Modeling of Passive Confinement of Concrete using Rigid Body Spring Model Rodolfo Mendoza Jr.1, Yoshihito Yamamoto2, Hikaru Nakamura3, and Taito Miura4 1
Ph.D. student, Department of Civil Engineering, Nagoya University Associate Professor, Department of Civil Engineering, Nagoya University 3 Professor, Department of Civil Engineering, Nagoya University 4 Assistant Professor, Department of Civil Engineering, Nagoya University
2
Tel: +81-52-789-5111, Email:
[email protected]
ABSTRACT The ability of Rigid Body Spring Model (RBSM) in simulating the behavior of confined concrete is demonstrated. First, the validation of the model in tri-axial test experiments is presented. The applicability of the model to simulate the behavior of concrete under passive confinement is then evaluated. Two types of passive confinement are presented: (1) passive confinement provided by lateral reinforcing bars termed as partial confinement and (2) the passive confinement provided by circular steel tube termed as full confinement. In the partial confinement case, lateral steel reinforcements in the form of circular steel ties or hoops are modeled using fiber beam elements. For the full confinement case, a geometrically nonlinear shell element with finite displacement and finite rotation feature was used to model the circular steel tube. The capability of RBSM to reproduce the softening behavior of concrete in low confinement case and hardening behavior in high confinement case is presented through verification with analytical confined concrete models and validation with published test results. A good agreement between reference and simulation results was obtained. Keywords: Rigid Body Spring Model, confined concrete, beam elements, nonlinear shell elements
1. INTRODUCTION The use of steel lateral confinements adds a certain level of ductility to the brittle behavior of concrete—in which the amount of added ductility depends largely on the volumetric ratio of lateral steel reinforcements and the distribution of confinement. The effectiveness of confinement depends on how much concrete is being confined by the lateral reinforcement. The use of lateral steel reinforcements in the form of hoops or spirals provides a non-uniform confinement, in which the concrete near the hoops is well confined while concrete in between hoops is unconfined. This type of passive confinement is termed here as partial confinement. If circular steel tube, on the other hand, is used as the lateral confinement, the entire concrete is fully confined and therefore this type of confinement is termed hereafter as full confinement. For partially confined concrete columns, Mander et al. (1988b) have shown that increasing the volumetric ratio, s , results in an increased in the capacity of the member. On the other hand, the strain at peak decreases with increasing lateral reinforcement ratio. It was also shown that the slope of the post-peak softening curve observed in uni-axially compressed columns decreases with increased lateral reinforcement ratio. Caner and Bazant (2002) investigated the amount of lateral steel confinement needed to suppress the post-peak softening of concrete in compression. In their study, lateral confinement is provided through the use of circular steel tube which confines the entire concrete section. They concluded that a reinforcement ratio (in their study defined as the ratio of steel tube volume to the total volume
of concrete) of at least 14% is needed to eliminate the strain-softening response in the axial load-deflection curve of concrete. A ratio of 8.0% was recommended where mild softening is allowed. Numerical methods for columns laterally confined by steel reinforcements has been largely based on analytical equations (e.g., Mander et al. (1988a), Saatcioglu and Razvi (1992), Akiyama et al. (2010), Sakai (2002)) . For steel-tube confined concrete composites, fiber and finite element models were introduced mainly to provide an accurate prediction of strength and deformation capacity of the composite member or to supplement experimental test results in order to investigate other parameters that may affect the behavior of the composite (e.g., Liang and Fragomeni (2009), Denavit and Hajjar (2012), Johannson and Gylltoft (2002)). In this study, we present the Rigid Body Spring Model (RBSM) as the uniform numerical method to model the behavior of concrete under various types of confinement. In modeling the partial passive confinement provided by lateral steel ties, fiber beams models were used to model the lateral steel reinforcements. For the case of full passive confinement, a nonlinear shell finite element with finite displacement and finite rotation capability was used to model the confining steel tube. The performance of RBSM in coupled with fiber beam model and nonlinear shell model is tested through comparison with analytical confined concrete models and test results of concrete-filled steel tubes (CFST). A discussion on the effect of the type of confinement in terms strength and ductility of uniaxially compressed columns is provided.
2. MODELING CONCRETE USING RBSM In this study, concrete is modeled using 3D-RBSM which was proven effective in simulating the salient features of concrete such as cracking, localization of deformation, and strain softening behavior (Yamamoto et al., 2008, 2014). The RBSM utilized in our study is based on a meso-scale, single-phase averaging model of concrete. The model has been successfully applied to investigate various concrete engineering problems—ranging from corrosioninduced cracking in concrete (Qiao et al., 2016) to the failure behavior of concrete plates subjected to rigid projectile impact (Yamamoto et al., 2016). The general concept of modeling concrete using RBSM can be summarized into four stages. First, the target or objective of the simulation is defined followed by the setting of an observation scale (e.g., macro, meso, and micro scale). Concrete constitutive models are then assigned directly into the normal and tangential springs of RBSM particles—since the response of the springs dictates the interaction of the particles instead of their internal behavior. In this study, we adopted the tension, compression, and shear constitutive models developed by Yamamoto et al. (2008, 2014). Lastly, the parameters from these constitutive models are calibrated using uni-axial and multi-axial tests performed on concrete specimens. The calibration process for the RBSM formulation used in this study is described in Yamamoto et al. (2008). 3. MODELING PASSIVE CONFINEMENT USING LATERAL STEEL TIES/HOOPS A representation of the passive confinement provided by steel lateral ties/hoops and the geometric description of linking the fiber beam elements with RBSM particles is provided in Figure 1. In this coupled modeling approach, the load-transfer mechanism between RBSM and beam elements is provided by zero-length link/interface elements. The concept was initially introduced by Bolander and Saito (1998) for two-dimensional RBSM elements and was extended to 3D-RBSM by Yamamoto (2008, 2010). Each beam element nodes consists of three translation and three rotational degrees of freedom. The beam element is discretized into several fibers (shown in Figure 2a) running longitudinally along the reinforcement length.
Link/interface element
RBSM Beam element （concrete） （steel rebar） Representation of partial confinement provided by steel ties
RBSM-Fiber beam model
Figure 1. RBSM-fiber beam model
The relative displacements between RBSM and beam element are evaluated based on the displacement of RBSM centroid and beam element nodes. A bilinear kinematic hardening material model was adopted for beam elements in which the hardening coefficient is taken as 1/100 as shown in Figure 2b.
4
s
2
max
fy
0
E -2
sh
-4 -4
(a) Fiber discretization of -2 0 2 4 beam element
s
(b) Material nonlinear model for beam element
0.1 max
s1
s2
s
(c) Bond-slip relation for link/interface element
Figure 2. Beam and link element models
The initiation of cracks is highly affected by the interaction of concrete and steel, i.e., RBSMbeam interaction. In order to properly account for this behavior, a bond stress-slip relation was introduced into the material model of link elements oriented along the reinforcements. The bond-slip relation is given in Figure 2c. Detailed discussion of this model can be found in Yamamoto (2008).
4. MODELING PASSIVE CONFINEMENT USING STEEL TUBE Full passive confinement of concrete can be achieved through the use of steel tube sections— where the steel laterally confines the entire concrete section. A 3D degenerated nonlinear shell element is used and coupled with RBSM to form the steel-tube confined concrete composite. Similar to RBSM-beam model, interface/link elements were also introduced to connect the RBSM and shell element. A representation of this coupled model is shown in Figure 3.
Shell FEM
Link Elements
RBSM
Interface/link elements
Connect to other RBSM element
0V 2K
0V 3K
0V 1K x ,y ,z g
g
g
G1 G3 G2
Rigid Voronoi element
Shell element
Connect to other RBSM element
Contact Area=
RBSM surface area no. of shell nodes
Coupled RBSM-Shell FEM
Figure 3. RBSM-shell model
Interface/link elements connect the peripheral RBSM surfaces to shell elements which are pre-assigned on every shell nodes, and therefore, are regularly spaced as can be observed in Figure 3. The shell element we utilized is a four-node, degenerated, iso-parametric shell element. The element is based on the work of Noguchi and Hisada (1995) in which an integrated FEM approach for total and updated Lagrangian method was used in the formulation. The shear locking problem commonly encountered in shell elements is addressed using a selective reduced integration scheme (.i.e., the transverse shear strain are evaluated only in the element midpoint). Material nonlinearity is modeled using Von Mises yield criterion with isotropic hardening. The RBSM-to-shell connection is shown in Figure 4. As mentioned, interface elements are pre-assigned on every shell nodes and are mapped on RBSM element surfaces that are in-contact with shell elements. A complete description of the interface element formulation can be found in Mendoza et al. (2017). The interface element spans from the mid-surface of shell to the selected computational point in the RBSM particle as depicted in Figure 4b.
Steel Tube Three interface elements per shell node: • Normal • Shear (tangent to vertical) • Shear (tangent to tube circumference)
Shell element mid-surface
V3k
x ks , y ks , z ks Concrete Core
r 3 1.0
xg , y g , z g
r 3 1.0
Imaginary separation h
(a) Section showing RBSM-to-shell connection
(b) Definition of spring interface length
Figure 4. Description of RBSM-shell connection
Constitutive models for interface elements between RBSM and shell are shown in Figure 5. These models represent normal and frictional interaction between concrete and steel. For
simplicity, bond behavior (such as those due to cohesion and adhesion) was not modeled as frictional resistance relies solely on the compression contact between RBSM and shell. The relation between the normal and shear springs is provided by the Mohr-Coulomb yield criterion. Friction coefficient parameter was found to be 0.25 from a separate analysis. A bilinear hardening model was adopted for the shear spring interface with an assumed hardening coefficient of 1/100. The RBSM and shell elements were allowed to separate when normal spring is under tension. Fv , Fc Fvcr , Fccr Fn Fvcr , Fccr H
n
K v ,c
Fn
Kn slip Mohr-Coulomb model
Normal spring
Fcr
slip
Shear springs
Figure 5. Constitutive models for interface elements in RBSM-shell model
5. CONCRETE UNDER ACTIVE CONFINEMENT For the RBSM to be effective in simulating passive confinement, the performance of the model—as a sole model for concrete without being coupled with beam or shell element— must be validated. The validation performed by Yamamoto et al. (2008) is reproduced to show that the RBSM can well simulate the behavior of concrete under active confinement. The results from the test performed by Kotsovos and Newman (1978) were used for comparison. The specimen was subjected to increasing hydrostatic pressure ranging from 18 MPa to a maximum pressure of 70 MPa or around 1.5 times the concrete compressive strength value. 100 mm -300
l側圧70MPa = 70MPa
(N/mm2)2) Stress 応力 (N/mm
側圧51MPa l = 51MPa
-200
250 mm
側圧35MPa l = 35MPa 側圧18MPa l = 18MPa
-100 解析結果 実験結果
0
0.05
0 Strain 軸ひずみ
-0.05
Figure 6. Validation of RBSM for Active Confinement
By comparing the test results with the RBSM simulation results in Figure 6, we can see that the RBSM can well simulate the stress-strain behavior of concrete under active confinement. This is generally true for all the levels of confinement investigated.
6. CONCRETE UNDER PARTIAL PASSIVE CONFINEMENT The RBSM-beam model was already used to investigate the uniaxial compression failure of concrete laterally confined by steel reinforcements (Yamamoto et al. 2010). In this section, the ability of RBSM to simulate the confinement provided by lateral steel reinforcements in the form of circular steel hoops is presented. Here, well-known analytical models for confined concrete are used as reference solutions. The analytical solutions are based on the study of Mander et al. (1988a), Saatcioglu and Razvi (1992), Sakai and Kawashima (2002), and Akiyama et al. (2010). In this evaluation, the varied parameters are the volumetric ratio of lateral steel reinforcements and the spacing of lateral ties. The simulation models are shown in Figure 7.
s
s
RBSM-beam
s
s
Lateral ties with increasing spacing modeled using beam elements
Figure 7. Simulation models for partial passive confinement
Figure 8 shows the comparison of results from various confined concrete models with the simulation results using RBSM-beam model. The graphs show the relation of the normalized (with the unconfined compressive strength) confined compressive strength of column, and the tie spacing over section width (diameter) ratio. The three graphs represent three levels of lateral confinement expressed in terms of increasing lateral steel reinforcement ratio ranging from 0.5% to 2.0 %. Among the analytical models, the predictions of RBSM-beam model are close to the estimates using equations provided by the Mander et al. (1988a) and Akiyama et al. (2010). It is clear from the results that the axial capacity of the column increases as the ratio of lateral reinforcement increases—which is consistent with the observations of Mander et al. (1988b). In terms of the effect of tie spacing, columns with closely spaced lateral ties showed higher axial capacity. This is true for all the three cases of lateral reinforcement ratio. RBSM-beam model 2
Mander et al. 2
Saatcioglu & Razvi
ρs=1.0%
1
1 0
1.5
ρs=2.0%
fcc' / fco'
fcc' / fco'
fco' fcc' // fco’ fcc’
ρs=0.5%
1.5
0.5
s/d s/d
1
Akiyama et al.
Sakai 2
1.5
0
1.5
1 0.5
1
s/d s/d
1.5
0
0.5
s/d s/d
1
Figure 8. Comparison of compressive strength predictions under increasing hoop spacing (partial passive confinement)
1.5
The effect of decreasing the hoop spacing is numerical evaluated. The reference solutions are based on the analytical equations provided by Akiyama et al. (2010). Again, three levels of steel reinforcement ratio were investigated, i.e. from 0.5% to 2.0%. The investigated hoop spacings are from as distant as 320 mm to as close as 40 mm. The vertical axis in the graphs represents the ratio between the axial stress and the confined concrete strength of concrete; the horizontal axis refers to the difference between the axial strain at peak and the axial strain corresponding to the confined concrete strength. The results show that higher ductility can be achieved if closely spaced hoops are used. This increased in ductility due to decreasing lateral spacing is however reduced with decreasing steel reinforcement ratio. These observations were well simulated by the RBSM-beam model. RBSM-beam model
1
0.6 0.4 0.2 0
s=40mm s=80mm s=160mm s=320mm 0.01
0.8
σc / fcc'
s=80mm s=160mm s=320mm
0.6 0.4 0.2
ρs=0.5%
-cccc εcc－ε
0.02 0
s=40mm s=80mm s=160mm s=320mm ρs=1.0% 0.01
ε c－ε cc c-cc
σc / fcc'
s=40mm s=80mm s=160mm s=320mm
s=40mm 0.8
σ ' cc//fccfcc’
Akiyama et al.
1
1
s=40mm s=80mm
0.8 0.6
s=160mm 0.4 s=40mm s=80mm 0.2 s=160mm s=320mm
0.02 0
s=320mm
ρs=2.0%
0.01
0.02
εcc－ε -cccc
Figure 9. Comparison of post-peak softening predictions under increasing hoop spacing (partial passive confinement)
7. CONCRETE UNDER FULL PASSIVE CONFINEMENT We present in this section the ability of RBSM in coupled with nonlinear shell element in simulating the behavior of concrete under full passive confinement. Previous validations of the RBSM-shell model can be found in Mendoza et al. (2016, 2018). Here, we present the ability of the model to capture the brittle-to-ductile transition behavior of steel-tube confined concrete columns with increasing confinement. The confinement is provided circular steel tube. Test results performed by O’Shea and Bridge (1997) on CFST were used for comparison. The test specimens are almost geometrically the same except for the thickness of steel tube. The properties of specimens are given in Table 1.
Specimen label S10CS50A S12CS50A S20CS50A
Table 1. Properties of simulated CFST specimens D(mm) T(mm) L(mm) D/T ratio f’c (MPa) 190.0 0.86 659.0 220 41.0 190.0 1.13 664.5 168 41.0 190.0 1.94 663.5 98 41.0
fy (MPa) 210.7 185.7 256.4
The lateral steel percentage ratio of simulated specimens ranges from 1.80% to 4.0%. For all the specimens, the load was applied simultaneously on both concrete and steel tube. Figure 10 shows the comparison of test and simulation results of RBSM-shell model. The graphs show the capability of RBSM-shell model in simulating the behavior of concrete under full passive confinement. The post-peak softening of the axial stress-strain curve at 1.80% confinement and the post-peak hardening response at 4.0% confinement were well simulated by the model.
RBSM-Shell 60
60
fcc’
50
S10CS50A
50
Test - O’Shea and Bridge 60
S12CS50A
50
40
40
40
30
30
30
20
20
20
10 0
s 1.8%
0.25
0.5
0.75
Axial strain (%)
1
10 1.25 0
s 2.4%
0.25
0.5
0.75
1
Axial strain (%)
S20CS50A
10 1.25 0
s 4.0%
0.25
0.5
0.75
1
1.25
Axial strain (%)
Figure 10. Comparison of axial stress-strain curves under increasing steel tube thickness (full passive confinement)
The predicted strengths were likewise generally in good agreement with those obtained from the experiments. The axial strains corresponding to peak stress were also well simulated by the model. From these results, we can say that the brittle behavior of concrete characterized by the post-peak softening of the stress-strain curve was partially suppressed at a steel ratio of 4.0%, although other authors—e.g., Caner and Bazant (2002)—suggests that full suppression of post-peak softening should mean a hardening post-peak response of the stress-strain curve which could be achieve at a lateral reinforcement ratio of at least greater than 14.0%.
8. CONCLUSION The performance of RBSM in simulating the behavior of concrete under various types and levels of confinement was demonstrated. The presented results showed that the RBSM can be used as a universal numerical tool for studying the effect of confinement in concrete. With the observed agreement between reference and simulation results, one can use the model with confidence to compare the merits of using steel tube as the main lateral reinforcement over the commonly used steel lateral ties/hoops. An important attribute of using a model coupled with RBSM is that it provides not only a quantitative evaluation of concrete strength and deformation but also a qualitative analysis of concrete behavior—such as the evaluation of cracking and localization of deformation which can be used to study damage propagation and distribution in concrete. In the context of restorability assessment, the model can be utilized in the performance evaluation of existing concrete structures retrofitted by steel jacketing techniques. In those cases, the evaluation of confinement provided by lateral hoops and the added confinement of steel tube can be assessed both qualitatively and quantitatively using the RSBM-beam and RBSM-shell models presented.
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Kotsovos, M.D., and Newman, J.B., 1978. Generalized stress-strain relations for concrete. Journal of Engineering Mechanics, Vol.104(4), 845-856. Liang, Q.Q., Fragomeni, S., 2009. Nonlinear analysis of circular concrete-filled steel tubular short columns under axial loading. Journal of Constructional Steel Research, Vol. 65, 2186-2196. Mander, J.B., Priestley, M.J.N., Park, R. 1988a. Theoretical Stress-Strain Model for Confined Concrete. Journal of Structural Engineering, Vol.114, No.8, pp.1804-1826. Mander, J.B., Priestley, M.J.N., Park, R. 1988b. Observed Stress-Strain Behavior of Confined Concrete. Journal of Structural Engineering, Vol.114, No.8, pp.1827-1849. Mendoza, R.J., Yamamoto, Y., Nakamura, H., and Miura, T., 2016. Modeling of Axially Loaded CFT using Mixed 3D-Rigid Body Spring Model and Geometrically Nonlinear Shell. Proceedings of the 18th JSCE International Summer Symposium, Japan Society of Civil Engineers. Mendoza, R.J., Yamamoto, Y., Nakamura, H., and Miura, T., 2018. Numerical simulation of compressive failure behaviors of concrete-filled steel tube using coupled discrete model and shell finite element. High Tech Concrete: Where Technology and Engineering Meet, Springer International Publishing AG. Mendoza, R.J., Yamamoto, Y., Nakamura, H., and Miura, T., 2017. Modeling of composite action in concrete-filled steel tubes using coupled RBSM and Shell FEM. Proceedings of the Japan Concrete Institute, Vol. 39. Noguchi, H., and Hisada, T. 1995. Integrated FEM formulation for total/updated Lagrangian Method in Geometrically Nonlinear Problems. JSME International Journal, Vol. 38, No.1. O’Shea M.D., and Bridge, R.Q., 2000. Design of Circular Thin-walled Concrete-filled Steel Tubes. Journal of Structural Engineering, Vol.126(11), 1295-1303. Qiao, D., Nakamura, H., Yamamoto, Y., and Miura, T., 2016. Crack patterns of concrete with single rebar subjected to non-uniform and localized corrosion. Journal of Construction Building and Materials, Vol. 116, 366-377. Saatcioglu M., and Razvi, S., 1992. Strength and Ductility of confined concrete. Journal of Structural Mechanics, Vol.118, pp.1590-1607. Sakai, J. and Kawashima, K., 2002. Effect of Spacing of Tie Reinforcement and Cross Ties on Lateral Confinement of Concrete. Proceedings of Japan Society of Civil Engineers, 717/I-61, 91-106. (in Japanese). Yamamoto, Y., Nakamura, H., Kuroda, I., and Furuya, N., 2008. Analysis of compression failure of concrete by three-dimensional rigid body spring model. Proceedings of Japan Society of Civil Engineers, E-64, 612-630. (in Japanese). Yamamoto, Y., Nakamura, H., Kuroda, I., and Furuya, N., 2010. Analysis of uniaxial compression failure of confined concrete by three dimensional rigid body spring model. Proceedings of Japan Society of Civil Engineers, E-66-4,433-451. (in Japanese). Yamamoto, Y., Nakamura, H., Kuroda, I., and Furuya, N., 2014. Crack propagation analysis of reinforced concrete wall under cyclic loading using RBSM. European Journal of Civil Engineering, Vol. 18 (7), 780-792. Yamamoto, Y., Okazaki, S., and Nakamura, H., 2016. Crack propagation and local failure simulation of reinforced concrete subjected to projectile impact using RBSM. Proceedings of the 35th International Conference on Ocean, Offshore and Artic Engineering (OMAE 2016).
