NONLINEAR CYCLIC TRUSS MODEL FOR SHEAR-CRITICAL REINFORCED CONCRETE COLUMNS

Transcription

1 NONLINEAR CYCLIC TRUSS MODEL FOR SHEAR-CRITICAL REINFORCED CONCRETE COLUMNS S.C. Girgin 1, Y. Lu 2 and M. Panagiotou 3 1 Researh Assistant, Civil Eng. Department, Dokuz Eylul University, Izmir 2 Graduate Student, Civil and Environmental Eng. Department, University of California, Berkeley 3 Assistant Professor Dotor, Civil and Environmental Eng. Department, University of California, Berkeley sadik.girgin@deu.edu.tr ABSTRACT: This study presents a nonlinear truss modeling approah for shear-ritial reinfored onrete olumns subjeted to yli loading. Nonlinear steel and onrete truss elements are used to represent steel reinforement and onrete areas, respetively, in the vertial and horizontal diretions. Nonlinear onrete trusses are used in the diagonals, aounting for the biaxial effet on the ompression behavior. Tension stiffening and softening effets are modeled for all onrete truss elements, aounting for mesh size effets and frature energy, and the effets of strain penetration are modeled. Flexure-shear interation is modeled expliitly through the oupling of these elements. The model is validated by omparing experimentally measured and numerially omputed response for two shear-ritial olumn speimens, both haraterized by signifiant flexure-shear interation effets and softening of the onrete in the diagonal diretion. The overall fore-deformation response is presented inluding signifiant strength degradation. In addition, the effets of the diagonal truss angle and onrete biaxial relationship on the response are studied. KEYWORDS: Nonlinear, truss model, yli, shear-ritial olumns, flexure-shear interation. 1. INTRODUCTION There has been a great deal of experimental researh on seismi behavior reinfored onrete olumns subjeted to yli reversals for deades. Reinfored onrete olumns have been reported to have shear failures and flexure-shear failures espeially in non seismially designed RC strutures during previous earthquakes. Flexure-shear interation ours as a result of oupling of nonlinearities due to axial, flexural, and shear fores. The omputation of nonlinear yli response of reinfored onrete olumns with flexureshear interation is still a hallenging problem. Modeling approahes for RC olumns onsidering flexure-shear interation for reinfored onrete olumns an be ategorized as (i) lumped plastiity models, (ii) fiber - setion beam - olumn element models, (iii) maro models, (iv) truss or strut-and-tie models. Lumped plastiity models use nonlinear hystereti rules for shear and flexure springs loated at the element ends to aount for nonlinear flexure-shear interation behavior (Lee and Elnashai 2001, Xu and Zhang 2011). Fiber element models have been used extensively sine 1970s. Mostafaei and Vehio (2008) developed a uniaxial shear flexure model for an element under monotoni loading onsidering average axial strains and average onrete ompression strains. Ceresa et al. (2009) presented a two dimensional Timoshenko fiber beam-olumn element and implemented to develop a flexure-shear model for RC beam olumn (frame) elements under yli loading. 1

2 Maro model presented by Mergos and Kappos (2008) onsidered shear strength degradation of a beamolumn element with two distributed flexibility sub-elements oupling onsidering shear and flexure deformations. Lodhi and Sezen (2012) proposed a proedure with two maro models for flexure and shear deformations ombined due to the dominant failure mode. Truss or strut-and-tie models have been proposed in the literature to apture the strength and stiffness harateristis of RC members during yli reversals. Kim and Mander (1999) improved a truss model for monotoni and yli behavior of RC olumns. Miki and Niwa (2004) proposed a three dimensional lattie model for biaxial responses of RC members. They used truss members and arh members spanning from top to bottom of the olumns to represent shear resisting mehanism of RC olumns. Park and Eom (2007) improved a truss model onsisting omposite elements of onrete and rebar for RC members. These models didn t aount for the mesh size effets for onrete material models. Zimmerman et al. (2013) used nonlinear truss model for numerial modeling for shear failure of RC olumns with onsidering strain penetration effets. This model didn t onsider biaxial effet for onrete diagonals in ompression. Panagiotou et al. (2011) presented a nonlinear yli truss modeling approah inluding flexure-shear interation (FSI) for plane stress RC members subjeted to yli loading. Lu and Panagiotou (2013) developed a nonlinear beam truss modeling approah onsidering flexure-shear interation for non-planar RC walls subjeted to uni-axial or multi-axial loading. These models aounted for biaxial effet for onrete diagonals in ompression, mesh size effets for onrete and used a parallel angle model. This study presents a nonlinear truss modeling approah for shear-ritial reinfored onrete olumns subjeted to yli loading. Nonlinear onrete truss elements aounts for tension softening for the biaxial effet on the ompression behavior in the diagonal diretion and tension stiffening in the vertial and horizontal diretions developed by Lu and Panagiotou (2013) are used. In addition, this model onsiders strain penetration effets resulted by longitudinal reinforement slip from anhorage of olumn to foundation. Flexure-shear interation is modeled expliitly through the oupling of these elements. The model is validated by omparing measured and omputed responses of two RC olumns tested and both haraterized by signifiant flexure-shear interation effets. 2. NONLINEAR TRUSS MODELING APPROACH The truss modeling approah in this paper uses nonlinear truss elements in the vertial, horizontal and diagonal diretions. Nonlinear onrete truss elements aounts for tension softening for the biaxial effet on the ompression behavior in the diagonal diretion and tension stiffening in the vertial and horizontal diretions are used. The RC olumn setion onsidered in non seismially designed RC strutures is shown in Figure 1(a). The lear height of the olumn is H. Column setion width and height is B and H, respetively. Figure 1(b) shows the truss model of the olumn. Vertial and horizontal truss elements representing reinforing bars and onrete, and their effetive areas are depited in Figure 1 (-d) referring to sub-segment A. The diagonal truss elements representing onrete only are shown in Figure 1 (e). The ross setion area of the diagonal truss elements is the produt of the setion width Band effetive width b eff of the olumn. Strain penetration effets due to anhorage deformations are onsidered using truss elements for reinforement and onrete with a length of Lsp 16dbl is assumed for all ase studies, in whih dbl is the longitudinal bar diameter (Figure 1b). Strain penetration onrete truss elements have larger areas than the vertial onrete truss elements to aount for the stiffness of the beams, while the area of steel truss elements are same with the vertial steel trusses. The diagonal truss angle d ranged between 42 0 and 52 0 in the ase studies and the effet of d is onsidered in the Disussion setion. 2

3 3. CONSTITUTIVE STRESS-STRAIN RELATIONSHIPS 3.1. Reinforing Steel Material Model A number of Giuffré-Menegotto-Pinto (GMP) steel material models are used to define the stress-strain relationship used for the reinforing steel.a single GMP model is shown in Figure 2, where f y is the yield strength, y the orresponding yield strain, E s the elasti modulus, and B s the post-yield hardening ratio the monotoni envelope for this material model is bilinear and a multi-linear envelope is readily ahieved by a parallel model. For the ase studies presented in this paper, five to six GMP models are used in eah parallel steel material model and the material parameters are hosen to math the experimentally reported steel behavior. The parameters ontrolling the transition between elasti and plasti branhes (R 0, R 1, R 2 ) as defined in Filippou et al. (1983), are: R 0 =15, R 1 = 0.9, R 2 = 0.15 for all ase studies. The effet of rebar bukling is not onsidered. Figure 1. Nonlinear truss modeling approah for a RC olumn speimen Figure 2. (a) Stress-strain relationship of the GMP steel material model 3

4 3.2. Conrete Model for Vertial and Horizontal Truss Elements The ompressive stress-strain relation is based on the Fujii onrete model (Hoshikuma et al. 1997). The stress-strain relation for onrete developed by Lu and Panagiotou (2013) is used for the models and shown in Figure 3, where f ' is the ompressive strength of unonfined onrete ourring at strain 0 0.2%. ' The initial onrete modulus was E 2 f / 0. The value of u aounted for mesh-size effets base on the notion of onrete frature energy in ompression (Bazant and Planas 1998). The referene length (L R ) based on the panel studies in Vehio and Collins (1986) is taken and for L L R 600 mm, u 0.4%. The tension stress-strain relationship during loading is linear until it reahes the tensile strength of onrete f t 0.33 f in MPa. After this point, the onrete softens in aordane to the tension stiffening equation ' of Stevens et al. (1991), whih has parameters for the bar diameter and steel ratio in the diretion of the bar. Upon unloading from a ompressive strain, the tangent modulus is Eu 0.5 E 0.5 ( f / ) until reahing zero stress, whih then reloads linearly to the point with the largest tensile strain that ourred before. The unloading from a tensile strain is linear with a tangent modulus E until reahing zero stress. After this, the material loads in ompression and targets a stress equal to at zero strain with Conrete Model for Diagonal Truss Elements The onrete material model used for the diagonal truss elements aounts for the bi-axial strain field on the onrete ompressive behavior as desribed by Vehio and Collins (1986). For truss element e 1 extending from node 1 to node 2 [shown in Figure 4(a)], the normal strain, ε n, is omputed using the zero-stiffness gauge element extending from the mid-length of the element to nodes 3 and 4, g 1 and g 2, respetively. The instantaneous ompressive stress of element e 1 is multiplied by the fator β determined from the instantaneous normal strain n, whih is the average of the strain measured with the gauge elements g 1 and g 2. When n 0, the relationship between β and n is tri-linear, as shown in Figure 4(b). ft Figure 3. Stress-strain relationship for onrete material models 4

5 For this study, the relation between β and n depends on the length of the gauge elements, as first proposed by Panagiotou et al. (2013). Here, int (600 / L g ) 1% and res (600 / Lg ) 2.5%, where L g is the total length of gauge elements g 1 and g 2, as shown in Figure 4(a). The value of β int = 0.3 and β res = 0.1 was hosen to be similar to that developed by Vehio and Collins (1986). Conrete model for diagonals in tension has ft, res 0.02 ft at tensile strain t, res. Element length effet is aounted for t, res, and t, res 75 r ( LR / LD ) as a funtion of diagonal element length ( L D ) for LR 600 mm. Figure 4. (a) Truss element aounting for biaxial effets in the ompressive stress-strain behavior of onrete. (b) Relation between onrete ompressive stress redution fator, β, and normal strain, n. 4. MODEL VALIDATION The omputer program OpenSees (MKenna et al. 2000) was used for all the ase studies in this paper. The existing parallel material and GMP steel model, Steel02, was used for the parallel steel model. Both the uniaxial and bi-axial onrete models as well as the four-node truss element used in the diagonals, as desribed above, were implemented in Opensees by Lu and Panagiotou (2013). The response was validated with the test data of two olumns and omputed using a displaement ontrol algorithm with uniaxial loading Case Study 1 - Sezen & Moehle (2006) In this ase study, a square setion RC olumn alled Speimen 1 was onsidered (Figure 5). The longitudinal and the volumetri transverse reinforement ratio was l 3.7% and t 0.17%, respetively. Conrete and steel material properties are listed in Figure 5. Aspet ratio ( a/ d ) of the olumn was 3.4. Constant axial load was applied to olumn speimen during the yli load reversals. The axial load ratio of the olumn N / f' Ag 0.15 where N is the vertial load applied to the setion and Ag is the gross setion area. Lateral displaements were applied to the speimen in terms of nominal yield displaement ( ) as y / 4, y / 2, 2 y, and 3 y,until failure, with three yles at eah amplitude. The drift ratio Θ is defined as Δ/ H, where Δ is the lateral displaement and H is the height of the olumn where the load is applied. Sezen and Moehle (2006) reported that for Speimen 1, after spalling of over onrete at 1.8 % drift ratio (first yle), diagonal raks at the mid-height of the olumn took plae at the subsequent yle and strength degradation started during the experimental study. After 2.7% drift ratio, the strength degraded nearly to zero while arrying the axial load. y 5

6 M1 truss model was developed onsidering atual positions of the stirrups for Speimen 1 is given in Figure 5 (b-). Vertial, horizontal and diagonal truss element areas are given in the table. The lateral fore displaement responses for experimentally measured and ylially and monotonially omputed using M1 model are ompared in Figure 7(a). The failure mode observed in the ase study is aptured with the diagonal onrete rushing. The peak strength in Model M1 is 1.08 times the experimentally measured, and the failure happened at 1.8 % drift ratio. The effet of diagonal truss is presented in the Disussion setion. Figure 5. Case Study 1 Sezen & Moehle (2006) Speimen Case Study 2 Priestley et al. (1994) Case study 2 is a retangular RC olumn, alled R1A, with an aspet ratio M / VD 2.0, see Figure 6 (a). The longitudinal and the volumetri transverse reinforement ratio was l 2.52% and t 0.083%, respetively. Conrete and steel material properties are listed in Figure 6. Constant axial load was applied to olumn speimen during the yli load reversals. The axial load ratio of the olumn N / f ' A where N is the vertial load applied to the setion and Ag is the gross setion area. The olumn was subjeted to and initial fore ontrolled stage, following a displaement ontrolled loading pattern in terms of displaement dutility fators. The drift ratio Θ is defined as Δ/ H, where Δ is the lateral displaement and H is the height of the olumn where the load is applied. R1A speimen showed limited dutile response until the initial yles at 0.8% drift ratio. g 6

7 Lateral Fore (kn) 2 nd Turkish Conferene on Earthquake Engineering and Seismology TDMSK At third yles of 1.4 % drift ratio rapid strength degradation started with a major diagonal rak in the lower portion of the olumn (Priestley et al., 1994). M2 truss model was developed due to atual positions of the stirrups for R1A test unit is given in Figure 6 (b-). Vertial, horizontal and diagonal truss element areas are given in the table. The lateral fore lateral displaement responses for experimentally measured and ylially and monotonially omputed using M2 model are ompared in Figure 7(b). The failure mode observed in the ase study is aptured with the diagonal onrete rushing. The peak strength in Model M2 is 1.06 times the experimentally measured, and the failure happened at 0.8 % drift ratio. Figure 6. Case Study 2 Priestley et al (1994) R1A test unit. Drift Ratio (%) Drift Ratio (%) Experimental M1 yli M1 monotoni Lateral displaement (mm) -250 Experimental -500 M2 yli M2 monotoni Lateral displaement (mm) First rushing of onrete diagonals Figure 7. Comparison of experimentally measured and omputed responses for (a) Case Study 1 and (b) Case Study 2. 7

8 Lateral Fore (kn) Lateral Fore (kn) 2 nd Turkish Conferene on Earthquake Engineering and Seismology TDMSK DISCUSSION It is shown in the previous setion that and numerially omputed responses using nonlinear truss modeling approah give very satisfatory results in omparison with experimental data in ase studies. In this setion the effet of angle of inlination of diagonal truss elements on the response are studied. In Case Study 1, M1-1 and M1-2 models have different diagonal truss angles about 42 0 and 52 0 are onsidered. For the truss models, diagonal elements e 1 and e 2 with initial rushing during analyses are shown in Figure 5(b). The omputed yli and monotoni load- displaement responses for the two models are given in in Figure 8. The results show that as the diagonal truss angle inreases, the initial rushing of diagonal onrete truss elements our at smaller drift ratios (Figure 8). For model M1-1 rushing of diagonal elements our at 1.8 %, while for model M2-2 it happens at 2% drift ratio. Drift Ratio (%) (a) 200 Drift Ratio (%) (b) Experimental M1-1 yli M1-1 monotoni Lateral displaement (mm) -200 Experimental M1-2 yli M1-2 monotoni Lateral displaement (mm) Drift Ratio (%) () M1-1 M Lateral displaement (mm) First rushing of onrete diagonals Figure 8. The omparison of response with different diagonal angles with experimental results for models (a) M1-1, (b) M1-2, () Effet of angle of diagonal truss elements for Case Study 1. 8

9 6. CONCLUSIONS This paper desribes a nonlinear truss modeling approah for RC olumns suffering from failures resulted by flexure-shear interation. The model aounts for the mesh size effets for onrete stress strain relationship, biaxial effet on ompression behavior for diagonal truss elements and strain penetration effets. The model is verified by omparison of omputed and experimentally measured responses for two RC olumns, both haraterized by signifiant flexure-shear interation effets and softening of the onrete in the diagonal diretion. The main onlusions drawn in the study: 1. The nonlinear yli truss models omputes satisfatorily the post-raking yli fore displaement response and overall response of RC olumns subjeted to yli loading. 2. The model ompute rushing of onrete diagonals for Case Study 1 and Case Study 2 in a good agreement with the test results. 3. The effet of angle of inlination for diagonal truss elements is investigated for Case Study 1. The results show that as the diagonal truss angle inreases, the initial rushing of diagonal onrete truss elements our at smaller drift ratios. REFERENCES Bazant, Z. P. and Planas, J. (1998). Frature and size effet in onrete and other quasibrittle materials. Boa Raton, FL: CRC Press. Filippou, F. C., Popov, E. P., and Bertero, V. V. (1983). Effets of bond deterioration on hystereti behavior of reinfored onrete joints. Report EERC 83-19, Earthquake Engineering Researh Center, University of California, Berkeley. Hoshikuma, J., Kawashima, K., Nagaya,K., and Taylor, A. W. (1997). Stress- strain model for onfined reinfored onrete in bridge piers, Journal of Strutural Engineering, 123(5), Kim, J. H., and Mander, J. B. (1999). Truss modeling of reinfored onrete shear flexure behaviour, MCEER Report , University at Buffalo, State University of New York. Lee, D. H., and Alnashai, A. S. (2001). Seismi analysis of RC bridge olumns with flexure-shear interation, Journal of Strutural Engineering, 127(5), 546:553. Lodhi, M. S., and Sezen, H. (2012). Estimation of monotoni behavior of reinfored onrete olumns onsidering shear-flexure-axial load interation, Earthquake Engineering and Strutural Dynamis, 41: Lu, Y. and Panagiotou, M. (2013). Three-dimensional yli beam-truss model for non-planar reinfored onrete walls, Journal of Strutural Engineering, aepted for publiation. MKenna, F., Fenves, G. L., Sott, M. H., and Jeremi, B. (2000). Open system for earthquake engineering simulation. Mergos PE, Kappos AJ. A (2008). A distributed shear and flexural flexibility model with shear flexure interation for R/C members subjeted to seismi loading. Earthquake Engineering and Strutural Dynamis, 37: Miki, T., and Niwa, J. (2004). Nonlinear analysis of RC strutural members using 3D lattie model, Journal of Advaned Conrete Tehnology, 2(3), Mostafaei, H., and Vehio, F. J. (2008). Uniaxial shear-flexure model for reinfored onrete elements, Journal of Strutural Engineering, 134(9),

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