Numerical Analysis on the High-Strength Concrete Beams Ultimate Behaviour

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1 IOP Conferene Series: Materials Siene and Engineering PAPER OPEN ACCESS Numerial Analysis on the High-Strength Conrete Beams Ultimate Behaviour To ite this artile: Piotr Smarzewski and Adam Stolarski 2017 IOP Conf. Ser.: Mater. Si. Eng View the artile online for updates and enhanements. Related ontent - Influene of steel fibers on the shear and flexural performane of high-strength onrete beams tested under blast loads O Algassem, Y Li and H Aoude - Review on fatigue behavior of highstrength onrete after high temperature Dongfu Zhao, Penghe Jia and Haijing Gao - Fatigue Behavior of Steel Fiber Reinfored High-Strength Conrete under Different Stress Levels Chong Zhang, Danying Gao and Zhiqiang Gu Reent itations - Dynami Relaxation Method for Load Capaity Analysis of Reinfored Conrete Elements Anna Szzeniak and Adam Stolarski This ontent was downloaded from IP address on 05/04/2019 at 22:56

2 Numerial Analysis on the High-Strength Conrete Beams Ultimate Behaviour Piotr Smarzewski 1, Adam Stolarski 2 1 Lublin University of Tehnology, 40 Nadbystrzyka, Lublin, Poland 2 Military University of Tehnology, 2 Kaliskiego, Warsaw-49, Poland p.smarzewski@pollub.pl Abstrat. Development of tehnologies of high-strength onrete (HSC) beams prodution, with the aim of reating a seure and durable material, is losely linked with the numerial models of real objets. The three-dimensional nonlinear finite element models of reinfored high-strength onrete beams with a omplex geometry has been investigated in this study. The numerial analysis is performed using the ANSYS finite element pakage. The ar-length (A-L) parameters and the adaptive desent (AD) parameters are used with Newton-Raphson method to trae the omplete load-defletion urves. Experimental and finite element modelling results are ompared graphially and numerially. Comparison of these results indiates the orretness of failure riteria assumed for the high-strength onrete and the steel reinforement. The results of numerial simulation are sensitive to the modulus of elastiity and the shear transfer oeffiient for an open rak assigned to high-strength onrete. The full nonlinear load-defletion urves at mid-span of the beams, the development of strain in ompressive onrete and the development of strain in tensile bar are in good agreement with the experimental results. Numerial results for smeared rak patterns are qualitatively agreeable as to the loation, diretion, and distribution with the test data. The model was apable of prediting the introdution and propagation of flexural and diagonal raks. It was onluded that the finite element model aptured suessfully the inelasti flexural behaviour of the beams to failure. 1. Introdution Improved performane omputing systems and the possibility of their use in the design of engineering strutures enourages intensive development of numerial methods for the analysis of stati and dynami strutural behaviour. Numerial methods are the only way to ahieve useful solutions to omplex spatial strutures made with materials behaving nonlinearly like onrete. Conrete is a ommonly used building material throughout the world. Material is brittle under low onfining pressure and weak in tension. Development of modern ivil engineering onstrution has aused demand for new types of onrete whih are required to possess improved strength, toughness, and durability [1]. High-strength onrete an be designed to have a higher workability, and a higher mehanial properties ompared to the traditional onrete [2 4]. The use of high-strength onrete in the building industry will ontinue to grow. The effet of onrete ompressive strength and flexural tensile reinforement ratio on load-defletion behaviour and dutility of reinfored high-strength onrete beams has been reported by several previous investigators [5 8]. Reinfored HSC members required not only experimental testing, but also finite element modelling of the failure behaviour. Few researhers studied the finite element modelling of reinfored onrete beams [9 11] by using the Content from this work may be used under the terms of the Creative Commons Attribution 3.0 liene. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal itation and DOI. Published under liene by Ltd 1

3 ANSYS finite element pakage. However, the available publiations on the finite element investigation of reinfored high-strength onrete beams are still limited. The subjet of the paper is the reinfored onrete beams onsisting of reinforement steel bars distributed disretely in the high-strength onrete. The aim of the work is to model the deformation proesses and the failure behaviour of reinfored onrete beams loaded statially taking into aount the physial and mehanial nonlinearity of strutural materials: onrete and reinforement steel. Finite element solutions for HSC beams were obtained by using ANSYS [12]. Experimental results of the work desribed in [6] and finite element modelling results are ompared graphially and numerially. 2. Modelling of materials 2.1. Modelling of high-strength onrete The desription of limit state of onrete under stati loading has been the subjet of numerous publiations. The equations of the limit surfae for onrete are desribed in the paper [13]. The proposed limit surfae equations are depended on the first invariant of the stress tensor and the seond, and the third invariant of the stress deviator. This desription allows the most faithful approximation of onrete experimental results in omplex stress state. In this paper, the limit surfae equation is depended on five stress parameters, in aordane with [14]. Moreover, the limit surfae evolution law is introdued. The failure riterion of onrete in a omplex stress state is desribed in the following equation F / f S, 1,...,4 (1) in whih: F α the funtion of σ xp, σ yp, σ zp normal stresses onditions in the diretion of the Cartesian oordinate system xyz, S α failure surfae dependent on the prinipal stresses σ 1, σ 2, σ 3, where: σ 1 = max(σ xp, σ yp, σ zp ), σ 3 = min(σ xp, σ yp, σ zp ), σ 1 σ 2 σ 3, and strength parameters: f uniaxial ompressive strength ausing rushing, f t uniaxial tensile strength, f b ultimate biaxial ompressive strength, f 1 ultimate ompressive strength for a biaxial ompression state superimposed in hydrostati stress state σ ha, f 2 ultimate ompressive strength for a uniaxial ompression state superimposed in hydrostati stress state σ ha. The failure of onrete is defined by four domains of stresses α: ompression-ompressionompression state α = 1; 0 σ 1 σ 2 σ 3, tension-ompression-ompression state α = 2; σ 1 0 σ 2 σ 3, tension-tension-ompression state α = 3; σ 1 σ 2 0 σ 3, and tension-tension-tension state α = 4; σ 1 σ 2 σ 3 0. The failure surfae is depended on the radii r t, r and the strength parameters: ξ and angle θ. Deviatory setional radius r t is determined by the parameters a 0, a 1, a 2 seleted in order that f t, f b, f 1 lie on the limit surfae. All the mentioned parameters are given in [11]. The limit surfae with an evolution law is used as a riterion of onrete failure aording to the following interpretation. The material is damaged if equation 1 is fulfilled. The state of failure an be distinguished as the state of raking if any prinipal stress is tensile. Otherwise it an be distinguished as the state of rushing if all prinipal stresses are ompressive. The evolution of limit surfae is determined by the proposal of hardening/softening law shown in figure 1. The stress-strain relationship for the uniaxial ompression is onfirmed by the experimental observations showing muh larger the limit strain of onrete in the struture than in the sample. We observed that the use of the stress-strain relation based on Model Code 90, leads to a signifiant derease in the ultimate defletion of beam. Moreover, experimental results indiated that a low dutility for the high-strength onrete in the strutural members is not justified, beause the strains reahed up to 6 12 [6,7]. Linear stress-strain relation for uniaxial ompression at the level of 0.3f was assumed. Then begins the region of elasti-plasti strengthening with linear inrease in stress up to the uniaxial ompressive strength f. After this, the stresses in onrete derease to 0.8f at the limit strain ε u. Strain ε 1 = 6 at f and the limit ompression strains, ε u = 12 were assumed. Stress-strain urve for tensile onrete is linear up to the tensile strength f t (see figure 1b). It is assumed that the tensile modulus of elastiity is equal to the ompressive one. After reahing f t, the tensile stress suddenly dereases to a value T f t. 2

4 Parameter T should be hosen from the range 0.6 T 1. The stiffening effet shows a gradual derease of tensile strength to zero, is desribed by the strain equal to 0.8 if T = 0.6, or 1.4 if T = 1. Figure 1. Stress-strain relation for HSC (a) uniaxial ompression, (b) uniaxial tension The hexahedral elements were applied for the onrete. Finite element is defined by the isotropi properties of the material, by eight nodes with three degrees of freedom in eah of them, and by the displaements of nodes in three-dimensional orthogonal loal oordinate system. In eah finite element, the strain and stress is alulated at all points of numerial integration. Smeared rak model provides a desription of the raking at any point of numerial integration in three diretions perpendiular to the prinipal stresses. Crak formation is desribed by the model of onrete. In the graphial representation the results are presented in the form of a irular raking outline in the diretion perpendiular to the prinipal stress. Subsequently, tangential stresses to the plane of the first rak may produe the seond, and the third rak, whih may emerge at the point of numerial integration in the diretion perpendiular to the ausative omponent orresponding to prinipal stress. In the state of raking and rushing the stiffness matrix of finite element is adapted to the state of the damage. The parameter β t is introdued for reduing shear transfer ausing slip in the plane perpendiular to the rak surfae. The relationship between stress and strain of the raked onrete in one plane is desribed by the following stiffness matrix Ht 1 / E /1 / E 0 / 1 1/ k D (2) t t Graphial interpretation of the softening parameter H t and the multiplier T for tensile stress relaxation is shown in figure 1b. More details an be find in [12] Modelling of steel The elasti-plasti model with linear hardening, and the same stress-strain relation for the tension and ompression is applied for the reinforing bars, figure 2. 3

5 Figure 2. Stress-strain relation for steel reinforement. Spatial spar element, with two nodes and three degrees of freedom in eah of them, was applied in the modelling of bars. Moreover, linear elasti model was assumed for the steel plates loated at the support and loaded points. The hexahedral elements were applied to model steel plates. Perfet bond between materials is assumed. The reinforement is onneted to onrete mesh nodes. Therefore, the onrete and the reinforement mesh share the same nodes and onrete oupies the same regions seized by rebars. 3. Numerial solutions of equilibrium equations 3.1. Newton-Raphson method with adaptive desent Newton-Raphson method is an iterative proess of solving nonlinear equations. The stiffness matrix and vetor of restoring loads are alulated on the basis of the displaement vetor. The stiffness matrix in Newton-Raphson method with adaptive desent modifiation is desribed as the sum of two matries [15] K T S 1 T i K K (3) where: K S seant stiffness matrix, K T tangent stiffness matrix, ξ adaptive desent parameter. The method is based on the agreement of the adaptive desent parameter ξ in the equilibrium iteration. The seant stiffness matrix is generated in the numerial method as an effet of the nonlinear plastiity, stiffness with large deformations, onrete rushing and post-raking stress relaxation Ar-length method In the ar-length method, the equation is dependent on load parameter λ [16] T a nr Ki Δui F F i (4) The vetor of inremental displaement Δu i onsists of two omponents I II Δui Δui Δu i (5) where: Δu I i displaement inrement vetor aused by a unit load parameter, Δu II i displaement inrement vetor in the Newton-Raphson algorithm. The inremental load parameter Δλ is searhed from the additional onstraints equation based on orthogonality insurane. It was found that the onvergene of solutions for default setting of onvergene tolerane limits was diffiult to ahieve due to the nonlinear behaviour of reinfored onrete. Therefore, tolerane limits for displaement were inreased to 5% in order to obtain the onvergene of the solutions. 4. Numerial results and disussion 4.1. Investigation objets 4

6 The dimensions for retangular beams tested by [6] were assumed for the numerial spatial beam models. Dimensions, reinforement and loading/support arrangements are shown in figure 3. Figure 3. Dimensions of beams with reinforement and loading/support arrangements (unit in mm) Figure 4. Modelling of support plate and load appliation plate (unit in mm) Half of the beam was modelled with regard to the longitudinal symmetry. Steel plates at the support were modelled as nodal imparting fores on rollers. Free rotation of the beam in the plane of bending was allowed. External fore is also applied through the steel plate. Uniform distribution of fores at the nodes in the diretion of the transverse symmetry axis of the steel plate was assumed. Boundary onditions are shown in figure Effet of elasti modulus and shear transfer oeffiient on load-defletion urves Compressive strength is neessary to determine other parameters of the nonlinear model of HSC. The omputations of model beam BP-1a made of onrete with a ompressive strength of 81.2 MPa were 5

7 performed to evaluate the effet of the elasti modulus (E ) and shear transfer oeffiient for an open rak (β t ) on the load-defletion urves. Different relationships between the ompressive strength and modulus of elastiity for HPC are shown in table 1. It may be noted that the alulated values of elasti modulus have a large satter of results. Table 1. Equations for determining modulus of elastiity of HSC. CEB-FIB 1/3 Equations E (MPa) for f = 81.2 MPa E 10 f CAN A23.3-M90 E 5 f ACI 363 E 3, 32 f 6, Prop. Kikizaki [17] E 3, 65 f Prop. Neville [17] E f Load-defletion urves at mid-span for beam BP-1a obtained in experimental test and in FEM analysis for different values of the elasti modulus are shown in figure 5a. It was found that the proposal of elasti modulus alulation aording with ACI 363 gives the best fit urve. Results of numerial analyses on impat of the shear transfer oeffiient for an open rak are presented in figure 5b. These values were hosen from the range of 0 to 0.5. Numerial alulations were performed using Newton- Raphson method with adaptive desent. For the oeffiient β t equal to 0.5 were obtained results similar to experimental data. (a) (b) Figure 5. Load-defletion urves at mid-span for BP-1a beam: (a) effet of elasti modulus (b) effet of shear transfer oeffiient 4.3. Material properties Uniaxial ompressive and tensile strengths of HSC were arried out on ube and ylindrial speimens. Modulus of elastiity are alulated on the basis of the experimentally determined uniaxial ompressive strength aording to ACI 363. Shear transfer oeffiient for open rak is estimated on the basis numerial analysis, see figure 5b. Properties of the steel bars were omputed on the basis of the axial tensile tests on steel bars ϕ16, ϕ10, ϕ6 mm [6]. The properties of HSC and steel bars are given for the beam models BP-1a/BP-2a, respetively. High-strength onrete is defined by the uniaxial ompressive strength f = 81.2/78.8 MPa, modulus of elastiity E = 36.8/36.4 GPa, uniaxial tensile strength f t = 5.23/4.57 MPa, Poisson s ratio ν = 0.15, 6

8 density ρ = 2600 kg/m 3, ompressive strain at the strength stress level σ 1 = 6, ultimate ompressive strain ε u = 12, shear transfer oeffiient for an open rak β t = 0.3, and shear transfer oeffiient for the losed rak β = 0.9. The appropriate material properties for the steel bars of ϕ16/ϕ10/ϕ6 mm diameters are as follows: modulus of elastiity E s = 196/194/201 GPa, yield stress f y = 437/420/353 MPa, uniaxial tensile strength f st = 713/624/466 MPa, ultimate strain at the yield stress ε su = 106/116/75, modulus of plasti deformation E T = /1792.1/ MPa, Poisson s ratio ν s = 0.3 and density ρ s = 7800 kg/m 3. Supporting and load transferring steel plates are defined by modulus of elastiity E s = 210 GPa, Poisson s ratio ν s = 0.3 and density ρ s = 7800 kg/m Load apaity and defletions Nonlinear load-defletion urves at mid-span for beams obtained in FEM analysis and test results are shown in figure 6. (a) (b) Figure 6. Load-defletion urves at mid-span for beam: (a) BP-1a, (b) BP-2a The stiffness alulated at the region of the elasti behaviour and post-raking was lose to test result. In the modelling of high-strength onrete beams less miro-raks were formed. Small differenes in the response of beams were identified during the rak propagation and yielding of reinforement steel. The load-defletion urves in the region of plasti strains show derease in stiffness. Both inremental-iterative methods give the numerial results qualitatively agreement with the tests results. Test results for reinfored high-strength onrete beams show that the effets of frature in the tensile regions are not ompensated by the elasti steel bars and ompressive onrete [5, 8]. Therefore, softening effets on the load-defletion urve for beam BP-1a, as a sudden load apaity drop, are observed. These results an be ahieved by ar-length algorithm whih enables to generate a omplete load-defletion path with loal and global softening stiffness. In addition, the ar-length method provides signifiant redution in omputation time Strains and rak distributions Load-strain urves for ompressive onrete at the mid-span of BP-beams were analyzed. The development of ompressive strain of the onrete is shown in figure 7. Curves obtained from the arlength solutions for the beam BP-1a with a low reinforement ratio showed a slight derease in the fore due to the appearane of raks in onrete. 7

9 (a) (b) Figure 7. Curves of load-strain in ompressive onrete at mid-span for beam: (a) BP-1a, (b) BP-2a Strains of the tensile bar under the load at the mid-span of the beam are reorded and shown in figure 8. In the ase of experimental beam BP-1a we an see the desending branhes of the urves. This reflets that the experimental beams were unloaded. (a) (b) Figure 8. Curves of load-strain in tensile bar at mid-span for beam: (a) BP-1a, (b) BP-2a The numerial results are almost equal to the experimental results in the linear-elasti region. Some inompatibilities of strain development in ompressive onrete and in tensile reinforement are observed in elasti-plasti region. The numerial smeared raks patterns for the BP-beams left side against the experimental raks distributions, are shown in figure 9. Numerial results are onsistent with experimental results as regards the loation, diretion and onentration. Ar-length method provides improved rak patterns obtained in the numerial solutions. Both the numerial and test results indiate that the rak distributions orrespond to the arrangement of stirrups. 8

10 Figure 9. Experimental and numerial rak patterns of BP-beams 5. Conlusions The full path of load-displaement with a loal and a global softening an be obtained by using the inremental-iterative ar-length method. Moreover, this method is highly effiient. Changeable load step and properly set up the ar-length parameters provide faster numerial omputation, and additionally guarantee aurate solutions. Additionally, steel plates plaed at the support and at the loaded point are very important for mapping the boundary onditions during the experimental tests. In this study the usefulness of the ar-length method was verified on spatial models of the reinfored high-strength onrete beams. The results obtained from the numerial and experimental analysis are ompared to eah other. It is seen that the finite element failure behaviour indiates a good agreement with the test failure behaviour. Aknowledgment This work was finanially supported by Ministry of Siene and Higher Eduation Poland, within the statutory researh number S/15/B/1/2017. Referenes [1] V. Afroughsabet, T. and Ozbakkaloglu, Mehanial and durability properties of high-strength onrete ontaining steel and polypropylene fibers, Constr. Build. Mater., vol. 94, pp , [2] L. Biolzi, G. L. Guerrini, and G. Rosati, Overall strutural behavior of high strength onrete speimens, Constr. Build. Mater., vol. 11, pp ,

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