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Hysteretic model for concrete under cyclic tension and tension-compression reversals
Engineering Structures 163 (2018) 388–395
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Hysteretic model for concrete under cyclic tension and tension-compression reversals Pei Zhang, Qingwen Ren, Dong Lei
T
⁎
College of Mechanics and Materials, Hohai University, Nanjing 210098, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Hysteretic model Concrete Cyclic tension Stress reversals Seismic response
A hysteretic model for concrete structure subjected to cyclic tension and tension–compression reversals is presented. The proposed model was intended to predict the complex hysteretic behavior of concrete under cyclic loading in a simple and practical way. Based on the analysis of the characteristic hysteretic behavior of concrete, the residual deformation in tension was considered principally due to the incomplete closure of the opening cracks. The mechanism for the hysteretic behavior of concrete under tension–compression reversals was suggested as the crack closing and opening. Considering the application within different numerical approaches, dimensionless stress-deformation coordinates was adopted to perform the hysteretic model. The unloading and reloading paths have been derived from the crack closing and opening mechanism and were represented as straight lines in the model. Partial unloading and reloading were considered in both cyclic tension and tension–compression reversals. The proposed model has been validated by comparison with available experimental results and the seismic response of a SDOF system with the hysteretic model has been analysed.
1. Introduction The safety assessment of concrete structures subjected to cyclic loading such as seismic excitation requires realistic constitutive models to reproduce the real behavior of the materials. Because of the low tensile strength, the concrete subjected to seismic load usually presents softening behavior in tension and hysteretic behavior in tension-compression reversals. As a result, the hysteretic model for concrete plays a significant role in determining the seismic responses of concrete structures including the deformation and energy evolution. However, for lack of experimental data, there are few specialized researches on the modeling of hysteretic behavior for concrete under cyclic tension and tension-compression reversals. Most existing models for concrete considering the cyclic loading in tension assumed linear unloading-reloading paths without hysteretic energy dissipation [1–4]. Some authors (Vecchio and Palermo [5], Reinhardt et al. [6], Yankelevsky and Reinhardt [7], Chang and Mander [8]) have proposed more advanced models considering the complete and partial unloading-reloading hysteretic behaviors with different modeling approaches. Vecchio and Palermo [5] presented constitutive formulations for concrete subjected to reversed cyclic loading consistent with a compression field approach. The model was built upon the preliminary work presented by Vecchio [9] and intended to apply in the context of
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smeared rotating cracks. The hysteretic rules used in cyclic tension followed the philosophy for concrete in compression. The unloading path was modeled with a Ramberg–Osgood formulation and the reloading path was modeled as a straight line with degrading reloading stiffness. A plastic offset during complete unloading in tension has been defined in the model and formulated based on the test data from Yankelevsky and Reinhardt [7] and Gopalaratnam [10]. Although the crack-closing process in compression loading has been described with a linear formulation in the literature, the hysteretic behavior in tensioncompression reversals has been neglected. Reinhardt et al. [6] proposed a relationship between the stress and crack opening displacement for concrete in the tension and compression region. The total deformation during cyclic load was split up into a crack opening displacement part and a strain part consisting of an elastic strain and an irreversible strain, and the irreversible strain part was neglected in the model. As a result, the uncracked material behaved in a liner manner and all nonlinearities were comprised in the crack. The model can be applied in numerical simulation through a discrete crack approach or smeared crack approach. Straight lines were used as the unloading and reloading paths. Because the crack opening displacement was assumed constant during the unloading process in tension, the hysteretic behavior in cyclic tension cannot be simulated by this model. Yankelevsky and Reinhardt [7] developed a stress versus total
Corresponding author. E-mail address: [email protected] (D. Lei).
https://doi.org/10.1016/j.engstruct.2018.02.051 Received 12 August 2017; Received in revised form 5 January 2018; Accepted 16 February 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.
Engineering Structures 163 (2018) 388–395
P. Zhang et al.
Nomenclature
u0
c f0
x 0 xclose
ft Fs k k0 kcrcl kcrop ktpre m Rs u u¨ g
viscous damping coefficient maximum of the earthquake induced resisting force in a linear system tensile strength of concrete resisting force elastic stiffness of SDOF system normalized elastic stiffness in compression normalized crack-closing stiffness normalized crack-opening stiffness normalized partial reloading stiffness quality strength reduction factor deformation relative to the ground ground acceleration
x res x tpre x tun y ytun ytpre ytun δt εt ζ ω
maximum of the earthquake induced deformation in a linear system normalized strain normalized strain corresponding to crack-closing compressive stress normalized residual tensile strain normalized partial reloading strain normalized unloading strain on tension envelope curve normalized stress normalized crack-closing compressive stress normalized partial reloading stress normalized unloading stress on tension envelope curve tensile displacement corresponding to the tensile strength tensile strain corresponding to the tensile strength damping ratio undamped elastic circular frequency
reversals is presented. Compared to previous ones, the model presents several advantages. It affords to consider the essential features of the complex hysteretic behavior of concrete in a simple and practical way. It can be used to simulate the complete or partial unloading and reloading behaviors for concrete under cyclic tension and tension-compression reversals. Straight lines are adopted to describe the unloading and reloading paths and several necessary hysteretic rules are proposed based on the mechanism of crack closing and opening. Furthermore, all the required input parameters can be obtained through conventional laboratory monotonic tension tests. The model has been validated by comparison with available experimental results in different cases and the analysis of seismic response based on the hysteretic model has been performed.
deformation relationship for concrete behavior in cyclic tension and compression. The model was based on a given experimental cyclic stress-deformation envelope, and has defined focal points which were used to reproduce the complete unloading-reloading cycles. The focal points governed the unloading and reloading curves either by rays transmitted from a certain focal point towards known points in the stress-deformation plane, or by their stress level. When all the focal points were located, the complete unloading-reloading curves can be produced through a simple graphical process. Although the model represented well the test results, the definition of focal point purely derived from the graphic feature needs more physical significance and the procedure determining the unloading and reloading curves is too complex. Chang and Mander [8] proposed a rule-based hysteretic model to simulate the hysteretic behavior of confined and unconfined concrete in both cyclic compression and tension for both ordinary as well as high strength concrete. Fifteen unloading and reloading paths determined by fifteen different rules were distinguished in the model. The fifteen paths were divided into three types: envelope curve, connecting curve and transition curve. The equation used by the authors for the unloading and reloading curves was a polynomial adjusted by a series of parameters: the slope at the origin and the slope at the end of each curve. To determine the parameters of cyclic curves for concrete in compression, statistical regression analysis was performed on the experimental data from Sinha et al. [11], Karsan and Jirsa [12], Spooner and Dougill [13], Okamoto et al. [14] and Tanigawa et al. [15]. The expressions proposed for compression have been modified by the authors for the condition of tension cyclic behavior. In the documented literatures, the most refereed experimental studies on the uniaxial tensile cyclic behavior of concrete are from Reinhardt [16], Cornelissen et al. [17] and Mazars et al. [18]. More recently, Nouailletas et al. [19] have performed direct cyclic tension tests on the concrete specimens, and the effect of crack reclosing on properties of concrete has been studied at the macroscale using the digital image correlation (DIC) technique. At present, the hysteretic model considering the hysteretic behavior during cyclic tension and tension-compression reversals is still rarely used in the seismic response analysis of concrete structure. The principal shortcoming of the available hysteretic models for concrete in the literatures is the complicated hysteretic rules applied to reproduce the unloading and reloading curves. These rules are usually derived from the geometrical properties of the cyclic stress-strain curves, which results in the lack of a clear physical meaning of the proposed model. A set of parameters is required to perform the complex rules and it reduces the applicability of the hysteretic model. In this paper, an efficient model capable of predicting the hysteretic behavior of concrete under cyclic tension and tension-compression
2. Characteristic behaviors of concrete in tension-compression reversals Before the introduction of the hysteretic model, it is necessary to describe the characteristic behaviors of concrete in tension-compression reversals. Direct tension cyclic tests with different stress ranges on a double-notched specimen have been performed by Reinhardt [16] in 1984 and the corresponding stress-deformation relationships were obtained. The deformation was defined as relative displacement and measured by four extensometers with 35 mm gauge length. One of the test results with complete hysteretic loops has been selected to study the feature of concrete behaviors in tension-compression reversals. Fig. 1 shows the reproduced three successive unloading-reloading cycles. The positive stress means the tension and the unloading is
Fig. 1. Three successive unloading-reloading cycles under tension-compression reversals.
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deformation coordinates is applied to present the proposed hysteretic model:
referring to the deformation decrease process. In the unloading stage, the specimen was unloaded from the tension softening branch to the compressive stress zone. Considerable residual deformation was observed when the tensile stress unloaded to zero, and it was recognized as the irreversible plastic strain by many researchers [20–22]. However, different opinions were proposed by some authors [5,19]. The residual deformation was reconsidered attributed to the cracked surfaces come into contact and do not realign due to shear slip along the cracked surfaces. And the evolution of the unloading curve was explained by friction phenomena generated by the mismatching discontinuity lips. In present paper, the generation of the residual deformation in tension is considered principally due to the crack incomplete closure rather than the plastic strain and the unloading process can essentially be regarded as a crack closure process. This interpretation can be verified by the fact that during the compression phase the residual strain progressively vanishes and the material recovers its initial stiffness (Fig. 1). The recovery of stiffness in compression has been accepted as the “unilateral effect” in plastic damage theory and it is actually caused by the complete closure of cracks. In the reloading stage, the deformation grew fast during the tension phase as shown in Fig. 1, and it can be deduced that the reloading curve is due to the reopening of the previous closing cracks. Fig. 2 shows the evolution of unloading curve stiffness during the three cycles normalized by the elastic stiffness. It can be seen that the evolution of unloading stiffness exhibits three different stages. The first stage corresponds to the initial unloading in tension where the stiffness is large and decreases in approximate linearity with the deformation. At the second stage, the stress unloads to zero and reverses to the compression zone. Although the stress direction changes, the stiffness remains constant in this stage, indicating the gradual closing process of the cracks. At the last stage, the stiffness increases rapidly with the decrease of deformation due to the growth of compression stress. Fig. 3 shows the evolution of the normalized reloading stiffness which can be divided into two stages. At the first stage, the compression stress reduces to zero and the stiffness decreases approximately linearly with the increase of deformation. At the second stage, the stiffness remains constant in a low level before the deformation reaches the previous unloading value. Based on the characteristic behaviors of concrete under tensioncompression reversals described above, some assumptions can be made as follows:
x=
ε δ σ = , y= εt δt ft
(1)
where y is the dimensionless stress normalized by the concrete tensile strength ft ; x is the dimensionless deformation normalized by the strain εt or displacement δt corresponding to the concrete tensile strength. Based on the normalized stress-deformation model, the corresponding constitutive relationships applicable for the discrete or smeared crack model can be easily obtained.
3.2. Tension envelope curve for concrete It is commonly accepted that the envelope curve for concrete subjected to axial cyclic load can be approximated by that under monotonic load. The envelope curve for concrete under monotonic tension has been studies by many researches and is usually expressed by a piecewise function consisting of an ascent branch before the peak and a descent branch after the peak. In the pre-peak branch, a linear elastic relationship represents well the concrete behavior and it has been accepted by most researchers. In the postpeak branch, several expressions were documented in the literature, including straight lines [23], polylinear curves [24], exponential curves [5], polynomial curves [25] or combinations of them [26]. The complete stress-deformation curve for concrete under monotonic tension has been experimentally investigated by Guo and Zhang [27] based on twenty-nine direct tension tests. A rational fraction expression for the descending branch was obtained in the literature and was used in the present model. The complete expression of the adopted tension envelope curve for concrete can be written in the dimensionless form as follows:
(x ⩽ 1) ⎧y = x x ⎨ y = αt (x − 1)1.7 + x (x ⩾ 1) ⎩
(2)
where αt is a materials parameter and determined by the concrete tensile strength as:
αt = 0.312ft2
1. The mechanism of the hysteretic behavior of concrete under tensioncompression reversals is the crack closing and opening. 2. The stiffness of unloading curve remains constant in the low tension and compression stress region. 3. The stiffness of reloading curve remains constant in the tension stress region.
(3)
Different tension envelope curves with various αt have been depicted as shown in Fig. 4. It can be seen that the concrete with high tensile strength exhibits sharp fall in the softening branch, which representing high brittleness.
These assumptions will be used to determine the unloading and reloading paths in the hysteretic model. 3. Proposed hysteretic model for concrete 3.1. Normalized coordinate system for present model The failure of concrete subjected to tensile force takes place in local fracture and is therefore a discontinuous phenomenon. The local crack in concrete can be modeled as a discrete crack or a smeared crack in numerical simulation and many constitutive relationships have been proposed for the application of various models [2–3,5–7]. However, the above-mentioned constitutive relationships usually adopted different forms of deformation, such as crack open displacement, relative displacement or strain, which limited the applicability of the corresponding models. In order to improve the model applicability, a dimensionless stress-
Fig. 2. The evolution of normalized unloading stiffness in three cycles.
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obtained to represent the relationship between x res and x tun as shown in Fig. 6. The complete reloading path from the stress-free point N is proposed as a straight line directed to the previous unloading point L as shown in Fig. 5. The change of reloading stiffness before returning to the envelop curve has been neglected for simplicity. Based on the geometric feature, the complete reloading stiffness ktre can be calculated as:
ktre =
ytun x tun−x res
(4)
Partial unloading and reloading under cyclic tension are also considered in the model. The partial reloading path starts from a partial reloading point ( x tpre , ytpre ) with nonzero stress and is directed to the previous unloading point on the envelop curve (Fig. 5). Based on the rule, the partial reloading stiffness ktpre can be calculated as:
ktpre =
Fig. 3. The evolution of normalized reloading stiffness in three cycles.
ytun −ytpre x tun−x tpre
(5)
The partial unloading activates when the deformation decreases during complete reloading or partial reloading process, and the partial unloading stiffness is suggested equal to the initial stiffness. 3.4. Crack closing and opening path in tension-compression reversals The hysteretic behavior for concrete in tension-compression reversals is represented by a crack-closing path and a crack-opening path as shown in Fig. 7. The crack-closing path depicted as line MP is determined by the residual deformation x res and crack-closing compressive stress ytun . The value of ytun represents the compressive stress level corresponding to the full recovery of compression stiffness or complete closure of the cracks. Based on the schematic, the stiffness of crackclosing path can be calculated as:
kcrcl =
Fig. 4. Tension envelope curves for concrete with different tensile strength.
0 yclose 0 xclose−x res
(6)
0 xclose
where is the deformation corresponding to the crack-closing compressive stress ytun and calculated as:
3.3. Unloading-reloading path in cyclic tension
0 0 xclose = yclose /k 0
As it was observed in many experiments [15,16,18,28], concrete subjected to cyclic tension loading exhibited hysteretic behaviors. In a typical cyclic test in tension, the stiffness of unloading curve was usually very high at the beginning and then dropped to a stable value as the tension stress decreased. Residual deformation can be observed if the tension stress unloaded to zero. When reversely reloaded in tension, the reloading stiffness exhibited almost constant and a hysteretic loop would be formed during the unloading-reloading process. In the present model, the hysteretic behavior is simulated by series straight lines whose stiffness is determined in a semi-empirical way. Several necessary hysteretic paths are proposed based on the assumptions summarized at the end of Section 2. Fig. 5 shows the possible unloading-reloading paths for concrete under cyclic tension. The complete unloading path from the unloading point L ( x tun , ytun ) on the envelop curve to the stress-free point N ( x res , 0) is consist of two segments: the unloading path LM with initial stiffness (equal to 1 in the normalized coordinates) followed by the unloading path MN with the crack-closing stiffness kcrcl . The crack-closing stiffness kcrcl is assumed as a constant during the unloading process. The value of kcrcl is determined by the residual deformation x res and crack-closing compressive stress ytun , and the computational procedure will be introduced in the next section. The residual deformation x res is considered dependent on the previous unloading deformation x tun , and the relationship has been determined in a semi empirical way from the test results [16,19]. Based on the regression analysis, a quadratic polynomial formula has been
(7)
where k 0 is the initial stiffness in compression and equals to 1 in the normalized coordinates. The value of crack-closing compressive stress ytun is suggested as 1/3 in present model based on the test results [16]. It is noteworthy that the value of ytun is not a constant, and presents a negative relationship with the damage of concrete in compression. Further discussion can be found in the documents [2,18]. The crack-opening path is consisted of the line PO in compression
Fig. 5. Unloading and reloading paths for concrete in cyclic tension.
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Fig. 6. Relationship between the residual deformation and the unloading deformation (NS – Narrow specimen; WS – Wide specimen).
Fig. 8. Comparison between model predictions and test result in case 1.
Fig. 9. Comparison between model predictions and test result in case 2.
Fig. 7. Unloading and reloading paths for concrete in tension-compression reversals.
Table 1 Mechanical properties of concrete in the literatures. Compressive strength (N/ mm2)
Splitting strength (N/ mm2)
Direct tensile strength (N/ mm2)
Young’s modulus (N/ mm2)
Reinhardt [16]
47.1 (6.0%)*
3.20 (9.4%)
3.20 (9.7%)
Cornelissen et al. [17] Nouailletas et al. [19]
48.6 (6.0%)
3.66 (8.3%)
2.43 (8.6%)
61.4 (3.5%)
4.90 (10.2%)
–
39,270 (8.5%) 22,420 (6.1%) 37,900 (15.3%)
* Relative standard deviation.
stress region with the initial stiffness and the line OL in tension stress region with the crack-opening stiffness kcrop . The crack opening mainly generates in the loading process in tension and the stiffness kcrop is calculated as follows:
kcrop =
ytun x tun
Fig. 10. Comparison between model prediction and test data from Cornelissen et al. [17].
cases have been depicted in Fig. 7. 4. Model verification with test results
(8) Several results of uniaxial cyclic tests with a variety of loading histories including both cyclic tension and tension-compression reversals have been compared with the predictions obtained by means of the presented model. The tests were carried out by Reinhardt [16], Cornelissen et al. [17] and Nouailletas et al. [19]. The mechanical properties of the concrete specimens used in the literatures are
Partial unloading and reloading from the crack-closing and crackopening path have been considered in the model. The suggested partial unloading and reloading stiffness in compression stress region is equivalent to the initial stiffness and the stiffness in tension stress region is equivalent to the complete reloading stiffness ktre . All possible 392
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another straight line based model which is known as the focal point model provided by Yankelevsky and Reinhardt [7] is presented. Fig. 8 shows the case of cyclic tension with the lower stress equal to 5% of the tensile strength. Six cycles are selected to perform the comparison. It can be seen that the tension envelope curve and hysteretic behavior have been simulated well by the proposed model. And compared with the focal point model, the hysteretic loops obtained by the proposed model are closer to the experimental results. Fig. 9 shows the case of tension-compression reversals and the lower stress is compressive and amounts to 15% of the tensile strength. For a clear comparison of the unloading and reloading paths between the test and model results, four cycles have been selected. The unloading and reloading curves in the same cycle present significant difference in this case and large hysteretic loops have been observed. It can be noticed that the simulated hysteretic loops of both models are slightly bigger than the test results. Table 2 summarizes the dissipated energy obtained with the proposed model and is compared against experimental results and numerical results obtained by Yankelevsky and Reinhardt [7]. The value of dissipated energy during each cycle is obtained by calculating the area of the corresponding hysteretic loop. The relative error between numerical results and test results has been calculated for the two models. The comparison shows that the proposed model presents a better total dissipated energy prediction than the focal point model by Yankelevsky and Reinhardt [7]. In order to validate the applicability of the proposed model, other experimental results by Cornelissen et al. [17] and Nouailletas et al. [19] are reproduced and compared with the model results. Figs. 10 and 11 exhibit the case of tension-compression reversal with the lower stress equivalent to the tensile strength. Large amounts of energy are dissipated during the tension-compression reversal process in this case. It can be seen that the unloading and reloading curves predicted by the proposed model present good agreement with the test results as a whole. There is some deviation in the curves of the high compression stress region and it may be attributed to the accumulated irreversible tension strain.
Fig. 11. Comparison between model prediction and test data from Nouailletas et al. [19].
Table 2 Dissipated energy of the models and experiments. Cycle number
Experiment (10−3 N/mm)
Present model
Yankelevsky and Reinhardt
(10−3 N/mm)
Error (%)
(10−3 N/ mm)
Error (%)
Case 1 – Fig. 8 1 1.432 2 1.631 3 1.897 4 1.769 5 1.712 6 0.799 Total 9.241
0.847 1.410 1.717 1.686 1.632 1.178 8.470
−40.9 −13.6 −9.5 −4.7 −4.7 47.4 −8.3
1.323 2.039 2.195 2.151 1.793 1.008 10.510
−7.6 25.0 15.7 21.5 4.7 26.1 13.7
Case 2 – Fig. 9 1 1.833 2 3.851 3 5.426 4 9.888 Total 20.998
2.230 4.756 7.656 12.335 26.977
21.7 23.5 41.1 24.7 28.5
3.255 5.074 7.790 11.884 28.003
77.6 31.8 43.6 20.2 33.4
5. Seismic response analysis based on the hysteretic model The concrete structure subjected to seismic excitation usually presents hysteretic behavior in tension-compression reversals. However, the hysteretic model is still rarely used in predicting the seismic response of concrete structure and the research of the effect of hysteretic model on the seismic response result is lacking. In this section, seismic response analysis of a single degree of freedom (SDOF) system based on the proposed hysteretic model is performed. The motion equation in a dimensionless coordinates has been derived firstly based on the normalized deformation and resisting force. For a nonlinear SDOF system, the equation of motion is:
mu¨ + cu̇ + Fs = −mu¨ g
(9)
where m is the mass, c is the viscous damping coefficient, Fs is the resisting force, u is the deformation relative to the ground, and u¨ g is the acceleration of the ground. Divide the Eq. (9) by m to obtain:
u¨ + 2ωζu̇ +
ω2Fs = −u¨ g k
(10)
where k is the elastic stiffness of the system, ω = k / m is the undamped elastic circular frequency and ζ = c /2mω is the damping ratio. For concrete materials, ut is the deformation corresponding to the tensile strength ft , let:
Fig. 12. The feature of the damage and plastic-damage models in normalized coordinates.
summarized in Table 1. The original experimental curves are reproduced in the stress-displacement plane with gray dashed lines as shown in Figs. 8–11. The prediction curves for different tests are obtained by substituting the corresponding displacement history and material parameters into the proposed model. A comparison with
u=
u ut = xut ut
u̇ =
u̇ ̇ t ut = xu ut
Eq. (10) can be rewritten as: 393
u¨ =
u¨ ut = xu ¨ t ut
(11)
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Fig. 13. The EI Centro ground motion.
Fig. 14. The deformation time history curves of the three models.
ut x¨ + 2ωζut x ̇ +
ω2Fs = −u¨ g k
(12)
Fig. 16. The dissipation energy evolution of the hysteretic model and damage model.
And then divide the Eq. (12) by ut to obtain:
x¨ + 2ωζx ̇ +
ω2y
=−
x¨ + 2ωζx ̇ + ω2y = −
u¨ g ut
(15)
(13) It is clear from Eq. (15) that the seismic response of a SDOF system is determined by the parameter ω , ζ and Rs , considering the u 0 is determined by ω and ζ . For the linearly elastic systems, the Eq. (15) can be rewritten as:
where x = u/ ut is the normalized deformation, and y = Fs / kut = σs / ft is the normalized resisting force or stress as described above. For the nonlinear response analysis, the softening is considered and the strength reduction factor Rs is defined as:
x¨ + 2ωζx ̇ + ω2x = −
ft
u Rs = = t f0 u0
u¨ g Rs u 0
(14)
u¨ g u0
(16)
Based on the normalized motion equation, comparison of seismic response between different models can be carried out. Seismic response of a SDOF system with the proposed hysteretic model has been calculated and compared with the result of a damage model and plastic-damage model. The basic feature of the damage and plastic-damage models provided with the same tension envelope curve of the proposed model is presented in a normalized coordinates as shown in Fig. 12. For the damage model, there is no residual deformation when the tension stress unloads to zero. For the plastic-
where f0 and u 0 are the peak values of the earthquake induced resisting force and deformation, respectively, in the corresponding linear system. Rs is equal to 1 for linearly elastic systems and Rs less than 1 implies that the system is not strong enough to remain elastic during the ground motion. Such a system will soften and deform into the inelastic range. Given the value of Rs , the tensile strength ft and corresponding deformation ut can be determined. Substituting Eq. (14) into Eq. (13) gives:
Fig. 15. The normalized force-deformation responses: (a) plastic-damage model; (b) damage model; (c) hysteretic model.
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the seismic duration on the total dissipation energy, which is significant in evaluating the cumulative seismic damage of concrete structures.
damage model, residual deformation produces with the complete unloading in tension and remains constant during compression loading. The relationship between the residual deformation and unloading deformation in the plastic-damage model is assumed to be the same as that in the proposed model. Different from the hysteretic model, the reloading path is considered identical with the unloading path in the damage and plastic-damage models. Analysis is performed based on the SDOF system with parameters ω = 4π , ζ = 5% and Rs = 0.5. The selected seismic excitation is EI Centro ground motion as shown in Fig. 13. Substituting the kinetic parameters and normalized constitutive relationship into the Eq. (15), and solving it with the Newmark method, the time history of normalized deformation and dissipated energy can be obtained. Fig. 14 exhibits the normalized deformation response of the three models. The deformation time history of the proposed model is similar to that of the damage model, while the plastic-damage model presents a considerable irreversible deformation due to the assumption of plastic strain. The maximum deformation response of the plastic-damage model is about 1.39 times that of other two models. Fig. 15 displays the normalized force-deformation time history of the three models. For the plastic-damage model (Fig. 15(a)), it can be noticed that the plastic strain accumulates during the tension-compression reversals and the stiffness recovers as soon as the stress enters into compressive region. A number of hysteretic loops are observed in the result of the hysteretic model (Fig. 15(c)). However, no hysteretic loop is observed during the tension-compression reversals in the results of the damage and plastic-damage models (Fig. 15(a) and (b)). The amount of energy dissipated during seismic excitation plays a significant role in the seismic damage assessment of concrete structure. Fig. 16 shows the dissipation energy evolution history of the proposed model and damage model. It can be seen that the total dissipation energy of the hysteretic model is almost three times that of the damage model. The additional energy is obviously caused by the hysteretic loops observed in Fig. 15(c). The seismic acceleration time history curve has been reproduced in Fig. 16 for further analysis. It can be seen that the energy dissipated by the damage model remains constant after the peak value of the seismic acceleration, which means that the post-peak seismic load has no effect on the final result of the model. For the hysteretic model, the dissipated energy continues to increase after the peak value of seismic acceleration and the whole seismic load history can be considered in the form of accumulated dissipation energy. It can be concluded that compared with the damage model, the proposed hysteretic model is able to consider the effect of the seismic duration on the total dissipation energy, and it is significant in evaluating the cumulative damage of concrete structure subjected to the seismic load, especially the earthquake with long duration.
Acknowledgements This study is supported by the National Natural Science Foundation of China (51739006, U1765204, 51679078) and Postgraduate Research & Practice Innovation Program of Jiangsu Province of China (Project number: 2017B661X14). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.engstruct.2018.02.051. References [1] Scott BD. Stress-strain behavior of concrete by overlapping hoops at low and high strain rates. ACI J 1982;79(1):13–27. [2] Sima JF, Roca P, Molins C. Cyclic constitutive model for concrete. Eng Struct 2008;30(3):695–706. [3] Foster SJ, Marti P. Cracked membrane model: finite element implementation. J Struct Eng 2003;129(9):1155–63. [4] Yassin MHM. Nonlinear analysis of prestressed concrete structures under monotonic and cyclic loads. Dissertation. Berkeley, California: University of California; 1994. [5] Vecchio FJ, Palermo D. Compression field modeling of reinforced concrete subjected to reversed loading: verification. ACI Struct J 2004;100(5):155–64. [6] Reinhardt HW, Cornelissen HAW, Hordjil DA. Tensile tests and failure analysis of concrete. J Struct Eng 1986;112(11):2462–77. [7] Yankelevsky DZ, Reinhardt HW. Uniaxial behavior of concrete in cyclic tension. J Struct Eng 1989;115(1):166–82. [8] Chang GA, Mander JB. Seismic energy based fatigue damage analysis of bridge columns: Part 1 – Evaluation of seismic capacity; 1994. [9] Vecchio FJ. Towards cyclic load modeling of reinforced concrete. ACI Struct J 1999;96(2):132–202. [10] Gopalaratnam. Softening response of plain concrete in direct tension. ACI Mater J 1985;82(3):310–23. [11] Sinha BP, Gerstle KH, Tulin LG. Stress-strain relations for concrete under cyclic loading. Journal Proceedings 1964;61(2):195–212. [12] Karsan ID, Jirsa JO. Behavior of concrete under compressive loading. J Struct Div 1969;95(12):2543–64. [13] Spooner DC, Young AG, Dougill JW. A quantitative assessment of damage sustained in concrete during compressive loading. Mag Concr Res 1976;28(96):168–9. [14] Okamoto S, Shiomi S, Yamabe K. Earthquake resistance of prestressed concrete structures. Proc Annual Convention AIJ 1976:1251–2. [15] Tanigawa Y, Uchida Y. Hysteretic characteristics of concrete in the domain of high compressive strain. Proc Annual Convention AIJ 1979:449–50. [16] Reinhardt HW. Fracture mechanics of an elastic softening material like concrete. Delft University of Technology; 1984(2). [17] Cornelissen HAW, Hordijk DA, Reinhardt HW. Experimental determination of crack softening characteristics of normalweight and lightweight concrete. Delft University of Technology; 1986(2). [18] Mazars J, Berthaud Y, Ramtani S. The unilateral behaviour of damaged concrete. Eng Fract Mech 1990;35(4):629–35. [19] Nouailletas O, Borderie CL, Perlot C, et al. Experimental study of crack closure on heterogeneous quasi-brittle material. J Eng Mech 2015;141(11):04015041. [20] Légeron F, Paultre P, Mazars J. Damage mechanics modeling of nonlinear seismic behavior of concrete structures. J Struct Eng 2005;131(6):946–55. [21] Wu JY, Li J, Rui F. An energy release rate-based plastic-damage model for concrete. Int J Solids Struct 2006;43(3):583–612. [22] Lee J, Fenves GL. Plastic-damage model for cyclic loading of concrete structures. J Eng Mech 1998;124(8):892–900. [23] Bažant Zdeněk P, Oh BH. Crack band theory for fracture of concrete. Matériaux Et Construction 1983;16(3):155–77. [24] Rots JG, Nauta PG, Kuster GMA, et al. Smeared crack approach and fracture localization in concrete. Delft University of Technology; 1985(1). [25] Lin CS, Scordelis AC. Nonlinear analysis of RC shells of general form. J Geol Soc Jpn 1975;101(12):965–70. [26] Cornelissen HAW. Experiments and theory for the application of fracture mechanics to normal and lightweight concrete. Fracture Toughness Fracture Energy Concr 1986:565–75. [27] Guo Z, Zhang X. Experimental investigation of complete stress-deformation curves of concrete in tension. J Build Struct 1988;84(4):278–85. [28] Chen X, Xu L, Bu J. Experimental study and constitutive model on complete stressstrain relations of plain concrete in uniaxial cyclic tension. KSCE J Civ Eng 2015;79(1):1–7.
6. Conclusion This paper presented a comprehensible model for predicting the hysteretic behavior of concrete under cyclic tension and tension-compression reversals. The model is introduced in the frame of a dimensionless stress-deformation coordinates. Complete and partial unloading and reloading paths are considered based on the crack closingopening mechanism. The model can be used to predict the complex hysteretic behaviors of concrete under cyclic tension and tensioncompression reversals in a simple and practical way. The proposed model has been validated by comparison with available experiments and shows satisfactory agreement with the experimental results in all cases. Comparisons with the focal point model have been carried out and the proposed model presents improvement on the total dissipation energy prediction. Through the seismic response analysis, the proposed model proved to be capable to consider the effect of
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Hysteretic model for concrete under cyclic tension and tension-compression reversals Pei Zhang, Qingwen Ren, Dong Lei
T
⁎
College of Mechanics and Materials, Hohai University, Nanjing 210098, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Hysteretic model Concrete Cyclic tension Stress reversals Seismic response
A hysteretic model for concrete structure subjected to cyclic tension and tension–compression reversals is presented. The proposed model was intended to predict the complex hysteretic behavior of concrete under cyclic loading in a simple and practical way. Based on the analysis of the characteristic hysteretic behavior of concrete, the residual deformation in tension was considered principally due to the incomplete closure of the opening cracks. The mechanism for the hysteretic behavior of concrete under tension–compression reversals was suggested as the crack closing and opening. Considering the application within different numerical approaches, dimensionless stress-deformation coordinates was adopted to perform the hysteretic model. The unloading and reloading paths have been derived from the crack closing and opening mechanism and were represented as straight lines in the model. Partial unloading and reloading were considered in both cyclic tension and tension–compression reversals. The proposed model has been validated by comparison with available experimental results and the seismic response of a SDOF system with the hysteretic model has been analysed.
1. Introduction The safety assessment of concrete structures subjected to cyclic loading such as seismic excitation requires realistic constitutive models to reproduce the real behavior of the materials. Because of the low tensile strength, the concrete subjected to seismic load usually presents softening behavior in tension and hysteretic behavior in tension-compression reversals. As a result, the hysteretic model for concrete plays a significant role in determining the seismic responses of concrete structures including the deformation and energy evolution. However, for lack of experimental data, there are few specialized researches on the modeling of hysteretic behavior for concrete under cyclic tension and tension-compression reversals. Most existing models for concrete considering the cyclic loading in tension assumed linear unloading-reloading paths without hysteretic energy dissipation [1–4]. Some authors (Vecchio and Palermo [5], Reinhardt et al. [6], Yankelevsky and Reinhardt [7], Chang and Mander [8]) have proposed more advanced models considering the complete and partial unloading-reloading hysteretic behaviors with different modeling approaches. Vecchio and Palermo [5] presented constitutive formulations for concrete subjected to reversed cyclic loading consistent with a compression field approach. The model was built upon the preliminary work presented by Vecchio [9] and intended to apply in the context of
⁎
smeared rotating cracks. The hysteretic rules used in cyclic tension followed the philosophy for concrete in compression. The unloading path was modeled with a Ramberg–Osgood formulation and the reloading path was modeled as a straight line with degrading reloading stiffness. A plastic offset during complete unloading in tension has been defined in the model and formulated based on the test data from Yankelevsky and Reinhardt [7] and Gopalaratnam [10]. Although the crack-closing process in compression loading has been described with a linear formulation in the literature, the hysteretic behavior in tensioncompression reversals has been neglected. Reinhardt et al. [6] proposed a relationship between the stress and crack opening displacement for concrete in the tension and compression region. The total deformation during cyclic load was split up into a crack opening displacement part and a strain part consisting of an elastic strain and an irreversible strain, and the irreversible strain part was neglected in the model. As a result, the uncracked material behaved in a liner manner and all nonlinearities were comprised in the crack. The model can be applied in numerical simulation through a discrete crack approach or smeared crack approach. Straight lines were used as the unloading and reloading paths. Because the crack opening displacement was assumed constant during the unloading process in tension, the hysteretic behavior in cyclic tension cannot be simulated by this model. Yankelevsky and Reinhardt [7] developed a stress versus total
Corresponding author. E-mail address: [email protected] (D. Lei).
https://doi.org/10.1016/j.engstruct.2018.02.051 Received 12 August 2017; Received in revised form 5 January 2018; Accepted 16 February 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.
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Nomenclature
u0
c f0
x 0 xclose
ft Fs k k0 kcrcl kcrop ktpre m Rs u u¨ g
viscous damping coefficient maximum of the earthquake induced resisting force in a linear system tensile strength of concrete resisting force elastic stiffness of SDOF system normalized elastic stiffness in compression normalized crack-closing stiffness normalized crack-opening stiffness normalized partial reloading stiffness quality strength reduction factor deformation relative to the ground ground acceleration
x res x tpre x tun y ytun ytpre ytun δt εt ζ ω
maximum of the earthquake induced deformation in a linear system normalized strain normalized strain corresponding to crack-closing compressive stress normalized residual tensile strain normalized partial reloading strain normalized unloading strain on tension envelope curve normalized stress normalized crack-closing compressive stress normalized partial reloading stress normalized unloading stress on tension envelope curve tensile displacement corresponding to the tensile strength tensile strain corresponding to the tensile strength damping ratio undamped elastic circular frequency
reversals is presented. Compared to previous ones, the model presents several advantages. It affords to consider the essential features of the complex hysteretic behavior of concrete in a simple and practical way. It can be used to simulate the complete or partial unloading and reloading behaviors for concrete under cyclic tension and tension-compression reversals. Straight lines are adopted to describe the unloading and reloading paths and several necessary hysteretic rules are proposed based on the mechanism of crack closing and opening. Furthermore, all the required input parameters can be obtained through conventional laboratory monotonic tension tests. The model has been validated by comparison with available experimental results in different cases and the analysis of seismic response based on the hysteretic model has been performed.
deformation relationship for concrete behavior in cyclic tension and compression. The model was based on a given experimental cyclic stress-deformation envelope, and has defined focal points which were used to reproduce the complete unloading-reloading cycles. The focal points governed the unloading and reloading curves either by rays transmitted from a certain focal point towards known points in the stress-deformation plane, or by their stress level. When all the focal points were located, the complete unloading-reloading curves can be produced through a simple graphical process. Although the model represented well the test results, the definition of focal point purely derived from the graphic feature needs more physical significance and the procedure determining the unloading and reloading curves is too complex. Chang and Mander [8] proposed a rule-based hysteretic model to simulate the hysteretic behavior of confined and unconfined concrete in both cyclic compression and tension for both ordinary as well as high strength concrete. Fifteen unloading and reloading paths determined by fifteen different rules were distinguished in the model. The fifteen paths were divided into three types: envelope curve, connecting curve and transition curve. The equation used by the authors for the unloading and reloading curves was a polynomial adjusted by a series of parameters: the slope at the origin and the slope at the end of each curve. To determine the parameters of cyclic curves for concrete in compression, statistical regression analysis was performed on the experimental data from Sinha et al. [11], Karsan and Jirsa [12], Spooner and Dougill [13], Okamoto et al. [14] and Tanigawa et al. [15]. The expressions proposed for compression have been modified by the authors for the condition of tension cyclic behavior. In the documented literatures, the most refereed experimental studies on the uniaxial tensile cyclic behavior of concrete are from Reinhardt [16], Cornelissen et al. [17] and Mazars et al. [18]. More recently, Nouailletas et al. [19] have performed direct cyclic tension tests on the concrete specimens, and the effect of crack reclosing on properties of concrete has been studied at the macroscale using the digital image correlation (DIC) technique. At present, the hysteretic model considering the hysteretic behavior during cyclic tension and tension-compression reversals is still rarely used in the seismic response analysis of concrete structure. The principal shortcoming of the available hysteretic models for concrete in the literatures is the complicated hysteretic rules applied to reproduce the unloading and reloading curves. These rules are usually derived from the geometrical properties of the cyclic stress-strain curves, which results in the lack of a clear physical meaning of the proposed model. A set of parameters is required to perform the complex rules and it reduces the applicability of the hysteretic model. In this paper, an efficient model capable of predicting the hysteretic behavior of concrete under cyclic tension and tension-compression
2. Characteristic behaviors of concrete in tension-compression reversals Before the introduction of the hysteretic model, it is necessary to describe the characteristic behaviors of concrete in tension-compression reversals. Direct tension cyclic tests with different stress ranges on a double-notched specimen have been performed by Reinhardt [16] in 1984 and the corresponding stress-deformation relationships were obtained. The deformation was defined as relative displacement and measured by four extensometers with 35 mm gauge length. One of the test results with complete hysteretic loops has been selected to study the feature of concrete behaviors in tension-compression reversals. Fig. 1 shows the reproduced three successive unloading-reloading cycles. The positive stress means the tension and the unloading is
Fig. 1. Three successive unloading-reloading cycles under tension-compression reversals.
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deformation coordinates is applied to present the proposed hysteretic model:
referring to the deformation decrease process. In the unloading stage, the specimen was unloaded from the tension softening branch to the compressive stress zone. Considerable residual deformation was observed when the tensile stress unloaded to zero, and it was recognized as the irreversible plastic strain by many researchers [20–22]. However, different opinions were proposed by some authors [5,19]. The residual deformation was reconsidered attributed to the cracked surfaces come into contact and do not realign due to shear slip along the cracked surfaces. And the evolution of the unloading curve was explained by friction phenomena generated by the mismatching discontinuity lips. In present paper, the generation of the residual deformation in tension is considered principally due to the crack incomplete closure rather than the plastic strain and the unloading process can essentially be regarded as a crack closure process. This interpretation can be verified by the fact that during the compression phase the residual strain progressively vanishes and the material recovers its initial stiffness (Fig. 1). The recovery of stiffness in compression has been accepted as the “unilateral effect” in plastic damage theory and it is actually caused by the complete closure of cracks. In the reloading stage, the deformation grew fast during the tension phase as shown in Fig. 1, and it can be deduced that the reloading curve is due to the reopening of the previous closing cracks. Fig. 2 shows the evolution of unloading curve stiffness during the three cycles normalized by the elastic stiffness. It can be seen that the evolution of unloading stiffness exhibits three different stages. The first stage corresponds to the initial unloading in tension where the stiffness is large and decreases in approximate linearity with the deformation. At the second stage, the stress unloads to zero and reverses to the compression zone. Although the stress direction changes, the stiffness remains constant in this stage, indicating the gradual closing process of the cracks. At the last stage, the stiffness increases rapidly with the decrease of deformation due to the growth of compression stress. Fig. 3 shows the evolution of the normalized reloading stiffness which can be divided into two stages. At the first stage, the compression stress reduces to zero and the stiffness decreases approximately linearly with the increase of deformation. At the second stage, the stiffness remains constant in a low level before the deformation reaches the previous unloading value. Based on the characteristic behaviors of concrete under tensioncompression reversals described above, some assumptions can be made as follows:
x=
ε δ σ = , y= εt δt ft
(1)
where y is the dimensionless stress normalized by the concrete tensile strength ft ; x is the dimensionless deformation normalized by the strain εt or displacement δt corresponding to the concrete tensile strength. Based on the normalized stress-deformation model, the corresponding constitutive relationships applicable for the discrete or smeared crack model can be easily obtained.
3.2. Tension envelope curve for concrete It is commonly accepted that the envelope curve for concrete subjected to axial cyclic load can be approximated by that under monotonic load. The envelope curve for concrete under monotonic tension has been studies by many researches and is usually expressed by a piecewise function consisting of an ascent branch before the peak and a descent branch after the peak. In the pre-peak branch, a linear elastic relationship represents well the concrete behavior and it has been accepted by most researchers. In the postpeak branch, several expressions were documented in the literature, including straight lines [23], polylinear curves [24], exponential curves [5], polynomial curves [25] or combinations of them [26]. The complete stress-deformation curve for concrete under monotonic tension has been experimentally investigated by Guo and Zhang [27] based on twenty-nine direct tension tests. A rational fraction expression for the descending branch was obtained in the literature and was used in the present model. The complete expression of the adopted tension envelope curve for concrete can be written in the dimensionless form as follows:
(x ⩽ 1) ⎧y = x x ⎨ y = αt (x − 1)1.7 + x (x ⩾ 1) ⎩
(2)
where αt is a materials parameter and determined by the concrete tensile strength as:
αt = 0.312ft2
1. The mechanism of the hysteretic behavior of concrete under tensioncompression reversals is the crack closing and opening. 2. The stiffness of unloading curve remains constant in the low tension and compression stress region. 3. The stiffness of reloading curve remains constant in the tension stress region.
(3)
Different tension envelope curves with various αt have been depicted as shown in Fig. 4. It can be seen that the concrete with high tensile strength exhibits sharp fall in the softening branch, which representing high brittleness.
These assumptions will be used to determine the unloading and reloading paths in the hysteretic model. 3. Proposed hysteretic model for concrete 3.1. Normalized coordinate system for present model The failure of concrete subjected to tensile force takes place in local fracture and is therefore a discontinuous phenomenon. The local crack in concrete can be modeled as a discrete crack or a smeared crack in numerical simulation and many constitutive relationships have been proposed for the application of various models [2–3,5–7]. However, the above-mentioned constitutive relationships usually adopted different forms of deformation, such as crack open displacement, relative displacement or strain, which limited the applicability of the corresponding models. In order to improve the model applicability, a dimensionless stress-
Fig. 2. The evolution of normalized unloading stiffness in three cycles.
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obtained to represent the relationship between x res and x tun as shown in Fig. 6. The complete reloading path from the stress-free point N is proposed as a straight line directed to the previous unloading point L as shown in Fig. 5. The change of reloading stiffness before returning to the envelop curve has been neglected for simplicity. Based on the geometric feature, the complete reloading stiffness ktre can be calculated as:
ktre =
ytun x tun−x res
(4)
Partial unloading and reloading under cyclic tension are also considered in the model. The partial reloading path starts from a partial reloading point ( x tpre , ytpre ) with nonzero stress and is directed to the previous unloading point on the envelop curve (Fig. 5). Based on the rule, the partial reloading stiffness ktpre can be calculated as:
ktpre =
Fig. 3. The evolution of normalized reloading stiffness in three cycles.
ytun −ytpre x tun−x tpre
(5)
The partial unloading activates when the deformation decreases during complete reloading or partial reloading process, and the partial unloading stiffness is suggested equal to the initial stiffness. 3.4. Crack closing and opening path in tension-compression reversals The hysteretic behavior for concrete in tension-compression reversals is represented by a crack-closing path and a crack-opening path as shown in Fig. 7. The crack-closing path depicted as line MP is determined by the residual deformation x res and crack-closing compressive stress ytun . The value of ytun represents the compressive stress level corresponding to the full recovery of compression stiffness or complete closure of the cracks. Based on the schematic, the stiffness of crackclosing path can be calculated as:
kcrcl =
Fig. 4. Tension envelope curves for concrete with different tensile strength.
0 yclose 0 xclose−x res
(6)
0 xclose
where is the deformation corresponding to the crack-closing compressive stress ytun and calculated as:
3.3. Unloading-reloading path in cyclic tension
0 0 xclose = yclose /k 0
As it was observed in many experiments [15,16,18,28], concrete subjected to cyclic tension loading exhibited hysteretic behaviors. In a typical cyclic test in tension, the stiffness of unloading curve was usually very high at the beginning and then dropped to a stable value as the tension stress decreased. Residual deformation can be observed if the tension stress unloaded to zero. When reversely reloaded in tension, the reloading stiffness exhibited almost constant and a hysteretic loop would be formed during the unloading-reloading process. In the present model, the hysteretic behavior is simulated by series straight lines whose stiffness is determined in a semi-empirical way. Several necessary hysteretic paths are proposed based on the assumptions summarized at the end of Section 2. Fig. 5 shows the possible unloading-reloading paths for concrete under cyclic tension. The complete unloading path from the unloading point L ( x tun , ytun ) on the envelop curve to the stress-free point N ( x res , 0) is consist of two segments: the unloading path LM with initial stiffness (equal to 1 in the normalized coordinates) followed by the unloading path MN with the crack-closing stiffness kcrcl . The crack-closing stiffness kcrcl is assumed as a constant during the unloading process. The value of kcrcl is determined by the residual deformation x res and crack-closing compressive stress ytun , and the computational procedure will be introduced in the next section. The residual deformation x res is considered dependent on the previous unloading deformation x tun , and the relationship has been determined in a semi empirical way from the test results [16,19]. Based on the regression analysis, a quadratic polynomial formula has been
(7)
where k 0 is the initial stiffness in compression and equals to 1 in the normalized coordinates. The value of crack-closing compressive stress ytun is suggested as 1/3 in present model based on the test results [16]. It is noteworthy that the value of ytun is not a constant, and presents a negative relationship with the damage of concrete in compression. Further discussion can be found in the documents [2,18]. The crack-opening path is consisted of the line PO in compression
Fig. 5. Unloading and reloading paths for concrete in cyclic tension.
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Fig. 6. Relationship between the residual deformation and the unloading deformation (NS – Narrow specimen; WS – Wide specimen).
Fig. 8. Comparison between model predictions and test result in case 1.
Fig. 9. Comparison between model predictions and test result in case 2.
Fig. 7. Unloading and reloading paths for concrete in tension-compression reversals.
Table 1 Mechanical properties of concrete in the literatures. Compressive strength (N/ mm2)
Splitting strength (N/ mm2)
Direct tensile strength (N/ mm2)
Young’s modulus (N/ mm2)
Reinhardt [16]
47.1 (6.0%)*
3.20 (9.4%)
3.20 (9.7%)
Cornelissen et al. [17] Nouailletas et al. [19]
48.6 (6.0%)
3.66 (8.3%)
2.43 (8.6%)
61.4 (3.5%)
4.90 (10.2%)
–
39,270 (8.5%) 22,420 (6.1%) 37,900 (15.3%)
* Relative standard deviation.
stress region with the initial stiffness and the line OL in tension stress region with the crack-opening stiffness kcrop . The crack opening mainly generates in the loading process in tension and the stiffness kcrop is calculated as follows:
kcrop =
ytun x tun
Fig. 10. Comparison between model prediction and test data from Cornelissen et al. [17].
cases have been depicted in Fig. 7. 4. Model verification with test results
(8) Several results of uniaxial cyclic tests with a variety of loading histories including both cyclic tension and tension-compression reversals have been compared with the predictions obtained by means of the presented model. The tests were carried out by Reinhardt [16], Cornelissen et al. [17] and Nouailletas et al. [19]. The mechanical properties of the concrete specimens used in the literatures are
Partial unloading and reloading from the crack-closing and crackopening path have been considered in the model. The suggested partial unloading and reloading stiffness in compression stress region is equivalent to the initial stiffness and the stiffness in tension stress region is equivalent to the complete reloading stiffness ktre . All possible 392
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another straight line based model which is known as the focal point model provided by Yankelevsky and Reinhardt [7] is presented. Fig. 8 shows the case of cyclic tension with the lower stress equal to 5% of the tensile strength. Six cycles are selected to perform the comparison. It can be seen that the tension envelope curve and hysteretic behavior have been simulated well by the proposed model. And compared with the focal point model, the hysteretic loops obtained by the proposed model are closer to the experimental results. Fig. 9 shows the case of tension-compression reversals and the lower stress is compressive and amounts to 15% of the tensile strength. For a clear comparison of the unloading and reloading paths between the test and model results, four cycles have been selected. The unloading and reloading curves in the same cycle present significant difference in this case and large hysteretic loops have been observed. It can be noticed that the simulated hysteretic loops of both models are slightly bigger than the test results. Table 2 summarizes the dissipated energy obtained with the proposed model and is compared against experimental results and numerical results obtained by Yankelevsky and Reinhardt [7]. The value of dissipated energy during each cycle is obtained by calculating the area of the corresponding hysteretic loop. The relative error between numerical results and test results has been calculated for the two models. The comparison shows that the proposed model presents a better total dissipated energy prediction than the focal point model by Yankelevsky and Reinhardt [7]. In order to validate the applicability of the proposed model, other experimental results by Cornelissen et al. [17] and Nouailletas et al. [19] are reproduced and compared with the model results. Figs. 10 and 11 exhibit the case of tension-compression reversal with the lower stress equivalent to the tensile strength. Large amounts of energy are dissipated during the tension-compression reversal process in this case. It can be seen that the unloading and reloading curves predicted by the proposed model present good agreement with the test results as a whole. There is some deviation in the curves of the high compression stress region and it may be attributed to the accumulated irreversible tension strain.
Fig. 11. Comparison between model prediction and test data from Nouailletas et al. [19].
Table 2 Dissipated energy of the models and experiments. Cycle number
Experiment (10−3 N/mm)
Present model
Yankelevsky and Reinhardt
(10−3 N/mm)
Error (%)
(10−3 N/ mm)
Error (%)
Case 1 – Fig. 8 1 1.432 2 1.631 3 1.897 4 1.769 5 1.712 6 0.799 Total 9.241
0.847 1.410 1.717 1.686 1.632 1.178 8.470
−40.9 −13.6 −9.5 −4.7 −4.7 47.4 −8.3
1.323 2.039 2.195 2.151 1.793 1.008 10.510
−7.6 25.0 15.7 21.5 4.7 26.1 13.7
Case 2 – Fig. 9 1 1.833 2 3.851 3 5.426 4 9.888 Total 20.998
2.230 4.756 7.656 12.335 26.977
21.7 23.5 41.1 24.7 28.5
3.255 5.074 7.790 11.884 28.003
77.6 31.8 43.6 20.2 33.4
5. Seismic response analysis based on the hysteretic model The concrete structure subjected to seismic excitation usually presents hysteretic behavior in tension-compression reversals. However, the hysteretic model is still rarely used in predicting the seismic response of concrete structure and the research of the effect of hysteretic model on the seismic response result is lacking. In this section, seismic response analysis of a single degree of freedom (SDOF) system based on the proposed hysteretic model is performed. The motion equation in a dimensionless coordinates has been derived firstly based on the normalized deformation and resisting force. For a nonlinear SDOF system, the equation of motion is:
mu¨ + cu̇ + Fs = −mu¨ g
(9)
where m is the mass, c is the viscous damping coefficient, Fs is the resisting force, u is the deformation relative to the ground, and u¨ g is the acceleration of the ground. Divide the Eq. (9) by m to obtain:
u¨ + 2ωζu̇ +
ω2Fs = −u¨ g k
(10)
where k is the elastic stiffness of the system, ω = k / m is the undamped elastic circular frequency and ζ = c /2mω is the damping ratio. For concrete materials, ut is the deformation corresponding to the tensile strength ft , let:
Fig. 12. The feature of the damage and plastic-damage models in normalized coordinates.
summarized in Table 1. The original experimental curves are reproduced in the stress-displacement plane with gray dashed lines as shown in Figs. 8–11. The prediction curves for different tests are obtained by substituting the corresponding displacement history and material parameters into the proposed model. A comparison with
u=
u ut = xut ut
u̇ =
u̇ ̇ t ut = xu ut
Eq. (10) can be rewritten as: 393
u¨ =
u¨ ut = xu ¨ t ut
(11)
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Fig. 13. The EI Centro ground motion.
Fig. 14. The deformation time history curves of the three models.
ut x¨ + 2ωζut x ̇ +
ω2Fs = −u¨ g k
(12)
Fig. 16. The dissipation energy evolution of the hysteretic model and damage model.
And then divide the Eq. (12) by ut to obtain:
x¨ + 2ωζx ̇ +
ω2y
=−
x¨ + 2ωζx ̇ + ω2y = −
u¨ g ut
(15)
(13) It is clear from Eq. (15) that the seismic response of a SDOF system is determined by the parameter ω , ζ and Rs , considering the u 0 is determined by ω and ζ . For the linearly elastic systems, the Eq. (15) can be rewritten as:
where x = u/ ut is the normalized deformation, and y = Fs / kut = σs / ft is the normalized resisting force or stress as described above. For the nonlinear response analysis, the softening is considered and the strength reduction factor Rs is defined as:
x¨ + 2ωζx ̇ + ω2x = −
ft
u Rs = = t f0 u0
u¨ g Rs u 0
(14)
u¨ g u0
(16)
Based on the normalized motion equation, comparison of seismic response between different models can be carried out. Seismic response of a SDOF system with the proposed hysteretic model has been calculated and compared with the result of a damage model and plastic-damage model. The basic feature of the damage and plastic-damage models provided with the same tension envelope curve of the proposed model is presented in a normalized coordinates as shown in Fig. 12. For the damage model, there is no residual deformation when the tension stress unloads to zero. For the plastic-
where f0 and u 0 are the peak values of the earthquake induced resisting force and deformation, respectively, in the corresponding linear system. Rs is equal to 1 for linearly elastic systems and Rs less than 1 implies that the system is not strong enough to remain elastic during the ground motion. Such a system will soften and deform into the inelastic range. Given the value of Rs , the tensile strength ft and corresponding deformation ut can be determined. Substituting Eq. (14) into Eq. (13) gives:
Fig. 15. The normalized force-deformation responses: (a) plastic-damage model; (b) damage model; (c) hysteretic model.
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the seismic duration on the total dissipation energy, which is significant in evaluating the cumulative seismic damage of concrete structures.
damage model, residual deformation produces with the complete unloading in tension and remains constant during compression loading. The relationship between the residual deformation and unloading deformation in the plastic-damage model is assumed to be the same as that in the proposed model. Different from the hysteretic model, the reloading path is considered identical with the unloading path in the damage and plastic-damage models. Analysis is performed based on the SDOF system with parameters ω = 4π , ζ = 5% and Rs = 0.5. The selected seismic excitation is EI Centro ground motion as shown in Fig. 13. Substituting the kinetic parameters and normalized constitutive relationship into the Eq. (15), and solving it with the Newmark method, the time history of normalized deformation and dissipated energy can be obtained. Fig. 14 exhibits the normalized deformation response of the three models. The deformation time history of the proposed model is similar to that of the damage model, while the plastic-damage model presents a considerable irreversible deformation due to the assumption of plastic strain. The maximum deformation response of the plastic-damage model is about 1.39 times that of other two models. Fig. 15 displays the normalized force-deformation time history of the three models. For the plastic-damage model (Fig. 15(a)), it can be noticed that the plastic strain accumulates during the tension-compression reversals and the stiffness recovers as soon as the stress enters into compressive region. A number of hysteretic loops are observed in the result of the hysteretic model (Fig. 15(c)). However, no hysteretic loop is observed during the tension-compression reversals in the results of the damage and plastic-damage models (Fig. 15(a) and (b)). The amount of energy dissipated during seismic excitation plays a significant role in the seismic damage assessment of concrete structure. Fig. 16 shows the dissipation energy evolution history of the proposed model and damage model. It can be seen that the total dissipation energy of the hysteretic model is almost three times that of the damage model. The additional energy is obviously caused by the hysteretic loops observed in Fig. 15(c). The seismic acceleration time history curve has been reproduced in Fig. 16 for further analysis. It can be seen that the energy dissipated by the damage model remains constant after the peak value of the seismic acceleration, which means that the post-peak seismic load has no effect on the final result of the model. For the hysteretic model, the dissipated energy continues to increase after the peak value of seismic acceleration and the whole seismic load history can be considered in the form of accumulated dissipation energy. It can be concluded that compared with the damage model, the proposed hysteretic model is able to consider the effect of the seismic duration on the total dissipation energy, and it is significant in evaluating the cumulative damage of concrete structure subjected to the seismic load, especially the earthquake with long duration.
Acknowledgements This study is supported by the National Natural Science Foundation of China (51739006, U1765204, 51679078) and Postgraduate Research & Practice Innovation Program of Jiangsu Province of China (Project number: 2017B661X14). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.engstruct.2018.02.051. References [1] Scott BD. Stress-strain behavior of concrete by overlapping hoops at low and high strain rates. ACI J 1982;79(1):13–27. [2] Sima JF, Roca P, Molins C. Cyclic constitutive model for concrete. Eng Struct 2008;30(3):695–706. [3] Foster SJ, Marti P. Cracked membrane model: finite element implementation. J Struct Eng 2003;129(9):1155–63. [4] Yassin MHM. Nonlinear analysis of prestressed concrete structures under monotonic and cyclic loads. Dissertation. Berkeley, California: University of California; 1994. [5] Vecchio FJ, Palermo D. Compression field modeling of reinforced concrete subjected to reversed loading: verification. ACI Struct J 2004;100(5):155–64. [6] Reinhardt HW, Cornelissen HAW, Hordjil DA. Tensile tests and failure analysis of concrete. J Struct Eng 1986;112(11):2462–77. [7] Yankelevsky DZ, Reinhardt HW. Uniaxial behavior of concrete in cyclic tension. J Struct Eng 1989;115(1):166–82. [8] Chang GA, Mander JB. Seismic energy based fatigue damage analysis of bridge columns: Part 1 – Evaluation of seismic capacity; 1994. [9] Vecchio FJ. Towards cyclic load modeling of reinforced concrete. ACI Struct J 1999;96(2):132–202. [10] Gopalaratnam. Softening response of plain concrete in direct tension. ACI Mater J 1985;82(3):310–23. [11] Sinha BP, Gerstle KH, Tulin LG. Stress-strain relations for concrete under cyclic loading. Journal Proceedings 1964;61(2):195–212. [12] Karsan ID, Jirsa JO. Behavior of concrete under compressive loading. J Struct Div 1969;95(12):2543–64. [13] Spooner DC, Young AG, Dougill JW. A quantitative assessment of damage sustained in concrete during compressive loading. Mag Concr Res 1976;28(96):168–9. [14] Okamoto S, Shiomi S, Yamabe K. Earthquake resistance of prestressed concrete structures. Proc Annual Convention AIJ 1976:1251–2. [15] Tanigawa Y, Uchida Y. Hysteretic characteristics of concrete in the domain of high compressive strain. Proc Annual Convention AIJ 1979:449–50. [16] Reinhardt HW. Fracture mechanics of an elastic softening material like concrete. Delft University of Technology; 1984(2). [17] Cornelissen HAW, Hordijk DA, Reinhardt HW. Experimental determination of crack softening characteristics of normalweight and lightweight concrete. Delft University of Technology; 1986(2). [18] Mazars J, Berthaud Y, Ramtani S. The unilateral behaviour of damaged concrete. Eng Fract Mech 1990;35(4):629–35. [19] Nouailletas O, Borderie CL, Perlot C, et al. Experimental study of crack closure on heterogeneous quasi-brittle material. J Eng Mech 2015;141(11):04015041. [20] Légeron F, Paultre P, Mazars J. Damage mechanics modeling of nonlinear seismic behavior of concrete structures. J Struct Eng 2005;131(6):946–55. [21] Wu JY, Li J, Rui F. An energy release rate-based plastic-damage model for concrete. Int J Solids Struct 2006;43(3):583–612. [22] Lee J, Fenves GL. Plastic-damage model for cyclic loading of concrete structures. J Eng Mech 1998;124(8):892–900. [23] Bažant Zdeněk P, Oh BH. Crack band theory for fracture of concrete. Matériaux Et Construction 1983;16(3):155–77. [24] Rots JG, Nauta PG, Kuster GMA, et al. Smeared crack approach and fracture localization in concrete. Delft University of Technology; 1985(1). [25] Lin CS, Scordelis AC. Nonlinear analysis of RC shells of general form. J Geol Soc Jpn 1975;101(12):965–70. [26] Cornelissen HAW. Experiments and theory for the application of fracture mechanics to normal and lightweight concrete. Fracture Toughness Fracture Energy Concr 1986:565–75. [27] Guo Z, Zhang X. Experimental investigation of complete stress-deformation curves of concrete in tension. J Build Struct 1988;84(4):278–85. [28] Chen X, Xu L, Bu J. Experimental study and constitutive model on complete stressstrain relations of plain concrete in uniaxial cyclic tension. KSCE J Civ Eng 2015;79(1):1–7.
6. Conclusion This paper presented a comprehensible model for predicting the hysteretic behavior of concrete under cyclic tension and tension-compression reversals. The model is introduced in the frame of a dimensionless stress-deformation coordinates. Complete and partial unloading and reloading paths are considered based on the crack closingopening mechanism. The model can be used to predict the complex hysteretic behaviors of concrete under cyclic tension and tensioncompression reversals in a simple and practical way. The proposed model has been validated by comparison with available experiments and shows satisfactory agreement with the experimental results in all cases. Comparisons with the focal point model have been carried out and the proposed model presents improvement on the total dissipation energy prediction. Through the seismic response analysis, the proposed model proved to be capable to consider the effect of
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