advertisement

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/313828611 Simpliﬁed Damage Plasticity Model for Concrete Article in Structural Engineering International · February 2017 DOI: 10.2749/101686616X1081 CITATIONS READS 234 41,203 5 authors, including: Milad Hafezolghorani Esfahani Farzad Hejazi Universiti Putra Malaysia University of the West of England, Bristol 13 PUBLICATIONS 269 CITATIONS 224 PUBLICATIONS 1,490 CITATIONS SEE PROFILE Ramin Vaghei 20 PUBLICATIONS 427 CITATIONS SEE PROFILE Keyhan Karimzadeh UNSW Sydney 8 PUBLICATIONS 282 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Design and Construction of 65m Span Post-tensioned Ultra High Performance Fiber-Reinforced Concrete (UHPFRC) Composite Bridge View project Design of 65m single span integral UHPFRC bridge located in Malaysia View project All content following this page was uploaded by Farzad Hejazi on 05 June 2017. The user has requested enhancement of the downloaded file. Simpliﬁed Damage Plasticity Model for Concrete Milad Hafezolghorani, PhD Candidate; Farzad Hejazi, Senior Lecturer; Ramin Vaghei, PhD; Mohd Saleh Bin Jaafar, Prof.; Keyhan Karimzade, Msc, Department of Civil Engineering, Faculty of Engineering, Universiti Putra Malaysia, Selangor, Malaysia. Contact: farzad@upm.edu.my DOI: 10.2749/101686616X1081 Abstract The past several years have witnessed an increase in research on the nonlinear analysis of the structures made from reinforced concrete. Several mathematical models were created to analyze the behavior of concrete and the reinforcements. Factors including inelasticity, time dependence, cracking and the interactive effects between reinforcement and concrete were considered. The crushing of the concrete in compression and the cracking of the concrete in tension are the two common failure modes of concrete. Material models were introduced for analyzing the behavior of unconﬁned concrete, and a possible constitutive model was the concrete damage plasticity (CDP) model. Due to the complexity of the CDP theory, the procedure was simpliﬁed and a simpliﬁed concrete damage plasticity (SCDP) model was developed in this paper. The SCDP model was further characterized in tabular forms to simulate the behavior of unconﬁned concrete. The parameters of the concrete damage plasticity model, including a damage parameter, strain hardening/softening rules, and certain other elements, were presented through the tables shown in the paper for concrete grades B20, B30, B40 and B50. All the aspects were discussed in relation to the effective application of a ﬁnite element method in the analysis. Finally, a simply supported prestressed beam was analyzed with respect to four different concrete grades through the ﬁnite element program. The results showed that the proposed model had good correlation with prior arts and empirical formulations. Keywords: concrete damage plasticity; concrete failure; unconﬁned concrete; ﬁnite element analysis; crack; crush. Introduction Finite element method based analysis of reinforced concrete structures has signiﬁcantly developed since 1970. Researchers have tried to analyze the behavior of concrete and have published several reports and technical reports on this subject. However, the behavior of concrete is complex, and many parameters must be considered for its analysis. Concrete is composed of qualitatively and quantitatively different types of materials. These materials exhibit different properties in terms of tension and compression. The structural mechanics of concrete structures is quite important, and concrete identiﬁcation parameters including the non-linear stress–strain relation of the concrete under imposing stress conditions and strain Peer-reviewed by international experts and accepted for publication by SEI Editorial Board Paper received: April 8, 2015 Paper accepted: December 17, 2015 68 Scientiﬁc Paper hardening/softening make the behavior of concrete more complicated. Hence, it becomes difﬁcult to determine damage in concrete. Constitutive models are used for this purpose, and an example of these models is the concrete damage plasticity (CDP) model. This model uses the ﬂow theory of plasticity and damage mechanics to analyze the concrete structures.1–4 Previous Studies Concrete damage plasticity is widely recognized as a precise and practical constitutive model to simulate concrete behavior. Carol et al.5 analyzed different combinations of plasticity and damage. Gatuingt and PijaudierCabot,6 and Kratzig and Polling7 used isotropic damage in their research, and elaborated on several types of plasticity combinations. Models involving the development of stress related plasticity in the effective stress space were introduced by some researchers.8–10 Lubliner et al.,11 Ananiev and Ozbolt,12 and Imran and also considered Pantazopoulou13 models that involved the formulation of plasticity in the nominal stress space. Grassel and Jirasek14 introduced different combinations of plasticity and damage applied to the concrete failure models. They analyzed the local uniqueness conditions of two combinations of strain scalar damage and stress based plasticity types. Furthermore, the triaxial damage plastic model accounted for the failure of concrete. A three dimensional interface model was presented by Grassl and Rempling.15 The combination of damage mechanics and the theory of plasticity were used as a basis for this model, and it enabled the researchers to vary the ratio of the permanent and total inelastic displacements. Yu et al.16 introduced a modiﬁed plastic damage model. The theoretical framework of the CDPM was used as a basis for this model, and the conﬁned concrete was modeled by ABAQUS with the conditions of nonuniform conﬁnement. The Lubliner yield criterion was used for the triaxial compression stress states by Zhang et al.17 The Lubliner criterion was reworked, improving its limits, and hence many stress states in engineering structures could be accounted for by this revised version of the Lubliner criterion. Taddei et al.18 proposed three-dimensional ﬁnite element models for unreinforced and reinforced walls panels based on concrete damage plasticity constitutive law. Grassel et al.19 utilized a constitutive model for concrete structures subjected to multiaxial and rate-dependent loading by combining an effective stress based plasticity model with an isotropic damage model based on plastic and elastic strain measures. Larsson et al.20 examined the laterally loaded lime-cement columns in a shear box and used a damage plasticity model to numerically analyze the columns. The model accounted for the stiffness degradation of the columns. Ming21 analyzed the basic Structural Engineering International Nr. 1/2017 concepts and properties of the concrete damage plasticity model using inferential reasoning and analysis. Computational techniques were used to determine the plastic factors for the concrete damage in ABAQUS. Zhang and Li22 introduced calibration techniques for concrete, under uniform and non-uniform conﬁnement, based on Lubliner theory through three dimensional simulation. Grassl et al.23 used a combination of plasticity and damage mechanics, to create a constitutive model that was employed to study the failure of concrete structures. The model was designed to analyze the properties of the failure process in concrete structures experiencing multi-axial loading. The plasticity model based on effective stress and the plastic and elastic strain measures based damage models were used for this purpose. Vaghei et al.24 proposed the ﬁnite element model to develop a three dimensional version addressing precast walls and the connection. Shang et al.25 studied the torque of the RC girder. They applied the ﬁnite element analysis software ABAQUS to concrete damaged plasticity model as a constitutive model for the analysis of concrete material. The outcomes from other experiments were used to validate the results of the ﬁnite element analysis. Tao et al.26 utilized a single, robust ﬁnite element (FE) model to study the bond behavior subjected to shear. Tiwari et al.27 investigated underground tunnels with curved alignment in the longitudinal direction subjected to blast loading. He observed that stress, deformation and damage responses of tunnel lining through three dimensional ﬁnite element simulations using concrete damage plasticity theory. Extant research indicates that it is quite complicated to represent reinforced concrete behavior using concrete damaged plasticity models as a constitutive law. This also may not be fully understood by prospective researchers. In the present study, the procedure of concrete damage plasticity theory was simpliﬁed and characterized in tabulated forms. The ﬁndings were formulated and presented in a tabular format for four common concrete grades, while concrete parameters could be extended to other concrete grades. Numerical analysis was then conducted to investigate the SCDP model, which was implemented in the ABAQUS ﬁnite element software in order to easily understand the mechanical behavior of concrete. The SCDP model was then veriﬁed against the current concrete damage plasticity theory and empirical formulations. Damage Plasticity Constitutive Model The isotropic damaged elasticity and the isotropic tensile and compressive plasticity were used in the concrete damaged plasticity model to study the behavior of concrete in a non-elastic manner. The total strain tensor ε was comprised of the elastic part εel and the plastic part εpl. ε = ε el + ε pl σ = D el : ε − ε pl σ = Del0 : ε −ε pl ð1Þ ð3Þ D el = ð1 −dÞDel0 ð4Þ ð2Þ The nominal stress with the degraded elastic tensor from (4) could be rewritten as follows: ð5Þ σ = ð1 −dÞDel0 : ε −ε pl The damage plasticity constitutive model was based on the following stress–strain relationship: σ = ð1 −dÞ:σ ! σ = ð1 −dt Þσ t + ð1 −dc Þσ c ð6Þ where dt and dc were two scalar damage variables, ranging from 0 (undamaged) to 1 (fully damaged).23 The damage model used for concrete was based on plasticity and considered the failure process of tensile cracking and compressive crushing. Isotropic hardening variables were expressed by inelastic compression , h and cracking strain εck, h , strain εin t c which include the plastic hardening strain εpl,h plus the residual strain due to damages. pl, h ε pl, h = εtpl, h ; ε pl = h ε pl, h , σ : ε_ pl , εc ε_ = ε_ el + ε_ pl ð7Þ Hardening variables were used to control the development of the yield or failure of the surface. These variables were connected to the processes of tension and compression loading. Structural Engineering International Nr. 1/2017 The behavior of the concrete was explained by the assumption that concrete damage plasticity utilized the yield function, f ε pl, h , σ , which represented the yield surface in effective stress space to determine the states of damage or failure.11 In the concrete damage plasticity model, the ﬂow rule was deﬁned as follows: _ ε_ pl = λ: ∂Gðσ Þ ∂σ ð8Þ In the concrete damage plasticity model, the ﬂow rule was non-associated. This meant that the yield function f ε pl, h , σ and the plastic potential gp did not coincide, and, therefore, the direction of the plastic ﬂow ∂G∂σðσ Þ was not normal to the yield surface. The plastic potential was deﬁned in effective stress space. However, in this study, due to the complex degradation mechanism of the uniaxial cyclic behavior of concrete (opening and closing of formed micro-cracks), the uniaxial response of concrete was investigated. Figure 1 indicated that the uniaxial compressive and tensile response of concrete was assumed to be inﬂuenced by damaged plasticity, and this assumption formed the basis of the model. The uniaxial compressive and tensile responses of concrete with respect to the concrete damage plasticity model subjected to compression and tension load were given by: pl, h ð9Þ σ t = ð1 − dt ÞE0 εt − εt , h σ c = ð1 − dc ÞE0 εc − εpl ð10Þ c Given the nominal uniaxial stress, the effective uniaxial stress σ t and σ c were derived as follows: σt pl, h ð11Þ = E 0 ε t − εt σt = ð1 −dt Þ σc , h = E0 εc − εpl σc = ð12Þ c ð1 −dc Þ where compressive strain εc equalled , h + εel , and tensile strain ε equalεpl t c c , h + εel . led εpl t t Simpliﬁed Concrete Damage Plasticity The values of the hardening and softening variables were used for the Scientiﬁc Paper 69 (a) (b) c c t B cu t0 cu c0 E0 (1 – dt) E0 (1 – dc) E0 0.5cu C 0.2cu E0 A E0 E0 "c in,h "c " " pl,h c el c "t ck,h "t "tel pl,h t " F ig . 1: Response of concrete to a uniaxial loading condition: (a) Compression, (b) Tension.11 (Units: [–]) determination of the cracking and crushing trends, respectively. They were responsible for the loss of the elastic stiffness and the development of the yield surface. The damage states in compression and tension were characterized independently by two hardening variables. These were indicated by εcpl, h and εtpl, h , which referred to equivalent plastic strains in tension and compression, respectively. where σ c and εc were nominal compressive stress and strain, respectively, and σ cu and ε’c were ultimate compressive strength and the strain of In concrete damage plasticity models, the plastic hardening strain in compression εcpl, h played a key role in ﬁnding the relation between the damage parameters and the compressive strength of concrete (see Fig. 1a) as follows: σ c = ð1 −dc ÞE0 εc − εcpl, h ð13Þ 8 σc in, h > < εc = εc − E 0 σc 1 > pl , h : εc = εc − E0 1 − dc ð14Þ dc σ c ð1 −dc Þ E0 ð15Þ εcpl, h = εcin, h − Generally, uniaxial compressive behavior could be characterized by either experimental tests or existing constitutive models, such as those proposed by Hognestad28 and Kent et al.29 for unconﬁned concrete. However, the present study employed the Kent and Park parabolic constitutive model for unconﬁned concrete, which was expressed by the following equation: " # εc εc 2 ð16Þ σ c = σ cu 2 ’ − ’ εc εc 70 Scientiﬁc Paper "0 .5 " "0.8 F i g. 2: Kent and Park model for conﬁned and unconﬁned concrete..29 (Units: [–]) the unconﬁned cylinder specimen, respectively. Park30 reported that ε’c equaled 0.002, and this value was also assumed in this study. Figure 2 indicated a parabolic increasing trend (A–B) for the hardening stage, while a linear behavior (B–C) was observed for the Plasticity parameters Material’s parameters B20 Concrete elasticity E (GPa) Uniaxial Compressive Behavior 0.002 D 21.2 0.2 Concrete compressive behavior Yield stress (MPa) Inelastic strain Dilation angle 31 Eccentricity 0.1 fb0/fc0 1.16 K 0.67 Viscosity parameter 0 Concrete compression damage Damage parameter C Inelastic strain 10.2 0 0 0 12.8 7.73585E-05 0 7.73585E-05 15 0.000173585 0 0.000173585 16.8 0.000288679 0 0.000288679 18.2 0.000422642 0 0.000422642 19.2 0.000575472 0 0.000575472 19.8 0.00074717 0 0.00074717 20 0.000937736 0 0.000937736 19.8 0.00114717 0.01 0.00114717 19.2 0.001375472 0.04 0.001375472 18.2 0.001622642 0.09 0.001622642 16.8 0.001888679 0.16 0.001888679 15 0.002173585 0.25 0.002173585 12.8 0.002477358 0.36 0.002477358 10.2 0.0028 0.49 0.0028 7.2 0.003141509 0.64 0.003141509 0.81 0.003501887 3.8 0.003501887 Concrete tensile behavior Yield stress (MPa) 2 0.02 Cracking strain Concrete tension damage Damage parameter T Cracking strain 0 0 0 0.000943396 0.99 0.000943396 Table 1: Material properties for concrete with SCDP model in class B20 Structural Engineering International Nr. 1/2017 Plasticity parameters Material’s parameters B30 Concrete Elasticity E (GPa) 26.6 0.2 Concrete compressive behavior Yield stress (MPa) Inelastic strain Dilation angle 31 Eccentricity 0.1 fb0/fc0 1.16 K 0.67 Viscosity parameter 0 Concrete compression damage Damage parameter C Inelastic strain (modulus of elasticity, E0) due to the damage when plastic strains increased in brittle materials (such as concrete and concrete-like material) as shown in Fig. 1a. The damage parameter (dc) was 0 at the maximum compressive stress, and thereafter, it began to decrease and continued decreasing until 0.8 was reached with respect to 20% remaining strength in large strains. 15.3 0 0 0 Uniaxial Tensile Behavior 19.2 4.8249E-05 0 4.8249E-05 22.5 0.000119844 0 0.000119844 25.2 0.000214786 0 0.000214786 27.3 0.000333074 0 0.000333074 28.8 0.000474708 0 0.000474708 29.7 0.000639689 0 0.000639689 30 0.000828016 0 0.000828016 29.7 0.001039689 0.01 0.001039689 28.8 0.001274708 0.04 0.001274708 In concrete damage plasticity models, the plastic hardening strain in tension εtpl, h was derived (see Fig. 1b as follows: pl, h ð19Þ σ t = ð1 − dt ÞE0 εt − εt 8 ck, h σt > = εt − < εt E0 ð20Þ σt 1 pl, h > : εt = εt − E0 1 − dt 27.3 0.001533074 0.09 0.001533074 25.2 0.001814786 0.16 0.001814786 22.5 0.002119844 0.25 0.002119844 19.2 0.002448249 0.36 0.002448249 15.3 0.0028 0.49 0.0028 10.8 0.003175097 0.64 0.003175097 0.81 0.003573541 5.7 0.003573541 Concrete tensile behavior Yield stress (MPa) 3 0.03 Cracking strain pl, h εt Concrete tension damage Damage parameter T Cracking strain 0 0 0 0.001167315 0.99 0.001167315 Table 2: Material properties for concrete with SCDP model in class B30 softening stages of conﬁned and unconﬁned concretes. The softening phase continued until 20% of the unconﬁned cylinder compressive strength (Point C) was reached; that is, the stress value was not allowed to continue to decrease, and perfect plastic behavior was assumed following the softening trend (C–D). For simplicity, the entire constitutive model was assumed to be a parabolic curve. Equation (16) assumed a nonlinear behavior for concrete from the beginning to the end. However, deﬁning the behavior of concrete up to 40% of its strength in the elastic phase was important in determining the effective elastic modulus. In other words, the constitutive model came into effect when the compressive strength was 60% of the concrete compressive strength. According to Figs. 1a and 2, inelastic hardening strain in compression, εcin, h was derived as follows: εcin, h = εc − σc E0 ð17Þ Cyclic behavior contributed to concrete behavior, which was deﬁned by effective parameters, including damage in compression and damage in tension. Compression damage (dc) was based on inelastic hardening strain in compression εcin, h that controlled the unloading curve slope. Given that dc increased with respect to an increase in εcin, h , it could be expressed as follows: dc = 1 − σc σ cu ð18Þ The tangent of the curve decreased with respect to the initial tangent Structural Engineering International Nr. 1/2017 ck, h = εt − dt σ t ð1 −dt Þ E0 ð21Þ Although concrete had many constitutive models in the tension phase, there were no signiﬁcant differences in their results due to the brittle behavior of concrete. Engineers seldom deﬁne the tension behavior of concrete in numerical modeling, mainly because the interaction between reinforcement and concrete is simpliﬁed and because certain changes in the stress–strain relation of concrete in the tension phase must be made to consider bar slips in concrete. Among the constitutive models for the tension phase considering tensile strength, 7% to 10% of maximum compressive strength σ cu was chosen as a tensile strength, σ t0, and in this study, the maximum value was taken (i.e. σ t0 = 0.1σ cu). In this paper, 1% of the tensile strength was considered during the analysis regardless of the realistic condition to prevent numerical instability. In contrast, correspondence strain value, where stress is 1% of the ultimate tensile strength, was taken as 10 times the percentage of the strain, in which stress was equal to ultimate tensile strength. Figure 1b showed that with a further increase in the hardening cracking ck, h the tension damage constrain, εt tinued to increase, and this could be expressed as follows: dt = 1 − σt σ t0 ð22Þ Scientiﬁc Paper 71 Application of SCDP Parameters in FE Programs Poisson’s ratio range for concrete was between 0.1 and 0.2. Poisson’s ratio, elasticity modulus, and stress–strain curve of concrete in compression and tension were related to loading history, which resulted in strain differences. The dilation angle was equal to volume strain over shear strain. The dilation angle for concrete was usually 20 to 40 , which affected material ductility. Consequently, the dilation angle had considerable effects on the entire model. An increase in the dilation angle increased the system ﬂexibility. From a practical viewpoint, the internal dilation angle depended on certain parameters, including plastic strain and conﬁned pressure. An increase in plastic strain and conﬁned pressure decreased the internal dilation angle. The material has a constant dilation angle for a large range of pressure stresses used for the conﬁnement of the material. The default ﬂow potential eccentricity was ε = 0.1, and by raising the value of , the curvature of the ﬂow potential increased. If the default ﬂow of potential eccentricity had a value much lower than the default value, there could be convergence problems when the conﬁning pressure is not high. The ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress was given by fb0/fc0, and its default value was 1.16 [34]. The ABAQUS software31 used a null default viscosity parameter so that the viscoplastic regularization did not occur. This parameter enhanced the convergence rate of the model when the softening process occurred, and it gave good results. In the FE program, the tension recovery parameter equal to zero denoted that this parameter contributed to the cyclic behavior and monitored the modulus of elasticity when compression behavior changed to tension behavior and vice versa. A value of zero for tension recovery implied that material tangent in the tension phase was completely affected by compression damages. However, compression recovery was taken as 1, which meant that tension damage did not affect the material tangent. All these assumptions were also adopted in reality. According to the aforementioned damage plasticity formulation derived in this section, four different concrete 72 Scientiﬁc Paper grades (20, 30, 40, and 50) were implemented within the framework in tabular format, that is, Tables 1, 2, 3, and 4, respectively. Accordingly, the hardening and softening rule as well as the evolution of the scalar damage variable for compression and tension were presented for concrete grades 20, 30, 40, and 50. The general framework of the damage plasticity formulation was clearly stated and could be extended to other concrete grades between B20 and B50. Numerical Computation The performance of the proposed constitutive model was further evaluated through the structural analysis of a simply supported partially prestressed beam, which was subject to self-weight and superimposed loading. Figure 3 shows a partially prestressed concrete beam that measured 16 m × 1 m × 0.6 m, and was modeled in ABAQUS. Twelve pieces of steel reinforcements sized D = 20 mm with Grade 460 Mpa and E = 200 GPa were placed in two rows with 100 mm spacing, horizontally and vertically. A prestressed tendon of grade 270 with seven wire strands (270 ksi = 1861 MPa; E = 196.5 GPa; and area of 1600 mm2) was placed at 200 mm eccentricity from the center. In this study, effective prestressing force (Pe = 2667.2 kN), self-weight, and superimposed dead load were applied to the simply supported partially prestressed beam. A concrete block and tendon were modeled by a threedimensional cube element (C3D8), and the steel reinforcement elements were 3D truss (T3D2). Plasticity parameters Material’s parameters B40 Concrete elasticity E (GPa) 30 0.2 Concrete compressive behavior Yield stress (MPa) Inelastic strain Dilation angle 31 Eccentricity 0.1 fb0/fc0 1.16 K 0.67 Viscosity parameter 0 Concrete compression damage Damage parameter C Inelastic strain 20.4 0 0 0 25.6 2.66667E-05 0 2.66667E-05 30 0.00008 0 0.00008 33.6 0.00016 0 0.00016 36.4 0.000266667 0 0.000266667 38.4 0.0004 0 0.0004 39.6 0.00056 0 0.00056 40 0.000746667 0 0.000746667 39.6 0.00096 0.01 0.00096 38.4 0.0012 0.04 0.0012 36.4 0.001466667 0.09 0.001466667 33.6 0.00176 0.16 0.00176 30 0.00208 0.25 0.00208 25.6 0.002426667 0.36 0.002426667 20.4 0.0028 0.49 0.0028 14.4 0.0032 0.64 0.0032 0.81 0.003626667 7.6 0.003626667 Concrete tensile behavior Yield stress (MPa) 4 0.04 Cracking strain Concrete tension damage Damage parameter T Cracking strain 0 0 0 0.001333333 0.99 0.001333333 Table 3: Material properties for concrete with SCDP model in class B40 Structural Engineering International Nr. 1/2017 Plasticity parameters Material’s parameters B50 Concrete elasticity E (GPa) 33.4 0.2 Concrete compressive behavior Yield stress (MPa) Dilation angle 31 Eccentricity 0.1 fb0/fc0 1.16 K 0.67 Viscosity parameter 0 Concrete compression damage Inelastic strain Damage parameter C Inelastic strain 25.5 0 0 0 32 5.73819E-06 0 5.73819E-06 37.5 4.13628E-05 0 4.13628E-05 42 0.000106874 0 0.000106874 45.5 0.000202271 0 0.000202271 48 0.000327555 0 0.000327555 49.5 0.000482726 0 0.000482726 50 0.000667782 0 0.000667782 49.5 0.000882726 0.01 0.000882726 48 0.001127555 0.04 0.001127555 45.5 0.001402271 0.09 0.001402271 42 0.001706874 0.16 0.001706874 37.5 0.002041363 0.25 0.002041363 32 0.002405738 0.36 0.002405738 25.5 0.0028 0.49 0.0028 18 0.003224148 0.64 0.003224148 0.81 0.003678183 9.5 0.003678183 Concrete tensile behavior Yield stress (MPa) 5 0.05 beam failed when the damage in tension reached 0.99. Accordingly, the maximum compression stress (S, Mises) in concrete and the mid-span displacement of the beam subjected to self-weight, superimposed dead load, and prestressing load were determined. Figure 8 shows the value of the concrete stress and mid-span displacement of the simply supported partially prestressed beam. The plots demonstrate that raising the concrete grade increases the maximum compressive stress in concrete as well as the midspan displacement. The compressive stress of concrete for different classes, B20, B30, B40, and B50, reached 92.35%, 84.53%, 81.475%, and 77.94%, respectively, of their corresponding ultimate concrete strength. Furthermore, the mid-span displacement shows a gradual increase from 53.8 mm in B20 to 74.64 mm in B50. The force versus displacement graphs obtained in the service stage due to self-weight and superimposed dead loads are shown in Fig. 9. The capacity of the partially prestressed concrete beam in B20 is 28.20452, whereas the capacities of B30, B40 and B50 indicated increases of 48.2%, 119% and 172.7%, respectively. Concrete tension damage Cracking strain Damage parameter T Cracking strain 0 0 0 0.001494322 0.99 0.001494322 (a) (b) 600 m m 1m Table 4: Material properties for concrete with SCDP model in class B50 1000 mm 200 mm 16 m 6 ϕ20 @ 100 mm 0.6 m Fi g. 3: Schematic view of the partially prestressed concrete beam: (a) Isometric view, (b) Cross section The effect of the SCDP model on the concrete behavior was numerically investigated for four different prestressed concrete grades, namely, B20, B30, B40, and B50. Three key features, including damage in tension, Von Mises stress, and maximum displacement, were recorded and compared to determine the effect of the SCDP method. The results were plotted to interpret the effect of the SCDP method on the aforementioned partially prestressed concrete classes. The damage in tension, stress distribution, and mid-span deﬂection of the partially prestressed concrete beams are shown in Figs. 4–7 for the four concrete classes B20, B30, B40, and B50, respectively. As demonstrated in these ﬁgures, the prestressed concrete Structural Engineering International Nr. 1/2017 Validation of Simpliﬁed and Tabulated Concrete Damage Plasticity Model Validation of the simpliﬁed and tabulated concrete damage plasticity model was achieved by comparing the empirical formulations given in previous research with that presented in this paper. Table 5 indicated the results of the empirical formulation as well as the SCDP model for a simply supported beam in the service stage subjected to self weight and super imposed dead load. The results showed a good correlation between these two approaches. The discrepancy was observed in the analysis techniques as the Newton–Raphson method was incorporated in the ﬁnite element modeling, while the linear technique was employed in the empirical formulations. where C, I and W were the distance of bottom ﬁber to the neutral axis, moment of inertia and service load at failure, respectively. Deﬂection due to prestress load and service load were sequentially depicted by ΔP, ΔSL. Scientiﬁc Paper 73 (a) DAMAGET (Avg: 75%) +9.900e–01 +8.910e–01 +7.920e–01 +6.930e–01 +5.940e–01 +4.950e–01 +3.960e–01 +2.970e–01 +1.980e–01 +9.900e–02 +0.000e+00 (b) S, Mises (Avg: 75%) +1.847e+01 +1.663e+01 +1.479e+01 +1.295e+01 +1.111e+01 +9.277e+00 +7.439e+00 +5.601e+00 +3.763e+00 +1.925e+00 +8.737e–02 (c) U, U2 +0.000e+00 –5.385e+00 –1.077e+01 –1.616e+01 –2.154e+01 –2.693e+01 –3.231e+01 –3.770e+01 –4.308e+01 –4.847e+01 –5.385e+01 F ig . 4: The results of the analysis on concrete B20: (a) damage in tension, (b) stress distribution (MPa), and (c) displacement distribution (mm) (a) DAMAGET (Avg: 75%) +9.900e–01 +8.910e–01 +7.920e–01 +6.930e–01 +5.940e–01 +4.950e–01 +3.960e–01 +2.970e–01 +1.980e–01 +9.900e–02 +0.000e+00 (b) S, Mises (Avg: 75%) +2.536e+01 +2.283e+01 +2.031e+01 +1.778e+01 +1.526e+01 +1.274e+01 +1.021e+01 +7.687e+00 +5.163e+00 +2.639e+00 +1.147e–01 (c) U, U2 +0.000e+00 –5.986e+00 –1.197e+01 –1.796e+01 –2.394e+01 –2.993e+01 –3.591e+01 –4.190e+01 –4.789e+01 –5.387e+01 –5.986e+01 F ig . 5: The results of the analysis on concrete B30: (a) damage in tension, (b) stress distribution (MPa), and (c) displacement distribution (mm) (a) DAMAGET (Avg: 75%) +9.900e–01 +8.910e–01 +7.920e–01 +6.930e–01 +5.940e–01 +4.950e–01 +3.960e–01 +2.970e–01 +1.980e–01 +9.900e–02 +0.000e+00 (b) S, Mises (Avg: 75%) +3.259e+01 +2.935e+01 +2.610e+01 +2.286e+01 +1.961e+01 +1.637e+01 +1.312e+01 +9.877e+00 +6.632e+00 +3.387e+00 +1.420e–01 (c) U, U2 +0.000e+00 –6.983e+00 –1.397e+01 –2.095e+01 –2.793e+01 –3.492e+01 –4.190e+01 –4.888e+01 –5.587e+01 –6.285e+01 –6.983e+01 F i g. 6: The results of the analysis on concrete B40: (a) damage in tension, (b) stress distribution (MPa), and (c) displacement distribution (mm) 74 Scientiﬁc Paper Structural Engineering International Nr. 1/2017 (a) DAMAGET (Avg: 75%) +9.900e–01 +8.910e–01 +7.920e–01 +6.930e–01 +5.940e–01 +4.950e–01 +3.960e–01 +2.970e–01 +1.980e–01 +9.900e–02 +0.000e+00 (b) S, Mises (Avg: 75%) +3.897e+01 +3.509e+01 +3.121e+01 +2.733e+01 +2.344e+01 +1.956e+01 +1.568e+01 +1.180e+01 +7.924e+00 +4.044e+00 +1.638e–01 (c) U, U2 +0.000e+00 –7.464e+00 –1.493e+01 –2.239e+01 –2.986e+01 –3.732e+01 –4.478e+01 –5.225e+01 –5.971e+01 –6.718e+01 –7.464e+01 Fi g. 7: The results of the analysis on concrete B50: (a) damage in tension, (b) stress distribution (MPa), and (c) displacement distribution (mm) 18.47 Subscripts “H” and “FE” denoted the contributions made by the empirical formulation and ﬁnite element modeling, respectively. 74.64 69.83 59.7774 53.8059 38.97 32.59 25.36 Mid Span Displacement (mm) B20 Maximum Principal Stress (Mpa) B30 B40 B50 Fi g . 8: Concrete stress and mid-span displacement of the simply supported prestressed concrete beam with different concrete grades 2000 B20 B30 B40 B50 1800 Reaction force (kN) 1600 1400 1200 1000 800 600 400 200 0 0 10 20 30 40 50 60 70 80 Mid span displacement (mm) Fi g . 9: Pushover curve of different concrete strength classes ΔP* (mm) ΔSL** (mm) ΔH*** (mm) ΔFE,SCDP (mm) σH**** (Mpa) σFE, (Mpa) Grade C (mm) I (mm4) B20 488.1 5.032 × 1010 78 −16.0 62.4 46.4 53.8059 24.3 18.47 B30 490.5 5.029 × 1010 101.5 −12.8 64.7 51.9 59.7774 31.9 25.36 B40 491.5 5.028 × 1010 128.8 −11.3 72.9 61.6 69.83 40.7 32.59 492.4 5.027 × 10 154.2 −10.2 78.3 68.1 74.64 48.9 38.97 B50 W (N/mm) Furthermore, SCDP in tabular form was veriﬁed through comparison with ﬁnite element models of the available CDP models given in extant literature. Two comparisons were carried out for concrete strength classes of 30 MPa and 50 MPa. Von Mises stress, damage in tension and midspan displacement were obtained by utilizing nonlinear ﬁnite element techniques of a partially prestressed concrete beam as presented in Figs. 10 and 10. The FE analysis incorporated the concrete damage plasticity model proposed by Tiwari27 and Jankowiak32 for concrete strength class of 30 MPa and 50 MPa. It should be noted that the model proposed by Tiwari assumed that the simple supported beam failed either the damage in tension reaching 0.82 or the concrete exceeded the maximum compressive stress. 10 SCDP 2 *ΔP = − PeL 8EI 5WL4 **ΔSL = 384EI ***ΔH = ΔP + ΔSL **** P Pe σH = A − Zt + MZSLt Table 5: concrete stress and mid-span displacement in the SCDP and empirical formulation Structural Engineering International Nr. 1/2017 Scientiﬁc Paper 75 DAMAGET (Avg: 75%) +8.254e-01 +7.428e-01 +6.603e-01 +5.778e-01 +4.952e-01 +4.127e-01 +3.302e-01 +2.476e-01 +1.651e-01 +8.254e-02 +0.000e+00 (a) S, Mises (Avg: 75%) +2.640e+01 +2.377e+01 +2.114e+01 +1.852e+01 +1.589e+01 +1.326e+01 +1.063e+01 +7.997e+00 +5.367e+00 +2.738e+00 +1.080e-01 (b) U, U2 +0.000e+00 –6.526e+00 –1.305e+01 –1.958e+01 –2.610e+01 –3.263e+01 –3.915e+01 –4.568e+01 –5.221e+01 –5.873e+01 –6.526e+01 (c) F ig . 10 : The results of the analysis on concrete B3027 : (a) damage in tension, (b) stress distribution (MPa), and (c) displacement distribution (mm) (a) DAMAGET (Avg: 75%) +9.900e–01 +8.910e–01 +7.920e–01 +6.930e–01 +5.940e–01 +4.950e–01 +3.960e–01 +2.970e–01 +1.980e–01 +9.900e–02 +0.000e+00 (b) S, Mises (Avg: 75%) +3.231e+01 +2.910e+01 +2.588e+01 +2.266e+01 +1.944e+01 +1.623e+01 +1.301e+01 +9.791e+00 +6.573e+00 +3.355e+00 +1.378e–01 (c) U, U2 +0.000e+00 –6.807e+00 –1.361e+01 –2.042e+01 –2.723e+01 –3.404e+01 –4.084e+01 –4.765e+01 –5.446e+01 –6.126e+01 –6.807e+01 F ig . 11 : The results of the analysis on concrete B5032: (a) damage in tension, (b) stress distribution (MPa), and (c) displacement distribution (mm) With respect to the results obtained and summarized in Table 6, it was observed that the compressive stress and mid-span displacement values of the available concrete damage plasticity models and SCDP model had good agreement. Figure 11 clearly demonstrated a reasonable correlation between the CDP models Tiwari Jankowiak SCDP SCDP model and the studied FE models. The maximum reaction force of the beam with the Tiwari and SCDP models in concrete class B30 showed a difference of approximately 4.6%. However, the discrepancy in the concrete class of B50 between Jankowiak model and SCDP model was approximately 16%. Compressive stress (MPa)B30 Compressive stress (MPa)B50 Mid span displacement (mm) B30 Mid span displacement (mm) B50 26.4 – 65.26 – – 32.3 – 68.07 25.36 38.97 59.86 74.64 Table 6: Comparison of results between concrete stress and mid-span displacement 76 Scientiﬁc Paper Conclusions This paper developed the present damage plasticity model (SCDP). This model simpliﬁed the procedure of existing damage plasticity model called CDP. It combined a stress-based plasticity part with a strain-based damage mechanics model for the unconﬁned prestressed concrete beam based on a tabular format for four different concrete grades (B20, B30, B40, and B50). Accordingly, the ﬁndings in this paper indicated the following conclusions from the simpliﬁcation provided by the proposed tabulated concrete damage plasticity model: • Due to its simplicity, the SCDP model was a suitable solution to Structural Engineering International Nr. 1/2017 Reaction force (kN) 2000 1800 1600 1400 1200 1000 800 600 400 200 0 B30 B30 (Tiwari 2015) B50 B50 (JANKOWIAK 2005) [2] Kang HD. Triaxial Constitutive Model for Plain and Reinforced Concrete Behavior University of Colorado: Boulder, 1997. [3] Grassl P, Lundgren K, & Gylltoft K. Concrete in compression: a plasticity theory with a novel hardening law. Int. J. Solids Struct. 2002; 39(20): 5205–5223. [4] Chen AC, & Chen WF. Constitutive relations for concrete. J. Eng. Mech. 1975; 101(4): 465–481. 0 10 20 30 40 50 60 70 80 Mid span displacement (mm) Fi g. 12: Validation of force versus displacement curves of the SCDP and available CDP models.27,32 model the cracking and the crushing of concrete. Thus, a 3D nonlinear FE model was developed to assess the performance of concrete with the SCDP model. All possible nonlinearities, namely, material and geometric, were considered in the developed FE model. • The presented 3D nonlinear FE model was then used to model different concrete classes. The developed FE model successfully predicted the concrete damage caused by tension, compressive stress in concrete, and mid-span displacement in the prestressed concrete beam. • The model realistically described the transition from tensile to compressive failure. This ﬁnding was achieved through the introduction of two separate isotropic damage variables for tension and compression. • The proposed three-dimensional FE model provided the framework to develop a realistic model to determine the behavior of the simply supported partially prestressed beam. Then, the comparison of the results between the SCDP model and available CDP models including, the two different concrete damage plasticity models, namely, the Tiwari27 and the Jankowiak and Lodygowski32 models, showed that the present CDP model had a good correlation with these CDP models. εpl Del Del0 σ σ d dt Nomenclature This work was supported by the University Putra Malaysia under Putra grant No. 9438709. This support is gratefully acknowledged. E Ec Es Ep ν εel young’s modulus concrete young’s modulus steel young’s modulus Prestressing steel young’s modulus poisson’s ratio elastic strain tensor dc σ cu ε’c εpl c εpl εpl t ,h εin c pl,h ε ck, h εt ,h εpl c pl, h εt :ε: ε_ el ε_ pl :λ: gp ΔFE σFE plastic strain tensor degraded elastic tensor initial elastic tensor nominal stress effective stress scalar damage variable scalar tension damage variable scalar compression damage variable ultimate compressive strength of unconﬁned cylinder strain of unconﬁned cylinder at ultimate compressive strength compressive plastic strain tensor plastic strain tensor tensile plastic strain tensor inelastic compression strain plastic hardening strain cracking strain plastic hardening strain in compression plastic hardening strain in tension total strain rate total strain rate in elastic part total strain rate in plastic part rate of plastic multiplier plastic potential mid span deﬂection due to FE model max concrete compressive stress due to FE model Acknowledgements References [1] Pekau O, & Zhang Z. Strain-space cracking model for concrete and its application. Comput. Struct. 1994; 51(2): 151–162. Structural Engineering International Nr. 1/2017 [5] Carol I, Rizzi E, & Willam K. On the formulation of anisotropic elastic degradation. I. Theory based on a pseudo-logarithmic damage tensor rate. Int. J. Solids Struct. 2001; 38(4): 491–518. [6] Gatuingt F, & Pijaudier-Cabot G. Coupled damage and plasticity modelling in transient dynamic analysis of concrete. Int. J. Numer. Anal. Methods Geomech. 2002; 26(1): 1–24. [7] Krätzig WB, & Pölling R. An elasto-plastic damage model for reinforced concrete with minimum number of material parameters. Comput. Struct. 2004; 82(15): 1201–1215. [8] Ju J. On energy-based coupled elastoplastic damage theories: constitutive modeling and computational aspects. Int. J. Solids Struct. 1989; 25(7): 803–833. [9] Lee J, & Fenves GL. Plastic-damage model for cyclic loading of concrete structures. J. Eng. Mech. 1998; 124(8): 892–900. [10] Jason L, Huerta A, Pijaudier-Cabot G, & Ghavamian S. An elastic plastic damage formulation for concrete: application to elementary tests and comparison with an isotropic damage model. Comput. Methods Appl. Mech. Eng. 2006; 195(52): 7077–7092. [11] Lubliner J, Oliver J, Oller S, & Onate E. A plastic-damage model for concrete. Int. J. Solids Struct. 1989; 25(3): 299–326. [12] Ananiev S, Ozbolt J. Plastic-damage model for concrete in principal directions. arXiv preprint 2007;0704.2662. [13] Imran I, & Pantazopoulou S. Plasticity model for concrete under triaxial compression. J. Eng. Mech. 2001; 127(3): 281–290. [14] Grassl P, & Jirásek M. Plastic model with non-local damage applied to concrete. Int. J. Numer. Anal. Methods Geomech. 2006; 30(1): 71–90. [15] Grassl P, & Rempling R. A damageplasticity interface approach to the meso-scale modelling of concrete subjected to cyclic compressive loading. Eng. Fract. Mech. 2008; 75(16): 4804–4818. [16] Yu T, Teng J, Wong Y, & Dong S. Finite element modeling of conﬁned concrete-II: plastic-damage model. Eng. Struct. 2010; 32(3): 680–691. [17] Zhang J, Zhang Z, & Chen C. Yield criterion in plastic-damage models for concrete. Acta Mech. Solida Sin. 2010; 23(3): 220–230. [18] Taddei F, Reindl L, Park J, Butenweg C, & Karadogan F. Numerical investigation of AAC wall panels based on the damage plasticity constitutive law. Cement Wapno Beton 2011; 7: 86–91. Scientiﬁc Paper 77 [19] Grassl P, Nystrom U, Rempling R, Gylltoft K. A damage-plasticity model for the dynamic failure of concrete. arXiv preprint 2011; arXiv:11031288. [20] Larsson S, Malm R, Charbit B, & Ansell A. Finite element modelling of laterally loaded lime–cement columns using a damage plasticity model. Comput. Geotech. 2012; 44: 48–57. [21] Ming C. Research on damage plastic calculation method of ABAQUS concrete damaged plasticity model. Transport. Stand. 2012; 2: 51–54. damage-plasticity approach to modelling the failure of concrete. Int. J. Solids Struct. 2013; 50(24): 3805–3816. [24] Vaghei R, Hejazi F, Taheri H, Jaafar MS, & Ali AAA. Evaluate performance of precast concrete wall to wall connection. APCBEE Procedia 2014; 9: 285–290. [25] Shang K, Guo Q, He Z, & Ye Y. A study of the torsional behavior of reinforced concrete trough girder based on ABAQUS. CICTP 2012; 10: 3029–3035. [22] Zhang J, & Li J. Investigation into Lubliner yield criterion of concrete for 3D simulation. Eng. Struct. 2012; 44: 122–127. [26] Tao Y, & Chen J. Concrete damage plasticity model for modeling FRP-to-concrete bond behavior. J. Compos. Construct. 2014; 19(1): 04014026. [23] Grassl P, Xenos D, Nyström U, Rempling R, & Gylltoft K. CDPM2: a [27] Tiwari R, Chakraborty T, & Matsagar V. Dynamic analysis of curved tunnels subjected to 78 Scientiﬁc Paper View publication stats internal blast loading. Adv. Struct. Eng. 2015; 1: 405–415. [28] Hognestad E. Study of combined bending and axial load in reinforced concrete members University of Illinois: Urbana, 1951. [29] Kent DC, & Park R. Flexural members with conﬁned concrete. J. Struct. Div. 1971; 97(7): 1969–1990. [30] Park R. Reinforced concrete structures John Wiley & Sons: New York, 1975. [31] Simulia D. ABAQUS 6.11 analysis user’s manual. ABAQUS 611 Documentation, 2011. [32] Jankowiak T, & Lodygowski T. Identiﬁcation of parameters of concrete damage plasticity constitutive model. Found. Civil Environ. Eng. 2005; 6: 53–69. Structural Engineering International Nr. 1/2017