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Experimental and numerical studies on hysteretic behavior of all-steel bamboo-shaped energy dissipaters
Engineering Structures 165 (2018) 38–49
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Experimental and numerical studies on hysteretic behavior of all-steel bamboo-shaped energy dissipaters ⁎
T
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Chun-Lin Wang , Ye Liu , Li Zhou Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education, Southeast University, Nanjing 210096, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Energy dissipaters Steel Low-cycle fatigue Deformation pattern Failure modes
Energy dissipaters constructed in precast structures play an important fuse-type role in concentrating damage and protecting the primary structure. The stable hysteretic behavior, easy fabrication and low cost are expected characteristics for high-performance energy dissipater. In this paper, an all-steel bamboo-shaped energy dissipater consisting of an inner bamboo-shaped core and an outer restraining tube was developed, which aimed at guaranteeing the performance of the energy dissipater under relatively high strain amplitude and avoiding the adverse effect of grouting and welding. Parametric studies on geometrical variables were performed to investigate the low-cycle fatigue behaviors and deformation patterns of the proposed bamboo-shaped dissipaters. Test results showed that all-steel bamboo-shaped dissipaters showed stable hysteretic curves and no local or overall buckling were observed. Failure modes of bamboo-shaped energy dissipater were affected by the lateral deformation resulted from bending, stress concentration around the fillet and the torsion in the segment. The torsion, contact conditions and the wavelength were discussed via finite element analyses and theoretical derivations.
1. Introduction
local torsional buckling [8] on the low-cycle fatigue performance of BRBs was carefully investigated and analyzed. All achievements obtained in working mechanism of BRB are available in developing the new all-steel BRB. However, the previously studied BRB had a bearing capacity more than 500 kN. When the high bearing capacity BRBs are applied in the precast prestressed concrete frame, a large amount of prestress will be necessary to re-center the structure to its original position [9]. Therefore, the development of relatively smaller energy dissipating bars is necessary. The smaller energy dissipating bars with lower bearing capacity have two advantages compared with conventional large BRBs: (1) Use a small amount of prestressed tendons while sustaining the selfcentering property and (2) Spare enough space for architectural design. As shown in Fig. 1(a), one type of energy dissipating bar with similar working mechanism of BRB has been recently developed [10], where the yielding segment was formed through weakening the cross section of the steel core. The epoxy or grout was filled into the gap between the yielding segment and the outer tube to constrain the lateral deformation of the steel core and prevent the buckling of the steel core. The application of this type of energy dissipating bars has been widely detailed and tested for different laminated veneer lumber (LVL) structural systems, such as LVL beam-column joints [11] and posttensioned timber walls [12]. Except for timber structure, energy dissipaters shown
In China, the precast concrete structure, a substitute for the cast-inplace concrete structure, has been recently favored by the government’s policies as well as by engineers and researchers, which is beneficial to protect environment and reduce labor cost in the construction. After the destructive 2008 WenChuan and 2010 YuShu earthquakes in China, how to efficiently apply precast structures in strong seismic regions through improving its seismic performance has been one of the major concerns. The metallic-yielding energy dissipation devices acting as ductile fuses are often employed in the precast prestressed concrete structures to increase the energy dissipation capacity [1]. In such structure, the precast concrete beams and columns are assembled together using prestressed tendons, and the prestressed tendons are designed to remain elastic during seismic loading to provide the structural self-centering capability. As a popular type of steel energy-dissipating damper, the bucklingrestrained brace (BRB), shows excellently stable hysteretic performance [2–4]. A typical BRB consists of a core member and an outer restraining member, where the former bears axial forces and the latter prevents the buckling of the core. To deepen the understanding of the working mechanism of BRBs in component level, a series work including the effect of weld of the rib [5], stopper [6], unbonding material [7] and
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Corresponding authors at: School of Civil Engineering, Southeast University, Nanjing 210096, China. E-mail addresses: [email protected] (C.-L. Wang), [email protected] (Y. Liu).
https://doi.org/10.1016/j.engstruct.2018.02.078 Received 2 November 2017; Received in revised form 24 February 2018; Accepted 25 February 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.
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the relative movement between the core and the tube along the longitudinal direction. The red paint is sprayed on the surface of segments. If any possible contact occurs between segments and the tube, the contact conditions can be apparently shown by the scratches of the red paint. From the cross-section details of the SBED specimen shown in Fig. 2(d), it is demonstrated that the core is surrounded by the tube, and small gaps, d1 and d2, are respectively provided between the slub and the tube and between the segment and the tube. The Q235b steel bars were adopted in the fabrication of the bamboo-shaped cores via computer numerical control lathe processing. The average material property obtained from coupon tests listed in Table 1. Measured geometric dimensions of SBED specimens are given in Table 2. 2.3. Preliminary design Fig. 1. Configuration of different dissipaters.
An appropriately designed SBED specimen is expected to have such characteristic: failure concentrated in segments and no damage observed in slubs and transitional zones, which means the slubs and transitional zones are ‘overstrength’ compared with segments. By adjusting ratio of the sectional area of slubs Asl to the sectional area of segments Ase, the slubs can be designed elastic and the segments are able to enter plasticity. The fulfillment of Eq. (1) ensures an appropriately designed SBED specimen.
in Fig. 1(a) were also widely tested in concrete structural systems including rocking walls [13] and bridge piers [14]. However, only limited literature was referred to the behavior of energy dissipaters [10]. Problems of previously studied energy dissipaters cannot be neglected, which are addressed as difficult grouting, unsymmetric hysteretic performance and limited configurations in structures.
Asl / Ase = (dsl / dse )2 ⩾ σu/ σy 2. Proposed devices
(1)
where σu/σy is computed as 1.46 from Table 1, and values of (dsl/dse)2 for all SBED specimens given in Table 2 satisfies Eq. (1). To avoid the squeeze between the slubs and the inner surface of the tube, the incremental diameter of the slub after loaded, Δd, should be limited less than the gap provided between slubs and the tube, d1. Taking Poisson’s ratio, υ, into consideration, Eq. (2) as follows should be satisfied:
2.1. Concept of a new energy dissipater These problems addressed above are urged to be solved for expanding the application of energy dissipaters in precast structures. As demonstrated in Fig. 1(b), a new fuse-type energy dissipating bar was proposed recently. The working mechanism of the proposed energy dissipater can be idealized as a laterally restrained bar loaded under compression, where lateral supports externalized as slubs. The components between slubs, called segments, can be ideally viewed as short bar, which is favorable to prevent buckling of the segment by adjusting the length of the segment. The application of the new fuse-type energy dissipater can be extended to both compression and tension range, due to the prevention of the buckling. The shape of the core of the energy dissipater is similar to the bamboo, so the new energy dissipating bar is named as bamboo-shaped energy dissipater (BED) hereinafter. Considering the advantages such as low density, high strength-toweight ratio, excellent formability and recyclability of aluminum alloy, the aluminum alloy BED was previously studied [15]. The experimental results showed that the unsatisfactory hysteretic behavior of the aluminum alloy BED was obtained under relatively large axial deformation, while the similar results were proved in other literature [16]. To develop BED specimens with better hysteretic behavior under relatively large strain amplitude, the all-steel BED specimens were investigated. A series of tests including 12 all-steel BED (SBED) specimens was performed to compare the key design parameters and address the low-cycle fatigue performance and deformation patterns under relatively large nominal axial deformation. The details of specimens, the test set-up and the test results are summarized as follows.
Δd = 0.5dsl ε1 = 0.5dsl (−υε2) < d1
(2)
where ε1 and ε2 are strains in the radial and longitudinal directions of slubs, respectively. In the experimental protocol, the design of slubs satisfied the requirement of Eq. (2). Besides, two 50-mm fixed ends in each SBED specimen are prepared for actuators to achieve enough holding force. All tubes made of Q235b steel have 20.0-mm inner diameter (din) and 30.0-mm external diameter (dex). The design of tubes satisfied the requirement of the overall buckling proposed by Usami et al. [17]. 2.4. Nomenclature The first part of the label represents geometric dimensions of the SBED specimen. L refers to the length of the segment and S means the length of the slub. The second part shows the adopted testing protocol. C1, C2, C3 and C4 respectively refer to a 1%, 2%, 3% and 4% constant strain amplitude loading, while V1 and V2 indicate two different variable strain amplitude loading methods (detailed in Section 3). All SBED specimens have four segment and three slubs. For instance, L40S20-C1 means that the SBED specimen has four 40-mm segments, three 20-mm slubs and is tested under 1% constant strain amplitude. However, L40S5-V1 indicates that the SBED specimen has four 40-mm segments, two 5-mm side slubs, one 10-mm middle slub and is tested under variable strain amplitude VSA1. The middle slub in specimen L40S5-V1 expanded to 10 mm is designed to prevent damage caused by the stress concentration around the opening from happening before the failure of the segments.
2.2. Configuration and dimensions Referring to Fig. 2, a typical SBED specimen consists of an inner bamboo-shaped core and an outer restraining tube. The core is composed of a succession of slubs, segments and transition zones depicted in Fig. 2(b). The transition zone is set between the segment and the fixed end to spare enough space for the actuator moving back and forth without touching the tube. In the middle slub, an opening with a radius of 1.5 mm is prepared for the stopper which is made of a nail with a radius of 1.4 mm and length of 40 mm. The stopper is added to control
3. Test setup and loading patterns The test setup is shown in Fig. 3 and a SBED specimen was installed in the hydraulic servo universal testing machine MTS 810, which is capable of producing up to 250 kN loading and maximum displacement 39
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Fig. 2. Details of SBED specimen.
Table 1 Material property of Q235b steel. Material
Young’s modulus E (GPa)
Yield stress σy (MPa)
Yield strain εy (%)
Ultimate strength σu (MPa)
Ultimate strain εu (%)
Q235b
208.4
284
0.136
416
39.43
of ± 300 mm. During a typical test, the axial displacement and force of the actuator were automatically collected by a digital data acquisition system. In the present study, three different low-cycle loading patterns illustrated in Fig. 4 were applied to SBED specimens. All loading patterns were controlled by the nominal axial strain of the core which is derived from the axial displacement of the actuator divided by total length of segments, Lse,t. As shown in Fig. 4, the strain amplitude, Δε, is the absolute value of the peak or valley strain in each cycle. As shown in Fig. 4(a), a constant strain amplitude (CSA) was imposed cyclically to the SBED specimen until failure. Four cycles of strain amplitude 0.08% were firstly employed as an evaluated procedure for testing the SBED specimen and the equipment system. For this reason, counting of constant strain amplitude cycles started subsequently. Four different CSAs including 1%, 2%, 3% and 4% were adopted. There are two different variable strain amplitude (VSA) loading patterns adopted in the present study. The first VSA (VSA1) is shown in Fig. 4(b). Four cycles of 0.08% were imposed to the SBED specimen at first. Then, the strain amplitude of each two cycles increased from 0.5% to 4.0% with an increment of 0.5% and maintained the strain amplitude 3.0% until failure. Some SBED specimens were tested under the second VSA (VSA2), as shown in Fig. 4(c). In VSA2, the strain amplitude of each two cycles increased from 0.5% to 3.0% with an increment of 0.5% and then
Fig. 3. Picture of testing setup.
maintained the strain amplitude 2.0% until failure. A slip between the tested specimen and the clampers of the actuator was observed in the test; and therefore, a calibration test (see Fig. 5) was designed to capture the relationship between the force of the
Table 2 Measured geometric dimensions of SBED specimens. No.
Specimens
dsl (mm)
Lsl,1/Lsl,2 (mm)
dse (mm)
Lse (mm)
Ltr (mm)
Ltotal (mm)
Lct (mm)
d1 (mm)
(dsl/dse)2
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12
L40S20-C1 L40S20-C2 L40S20-C3 L40S20-C4 L60S20-C2 L60S20-C3 L40S5-V1 L60S5-V1 L80S5-V1 L40S20-V1 L60S20-V2 L70S20-V2
18.9 19 18.9 19.1 19 19 19 19 19 18.9 18.6 19
20.1/20.1 19.8/19.8 19.6/19.6 19.9/19.9 20.0/20.0 19.9/19.9 10.0/5.0 10.0/5.0 10.0/5.0 19.8/19.8 20.2/20.2 20.6/20.2
14 14 13.8 13.9 13.9 14 14 14.5 14 14.1 14.1 13.8
40.2 40.3 40.4 40.1 60 60 40.5 60.4 80.1 40.5 60.4 69.4
30.2 29.8 30.1 29.9 29.9 29.8 25.1 25.2 24.9 29.8 29.9 29.8
281.5 280.2 280.6 279.9 359.8 359.3 232.2 312 390.2 281 362 399
260.0 260.0 260.0 260.0 340.0 340.0 210.0 290.0 370.0 260.0 340.0 380.0
0.55 0.50 0.55 0.45 0.55 0.55 0.50 0.50 0.50 0.55 0.70 0.50
1.82 1.84 1.88 1.89 1.87 1.84 1.84 1.72 1.84 1.80 1.74 1.90
Note: dsl and Lsl are the diameter and length of the slub respectively; dse and Lse are the diameter and length of the segment; Ltr is the length of the transition zone; Ltotal is the total length of the core (excluding two fixed ends); Lct is the length of the tube; d1 is the calculated gap width between the slub and the tube. The symbols listed in the table are illustrated in Fig. 2.
40
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Fig. 4. Loading patterns: (a) CSA, (b) VSA1 and (c) VSA2.
Fig. 5. Slip-force relationship calibration test.
uc = u−P / K c
actuator and the slip value, which was influenced by the size of the clampers, the length and the diameter of the fixed end, and the material of the SBED specimen. To obtain the force-slip relationship for the tested SBED specimen, all parameters mentioned above were kept the same in both of the quasit-static and calibration tests. The total length of the calibrated bar excluding two fixed ends, Lt, was 200 mm and the length of the calibrated segment, Lc, was 100 mm. A 100-mm extensometer was employed over the length of the calibrated segment. Based on the measured force, Pc, and the measured average strain of the calibrated segment from the extensometer, εave, the relationship of the force and the slip value was computed as expressed in Eq. (3)
Pc = K c (Δt −εave × Lt )
(4)
where u and P are the displacement and the force of the actuator, respectively. 4. Experimental results and discussions 4.1. Low-cycle fatigue life The experimental stress-strain hysteresis curves after calibration are presented in Fig. 6. The tensile state of the SBED specimen is displayed in the positive direction. The abscissa is the nominal axial strain of the bamboo-shaped core defined as the calibrated displacement of the SBED specimen divided by the total length of segments, while the ordinate is the nominal axial stress derived from the axial force of the actuator divided by Ase. All tested SBED specimens demonstrate stable and repeated hysteretic capacity, without any local and overall
(3)
where Δt is displacement of the actuator; Kc is defined as the slip stiffness; εave was assumed constant along the calibrated bar. Then, the calibrated displacement of the SBED specimen, uc, was calibrated as 41
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Fig. 6. Stress-strain hysteresis curves of SBED specimens.
L80S5-V1 with four 80-mm segments were tested and compared. The numbers of cycles at additional constant strain amplitude, ni, were 14, 14 and 12 for the three SBED specimens, respectively. Similar low-cycle fatigue life were observed in specimens L40S5-V1 and L60S5-V1, which showed that the low-cycle fatigue life of SBED specimens had little relationship with the lengths of the segments for SBED specimens with 5-mm slubs and the stress concentration around the fillet accounted for the failure mechanism (see details in Section 4.3). However, a decrease of low-cycle fatigue life for specimen L80S5-V1 was observed due to torsion (see details in Sections 4.3 and 5.2), compared with L40S5-V1 and L60S5-V1. Furthermore, the effect of the different lengths of segments on the low-cycle fatigue life of SBED specimens with 20-mm slubs was
buckling during loading histories even at the maximum strain amplitude. Test results of all SBED specimens are summarized in Table 3. From the comparison between specimens S1-S4, including L40S20C1 with Nf of 114, L40S20-C2 with Nf of 47, L40S20-C3 with Nf of 29, L40S20-C4 with Nf of 11, the low-cycle fatigue life of SBED specimens in terms of the number of the failure cycles, Nf, were found to be related with the amplitudes of the constant strain. The increasing constant strain amplitude decreased the low-cycle fatigue cycles of SBED specimens. Two additional tests on L60S20-C2 with Nf of 39 and L60S20-C3 with Nf of 18 also addressed the same conclusion. To evaluate the effect of the different lengths of segments on the low-cycle fatigue life of SBED specimens with 5-mm slubs, L40S5-V1 with four 40-mm segments, L60S5-V1 with four 60-mm segments and 42
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Table 3 Test results of SBED specimens.
Table 4 Comparison between Kt and Kexp.
No.
Specimen
Δε (%)
Nf
ni
CPD (%)
Contact condition
Loading pattern
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12
L40S20-C1 L40S20-C2 L40S20-C3 L40S20-C4 L60S20-C2 L60S20-C3 L40S5-V1 L60S5-V1 L80S5-V1 L40S20-V1 L60S20-V2 L70S20-V2
1 2 3 4 2 3 – – – – – –
114 47 29 11 39 18 – – – – – –
– – – – – – 14 14 12 24 34 28
2612.2 2444.2 2418.1 1218.7 2081.8 1476.7 2031.7 2080.4 1942.2 2839.8 2226.4 1969.1
No No No No Yes Yes Yes Yes Yes Yes Yes Yes
CSA CSA CSA CSA CSA CSA VSA1 VSA1 VSA1 VSA1 VSA2 VSA2
Stiffness
Kt (N/mm) Kexp (N/mm) ΔK = 100 |Kexp − Kt|/ Kexp (%)
and Δy is the yielding deformation.
discussed by the comparison between specimens L60S20-V2 with four 60-mm segments and L70S20-V2 with four 70-mm segments. The number of cycles at additional constant strain amplitude, ni, was 34 for specimen L60S20-V2 and 28 for specimen L70S20-V2. The low-cycle fatigue cycles of SBED specimens with 20-mm slubs decreased with the increase of the length of the segment. When L40S5-V1 with ni of 14 and L40S20-V1 with ni of 24 were compared, it is found that the low-cycle fatigue life of SBED specimens with the same length of segment increased with the length of slubs. This is caused by the different rotation capacity of the 5-mm slubs and 20mm slubs [15]. The free rotation of slubs occurs when din is more than the diagonal length of the slub and the restrained rotation of slubs appears at the situation of din less than the diagonal length of the slub. The weaker rotation capacity of 20-mm slubs decreased the lateral deformation of the segments (see details in Section 4.3) and improved the low-cycle fatigue life of the SBED specimen. 4.2. Analytical stiffness model To compute the initial elastic stiffness of SBED specimens, an inseries stiffness model was proposed shown in Fig. 7. As expressed in Eq. (5), the initial elastic stiffness, Kt, is evaluated as the sum of the individual stiffness of the in-series slubs, segments and transitional zones.
1 (n1/ Ksl,1 + n2/ Ksl,2 + m / Kse + 2/ Ktr )
L40S20-C3
L60S20-C3
L40S5-V1
L60S5-V1
L80S5-V1
140,772 131,382 7.2
85,070 94,717 10.2
160,072 151,281 5.8
114,009 105,098 8.5
88,533 83,086 6.6
SBED specimens. As shown in Fig. 8(a)–(d), the lateral deformation of SBED specimens increased with the constant strain amplitude. The shape of the specimen L40S20-C1 nearly kept in a straight line after loaded while the specimen L40S20-C4 exhibited a relatively severer deformation. No contact between segments and the restraining tube was observed in SBED specimens with 40-mm segments and 20-mm slubs under constant strain amplitudes of 1%, 2%, 3% and 4%. When comparing Fig. 8(e) and (f), the lateral deformation of specimen L40S20-V1 with larger slubs was smaller than that of specimen L40S5V1 with smaller slubs. It proved that the weaker rotation capacity of slubs decreased the lateral deformation of the segments. As shown in Fig. 8(e), (g) and (h), the contact in L40S5-V1 was viewed as the point contact and the contact conditions in L60S5-V1 and L80S5-V1 were the line contact, while the contact areas of the three SBED specimens gradually increased with the length of segments. Especially, a torsional deformation was observed in specimen L80S5-V1 via the scratches of the red paint on the surface of segments. The failure modes of SBED specimens were affected by the lateral deformation resulted from bending, stress concentration around the fillet and the torsion in the segment. Specimens L40S20-C1, L40S20-C2, L40S20-C3 and L40S20-C4 failed around the wave peak depicted in Fig. 8(a)–(d), where the high cumulative plastic strain existed. From the local enlarged drawings of the four specimens, the fracture surfaces developed from relatively smooth and vertical to rough and oblique with the increase of strain amplitude, which means that the fracture modes evolved from tensile failure to shear failure. The failure positions of specimens L40S5-V1, L40S20-V1 and L60S5-V1 were around the fillet shown in Fig. 8(e)–(g), where the intense stress concentration existed. The smoother fillets are required to avoid such failure caused by stress concentration. The specimen L80S5-V1 finally failed at the position of scratches caused by the torsion in segment, as shown in Fig. 8(h).
Note: Nf is number of failure cycles, as shown in Fig. 4(a); ni is number of cycles at additional constant strain amplitude, as shown in Fig. 4(b)and (c); CPD is cumulative inelastic deformation [18] = ∑i |Δpi |/Δy , where |Δpi | is the plastic deformation in ith cycle
Kt =
Specimen
4.4. Compression-strength adjustment factor
(5)
where Ksl,i equal to EAsl/Lsl,i is the elastic stiffness of the slub, and i is 1 for the middle slub and 2 for the side slub; Kse equal to EAse/Lse is the elastic stiffness of the segment; Ktr equal to EAtr/Ltr is the elastic stiffness of the transition zone; n1 and n2 is the number of the middle slub and side slubs, respectively; m is the number of segments. As listed in Table 4, the accuracy of Eq. (5) was verified by the comparison between the analytical elastic stiffness, Kt, and measured elastic stiffness, Kexp.
The compression-strength adjustment factor, β, was affected by the contact force and the friction force between the bamboo-shaped core and the restraining tube, which is defined as the ratio of the maximum compression force to the maximum tension force in one hysteresis loop [18]. The calculated β of representative SBED specimens is shown in Fig. 9. All β values of SBED specimens are below 1.3 satisfying the requirement in AISC 341-10 [18]. As shown in Fig. 9(a), the variation of β values was divided into three stages:
4.3. Deformation and failure modes The failure pictures of SBED specimens shown in Fig. 8 demonstrated failure positions, contact conditions and deformation patterns of
(1) Initial adjustment stage. The β values were relatively high and decreased at the first few cycles due to the cyclic hardening effect [7]. In this period, the cyclic stress amplitude increased considerably with the loading cycles until the formation of a static hysteresis loop. However, the first β value less than 1.0 in specimen L40S20C1 was resulted from the relatively high upper yield force. (2) Stable stage. When the strain amplitude was relatively high (e.g., 3% and 4%), the β values increased with the loading history because of the increasing friction forces, which were resulted from the
Fig. 7. In-series stiffness model.
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Fig. 8. Deformation and failure modes of SBED specimens.
(CSA) stage in VSA1, β values of specimens L40S5-V1, L60S5-V1 and L80S5-V1 increased with the loading cycles. Based on the analysis of β values’ variation, recommended β values can be provided for different strain amplitudes as follows: The recommended β value for SBED specimens is 1.0 for strain amplitude below 2% and 1.2 for strain amplitude at 4%. Under the strain amplitude between 2% and 4%, the recommended β value can be obtained by linear interpolation. When SBED specimens are adopted in some practical project, β values of the applied SBED specimens can be determined based on the actual required strain amplitude in the project; and therefore, the design SBED strength in compression can be thus obtained from Eq. (6) [18]:
severer contact between slubs and the tube. When the strain amplitude was relatively low (e.g., 1% and 2%), the β values kept almost unchanged. The nearly straight lateral deformation of the bamboo-shaped core in specimen L40S20-C1 caused small contact and friction forces between slubs and the tube, and accounted for β values of specimen L40S20-C1 almost 1.0 after stabilization; (3) A sharp increase of β values in some specimens like L40S20-C2, L40S20-C3 and L40S20-C4 at the last loading step occurred because of a distinct decrease of the maximum tension force before failure. In Fig. 9(b), β values at the variable strain amplitude (VSA) stage in VSA1 varied with fluctuation. During the constant strain amplitude
Fig. 9. Compression-strength adjustment factors of SBED specimens: (a) under CSA and (b) under VSA1.
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respectively). The segments were numbered as N1, N2, N3 and N4 to clearly analyze the torsion in each segment, as shown in Fig. 12. The positive torsion was regarded as the clockwise direction. The torsion angles of specimens L40S5-V1 and L80S5-V1 are presented in Fig. 13 to evaluate the relationship between the lengths of segments and the torsion angles. The torsion angle of specimen L40S5-V1 was negligible compared with that of specimen L80S5-V1, which agreed well with the test observations of specimens L40S5-V1 and L80S5-V1 in Fig. 8(e) and (h). As depicted in Fig. 13, the torsion angle varied with the distance to the zero position in specimen L80S5-V1, and the maximum torsion angle was found in segment N2 with a value of 0.27 rad. From the shape of the torsion curve in Fig. 13, the torsion of the bamboo-shaped core was almost symmetric with the middle slub.
Fig. 10. Finite element model for SBED specimen.
Fca = βωFy
(6)
where ω is the strain hardening adjustment factor; Fy is the axial yield strength of SBED specimen, which is established using σy determined from coupon tests. ω is estimated by the ratio of σu to σy from coupon tests, which is 1.46 for the Q235b material adopted in the present study.
5.3. Effect of cumulative plastic deformation As discussed in Section 4.3, some SBED specimens failed around fillets due to intense stress concentration, while the failure mechanism of some other SBED specimens including specimens L40S20-C1, L40S20-C2, L40S20-C3 and L40S20-C4 could be explained as the effect of the cumulative plastic deformation. As shown in Fig. 14, the cumulative plastic strain contours are plotted for specimens L40S20-C3 and L40S20-C4 at the maximum compressive strain of 11th cycle. The maximum cumulative plastic strains of specimens L40S20-C3 and L40S20-C4 were 3.084 and 3.985, respectively. The higher cumulative plastic strains could result in earlier failure of the SBED specimen, so that the higher low-cycle fatigue life of specimen L40S20-C3 than specimen L40S20-C4 was explained in terms of the cumulative plastic strain. The positions of the maximum cumulative plastic strains of specimens L40S20-C3 and L40S20-C4 were exactly the same with the observed failure positions of the two specimens depicted in Fig. 8(c) and (d). The failure of the SBED specimen initiated from the concave where the cumulative plastic strain was maximum and gradually developed to full section. The maximum cumulative plastic strain of specimen L60S20-C3 extracted from the numerical analyses was 3.855, which is more than that of specimen L40S20-C3 but less than that of specimen L40S20-C4. It agreed well with the test result that the lowcycle fatigue life of specimen L60S20-C3 was worse than specimen L40S20-C3 but better than specimen L40S20-C4.
5. Numerical discussions 5.1. Validation of finite element model The finite element analyses via ABAQUS [19] were conducted to investigate the torsion deformation, failure mechanism and contact conditions of SBED specimens. The established model for the SBED specimen is presented in Fig. 10. The C3D8R (8-node reduced integrated) elements were assigned to the SBED specimen. A calibrated combined hardening model was adopted for the bamboo-shaped core and the tube was considered elastic. Calibrated parameters of the combined hardening model are listed in Table 5. The evolution law of the combined hardening model [19] can be described as a nonlinear kinematic hardening component and an isotropic hardening component. The nonlinear kinematic hardening component is concerned with Ci and γi and the isotropic hardening behavior of the model defines values of Q∞ and b. The interaction property employed in the model included tangential and normal behaviors to transmit shear forces and normal forces. The isotropic directionality and friction coefficient of 0.1 [20] were considered in the tangential behavior and the hard contact was adopted in the normal behavior. The initial geometric imperfection with magnitude of 1/1000 of the length of the core was applied to the SBED specimen by perturbations in the geometry. All degrees of freedom of two ends of the tube were constrained. The y-direction constraint of the bambooshaped core was released to apply axial displacements while other degrees of freedom were constrained. The displacements employed in finite element analyses were the same with the calibrated displacement in tests. The comparisons between the results of tests and finite element analyses chosen from representative specimens are presented in Fig. 11. The numerical elastic stiffness and hysteresis loops matched well with the experimental results.
5.4. Evolution of contact The contact stress contours of specimen are presented in Fig. 15. The positions with contact stress above zero obtained from numerical analyses agreed well with the contact positions observed in both specimens L60S5-V1 and L80S5-V1. A gradual contact phenomenon was observed in the analyses of contact conditions for SBED specimens. The contact stress contours of specimen L60S20-C3 at the maximum compressive strain of different loading cycles are shown in Fig. 16. No contact stress was observed in segments of specimen L60S20-C3 in the first cycle, and then the contact between the segment and the tube gradually occurred with the progress of loading cycles. The gradual contact was mainly caused by the cumulative plastic deformation in segments. A displacement-based model [15] including a segment and two halves of slubs was proposed to explore the relationship between the maximum lateral displacement of the segment and the length of the segment under the relatively low strain amplitude in the previous study. Both of the initial geometrical imperfection of the segment and the rotation of the slub have been considered in the proposed model. The maximum lateral displacement of the segment caused from the rotation angle of the slub is α(Lse + Lsl,2)/π when the rotation angle of the slub is α. The maximum initial geometrical imperfection is assumed to be 0.001Lse. The maximum lateral displacement of the segment under a certain axial load F was evaluated by Eq. (7) [15], where the loading and unloading responses were taken into account under the relatively
5.2. Torsion of segment The displacement contours for specimens L40S5-V1 and L80S5-V1 are plotted at the at the last cycle before failure (ni = 14 and 12, Table 5 Calibration parameters of combined hardening model for Q235b steel. Material
σy
Q∞
b
C1
γ1
C2
γ2
C3
γ3
Q235b
284
10
1.2
100,000
2500
7500
100
500
0
Note: Q∞ is the maximum change in the size of the yield surface; b defines the rate at which the size of the yield surface changes as plastic straining develops; Ci and γi are parameters related with backstresses.
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Fig. 11. Comparisons between results of tests and finite element analyses.
Fig. 12. Computational graph of torsion angle: (a) specimen L40S5-V1 and (b) specimen L80S5-V1.
factor, adopted as 0.5. To analytically judge the contact conditions of the SBED specimen loaded under the relatively high strain amplitude, a modified displacement-based judgement model was developed considering the effect of the cumulative plastic deformation. For SBED specimens loaded under relatively larger strain amplitude, the full section of the segment gradually entered plasticity with the loading histories. The equivalent flexural modulus Er in Eq. (7) was then replaced with the tangent modulus Et approximated as 0.02E for simplification [8,21] in Eq. (8).
Llat ,u =
low strain amplitude.
(
α
α
π 2Er Ise 0.001Lse + π Lse + π Lsl,2
)
π 2Er Ise−F (μLse )2
(7)
where Er equal to 8EEt/(E + Et ) is equivalent flexural modulus [15]; Ise is the moment of inertia of the segment; μ is the effective length 1/3
α
α
π 2Et Ise−F (μLse )2
) (8)
The lower and upper bounds of the maximum lateral displacement of the segment were then obtained from Eq. (7) and Eq. (8), respectively. As computed in Table 6, the lower bound of the maximum lateral displacement of the 60-mm segment is 1.63 mm while the upper bound is 4.7 mm. The calculation result from the lower bound of the maximum lateral displacement of the segment showed that no contact occurred. It explained the contact condition in the first cycle of specimen L60S20C3, where the loading and unloading responses should be considered. The contact was expected based on the computation of the upper bound with consideration of full-section plasticity, which agreed well with results of test and finite element analysis at later loading cycles. The numerical analyses of contact conditions for SBED specimens with 40-mm segments under different constant strain amplitudes were
Fig. 13. Torsion angles of SBED specimens.
Llat ,l =
(
π 2Et Ise 0.001Lse + π Lse + π Lsl,2
1/3 3
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Fig. 14. Cumulative plastic strain contours: (a) specimen L40S20-C3 and (b) specimen L40S20-C4.
Fig. 15. Contact stress contours and failure photos: (a) specimen L60S5-V1 and (b) specimen L80S5-V1.
Fig. 16. Contact evolution of SBED specimen L60S20-C3.
[22], expressed in Eq. (9).
Table 6 Lower and upper bounds of segment’s lateral displacement. Lse (mm)
40
60
Llat,l/Contact Llat,u/Contact
1.08/No 1.46/No
1.63/No 4.7/Yes
1 + 2n w ⩽
PL2 ⩽ 1 + 4n w 4π 2EI
(9)
where the values of nw in the left and right sides of the inequality equation represent the upper and lower bounds of the wave number, respectively; L is the length of the constrained column; I is the moment of inertia of the constrained column. Eq. (9) was extended to the elasticplastic behavior by replacing E with the tangent modulus Et based on the elastic-plastic buckling theory proposed by Shanley [23]. Then, the wavelength formed in an individual segment is estimated in Eq. (10).
conducted to further verify the correctness of Eq. (7) and Eq. (8), as shown in Fig. 17. The numerical results of specimens L40S20-C3 and L40S20-C4 were taken at the maximum compressive strain amplitude of 29th cycle and 11st cycle, respectively. No contact was observed in the numerical results of specimens L40S20-C3 and L40S20-C4 during the whole loading histories, which agreed well with the test observations discussed in Section 4.3. The analytical lower and upper bounds of the lateral displacement of the SBED specimen with 40-mm segments listed in Table 6 were 1.08 mm and 1.46 mm respectively, both of which were less than the gap width, d2, between segments and the tube. The prediction of contact conditions from the lower and upper bounds of the lateral segment’s displacement for specimens L40S20-C3 and L40S20C4 satisfied well with the finite element analyses and test observations. Therefore, the lower bound of the maximum lateral displacement of the segment expressed in Eq. (7) is recommended to be adopted in predicting the initial contact conditions for SBED specimens, while the upper bound expressed in Eq. (8) is adopted to predict the final contact conditions for SBED specimens.
Lse 1 int ⎛ 2
⎝
Lse
⩽ lw ⩽
2 PLse
1 −2⎞ 4π 2Et Ise ⎠
1 int ⎛ 4
⎝
2 PLse
4π 2Et Ise
1
−4⎞ ⎠
(10)
where the ‘int’ is a function to discard all numbers after the decimal point and leave the integer; lw is the high-mode buckling wavelength of the segment. The lower bound of lw [24] is adopted to evaluate the development of the wave in SBED specimens, as expressed in Eq. (11).
Lse
lw = 1
int ⎛ 2 ⎝
2 PLse
4π 2Et Ise
1
−2⎞ ⎠
(11)
To form at least a full wave in an individual segment, the minimum Lse should be 210 mm computed from Eq. (12), where the value of P was taken as 63.1 kN from the maximum force in specimen L60S5-V1.
5.5. Estimation of wavelength
1 2
The wave number, nw, of a bi-laterally constrained elastic column was estimated by solving the fourth order, linearized differential equation of an Euler beam under increasing axial compression load
PLse2 1 − ⩾1 4π 2Et Ise 2
(12)
The wavelength would remain nearly unchanged even if the subsequently applied compressive strain was smaller than the previous one
Fig. 17. Contact evolution: (a) specimen L40S20-C3 and (b) specimen L40S20-C4.
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Fig. 18. The development of waves in different lengths of segments (lateral displacement magnified by 4).
Fig. 19. Effect of side slubs on wave development (lateral displacement magnified by 4).
[7]. A series of finite element analyses was then conducted to verify the relationship between the length of the segment and the development of waves described in Eq. (11), where all finite element cases were loaded to the maximum strain amplitude (4%) of loading pattern VSA1 depicted in Fig. 4(b). The nomenclature of numerical cases in Figs. 18 and 19 was the same with that of SBED specimens. Numerical results are shown in Fig. 18. Until the length of the segment increased up to 160 mm in Fig. 18(b), a fully developed wave was observed in an individual segment between two adjacent slubs. The analytical minimum value from Eq. (12) to form a full-length is more than the exact value obtained from finite element analyses. This difference was mainly because the derivation of Eq. (11) did not take the rotation of slubs into consideration. A rotation adjustment factor, Cr, considering the rotation of slubs was then introduced based on the test results and finite element analyses, expressed in Eq. (13).
SBED specimens were compared to provide an effective way to fabricate the new SBED. The main results are summarized as follows: 1. On the basis of the test results, all SBED specimens show stable and repeated hysteresis loops without any local and overall buckling even at the maximum strain amplitude. Geometrical parameters, including the length of slub and segment, have significant influence on the hysteretic behaviors and deformation patterns of SBED. 2. The proposed in-series stiffness model can be effectively applied to evaluate the elastic stiffness of SBED and the displacement-based judgement model is capable of defining the contact conditions of SBED. 3. The failure modes of SBED are affected by the lateral deformation resulted from bending, stress concentration around the fillet and the torsion in the segment. The stress concentration and torsion are expected not to happen in SBED; therefore, special details are required to reduce the stress concentration by smoothing the fillets, and the length of the segment should be controlled to limit the torsion. 4. Based on the evolution law of the compression-strength adjustment factor β, the recommended value of β for SBED varies with the applied strain amplitude, where 1.0 for strain amplitude below 2% and 1.2 for strain amplitude at 4%. For the strain amplitude between 2% and 4%, the linear interpolation can be employed to obtain the recommended β value. 5. A calibrated theoretical derivation considering the effect of the slubs’ rotation is proposed to evaluate the wave development in segments. The 5-mm side slubs are proved to have no influence on the wave formation but reduce the lateral deformation of the segment.
Lse
lw = 1 int ⎛⎜Cr ⎛ 2
⎝
⎝
2 PLse
4π 2Et Ise
− 2 ⎞⎞ ⎠⎠ 1
⎟
(13)
where Cr is calibrated as 1.56 for 20-mm slubs. Based on Eq. (13), two full waves can be formed in an individual segment with length more than 250 mm. As verified in Fig. 18(d), two full waves were observed in numerical case L280S20-4% with 280-mm segments, which agreed well with the analytical result based on Eq. (13). The development of waves in SBED specimens with 5-mm side slubs were also investigated based on test results and finite element analyses. The wave development of specimen L80S5-V1 (see Fig. 10(h) and Fig. 19(a)) and case L160S20-4% in Fig. 19(b) was almost same. The lateral deformation of segments observed in specimen L80S5-V1 was reduced compared with case L160S20-4%, which is resulted from the existence of 5-mm side slubs capable of reducing the lateral deformation of segments.
Acknowledgements The authors would like to acknowledge financial supports from the National Key Research and Development Program of China (2016YFC0701400), National Natural Science Foundation of China (51508363, 51678138), Natural Science Foundation of Jiangsu Province of China (BK20151407), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
6. Conclusions In the present study, 12 all-steel bamboo-shaped energy dissipaters in all were tested to investigate hysteretic behaviors and deformation patterns of SBED and develop high-performance energy dissipaters under relatively high strain amplitude. Besides, design parameters of 48
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Appendix A. Supplementary material
Eng 2016;142:04016134. [11] Palermo A, Pampanin S, Buchanan A, Newcombe M. Seismic design of multi-storey buildings using laminated veneer lumber (LVL). New Zealand Society for earthquake engineering conference, Wairakei Resort, Taupo, New Zealand; 2005. [12] Francesco Sarti, Alessandro Palermo, Stefano Pampanin. Quasi-static cyclic testing of two-thirds scale unbonded posttensioned rocking dissipative timber walls. J Struct Eng 2015;142:E4015005. [13] Marriott D, Pampanin S, Bull D, Palermo A. Dynamic testing of precast, post-tensioned rocking wall systems with alternative dissipating solutions. New Zealand Society for earthquake engineering conference, Bayview Wairakei Resort, New Zealand; 2008. p. 39–54. [14] Palermo A, Pampanin S, Marriott D. Design, modeling, and experimental response of seismic resistant bridge piers with posttensioned dissipating connections. J Struct Eng 2007;133:1648–61. [15] Chun-Lin Wang, Ye Liu, Zhou Li, Guangren Zhou, Zhitao Lu. Concept and performance testing of an aluminum alloy bamboo-shaped energy dissipater. Struct Design Tall Spec Build. 2017:e1444. [16] Chun-Lin Wang, Tsutomu Usami, Jyunki Funayama, Fumiaki Imase. Low-cycle fatigue testing of extruded aluminium alloy buckling-restrained braces. Eng Struct 2013;46:294–301. [17] Tsutomu Usami, Chun-Lin Wang, Jyunki Funayama. Developing high-performance aluminum alloy buckling-restrained braces based on series of low-cycle fatigue tests. Earthquake Eng Struct Dyn 2012;41:643–61. [18] Construction American Institute of Steel. ANSI/AISC 341-10 seismic provisions for structural steel buildings. American Institute of Steel Construction, Chicago, IL; 2010. [19] Hibbitt, Karlsson, Sorensen. ABAQUS/standard user's manual: Hibbitt, Karlsson & Sorensen; 2001. [20] Hoveidae N, Rafezy B. Overall buckling behavior of all-steel buckling restrained braces. J Constr Steel Res 2012;79:151–8. [21] An-Chien Wu, Pao-Chun Lin, Keh-Chyuan Tsai. High-mode buckling responses of buckling-restrained brace core plates. Earthquake Eng Struct Dyn 2014;43:375–93. [22] Herzl Chai. The post-buckling response of a bi-laterally constrained column. J Mech Phys Solids 1998;46:1155–81. [23] Shanley FR. Inelastic column theory. J Aeronaut Sci. 1947;14:261–7. [24] Robert Tremblay, André Filiatrault, Peter Timler, Michel Bruneau. Performance of steel structures during the 1994 Northridge earthquake. Can J Civ Eng 1995;22:338–60.
Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.engstruct.2018.02.078. References [1] Pampanin Stefano, Amaris Alejandro, Akguzel Umut, Palermo Alessandro. Experimental investigations on high-performance jointed ductile connections for precast frame systems. First European conference on earthquake engineering and seismology, Geneva, Switzerland; 2006. p. 2038–48. [2] Ziqin Jiang, Yanlin Guo, Bohao Zhang, Xuqiao Zhang. Influence of design parameters of buckling-restrained brace on its performance. J Constr Steel Res 2015;105:139–50. [3] Zhao Junxian, Wu Bin, Li Wei, Ou Jinping. Local buckling behavior of steel angle core members in buckling-restrained braces: Cyclic tests, theoretical analysis, and design recommendations. Eng Struct 2014;66:129–45. [4] Liang-Jiu Jia, Hanbin Ge, Maruyama Rikuya, Shinohara Kazuki. Development of a novel high-performance all-steel fish-bone shaped buckling-restrained brace. Eng Struct 2017;138:105–19. [5] Chun-Lin Wang, Tsutomu Usami, Jyunki Funayama. Improving Low-Cycle Fatigue Performance of High-Performance Buckling-Restrained Braces by Toe-Finished Method. J Earthquake Eng 2012;16:1248–68. [6] Chun-Lin Wang, Tsutomu Usami, Jyunki Funayama. Evaluating the influence of stoppers on the low-cycle fatigue properties of high-performance buckling-restrained braces. Eng Struct 2012;41:167–76. [7] Quan Chen, Chun-Lin Wang, Shaoping Meng, Bin Zeng. Effect of the unbonding materials on the mechanic behavior of all-steel buckling-restrained braces. Eng Struct 2016;111:478–93. [8] Chun-Lin Wang, Tao Li, Chen Quan Wu, Jing Ge Hanbin. Experimental and theoretical studies on plastic torsional buckling of steel buckling-restrained braces. Adv Struct Eng 2014;17:871–80. [9] Ekkachai Yooprasertchai, Hadiwijaya Irma J, Warnitchai Pennung. Seismic performance of precast concrete rocking walls with buckling restrained braces. Mag Concr Res 2015;68:462–76. [10] Francesco Sarti, Alessando Palermo, Stefano Pampanin. Fuse-Type External Replaceable Dissipaters: Experimental Program and Numerical Modeling. J Struct
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Experimental and numerical studies on hysteretic behavior of all-steel bamboo-shaped energy dissipaters ⁎
T
⁎
Chun-Lin Wang , Ye Liu , Li Zhou Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education, Southeast University, Nanjing 210096, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Energy dissipaters Steel Low-cycle fatigue Deformation pattern Failure modes
Energy dissipaters constructed in precast structures play an important fuse-type role in concentrating damage and protecting the primary structure. The stable hysteretic behavior, easy fabrication and low cost are expected characteristics for high-performance energy dissipater. In this paper, an all-steel bamboo-shaped energy dissipater consisting of an inner bamboo-shaped core and an outer restraining tube was developed, which aimed at guaranteeing the performance of the energy dissipater under relatively high strain amplitude and avoiding the adverse effect of grouting and welding. Parametric studies on geometrical variables were performed to investigate the low-cycle fatigue behaviors and deformation patterns of the proposed bamboo-shaped dissipaters. Test results showed that all-steel bamboo-shaped dissipaters showed stable hysteretic curves and no local or overall buckling were observed. Failure modes of bamboo-shaped energy dissipater were affected by the lateral deformation resulted from bending, stress concentration around the fillet and the torsion in the segment. The torsion, contact conditions and the wavelength were discussed via finite element analyses and theoretical derivations.
1. Introduction
local torsional buckling [8] on the low-cycle fatigue performance of BRBs was carefully investigated and analyzed. All achievements obtained in working mechanism of BRB are available in developing the new all-steel BRB. However, the previously studied BRB had a bearing capacity more than 500 kN. When the high bearing capacity BRBs are applied in the precast prestressed concrete frame, a large amount of prestress will be necessary to re-center the structure to its original position [9]. Therefore, the development of relatively smaller energy dissipating bars is necessary. The smaller energy dissipating bars with lower bearing capacity have two advantages compared with conventional large BRBs: (1) Use a small amount of prestressed tendons while sustaining the selfcentering property and (2) Spare enough space for architectural design. As shown in Fig. 1(a), one type of energy dissipating bar with similar working mechanism of BRB has been recently developed [10], where the yielding segment was formed through weakening the cross section of the steel core. The epoxy or grout was filled into the gap between the yielding segment and the outer tube to constrain the lateral deformation of the steel core and prevent the buckling of the steel core. The application of this type of energy dissipating bars has been widely detailed and tested for different laminated veneer lumber (LVL) structural systems, such as LVL beam-column joints [11] and posttensioned timber walls [12]. Except for timber structure, energy dissipaters shown
In China, the precast concrete structure, a substitute for the cast-inplace concrete structure, has been recently favored by the government’s policies as well as by engineers and researchers, which is beneficial to protect environment and reduce labor cost in the construction. After the destructive 2008 WenChuan and 2010 YuShu earthquakes in China, how to efficiently apply precast structures in strong seismic regions through improving its seismic performance has been one of the major concerns. The metallic-yielding energy dissipation devices acting as ductile fuses are often employed in the precast prestressed concrete structures to increase the energy dissipation capacity [1]. In such structure, the precast concrete beams and columns are assembled together using prestressed tendons, and the prestressed tendons are designed to remain elastic during seismic loading to provide the structural self-centering capability. As a popular type of steel energy-dissipating damper, the bucklingrestrained brace (BRB), shows excellently stable hysteretic performance [2–4]. A typical BRB consists of a core member and an outer restraining member, where the former bears axial forces and the latter prevents the buckling of the core. To deepen the understanding of the working mechanism of BRBs in component level, a series work including the effect of weld of the rib [5], stopper [6], unbonding material [7] and
⁎
Corresponding authors at: School of Civil Engineering, Southeast University, Nanjing 210096, China. E-mail addresses: [email protected] (C.-L. Wang), [email protected] (Y. Liu).
https://doi.org/10.1016/j.engstruct.2018.02.078 Received 2 November 2017; Received in revised form 24 February 2018; Accepted 25 February 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.
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the relative movement between the core and the tube along the longitudinal direction. The red paint is sprayed on the surface of segments. If any possible contact occurs between segments and the tube, the contact conditions can be apparently shown by the scratches of the red paint. From the cross-section details of the SBED specimen shown in Fig. 2(d), it is demonstrated that the core is surrounded by the tube, and small gaps, d1 and d2, are respectively provided between the slub and the tube and between the segment and the tube. The Q235b steel bars were adopted in the fabrication of the bamboo-shaped cores via computer numerical control lathe processing. The average material property obtained from coupon tests listed in Table 1. Measured geometric dimensions of SBED specimens are given in Table 2. 2.3. Preliminary design Fig. 1. Configuration of different dissipaters.
An appropriately designed SBED specimen is expected to have such characteristic: failure concentrated in segments and no damage observed in slubs and transitional zones, which means the slubs and transitional zones are ‘overstrength’ compared with segments. By adjusting ratio of the sectional area of slubs Asl to the sectional area of segments Ase, the slubs can be designed elastic and the segments are able to enter plasticity. The fulfillment of Eq. (1) ensures an appropriately designed SBED specimen.
in Fig. 1(a) were also widely tested in concrete structural systems including rocking walls [13] and bridge piers [14]. However, only limited literature was referred to the behavior of energy dissipaters [10]. Problems of previously studied energy dissipaters cannot be neglected, which are addressed as difficult grouting, unsymmetric hysteretic performance and limited configurations in structures.
Asl / Ase = (dsl / dse )2 ⩾ σu/ σy 2. Proposed devices
(1)
where σu/σy is computed as 1.46 from Table 1, and values of (dsl/dse)2 for all SBED specimens given in Table 2 satisfies Eq. (1). To avoid the squeeze between the slubs and the inner surface of the tube, the incremental diameter of the slub after loaded, Δd, should be limited less than the gap provided between slubs and the tube, d1. Taking Poisson’s ratio, υ, into consideration, Eq. (2) as follows should be satisfied:
2.1. Concept of a new energy dissipater These problems addressed above are urged to be solved for expanding the application of energy dissipaters in precast structures. As demonstrated in Fig. 1(b), a new fuse-type energy dissipating bar was proposed recently. The working mechanism of the proposed energy dissipater can be idealized as a laterally restrained bar loaded under compression, where lateral supports externalized as slubs. The components between slubs, called segments, can be ideally viewed as short bar, which is favorable to prevent buckling of the segment by adjusting the length of the segment. The application of the new fuse-type energy dissipater can be extended to both compression and tension range, due to the prevention of the buckling. The shape of the core of the energy dissipater is similar to the bamboo, so the new energy dissipating bar is named as bamboo-shaped energy dissipater (BED) hereinafter. Considering the advantages such as low density, high strength-toweight ratio, excellent formability and recyclability of aluminum alloy, the aluminum alloy BED was previously studied [15]. The experimental results showed that the unsatisfactory hysteretic behavior of the aluminum alloy BED was obtained under relatively large axial deformation, while the similar results were proved in other literature [16]. To develop BED specimens with better hysteretic behavior under relatively large strain amplitude, the all-steel BED specimens were investigated. A series of tests including 12 all-steel BED (SBED) specimens was performed to compare the key design parameters and address the low-cycle fatigue performance and deformation patterns under relatively large nominal axial deformation. The details of specimens, the test set-up and the test results are summarized as follows.
Δd = 0.5dsl ε1 = 0.5dsl (−υε2) < d1
(2)
where ε1 and ε2 are strains in the radial and longitudinal directions of slubs, respectively. In the experimental protocol, the design of slubs satisfied the requirement of Eq. (2). Besides, two 50-mm fixed ends in each SBED specimen are prepared for actuators to achieve enough holding force. All tubes made of Q235b steel have 20.0-mm inner diameter (din) and 30.0-mm external diameter (dex). The design of tubes satisfied the requirement of the overall buckling proposed by Usami et al. [17]. 2.4. Nomenclature The first part of the label represents geometric dimensions of the SBED specimen. L refers to the length of the segment and S means the length of the slub. The second part shows the adopted testing protocol. C1, C2, C3 and C4 respectively refer to a 1%, 2%, 3% and 4% constant strain amplitude loading, while V1 and V2 indicate two different variable strain amplitude loading methods (detailed in Section 3). All SBED specimens have four segment and three slubs. For instance, L40S20-C1 means that the SBED specimen has four 40-mm segments, three 20-mm slubs and is tested under 1% constant strain amplitude. However, L40S5-V1 indicates that the SBED specimen has four 40-mm segments, two 5-mm side slubs, one 10-mm middle slub and is tested under variable strain amplitude VSA1. The middle slub in specimen L40S5-V1 expanded to 10 mm is designed to prevent damage caused by the stress concentration around the opening from happening before the failure of the segments.
2.2. Configuration and dimensions Referring to Fig. 2, a typical SBED specimen consists of an inner bamboo-shaped core and an outer restraining tube. The core is composed of a succession of slubs, segments and transition zones depicted in Fig. 2(b). The transition zone is set between the segment and the fixed end to spare enough space for the actuator moving back and forth without touching the tube. In the middle slub, an opening with a radius of 1.5 mm is prepared for the stopper which is made of a nail with a radius of 1.4 mm and length of 40 mm. The stopper is added to control
3. Test setup and loading patterns The test setup is shown in Fig. 3 and a SBED specimen was installed in the hydraulic servo universal testing machine MTS 810, which is capable of producing up to 250 kN loading and maximum displacement 39
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Fig. 2. Details of SBED specimen.
Table 1 Material property of Q235b steel. Material
Young’s modulus E (GPa)
Yield stress σy (MPa)
Yield strain εy (%)
Ultimate strength σu (MPa)
Ultimate strain εu (%)
Q235b
208.4
284
0.136
416
39.43
of ± 300 mm. During a typical test, the axial displacement and force of the actuator were automatically collected by a digital data acquisition system. In the present study, three different low-cycle loading patterns illustrated in Fig. 4 were applied to SBED specimens. All loading patterns were controlled by the nominal axial strain of the core which is derived from the axial displacement of the actuator divided by total length of segments, Lse,t. As shown in Fig. 4, the strain amplitude, Δε, is the absolute value of the peak or valley strain in each cycle. As shown in Fig. 4(a), a constant strain amplitude (CSA) was imposed cyclically to the SBED specimen until failure. Four cycles of strain amplitude 0.08% were firstly employed as an evaluated procedure for testing the SBED specimen and the equipment system. For this reason, counting of constant strain amplitude cycles started subsequently. Four different CSAs including 1%, 2%, 3% and 4% were adopted. There are two different variable strain amplitude (VSA) loading patterns adopted in the present study. The first VSA (VSA1) is shown in Fig. 4(b). Four cycles of 0.08% were imposed to the SBED specimen at first. Then, the strain amplitude of each two cycles increased from 0.5% to 4.0% with an increment of 0.5% and maintained the strain amplitude 3.0% until failure. Some SBED specimens were tested under the second VSA (VSA2), as shown in Fig. 4(c). In VSA2, the strain amplitude of each two cycles increased from 0.5% to 3.0% with an increment of 0.5% and then
Fig. 3. Picture of testing setup.
maintained the strain amplitude 2.0% until failure. A slip between the tested specimen and the clampers of the actuator was observed in the test; and therefore, a calibration test (see Fig. 5) was designed to capture the relationship between the force of the
Table 2 Measured geometric dimensions of SBED specimens. No.
Specimens
dsl (mm)
Lsl,1/Lsl,2 (mm)
dse (mm)
Lse (mm)
Ltr (mm)
Ltotal (mm)
Lct (mm)
d1 (mm)
(dsl/dse)2
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12
L40S20-C1 L40S20-C2 L40S20-C3 L40S20-C4 L60S20-C2 L60S20-C3 L40S5-V1 L60S5-V1 L80S5-V1 L40S20-V1 L60S20-V2 L70S20-V2
18.9 19 18.9 19.1 19 19 19 19 19 18.9 18.6 19
20.1/20.1 19.8/19.8 19.6/19.6 19.9/19.9 20.0/20.0 19.9/19.9 10.0/5.0 10.0/5.0 10.0/5.0 19.8/19.8 20.2/20.2 20.6/20.2
14 14 13.8 13.9 13.9 14 14 14.5 14 14.1 14.1 13.8
40.2 40.3 40.4 40.1 60 60 40.5 60.4 80.1 40.5 60.4 69.4
30.2 29.8 30.1 29.9 29.9 29.8 25.1 25.2 24.9 29.8 29.9 29.8
281.5 280.2 280.6 279.9 359.8 359.3 232.2 312 390.2 281 362 399
260.0 260.0 260.0 260.0 340.0 340.0 210.0 290.0 370.0 260.0 340.0 380.0
0.55 0.50 0.55 0.45 0.55 0.55 0.50 0.50 0.50 0.55 0.70 0.50
1.82 1.84 1.88 1.89 1.87 1.84 1.84 1.72 1.84 1.80 1.74 1.90
Note: dsl and Lsl are the diameter and length of the slub respectively; dse and Lse are the diameter and length of the segment; Ltr is the length of the transition zone; Ltotal is the total length of the core (excluding two fixed ends); Lct is the length of the tube; d1 is the calculated gap width between the slub and the tube. The symbols listed in the table are illustrated in Fig. 2.
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Fig. 4. Loading patterns: (a) CSA, (b) VSA1 and (c) VSA2.
Fig. 5. Slip-force relationship calibration test.
uc = u−P / K c
actuator and the slip value, which was influenced by the size of the clampers, the length and the diameter of the fixed end, and the material of the SBED specimen. To obtain the force-slip relationship for the tested SBED specimen, all parameters mentioned above were kept the same in both of the quasit-static and calibration tests. The total length of the calibrated bar excluding two fixed ends, Lt, was 200 mm and the length of the calibrated segment, Lc, was 100 mm. A 100-mm extensometer was employed over the length of the calibrated segment. Based on the measured force, Pc, and the measured average strain of the calibrated segment from the extensometer, εave, the relationship of the force and the slip value was computed as expressed in Eq. (3)
Pc = K c (Δt −εave × Lt )
(4)
where u and P are the displacement and the force of the actuator, respectively. 4. Experimental results and discussions 4.1. Low-cycle fatigue life The experimental stress-strain hysteresis curves after calibration are presented in Fig. 6. The tensile state of the SBED specimen is displayed in the positive direction. The abscissa is the nominal axial strain of the bamboo-shaped core defined as the calibrated displacement of the SBED specimen divided by the total length of segments, while the ordinate is the nominal axial stress derived from the axial force of the actuator divided by Ase. All tested SBED specimens demonstrate stable and repeated hysteretic capacity, without any local and overall
(3)
where Δt is displacement of the actuator; Kc is defined as the slip stiffness; εave was assumed constant along the calibrated bar. Then, the calibrated displacement of the SBED specimen, uc, was calibrated as 41
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Fig. 6. Stress-strain hysteresis curves of SBED specimens.
L80S5-V1 with four 80-mm segments were tested and compared. The numbers of cycles at additional constant strain amplitude, ni, were 14, 14 and 12 for the three SBED specimens, respectively. Similar low-cycle fatigue life were observed in specimens L40S5-V1 and L60S5-V1, which showed that the low-cycle fatigue life of SBED specimens had little relationship with the lengths of the segments for SBED specimens with 5-mm slubs and the stress concentration around the fillet accounted for the failure mechanism (see details in Section 4.3). However, a decrease of low-cycle fatigue life for specimen L80S5-V1 was observed due to torsion (see details in Sections 4.3 and 5.2), compared with L40S5-V1 and L60S5-V1. Furthermore, the effect of the different lengths of segments on the low-cycle fatigue life of SBED specimens with 20-mm slubs was
buckling during loading histories even at the maximum strain amplitude. Test results of all SBED specimens are summarized in Table 3. From the comparison between specimens S1-S4, including L40S20C1 with Nf of 114, L40S20-C2 with Nf of 47, L40S20-C3 with Nf of 29, L40S20-C4 with Nf of 11, the low-cycle fatigue life of SBED specimens in terms of the number of the failure cycles, Nf, were found to be related with the amplitudes of the constant strain. The increasing constant strain amplitude decreased the low-cycle fatigue cycles of SBED specimens. Two additional tests on L60S20-C2 with Nf of 39 and L60S20-C3 with Nf of 18 also addressed the same conclusion. To evaluate the effect of the different lengths of segments on the low-cycle fatigue life of SBED specimens with 5-mm slubs, L40S5-V1 with four 40-mm segments, L60S5-V1 with four 60-mm segments and 42
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Table 3 Test results of SBED specimens.
Table 4 Comparison between Kt and Kexp.
No.
Specimen
Δε (%)
Nf
ni
CPD (%)
Contact condition
Loading pattern
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12
L40S20-C1 L40S20-C2 L40S20-C3 L40S20-C4 L60S20-C2 L60S20-C3 L40S5-V1 L60S5-V1 L80S5-V1 L40S20-V1 L60S20-V2 L70S20-V2
1 2 3 4 2 3 – – – – – –
114 47 29 11 39 18 – – – – – –
– – – – – – 14 14 12 24 34 28
2612.2 2444.2 2418.1 1218.7 2081.8 1476.7 2031.7 2080.4 1942.2 2839.8 2226.4 1969.1
No No No No Yes Yes Yes Yes Yes Yes Yes Yes
CSA CSA CSA CSA CSA CSA VSA1 VSA1 VSA1 VSA1 VSA2 VSA2
Stiffness
Kt (N/mm) Kexp (N/mm) ΔK = 100 |Kexp − Kt|/ Kexp (%)
and Δy is the yielding deformation.
discussed by the comparison between specimens L60S20-V2 with four 60-mm segments and L70S20-V2 with four 70-mm segments. The number of cycles at additional constant strain amplitude, ni, was 34 for specimen L60S20-V2 and 28 for specimen L70S20-V2. The low-cycle fatigue cycles of SBED specimens with 20-mm slubs decreased with the increase of the length of the segment. When L40S5-V1 with ni of 14 and L40S20-V1 with ni of 24 were compared, it is found that the low-cycle fatigue life of SBED specimens with the same length of segment increased with the length of slubs. This is caused by the different rotation capacity of the 5-mm slubs and 20mm slubs [15]. The free rotation of slubs occurs when din is more than the diagonal length of the slub and the restrained rotation of slubs appears at the situation of din less than the diagonal length of the slub. The weaker rotation capacity of 20-mm slubs decreased the lateral deformation of the segments (see details in Section 4.3) and improved the low-cycle fatigue life of the SBED specimen. 4.2. Analytical stiffness model To compute the initial elastic stiffness of SBED specimens, an inseries stiffness model was proposed shown in Fig. 7. As expressed in Eq. (5), the initial elastic stiffness, Kt, is evaluated as the sum of the individual stiffness of the in-series slubs, segments and transitional zones.
1 (n1/ Ksl,1 + n2/ Ksl,2 + m / Kse + 2/ Ktr )
L40S20-C3
L60S20-C3
L40S5-V1
L60S5-V1
L80S5-V1
140,772 131,382 7.2
85,070 94,717 10.2
160,072 151,281 5.8
114,009 105,098 8.5
88,533 83,086 6.6
SBED specimens. As shown in Fig. 8(a)–(d), the lateral deformation of SBED specimens increased with the constant strain amplitude. The shape of the specimen L40S20-C1 nearly kept in a straight line after loaded while the specimen L40S20-C4 exhibited a relatively severer deformation. No contact between segments and the restraining tube was observed in SBED specimens with 40-mm segments and 20-mm slubs under constant strain amplitudes of 1%, 2%, 3% and 4%. When comparing Fig. 8(e) and (f), the lateral deformation of specimen L40S20-V1 with larger slubs was smaller than that of specimen L40S5V1 with smaller slubs. It proved that the weaker rotation capacity of slubs decreased the lateral deformation of the segments. As shown in Fig. 8(e), (g) and (h), the contact in L40S5-V1 was viewed as the point contact and the contact conditions in L60S5-V1 and L80S5-V1 were the line contact, while the contact areas of the three SBED specimens gradually increased with the length of segments. Especially, a torsional deformation was observed in specimen L80S5-V1 via the scratches of the red paint on the surface of segments. The failure modes of SBED specimens were affected by the lateral deformation resulted from bending, stress concentration around the fillet and the torsion in the segment. Specimens L40S20-C1, L40S20-C2, L40S20-C3 and L40S20-C4 failed around the wave peak depicted in Fig. 8(a)–(d), where the high cumulative plastic strain existed. From the local enlarged drawings of the four specimens, the fracture surfaces developed from relatively smooth and vertical to rough and oblique with the increase of strain amplitude, which means that the fracture modes evolved from tensile failure to shear failure. The failure positions of specimens L40S5-V1, L40S20-V1 and L60S5-V1 were around the fillet shown in Fig. 8(e)–(g), where the intense stress concentration existed. The smoother fillets are required to avoid such failure caused by stress concentration. The specimen L80S5-V1 finally failed at the position of scratches caused by the torsion in segment, as shown in Fig. 8(h).
Note: Nf is number of failure cycles, as shown in Fig. 4(a); ni is number of cycles at additional constant strain amplitude, as shown in Fig. 4(b)and (c); CPD is cumulative inelastic deformation [18] = ∑i |Δpi |/Δy , where |Δpi | is the plastic deformation in ith cycle
Kt =
Specimen
4.4. Compression-strength adjustment factor
(5)
where Ksl,i equal to EAsl/Lsl,i is the elastic stiffness of the slub, and i is 1 for the middle slub and 2 for the side slub; Kse equal to EAse/Lse is the elastic stiffness of the segment; Ktr equal to EAtr/Ltr is the elastic stiffness of the transition zone; n1 and n2 is the number of the middle slub and side slubs, respectively; m is the number of segments. As listed in Table 4, the accuracy of Eq. (5) was verified by the comparison between the analytical elastic stiffness, Kt, and measured elastic stiffness, Kexp.
The compression-strength adjustment factor, β, was affected by the contact force and the friction force between the bamboo-shaped core and the restraining tube, which is defined as the ratio of the maximum compression force to the maximum tension force in one hysteresis loop [18]. The calculated β of representative SBED specimens is shown in Fig. 9. All β values of SBED specimens are below 1.3 satisfying the requirement in AISC 341-10 [18]. As shown in Fig. 9(a), the variation of β values was divided into three stages:
4.3. Deformation and failure modes The failure pictures of SBED specimens shown in Fig. 8 demonstrated failure positions, contact conditions and deformation patterns of
(1) Initial adjustment stage. The β values were relatively high and decreased at the first few cycles due to the cyclic hardening effect [7]. In this period, the cyclic stress amplitude increased considerably with the loading cycles until the formation of a static hysteresis loop. However, the first β value less than 1.0 in specimen L40S20C1 was resulted from the relatively high upper yield force. (2) Stable stage. When the strain amplitude was relatively high (e.g., 3% and 4%), the β values increased with the loading history because of the increasing friction forces, which were resulted from the
Fig. 7. In-series stiffness model.
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Fig. 8. Deformation and failure modes of SBED specimens.
(CSA) stage in VSA1, β values of specimens L40S5-V1, L60S5-V1 and L80S5-V1 increased with the loading cycles. Based on the analysis of β values’ variation, recommended β values can be provided for different strain amplitudes as follows: The recommended β value for SBED specimens is 1.0 for strain amplitude below 2% and 1.2 for strain amplitude at 4%. Under the strain amplitude between 2% and 4%, the recommended β value can be obtained by linear interpolation. When SBED specimens are adopted in some practical project, β values of the applied SBED specimens can be determined based on the actual required strain amplitude in the project; and therefore, the design SBED strength in compression can be thus obtained from Eq. (6) [18]:
severer contact between slubs and the tube. When the strain amplitude was relatively low (e.g., 1% and 2%), the β values kept almost unchanged. The nearly straight lateral deformation of the bamboo-shaped core in specimen L40S20-C1 caused small contact and friction forces between slubs and the tube, and accounted for β values of specimen L40S20-C1 almost 1.0 after stabilization; (3) A sharp increase of β values in some specimens like L40S20-C2, L40S20-C3 and L40S20-C4 at the last loading step occurred because of a distinct decrease of the maximum tension force before failure. In Fig. 9(b), β values at the variable strain amplitude (VSA) stage in VSA1 varied with fluctuation. During the constant strain amplitude
Fig. 9. Compression-strength adjustment factors of SBED specimens: (a) under CSA and (b) under VSA1.
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respectively). The segments were numbered as N1, N2, N3 and N4 to clearly analyze the torsion in each segment, as shown in Fig. 12. The positive torsion was regarded as the clockwise direction. The torsion angles of specimens L40S5-V1 and L80S5-V1 are presented in Fig. 13 to evaluate the relationship between the lengths of segments and the torsion angles. The torsion angle of specimen L40S5-V1 was negligible compared with that of specimen L80S5-V1, which agreed well with the test observations of specimens L40S5-V1 and L80S5-V1 in Fig. 8(e) and (h). As depicted in Fig. 13, the torsion angle varied with the distance to the zero position in specimen L80S5-V1, and the maximum torsion angle was found in segment N2 with a value of 0.27 rad. From the shape of the torsion curve in Fig. 13, the torsion of the bamboo-shaped core was almost symmetric with the middle slub.
Fig. 10. Finite element model for SBED specimen.
Fca = βωFy
(6)
where ω is the strain hardening adjustment factor; Fy is the axial yield strength of SBED specimen, which is established using σy determined from coupon tests. ω is estimated by the ratio of σu to σy from coupon tests, which is 1.46 for the Q235b material adopted in the present study.
5.3. Effect of cumulative plastic deformation As discussed in Section 4.3, some SBED specimens failed around fillets due to intense stress concentration, while the failure mechanism of some other SBED specimens including specimens L40S20-C1, L40S20-C2, L40S20-C3 and L40S20-C4 could be explained as the effect of the cumulative plastic deformation. As shown in Fig. 14, the cumulative plastic strain contours are plotted for specimens L40S20-C3 and L40S20-C4 at the maximum compressive strain of 11th cycle. The maximum cumulative plastic strains of specimens L40S20-C3 and L40S20-C4 were 3.084 and 3.985, respectively. The higher cumulative plastic strains could result in earlier failure of the SBED specimen, so that the higher low-cycle fatigue life of specimen L40S20-C3 than specimen L40S20-C4 was explained in terms of the cumulative plastic strain. The positions of the maximum cumulative plastic strains of specimens L40S20-C3 and L40S20-C4 were exactly the same with the observed failure positions of the two specimens depicted in Fig. 8(c) and (d). The failure of the SBED specimen initiated from the concave where the cumulative plastic strain was maximum and gradually developed to full section. The maximum cumulative plastic strain of specimen L60S20-C3 extracted from the numerical analyses was 3.855, which is more than that of specimen L40S20-C3 but less than that of specimen L40S20-C4. It agreed well with the test result that the lowcycle fatigue life of specimen L60S20-C3 was worse than specimen L40S20-C3 but better than specimen L40S20-C4.
5. Numerical discussions 5.1. Validation of finite element model The finite element analyses via ABAQUS [19] were conducted to investigate the torsion deformation, failure mechanism and contact conditions of SBED specimens. The established model for the SBED specimen is presented in Fig. 10. The C3D8R (8-node reduced integrated) elements were assigned to the SBED specimen. A calibrated combined hardening model was adopted for the bamboo-shaped core and the tube was considered elastic. Calibrated parameters of the combined hardening model are listed in Table 5. The evolution law of the combined hardening model [19] can be described as a nonlinear kinematic hardening component and an isotropic hardening component. The nonlinear kinematic hardening component is concerned with Ci and γi and the isotropic hardening behavior of the model defines values of Q∞ and b. The interaction property employed in the model included tangential and normal behaviors to transmit shear forces and normal forces. The isotropic directionality and friction coefficient of 0.1 [20] were considered in the tangential behavior and the hard contact was adopted in the normal behavior. The initial geometric imperfection with magnitude of 1/1000 of the length of the core was applied to the SBED specimen by perturbations in the geometry. All degrees of freedom of two ends of the tube were constrained. The y-direction constraint of the bambooshaped core was released to apply axial displacements while other degrees of freedom were constrained. The displacements employed in finite element analyses were the same with the calibrated displacement in tests. The comparisons between the results of tests and finite element analyses chosen from representative specimens are presented in Fig. 11. The numerical elastic stiffness and hysteresis loops matched well with the experimental results.
5.4. Evolution of contact The contact stress contours of specimen are presented in Fig. 15. The positions with contact stress above zero obtained from numerical analyses agreed well with the contact positions observed in both specimens L60S5-V1 and L80S5-V1. A gradual contact phenomenon was observed in the analyses of contact conditions for SBED specimens. The contact stress contours of specimen L60S20-C3 at the maximum compressive strain of different loading cycles are shown in Fig. 16. No contact stress was observed in segments of specimen L60S20-C3 in the first cycle, and then the contact between the segment and the tube gradually occurred with the progress of loading cycles. The gradual contact was mainly caused by the cumulative plastic deformation in segments. A displacement-based model [15] including a segment and two halves of slubs was proposed to explore the relationship between the maximum lateral displacement of the segment and the length of the segment under the relatively low strain amplitude in the previous study. Both of the initial geometrical imperfection of the segment and the rotation of the slub have been considered in the proposed model. The maximum lateral displacement of the segment caused from the rotation angle of the slub is α(Lse + Lsl,2)/π when the rotation angle of the slub is α. The maximum initial geometrical imperfection is assumed to be 0.001Lse. The maximum lateral displacement of the segment under a certain axial load F was evaluated by Eq. (7) [15], where the loading and unloading responses were taken into account under the relatively
5.2. Torsion of segment The displacement contours for specimens L40S5-V1 and L80S5-V1 are plotted at the at the last cycle before failure (ni = 14 and 12, Table 5 Calibration parameters of combined hardening model for Q235b steel. Material
σy
Q∞
b
C1
γ1
C2
γ2
C3
γ3
Q235b
284
10
1.2
100,000
2500
7500
100
500
0
Note: Q∞ is the maximum change in the size of the yield surface; b defines the rate at which the size of the yield surface changes as plastic straining develops; Ci and γi are parameters related with backstresses.
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Fig. 11. Comparisons between results of tests and finite element analyses.
Fig. 12. Computational graph of torsion angle: (a) specimen L40S5-V1 and (b) specimen L80S5-V1.
factor, adopted as 0.5. To analytically judge the contact conditions of the SBED specimen loaded under the relatively high strain amplitude, a modified displacement-based judgement model was developed considering the effect of the cumulative plastic deformation. For SBED specimens loaded under relatively larger strain amplitude, the full section of the segment gradually entered plasticity with the loading histories. The equivalent flexural modulus Er in Eq. (7) was then replaced with the tangent modulus Et approximated as 0.02E for simplification [8,21] in Eq. (8).
Llat ,u =
low strain amplitude.
(
α
α
π 2Er Ise 0.001Lse + π Lse + π Lsl,2
)
π 2Er Ise−F (μLse )2
(7)
where Er equal to 8EEt/(E + Et ) is equivalent flexural modulus [15]; Ise is the moment of inertia of the segment; μ is the effective length 1/3
α
α
π 2Et Ise−F (μLse )2
) (8)
The lower and upper bounds of the maximum lateral displacement of the segment were then obtained from Eq. (7) and Eq. (8), respectively. As computed in Table 6, the lower bound of the maximum lateral displacement of the 60-mm segment is 1.63 mm while the upper bound is 4.7 mm. The calculation result from the lower bound of the maximum lateral displacement of the segment showed that no contact occurred. It explained the contact condition in the first cycle of specimen L60S20C3, where the loading and unloading responses should be considered. The contact was expected based on the computation of the upper bound with consideration of full-section plasticity, which agreed well with results of test and finite element analysis at later loading cycles. The numerical analyses of contact conditions for SBED specimens with 40-mm segments under different constant strain amplitudes were
Fig. 13. Torsion angles of SBED specimens.
Llat ,l =
(
π 2Et Ise 0.001Lse + π Lse + π Lsl,2
1/3 3
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Fig. 14. Cumulative plastic strain contours: (a) specimen L40S20-C3 and (b) specimen L40S20-C4.
Fig. 15. Contact stress contours and failure photos: (a) specimen L60S5-V1 and (b) specimen L80S5-V1.
Fig. 16. Contact evolution of SBED specimen L60S20-C3.
[22], expressed in Eq. (9).
Table 6 Lower and upper bounds of segment’s lateral displacement. Lse (mm)
40
60
Llat,l/Contact Llat,u/Contact
1.08/No 1.46/No
1.63/No 4.7/Yes
1 + 2n w ⩽
PL2 ⩽ 1 + 4n w 4π 2EI
(9)
where the values of nw in the left and right sides of the inequality equation represent the upper and lower bounds of the wave number, respectively; L is the length of the constrained column; I is the moment of inertia of the constrained column. Eq. (9) was extended to the elasticplastic behavior by replacing E with the tangent modulus Et based on the elastic-plastic buckling theory proposed by Shanley [23]. Then, the wavelength formed in an individual segment is estimated in Eq. (10).
conducted to further verify the correctness of Eq. (7) and Eq. (8), as shown in Fig. 17. The numerical results of specimens L40S20-C3 and L40S20-C4 were taken at the maximum compressive strain amplitude of 29th cycle and 11st cycle, respectively. No contact was observed in the numerical results of specimens L40S20-C3 and L40S20-C4 during the whole loading histories, which agreed well with the test observations discussed in Section 4.3. The analytical lower and upper bounds of the lateral displacement of the SBED specimen with 40-mm segments listed in Table 6 were 1.08 mm and 1.46 mm respectively, both of which were less than the gap width, d2, between segments and the tube. The prediction of contact conditions from the lower and upper bounds of the lateral segment’s displacement for specimens L40S20-C3 and L40S20C4 satisfied well with the finite element analyses and test observations. Therefore, the lower bound of the maximum lateral displacement of the segment expressed in Eq. (7) is recommended to be adopted in predicting the initial contact conditions for SBED specimens, while the upper bound expressed in Eq. (8) is adopted to predict the final contact conditions for SBED specimens.
Lse 1 int ⎛ 2
⎝
Lse
⩽ lw ⩽
2 PLse
1 −2⎞ 4π 2Et Ise ⎠
1 int ⎛ 4
⎝
2 PLse
4π 2Et Ise
1
−4⎞ ⎠
(10)
where the ‘int’ is a function to discard all numbers after the decimal point and leave the integer; lw is the high-mode buckling wavelength of the segment. The lower bound of lw [24] is adopted to evaluate the development of the wave in SBED specimens, as expressed in Eq. (11).
Lse
lw = 1
int ⎛ 2 ⎝
2 PLse
4π 2Et Ise
1
−2⎞ ⎠
(11)
To form at least a full wave in an individual segment, the minimum Lse should be 210 mm computed from Eq. (12), where the value of P was taken as 63.1 kN from the maximum force in specimen L60S5-V1.
5.5. Estimation of wavelength
1 2
The wave number, nw, of a bi-laterally constrained elastic column was estimated by solving the fourth order, linearized differential equation of an Euler beam under increasing axial compression load
PLse2 1 − ⩾1 4π 2Et Ise 2
(12)
The wavelength would remain nearly unchanged even if the subsequently applied compressive strain was smaller than the previous one
Fig. 17. Contact evolution: (a) specimen L40S20-C3 and (b) specimen L40S20-C4.
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Fig. 18. The development of waves in different lengths of segments (lateral displacement magnified by 4).
Fig. 19. Effect of side slubs on wave development (lateral displacement magnified by 4).
[7]. A series of finite element analyses was then conducted to verify the relationship between the length of the segment and the development of waves described in Eq. (11), where all finite element cases were loaded to the maximum strain amplitude (4%) of loading pattern VSA1 depicted in Fig. 4(b). The nomenclature of numerical cases in Figs. 18 and 19 was the same with that of SBED specimens. Numerical results are shown in Fig. 18. Until the length of the segment increased up to 160 mm in Fig. 18(b), a fully developed wave was observed in an individual segment between two adjacent slubs. The analytical minimum value from Eq. (12) to form a full-length is more than the exact value obtained from finite element analyses. This difference was mainly because the derivation of Eq. (11) did not take the rotation of slubs into consideration. A rotation adjustment factor, Cr, considering the rotation of slubs was then introduced based on the test results and finite element analyses, expressed in Eq. (13).
SBED specimens were compared to provide an effective way to fabricate the new SBED. The main results are summarized as follows: 1. On the basis of the test results, all SBED specimens show stable and repeated hysteresis loops without any local and overall buckling even at the maximum strain amplitude. Geometrical parameters, including the length of slub and segment, have significant influence on the hysteretic behaviors and deformation patterns of SBED. 2. The proposed in-series stiffness model can be effectively applied to evaluate the elastic stiffness of SBED and the displacement-based judgement model is capable of defining the contact conditions of SBED. 3. The failure modes of SBED are affected by the lateral deformation resulted from bending, stress concentration around the fillet and the torsion in the segment. The stress concentration and torsion are expected not to happen in SBED; therefore, special details are required to reduce the stress concentration by smoothing the fillets, and the length of the segment should be controlled to limit the torsion. 4. Based on the evolution law of the compression-strength adjustment factor β, the recommended value of β for SBED varies with the applied strain amplitude, where 1.0 for strain amplitude below 2% and 1.2 for strain amplitude at 4%. For the strain amplitude between 2% and 4%, the linear interpolation can be employed to obtain the recommended β value. 5. A calibrated theoretical derivation considering the effect of the slubs’ rotation is proposed to evaluate the wave development in segments. The 5-mm side slubs are proved to have no influence on the wave formation but reduce the lateral deformation of the segment.
Lse
lw = 1 int ⎛⎜Cr ⎛ 2
⎝
⎝
2 PLse
4π 2Et Ise
− 2 ⎞⎞ ⎠⎠ 1
⎟
(13)
where Cr is calibrated as 1.56 for 20-mm slubs. Based on Eq. (13), two full waves can be formed in an individual segment with length more than 250 mm. As verified in Fig. 18(d), two full waves were observed in numerical case L280S20-4% with 280-mm segments, which agreed well with the analytical result based on Eq. (13). The development of waves in SBED specimens with 5-mm side slubs were also investigated based on test results and finite element analyses. The wave development of specimen L80S5-V1 (see Fig. 10(h) and Fig. 19(a)) and case L160S20-4% in Fig. 19(b) was almost same. The lateral deformation of segments observed in specimen L80S5-V1 was reduced compared with case L160S20-4%, which is resulted from the existence of 5-mm side slubs capable of reducing the lateral deformation of segments.
Acknowledgements The authors would like to acknowledge financial supports from the National Key Research and Development Program of China (2016YFC0701400), National Natural Science Foundation of China (51508363, 51678138), Natural Science Foundation of Jiangsu Province of China (BK20151407), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
6. Conclusions In the present study, 12 all-steel bamboo-shaped energy dissipaters in all were tested to investigate hysteretic behaviors and deformation patterns of SBED and develop high-performance energy dissipaters under relatively high strain amplitude. Besides, design parameters of 48
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Appendix A. Supplementary material
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