consumption of energy in production of cement and steel have required the use of structures using the full potential of these materials. Despite the design codes permitting the use of simplify models to calculate the reinforced concrete structures, only with the development of more sophisticated methods will be possible to obtain an optimized structure. Using an appropriate numerical method, that incorporates the nonlinearity of the physical material, will be possible to predict with more accuracy the structural behavior of structures under bending, identifying their state of cracking and deformation.

Since 1904, the finite difference method (FDM) has been used in solving differential equations for structural analysis of continuous beams or plates loaded in plane or out of plane [1]. As a computational method the FDM, together with finite element method (FEM), originated in the engineering literature in the mid 1950s. Many ideas and techniques are common to both methods, and in some cases they coincide. Nevertheless, with its more systematic use of the variational approach, its greater geometric flexibility, and the way it more easily lends itself to error analysis, the FEM has become the dominating approach both among numerical analysts and in applications [2]. However, recent studies have demonstrated the capability of the FDM in the analysis of reinforced concrete structures: structural analysis of slab [3], tensile members [4], frames [5] and columns [6] of reinforced concrete has been developed using physical or geometrical nonlinearity approach to evaluate the cracking, ultimate load, spalling, steelconcrete bond-slip, deflection and buckling. The results indicate that: i) the comparisons with experimental results confirm the validity of the method and ii) the finite difference analysis procedure is both fast-running and accurate and most suitable for design office application, combining the speed of analysis and the detail and accuracy of FE analysis [7].

An evolution of the finite difference method is the Finite Difference Energy Method (FDEM). Finite difference analysis divides a member into discrete segments to find an approximate solution to the differential equations of member motion. The segments in partial difference analysis are divided by nodes, with the displacement at each node expressed in terms of the difference in displacement of adjacent nodes. In FDEM these differences are used directly in equations associated to principle of virtual work or to principle of potential energy. This fact leads to some advantages over the traditional approach of the finite difference method with lower order derivatives and higher accuracy. With the variational principle, the boundary condition is taken care automatically and the matrix of equilibrium equations is symmetric (in the case of elements without damage), which yield considerable savings in computational effort. By relying on energy equations, the method has a similar structure with the finite element method (FEM): the discretization is conceived in terms of nodal degrees of freedom and it uses the same kinematic boundary conditions. One advantage of this method with respect to the FEM is a reduction of degrees of freedom by node what demonstrates the great potential to mechanical analysis of complex structures.

In fact, structural applications of FDEM have emerged in 1950s, with Houbolt [8], Griffin and Varga [9] and Bushnell [10]. In Brazil the FDEM has been used since 1980s to analyze of isotropic beams, plates and shells [11-14]. Recently the FDEM, in association with Damage Mechanics, was used to modeling of reinforced concrete beams and fiber concrete beams [15] resulting in good agreements with experimental results. This qualifies the model for analysis of more complex structures such as reinforced concrete slabs.

The application of nonlinear models for the study of reinforced concrete slabs is less reported in the technical literature, when compared with studies on beams. However, some important contributions to the theoretical prediction of the bending behavior of slabs can be verified as the association between elastoplastic model and FEM [16-17]. The use of Mazars [18] with Boundary Element Method [19] alone or combined with FEM [20] has also been reported. Fracture Mechanic model and FEM were used by Bandeira [21] to analyze reinforced concrete slabs with commercial software DIANA. All results indicate that, with the use of non-linear models for modeling the stress-strain behavior of concrete, it is possible to predict satisfactorily the structural behavior of reinforced concrete slabs under bending.

The main objective of this work is to verify the application of the FDEM to nonlinear analysis of reinforced concrete slabs using the Damage Model of Mazars [18]. Another nonlinear models, based at experimental results, has been used to describe the stress-strain behavior of concrete [22-24] but the application of Mazars's Model showed a good performance to analyze of reinforced concrete beams [25-26] and requires few experimental parameters. Two reinforced concrete slabs evaluated experimentally by Taylor et al. [27] and McNeice [28] were used to validate the proposed model.

2. Analytical formulation

2.1 Damage model

The damage model proposed by Mazars [18] has easy implementation, when compared with anysotropic model, for example. This is because the damage variable D is associated with the extension deformation of concrete according to the following hypotheses:

a) The damage process of concrete occurs in elastic state;

b) The concrete is considered isotropic before and after the damage;

c) It is supposed that the damage begins when the equivalent strain reaches a reference strain value ε_d0, determined in uni axial tensile tests in relation to the maximum stress.

In this analysis, the stress in a given main direction i(i= 1, 2 and 3) can be represented by positive and negative:

Thus the tensile and compressive strains can be represented us-where: ing Hooke's Law:

where u is the coefficient of Poisson of concrete.

The extension state is locally characterized by an equivalent strain that is expressed by:

Where ε_i , main strain components, and ε⁺_i+, the positive parts, with i = 1, 2 and 3, are defined by (4) and (5), respectively:

According Mazars [18] the damage process starts when the tensile strain exceeds the deformation ε_do, referred to the maximum stress of an uniaxial tension test (Equation 6).. As the concrete behaves differently in tension and compression, the damage variable D is obtained by combining properly the variables D_ct and D_cc, related to tension and compression, respectively, as follows:

where:

with

and: 0 < α_t< 1;0 < α_c< 1; e α_t+ α_c = 1.

D_ct and D_cc are defined as:

where A_t, B_t, A_c and B_c care experimental parameters defined related to stress-strain curves of concrete under tension or compression, respectively.

The constitutive relation of damaged concrete can defined thus as:

where E_c represents the elastic modulus of concrete undamaged.

2.2 Flexural model

The Classical laminated plate theory was used to model the reinforced concrete slab. In comparison, the discrete approach to modeling concrete and steel reinforcement include a large number of functions and material parameters and involve tedious programming and computational effort due higher degrees of freedom [17]. However, layered approach is more simple since the materials are represented by layers and the strain and stress of concrete and reinforcement through the thickness of the cross-section can be monitored progressively, in this manner providing an accurate representation of the structural behavior. Layered approach in FEM have been widely used to analysis of reinforced concrete structures [17, 24, 29-31] with relative success.

Similar to the Euler-Bernoulli beam theory and the plate theory, the classical laminated plate theory is only valid for thin laminates with small displacement in the transverse direction. It shares the same classical plate theory assumptions: [32-33]:

1) The layers are perfectly bonded together;

2) The laminate is thin compared to the lateral dimensions and is loaded in its plane;

3) A straight line originally normal to the undeformed mid-surface remains straight and rotates so as to remain straight and normal to the deformed mid-surface plane;

4) A line normal to the mid-surface of the plate is inextensible (does not stretch);

5) Linear geometric analysis is considered with smalls deflections and rotations when compared to unity. The Kirchhoff's thin plate approximations can be used here to derive relationships between in-plane and out-of-plane displacements, curvatures, twists, and strains;

6) The material of each layer is linearly elastic and has three planes of material symmetry (i.e, orthotropic). In fact, the classical plate theory usually assumes that the material is isotropic, while a fiber reinforced composite laminate with multiple layers may have more complicated stress-strain relations.

7) Loads are applied in mid-surface plane.

Based assumptions 1 to 5 the deformations u , v and w can be described entirely in terms of the deformation of the mid-surface plane u₀, v₀ and w₀ (Figure 1):

where and are rotations of a normal the mid-plane about the x and y axes, respectively.

Based in assumption 5, by substitution of these displacements relation into the strain-displacement equations of the classical theory of elasticity, the following relations are obtained:

Each lamina in the laminate is assumed to be in plane stress state so that the constitutive relation for a typical lamina k, assuming nonlinearity of material, can be written as:

The elements of [] are related to the engineering constants E, G and u, according assumption 6, as follows. Transformation of lamina stress and strain from the 1-2 axes to the rotated x -y axes (orientation θ) is shown here in matrix notation.

where:

E₁ is the longitudinal elastic modulus in direction x ;

E₂ is the longitudinal elastic modulus in direction y;

u₁₂ = u₂₁ is the major Poisson's ratio of layer associates to directions x and y ;

G₁₂ is the shear modulus associates to directions x and y ;

D represents the damage of material.

In this formulation, the loads and moments can be expressed in matrix form as (Figure 2):

where N_xx , N_yy and N_xyare in plane membrane forces per unit length (due to stretching of the plate mid surface), M_xx and M_yy are bending moments per unit length about the y and x axes, respectively, and M_xy are twisting moment per unit length (similar to torsion in a beam).

These matrices are termed: A_ij = extensional stiffness matrix; D_ij= bending stiffness matrix; B_ij= extension-bending coupling matrix. A convenient form for the [A], [B] and [D] matrices can be established by examining the position of the k^th lamina in Figure 2:

Can be observed in equation 17 that internal forces and moments are dependent of displacements, layer position and material properties. Therefore, the application of nonlinear models (damage model of concrete and plasticity of steel) affect directly the mechanical behavior of laminated plates.

The virtual work done by internal forces is expressed by:

where r₀ is the area of the mid-surface of the plate.

The virtual work done by external forces can be expressed by Eq. (20) considering loads acting on the domain and on the boundary of the plate (Figure 3 shows positive directions of loads) and according assumption 7:

which

is the transversal force per unit area acting on mid-surface of the plate; _xxxy and _xz are the forces per length unit along the edges x =0 = and xa, second to the directions x , y e z , respectively; M _xx is the bending moment per unit length, applied along the edges x = 0 and x = a;

_xxxy and _xz are the forces per length unit along the edges y_-=0 and y =b, according to the directions x , y e z , respectively; _yy is the Bending moment per unit length applied along the edges y =0 and y =b.

The moment of torsion was not considered in the external work since it is not usually applied in slabs, despite it appears the internal efforts.

3. Numerical formulation

After expressing the work done by internal forces as a function of displacement, by replacing the expressions of internal efforts (17a) and (17b) in (19), the numerical treatment according to EFDM can be applied.

First, it is necessary to introduce in equations of the δW_int and δW_ext the representations used in finite differences for the derivatives of the displacements. It is assumed, for evaluation these virtual works, that the plate is divided into sections of integration (area elements) in which all magnitudes involved are supposed constant. After determination of δW_inte δW_ext, by the summation of the contributions of the various sections of integration, a system of algebraic equations of equilibrium is obtained by application of principle of virtual work. The resolution of this system permits to obtain the numerical solution based in unknowns nodal displacements u₀ , v₀ e w₀ . To determination of the system of algebraic equations it is considered the kinematic condition of the boundary and others variation of displacements, arbitrary and independent of each other. In the numerical formulation developed in this work are used two types of representations for the derivatives of the displacements: centered representation and reduced representation. For this, consider the Figure 4, where f (x) represents the function u₀ , v₀ e w₀ and m is the point where the derivatives are evaluated (pivotal point). The nodal spacing is λ. First and second derivative of the function f (x) evaluated at point m are and .

In areas located along the edges of the board, the use of representation centered at the first derivative of u₀ and v₀ can cause singularity in the coefficient matrix. Therefore, for those derivatives is assumed to reduced representation, as suggested by Graça [14]. This representation can be defined by .

For the numerical discretization of the domain L_a× L _x of plate, the method provides a generation of sections of integration obtained from subdivisions in the x and y directions: n_x and n_y, respectively. Thus, an generic section of integration presents the rect-angular dimensions λ_xL_a|n_x and λ_y, with a total of (n_x +3)× (n_y +3) nodes and (n_x+1)× + 1 (n_y +1) area elements for ntegration, with nine different types of section, as shown in Figure 5. The area element type 1, 3, 7 and 9 are located in the corners of the plate and display area , while the element type 2, 4, 6 and 8, the area is . The remaining sections of integration (type 5) have area λ_x×λ_y, as illustrated in Figure 5.

The section of integration (see Figure 5) is composed of nine nodal points and for each of these points are associated with three degrees of freedom u₀, v₀ and w₀, which incorporates, for each area element a total of 27 displacements. Note that the second-order derivatives of the displacements of the nodes, associated with the edges of the board, require external nodes, called virtual nodes.

More information about the sequence of development of the numerical treatment, according to EFDM, can be found in [13-15].

4. Comparisons with experimental results

Two slabs solved in the literature was resolved by the aid of the newly developed computer program: a two-way slab supported by its four edges and submitted to distributed loading [27]; a two-way slab supported by its four corner and submitted to concentrated load [28]. The numerical discretization used the geometry and loading symmetry, analyzing thus, a quarter of plate. The numerical results were compared with experimental load-deflection curves of slabs and rupture load.

4.1 Computer simulation of stress-strain curves

Table 1 shows the experimental dates of used concrete ( E_c, f_c, f_te u_c) and steel ( E_s e f_y ) and the parameter values associated with damage model. Based on dates, the theoretical stress-strain curves (Figure 6) were obtained using equations (10) and (11).

Initially, the strain value was determined. After this, appropriate constants A_t, B_t, A_c and B_c. are tested until to obtain a good agreement between experimental and theoretical values. The obtained tensile parameter A_t shown similarity with values indicated by Lemaitre e Mazars [34] and Challamel [35]. How the experimental stress-strain curves were not obtained by authors, the compressive parameters A_c and B_c were changed until to obtain a same experimental compressive strength f_c and a peak strain between 2‰ and 3‰. Similar procedure was used by many researchers [18, 19, 23, 25, 34, 36].

Reinforcing steel was modeled as an uniaxial layer considered as an one-dimensional material in the reinforcement direction. The center of steel layer coincides with reinforcement center of experimental slabs and has an equivalent area. An elastoplastic model was used and three parameters are necessary as input data to define the steel model: the initial Young's modulus E_s, the yield stress f_y and the ultimate strain ε_su (adopted as 10‰).

Figure 7 shows the geometry, structure and reinforcement characteristics of RC-slab experimented by Taylor et al [27]. The slab, supported by its four edges and subjected to a distributed load at the top, has a square shape with 915 mm side length and 51 mm thickness.

One very important feature of non-linear models applied to the analysis of structural reinforced concrete elements is the problem of mesh size. In this example, two numerical parameters were tested. The ideal discretization was obtained after several tests and resulted in a plate with 8 x 8 subdivisions. The cross section was divided into 10, 15, 22 and 36 concrete and steel layers superimposed.

The load-deflection curve, calculated with 8 x 8 subdivisions and 15 layers is depicted in Fig. 8, together with the experimental results. Can be observed the presence of three numerical curves resultant of application, or not, of different damage parameter to shear modulus G₁₂:

1) G₁₂ = , where E₁ = E_c and ; u = u_c;

2) G₁₂ = , where E₁ = E_c and ; u = u_c and D_c = a, D_ct +a D_cc;

3) G₁₂ = , where E₁ = E_c and ; u = u_c.

It was observed a great influence of variation of transversal modulus on load-deflection behaviour of slabs. The adoption of undamaged modulus (situation 1) results in the more stiffness solution where the loss of load after the concrete cracking cannot to be adequately modeled. However, the application of damage coefficient The results illustrated in Fig. 19 indicate in a numerical curve with lower stiffness, after cracking, than experimental curve. In fact, the main factor that affect the loss of transversal stiffness of concrete is the cracking of slab in region under traction. In situation 3 the isotropic damage of G₁₂ is associated with the D_ct parameter and a best agreement between experimental and numerical results is observed. The fail of slab was characterized by yielding of steel of reinforcement (ε_s> 10‰).

In order to study the influence of layer, the example was analyzed by three other meshes, composed of 10, 22 and 36 layers for the cross section. The results illustrated in Fig. 12 indicate, for the presented example, that the increasing of the number of layers from 15 to 36 produced almost the same results for the load-displacement curve. So, it is important to point out that 15 layers for the cross section is really enough, even when the non-linearity effects are very significant.

The load-deflection relationship obtained by this model is again shown in Fig. 10 in addition to results presented by Jiang e Mirza [16]. This last model is based in Finite Element Method, Plastic Model to concrete and Classic Theory of Plates. The obtained results are very close to the numerical ones, showing the good accuracy of the present model with a similar degree of freedom.

4.3 Two-way slab supported by its four corners

The second example examined (tested by McNeice [28]) consisted of square corner-supported slab subjected to a point load applied at the center. The two-way slab was 915x915 mm square and 44 mm thick, and reinforced with an orthogonal mesh giving a reinforcement ratio of 0.85%. Geometric properties, reinforcement and material parameters are shown in Fig. 11. Material parameters are shown in Table 1. The transversal modulus was damage according situation 3 indicated in the last item.

In representing this specimen, a 14 x 14 subdivision (15 x 15 area elements) was used to model one-quarter of the slab. This example also was analyzed by meshes composed of 10, 15, 22 and 36 layers for the cross section. The results illustrated in Fig. 12 indicate reasonably accurate predictions of load-deflection curve and they confirm the use of 15 layers as the better solution. The modification of stiffness, after cracking of concrete, is resultant of numerical application of damage to concrete. In ultimate load it is verified a large concrete strain (ε_c 5.7‰) and a steel strain in elastic regime (ε_s 10‰) that indicate a rupture of slab by crushing of concrete under compression.

The McNiece slab is one often used as a benchmark for calibrating nonlinear analyses. Comparison of present model with another numerical model are shown in Figure 13. Kratz e Polling [37] used elastoplastic damage model to represent the stress-strain behavior of concrete under tension and compression in a Finite Element Method (FEM) program. Crisfield [38] evaluated the combination of FEM, using square elements, with linear softening behavior of concrete under traction. The criterion of Von Mises was adopted as the failure criterion for the concrete under compression. Doulah e Kabir [39] used FEM (square element with 4 degrees of freedom) and Classic Theory of Plates of Mindlin. To concrete behavior was adopted the linear softening in traction and bi-lin ear curve in compression. Zhang et al [17] used FEM (square element with 24 degrees of freedom) associated with Classic Theory of Plates of Mindlin and Reissner. Concrete was modeled as elastic linear under compression and to the tensile behavior under traction was used a new model that consider the crack ing of concrete in orthogonal direction with main direction. It is observed a better agreement with Krazt and Polling model while another models shown a more stiffness solution after first crack and rupture load very different of the experimental results. Thus the damage model appears as a good alternative for modeling concrete behavior.

5. Conclusion

A new model was presented in this work to predict the flexural behavior of reinforced concrete slabs. Combining the Finite Difference Energy Method with Classical Theory of Laminates it was possible to perform a nonlinear analyses by application of Damage Model, proposed by Mazars, to represent the concrete behavior under tracion and compression.

The results indicate the necessity of the application of tensile damage parameter ( D_ct) to reduces the transversal elastic modulus after cracking of concrete. In fact, how the evaluation of reinforced concrete slab is a bidimensional problem, the cracking affect the properties in all directions.

The comparison with experimental results demonstrate the efficiency of method to describe the load-deflection behavior and to determine the rupture load. Comparing with other numerical solutions, using FEM, it is observed a higher accuracy of proposed model, mainly considering the low discretization necessary.

6. Aknowledgements

The authors would like to thank the CNPq and FAPESB (PPP 010/2010) and CAPES for financial support.

7. References

Received: 25 Sep 2013

Accepted: 11 Jul 2014

Available Online: 01 Dec 2014

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Publication Dates

• Publication in this collection
  11 Dec 2014
• Date of issue
  Dec 2014

History

• Accepted
  11 July 2014
• Received
  25 Sept 2013