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Hysteretic model of Y-type perfobond rib connectors with large number of ribs
Journal of Constructional Steel Research xxx (xxxx) xxx
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Hysteretic model of Y-type perfobond rib connectors with large number of ribs Dae-Yoon Kim*, Munkhtulga Gombosuren, Oneil Han, Sang-Hyo Kim Department of Civil and Environmental Engineering, Yonsei University, N504, 1st Engineering Building, 50 Yonsei-ro, Seodaemun-gu, Seoul, 03722, South Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 August 2019 Received in revised form 18 October 2019 Accepted 20 October 2019 Available online xxx
This study analyzes the shear and hysteresis behavior of Y-type perfobond rib shear connector and suggests a hysteretic model. Y-type perfobond rib shear connectors with different number of ribs (4 ribs, 6 ribs, and 8 ribs) are considered to evaluate energy dissipation. The aim of this study is to do estimation on hysteretic behavior for large number of ribs, as the size of large number of ribs makes it difficult to proceed with an experiment. Numerical analysis and experimental study are conducted to compare the shear strength and energy dissipation of the specimens. In the experimental study, monotonic loading test and cyclic loading test are performed to investigate the hysteresis behavior of Y-type perfobond rib shear connectors. Cyclic loading test is performed in accordance with European Convention for Constructional Steelwork (ECCS) reference which is a procedure of test for assessing the behavior of structural elements under cyclic loadings. Numerical analysis is conducted since the authentication of Y-type perfobond rib shear connector should be modelled to find the shear resistance of each specimen. For hysteretic model of Ytype perfobond rib shear connector, Bouc-Wen-Baber-Noori (BWBN) model is adopted. Finally, the BWBN model parameters are predicted for shear connectors with large number of ribs based on the experimental data of shear connectors with small number of ribs. The slip amplitude is separated into small slips (1e4 mm) and large slips (over 4 mm) in order to improve the accuracy of the energy dissipation prediction. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Composite structure Y-type perfobond rib shear connector Cyclic loading Hysteresis analysis Bouc-Wen-Baber-Noori (BWBN) Energy dissipation
1. Introduction Nowadays, composite structures have been widely used in the construction industry due to their cost-effectiveness and structural efficiency. In seismic zone, the role of the shear connector is very crucial to distribute seismic loading, and to increase the resisting capacity of structure. The resisting capacity of structure might be drastically reduced under repeated seismic loadings. To investigate this deterioration, the hysteretic performance of structure should be determined. The energy dissipated capacity of a structure is one of the significant factor in order to maintain its structural performance under seismic loading condition. Many researchers suggested shear connectors which can be effective to resist shear and fatigue failure. The stud type shear connector [1] and flat type shear
* Corresponding author. E-mail addresses: [email protected] (D.-Y. Kim), [email protected] (M. Gombosuren), [email protected] (O. Han), [email protected] (S.-H. Kim).
connector [2] are widely used in construction field. Due to the workability of headed-stud-type connector, it is popular in construction field. However, it has disadvantages in the way that fatigue at connector roots and weakness under large slips cause failure [3,4]. Perfobond rib shear connector is another alternative of rigid shear connector, consisting of a flat steel plate with dowel holes. It provides preeminent shear resistance of end bearing effect, dowel resistance and the resistance of transverse rebar placed through dowel holes [3,5e11]. Even if it has a prominent performance on shear resistance, it also has weakness on brittle fracture [3,5,12]. Classen et al. conducted experimental and analytical analysis for composite dowels with puzzle (PZ) or clothoid (CL) shape and pry out failure and concluded that the most important parameter influencing the deformation capacity is the concrete cover [13]. Kopp et al. presented the design formulae for longitudinal static shear resistance, verifications for beam-type section for composite dowels with PZ and CL shape [14]. Furthermore, the fatigue resistance of composite dowel shear connectors was evaluated and CL shape exhibits significantly better fatigue resistance than PZ shape
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Fig. 1. Dimensions of Y-type perfobond rib shear connector (unit: mm). (a) Specimen with four-rib system (Y-4). (b) Specimen with six-rib system (Y-6). (c) Specimen with eight-rib system (Y-8).
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[15]. The performance of composite dowel shear connector under cyclic loading was also verified by Classen et al. [16]. Additionally, composite dowels can be used for the connection of ultra-highperformance concrete slabs and high-strength steel members researched by Classen et al. [17]. Recently, the Y-type perfobond rib shear connector with excellent shear strength and ductility was developed by Kim et al. [18]. This shear connector was initially designed for highway composite bridges. Kim et al. improved the shear resistance equation for Ytype perfobond rib shear connector by considering various design variables, including the Y-rib dimensions, number of ribs, and concrete compressive strength [19,20]. Also, the effect of doublerow Y-type perfobond rib shear connectors based on push-out test was evaluated [21]. Later, conventional Y-type perfobond rib shear connector was modified in order to use it in general buildings. Kim et al. evaluated hysteretic performance of stubby Y-type perfobond rib and compared the structural performance with stud shear connectors [22,23]. The applicability of Y-type perfobond rib shear connector was also verified through the behavior evaluation of partially corrugated steel web PSC girder with Y-type perfobond rib shear connector [24] and FE analysis model for concrete steel composite girder with headed studs and Y-type shear connectors [25], respectively. In the results, Y-type shear connector shows the advantages over the headed stud shear connectors, such as the shear resistance, the ductility, the increased load-carring capacity of composite girder [25]. This study is mainly focused on the shear strength and hysteresis behavior of stubby Y-type perfobond rib shear connector. Monotonic and cyclic loading tests are conducted on three different specimens depending on the number of ribs (4, 6 and 8 ribs). From monotonic loading tests, the displacement ductility and shear resistance are analyzed, while the energy dissipation is analyzed by cyclic loading tests. Also, numerical analysis is performed by using
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ABAQUS in order to compare with monotonic loading test results. The monotonic loading test for eight ribs is not implemented, so numerical analysis is performed for a specimen with eight ribs to predict the maximum shear strength. In order to show the hysteresis behavior of structural systems, the Bouc-Wen-Baber-Noori (BWBN) model is chosen. There are numerous studies focusing on hysteresis behavior of structure. The BWBN model was first suggested by Bouc [26], Wen [27], Barber and Noori [28]. This model can estimate more accurate results and describe pinching behavior [29]. BWBN parameters are obtained for hysteresis behavior of Y-type perfobond rib shear connector by using monotonic loading and cyclic loading test results. From monotonic loading tests, shape parameters are obtained, while parameters of stiffness degradation, strength degradation, and pinching behavior are determined by cyclic loading test results. Finally, BWBN parameters are predicted for Y-type perfobond shear connectors with 8, 10 and 12 ribs based on the experimental results and numerical analysis.
Table 2 Slip occurred in each cycle during the cyclic loading test. Cycle No.
Slip (mm)
Note
Cycle No.
Slip (mm)
Note
1 2 3 4 5 6 7 8 9
±0.02 ±0.03 ±0.04 ±0.06 ±0.09 ±0.12 ±0.18 ±0.27 ±0.4
Initial loads
10 11 12 13 14 15 16
±1 ±2 ±3 ±4 ±8 ±8 ±8
Cyclic loads
Table 1 Structural steel properties. Type
Yield strength (MPa)
Ultimate strength (Mpa)
Elongation (%)
Young’s modulus (Mpa)
Application
SS400 SM490 SD400
389 452 400
451 560 639
25.0 19.0 17.3
210,000 210,000 210,000
Y-type perfobond rib H-beam Transverse rebar (D13)
Fig. 2. Cyclic loading test set-up.
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2. Experimental analysis 2.1. Test specimen The dimensions of Y-type perfobond rib shear connector is shown in Fig. 1. Y-rib have a thickness of 8 mm, the height of 50 mm, and a dowel diameter of 30 mm. Total lengths of specimens are 980 mm, 1180 mm, 1380 mm for four, six and eight ribs, respectively. The material properties of concrete and steel used in all the specimens are as follows: design strength of the concrete is 35 MPa, steel is determined based on the Korea Highway Bridge Specifications (KHBS 2010) [30] which are shown in Table 1. Styrofoam is attached on the bottom end of Y-ribs in order to eliminate the end-bearing effect. Before the experiment, cylinder compressive strength tests of concrete are analyzed. The average compressive strength of the concrete cylindrical specimen before the test is measured at 33.4 MPa. Six specimens (Y-4-M, Y-6-M) for monotonic loading tests and nine specimens (Y-4-C, Y-6-C and Y-8-C) for cyclic loading tests have been tested. 2.2. Test procedure Fig. 3. Cyclic loading history.
Monotonic loading tests are conducted on Y-type perfobond rib shear connector with four and six ribs. All tests are conducted with
Fig. 4. Monotonic and cyclic loading test results. (a) Y-4-M/C specimen. (b) Y-6-M/C specimen. (c) Y-8-C specimen.
Fig. 5. Energy dissipation by Y-type perfobond rib shear connector. (a) Cumulated energy dissipation. (b) Energy dissipation at each cycle.
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a universal testing machine (UTM) which has a capacity of 1,000 kN. A monotonic load was applied at a loading rate of 0.02 mm/s until displacement on the UTM reached 50 mm. The cyclic loading tests are performed in accordance with European Convention for Constructional Steelwork (ECCS) [31] which represents a procedure of test for assessing the behavior of structural elements under cyclic loadings. The specimens with four, six, and eight ribs are tested using the same testing machine with monotonic loading tests. Fig. 2 shows test set-up of the Y-type perfobond rib shear connector with different number of ribs. Cyclic loading condition for the Y-type perfobond rib shear connector consisted of initial cyclic loading and cyclic loading. Initial cyclic loading condition is ranged between 0 to ±0.5 mm slip. The cyclic load is applied from ±1 mm until ±16 mm slip. For ±8 and ± 16 slips, the cycles reiterate three times. Cyclic loading conditions are shown in Table 2 and Fig. 3. BWBN model for 16 mm slip is difficult to calibrate since it has very large slip which cannot easily occur. Therefore, experimental results are shown until 16 mm slip, but BWBN model is suggested until 8 mm slip.
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2.3. Test results The monotonic and cyclic loading test results of Y-type perfobond rib shear connector are shown in Fig. 4. In push-out tests, the maximum shear resistances of Y-4-M specimen (468.7 kN) and Y-6M (776.2 kN) are turned out. The shear strength of Y-6-M specimen increases by 65% compared to Y-4-M specimen as the number of ribs are increased. The cumulated energy dissipation and energy dissipation at each cycle of Y-type perfobond rib shear connector with different number of ribs are shown in Fig. 5. Total cumulative energy of Y-4-C specimen is 19,886 kN mm, Y-6-C specimen is 29,520 kN mm, Y-8-C specimen is 30,963 kN mm. The amount of total energy dissipated
Fig. 6. Stiffness degradation due to the cyclic loading.
Table 3 Concrete damaged plasticity (CDP) model. Dilation angle
Eccentricity
fbo fco
K
Viscosity parameter
36
0.1
1.16
0.667
0
Fig. 7. Stress-strain relationships of concrete. (a) Tensile behavior (b) Compressive behavior.
Fig. 8. Stress-strain relationships of steel and reinforcement. (a) Tensile behavior (b) Compressive behavior.
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increases by 48% for Y-6-C specimen, by 56% for Y-8-C specimen compared to Y-4-C specimen. Shear resistance per a rib is decreased with increasing number of ribs. This effect is already explained by Classen that the different stresses in the shear connector in the edge zones and in the middle of the group [32]. Fig. 6 shows the stiffness degradation during each cycle of the Ytype perfobond rib shear connector with different number of ribs. The initial stiffness of each specimen is measured to be 199 kN/mm for the Y-4-C specimen, 245 kN/mm for the Y-6-C specimen and 264 kN/mm for the Y-8-C specimen. Residual stiffness drops to approximately 80% of initial stiffness at 11th cycle for Y-4-C specimen, 12th cycle for Y-6-C and Y-8-C specimens. Until 14th cycle, Y6-C specimen and Y-8-C specimen follow a similar decreasing pattern. After the 14th cycle, Y-6-C specimen decreases dramatically. The residual stiffness of every specimen drops to approximately 8% of the initial stiffness at 19th cycle.
Fig. 9. Boundary condition and loading rate.
3. Numerical analysis 3.1. Numerical modelling Y-type perfobond rib shear connector with a different number of ribs is analyzed as a finite element model (FEM) by using commercial software ABAQUS. Finite element models of Y-type perfobond rib shear connectors are shown in Fig. 10. Concrete is considered as of compressive strength of 35 MPa, Poisson’s ratio of 0.2 and elastic modulus of 20,580 MPa. Concrete damaged plasticity (CDP) model which shows general capability for modelling concrete is used as material model of concrete in the analysis, and is listed in Table 3. Stress-strain relationship of concrete is shown in Fig. 7. Additionally, isotropic hardening model is used for describing the material property of steel. Fig. 8 illustrates stress-strain relationship of the structural steel and reinforcement used in FEA [33,34]. Also, Poisson’s ratio is 0.3 and Young’s modulus is 210 GPa as well as 200GPa for steel material. Boundary condition and loading rate of FEM model is shown in Fig. 9.
Fig. 11. Load-slip curve of Y-type perfobond rib shear connector by numerical analysis.
Fig. 10. Finite element model of shear connectors with various number of ribs. (a) Y-4 specimen (b) Y-6 specimen (c) Y-8 specimen.
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4. Hysteretic models of Y-type perfobond rib shear connectors with different number of ribs
3.2. Numerical analysis results Load-slip curves of Y-type perfobond rib shear connector with four, six and eight ribs are shown in Fig. 11. The maximum shear strengths of Y-4, Y-6, and Y-8 specimens are 470.9 kN, 745.1 kN and 915.8 kN, respectively. As shown in Fig. 12, load-slip curves of monotonic loading tests and numerical analysis are compared. The numerical modelling results show similar shear resistance with the experimental results.
The hysteretic performance of Y-type perfobond rib shear connector with the different number of ribs is analyzed by using experimental results and proposed hysteretic model. In order to show the hysteresis behavior of structural systems, the Bouc-WenBaber-Noori (BWBN) model is chosen. BWBN model is one of the most effective method as it can clearly show the effect of stiffness degradation, strength degradation, and pinching. Bouc-Wen model which is used to describe non-linear hysteretic system was introduced by Bouc [26] and Wen [27] extended
Fig. 12. Comparison of load-slip curves between experimental and numerical analysis results. (a) Load-slip curves of Y-4 specimen (b) Load slip curves of Y-6 specimen.
Fig. 13. Flowchart for determination of BWBN model parameters.
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the model to produce a variety of hysteretic patterns. This model is able for taking, in analytical form, a range of hysteric cycle shapes matching the behavior of a vast class of hysteric systems. The BoucWen approach apply compact equations which are well convenient
for structural analysis (stability, performances) or synthesis. Then Barber and Noori [28] improved the model to include strength, stiffness and pinching degradation effects. The convenience to use BWBN model is that it can be utilized for an extremely wide range
Fig. 14. Hysteresis behavior of the Y-4-C specimen. (a) Normalized force during each loading step. (b) Hysteresis slip. (c) Cumulated energy ratio at each cycle.
Fig. 15. Hysteresis behavior of the Y-6-C specimen. (a) Normalized force during each loading step. (b) Hysteresis slip. (c) Cumulated energy ratio at each cycle.
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of applications, and can show pinching behavior of the structure. In BWBN model, the inelastic restoring force of a structure with hysteresis behavior is expressed by the sum of elastic and hysteric term which is shown in equation (1).
Rðu; zÞ ¼ ak0 u þ ð1 aÞk0 z
(1)
where R is the total restoring force, u is the translational displacement (elastic component), z is the hysteric displacement (hysteric component), k0 is the initial stiffness, and ⍺ is the postyield stiffness ratio. The total restoring force (R) and hysteric displacement (z), which result from the translational displacement (u), are calculated via Newton-Raphson method [29]. In order to calculate an error of the loading step, weight factors were applied to equation (2). As a result, equation (2) shows that the difference of normalized force giving the root means square error (RMSE) of the normalized force based on the hysteresis loop and the experimentally obtained data. The flowchart of the procedure for the parameter calibration of BWBN model is shown in Fig. 13. 2 0:5
N 1 X Exp εF ¼ f wðF i F BWBN Þ g i N i¼1
(2)
where εF is the RMSE of the normalized force, N is the number of Exp
loading steps, F i
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Table 4 Energy dissipation from experiment and BWBN model. Specimen
Y-4 Y-6 Y-8
Energy dissipation (kN mm) EXP
BWBN
Error
8,472.9 12,769.6 11,524.7
9,296.3 12,797.4 12,562.0
9.7% 0.2% 9.0%
Table 5 BWBN model parameters (1e8 mm slips). Parameter
a b g k n
dn dh xs q p
j dj l Error
Specimen Y-4-M/C
Y-6-M/C
Y-8-M/C
) ) ) ) 1.200 ) ) ) ) ) ) )
0.015 1.500 0.500 1.000 2.160 0.080 0.230 0.934 0.060 1.700 0.100 0.010
/ / / / 2.580 / / / / / / /
) 9.7%
0.800 0.2%
/ 9.0%
is the normalized force by the experimental
Fig. 16. Hysteresis behavior of the Y-8-C specimen. (a) Normalized force during each loading step. (b) Hysteresis slip. (c) Cumulated energy ratio at each cycle.
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result, and F BWBN is the force predicted by the BWBN model, w is i the weight factor considering the amount of energy dissipated. The hysteresis behavior of Y-4-C, Y-6-C and Y-8-C specimens are shown from Figs. 14-16. In addition, energy dissipations for each specimen is shown in Table 4. The total amount of energy dissipation of Y-4-C is 8,472.9 kN mm for experimental test results, and is 9,296.3 kN mm for BWBN model at 8 mm slip. The BWBN model results are 9.7% higher than experimental results in terms of total energy dissipation. The total amount of energy dissipations of Y-6-
Table 6 Maximum shear resistance of each specimen by numerical analysis. Specimen
Maximum shear resistance (MSR) (kN)
Rate of increase (ROI)
Y-4 Y-6 Y-8 Y-10 Y-12
470.9 745.0 915.8 1,125.4 1,251.4
e 1.60 1.94 2.38 2.65
Table 7 Value estimation for parameter n.
M/C is 12,769.6 kN mm for experimental results, and 12,797.4 kN mm for BWBN model at 8 mm slip as shown in Table 4. The BWBN model shows 0.2% higher energy dissipation compare with experimental result. The total amount of energy dissipation of Y-8-C is 11,524.7 kN mm for experimental test results, and 12,562.0 kN mm for BWBN model at 8 mm slip. The BWBN model shows 9.0% higher energy dissipation compare with experimental test result. Table 5 shows BWBN model parameters for each specimen. Among 13 parameters in BWBN, 12 parameters are fixed. Only parameter n is variable. n shows an increasing pattern depending on the increasing number of ribs. In BWBN parameters, parameters a, b, g, n and k are shape parameters which are related to initial stiffness and basic shape of the hysteresis loop. The deterioration in structural performance can be determined by stiffness degradation parameter dn and strength degradation parameter dn. Finally, pinching behavior can be determined by parameters xs , q, p, j, dj and l. 5. BWBN model estimation for Y-type perfobond rib shear connectors 5.1. The estimation of BWBN parameters
Specimen
n
Rate of increase (ROI)
Y-4 Y-6 Y-8 Y-10 Y-12
1.20 2.16 2.62 3.21 3.57
e 1.80 2.18 2.67 2.98
Here, BWBN model parameters of Y-type perfobond rib shear connector with 8, 10 and 12 ribs are estimated. It is expected that the hysteresis behavior of Y-type perfobond rib shear connector with a large number of ribs can be predicted based on specimens with small number of ribs. Through the calibration of BWBN model parameters which
Fig. 17. Hysteresis behavior of the Y-8-C specimen (1e8 mm slips) (estimated). (a) Normalized force during each loading step. (b) Hysteresis slip. (c) Cumulated energy ratio at each cycle.
Please cite this article as: D.-Y. Kim et al., Hysteretic model of Y-type perfobond rib connectors with large number of ribs, Journal of Constructional Steel Research, https://doi.org/10.1016/j.jcsr.2019.105818
D.-Y. Kim et al. / Journal of Constructional Steel Research xxx (xxxx) xxx Table 8 Energy dissipations in experiment and BWBN model. Specimen
Y-4 Y-6 Y-8
Energy dissipation (kN mm) EXP
BWBN
Error
8,472.9 12,769.6 11,524.7
9,296.3 12,797.4 12,669.7
9.7% 0.2% 9.9%
Table 9 BWBN model parameters (1e8 mm slips) (estimated for 8 ribs). Parameter
a b g k n
dn dh xs q p
j dj l Error
Specimen Y-4-M/C
Y-6-M/C
Y-8-M/C
) ) ) ) 1.200 ) ) ) ) ) ) )
0.015 1.500 0.500 1.000 2.160 0.080 0.230 0.934 0.060 1.700 0.100 0.010
/ / / / 2.620 / / / / / / /
) 9.7%
0.800 0.2%
/ 9.9%
11
represents in Table 5, only parameter n is changed, while the other parameters are fixed. The effect of parameter n is directly related to the maximum shear resistance of specimens. When the shear strength of specimen increases, parameter n should be also increased. Table 6 shows maximum shear resistance of specimens with an increasing number of ribs based on numerical analysis results. To estimate parameter n for large number of ribs, the rates of increase on maximum shear resistance and parameter of small number of ribs is used. Parameter n can be estimated by equating two different ratios. The ratio of ROI for MSR (Y-4 to Y-6) to ROI for n (Y-4 to Y-6) is equal to the ratio of ROI for MSR (Y-4 to Y-x) to ROI for n (Y-4 to Y-x). ROI for MSR (Y-4 to Y-6) means rate of increase on maximum shear resistance from Y-4 to Y-6 which is 1.60 as shown in Table 6. Also, ROI for n (Y-4 to Y-6) means rate of increase on parameter n from Y4 to Y-6 which is 1.80 as shown in Table 7. Here, MSR for Y-x (x ribs) can be defined from analysis model, so parameter n for x ribs can be determined. 5.2. Parameter estimation for overall range (1e8 mm) The results of 4 and 6 ribs are the same as in Figs. 14e15. Instead, parameter n for eight ribs is predicted from 2.58 to 2.62 which increases the error of energy dissipation to 9.9%. The hysteresis behavior of the Y-8-C specimen is shown in Fig. 17. Additionally, the energy dissipations and BWBN parameters are shown in Table 8 and Table 9, respectively.
Fig. 18. Hysteresis behavior of Y-4-C specimen (1e4 mm slips). (a) Hysteresis slip (b) Normalized force.
Fig. 19. Hysteresis behavior of Y-6-C specimen (1e4 mm slips). (a) Hysteresis slip (b) Normalized force.
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Fig. 20. Hysteresis behavior of Y-8-C specimen (1e4 mm slips). (a) Hysteresis slip (b) Normalized force.
5.3. Parameter prediction in small slip range (1e4 mm)
Table 10 Energy dissipations in experimental results and BWBN model. Specimen
Y-4 Y-6 Y-8
Energy dissipation (kN mm) (1e4 mm slips)
Energy dissipation (kN mm) (8 mm slips)
EXP
BWBN
Error
EXP
BWBN
Error
2,142.1 2,498.5 1,996.5
2,175.5 2,501.7 2,160.0
1.5% 0.1% 8.1%
6,330.7 10,271.1 9,528.1
6,865.6 10,280.7 10,382.6
8.4% 0.1% 8.9%
Since the structural behavior under cyclic loading is non-linear, the estimation of BWBN for overall range shows error. To reduce the error of energy dissipation, the slip range is divided into two parts. One is small slip range (1e4 mm). The other is large slip range (8 mm). Like above, the other parameters except for n are fixed, and n for 8,10 and 12 ribs are predicted. Each slip applies one fully round
Fig. 21. Hysteresis behavior of Y-4-C specimen (8 mm slips). (a)_Hysteresis slip (b) Normalized force.
Fig. 22. Hysteresis behavior of Y-6-C specimen (8 mm slips). (a)_Hysteresis slip (b) Normalized force.
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cyclic loading up to 4 mm slip. Hysteresis behaviors for each specimen are shown from Figs. 18-20. In terms of an error of energy dissipation, Y-4, Y-6, and Y-8 are estimated as 1.50%, 0.10%, and 8.10% respectively as shown in Table 10. The BWBN model parameter for small slips (1e4 mm) are indicated in Table 11.
loadings. The hysteresis behaviors of each specimen at 8 mm slips are shown from Figs. 21-23. For an error of energy dissipation, Y-4, Y-6, and Y-8 are evaluated as 8.40%, 0.10%, and 8.90% respectively as shown in Table 10. The BWBN model parameter for large slips (8 mm) are indicated in Table 11.
5.4. Parameter estimation in large slip range (8 mm)
5.5. Combination of small (1e4 mm) and large (8 mm) slip ranges Finally, the predicted hysteresis loops of small and large slips are
In terms of 8 mm slip, it is applied three times fully round cyclic
Fig. 23. Hysteresis behavior of Y-8-C specimen (8 mm slips). (a) Hysteresis slip (b) Normalized force.
Table 11 BWBN model parameter for small slips (1e4 mm) and large slip (8 mm). Parameter
a b g k n
dn dh xs q p
j dj l Error
Specimen (1e4 mm slips)
Specimen (8 mm slips)
Y-4
Y-6
Y-8
Y-10
Y-12
Y-4
Y-6
Y-8
Y-10
Y-12
) ) ) ) 1.160 ) ) ) ) ) ) )
) ) ) ) 2.380 ) ) ) ) ) ) )
0.015 1.500 0.500 1.000 2.930 0.040 0.200 0.980 0.060 2.200 0.124 0.010
/ / / / 3.590 / / / / / / /
/ / / / 3.990 / / / / / / /
) ) ) ) 1.060 ) ) ) ) ) ) )
) ) ) ) 2.180 ) ) ) ) ) ) )
0.015 1.500 0.500 1.000 2.680 0.055 0.200 0.938 0.068 1.700 0.100 0.010
/ / / / 3.350 / / / / / / /
/ / / / 3.590 / / / / / / /
) 1.5%
) 0.1%
0.800 8.1%
/
/
) 8.4%
) 0.1%
0.800 8.9%
/
/
Fig. 24. Hysteresis behavior of Y-4-C specimen (1e4 & 8 mm slips). (a) Hysteresis slip (b) Cumulated energy dissipation at each cycle.
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Fig. 25. Hysteresis behavior of Y-6-C specimen (1e4 & 8 mm slips). (a) Hysteresis slip (b) Cumulated energy dissipation at each cycle.
Fig. 26. Hysteresis behavior of Y-8-C specimen (1e4 & 8 mm slips). (a) Hysteresis slip (b) Cumulated energy dissipation at each cycle.
Table 12 Energy dissipations of experiment and combined BWBN model. Specimen
Energy dissipation (kN mm) EXP
BWBN
Error
Y-4 Y-6 Y-8
8,472.8 12,769.6 11,524.7
9,040.9 12,782.5 12,486.2
6.7% 0.1% 8.3%
combined in order to compare with calibrated BWBN model results in Table 5. As shown from Figs. 24-26, the hysteresis slip and energy dissipation of Y-4, Y-6, and Y-8 are represented. The errors of energy dissipation are evaluated as 6.70%, 0.10% and 8.30%, respectively in Table 12. 5.6. Comparison of BWBN models for overall range (1e8 mm) and divided range (1e4 mm & 8 mm) The separated BWBN model parameters shows less error in terms of energy dissipation compared to the BWBN model which is not separated. In Tables 8 and 12, the error decreases by 3.0%, 0.1% and 1.6% for Y-4, Y-6 and Y-8, respectively.
propose the procedure to set up the hysteresis models for the shear connectors with large number of ribs has been found, in which the BWBN model developed with the experimental data for the shear connectors with small number of ribs can be extended to the BWBN model for large number of ribs. In the proposed procedure, 12 parameters among 13 parameters required in BWBN model are selected based on the BWBN model developed for the shear connectors with small number of ribs. The remaining one parameter, n, can be evaluated based on the static shear resistance capacity of the shear connector with large number of ribs. The static capacity of the shear connector with large number of ribs may be obtained from either experiment or numerical simulation. Following the proposed procedure the experimental hysteresis loading experiments with quite large specimens with many ribs, which require much experimental efforts, can be skipped. The proposed procedure has been evaluated by developing the BWBN model for the 8-rib shear connector from the BWBN models for 4 ribs and 6 ribs, and it is found that the error raised from the proposed procedure may be adopted within the general tolerance ranges, understanding that the BWBN models developed based on the real experimental hysteresis results cannot match the real hysteresis behaviors.
6. Concluding remarks
Acknowledgments
The hysteresis behaviors of Y-type perfobond rib shear connectors with various number of ribs have been investigated, from 4rib shear connector to 8-rib shear connector. Based on the developed hysteresis models based on BWBN model, the possibility to
This research was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (No. 20174030201480).
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Hysteretic model of Y-type perfobond rib connectors with large number of ribs Dae-Yoon Kim*, Munkhtulga Gombosuren, Oneil Han, Sang-Hyo Kim Department of Civil and Environmental Engineering, Yonsei University, N504, 1st Engineering Building, 50 Yonsei-ro, Seodaemun-gu, Seoul, 03722, South Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 August 2019 Received in revised form 18 October 2019 Accepted 20 October 2019 Available online xxx
This study analyzes the shear and hysteresis behavior of Y-type perfobond rib shear connector and suggests a hysteretic model. Y-type perfobond rib shear connectors with different number of ribs (4 ribs, 6 ribs, and 8 ribs) are considered to evaluate energy dissipation. The aim of this study is to do estimation on hysteretic behavior for large number of ribs, as the size of large number of ribs makes it difficult to proceed with an experiment. Numerical analysis and experimental study are conducted to compare the shear strength and energy dissipation of the specimens. In the experimental study, monotonic loading test and cyclic loading test are performed to investigate the hysteresis behavior of Y-type perfobond rib shear connectors. Cyclic loading test is performed in accordance with European Convention for Constructional Steelwork (ECCS) reference which is a procedure of test for assessing the behavior of structural elements under cyclic loadings. Numerical analysis is conducted since the authentication of Y-type perfobond rib shear connector should be modelled to find the shear resistance of each specimen. For hysteretic model of Ytype perfobond rib shear connector, Bouc-Wen-Baber-Noori (BWBN) model is adopted. Finally, the BWBN model parameters are predicted for shear connectors with large number of ribs based on the experimental data of shear connectors with small number of ribs. The slip amplitude is separated into small slips (1e4 mm) and large slips (over 4 mm) in order to improve the accuracy of the energy dissipation prediction. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Composite structure Y-type perfobond rib shear connector Cyclic loading Hysteresis analysis Bouc-Wen-Baber-Noori (BWBN) Energy dissipation
1. Introduction Nowadays, composite structures have been widely used in the construction industry due to their cost-effectiveness and structural efficiency. In seismic zone, the role of the shear connector is very crucial to distribute seismic loading, and to increase the resisting capacity of structure. The resisting capacity of structure might be drastically reduced under repeated seismic loadings. To investigate this deterioration, the hysteretic performance of structure should be determined. The energy dissipated capacity of a structure is one of the significant factor in order to maintain its structural performance under seismic loading condition. Many researchers suggested shear connectors which can be effective to resist shear and fatigue failure. The stud type shear connector [1] and flat type shear
* Corresponding author. E-mail addresses: [email protected] (D.-Y. Kim), [email protected] (M. Gombosuren), [email protected] (O. Han), [email protected] (S.-H. Kim).
connector [2] are widely used in construction field. Due to the workability of headed-stud-type connector, it is popular in construction field. However, it has disadvantages in the way that fatigue at connector roots and weakness under large slips cause failure [3,4]. Perfobond rib shear connector is another alternative of rigid shear connector, consisting of a flat steel plate with dowel holes. It provides preeminent shear resistance of end bearing effect, dowel resistance and the resistance of transverse rebar placed through dowel holes [3,5e11]. Even if it has a prominent performance on shear resistance, it also has weakness on brittle fracture [3,5,12]. Classen et al. conducted experimental and analytical analysis for composite dowels with puzzle (PZ) or clothoid (CL) shape and pry out failure and concluded that the most important parameter influencing the deformation capacity is the concrete cover [13]. Kopp et al. presented the design formulae for longitudinal static shear resistance, verifications for beam-type section for composite dowels with PZ and CL shape [14]. Furthermore, the fatigue resistance of composite dowel shear connectors was evaluated and CL shape exhibits significantly better fatigue resistance than PZ shape
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Fig. 1. Dimensions of Y-type perfobond rib shear connector (unit: mm). (a) Specimen with four-rib system (Y-4). (b) Specimen with six-rib system (Y-6). (c) Specimen with eight-rib system (Y-8).
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[15]. The performance of composite dowel shear connector under cyclic loading was also verified by Classen et al. [16]. Additionally, composite dowels can be used for the connection of ultra-highperformance concrete slabs and high-strength steel members researched by Classen et al. [17]. Recently, the Y-type perfobond rib shear connector with excellent shear strength and ductility was developed by Kim et al. [18]. This shear connector was initially designed for highway composite bridges. Kim et al. improved the shear resistance equation for Ytype perfobond rib shear connector by considering various design variables, including the Y-rib dimensions, number of ribs, and concrete compressive strength [19,20]. Also, the effect of doublerow Y-type perfobond rib shear connectors based on push-out test was evaluated [21]. Later, conventional Y-type perfobond rib shear connector was modified in order to use it in general buildings. Kim et al. evaluated hysteretic performance of stubby Y-type perfobond rib and compared the structural performance with stud shear connectors [22,23]. The applicability of Y-type perfobond rib shear connector was also verified through the behavior evaluation of partially corrugated steel web PSC girder with Y-type perfobond rib shear connector [24] and FE analysis model for concrete steel composite girder with headed studs and Y-type shear connectors [25], respectively. In the results, Y-type shear connector shows the advantages over the headed stud shear connectors, such as the shear resistance, the ductility, the increased load-carring capacity of composite girder [25]. This study is mainly focused on the shear strength and hysteresis behavior of stubby Y-type perfobond rib shear connector. Monotonic and cyclic loading tests are conducted on three different specimens depending on the number of ribs (4, 6 and 8 ribs). From monotonic loading tests, the displacement ductility and shear resistance are analyzed, while the energy dissipation is analyzed by cyclic loading tests. Also, numerical analysis is performed by using
3
ABAQUS in order to compare with monotonic loading test results. The monotonic loading test for eight ribs is not implemented, so numerical analysis is performed for a specimen with eight ribs to predict the maximum shear strength. In order to show the hysteresis behavior of structural systems, the Bouc-Wen-Baber-Noori (BWBN) model is chosen. There are numerous studies focusing on hysteresis behavior of structure. The BWBN model was first suggested by Bouc [26], Wen [27], Barber and Noori [28]. This model can estimate more accurate results and describe pinching behavior [29]. BWBN parameters are obtained for hysteresis behavior of Y-type perfobond rib shear connector by using monotonic loading and cyclic loading test results. From monotonic loading tests, shape parameters are obtained, while parameters of stiffness degradation, strength degradation, and pinching behavior are determined by cyclic loading test results. Finally, BWBN parameters are predicted for Y-type perfobond shear connectors with 8, 10 and 12 ribs based on the experimental results and numerical analysis.
Table 2 Slip occurred in each cycle during the cyclic loading test. Cycle No.
Slip (mm)
Note
Cycle No.
Slip (mm)
Note
1 2 3 4 5 6 7 8 9
±0.02 ±0.03 ±0.04 ±0.06 ±0.09 ±0.12 ±0.18 ±0.27 ±0.4
Initial loads
10 11 12 13 14 15 16
±1 ±2 ±3 ±4 ±8 ±8 ±8
Cyclic loads
Table 1 Structural steel properties. Type
Yield strength (MPa)
Ultimate strength (Mpa)
Elongation (%)
Young’s modulus (Mpa)
Application
SS400 SM490 SD400
389 452 400
451 560 639
25.0 19.0 17.3
210,000 210,000 210,000
Y-type perfobond rib H-beam Transverse rebar (D13)
Fig. 2. Cyclic loading test set-up.
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2. Experimental analysis 2.1. Test specimen The dimensions of Y-type perfobond rib shear connector is shown in Fig. 1. Y-rib have a thickness of 8 mm, the height of 50 mm, and a dowel diameter of 30 mm. Total lengths of specimens are 980 mm, 1180 mm, 1380 mm for four, six and eight ribs, respectively. The material properties of concrete and steel used in all the specimens are as follows: design strength of the concrete is 35 MPa, steel is determined based on the Korea Highway Bridge Specifications (KHBS 2010) [30] which are shown in Table 1. Styrofoam is attached on the bottom end of Y-ribs in order to eliminate the end-bearing effect. Before the experiment, cylinder compressive strength tests of concrete are analyzed. The average compressive strength of the concrete cylindrical specimen before the test is measured at 33.4 MPa. Six specimens (Y-4-M, Y-6-M) for monotonic loading tests and nine specimens (Y-4-C, Y-6-C and Y-8-C) for cyclic loading tests have been tested. 2.2. Test procedure Fig. 3. Cyclic loading history.
Monotonic loading tests are conducted on Y-type perfobond rib shear connector with four and six ribs. All tests are conducted with
Fig. 4. Monotonic and cyclic loading test results. (a) Y-4-M/C specimen. (b) Y-6-M/C specimen. (c) Y-8-C specimen.
Fig. 5. Energy dissipation by Y-type perfobond rib shear connector. (a) Cumulated energy dissipation. (b) Energy dissipation at each cycle.
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a universal testing machine (UTM) which has a capacity of 1,000 kN. A monotonic load was applied at a loading rate of 0.02 mm/s until displacement on the UTM reached 50 mm. The cyclic loading tests are performed in accordance with European Convention for Constructional Steelwork (ECCS) [31] which represents a procedure of test for assessing the behavior of structural elements under cyclic loadings. The specimens with four, six, and eight ribs are tested using the same testing machine with monotonic loading tests. Fig. 2 shows test set-up of the Y-type perfobond rib shear connector with different number of ribs. Cyclic loading condition for the Y-type perfobond rib shear connector consisted of initial cyclic loading and cyclic loading. Initial cyclic loading condition is ranged between 0 to ±0.5 mm slip. The cyclic load is applied from ±1 mm until ±16 mm slip. For ±8 and ± 16 slips, the cycles reiterate three times. Cyclic loading conditions are shown in Table 2 and Fig. 3. BWBN model for 16 mm slip is difficult to calibrate since it has very large slip which cannot easily occur. Therefore, experimental results are shown until 16 mm slip, but BWBN model is suggested until 8 mm slip.
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2.3. Test results The monotonic and cyclic loading test results of Y-type perfobond rib shear connector are shown in Fig. 4. In push-out tests, the maximum shear resistances of Y-4-M specimen (468.7 kN) and Y-6M (776.2 kN) are turned out. The shear strength of Y-6-M specimen increases by 65% compared to Y-4-M specimen as the number of ribs are increased. The cumulated energy dissipation and energy dissipation at each cycle of Y-type perfobond rib shear connector with different number of ribs are shown in Fig. 5. Total cumulative energy of Y-4-C specimen is 19,886 kN mm, Y-6-C specimen is 29,520 kN mm, Y-8-C specimen is 30,963 kN mm. The amount of total energy dissipated
Fig. 6. Stiffness degradation due to the cyclic loading.
Table 3 Concrete damaged plasticity (CDP) model. Dilation angle
Eccentricity
fbo fco
K
Viscosity parameter
36
0.1
1.16
0.667
0
Fig. 7. Stress-strain relationships of concrete. (a) Tensile behavior (b) Compressive behavior.
Fig. 8. Stress-strain relationships of steel and reinforcement. (a) Tensile behavior (b) Compressive behavior.
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increases by 48% for Y-6-C specimen, by 56% for Y-8-C specimen compared to Y-4-C specimen. Shear resistance per a rib is decreased with increasing number of ribs. This effect is already explained by Classen that the different stresses in the shear connector in the edge zones and in the middle of the group [32]. Fig. 6 shows the stiffness degradation during each cycle of the Ytype perfobond rib shear connector with different number of ribs. The initial stiffness of each specimen is measured to be 199 kN/mm for the Y-4-C specimen, 245 kN/mm for the Y-6-C specimen and 264 kN/mm for the Y-8-C specimen. Residual stiffness drops to approximately 80% of initial stiffness at 11th cycle for Y-4-C specimen, 12th cycle for Y-6-C and Y-8-C specimens. Until 14th cycle, Y6-C specimen and Y-8-C specimen follow a similar decreasing pattern. After the 14th cycle, Y-6-C specimen decreases dramatically. The residual stiffness of every specimen drops to approximately 8% of the initial stiffness at 19th cycle.
Fig. 9. Boundary condition and loading rate.
3. Numerical analysis 3.1. Numerical modelling Y-type perfobond rib shear connector with a different number of ribs is analyzed as a finite element model (FEM) by using commercial software ABAQUS. Finite element models of Y-type perfobond rib shear connectors are shown in Fig. 10. Concrete is considered as of compressive strength of 35 MPa, Poisson’s ratio of 0.2 and elastic modulus of 20,580 MPa. Concrete damaged plasticity (CDP) model which shows general capability for modelling concrete is used as material model of concrete in the analysis, and is listed in Table 3. Stress-strain relationship of concrete is shown in Fig. 7. Additionally, isotropic hardening model is used for describing the material property of steel. Fig. 8 illustrates stress-strain relationship of the structural steel and reinforcement used in FEA [33,34]. Also, Poisson’s ratio is 0.3 and Young’s modulus is 210 GPa as well as 200GPa for steel material. Boundary condition and loading rate of FEM model is shown in Fig. 9.
Fig. 11. Load-slip curve of Y-type perfobond rib shear connector by numerical analysis.
Fig. 10. Finite element model of shear connectors with various number of ribs. (a) Y-4 specimen (b) Y-6 specimen (c) Y-8 specimen.
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4. Hysteretic models of Y-type perfobond rib shear connectors with different number of ribs
3.2. Numerical analysis results Load-slip curves of Y-type perfobond rib shear connector with four, six and eight ribs are shown in Fig. 11. The maximum shear strengths of Y-4, Y-6, and Y-8 specimens are 470.9 kN, 745.1 kN and 915.8 kN, respectively. As shown in Fig. 12, load-slip curves of monotonic loading tests and numerical analysis are compared. The numerical modelling results show similar shear resistance with the experimental results.
The hysteretic performance of Y-type perfobond rib shear connector with the different number of ribs is analyzed by using experimental results and proposed hysteretic model. In order to show the hysteresis behavior of structural systems, the Bouc-WenBaber-Noori (BWBN) model is chosen. BWBN model is one of the most effective method as it can clearly show the effect of stiffness degradation, strength degradation, and pinching. Bouc-Wen model which is used to describe non-linear hysteretic system was introduced by Bouc [26] and Wen [27] extended
Fig. 12. Comparison of load-slip curves between experimental and numerical analysis results. (a) Load-slip curves of Y-4 specimen (b) Load slip curves of Y-6 specimen.
Fig. 13. Flowchart for determination of BWBN model parameters.
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the model to produce a variety of hysteretic patterns. This model is able for taking, in analytical form, a range of hysteric cycle shapes matching the behavior of a vast class of hysteric systems. The BoucWen approach apply compact equations which are well convenient
for structural analysis (stability, performances) or synthesis. Then Barber and Noori [28] improved the model to include strength, stiffness and pinching degradation effects. The convenience to use BWBN model is that it can be utilized for an extremely wide range
Fig. 14. Hysteresis behavior of the Y-4-C specimen. (a) Normalized force during each loading step. (b) Hysteresis slip. (c) Cumulated energy ratio at each cycle.
Fig. 15. Hysteresis behavior of the Y-6-C specimen. (a) Normalized force during each loading step. (b) Hysteresis slip. (c) Cumulated energy ratio at each cycle.
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of applications, and can show pinching behavior of the structure. In BWBN model, the inelastic restoring force of a structure with hysteresis behavior is expressed by the sum of elastic and hysteric term which is shown in equation (1).
Rðu; zÞ ¼ ak0 u þ ð1 aÞk0 z
(1)
where R is the total restoring force, u is the translational displacement (elastic component), z is the hysteric displacement (hysteric component), k0 is the initial stiffness, and ⍺ is the postyield stiffness ratio. The total restoring force (R) and hysteric displacement (z), which result from the translational displacement (u), are calculated via Newton-Raphson method [29]. In order to calculate an error of the loading step, weight factors were applied to equation (2). As a result, equation (2) shows that the difference of normalized force giving the root means square error (RMSE) of the normalized force based on the hysteresis loop and the experimentally obtained data. The flowchart of the procedure for the parameter calibration of BWBN model is shown in Fig. 13. 2 0:5
N 1 X Exp εF ¼ f wðF i F BWBN Þ g i N i¼1
(2)
where εF is the RMSE of the normalized force, N is the number of Exp
loading steps, F i
9
Table 4 Energy dissipation from experiment and BWBN model. Specimen
Y-4 Y-6 Y-8
Energy dissipation (kN mm) EXP
BWBN
Error
8,472.9 12,769.6 11,524.7
9,296.3 12,797.4 12,562.0
9.7% 0.2% 9.0%
Table 5 BWBN model parameters (1e8 mm slips). Parameter
a b g k n
dn dh xs q p
j dj l Error
Specimen Y-4-M/C
Y-6-M/C
Y-8-M/C
) ) ) ) 1.200 ) ) ) ) ) ) )
0.015 1.500 0.500 1.000 2.160 0.080 0.230 0.934 0.060 1.700 0.100 0.010
/ / / / 2.580 / / / / / / /
) 9.7%
0.800 0.2%
/ 9.0%
is the normalized force by the experimental
Fig. 16. Hysteresis behavior of the Y-8-C specimen. (a) Normalized force during each loading step. (b) Hysteresis slip. (c) Cumulated energy ratio at each cycle.
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10
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result, and F BWBN is the force predicted by the BWBN model, w is i the weight factor considering the amount of energy dissipated. The hysteresis behavior of Y-4-C, Y-6-C and Y-8-C specimens are shown from Figs. 14-16. In addition, energy dissipations for each specimen is shown in Table 4. The total amount of energy dissipation of Y-4-C is 8,472.9 kN mm for experimental test results, and is 9,296.3 kN mm for BWBN model at 8 mm slip. The BWBN model results are 9.7% higher than experimental results in terms of total energy dissipation. The total amount of energy dissipations of Y-6-
Table 6 Maximum shear resistance of each specimen by numerical analysis. Specimen
Maximum shear resistance (MSR) (kN)
Rate of increase (ROI)
Y-4 Y-6 Y-8 Y-10 Y-12
470.9 745.0 915.8 1,125.4 1,251.4
e 1.60 1.94 2.38 2.65
Table 7 Value estimation for parameter n.
M/C is 12,769.6 kN mm for experimental results, and 12,797.4 kN mm for BWBN model at 8 mm slip as shown in Table 4. The BWBN model shows 0.2% higher energy dissipation compare with experimental result. The total amount of energy dissipation of Y-8-C is 11,524.7 kN mm for experimental test results, and 12,562.0 kN mm for BWBN model at 8 mm slip. The BWBN model shows 9.0% higher energy dissipation compare with experimental test result. Table 5 shows BWBN model parameters for each specimen. Among 13 parameters in BWBN, 12 parameters are fixed. Only parameter n is variable. n shows an increasing pattern depending on the increasing number of ribs. In BWBN parameters, parameters a, b, g, n and k are shape parameters which are related to initial stiffness and basic shape of the hysteresis loop. The deterioration in structural performance can be determined by stiffness degradation parameter dn and strength degradation parameter dn. Finally, pinching behavior can be determined by parameters xs , q, p, j, dj and l. 5. BWBN model estimation for Y-type perfobond rib shear connectors 5.1. The estimation of BWBN parameters
Specimen
n
Rate of increase (ROI)
Y-4 Y-6 Y-8 Y-10 Y-12
1.20 2.16 2.62 3.21 3.57
e 1.80 2.18 2.67 2.98
Here, BWBN model parameters of Y-type perfobond rib shear connector with 8, 10 and 12 ribs are estimated. It is expected that the hysteresis behavior of Y-type perfobond rib shear connector with a large number of ribs can be predicted based on specimens with small number of ribs. Through the calibration of BWBN model parameters which
Fig. 17. Hysteresis behavior of the Y-8-C specimen (1e8 mm slips) (estimated). (a) Normalized force during each loading step. (b) Hysteresis slip. (c) Cumulated energy ratio at each cycle.
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D.-Y. Kim et al. / Journal of Constructional Steel Research xxx (xxxx) xxx Table 8 Energy dissipations in experiment and BWBN model. Specimen
Y-4 Y-6 Y-8
Energy dissipation (kN mm) EXP
BWBN
Error
8,472.9 12,769.6 11,524.7
9,296.3 12,797.4 12,669.7
9.7% 0.2% 9.9%
Table 9 BWBN model parameters (1e8 mm slips) (estimated for 8 ribs). Parameter
a b g k n
dn dh xs q p
j dj l Error
Specimen Y-4-M/C
Y-6-M/C
Y-8-M/C
) ) ) ) 1.200 ) ) ) ) ) ) )
0.015 1.500 0.500 1.000 2.160 0.080 0.230 0.934 0.060 1.700 0.100 0.010
/ / / / 2.620 / / / / / / /
) 9.7%
0.800 0.2%
/ 9.9%
11
represents in Table 5, only parameter n is changed, while the other parameters are fixed. The effect of parameter n is directly related to the maximum shear resistance of specimens. When the shear strength of specimen increases, parameter n should be also increased. Table 6 shows maximum shear resistance of specimens with an increasing number of ribs based on numerical analysis results. To estimate parameter n for large number of ribs, the rates of increase on maximum shear resistance and parameter of small number of ribs is used. Parameter n can be estimated by equating two different ratios. The ratio of ROI for MSR (Y-4 to Y-6) to ROI for n (Y-4 to Y-6) is equal to the ratio of ROI for MSR (Y-4 to Y-x) to ROI for n (Y-4 to Y-x). ROI for MSR (Y-4 to Y-6) means rate of increase on maximum shear resistance from Y-4 to Y-6 which is 1.60 as shown in Table 6. Also, ROI for n (Y-4 to Y-6) means rate of increase on parameter n from Y4 to Y-6 which is 1.80 as shown in Table 7. Here, MSR for Y-x (x ribs) can be defined from analysis model, so parameter n for x ribs can be determined. 5.2. Parameter estimation for overall range (1e8 mm) The results of 4 and 6 ribs are the same as in Figs. 14e15. Instead, parameter n for eight ribs is predicted from 2.58 to 2.62 which increases the error of energy dissipation to 9.9%. The hysteresis behavior of the Y-8-C specimen is shown in Fig. 17. Additionally, the energy dissipations and BWBN parameters are shown in Table 8 and Table 9, respectively.
Fig. 18. Hysteresis behavior of Y-4-C specimen (1e4 mm slips). (a) Hysteresis slip (b) Normalized force.
Fig. 19. Hysteresis behavior of Y-6-C specimen (1e4 mm slips). (a) Hysteresis slip (b) Normalized force.
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Fig. 20. Hysteresis behavior of Y-8-C specimen (1e4 mm slips). (a) Hysteresis slip (b) Normalized force.
5.3. Parameter prediction in small slip range (1e4 mm)
Table 10 Energy dissipations in experimental results and BWBN model. Specimen
Y-4 Y-6 Y-8
Energy dissipation (kN mm) (1e4 mm slips)
Energy dissipation (kN mm) (8 mm slips)
EXP
BWBN
Error
EXP
BWBN
Error
2,142.1 2,498.5 1,996.5
2,175.5 2,501.7 2,160.0
1.5% 0.1% 8.1%
6,330.7 10,271.1 9,528.1
6,865.6 10,280.7 10,382.6
8.4% 0.1% 8.9%
Since the structural behavior under cyclic loading is non-linear, the estimation of BWBN for overall range shows error. To reduce the error of energy dissipation, the slip range is divided into two parts. One is small slip range (1e4 mm). The other is large slip range (8 mm). Like above, the other parameters except for n are fixed, and n for 8,10 and 12 ribs are predicted. Each slip applies one fully round
Fig. 21. Hysteresis behavior of Y-4-C specimen (8 mm slips). (a)_Hysteresis slip (b) Normalized force.
Fig. 22. Hysteresis behavior of Y-6-C specimen (8 mm slips). (a)_Hysteresis slip (b) Normalized force.
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13
cyclic loading up to 4 mm slip. Hysteresis behaviors for each specimen are shown from Figs. 18-20. In terms of an error of energy dissipation, Y-4, Y-6, and Y-8 are estimated as 1.50%, 0.10%, and 8.10% respectively as shown in Table 10. The BWBN model parameter for small slips (1e4 mm) are indicated in Table 11.
loadings. The hysteresis behaviors of each specimen at 8 mm slips are shown from Figs. 21-23. For an error of energy dissipation, Y-4, Y-6, and Y-8 are evaluated as 8.40%, 0.10%, and 8.90% respectively as shown in Table 10. The BWBN model parameter for large slips (8 mm) are indicated in Table 11.
5.4. Parameter estimation in large slip range (8 mm)
5.5. Combination of small (1e4 mm) and large (8 mm) slip ranges Finally, the predicted hysteresis loops of small and large slips are
In terms of 8 mm slip, it is applied three times fully round cyclic
Fig. 23. Hysteresis behavior of Y-8-C specimen (8 mm slips). (a) Hysteresis slip (b) Normalized force.
Table 11 BWBN model parameter for small slips (1e4 mm) and large slip (8 mm). Parameter
a b g k n
dn dh xs q p
j dj l Error
Specimen (1e4 mm slips)
Specimen (8 mm slips)
Y-4
Y-6
Y-8
Y-10
Y-12
Y-4
Y-6
Y-8
Y-10
Y-12
) ) ) ) 1.160 ) ) ) ) ) ) )
) ) ) ) 2.380 ) ) ) ) ) ) )
0.015 1.500 0.500 1.000 2.930 0.040 0.200 0.980 0.060 2.200 0.124 0.010
/ / / / 3.590 / / / / / / /
/ / / / 3.990 / / / / / / /
) ) ) ) 1.060 ) ) ) ) ) ) )
) ) ) ) 2.180 ) ) ) ) ) ) )
0.015 1.500 0.500 1.000 2.680 0.055 0.200 0.938 0.068 1.700 0.100 0.010
/ / / / 3.350 / / / / / / /
/ / / / 3.590 / / / / / / /
) 1.5%
) 0.1%
0.800 8.1%
/
/
) 8.4%
) 0.1%
0.800 8.9%
/
/
Fig. 24. Hysteresis behavior of Y-4-C specimen (1e4 & 8 mm slips). (a) Hysteresis slip (b) Cumulated energy dissipation at each cycle.
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D.-Y. Kim et al. / Journal of Constructional Steel Research xxx (xxxx) xxx
Fig. 25. Hysteresis behavior of Y-6-C specimen (1e4 & 8 mm slips). (a) Hysteresis slip (b) Cumulated energy dissipation at each cycle.
Fig. 26. Hysteresis behavior of Y-8-C specimen (1e4 & 8 mm slips). (a) Hysteresis slip (b) Cumulated energy dissipation at each cycle.
Table 12 Energy dissipations of experiment and combined BWBN model. Specimen
Energy dissipation (kN mm) EXP
BWBN
Error
Y-4 Y-6 Y-8
8,472.8 12,769.6 11,524.7
9,040.9 12,782.5 12,486.2
6.7% 0.1% 8.3%
combined in order to compare with calibrated BWBN model results in Table 5. As shown from Figs. 24-26, the hysteresis slip and energy dissipation of Y-4, Y-6, and Y-8 are represented. The errors of energy dissipation are evaluated as 6.70%, 0.10% and 8.30%, respectively in Table 12. 5.6. Comparison of BWBN models for overall range (1e8 mm) and divided range (1e4 mm & 8 mm) The separated BWBN model parameters shows less error in terms of energy dissipation compared to the BWBN model which is not separated. In Tables 8 and 12, the error decreases by 3.0%, 0.1% and 1.6% for Y-4, Y-6 and Y-8, respectively.
propose the procedure to set up the hysteresis models for the shear connectors with large number of ribs has been found, in which the BWBN model developed with the experimental data for the shear connectors with small number of ribs can be extended to the BWBN model for large number of ribs. In the proposed procedure, 12 parameters among 13 parameters required in BWBN model are selected based on the BWBN model developed for the shear connectors with small number of ribs. The remaining one parameter, n, can be evaluated based on the static shear resistance capacity of the shear connector with large number of ribs. The static capacity of the shear connector with large number of ribs may be obtained from either experiment or numerical simulation. Following the proposed procedure the experimental hysteresis loading experiments with quite large specimens with many ribs, which require much experimental efforts, can be skipped. The proposed procedure has been evaluated by developing the BWBN model for the 8-rib shear connector from the BWBN models for 4 ribs and 6 ribs, and it is found that the error raised from the proposed procedure may be adopted within the general tolerance ranges, understanding that the BWBN models developed based on the real experimental hysteresis results cannot match the real hysteresis behaviors.
6. Concluding remarks
Acknowledgments
The hysteresis behaviors of Y-type perfobond rib shear connectors with various number of ribs have been investigated, from 4rib shear connector to 8-rib shear connector. Based on the developed hysteresis models based on BWBN model, the possibility to
This research was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (No. 20174030201480).
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Please cite this article as: D.-Y. Kim et al., Hysteretic model of Y-type perfobond rib connectors with large number of ribs, Journal of Constructional Steel Research, https://doi.org/10.1016/j.jcsr.2019.105818