Numerical Simulation of Reinforced Concrete Members with Reinforcing Bar Buckling Sadık Can Girgin1*, Ioannis Koutromanos2, Mohammedreza Moharrami2 1
Department of Civil Engineering, Dokuz Eylul University, İzmir, Turkey, *e-mail:
[email protected] Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA 2
Abstract Modern reinforced concrete (RC) frame members are designed to develop inelastic flexural deformations in their end regions. The occurrence of longitudinal bar buckling in inelastic regions can crucially affect the ductility of such members. The performance assessment for RC structures should rely on analytical models which can account for the effect of rebar buckling on the nonlinear response. This study extends a previously proposed nonlinear truss modeling approach and nonlinear beam approaches for modeling RC elements whose response is affected by rebar buckling. The truss elements representing to reinforcing steel are provided with a uniaxial material model which can explicitly account for inelastic buckling and rupture of rebars. The modeling approach is validated using the results of experimental tests on RC beam and column specimens. Keywords: reinforced concrete, rebar buckling, beam model, nonlinear truss model
1 Introduction Reinforced concrete (RC) frames in earthquake prone regions are designed to resist flexural and shear deformations due to cyclic effects. Also lateral deformations due to buckling may occur resulting from high compressive strains in the end regions of members. Analytical prediction of ductility of RC elements can be overestimated if buckling of rebars is not accounted for in the numerical models. Dhakal and Maekawa (2002) proposed a fiber-based analytical model and validated the model by single rebar tests with different slenderness ratios (Lb/dbl), where Lb and dbl are buckling length and longitudinal bar diameter, respectively. Fragiadakis et al. (2007) proposed modifications for Monti-Nuti model with the assumption that buckling occurs between two consecutive stirrups. However, experimental studies showed that buckling length can span a larger length between several stirrups (Massone and Lopez, 2014). Zong et al (2013) proposed beamon-springs model with initial imperfections of longitudinal rebar for modeling circular RC columns. Massone and Lopez (2014) proposed a concentrated plasticity fiber model with four plastic hinges considering initial
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imperfections and validated the model with three column tests. A hybrid finite element method with fiber-based and finite element model with solid elements to predict reinforcement buckling was proposed by Feng et al. (2015). Longitudinal rebars with stirrups were modeled subjected to strain histories from previous tests and load history effect was captured. This study extends truss-based and nonlinear beam modeling approaches for modeling reinforced concrete (RC) elements with significant reinforcing bar buckling. Reinforcing bars susceptible to buckling are modeled explicitly with a uniaxial material model accounting for inelastic buckling and rupture proposed by proposed by Kim and Koutromanos (2016). The reinforcing material model was validated by a single rebar test. Also, experimental and numerical cyclic response of one RC beam and a column with significant buckling of the reinforcing bars are validated by using truss-based and fiber-based models, respectively.
2 Numerical Models 2.1 Nonlinear Truss Model Nonlinear truss modeling approach is able to capture shear or flexure-shear interaction effects explicitly to simulate cyclic response of RC walls (Lu and Panagiotou, 2014). Nonlinear truss model here uses nonlinear truss elements in the vertical, horizontal, and diagonal directions to model the concrete and steel reinforcement in each direction. Nonlinear concrete truss elements account for biaxial effects in the diagonal direction and tension stiffening in the vertical and horizontal directions. A cantilever beam is shown in Figure 1a with section dimensions of width (B) and height (H) . Truss model of the beam is shown in Figure 1(b) where Hm, the distance between top and bottom longitudinal rebars and L is the shear span. Anchorage deformations are considered by using additional truss elements within the length (La) in the truss model. La was calculated as La = sy/εy where sy is total slip at the yield (Zhao and Srithraran, 2007). Concrete truss elements had zero-tensile strength within the region (Moharrami et al., 2015).
(a) (b) Figure 1. (a) Details of a cantilever beam, (b) Nonlinear truss model The inclination angle of diagonal elements (θd) can be established between 310 and 590. In this study θd was considered as 430 angle. Vertical and horizontal truss elements were placed and their effective areas are depicted in Figure 2 (a-c) where s and h are spacing between vertical and horizontal elements, respectively. Figure 2 (d) shows the diagonal cross-section and the area of diagonal elements is the product of beam width (B) and effective width (beff).
(a)
(b)
(c)
(d)
Figure 2. Concrete and steel truss section effective areas for truss model
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2.2 Nonlinear Beam Models Nonlinear beam models include force-based beam-column elements with uniaxial concrete and steel material properties. Stresses are assumed to be constant between integration points along the beam-column element length. Numerical model of a cantilever column with three force-based beam-column elements is shown in Figure 3. OpenSees (McKenna et al. 2015) nonlinear force-based beam-column elements with Gauss-Lobatto quadrature and two integration points and corotational transformation were used. Fiber-based model herein considered the effects of slip of longitudinal rebars from anchorage of column to foundation by adding steel and concrete truss elements within the length (La).
Figure 3. Fiber-based numerical simulation model of a cantilever column
3 Material Models 3.1 Model for Reinforcing Steel A uniaxial material model capable of representing buckling and fracture of reinforcing bars was developed by Kim and Koutromanos (2016) enhancing the material model by Dodd and Restrepo (1995). After onset of buckling, a reduction coefficient was introduced in the enhanced material model in the core hysteretic law to multiply the stresses. Monotonic stress-strain relationship of the material model is shown in Figure 4, where fy, εy; fsh1, εsh1 and fu, εu are yield, intermediate and ultimate stresses and strains, respectively. The enhanced material model considers the effect of rebar buckling. Material model was validated with a cyclic uniaxial test conducted by Ma et al. (1976) as shown in Figure 4b. Mechanical properties of 19 mm rebar specimen were fy = 455 MPa; fu = 655 MPa; Es = 199955 MPa; εsh = 0.014; εu = 0.12. Slenderness ratio of the specimen was 4.4. The computed response and measured test results for Specimen 3 and numerical model matched closely with the experimentally measured response.
(a) (b) Figure 4. (a) Monotonic stress-strain relationship for steel model (Kim and Koutromanos, 2016), (b) Results of analysis for the specimen by Ma et al (1976).
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Inelastic bar buckling of the beam reinforcing bars are considered in the models by assigning slenderness ratio (Lb/dbl) in the uniaxial steel material for top and bottom rebars. Buckling mode (n) and buckling length (Lb) of longitudinal rebars were determined due to the procedure outlined in Dhakal and Maekawa (2002) summarized in Figure 5.
Figure 5. Flow chart for buckling length determination (adapted from Dhakal and Maekawa, 2002)
3.2 Model for Concrete The stress-strain law for concrete is the one by Lu and Panagiotou (2014), which is schematically presented in Figure 6. The initial concrete modulus was = 5000 (MPa). The softening part in compression was adjusted using the method by Lu and Panagiotou (2014). For horizontal and vertical concrete truss elements, tensile strength was = 0.33 (MPa) with a concrete softening portion in accordance with tension stiffening of Stevens et al. (1991).
Figure 6. Stress-strain relationship of concrete material models (from Moharrami et al., 2015).
4 Model Validation The computer program OpenSees (McKenna et al. 2015) was used for all the analyses described in this paper. The response was computed using a static displacement control algorithm with uniaxial loading, as described in each of the models. Newton with Initial Tangent were used in the solution algorithms. For each iteration, the residual of error was computed using Energy Increment, and the relative tolerance was equal to 10-3.
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4.1 Beam Test by Visnjic et al. The nonlinear truss modeling approach is validated using the results of the experimental tests on a RC beam specimen tested by Visnjic et al. (2016). Beam #2 specimen with section dimensions of 1219 mm height × 762 mm wide is shown in Figure 7 and was subjected to vertical tip displacements. The beam had longitudinal tension reinforcement corresponding to a longitudinal steel ratio ρl,t = 0.56 at the top and bottom of the beam, as well as shear reinforcement (hoops) at spacing s = 279 mm for Beam #2. The volumetric confinement reinforcement ratio was ρs = 0.51% for Beam#2. Concrete and reinforcing steel material properties are given in Table 1.
Figure 7. Section reinforcing details of Beam#2 specimen (adapted from Visnjic et al., 2016) Table 1. Material properties of Beam specimen Concrete
Reinforcing Steel
fc' (MPa)
εo
ft (MPa)
42
0.3%
2.14
fy (MPa)
fu (MPa)
εy
εsh
εu
#5
455
706
0.23%
-
17%
#11
480
685
0.3%
1.1%
23%
Positive displacement is in the downward direction. The drift ratio θ is defined as Δ/H, where Δ is the vertical displacement and H is the length along the beam where the load is applied. During the beam test, it was reported that flexural failures with buckling of rebars were observed for RC beam. For Beam#2, the initiation of buckling was noticeable for the cycle at θ = 2.9%, excessive buckling in the top rebars was observed during negative displacements of the cycle at θ = 3.9%. Following that, buckling of the bottom rebar was observed during the cycle at θ = 5.4%. Nonlinear truss model is developed with 430 truss angles. Top and bottom horizontal concrete and steel (Ach1, Ash1), inner horizontal concrete and steel (Ach2, Ash2), vertical concrete and steel (Acv, Asv) and diagonal (Ad) truss element areas of truss models are shown in Table 2. Truss element lengths (La) were calculated as 13dbl and 15dbw, where dbl (35.8 mm) and dbw (12.7 mm) are longitudinal and web reinforcement diameters, respectively. Slenderness ratio of the longitudinal rebars (Lb/dbl) was calculated as 16.9 for Beam #2 using the procedure outlined in Dhakal and Maekawa (2002). Inelastic material model including buckling was used by assigning slenderness ratios for the top and bottom two truss elements at the vicinity of the anchorage block. Table 2. Vertical, horizontal and diagonal truss element areas Ach1 (mm2) 34138
Ash1 (mm2) 5036
Ach2 (mm2) 68275
Ash2 (mm2) 94
Acv (mm2) 72580.5
Asv (mm2) 395
Ad (mm2) 49730
Beam#2 truss model showed a good agreement with the experimental results. The applied force – drift ratio of experimentally measured and numerical cyclic response for beam is compared in Figure 8a. Figure 8b shows analytical damage patterns at strength degradation initiated.
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4000 3000
Moment (kNm)
2000 1000 0 -1000 -2000 -3000 -4000 -8
Beam #2 B Measured -6
-4
-2 0 2 Drift ratio (%)
4
6
8
(a)
(b) Figure 8. (a) Comparison of measured and numerical model responses, (b) Analytical damage pattern at strength degradation initiated.
4.2 Column Test by Saatcioglu and Grira The nonlinear beam modeling method is validated using the results of the experimental tests on a RC column specimen tested by Saatcioglu and Grira (1999). Concrete and reinforcing steel material properties for BG2 column specimen are given in Table 3. Reinforcing details of the test specimen are shown in Figure 9. The transverse reinforcement ratio of BG2 specimen for all directions is 0.02. Constant axial load was applied to column specimen during cyclic load reversals. The axial load ratio of the column was (P/Ag fc’) 39% where P is the axial load applied to the section and Ag is the gross section. During the test, abrupt failure occurred due to longitudinal bar buckling at 5% drift ratio. Table 3. Material properties of BG2 specimen Concrete
Reinforcing Steel
fc' (MPa)
εo
ft (MPa)
34
0.21%
1.92
Rebar Diameter (mm) 9.53 19.5
fy (MPa)
fu (MPa)
εy
εsh
εu
570 452
686 656
0.27% 0.2%
0.6% 0.01
5% 10%
The stress-strain relation of cover concrete has unconfined concrete material model properties with compressive strength (fc’) occurring at ε0 = 0.2% strain. Compressive strength decreases linearly at εu = 0.5% strain for unconfined concrete. The stress-strain relation of confined concrete properties with compressive strength (fcc) at εc0 was calculated as 54 MPa at 0.36% strain using Mander concrete model (1988). Slenderness ratio (Lb/dbl) was calculated as 9.7 for longitudinal rebars of Specimen BG2. Comparisons of computed and experimental responses are shown in Figure 10a. The stress-strain responses of the cover and core concrete at base section are shown in Figure 10b. Strength degradation was computed due to onset of buckling of longitudinal rebars as shown in Figure 10c.
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Figure 9. Section reinforcing details of BG2 column specimen (adapted from Saatcioglu and Grira, 1999).
Figure 10. (a) Comparison of measured and numerical model responses of BG2 column specimen, computed stress-strains for (b) concrete, (c) steel fibers.
4 Conclusions This study has used a recently developed uniaxial constitutive model for reinforcing steel, to account for the effect of inelastic buckling in RC members. The steel material law has been combined with an existing uniaxial model for concrete and implemented in nonlinear truss models and beam models. The modeling tools have been validated with the results of experimental tests. Further validation of the proposed analysis tools will ultimately allow their use for the systematic assessment of the performance of RC components and systems.
Acknowledgements The work presented in this paper was conducted during the visit of the first author as a post-doctoral scholar to the Virginia Polytechnic Institute and State University. This visit was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) 2219 Program.
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References Bazant, Z. P. and Planas, J. (1998). Fracture and size effect in concrete and other quasibrittle materials. Boca Raton, FL: CRC Press. Dhakal, R.P. and Maekawa,K. (2002). Modeling for postyield buckling of reinforcement. ASCE Journal of Structural Engineering, 128(9), 1139-1147. Dodd, L.L., and Restrepo-Posada, J.I. (1995). Model for predicting cyclic behavior of reinforcing steel. ASCE Journal of Structural Engineering, 121(3), 433-445. Feng, Y., Kowalsky, M.J.,and Nau, J. M. (2015). Finite-element method to predict reinforcing bar buckling in RC structures. ASCE Journal of Structural Engineering, 141(5). Fragiadakis, M., Pinho, R., and Antoniou,S. (2007). Modeling inelastic buckling of reinforcing bars under earthquake loading. ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Greece. Kim, S.E. and Koutromanos I. (2016). Constitutive model for reinforcing steel under cyclic loading, Journal of Structural Engineering, Accepted for publication. Lu, Y. and Panagiotou, M. (2014). Three-dimensional cyclic beam-truss model for non-planar reinforced concrete walls, Journal of Structural Engineering, 140(3). Ma, S.M., Bertero, V. V.,and Popov, E. ( 1976). Experimental and analytical studies on the hysteretic behavior of reinforced concrete rectangular and T-beams, EERC Report 76/02 Earthquake Engineering Research Center, University of California, Berkeley. Mander, J.B., Priestley, M.J.N., ve Park, R. (1988). Theoretical stress-strain model for confined concrete, Journal of Structural Division (ASCE), 114 (8), 1804-1826. Massone, L.M., and Lopez, E. E. (2014). Modeling of reinforcement global buckling in RC elements. Engineering Structures, 59, 484-494. McKenna, F., Fenves, G. L., Scott, M. H., and Jeremic, B. (2015). Open system for earthquake engineering simulation. http://opensees.berkeley.edu Moharrami M., Koutromanos I., Panagiotou M., Girgin S.C. (2015). Analysis of shear-dominated RC columns using the nonlinear truss analogy, Earthq. Eng. Struct. Dyn., 44(5), 677-694. Stevens NJ, Uzumeri SM, Collins MP, Will TG. (1991) Constitutive model for reinforced concrete finite element analysis, ACI Structural Journal, 88(1):49–59. Vecchio, F. G. and Collins, M.P. (1986). The modified compression field theory for reinforced concrete elements subjected to shear, Journal of the American Concrete Institute, 83(2), 219-231. Visnjic, T., Antonellis, G., Panagiotou, M. and Moehle, J. P. (2016). Large reinforced concrete special moment frames under simulated seismic loading. ACI Structural Journal, 113(3), 469-480. Saatcioglu, M., and Grira M. (1999). Confinement of Reinforced Concrete Columns with Welded Reinforcement Grids, ACI Structural Journal, 96 (2), 248-258. Zhao J, Sritharan S. (2007). Modeling of strain penetration effects in fiber-based analysis of reinforced concrete structures, ACI Structural Journal; 96(1): 29-39. Zong, Z., Kunnath, S., and Monti, G. (2013). Simulation of reinforcing bar buckling in circular reinforced concrete columns. ACI Structural Journal; 110(4):607-616.
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