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materials Article Deformation Analysis of Reinforced Beams Made of Lightweight Aggregate Concrete 1 , 2 3 1 Darius Bacinskas * , Deividas Rumsys , Aleksandr Sokolov and Gintaris Kaklauskas Department of Reinforced Concrete Structures and Geotechnics, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania; [email protected] SRP Projektas Ltd., Savanoriu ˛ pr. 176c, LT-03154 Vilnius, Lithuania; [email protected] Laboratory of Innovative Building Structures, Vilnius Gediminas Technical University, Sauletekio av. 11, LT-10223 Vilnius, Lithuania; [email protected] * Correspondence: [email protected]; Tel.: +370-674-23-717 Received: 20 November 2019; Accepted: 17 December 2019; Published: 19 December 2019 Abstract: In the present trend of constructing taller and longer structures, the application of lightweight aggregate concrete is becoming an increasingly important advanced solution in the modern construction industry. In engineering practice, the analysis of lightweight concrete elements is performed using the same algorithms that are applied for normal concrete elements. As an alternative to traditional engineering methods, nonlinear numerical algorithms based on constitutive material models may be used. The paper presents a comparative analysis of curvature calculations for ﬂexural lightweight concrete elements, incorporating analytical code methods EN 1992-1 and ACI 318-19, as well as a numerical analysis using the constitutive model of cracked tensile lightweight concrete recently proposed by the authors. To evaluate the adequacy of the theoretical predictions, experimental data of 51 lightweight concrete beams of ﬁve dierent programs reported in the literature were collected. A comparison of theoretical and experimental results showed that the most accurate predictions are obtained using numerical analysis and the constitutive model proposed by the authors. In the future, the latter algorithm can be used as a reliable tool for improving the design standard methods or numerical modeling of lightweight concrete elements subjected to short-term loading. Keywords: lightweight aggregate concrete; reinforced concrete; ﬂexural elements; curvature; short-term loading; tension stiening; constitutive model; numerical modeling 1. Introduction Concrete has become the most widely used construction material worldwide. Moreover, concrete is the most widely used synthetic material. Compared with other materials, only water is used in greater quantities. Over the past 30 years, concrete production has increased by a factor of >3 times to 3 3 approximately 3.8 billion m per year [1]. This represents >1 m per person per year worldwide [2]. Compared with other traditional construction materials (e.g., steel, timber, polymers, and aluminum), the amount of concrete production is twice that of other traditional materials combined. By 2050, world concrete production is projected to be a factor of four higher than the 1990 level [3]. The increasing amount of concrete production leads to a rising demand for innovative solutions for concrete structures and their implementation in real construction projects [4]. In the present trend of constructing taller and longer structures, the application of lightweight aggregate concrete is becoming an increasingly important advanced solution in the modern construction industry. Numerous studies around the world have been dedicated to research in the ﬁeld of lightweight aggregate concrete. Consequently, various concrete mixtures with dierent mechanical properties have been proposed [5–10]. However, in most cases, traditional studies usually address the optimization of Materials 2020, 13, 20; doi:10.3390/ma13010020 www.mdpi.com/journal/materials Materials 2020, 13, 20 2 of 13 concrete properties with respect to one or more aspects, such as microstructure, mechanical resistance, and durability [6–9]. Consequently, the obtained ﬁndings do not lead to the ﬁnal expected eect [5]. The improvement of selected material properties is accompanied by changes in other important parameters [9]. Moreover, experimental results are usually achieved by testing on small-scale specimens. Despite the fact that standardized techniques for material testing are usually applied, the obtained results and material models sometimes do not reﬂect the real mechanical behavior of large-scale load-bearing structures [11,12]. The application of advanced concrete mixes for structural members must be analyzed in an integral way—starting from the optimal composition test and ending with the evaluation of structural behavior of large-scale prototype members subjected to real operating conditions and external mechanical loading [5]. In current engineering practice, for the limit state analysis of reinforced lightweight concrete and normal concrete elements, the same algorithms are applied [13,14]. The inﬂuence of lightweight aggregate concrete on the structural behavior is taken into account by introducing additional empirical coecients that depend on the concrete density. However, the obtained predictions of lightweight concrete members in most cases do not correspond to their real mechanical behavior—both crack width and deformation of lightweight concrete elements are underestimated [15], and the errors can reach 100%, particularly for the lightly reinforced concrete members [5]. These tendencies can be explained by the fact that lightweight concrete signiﬁcantly diers from normal concrete. In particular, the properties of lightweight concrete are highly dependent on the type, amount, and mechanical properties of the selected lightweight aggregates [10,16] as well as the technological aspects of concrete mix preparation [17,18]. Traditional engineering methods, which during many years have been developed to improve normal-weight concrete mixes, are usually insucient for evaluating these factors. Another important, though often neglected, aspect of the serviceability analysis can be attributed to the assessment of the restrained shrinkage-induced stress–strain state at the pre-loading stage [19–22]. Some researchers [14,21,23,24] note that early-age cracking of reinforced lightweight concrete elements occurs, in particular, because of shrinkage of concrete in combination with lower tensile strength. These eects are not taken into account in traditional engineering techniques. The application of performance-based design concepts in advanced structural engineering has increased the integration of alternative numerical methods in the design process of complex modern structures [22,25]. Adequate constitutive models representing the behavior of concrete and reinforcement, as well as their interaction, must be used in the following algorithms [26,27]. Numerous physical models have been proposed for the analysis of conventional reinforced concrete elements [19,28–30]. However, studies in the ﬁeld of constitutive modeling of lightweight aggregate concrete are insucient and still require a solution because advanced lightweight concrete is a relatively new material [31]. This paper presents a comparative analysis of curvature calculations for ﬂexural lightweight aggregate concrete elements, incorporating analytical code methods (EN 1992-1 (EC2) [32] and ACI 318-19 (ACI) [33]), as well as a numerical analysis using the constitutive model of cracked tensile lightweight concrete recently proposed by the authors [5,23]. To evaluate the adequacy of the theoretical predictions, experimental data of 51 lightweight concrete beams tested during ﬁve dierent programs were collected. The reinforcement ratio of the experimental beams ranged from 0.33% to 2.82%, the density ranged from 1651 to 2000 kg/m , and the compressive strength of concrete ranged from 20 to 70 MPa. A comparison of theoretical and experimental results showed that the most accurate predictions are obtained using numerical analysis and the constitutive model proposed by the authors. In the future, the latter algorithm can be used as a reliable tool for improving the design standard methods or numerical modeling of lightweight concrete elements subjected to short-term loading. Materials 2020, 13, 20 3 of 13 2. Calculation Methods Employed for Comparative Analysis 2.1. Eurocode 2 (EC2) According to EC2 [32] methodology, curvatures of reinforced lightweight concrete beams are calculated using the same relationships as for traditional reinforced concrete elements. The algorithm distinguishes two stages of deformation of reinforced concrete elements. In the ﬁrst stage (before cracking), the element behavior is fully elastic, and the curvature is calculated by applying the fundamental relationships of material mechanics. In the second stage (during which the element is fully cracked), tensile stresses are entirely carried by the tensile reinforcement. At this stage, the curvature is calculated using the geometric characteristics of the fully cracked cross section. The mean curvature at any intermediate stress–strain stage can be assessed by interpolation between values calculated for stages I and II. Using this concept, the tension-stiening eect is taken into account. The mean curve is calculated by the following formula: M M = (1 ) + , (1) E I E I lcm u lcm c where is the mean curvature of the cross section, M is the bending moment at the considered load level, I is the moment of inertia of the non-cracked cross section, and I is the moment of inertia of the u c fully cracked cross section. E is the average modulus of elasticity of lightweight aggregate concrete lcm calculated by using 0.3 2 E = 22( f /10) (/2200) , (2) lcm lcm where f is the average compressive strength of lightweight aggregate concrete, and is the density lcm of concrete. Here, is the interpolation coecient; if the cross section is not cracked, = 0; otherwise, it is calculated by using cr = 1 , (3) where M is the cracking moment, and is the coecient that takes into account the inﬂuence of cr the loading duration (short or long-term) as well as type of loading (static or cyclic) on the average deformations. The coecient is 1 and 0.5 for short-term static loads and long-term or cyclical loads, respectively. EC2 also provides an expression for the calculation of curvature caused by concrete shrinkage cs deformations: = " , (4) cs cs e where " is the free shrinkage deformation, is the ratio of reinforcement and concrete modulus cs e of elasticity (eective modular ratio), S is the ﬁrst moment of area of the reinforcement about the centroid of the section, and I is the second moment of the area of the section. The above relationship is commonly used to calculate long-term curvatures with curvature increases caused by shrinkage taken into account. However, EC2 does not provide any direct recommendations for short-term deformational analysis to evaluate the shrinkage eect in the pre-loading stage. As mentioned above, concrete free shrinkage is restrained by reinforcement, causing tension stresses in concrete even before loading. Depending on the shrinkage value and reinforcement ratio, this can signiﬁcantly decrease the cracking limit and can result in considerable prediction errors [20,21]. 2.2. ACI 318-19 (ACI) According to the ACI standard [33], the curvature of non-cracked cross-sectional elements is calculated using the following fundamental formula, considering elastic geometric and physical characteristics: = , (5) E I lc Materials 2020, 13, 20 4 of 13 where M is the maximum bending moment, I is the moment of inertia of the non-cracked gross section, and E is the modulus of elasticity of lightweight aggregate concrete calculated by the following lc formula: 1.5 E = 0.043 f , (6) lc where f is the compressive strength (in MPa). The eective moment of inertia of the cracked cross section is calculated by interpolation between the moments of inertia of the non-cracked (I ) and the fully cracked (I ) cross sections: g cr " # 3 3 M M cr cr I = I + 1 I I , (7) e g cr g M M where M is the cracking moment calculated as follows: cr f I r g M = , (8) cr where f is the modulus of rupture, and y is the distance from the centroid of the gross concrete section r t to the bottom tensile layer. The curvature of the cracked element is calculated with Formula (5) using the eective moment of inertia: = . (9) E I lc e 2.3. Numerical Method for Deformation Analysis Using a Tension-Stiening Model of Lightweight Aggregate Concrete As an alternative to design codes, numerical methods with incorporated constitutive models of materials can be used to assess nonlinear stress–strain behavior of reinforced concrete members. The current study applies the modiﬁed tension-stiening relationship originally proposed by Sokolov [29] Materials 2019, 12, x FOR PEER REVIEW 5 of 13 for traditional reinforced concrete. The modiﬁed and original models are presented in Figure 1a. σ 1 ct σ /f ct lct theoretical (ρ = 0.31 %) lct R 0.8 derived (ρ = 0.31 %) Proposed model theoretical (ρ = 0.45 %) 0.6 Sokolov (2010) derived (ρ = 0.45 %) 0.55f lct 0.4 0.2 ct ε /ε ct cr ε ε ε ε 1 cr 2 3 0 5 10 15 20 25 30 35 a) b) Figure 1. Tension–stiening model of structural lightweight concrete: (a) theoretical diagrams and (b) Figure 1. Tension–stiffening model of structural lightweight concrete: (a) theoretical diagrams and (b) normalized stress–strain diagrams obtained for the selected experimental beams [5]. normalized stress–strain diagrams obtained for the selected experimental beams [5]. Constitutive modeling techniques for deriving the modiﬁed model including applied experimental The proposed modified constitutive model (Figure 1a) is approximated by a three-curve results of ﬂexural reinforced lightweight concrete beams are discussed in more detail in references [5,23]. diagram. The ascending branch of the curve represents the elastic behavior of the reinforced concrete The basic aspects of the physical modeling are presented below. The methodology is based on the before cracking. Meanwhile, the horizontal line and descending branch describe the stages of crack layered section model, implying the successive application of the direct (curvature prediction) and formation and further development, respectively. According to [5], the ultimate tensile strength is σct inverse (constitutive modeling) approaches. The method proposed by Kaklauskas and Ghaboussi [34] = 0.55flct, where flct is the average tensile strength of lightweight aggregate concrete calculated was applied for constitutive modeling to obtain average stress and average strain diagrams for cracked according to the EC2 standard: tensile concrete. The mathematical algorithm of the applied inverse procedure is discussed in more detail in [20,29,35]. Experimental stress–strain diagrams representing the tension-stiening eect for 𝑓 = 𝑓 𝜂 , (10) 𝑡𝑐𝑚𝑙 𝑡𝑐𝑚 1 𝜂 = 0.40 + 0.60 𝜌 ⁄2200, (11) where is the density of lightweight aggregate concrete. 2⁄3 𝑓 = 0.30𝑓 for concrete grade ≤ 50/55. (12) 𝑡𝑐𝑚 𝑘𝑐𝑙 𝑓 + 8 𝑘𝑐𝑙 (13) 𝑓 = 2.12 [1 + ( )] for concrete grade > 50/55. 𝑡𝑐𝑚 The strain 1 corresponding to the ultimate tensile stress is determined by the following relationship: 𝜀 = 0.55𝜀 , (14) where εcr = flct/Elcm is the theoretical cracking strain corresponding to the tensile strength, and Elcm is the modulus of elasticity of concrete calculated according to EC2 depending on the compressive strength of concrete. The shape of the descending part of the diagram is described by the following formula [5,29]: 𝑐𝑡 𝜎 = 𝑓 (1 − 0.27 ln ( ) − 0.21𝜌 ), (15) 𝑡𝑐𝑙 𝑅 𝑐𝑟 where R is the reinforcement ratio [%]. The strain ε2 is calculated by using the relationship derived in Equation (15), and the ultimate tensile stress of concrete, σct = 0.55flct: 1.667−0.78𝜌 𝜀 = 𝜀 𝑒 . (16) The length of the descending branch is defined by the maximal strain ε3 corresponding to zero stress. This strain is calculated by the following formula: 3.7−0.78𝜌 𝜀 = 𝜀 𝑒 . (17) The stress–strain relations of the above constitutive law are shown in Figure 1b along with the stress–strain diagrams obtained from the experimental moment–curvature response. The comparison shows close agreement of the proposed model and the curves obtained from the tests. 𝑐𝑟 𝑐𝑟 𝑐𝑡 𝑐𝑟 𝐿𝐶 𝑙𝑛 𝐿𝐶 Materials 2020, 13, 20 5 of 13 ﬂexural members have been obtained by performing a three-step computation. The latter includes the elimination of the concrete shrinkage eect on the stress–strain behavior of reinforced concrete members before loading [20]. In the ﬁrst step, using an inverse procedure [34], average tensile stress–strain diagrams are obtained from experimental moment–curvature relationships. In the second step, the obtained curves are used in the direct approach with shrinkage deformations taken into account. By using the above technique, the modiﬁed moment-curvature diagrams for experimental specimens are obtained by eliminating the inﬂuence of shrinkage deformations. In the ﬁnal step, the modiﬁed moment–curvature diagrams are used again in the inverse algorithm. Consequently, tension–stiening diagrams with shrinkage eliminated are derived. Examples of the normalized tension–stiening diagrams obtained using the above procedure together with the proposed model are presented in Figure 1b. This ﬁgure represents stress–strain relations of nine experimental concrete beams reported in [5]. The beams were made of lightweight aggregate concrete with density = 1897–1959 kg/m and were having reinforcement ratio = 0.31% and 0.45%. Stress–strain relations were derived from the test moment-curvature diagrams. The proposed modiﬁed constitutive model (Figure 1a) is approximated by a three-curve diagram. The ascending branch of the curve represents the elastic behavior of the reinforced concrete before cracking. Meanwhile, the horizontal line and descending branch describe the stages of crack formation and further development, respectively. According to [5], the ultimate tensile strength is = 0.55f , ct lct where f is the average tensile strength of lightweight aggregate concrete calculated according to the lct EC2 standard: f = f , (10) ctm 1 lctm = 0.40 + 0.60/2200, (11) where is the density of lightweight aggregate concrete. 2/3 f = 0.30 f for concrete grade LC50/55. (12) ctm lck " !# f + 8 lck f = 2.12ln 1 + for concrete grade > LC50/55. (13) ctm The strain " corresponding to the ultimate tensile stress is determined by the following relationship: " = 0.55" , (14) cr where " = f /E is the theoretical cracking strain corresponding to the tensile strength, and E cr lct lcm lcm is the modulus of elasticity of concrete calculated according to EC2 depending on the compressive strength of concrete. The shape of the descending part of the diagram is described by the following formula [5,29]: ct = f 1 0.27 ln 0.21 , (15) ct R lct cr where is the reinforcement ratio [%]. The strain " is calculated by using the relationship derived in Equation (15), and the ultimate tensile stress of concrete, = 0.55f : ct lct 1.6670.78 " = " e . (16) 2 cr The length of the descending branch is deﬁned by the maximal strain " corresponding to zero stress. This strain is calculated by the following formula: 3.70.78 " = " e . (17) 3 cr Materials 2020, 13, 20 6 of 13 The stress–strain relations of the above constitutive law are shown in Figure 1b along with the Materials 2019, 12, x FOR PEER REVIEW 6 of 13 stress–strain diagrams obtained from the experimental moment–curvature response. The comparison shows close agreement of the proposed model and the curves obtained from the tests. A nonlinear numerical analysis was performed using the finite element software ATENA A nonlinear numerical analysis was performed using the ﬁnite element software ATENA (Cervenka (Cervenka Consulting Ltd., Prague, Czech Republic). Two-dimensional finite element models of Consulting Ltd., Prague, Czech Republic). Two-dimensional ﬁnite element models of experimental experimental reinforced concrete elements were created employing constitutive models for reinforced concrete elements were created employing constitutive models for compressive and tensile compressive and tensile concrete and reinforcement. The behavior of the reinforcement is concrete and reinforcement. The behavior of the reinforcement is represented by an elastic–plastic represented by an elastic–plastic model corresponding to the yield strength of steel and modulus of model corresponding to the yield strength of steel and modulus of elasticity. A linear elastic diagram elasticity. A linear elastic diagram was used to model the compressive concrete. The proposed was used to model the compressive concrete. The proposed constitutive model (Figure 1a) was used to constitutive model (Figure 1a) was used to describe the behavior of lightweight aggregate concrete describe the behavior of lightweight aggregate concrete in tension. The 3D Non Linear Cementitious in tension. The 3D Non Linear Cementitious 2 User material model (based on SBETA material model 2 User material model (based on SBETA material model oered by ATENA) was utilized. Concrete offered by ATENA) was utilized. Concrete without cracks is considered as isotropic and concrete without cracks is considered as isotropic and concrete with cracks as orthotropic body. Smeared with cracks as orthotropic body. Smeared crack and fracture mechanics approaches are combined in crack and fracture mechanics approaches are combined in ATENA to assess the nonlinear behavior of ATENA to assess the nonlinear behavior of reinforced concrete elements after cracking. In this study, reinforced concrete elements after cracking. In this study, the ﬁxed crack model was used. A fracture the fixed crack model was used. A fracture mechanics approach employed in ATENA for softening mechanics approach employed in ATENA for softening behavior is based on the crack band model. behavior is based on the crack band model. Such a model substantially reduces mesh sensitivity [36]. Such a model substantially reduces mesh sensitivity [36]. A typical ﬁnite element model including the A typical finite element model including the loading and support conditions of the test beams is loading presen and ted isupport n Figureconditions 2. of the test beams is presented in Figure 2. Symmetry axis P/2 P/2 Steel plate Steel plate a L – 2a a Figure 2. A typical finite element model and loading and support conditions of the test beams Figure 2. A typical ﬁnite element model and loading and support conditions of the test beams. The results of the nonlinear analysis strongly depend on the size of the finite mesh. Previous The results of the nonlinear analysis strongly depend on the size of the ﬁnite mesh. Previous studies [36,37] have shown that the accuracy of numerical analysis results obtained by ATENA is studies [36,37] have shown that the accuracy of numerical analysis results obtained by ATENA is sufficient using six finite elements per model height. According to the recommendations [36], the sucient using six ﬁnite elements per model height. According to the recommendations [36], the mesh mesh size was normalized by assuming a 20 mm characteristic finite element length. Such size was normalized by assuming a 20 mm characteristic ﬁnite element length. Such normalization normalization enables to eliminate the influence of the obtained results on the finite element mesh enables to eliminate the inﬂuence of the obtained results on the ﬁnite element mesh size. Isoparametric size. Isoparametric quadrilateral finite elements of eight degrees of freedom with four integration quadrilateral ﬁnite elements of eight degrees of freedom with four integration points were used to points were used to model the concrete beams. Reinforcement bars were modeled with truss finite model the concrete beams. Reinforcement bars were modeled with truss ﬁnite elements. It should also elements. It should also be emphasized that shrinkage deformations prior to short-term loading have be emphasized that shrinkage deformations prior to short-term loading have been taken into account been taken into account in the numerical analysis. Shrinkage was modeled as a prescribed in the numerical analysis. Shrinkage was modeled as a prescribed deformation aecting concrete deformation affecting concrete macroelements [37]. The modeling aspects are described in more macroelements [37]. The modeling aspects are described in more detail in reference [23]. detail in reference [23]. 3. Database of Experimental Results and Accuracy Analysis of Predictions 3. Database of Experimental Results and Accuracy Analysis of Predictions The database consists of data from 51 lightweight aggregate concrete ﬂexural elements obtained The database consists of data from 51 lightweight aggregate concrete flexural elements obtained from ﬁve dierent test programs reported by Carmo et al. [15], Sin et al. [14], Bernardo et al. [38], from five different test programs reported by Carmo et al. [15], Sin et al. [14], Bernardo et al. [38], Wu Wu et al. [39], and Vakhshouri [30]. The main characteristics of the ﬂexural elements are given in et al. [39], and Vakhshouri [30]. The main characteristics of the flexural elements are given in Tables Tables 1 and 2. The reinforcement percentage of the experimental beams ranged from 0.33% to 2.82%, 1 and 2. The reinforcement percentage of the experimental beams ranged from 0.33% to 2.82%, the the density ranged from 1651 to 2000 kg 3 /m , and the compressive strength of concrete ranged from density ranged from 1651 to 2000 kg/m , and the compressive strength of concrete ranged from 20 to 20 to 70.1 MPa. Table 2 includes the characteristics of tensile strength of concrete and shrinkage 70.1 MPa. Table 2 includes the characteristics of tensile strength of concrete and shrinkage deformations required for the numerical analysis. These characteristics were calculated according to deformations required for the numerical analysis. These characteristics were calculated according to the EC2 standard [32]. All the beams were tested under the four-point bending configuration (Figure 2) with the span and shear span parameters given in Tables 1. Materials 2020, 13, 20 7 of 13 the EC2 standard [32]. All the beams were tested under the four-point bending conﬁguration (Figure 2) with the span and shear span parameters given in Table 1. Table 1. Main geometrical characteristics of experimental beams. Number Span L , Shear Depth h, Width b, Reinforcement No. Reference of Beams m Span a, m mm mm Percentage , % 1 Carmo et al. [15] 13 2.80 1.0 270 120 0.53–2.82 2 Sin et al. [14] 18 2.80 1.0 300 150 0.69–2.27 3 Bernardo et al. [38] 14 2.40 0.8 300 150 0.38–2.69 4 Wu et al. [39] 3 4.00 1.4 400 250 0.33–1.310 5 Vakhshouri [30] 3 3.50 1.167 161 400 0.83 Total: 51 2.40–4.00 0.8–1.4 161–400 120–400 0.33–2.82 Table 2. Main material characteristics of experimental beams. Concrete Density , Compressive Tensile Strength Shrinkage Strain No. Reference kg/m Strength f , MPa f , MPa " , 10 lcm lctm shr 1 Carmo et al. [15] 1870–1900 37.0–70.0 2.84–4.37 313–395 2 Sin et al. [14] 1700–2000 25.1–70.1 1.72–4.17 141–175 3 Bernardo et al. [38] 1651–1953 20.0–55.0 1.36–3.78 249–388 4 Wu et al. [39] 1900 34.2 2.43 134 5 Vakhshouri [30] 2000 31.0 2.29 180 Total: 1651–2000 20.0–70.1 1.36–4.37 134–395 Comparison of the theoretical and experimental results of the selected eight beams is given in Figure 3. The theoretical analysis of the experimental beams was terminated at the load corresponding to the ultimate bending moment M calculated according to EC2 [32]. In the ﬁgure, corresponds Rm R to the reinforcement percentage, is the concrete density, and f is the compressive strength of lcm lightweight concrete. Results are compared at the service loading level taken as M = 0.6M (shown ser Rm in Figure 3 by red dashed line). The comparison of the experimental curvatures against the theoretical results predicted by EC2 and ACI codes as well as the numerical approach is shown in Figure 4. The predictions were made for all the beams at three dierent loading levels: 0.4M , 0.6M (M ), and 0.8M . The mean Rm Rm ser Rm value (x ), standard deviation ( ), and coecient of variation (V ) of the relative curvature ( / ) m exp std k th are shown at the bottom of each graph in this ﬁgure. The mean errors of 12.8%, 14.3%, and 14.4% are obtained for the EC2 standard at load levels of 0.4M , 0.6M , and 0.8M , respectively. The Rm Rm Rm computational errors are rather modest and do not depend on the load level. Slightly smaller mean errors of 11.7%, 12.9%, and 12.4% assessed at the same load levels are obtained by using the ACI method. The predictions of the numerical model resulted in 1.8%, 2.7%, and 1.6% mean curvature errors obtained at load levels of 0.4M , 0.6M , and 0.8M , respectively. It is important to note that Rm Rm Rm EC2 and ACI code methods produced predictions that were too sti. Materials 2020, 13, 20 8 of 13 Materials 2019, 12, x FOR PEER REVIEW 8 of 13 25 45 M, kNm M, kNm Beam B1-0.0-55 Beam B2-0.0-70 M = 12.36 kNm M = 23.9 kNm ser ser 30 EC2 EC2 ACI ACI Model Model 20 Test Test ρ = 0.53 % ρ = 1.077 % R R 3 3 ρ = 1870 kg/m ρ = 1900 kg/m f = 51.0 MPa 5 f = 70.0 MPa lcm κ, 1/m lcm κ, 1/m 0 0 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 0.025 (a) (b) 40 M, kNm M, kNm 60 Beam #5 Beam #6 M = 20.9 kNm ser EC2 M = 28.8 kNm ser 25 EC2 ACI ACI Model 20 30 Model Test Test ρ = 0.693 % R ρ = 1.043 % 10 3 ρ = 1860 kg/m ρ = 1860 kg/m f = 33.8 MPa f = 33.8 MPa 5 lcm κ, 1/m κ, 1/m lcm 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 (c) (d) 25 M, kNm 35 M, kNm Beam 1(22.0-0.38) Beam 1(49.2-0.55) M = 13.74 kNm ser M = 19.13 kNm ser EC2 EC2 ACI ACI Model Model Test Test ρ = 0.38 % 10 ρ = 0.55 % R R 3 3 ρ = 1667 kg/m ρ = 1827 kg/m f = 20.0 MPa f = 44.8 MPa κ, 1/m κ, 1/m lcm lcm 0 0,005 0,01 0,015 0,02 0 0,005 0,01 0,015 0,02 (e) (f) 160 35 M, kNm M, kNm Beam FL2 Beam LWC-4 M = 79.5 kNm 25 ser M = 17.0 kNm ser EC2 EC2 ACI ACI Model 15 Model Test Test ρ = 0.98 % ρ = 0.83 % R R 40 3 3 ρ = 1900 kg/m ρ = 2000 kg/m 20 f = 34.2 MPa f = 31.0 MPa lcm lcm κ, 1/m κ, 1/m 0 0,01 0,02 0,03 0,04 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 (g) (h) Figure 3. Comparison of theoretical and experimental moment-curvature diagrams: (a,b) Carmo et Figure 3. Comparison of theoretical and experimental moment-curvature diagrams: (a and b) Carmo al. [15]; (c,d) Sin et al. [14]; (e,f) Bernardo et al. [38]; (g) Wu et al. [39]; and (h) Vakhshouri [30]. et al. [15]; (c and d) Sin et al. [14]; (e and f) Bernardo et al. [38]; (g) Wu et al. [39]; and (h) Vakhshouri [30]. Materials 2020, 13, 20 9 of 13 Materials 2019, 12, x FOR PEER REVIEW 9 of 13 EC2 0.4M 0.6M 0.8M Rm Rm Rm 0.02 0.02 0.03 κ , 1/m κ , 1/m κ , 1/m th th th 0.015 0.015 0.02 0.01 0.01 0.01 0.005 0.005 κ , 1/m κ , 1/m κ , 1/m exp exp exp 0 0 0 0.005 0.01 0.015 0.02 0 0.01 0.02 0.03 0 0.005 0.01 0.015 0.02 x = 0. 872 σ = 0. 118 V = 0. 136 x = 0. 857 σ = 0. 092 V = 0. 107 x = 0. 856 σ = 0. 070 V = 0. 082 m std k m std k m std k ACI 0.4M 0.6M 0.8M Rm Rm Rm 0.02 0.02 0.03 κ , 1/m κ , 1/m κ , 1/m th th th 0.015 0.015 0.02 0.01 0.01 0.01 0.005 0.005 κ , 1/m κ , 1/m κ , 1/m exp exp exp 0 0 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 0 0.01 0.02 0.03 x = 0. 883 σ = 0. 159 V = 0. 180 x = 0. 871 σ = 0. 118 V = 0. 135 x = 0. 876 σ = 0. 081 V = 0. 092 m std k m std k m std k Model 0.4M 0.6M 0.8M Rm Rm Rm 0.02 0.02 0.03 κ , 1/m κ , 1/m κ , 1/m th th th 0.015 0.015 0.02 0.01 0.01 0.01 0.005 0.005 κ , 1/m κ , 1/m κ , 1/m exp exp exp 0 0 0 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 0 0.01 0.02 0.03 x = 0. 982 σ = 0. 137 V = 0. 140 x = 0. 973 σ = 0. 093 V = 0. 095 x = 0. 984 σ = 0. 075 V = 0. 076 m std k m std k m std k Figure 4. Comparison of experimental and theoretical curvatures at dierent load levels. Figure 4. Comparison of experimental and theoretical curvatures at different load levels. Figure 5 shows the scatter of the normalized curvature predictions for ranges of material and geometrical parameters such as reinforcement percentage , concrete density , compressive strength of concrete, f , and shrinkage deformation " . The latter results were obtained for all the beams lcm shr at a service load M = 0.6M . Figure 5 shows that none of the listed parameters, except concrete ser Rm density and shrinkage strain, signiﬁcantly aects the prediction accuracy in any of the methods. There is a general tendency that accuracy decreases with a rise in density and an increase in free shrinkage strain. Comparison of the results demonstrates that in all cases the proposed approach gives the most accurate predictions of mean normalized curvature as the respective trend-line approaches the unity line (shown as a black solid line). Materials 2020, 13, 20 10 of 13 Materials 2019, 12, x FOR PEER REVIEW 10 of 13 2 κ /κ 2 κ /κ th exp th exp EC2 EC2 ACI ACI 1.5 1.5 Model Model 1 1 0.5 0.5 ρ , % ρ, kg/m 0 0 0 0.5 1 1.5 2 2.5 3 1600 1675 1750 1825 1900 1975 2050 (a) (b) 2 2 κ /κ κ /κ th exp th exp EC2 EC2 ACI 1.5 1.5 ACI Model Model 0.5 0.5 f , MPa ε ×(– lcm shr 15 25 35 45 55 65 75 100 200 300 400 500 600 (c) (d) Figure 5. Relative curvatures estimate by dierent methods vs: (a) reinforcement ratio ; (b) density Figure 5. Relative curvatures estimate by different methods vs: (a) reinforcement ratio ρR; (b) density ; (c) compressive strength f ; and (d) deformation of shrinkage " . lcm shr ρ; (c) compressive strength flcm; and (d) deformation of shrinkage εshr 4. Conclusions 4. Conclusions A comparison analysis of theoretical and experimental results of deformations of reinforced lightweight concrete beams yields the following conclusions: A comparison analysis of theoretical and experimental results of deformations of reinforced 1. In engineering practice, the analysis of lightweight concrete elements is usually performed using lightweight concrete beams yields the following conclusions: the same algorithms used for normal concrete elements. The inﬂuence of lightweight concrete on 1. In engineering practice, the analysis of lightweight concrete elements is usually performed using the structural behavior is evaluated by additional density-dependent empirical coecients. This the same algorithms used for normal concrete elements. The influence of lightweight concrete prediction of the behavior of lightweight concrete often does not correspond to the real behavior on the structural behavior is evaluated by additional density-dependent empirical coefficients. of the structure. In many cases, the deformation of reinforced lightweight aggregate concrete This prediction of the behavior of lightweight concrete often does not correspond to the real elements is underestimated, and the resulting errors can reach 100%. behavior of the structure. In many cases, the deformation of reinforced lightweight aggregate 2. As an alternative to traditional engineering methods, nonlinear numerical algorithms based on concrete elements is underestimated, and the resulting errors can reach 100%. physical material models that reﬂect the behavior of elements at various stages of operation may 2. As an alternative to traditional engineering methods, nonlinear numerical algorithms based on be used. Although many physical models of concrete have been proposed for the prediction physical material models that reflect the behavior of elements at various stages of operation may of load carrying capacity and deformations in conventional reinforced concrete elements, there be used. Although many physical models of concrete have been proposed for the prediction of are no reliable physical models for the numerical analysis of reinforced lightweight aggregate load carrying capacity and deformations in conventional reinforced concrete elements, there are concrete elements. no reliable physical models for the numerical analysis of reinforced lightweight aggregate 3. The constitutive model of cracked tensile lightweight concrete earlier proposed by the authors concrete elements. wasused in a comparative deformation analysis of reinforced lightweight concrete beams. Stresses 3. The constitutive model of cracked tensile lightweight concrete earlier proposed by the authors characterizing the cracking limit were reduced in the modiﬁed model by considering the wasused in a comparative deformation analysis of reinforced lightweight concrete beams. characteristics of formation of lightweight concrete cracks. The proposed model is approximated Stresses characterizing the cracking limit were reduced in the modified model by considering by a three-curve diagram. The rising part of the curve describes the elastic behavior of concrete the characteristics of formation of lightweight concrete cracks. The proposed model is before cracking. The horizontal and descending parts of the curve describe the stages of formation approximated by a three-curve diagram. The rising part of the curve describes the elastic and development of cracks, respectively. behavior of concrete before cracking. The horizontal and descending parts of the curve describe 4. The adequacy of results obtained by design code techniques and numerical modeling method was the stages of formation and development of cracks, respectively. veriﬁed by employing experimental data of reinforced lightweight aggregate concrete elements 4. The adequacy of results obtained by design code techniques and numerical modeling method published in the literature. The data sample consisted of 51 ﬂexural elements obtained from ﬁve was verified by employing experimental data of reinforced lightweight aggregate concrete dierent test programs. Numerical analysis of experimental beams was performed using the elements published in the literature. The data sample consisted of 51 flexural elements obtained from five different test programs. Numerical analysis of experimental beams was performed Materials 2020, 13, 20 11 of 13 nonlinear ﬁnite element software ATENA and the constitutive model proposed by the authors to model the behavior of the cracked tensile concrete. 5. A comparison of theoretical and experimental results revealed that the most accurate calculation results were obtained by using the numerical model. At the service load level (M = 0.6M , ser Rm where M is the theoretical average bearing bending moment calculated by EC2), the mean Rm value of the relative curvature ( / ) obtained by using the numerical model was 0.973, and the exp th standard deviation was 0.093. By using the EC2 standard, the mean value of the relative curvature / was 0.857, and the standard deviation was 0.092. The mean and standard deviation th exp values of 0.871 and 0.118, respectively, were obtained by using the ACI standard method. The comparative analysis shows that EC2 and ACI code methods produced predictions that were too sti. 6. The inﬂuence of the reinforcement percentage , concrete density , compressive strength of concrete, f , and shrinkage deformation " has little eect on the mean curvatures predicted by lcm shr the code and numerical methods. There is a general tendency that the accuracy of the methods decreases with the rise in density and the increase in free shrinkage strain. Comparison of results demonstrates that in all cases the proposed approach gives the most accurate predictions of mean normalized curvature. 7. In the future, the proposed constitutive model of lightweight aggregate concrete together with numerical ﬁnite element algorithms can be used as a reliable tool for improving the design code techniques or for adequate numerical modeling of reinforced lightweight aggregate concrete elements under short-term loading. Author Contributions: D.B. and D.R. conceived the idea of the work. D.R. collected and analyzed the experimental data. D.B., D.R., G.K. and A.S. made the theoretical analysis of experimental beams, compared the theoretical and experimental results, wrote the paper. D.B. and G.K. concluded the main ﬁndings. Funding: The study was performed within the project No. 09.3.3-LMT-K-712-01-0145 that has received funding from European Social Fund under grant agreement with the Research Council of Lithuania (LMTLT). Acknowledgments: The authors gratefully acknowledge the ﬁnancial support provided by the European Social Fund under grant agreement with the Research Council of Lithuania (LMTLT) for the project No. 09.3.3-LMT-K-712-01-0145. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Mindess, S. Developments in the Formulation and Reinforcement of Concrete, 1st ed.; Woodhead Publishing: Cambridge, UK, 2008. 2. Karen, L.; Scrivener, R.; Kirkpatrick, J. Innovation in use and research on cementitious material. Cem. Concr. Res. 2008, 38, 128–136. [CrossRef] 3. Damtoft, J.S.; Lukasik, J.; Herfort, D.; Sorrentino, D.; Gartner, E.M. Sustainable development and climate change initiatives. Cem. Concr. Res. 2008, 38, 115–127. [CrossRef] 4. 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Published: Dec 19, 2019
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