Stress resultant nonlinear constitutive model for cracked reinforced concrete panels

Abstract

The paper proposes a novel stress resultant nonlinear constitutive model for Reinforced Concrete (RC) panels adapted to cyclic loadings. An analytical multi-scale analysis is applied by taking a concrete strut with embedded steel reinforcement between two consecutive cracks as representative volume element. Some suitable assumptions are adopted in order to incorporate the most important nonlinear phenomena characterizing reinforced concrete behavior: concrete damage, concrete cracking, bond-slip stress (at the origin of the tension stiffening effect) and steel yielding. The model is validated by comparison with experimental data concerning tension and tension-compression uniaxial tests on RC beams and a cyclic (non-reversing) shear test on an RC wall.

Introduction

Industrial buildings, in particular Nuclear Power Plants (NPP), have to fulfill severe structural requirements according to the modern design codes. The computational time required for nonlinear structural analyses of this type of large-dimension Reinforced Concrete (RC) facilities, sometimes necessary for their seismic assessment, is significant. However, the so-called global or effective modeling approaches can ensure numerical efficiency and robustness. These approaches are characterized by the use of relatively large size Finite Elements (FE) where the material model represents the reinforced concrete behavior as an equivalent homogeneous material, as opposed to approaches based on distinct concrete and steel modeling and the introduction of some kinematic and/or stress transfer conditions. In civil engineering, this type of global modeling strategy is usually coupled with linear elastic behavior assumptions.

Nevertheless, recent safety requirements for NPP have introduced the necessity of using more realistic models able to reproduce the actual nonlinear behavior of RC structures, both for static and dynamic load cases. In particular, these models should be able to take into account the cracking onset and its development, in order to correctly estimate the crack widths (and also spacing and direction) since engineering design standards provide some bounds to these values to fulfill prescribed aesthetic, durability and confinement (for NPP) conditions, or to ensure the strength of equipment anchorage. Concrete stiffness reduction (as a critical contribution to the dynamic structural behavior), steel yielding and permanent strains are other important parameters in structural design which should be accurately assessed.

Concerning RC panels, a relatively important number of global nonlinear models able to reproduce the development of cracks have been recently developed. They can be split into two categories.

On the one hand, the so-called phenomenological approaches describe cracking in RC panels adopting suitable assumptions or laws for all the physical phenomena governing the nonlinear structural response. Since they are developed and calibrated from an important number of experimental tests, they are the result of a deep understanding of the behavior of RC structures. Nevertheless, these approaches often lead to models that are only applicable for design cases similar to the experimental campaigns adopted for the calibration, usually only under monotonic loading conditions. Furthermore, their numerical implementation requires computational expensive iterations to satisfy both equilibrium and constitutive modeling conditions at the local scale and the link with the global scale is not explicitly described. The following constitutive models belong to this class. The Modified Compression Field Theory (MCFT) of Vecchio and Collins [30] is a fully rotating, smeared, stress-free crack model where equilibrium, compatibility and stress-strain relations are formulated in terms of average strains and stresses; an orthotropic concrete model with compression softening and tension stiffening effects is adopted. The Disturbed Stress Field Model (DSFM) of Vecchio [29] is an extension of the previous model allowing the principal stress and strain directions to be different by considering the tangential slip at cracks and the transmitted shear stress (aggregate interlock). The Cracked Membrane Model (CMM) developed by Kaufmann [16] and Kaufmann and Marti [17] and the Extended Cracked Membrane Model (ECMM) developed by Pimentel et al. [22] can be seen respectively as equivalent of MCFT and DSFM, respectively, when establishing equilibrium equations directly at cracks, since they both take into account concrete compression softening and tension stiffening, the former being a rotating crack approach and the latter a fixed crack one enabling stress transfer through cracks. The Softened Membrane Model (SMM) of Hsu and Zhu [15] applies the principles of Continuum Mechanics to the orthotropic continuous cracked (smeared) RC material. Finally, the PARC model developed by Belletti et al. [1] is developed from equilibrium of an RC strut between two consecutive cracks and accounts for dowel action, aggregate interlock, bridging effect, tension stiffening, and concrete softening in compression.

On the other hand, several constitutive models based on the global (or stress-resultant) modeling approach are directly formulated for an FE implementation at the structural element scale: the in-plane stress resultant $\mathit{N}$ is explicitly expressed as a function of the generalized membrane strains $\mathbf{∊}$ and $n$ internal variables $\underline{\alpha }=({\alpha }_1\text{,}\dots \text{,}{\alpha }_n)$ whose role is to reproduce the nonlinear cyclic response of RC panels. The corresponding evolution laws are formulated within the framework of the Thermodynamics of Irreversible Processes (TIP) (see e.g. [9], [10], [18], [23]), where the state of the material is defined by the Helmholtz free energy surface density ${\psi }^o(\mathbf{∊}\text{,}\underline{\alpha })$, whose gradient define the non-dissipative thermodynamic forces (index $\mathit{nd}$) by the state equations:

$$\mathit{N}={\mathit{N}}^{\mathit{nd}}≔\frac{\partial {\psi }^o(\mathbf{∊}\text{,}\underline{\alpha })}{\partial \mathbf{∊}}-q_i^d=q_i^{\mathit{nd}}≔\frac{\partial {\psi }^o(\mathbf{∊}\text{,}\underline{\alpha })}{\partial {\alpha }_i}$$

The index $d$ (hereinafter omitted) stands for dissipative thermodynamic forces (hereinafter thermodynamic forces).

By only considering isothermal transformations, the second principle of TIP, which ensures the condition of a nonnegative power dissipation surface density $\dot{D}$, can be expressed as:

$$\dot{D}=\mathit{N}:\dot{\mathbf{∊}}-{\dot{\psi }}^o(\mathbf{∊}\text{,}\underline{\alpha })=\mathit{N}:\dot{\mathbf{∊}}-\frac{\partial {\psi }^o}{\partial \mathbf{∊}}:\dot{\mathbf{∊}}-{\displaystyle \sum _i}\frac{\partial {\psi }^o}{\partial {\alpha }_i}·{\dot{\alpha }}_i={\displaystyle \sum _i}{q}_i·{\dot{\alpha }}_i⩾0$$

Moreover, in the considered models, the evolution of the internal variables is defined according to the Generalized Standard Materials Theory (GSMT) (see [13]), which allows a well-defined energetic characterization and entails that the time integration algorithm is associated with a well-posed minimization problem. According to this theory, the internal variable evolution is given by the normality rule:

$${\dot{\alpha }}_i={\dot{\lambda }}_i\frac{\partial f_i}{\partial q_i}$$

where the threshold functions $f_i$ depend in general on $f_i(q_i\text{,}\mathbf{∊}\text{,}\underline{\alpha }\text{;}\underline{\mu })$, with $\underline{\mu }$ a set of variables acting as parameters (without associated thermodynamic forces). Threshold functions $f_i$ are differentiable and convex with respect to $q_i$ for any $\mathbf{∊}\text{,}\underline{\alpha }\text{,}\underline{\mu }$ set. They define the elastic domain for each nonlinear mechanism by satisfying, together with their associated plastic multipliers ${\dot{\lambda }}_i$, the Kuhn-Tucker conditions:

$${\dot{\lambda }}_i⩾0f_i⩽0{\dot{\lambda }}_i{f}_i=0$$

This type of model formulation ensures a high degree of robustness and versatility to any dynamic load case that can occur during an RC building FE analysis and has shown good performances when applied to plain concrete [24]. GLRC_DM [20] and DHRC [5], [6] are two RC plate nonlinear constitutive models of this type implemented in the Code_Aster FE software [8]. The first one consists in a damage model while the latter couples irreversible strains (induced by steel rebar debonding) with concrete damage. Nevertheless, these constitutive models are not always able to reproduce accurately the nonlinear behavior for RC plates under severe load cases. In particular, their range of validity is limited to a moderate nonlinear response of RC plates, in the Serviceability Limit State domain, as defined in Eurocode2 (EC2) [3]. Therefore, they cannot give accurate results concerning the actual local cracking state of RC elements and the yielding of the steel reinforcement bars.

Moreover, it is noticed that GLRC_DM and DHRC models have been built using two different multi-scale analyses: a heuristic homogenization process for the former and a fully justified numerical averaging method applied on the mechanical fields on a Representative Volume Element (RVE) for the latter. The multi-scale analysis, often applied in civil engineering to derive equivalent RC plate constitutive relations, has been introduced in the literature for many decades and applied in different engineering fields. For example, the homogenization technique has been justified using an asymptotic expansion method on three-dimensional elasticity equations, leading to the well-known bi-dimensional linear plate theory (see [4]), assuming that both underlying small parameters (the ratio of the thickness over the plate dimensions, and the ratio of the heterogeneities size over the plate thickness) are of the same order. A first attempt to apply this method to an RC plate, limited to the linear elastic range, was proposed by Destuynder and Theodory [7]. The authors established the membrane and flexural equivalent stiffness tensors in terms of stress resultant in the RC plate, after solving the underlying linear elastic auxiliary problems on a periodic unit cell, or RVE, including both concrete and steel grids, and average value calculations.

The aim of this paper is to propose a novel stress resultant (global) constitutive model for RC panels suitable for a robust FE implementation and able to reproduce the nonlinear response under cyclic solicitations in the entire SLS domain. The onset of steel yielding, which usually characterizes the beginning of the Ultimate Limit State (ULS) of RC elements, is also modeled.

The previously mentioned phenomenological models are the result of a deep understanding of the behavior of RC. Therefore, they are a consistent physical basis for the development of the new model and their local scale descriptions focusing on local displacement and stress variables are adopted. Inspired by these models, the following four physical nonlinear phenomena are considered:

• Isotropic concrete damage in compression,

• Concrete cracking, considering both normal and tangential-to-the-crack displacements and stress transfer,

• Bond stress at the steel-concrete interface caused by their relative slip and causing the tension stiffening effect (e.g. [21]),

• Yielding of steel reinforcement bars occurring locally at cracks, where the maximum (in absolute value) steel stresses are reached.

The link between these local phenomena on the one hand and the global scale model on the other hand is made by means of a closed-form multi-scale analysis, adopting suitable assumptions. The principles of this multi-scale analysis applied on a cracked RC panel are presented in Section 2. The local scale description of the four nonlinear phenomena taken into account by the model is presented in Section 3.

With the previous elements, a closed-form multi-scale analysis is applied on an RC panel in the stabilized crack state configuration and is presented in Section 4. Then, Section 5 is devoted to the formulation of the general form of the obtained stress-resultant constitutive model in the well-defined theoretical framework of TIP and GSMT, which guarantees an efficient FE software implementation, adapted to all types of load paths (including cyclic ones).

Finally, in Section 6, the new model is applied to uniaxial pure tension and tension-compression tests on RC beams and to a global shear cyclic test (without inversion of the sign of the applied force) on a RC wall, in order to highlight its capacity to reproduce experimental results concerning both global (force-displacement) and local (crack widths) values.