Epistemic uncertainty of plane-stress continuum …Epistemic uncertainty of plane-stress continuum...

Epistemic uncertainty of plane-stress continuum concrete models

M. Chacón (1), J.C. de la Llera (1), M. Hube (1) and D. Celentano (2)

(1) Department of Structural and Geotechnical Engineering, Pontificia Universidad Católica de Chile, and

National Center for the Integrated Management of Natural Disaster (CIGIDEN) CONICYT/FONDAP/15110017 (2) Department of Mechanical Engineering, Pontificia Universidad Católica de Chile

Abstract

This article deals with the epistemic uncertainty of plane-stress finite element (FE) models. It is well known that the nonlinear response of reinforced concrete structures is strongly dependent on the stress-strain constitutive concrete models adopted. Herein, this uncertainty is evaluated using five different continuum constitutive concrete models, which are very popular in FE software. Because several of these models fail to converge in strain-softening regimes, modified models are proposed for their consistent plane-stress numerical implementation. The five concrete models studied are: (i) the hyperbolic Drucker-Prager (DPH) plastic model; (ii) the Lubliner-Lee-Fenves (LLF) plastic-damage model; (iii) the Wu-Li-Faría (WLF) model; (iv) the Faría-Oliver-Cervera (FOC) model; and (v) the total strain rotating (ROT) smeared-crack model. All models have been implemented in ANSYS software using user-material FORTRAN routines, which are adequate for shell-type finite elements. Results were validated using a set of experimental benchmark tests subjected to uniaxial and biaxial stress states under monotonic and cyclic loadings. Epistemic uncertainty in the model responses are qualitatively described and quantified as well.

Keywords: inelastic concrete models, plane-stress formulations, viscous-regularization.

1 Introduction

The use of nonlinear plane-stress constitutive models can predict with accuracy and speed the seismic response of reinforced concrete (RC) buildings. Finite element (FE) micro-models elaborated with shell elements allows simulating a detailed stress and strain field distribution under complex structural geometries and multi-axial loading conditions, which occurs in the case of RC shear wall buildings. Furthermore, given the benefits of the later advances in supercomputers, large building models can be developed to predict the seismic response with a considerable CPU time reduction and large amount of valuable data information. Also, nonlinear analysis of RC structures is mandatory to evaluate the inelastic response and identify potential damage zones in buildings. Because concrete is a brittle material that exhibits a strongly nonlinear response associated with the propagation of cracking, its correct modeling in practice is difficult and requires a robust numerical algorithms to achieve convergence. These analyses are critical given the brittle performance observed in RC shear wall buildings during recent earthquakes [1]. In the past thirty years, many two-dimensional (2D) and three-dimensional (3D) concrete constitutive models have been elaborated to describe the mechanical behavior of concrete under multi-axial stress loadings. Each formulation use different notations and inputs, which in some cases gives different responses and are difficult to compare among them. This class of uncertainty is called epistemic and is quantified in this paper.

XII Chilean Conference on Seismology and Earthquake Engineering ACHISINA2019 Valdivia, April 3 to, 5, 2019

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Five main groups of concrete models can be distinguished: (i) plastic models, which are based on the flow theory of plasticity to describe the plastic strains and hardening behavior, where a non-associated flow rule and a single- or multi-surface yield criterion is considered to distinguish between compression and tension behavior (Mohr-Coulomb, Drucker-Prager, and [2][3]); (ii) damage models, which are based on continuum damage mechanics (CDM) [4] to predict the stiffness degradation and strain-softening behavior due to micro-crack propagation. It can include scalar variables to characterize the isotropic damage process [5][6][7] or a tensor variables to describe the anisotropic damage effects [8]; (iii) plastic-damage models, which combine plasticity and CDM theories [7][9][10]; (iv) fracture models, based on the nonlinear fracture mechanics theory [11][12]; and (v) mixed models, that combine plastic-damage smeared crack or fracture-plastic models [13][14]. A plane-stress state is characterized by a zero normal stresses, which usually occur in walls, slabs and membranes. Multi-layered shell finite elements are commonly used to simulate a plane-stress condition due that gives an accurate distribution of in-plane and out-of-plane concrete stresses, with a reduced CPU time relative to solid finite elements [15]. Accuracy in these elements is strongly dependent on the numerical implementation of their materials [16]. For strain-driven models, the numerical implementation consists in: (i) an updated stress algorithm given a strain increment; and (ii) a consistent stiffness matrix according to the updated stress. Implicit integration schemes with return-mapping algorithms (RMA) are generally used for plastic and plastic-damage models, whereas explicit integration schemes are considered for damage and smeared crack models. Indeed, in the plane-stress case, an additional constraint is imposed to satisfy the condition of null out-of-plane stress. Generally, the algorithm used for the plastic component in the 3D-case is not valid, requiring specific algorithms for this case. The most popular approach is the plane stress-projected method, in which the plane-stress constraint is enforced within the equations at the Gauss point level [17][18] and that consider only the in-plane stress and strain components, leading a direct, exact, and efficient computationally procedure [19]. Most of these algorithms are implemented in FE software using local models, where the stress at each integration point is dependent only on their respective strain. Convergence problems arise in local models of materials with softening behavior due to strain-localization, leading to spurious mesh sensitivity and unreliable results [20]. Local models can be improved with different methods that include an intrinsic length scale in the continuum equations, such as higher-order gradients [21], non-local media [20], or rate-dependent terms [22]. Inner the possibility, rate-dependent approach is the most adequate and easier to implement for plastic and damage concrete models. They incorporate a numerical viscosity in the stress-strain constitutive equations, improving convergence greatly, but at the expense of a gradual over stress condition depending on the strain-rate increment. The objective of this paper is to quantify the epistemic uncertainty of inelastic concrete models by comparing the response of five different continuum constitutive concrete models. The paper presents in Section 2 a brief description of the set of concrete models considered. Then, numerical examples using basic benchmarks tests subject to monotonic and cyclic loading conditions under uniaxial and biaxial stresses are presented in Section 3. Finally, Section 4 measures the epistemic uncertainty of concrete models with a set of response parameters of the experimental tests of the Section 3.


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2 Inelastic concrete models

2.1 Review of theoretical formulations

This section summarized briefly the five inelastic concrete models considered in this article, showing the main equations and properties. Some modifications were incorporate to original models to improve their convergence. For the sake of simplicity, all equations are described in a vectorized and matrix format according to an adequate conversion of their respective tensors, considering only the components associated to in-plane behavior ((∙)��, (∙)��, (∙)��).

Fig. 1 -Yield criterion of DPH, LLF and WLF models: (a) 3D view in principal stress; (b) deviatoric π-plane; (c) tensile/compressive meridians in Rendulic plane; and (d) biaxial-stress plane. The following parameters are used. Common for all ��

�=6MPa, ��=20MPa, ��

� = 1.16��. For the LLF and WLF models �� = ��

�, �� = ��, �±=0 and ��=0.7.

Drucker-Prager Hyperbolic (DPH) model This plastic model, so-called the "Extended Drucker-Prager'' model was defined by Drucker and Prager and modified by [23][24]. Is a simplification of Mohr-Coulomb model and have been used as a first approximation to simulate concrete. Consider a yield surface with a cone shape to simulate the pressure-dependent behavior and the asymmetric tensile/compressive strength of concrete. Also, include a smoothed hyperbolic flow potential surface that controls the inelastic volumetric strain rate (dilatancy). This surface avoids the singularity at the cone’s apex present in the classical Drucker-Prager model, giving a unique flow direction and consequently improves the convergence in tensile regimes. Lubliner-Lee-Fenves (LLF) model This plastic-damage model, so-called ''Barcelona" model, was first developed by Lubliner et al. [9] and later improved by Lee & Fenves [10]. The model considers a yield criterion that simulates the multi-axial behavior of concrete (biaxial and triaxial), where two hardening scalar variables �± controls and distinguish the positive/negative evolution strength of the surface and couple the plastic and damage component. Also, include a damage component that differentiates the stiffness recovery from tension


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to compression load state and viceversa, respectively. Moreover, the plastic component assumes the so-called effective stress space plasticity, which is related to the effective stress vector �� and is dependent (coupled) of damage component [7]. Finally, for the sake of efficiency, the model is expressed in terms of the principal effective space.

Wu-Li-Faría (WLF) model This plastic-damage model was first developed by Faría & Oliver [6] and modified by Wu & Li [7]. Similar to the LLF model, the model assume the effective stress space plasticity related to the effective stress vector, but in this case is independent (decoupled) of damage component [7]. Due to this condition, has the advantage to include optionally the plastic component, conversely to the LLF model. Thus, two variants were developed for this model: one approach that includes the plastic and damage components (WLF) and other one with pure damage behavior (WLF0). Moreover, the effective stress vector �� is splitted into positive �� and negative �� parts, to account separately the cracking (tension) and shear (compression) damage mechanisms for degradation of concrete using the spectral decomposition [8]. Also, the model assumed a yield criterion, and a flow potential similar to the LLF model, and two hardening parameters �± to control the positive/negative plastic component. Conversely, for the damage component, consider the definition of an elasto-plastic HFE potential in order to satisfy the thermodynamic irreversible damage process. Faría -Oliver-Cervera (FOC) model This plastic-damage model was proposed by Faría and Oliver [6]. Take identical assumptions than the WLF model for the damage and viscous components, and use a simplified representation for the evolution of the plastic strain in an explicit scheme. Total strain rotating crack (ROT) model This smeared-crack model was first developed by Cope, Rao & Gupta [25][26], and enhanced by [27]. We include some modifications in the formulation that produce a simple and robust mathematical formulation. The model assume the so-called the "total strain'' formulation present in the hypo-elastic materials, i.e. that stress vector � depends only of total strain vector � and that a set of orthogonal crack planes rotates according to direction of principal strain vector ��. Also, the model is modified to include an irreversible damage process for each i-th principal strain direction ��̂, where a secant unloading to origin (no plastic strains) is considered. Moreover, the model assumes the principle of co-axiality, that is, the principal stress directions coincide with the principal strain directions. Finally, the model only includes uniaxial effects. Table 1 shown the constitutive equations considered to calculate the plastic component of the stress vector for each concrete model. Also, Table 2 includes the expressions to evaluate the damage and total (nominal) component of the stress vector. Additionally, all models, with the exception the DPH model, can include a strain-rate dependency with a visco-plastic model. This approach improves considerably the convergence in strain-softening regimes. In all cases, it is considered the Duvaut-Lions visco-plastic model [30] due to ease of implementation. More details of these equations are explained in [31]. The unique parameter that controls the strain-rate increment is the numerical viscosity �� and is equivalent to the relaxation time. Table 3 shown the main capabilities for described concrete models. Classification of models (plastic, damage, plastic-damage), strain-softening behavior, stress state effects (biaxial or triaxial), unilateral effect and strain-rate effect are mentioned. Also, the table lists the inelastic inputs parameters.


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Table 1– Constitutive equations of the plastic component of inelastic concrete models.

Table 2– Constitutive equations of the damage and total component of inelastic concrete models.

2.2 Numerical implementation and convergence issues

This section includes main steps used to numerical implementation and recommendations to get a stable convergence of concrete models.

To incorporate the above concrete models in a FE software it is required elaborate the numerical implementation of constitutive equations. In all cases, a strain-driven scheme is considered, with an algorithm to evaluate the update stress vector and internal state variables at each integration point. In the DPH, LLF and WLF models, the plastic component of models are evaluated with a backward Euler (implicit) scheme, where a Return-mapping algorithm composed of a trial elastic-predictor step and a plastic-corrector step is considered [29]. Conversely, in the FOC model the plastic component is evaluated in an explicit scheme. In all cases, the RMA solve the consistency operator through an iterative Newton's method. The election of an adequate initial value, non-zero derivatives, and a unique scalar variable to be solved rather than a system of equations, are critical for the convergence of this method. Specifically, for the plane-stress condition, the projected-return mapping algorithm is


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adopted as solution for the DPH, LLF and WLF models. Thus, an enforcement of the consistency condition is used to reduce the solution to a simple scalar nonlinear equation. Conversely, the damage component of all models is evaluated with an explicit scheme, with the exception of the LLF models, which require the simultaneous solution of both components.

Table 3– Properties of concrete models and their input parameters.

Additionally, for the numerical integration of models, the algorithmic consistent tangent matrix ��

��

is calculated, which are found by computing the derivatives of equations involved in the stress updated algorithm. For the developed models, the consistent operator can be obtained in explicit expressions, which are derived in detail by [28]. It is well known the sensitivity of this operator in the convergence of models at a finite element level. Thus, to improve the convergence we recommend the use of continuous and smooth derivatives (C1-class) for this operator, taking special attention in strain-softening regimes. Analogously, we recommend the use of smooth C2-class functions for the flow potential of the DPH model; the C1-class approximated Heaviside function in the yield criterion of the LLF and WLF models; and the use of C1-class functions for the uniaxial laws (� − �, � − �, or � −�) in all models. In addition, the use of any asymmetric Newton-Raphson solver is mandatory if the stiffness matrix is non-symmetric. They occur in the LLF and WLF models, which are very sensitive to this condition under multi-axial loadings. Moreover, the incorporation of numerical viscous-plastic model in the concrete models improves considerably the convergence in strain-softening regimes. The Duvaut-Lions [30] model is more appropriate for the regularization of the rate-independent damage and plastic-damage models, because the Perzyna model fails to converge to the rate-independent backbone model in some cases [10]. The LLF, WLF, FOC and ROT models can include a viscous-regularization in their formulation using Duvaut-Lions model. It is recommended to use a ratio of numerical viscosity to load step increment between 0.001 and 1.0 to get an adequate convergence without compromising accuracy in the response.

3 Numerical tests

In this section, a set of numerical examples are used to validate the capabilities of the constitutive concrete models described in Section 2. Taking the numerical algorithms detailed in [28], the five concrete models were implemented in the software ANSYS [23] throught user-material FORTRAN77 routines (USERMAT.f). These material routines works at Gauss integration point level of each finite element. Four experimental benchmark test were simulated with a single-element according to loading conditions: (i) cyclic uniaxial compression; (ii) monotonic biaxial; (iii) cyclic tension-compression; and (iv) strain-rate effect. All examples were modeled using 4-node isoparametric shell element (SHELL181) with six Degree Of Freedom (DOF) at each node using 2x2 Gauss integration scheme. Also,


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in all cases, excepting the biaxial test, a pure uniaxial stress state for the boundary constraints is imposed. For the numerical concrete models, an exponential relation for the DPH model is considered for the hardening cohesion law � given by

�(�) = �� + (�� − ��)�� ⁄ , (1)

where �� = �� and �� = ��/�� , with �� > 1, �� ≥ 1 experimental fitted parameters, and ��

concrete Young’s modulus. Also, the LLF, WLF, FOC, and ROT models assume an exponential relation for the positive/negative uniaxial stress laws given by

��(��) = ��

��, �� < ��

��1 − �� + ��, �� ≥ ��

�� (2)

��(��) = ��, �� < ��

�

��1 − �� + ��(��)�, �� ≥ ��

�, (3)

where �± = �/��±, ��

� = ��/��, ��

� = ��/��, with �� and �� are parameters calculated in an iterative process in order to dissipate a specific value of fracture energy ��

� under the stress-strain �� − �

curve (see [31]). Moreover, for the LLF model it is required define a positive/negative damage laws �±(�±), where is used an exponential relation given by

�±(�±) = 1 − ��±�±, (4)

with �± are experimental fitted parameters that control the unloading branch of entire response. Also, for the plastic component of the WLF model a bilinear relation for the positive/negative effective uniaxial stress �� [7] is defined as

��±(�±) = ��± + ��̅

±�±, (5)

where ��± are the positive/negative initial stress, which are chosen for convenience at the range

�� ∈ [0, ��

�] and �� ∈ [0, ��

�] , respectively, and ��̅± = ��

±��/(�� − ��±) , with ��

± their respective hardening slope. An adequate conversion among uniaxial laws required for each concrete model is generated, as explained in [31]. Table 4 list the material parameters adopted for each benchmark test. Additional parameters are listed in the figure of each example.

Table 4– List of parameters used in the concrete models.

Cyclic uniaxial tests Numerical concrete models are compared with cyclic uniaxial compressive loading-unloading and reloading experimental data of Karsan & Jirsa [32]. Fig.2 shown the response of the five concrete models under compressive loads.


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It can observed in all models, except for the DPH model, a fit well with the post-peak backbone response of experimental tests, where the WLF0 and ROT models gives the best approximation. Although, both models fail in the unloading branch, due that neglects the plastic strains. Also note that these models have identical responses them, although are elaborated with formulations completely different. In contrast, the unloading branch of the LLF, WLF and FOC models fits close to experimental response due that incorporate the plastic and damage components in their formulations.

Fig.2– Validation of concrete models under cyclic uniaxial compressive test of Karsan & Jirsa [32]: (a) DPH model; (b) LLF model; (c) WLF model; (d) FOC model; and (e) ROT model. The following additional parameters are used. For the DPH model: ��

�=3.48MPa, ��=12MPa, ��=1, ��=3; LLF model: ��=6500, ��=7500; and WLF model: ��

�=20MPa, ��=0.16��,

��=0.48��.

Monotonic biaxial tests All the concrete models, except the ROT model, are compared with monotonic biaxial test of Kupfer et al. [33]. This test is performed with a constant biaxial loading ratio of � = ��/��, where �� and �� are the stresses imposed. It is assumed a pure biaxial stress state for the boundary constraints. A stress-controlled test are performed up to reach the peak stress, with the exception of the uniaxial case (�=0) simulated with displacement-controlled. The inputs parameters are fitted with the cases �=0, 1 and 0.52 simultaneously. Fig. 3a shown the biaxial peak strength surface for the DPH, WLF0, WLF, FOC and LLF models under different combination of biaxial loading ratios � = ��/��. For the DPH model, the parameters � and � are fitted with tension/compression biaxial strength of concrete. It can observed, that all models fits close with the experimental results, especially in compression-compression (C-C) regime, due that are influenced by the Drucker-Prager yield criterion. The major differences among models occur in the tension-compression (T-C) regimes. The exception occur with the DPH model, where fit well only in the equal biaxial loading ratio �=1.


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Fig. 3– (a) Biaxial peak strength surface for the DPH, WLF0, WLF, FOC and LLF models and the biaxial test results of Kupfer et al. [33]; and (b) validation of the LLF, WLF0, WLF, FOC and ROT models under cyclic tension-compression tests of Mazars et al. [34]. The following additional parameters are used. For the DPH model: ��

�=3.5MPa and �� = ��

�; LLF model:

��=12000, ��=200; WLF model: ��=12MPa, ��

�=0.3��, and ��=0.4��; and the FOC model: ��=0.54 and ��=0.75.

Cyclic tension-compression test To validate the unilateral effect, the LLF, WLF0, WLF, FOC and ROT models are compared with the cyclic uniaxial test of Mazars et al. [34]. This test was first subjected to uniaxial tension followed by uniaxial compression in parallel directions. Fig.3b shown the axial stress �� vs axial strain �� of this models. It also included the response of the LLF model with three values of stiffness recovery factor ��

� (0, 0.5 and 1). It can notice that all models recovery the initial elastic stiffness once the load goes into the compression state (step 2 and 4). The exception occur, obviously, in the LLF model when ��

�=0.5 and 0, due that this parameter controls the value of recovery compression stiffness. Moreover, it is observed that all models, with the exception of the WLF0 and ROT models, take the compression backbone branch close to experimental data (step 4), due that include plastic strain in their formulations. In addition, its observed that the LLF, WLF and FOC models recovery the damaged stiffness obtained in the last cycle of tension (step 3) when the load goes from compression to tension state (step 6). This condition is also shared by the WLF0 and ROT models (not shown in the plot) and is so-called that the models have "damage memory", which is agree with the thermodynamic of irreversible process.

4 Epistemic uncertainty estimation

To measure the epistemic uncertainties on inelastic constitutive concrete models studied, five types of response parameters were considered: (1) peak stress �� and respective strain �� of the monotonic

stress-strain curve; (2) dissipated energy of the monotonic �̅� stress-strain curve; (3) dissipated energy of the first �̅��

last �̅��and total �̅� loading-unloading cycle of stress-strain curve, respectively;

(4) linearized least square stiffness of hardening branch ��, and softening branch ��, of monotonic


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stress-strain curve, respectively; and (5) first, ��, and last ��

linearized least square stiffnesses in

loading-unloading cycle, respectively. Fig. 4 illustrates these parameters for clarification.

Fig. 4– Definition of response parameters to measure epistemic uncertainty in inelastic concrete models.

The uncertainty of inelastic concrete models is measured as the ratio of simulated concrete models �� relative to experimental test results �� . The uncertainty of the ratios ��/�� is

characterized by its minimum, maximum values, and the standard deviation σ. Fig. 5 summarizes the box-plot of response parameters for the uniaxial, biaxial, and unilateral simulations, respectively. The box-plots considered hereafter contains a rectangle whose length is the difference between the first and third quartile, a median �̅ represented by an intermediate horizontal line, a mean represented by a rhombus, whiskers equivalent in width to two standard deviations (2σ), and outliers which fall outside the range (�̅ ± �). First, for the uniaxial compressive case, it can observed up to 47% less amount of energy dissipated by the first loading-unloading cycle �̅��

of numerical simulations than experimental test, with a

considerable uncertainty of up to σ=20.6%. Similarly, up to 40% more flexible is the first loading-unloading stiffness ��

, mainly due to WLF0 and ROT models, which unloads to origin. Conversely,

there are up to 1.32 times more energy dissipated by monotonic �̅� stress-strain curve by numerical simulations than in the experimental test, mainly due to DPH plastic model. Furthermore, a high variability exist for the loading-unloading stiffness of last cycle ��

, with σ=175.6%, leading the most

important source of epistemic uncertainty. This uncertainty is mainly due to the differences in the taxonomy of stress-strain constitutive concrete models considered (plastic, damage or plastic-damage). For the biaxial case, the box-plot for each variable correspond to combination of all stress ratios simulated � = ��/�� with �=0, 1, and 0.52. For one hand, a good fit adjust exist for all response variables measured, with a medium-level of uncertainty less than 30%. For other hand, a low-level of uncertainty is observed for the peak stress ��, with σ=4%, whereas a high uncertainty is observed in

the variables ��, �̅�, and ��, with a standard deviation of 83.7%, 58.9% and 55.2%, respectively.

Finally, for the uniaxial cyclic tension-compression (unilateral effect), the standard deviation of all parameters is less than 10%, with the exception of the variables �̅� and ��

, where σ=25.9% and

38.3%, respectively.


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Fig. 5– Response parameters of the numerical concrete models normalized by the experimental benchmark test results: (a) uniaxial cyclic compression; (b) biaxial monotonic; and (c) uniaxial cyclic tension-compression. Box-plot diagram (top); and maximum, minimum and standard deviation σ (%) (bottom). (In case c, values in parenthesis associated with the compressive load case.)

6 Conclusions

Epistemic uncertainty of five plane-stress constitutive concrete models has been studied by comparing model responses in simple benchmark examples. A brief description of these models in a common vectorized notation was presented, providing some recommendations for their numerical implementation and to improve their convergence. Moreover, numerical benchmark test examples under uniaxial, biaxial, and tension-compression stresses demonstrate the different capacities of the models considered. The epistemic uncertainty observed in the response of a concrete prism is enough to asses correctly the sensitivity of these concrete models. Thus, it is concluded that the unloading-loading linearized stiffness of the last cycle ��

for the uniaxial cyclic compression test, provide the most important

source of epistemic uncertainty, with a standard deviation of σ=175.6%. In contrast, low uncertainty was observed in peak stresses ��, for all test simulations, where σ=12.8%.

7 Acknowledgments

This research has been funded by Grants CONICYT/Fondecyt/1141187, CONICYT/FONDAP/15110017 (National Research Center for Integrated Natural Disaster Management, CIGIDEN), and CONICYT/Doctorado Nacional/ 2013.


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