Experimental performance of buckling-restrained braces with steel cores of H-section and half-wavelength evaluation of higher-order local buckling

Abstract

To prevent lower-order local buckling of H-section steel core, an improved type of buckling-restrained braces named buckling-restrained brace with H-section steel core was proposed by the authors. This article further investigates the effect of configuration details on seismic performance of buckling-restrained braces with H-section steel core and compares two half-wavelength calculation methods for higher-order local buckling of H-section steel core. First, quasi-static cyclic tests are described on two newly designed buckling-restrained braces with H-section steel cores and another buckling-restrained brace with flat steel core. Then, Bleich’s and Lundquist’s methods are reviewed for evaluating half wavelength of higher-order local buckling based on elastoplastic buckling theory of plates and compared with the test results of four buckling-restrained braces with H-section steel core including the two from a previous test. It is found from the test results that due to H-section steel core’s higher self-stability, the compression force fluctuation was not observed on the hysteretic loops of buckling-restrained brace with H-section steel core with even larger clearance but on the buckling-restrained brace with flat core. The buckling-restrained brace with H-section steel core was also advantageous over the buckling-restrained brace with flat core in terms of having lower compression strength adjustment factor β. A stopper in the middle of the core member and the gradual change of cross section of the core plate around the end of stiffeners could help to improve the fatigue performance of buckling-restrained braces. The test results also confirmed that Lundquist’s theory was more reliable for evaluating the half wavelength of higher-order local buckling for H-section steel core.

Keywords

buckling theory of plate buckling-restrained brace half wavelength higher-order local buckling H-section steel

Introduction

Buckling-restrained braces (BRBs) have been widely used to improve seismic performance of structures for their excellent energy dissipation capability and strong appeal to structural engineers with conventional building materials such as steel and concrete. A typical BRB mainly contains three parts: the inner core, the restraining member, and the unbonded material (or clearance) between them. The inner core is designed to resist the imposed external load and dissipate the input energy through its elastic–plastic deformation. The restraining member is used to restrict the lateral deformation of inner core to ensure that the inner core can yield rather than buckle in compression. The unbonded material (or clearance) is set to accommodate Poisson’s effect of inner core and reduce the friction between the core and restraining member.

For the steel core of BRBs, typical cross sections used in practice are rectangle, cruciform, H-section, and circular ring. Compared to flat or cruciform steel with the same cross-sectional area, the steel core with H-section is usually more stable on its own. Nagao et al. (1992a, 1992b) proposed BRBs with an H-section steel core restrained by reinforced concrete and discussed their global stability design criteria considering the stiffness and strength requirements of the restraining reinforced concrete. Suzuki et al. (1994) and Kouno et al. (1994) compared the seismic performance of BRBs with the H-section steel core covered by reinforced concrete, prestressed reinforced concrete, steel fiber concrete, or circular steel tube. Suzuki et al. (1995), Usami and Kaneko (2001), Usami et al. (2002), Iwata et al. (2000), Ju et al. (2009), and Kim et al. (2015) conducted research on the seismic behavior of BRBs with an H-section steel core restrained by a steel tube.Tuchida et al. (1999), Kobayashi et al. (1999), and Kaneko et al. (2000) investigated the mechanical behaviors and design method of BRBs whose restrainer was a square steel tube with tuning bolts. Narihara et al. (2004) and Suzuki et al. (2005) developed a simple type of BRBs with H-section steel core. The energy dissipation portion of the core member was fabricated by partly cutting the flanges of H-section steel, and the restraining members were two channel steels lying on each side of the web and fastened together by high-strength bolts. Oda and Usami (2010) proposed a novel type of BRB with the H-section steel core restrained by a steel tube assembled with two steel plates and two channel steels with high-strength bolts. The technology proposed by Oda and Usami (2010) can not only be used to manufacture new BRBs but also be applied to reinforce unconstrained H-section steels in existing structures. However, because the flanges and web were not well restrained, serious lower-order local buckling of H-section steel core occurred, and the ultimate deformation capacity of the specimen was limited. Configuration improvements over the BRBs proposed by Oda and Usami (2010) were made by Funayama et al. (2012), where two steel plates as the inner restrainers were welded perpendicularly to the back of the channel steel in the longitudinal direction. Once the restraining members were assembled, the flange of H-section steel core can be well restrained by the inner welded plate and the outer face plate. As a result, the ultimate deformation capacity of the specimens was effectively improved. To prevent lower-order local buckling of H-section steel core as observed by Oda and Usami (2010), the authors of this article also proposed an improved type of buckling-restrained braces with H-section steel core which was named HBRB (Li, 2011; Li et al., 2013). The configuration and assembling process of a typical HBRB are shown in Figure 1, where it is seen that the H-section steel core is sandwiched by two U-shaped steels and two restraining plates which are assembled with the fillers and high-strength bolts. Three HBRB specimens were tested in Li et al. (2013), and test results showed that the ultimate deformation capacity of HBRBs was higher than those in Oda and Usami (2010), and all the H-section steel cores successfully developed into the higher-order buckling mode.

Figure 1.

Configuration of HBRB.

When the H-section steel core develops into the higher-order buckling mode, local stability design and corresponding fatigue life analysis become critical to ensure satisfactory performance of HBRBs. For restraining member, it should have enough strength to resist the contact force caused by local buckling of the core; for the core itself, its total strain, that is, axial strain plus bending strain, should not be too large for the sake of fatigue life. It has been shown that the contact force and bending strain have direct links to half buckling wavelength of the inner core (Chou and Chen, 2010; Genna and Gelfi, 2012a, 2012b; Jiang et al., 2015; Lin et al., 2012; Matsui and Takeuchi, 2012; Takeuchi et al., 2007, 2010, 2012; Wu et al., 2014; Wu and Mei, 2015). Although the buckling mode and half wavelength of BRB’s core member have been studied in recent years, most of them were focused on BRBs with flat steel core (Genna and Gelfi, 2012a, 2012b; Jiang et al., 2015; Wu et al., 2014; Wu and Mei, 2015), and researches on section steel core were rather limited. Zhao et al. (2014) studied local buckling behavior of steel angle core and calculated half wavelength of local buckling based on Bleich’s buckling theory of plates where the elastoplastic plate was regarded as bidirectional orthotropic. But the problem is that Bleich’s formulation adopted by Zhao et al. (2014) has not been sufficiently validated through experimental data elsewhere, while the modified version of Bleich’s formula, which has been widely cited, did not include the information of half wavelength, although it was based on the test results. To provide recommendations for the design of plates used in aviation, Lundquist also proposed a buckling theory of plates based on large amount of plate tests, which offered another method for half-wavelength calculation of buckled plate.

This article is intended to further investigate the effect of configuration details on seismic performance of HBRBs and find out the half-wavelength evaluation method for higher-order local buckling of H-section steel core. First, quasi-static cyclic tests on two newly designed HBRB specimens and another specimen with flat steel plate core are described. Then, two methods for evaluating half wavelength of higher-order local buckling based on elastoplastic buckling theory of plates proposed by Bleich and Lundquist are reviewed, respectively, and the one presented by Lundquist is chosen for H-section steel core through comparison, which is validated by the test results in this article as well as those in Li et al. (2013).

Quasi-static test

Configuration of HBRB specimens

Figure 2 shows the configuration of four HBRB specimens labeled S1–S4, among which S4 differs from the other three mainly in that S4 had a stopper in the middle of the core. The core members of the HBRBs were fabricated by HW100 × 100 of hot rolled H-section steel with stiffening plates welded to flange edges as shown in Figure 3. Note that S1 and S2 were previously tested and denoted as HBRB2 and HBRB3 in Li et al. (2013), respectively, while S3 and S4 were newly tested specimens. To compare the seismic performance under relatively large clearance, a BRB employing flat steel plate core which is labeled S5 was also designed and tested, and the value of the clearance between inner core and restraining members was set close to that of HBRB. The configuration of S5 and its core member is presented in Figures 4 and 5, respectively. The geometric dimensions of all the five BRBs are summarized in Tables 1 and 2.

Figure 2.

Details of HBRB (S1–S4).

Figure 3.

Details of H-section steel core and restraining member.

Figure 4.

Details of BRB with flat steel core (S5).

Figure 5.

Details of flat steel core and restraining member.

Table 1.

Dimensions of H-section steel core.

Specimen	Core type	Steel mill	L _s (mm)	L _y (mm)	t _w (mm)	t _f (mm)	W (mm)	H (mm)	R (mm)	A _y (mm²)	A _s (mm²)
S1	A	Laiwu	150	1050	5.5	7.0	100	100	10	1959	4368
S2	A	Laiwu	180	1050	5.5	7.0	100	100	10	1959	4368
S3	A	Tangshan	180	1050	6.0	7.1	100	100	10	2021	4421
S4	B	Tangshan	180	1050	6.0	7.1	80	100	10	1737	4137
S5	–	Anshan	135	900	–	7.4	–	–	–	592	1552

L _s: length of stiffening segment at each end; L_y: length of yielding segment; t_w and t_f: thicknesses of web and flange, respectively; W and H: width and height of cross section of H-section steel, respectively; R: chamfering radius; A_y and A_s: cross-sectional areas of yielding and stiffening segments, respectively.

Table 2.

Dimensions of the restraining member.

Specimen	U-shaped steel				Restraining plate
Specimen	L _u1 (mm)	L _u2 (mm)	L _u (mm)	t _u (mm)	L _r (mm)	t _r (mm)
S1	130	150	1280	7.5	1280	9.6
S2	160	180	1340		1340
S3	160	180	1340		1340
S4	180	180	1350		1350
S5	–	–	1140		1140	19.8

L _u1 and L_u2: length of the stiffening segment at each end of the U-shaped steel; L_u and L_r: total length of the U-shaped steel and restraining plate, respectively; t_u and t_r: thicknesses of the U-shaped steel and restraining plate, respectively.

Material properties

Table 3 lists material properties of the core member for specimens S3–S5 based on tensile coupon tests.

Table 3.

Mechanical properties of steel cores.

Specimen	Location	Steel grade	f _y (MPa)	f _u (MPa)	Elongation (%)
S3, S4	Flange	Q235	278	418	29.2
S3, S4	Web	Q235	317	427	29.7
S5	Core plate	Q235	328	495	26.7

f _y: yield stress; f_u: ultimate stress.

Test setup and loading protocol

Component tests were conducted on the 2500-kN MTS electrohydraulic testing machine at the Structural and Seismic Testing Center, Harbin Institute of Technology. As shown in Figure 6, both ends of the specimen were firmly gripped by the hydraulic clamp of the testing machine. Reaction force and axial displacement of each specimen were measured by the built-in force and displacement sensors of the actuator. The specimens were loaded in a displacement control mode based on the loading protocol expressed in nominal strain of the core as shown in Figure 7, where the target core strain was 0.4%, 0.6%, 0.8%, 1.0%, 1.2%, 1.4%, 1.6%, 1.8%, 2.0%, 2.2%, 2.4%, 2.6%, 2.8% (with two cycles each except for 1.4% with six cycles), and 3.0% (with cyclic loading kept until the specimen failed). The nominal strain of the core was defined as the deformation of the BRB divided by the length of yielding segment L_y.

Figure 6.

Test setup.

Figure 7.

Loading protocol.

It is required by the Chinese technical specification JGJ 99-98 (1998) that the inter-story drift of a high-rise steel structure should be no larger than 1/70 under severe earthquake. According to the study by Iwata and Murai (2006), core strain of 1.4% approximately corresponds to the inter-story drift of 1/70 for a typical frame where the BRB is installed with a horizontal inclination of 45°, and the yielding length of the inner core is half of the brace’s total length. As a result, six loading cycles were applied at the target strain of 1.4% to investigate the seismic performance of each specimen under severe earthquake. In addition, given that the core strain demand of BRBs generally remains within the range of 1%–2% (Tremblay et al., 2006), 3% was selected as the maximum target core strain.

Test results

Hysteretic response

The hysteretic loops with nondimensional axial force P/P_y (load divided by the yield force of the specimen) versus nominal core strain ε are presented in Figure 8 for S3–S5, while those of S1 and S2 were included in Li et al. (2013). The load-bearing capacity of S3 and S4 declined on the tensile side of the second loop with target strain of 2.2% and 3.0%, respectively, while S5 failed on the fifth tension with target strain of 3%. Compared with the test results of Iwata et al. (2000), Ju et al. (2009), Tuchida et al. (1999), Kaneko et al. (2000), and Oda and Usami (2010), it can be found that the local stability performance and ultimate deformation capacity of HBRB are more desirable for the H-section steel core herein was well restrained by the restraining members.

Figure 8.

Hysteretic curves of specimens (a) S3, (b) S4, and (c) S5.

Obvious fluctuations can be observed on the hysteretic loops of specimen S5, which could be caused by the buckling of the core plate. In a displacement control mode, the axial force may jump back a little when the BRB develops into next higher buckling mode and then goes up until the next transition of buckling mode (Prasad, 1989; Zhao and Wu, 2009). Note that the total thickness of clearance along the weak axis of the core is 3.2 mm, as shown in Figure 4, which is quite large. This may explain why the compression force fluctuation herein appears more significant than many experimental results of BRBs with flat steel cores reported elsewhere such as in Mei (2015).

In contrast, with even larger clearance than S5 as can be seen from comparison between Figures 2 and 4, no obvious force fluctuations are found on the hysteretic loops of specimens S3 and S4. Such advantageous property may benefit from the higher stability of H-section steel than flat steel. Because the flange and web of H-section steel can provide restraining effect for each other, local stability performance of H-section steel can be much better than flat steel plate if the clearance between inner core and restraining members is the same in the two cases.

As to be mentioned in sections “Bleich’s theory” and “Lundquist’s theory” that the tangent modulus E_t and hence the modulus degradation factor η which equals to the ratio of tangent modulus E_t to elastic modulus E should be determined to calculate half wavelength. The tangent modulus E_t can be obtained from the tangent stiffness K_t of the specimen by

E_{t} = \frac{L_{y}}{A_{y} [1 / K_{t} - (2 L_{s}) / E A_{s}]}

(1)

where A_y, A_s, L_s, and L_y are defined in Table 1. To eliminate the effect of friction force between steel core and restraining member, the tangent stiffness K_t is determined from the data of hysteretic loops in tension (Zhao et al., 2014). Thus, obtained values of K_t and η are listed in Table 4, where ε_ultimate refers to the maximum nominal tensile strain that the core ever reached.

Table 4.

Modulus degradation factor.

Specimen	ε _ultimate (%)	K _t (kN/mm)	η (%)
S1	3.0	1.74	0.464
S2	2.4	1.52	0.410
S3	2.2	1.12	0.292
S4	3.0	0.95	0.288

Values of K_t and η at strain level of ε_ultimate are obtained based on the last hysteretic loop prior to failure.

Seismic performance evaluation

The mechanical properties of S3–S5 are evaluated primarily through four indices in this article, namely, elastic stiffness, yield force, compression strength adjustment factor β, and cumulative plastic ductility (CPD).

Elastic stiffness and yield force

With the yielding segment and stiffening segment of the core member in series, the initial elastic axial stiffness can be obtained as

\frac{1}{K_{ec}} = \frac{L_{y}}{E A_{y}} + \frac{2 L_{s}}{E A_{s}}

(2)

in which K_ec is the theoretical elastic stiffness of the specimen; L_y and L_s are the lengths of yielding segment and stiffening segment of the inner core, respectively; A_y is the cross-sectional area of yielding segment; and A_s is the cross-sectional area of stiffening segment. The calculated yield force and elastic stiffness are listed in Table 5. As shown in Table 5, the analytical elastic stiffness and yield force of each specimen agree well to the test values with the largest difference of 6.1%.

Table 5.

Mechanical properties of BRB specimens.

Specimen	Elastic stiffness (kN/mm)			Yield force (kN)			β			CPD
Specimen	K _et	K _ec	Error (%)	P _yt	P _yc	Error (%)	Strain 1.4%	2%	Ultimate	CPD
S3	315	333	5.4	616	585	5.3	1.09	1.25	1.30	561
S4	282	292	3.4	537	506	6.1	1.05	1.08	1.26	1015
S5	119	114	4.4	197	194	1.5	1.12	1.21	1.81	1226

K _et and P_yt: elastic stiffness and yield force obtained from test results, respectively; K_ec and P_yc: theoretical elastic stiffness and yield force, respectively; CPD: cumulative plastic ductility; BRB: buckling-restrained brace.

Compression strength adjustment factor β

At the same strain amplitude, the reaction force of BRB under compression is usually larger than that under tension. This phenomenon is mainly due to the existence of friction force caused by the interaction of the core and restraining members. Compression strength adjustment factor β is applied to evaluate such imbalance response of BRB as

β_{i} = \frac{C_{i, max}}{T_{i, max}}

(3)

where C_i,_max and T_i,_max represent the maximum compression and tension forces at the same strain amplitude of the ith loading cycle, respectively.

The maximum values of β at the different states are also listed in Table 5. It is seen that β of S5 is much larger than those of S3 and S4, which indicates a further advantage of H-section core over flat steel core besides the elimination of compression force fluctuation. In addition, β of S4 is smaller than that of S3, and even the maximum strain of S4 is greater than S3. This implies the positive effect of the stopper in the middle of the core.

CPD

CPD index is usually used to evaluate the cumulative deformation capacity of BRB, which can be obtained from

CPD = \sum [\frac{2 ({| δ_{t max} |}_{i} + {| δ_{c max} |}_{i})}{δ_{yc}} - 4]

(4)

in which ${| δ_{t max} |}_{i}$ and ${| δ_{c max} |}_{i}$ represent the maximum tension and compression axial deformations of the ith loading cycle, respectively, and δ_yc is the yielding deformation.

As shown in Table 5, the cumulative plastic deformation capacity of S3 is obviously much weaker than those of specimens S4 and S5. This, as well as smaller maximum strain of S3, is mainly due to the serious deformation concentration of the core member with the absence of the stopper in S3, which will be further discussed in section “Width change of core members.” Although S5 has the largest value of compression strength adjustment factor, it performs best in terms of the cumulative plastic deformation capacity. This may be attributed to the gradual change of the cross section of the core plate around the end of stiffener in S5 and hence the relief of stress concentration. Such technology was not applied to S3 and S4 for the convenience of fabrication.

Failure mode

All the specimens finally failed with the fracture of core member, and no global buckling or local buckling could be observed from restraining member. The failure modes of S3–S5 are shown in Table 6. The inner cores of all the three specimens totally fractured except for S4 with only one flange fractures completely. So, the restraining members can be disassembled without extra damage to the core for S3 and S5. In order to retain the original failure mode of end regions of S4, cutting position was chosen at the midspan of the HBRB (see the dotted yellow circle in Table 6) when disassembling S4. It can be found from Table 6 that the core plate of specimen S5 broke with the fracture located outside the stiffening region while S3 and S4 fractured just from the edge of stiffening region. It has been confirmed that the residual stress caused by welding as well as the stress concentration induced by the abrupt change of cross section will reduce the fatigue performance of the core plate (Chen et al., 2016; Zhao et al., 2011). For the core plate with abruptly changed cross section just as that of specimens S3 and S4, the adverse effects of residual stress and stress concentration will superpose at the end of the stiffeners. While for the core plate with gradually changed cross section as that of specimen S5, the stress concentration caused by the change of core plate can be mitigated, and the superposition of the residual stress and the stress concentration at the end of the stiffeners can be avoided. As mentioned in section “Seismic performance evaluation,” specimen S5 performed best in fatigue property of all the three specimens. Based on failure mode of the three specimens and given that the welding process of specimen S5 was the same as that of specimens S3 and S4, it is believed that the gradually changed cross section of the core plate around the end of stiffeners can help to improve the fatigue performance of BRBs. But S5 still broke around the end of the core where the cross section started to change. This might be attributed to the linear variation of the cross section, and the cross section change with a style of arc chamfering, as employed by Narihara et al. (2004), Suzuki et al. (2005), and Funayama et al. (2012), might be more effective to reduce the stress concentration.

Table 6.

Failure mode of BRB specimens.

Specimen	Failure mode
S3
S3	1. H-section steel core totally fractured at the edge of stiffening region on the right-hand side 2. Welds connecting H-section steel core and stiffeners were slightly torn open
S4
S4	1. The web and the flange on one side fractured at the edge of stiffening region; the flange on the other side did not break but obvious necking can be observed 2. Welds connecting H-section steel core and stiffeners were partly torn open
S5

BRB: buckling-restrained brace.

Width change of core members

The uniformness of deformation along the BRBs can be revealed from the transverse strain of the core at the different cross sections. The widths before test are seen in Figures 3 and 5 and Table 1, while those of flanges of H-steel and flat steel after test are shown in Figure 9. The unit is millimeter, and the numbers in dashed box are the maximum or minimum widths. From the width change, the transverse strain at the different cross sections can be evaluated. The observations from Figure 9 are as follows:

S3: The flanges expand in the upper end region and shrink in the middle and lower end region. The maximum expansion strain is 3.2%, and the maximum shrinkage strain is 9%.

S4: On the F2–F3 side, the flanges expand in both end regions and slightly narrow in the middle with the maximum expansion strain of 4.88% and the maximum shrinkage strain of 2.5%. On the side of F1–F4, flanges expand in the upper end region and narrow in the middle and lower end region. Section shrinkage in the lower end region is more serious than that in the middle. Maximum expansion and shrinkage strains on this side are 4.25% and 5%, respectively.

S5: The core plate expands in the end regions and narrows in the middle. The maximum expansion strains around upper end and lower end are 11.75% and 13.25%, respectively. In the region from the stopper to upper end, maximum shrinkage strain is 10%, and from the stopper to lower end, the value is 15%.

It is interesting that expansion and shrinkage of flanges can both be found in the end regions of the H-section steel cores. For specimen S3, such deformation pattern is believed to be caused mainly by the friction force between the H-section steel core and restraining members. Because there is no stopper in the middle of the steel core for specimen S3, the friction force was progressively transferred from the upper end to the lower end which made the actual axial compressive stress in the upper region of the core larger than that in the lower region, leading to larger transverse expansion of the upper region under compression. In the following tensile loading step, because the cross-sectional area of the upper region was larger than that of the lower region, the actual tensile stress in the lower region should be larger than that in the upper region resulting in that the shrinkage in the lower region was more significant. Under the growing cyclic load, such expansion and shrinkage effects would accumulate and finally led to the transverse deformation pattern as indicated in Figure 9(a). For specimen S4, the friction force transfer mode was different from that of specimen S3. Due to the existence of the stopper in the middle, the friction force was transferred from the two ends to the middle portion. So, the transverse expansion should be observed on flanges in both end regions while shrinkage in the middle portion. The most measured residual widths matched the expectations as shown in Figure 9(b), except that transverse shrinkage appeared on the flanges in the lower end region around the fracture location on the side of F1–F4. As mentioned in Table 6, in the fracture region of specimen S4, the web and the flanges on one side fractured while the flanges on the other side, that is, F1–F4, did not. After the flanges on the side of F2–F3 and the web fractured in tension, the imposed tensile deformation would be concentrated on the flanges on the side of F1–F4. According to the theory of Poisson’s effect, a plate will shrink in transverse when it is in tension. As a result, the flanges on the side of F1–F4 shrunk as shown in Figure 9(b) due to the concentration of tensile deformation.

Figure 9.

Width of steel core after test: (a) S3, (b) S4, and (c) S5.

The above observations indicate that the deformation concentration of the flat core plate is more serious than that of H-section steel. Serious deformation concentration of the core plate may help explain why the compression strength adjustment factor β of specimen S5 is larger than that of specimen S3 and S4. Between the two HBRBs, the maximum transverse strain of S3 is 9% with only ultimate axial strain of 2.2%, in contrast to S4 with the maximum transverse strain of 5% but the ultimate axial strain of 3%. It implies that setting stopper in the middle of the core member can help to mitigate the deformation concentration and hence improves the fatigue performance of BRBs.

Half wavelength of local buckling for H-section steel core

After the test, the restraining members were disassembled, and obvious higher-order local buckling of flanges and web for H-section steel core can be observed. Although the wavelength of the core varies with different amplitudes of compressive axial load, the wave shape will be retained after unloading because of residual plastic flexural strain. So, it has been a common practice to measure the half wavelength after tests (Genna and Gelfi, 2012a; Midorikawa et al., 2010, 2012; Zhao et al., 2014). The measurement of half wavelength of local buckling for S2 is shown in Figure 10, where the inflection points were marked white. The measured results are listed in Table 7. It can be seen that the half waves close to the two ends are apparently shorter than those in the middle portion which may be attributed to the uneven axial load imposed on H-section steel core along the longitude direction caused by the friction force between inner core and restraining member.

Table 7.

Half wavelength of local buckling for H-section steel core.

Location		Measured values of half wavelength from left end to fracture point at right end (mm)
	S1	F1: 48, 49, 72, 74, 95, 75, 62, 125, 50, 50, 50, 65, 50, 40, 35, 40, 53
		F2: 37, 68, 60, 85, 74, 90, 93, 100, 90, 50, 53, 72, 53, 50, 41, 30
		F3: 38, 72, 45, 35, 35, 35, 50, 50, 50, 40, 60, 40, 40, 50, 55, 70, 45, 55, 55, 65, 46, 36
		F4: 49, 48, 77, 55, 42, 35, 40, 54, 49, 40, 43, 68, 72, 51, 48, 54, 45, 38, 32, 30, 85
		W: 43, 61, 62, 81, 88, 90, 100, 92, 88, 36, 72, 35, 86, 35, 56
	S2	F1: 75, 35, 30, 31, 34, 47, 50, 60, 50, 57, 64, 68, 93, 60, 30, 40, 32, 35, 36, 37, 40
		F2: 30, 45, 60, 45, 55, 48, 72, 52, 56, 55, 43, 45, 49, 55, 68, 124, 48, 45, 34
		F3: 43, 40, 47, 52, 65, 55, 53, 52, 63, 76, 108, 60, 66, 79, 105, 65
		F4: 38, 34, 35, 37, 54, 41, 74, 53, 53, 48, 83, 69, 72, 62, 106, 61, 43, 50
		W: 55, 55, 51, 48, 52, 65, 65, 48, 68, 55, 104, 69, 62, 88, 125
	S3	F1: 30, 45, 90, 45, 32, 35, 38, 45, 43, 51, 46, 51, 45, 54, 48, 107, 57, 58, 37, 43, 35
		F2: 41, 35, 35, 53, 73, 47, 48, 48, 92, 61, 55, 52, 44, 55, 67, 83, 67, 30
		F3: 50, 30, 32, 41, 90, 51, 50, 55, 66, 86, 41, 42, 50, 52, 86, 55, 104, 43
		F4: 32, 40, 56, 95, 90, 58, 40, 42, 135, 71, 45, 78, 160, 36, 35, 37
		W: 42, 53, 95, 52, 86, 38, 62, 237, 38, 67, 50, 100, 90, 62
	S4	F1: 81, 40, 47, 40, 50, 49, 68, 50, 178, 43, 40, 44, 50, 41, 96, 37, 95
		F2: 38, 38, 31, 35, 53, 46, 86, 31, 52, 174, 43, 33, 44, 54, 51, 40, 39, 68, 47
		F3: 34, 39, 31, 35, 72, 39, 41, 32, 33, 33, 32, 160, 48, 37, 44, 49, 66, 40, 34, 38, 36, 46
		F4: 45, 31, 35, 60, 42, 50, 40, 75, 48, 167, 41, 35, 52, 45, 35, 93, 47, 57, 35
		W: 40, 63, 60, 43, 40, 42, 64, 55, 96, 96, 88, 90, 42, 52, 61, 52, 57

Figure 10.

Half-wavelength measurement of H-section steel core of specimen S2.

Half-wavelength evaluation of higher-order local buckling of H-section steel core

To calculate half wavelength of local buckling for H-section steel, elastoplastic buckling theory of plates needs to be used. Bleich’s theory is most widely adopted in stability design of structural plates. However, it should be noted that Bleich’s formula which can be used to obtain the half wavelength lacks sufficient experimental validation. For higher-order local buckling analysis of H-section steel, Lundquist’s formula which was proposed based on large amount of test results appears more reliable as can be seen in the following analysis of sections “Bleich’s theory” and “Lundquist’s theory.”

Analytical model

To calculate half wavelength of buckling of a plate, the boundary conditions should be determined first. For local buckling analysis of H-section steel in the higher-order buckling mode, the locally buckled flange or web plate can be separated from the entire plate with loading edges going through the inflection points as shown in Figure 11. Then, the analytical model can be simplified as a small rectangular plate with loading edges simply supported (Zhao et al., 2014). For edges without load imposed, considering that the restraining action from the flanges on the web is much stronger than the other way around, boundary conditions of “one edge simply supported, the other free” and “both edges fixed” are assumed for flange and web, respectively. For the separate portion of the core plate, the half wave number is apparently unity.

Figure 11.

Analytical model of local buckling for H-section steel.

Bleich’s theory

Based on the analyses conducted by Bleich (1952), if a plate buckles in elastic stage, the critical buckling stress σ_cr,e can be calculated as

σ_{cr, e} = k_{e} \times \frac{1}{12 (1 - υ^{2})} \times \frac{π^{2} E}{{(b / t)}^{2}}

(5)

in which k_e is the elastic buckling coefficient of plate and is expressed as

k_{e} = C_{1} + C_{2} \frac{a^{2}}{m^{2} b^{2}} + \frac{m^{2} b^{2}}{a^{2}}

(6)

where C₁ and C₂ are the coefficients only related to boundary conditions of plate as shown in Table 8; b, a, and t are the width, length, and thickness of plate, respectively; m is the number of buckling half waves; E is the elastic modulus; and υ is the elastic Poisson’s ratio.

Table 8.

Values of C₁ and C₂ under different boundary conditions (Bleich, 1952).

	One edge simply supported, the other free	Both edges fixed
C ₁	0.425	2.50
C ₂	0	5.00

By regarding elastoplastic plate as bidirectional orthotropic and assuming that the flexural rigidity along the loaded direction should be reduced by η, the torsional rigidity should be reduced by $\sqrt{η}$ , and the flexural rigidity along the unloaded direction needed no reduction, Bleich obtained the critical elastoplastic buckling stress of plate as

σ_{cr, p} = k_{p} \times \frac{1}{12 (1 - υ^{2})} \times \frac{π^{2} E}{{(b / t)}^{2}}

(7)

in which k_p is the elastoplastic buckling coefficient of plate and is expressed as

k_{p} = \sqrt{η} (C_{1} + C_{2} \frac{a^{2}}{m^{2} b^{2} \sqrt{η}} + \frac{m^{2} b^{2}}{a^{2}} \sqrt{η})

(8)

where η is the modulus degradation factor which equals to the ratio of tangent modulus E_t to elastic modulus E. Half wavelength of plate can be obtained based on equation (8), but equation (8) was not sufficiently validated through experimental data. Although the revised version of equation (8) which is expressed as

k_{p} = \sqrt{η} (C_{1} + 2 \sqrt{C_{2}})

(9)

has been validated by test results of long plates (Bleich, 1952) and widely adopted in stability design of structural plates (Chen, 2008; GB 50017-2003, 2003), it cannot be used for the calculation of half wavelength along the longitude direction of H-section steel because of the absence of half wave number m in equation (9).

Lundquist’s theory

Within elastic stage, the formula for critical buckling stress of plate proposed by Lundquist is the same as that proposed by Bleich. Beyond elastic stage, based on test results, Lundquist found that the elastoplastic buckling coefficient of plate k_p is related to elastoplastic buckling coefficient k_e by a constant dependent on η through (Lundquist and Stowell, 1942a, 1942b)

k_{p} = η^{*} \times k_{e} = η^{*} (C_{1} + C_{2} \frac{a^{2}}{m^{2} b^{2}} + \frac{m^{2} b^{2}}{a^{2}})

(10)

For the plate with one of the unloaded edges simply supported and the other free, if a/mb > 2.5

η^{*} = \frac{η + \sqrt{η}}{2}

(11)

otherwise

η^{*} = \frac{η + 3 \sqrt{η}}{4}

(12)

For the plate with both the unloaded edges fixed, the expression for η* is the same as equation (12). From equations (8) and (10), it can be observed that the formulae determining elastoplastic buckling coefficient of plate proposed by Bleich and Lundquist are quite different. Assuming one simply supported and one free for the two unloaded boundaries of the flange and both fixed for the web, the values of elastoplastic buckling coefficients of flange and web under different values of a/(mb) are compared in Figure 12, where η is adopted as 0.46% based on the test results of specimen S1 listed in Table 4. It is seen that the differences of the two theories are significant except for larger a/(mb) which characterizes longer flange of H-section steel. For k_p of the web, the difference of the two theories is even larger. In Figure 12(b), the two thick dots represent the minimum k_p of the two theories. For a given a/b, the number of half wave will increase, and hence, a/(mb) will reduce with increasing axial load, and k_p will increase accordingly. So, only the solid lines in Figure 12(b) have practical meaning, and the difference in k_p represented by the two solid lines is significant in the mutual range of meaningful a/(mb).

Figure 12.

Comparison of elastoplastic buckling coefficients: (a) flange and (b) web.

Given that equation (10) has been confirmed by test results and can be used to determine the half wavelength of higher-order buckling, Lundquist’s formula is adopted in this article. Considering a single half wave of plate and letting m = 1, we may obtain from equations (7) and (10) the half wavelength which is denoted by a′, for given b and σ_cr,p. It should be noted that for the web, there are two positive solutions of a′ to equations (7) and (10). The actual solution should be the smaller a′ because the half wavelength develops from it maximum value $a'_{cr}$ and reduces with increasing axial load.

It is important to mention here that η in equations (11) and (12) will be too large if it is determined from the stress–strain curve of the material test on the assumption that no deflection takes place until the critical stress is reached, and the effect of initial deflection must be considered separately. However, if η is determined from the component test, part, if not all, of this effect can be automatically considered (Lundquist, 1939). So, the factor η in this article was obtained based on the hysteretic curve of each specimen.

In addition, the effect of gap size is not directly considered in equation (10). The contact force between inner core and restraining member increases with the growth of gap size, which leads to a larger friction force. As a result, when the same ultimate strain is reached, a larger compressive axial load usually needs to be imposed on the BRB with a larger gap, resulting in smaller half wavelengths especially around the end regions of the core member. Therefore, if the half wavelengths are calculated based on measured reaction force corresponding to the ultimate compressive stress of each specimen, then the effects of gap size can be automatically taken into account.

Validation of half-wavelength formula

To calculate the half wavelength using equations (7) and (10), modulus degradation factor η, effective width b, and buckling stress of plate σ_cr,p need to be determined. The values of η at failure strain can be found in Table 4. For the hot rolled H-section steel, considering the chamfering between flange and web (GB 50017-2003, 2003; Zhao et al., 2014), their effective widths can be expressed as

b_{f} = \frac{(W - t_{w})}{2} - R

(13)

b_{w} = H - 2 t_{f} - 2 R

(14)

where W, H, R, t_w, and t_f are defined in Table 1. The buckling stress σ_cr,p is simply axial compressive stress from test results, which is expressed as

σ_{cr, p} = \frac{P_{c, \max}}{A_{y}}

(15)

where P_c,max is the maximum compressive load imposed on specimen which equals to 1123, 1028, 958, and 891 kN for specimens S1–S4, respectively. Then, with these parameters available, the theoretical values of the half wavelength based on formula proposed by Bleich and Lundquist can be obtained, and they are listed in Table 9.

Table 9.

Comparison of theoretical values and test results of half wavelength for HBRBs (mm).

	S1		S2		S3		S4
	Flange	Web	Flange	Web	Flange	Web	Flange	Web
$a'_{Bleich}$	10	–	10	–	9	–	11	–
$a'_{Lundquist}$	33	29	34	30	33	36	36	32
$a'_{Test}$	30	35	30	48	30	38	31	40

$a'_{Test}$ : minimum value of measured half wavelength of buckling for flange and web; –: no resolution; HBRB: buckling-restrained braces with H-section steel core.

The measured half wavelengths were unevenly distributed along the core as shown in Table 7. Because the smaller wavelength could induce lager bending strain and contact force which were more unfavorable, minimum measured values are considered and listed also in Table 9. It is seen from Table 9 that the results by Lundquist agree well with test results, while Bleich’s formula results in significant errors: they are only around one-third of the test results for the flange. For the web, minimum buckling load obtained according to Bleich’s theory is greater than the largest axial load during the test, indicating no buckling at all, which obviously contradicts the test results.

Another finding from Table 9 is that the measured half wavelengths of flange were smaller than those calculated with Lundquist’s formula, while for web, the measured data were larger. This phenomenon might be attributed to the uneven distribution of stress on flanges and web caused by the different friction forces from restraining member onto the flange and web. From visual inspection of the contact surfaces of steel core and restraining member after test, it was found that the friction surfaces on the flange appeared more polished than those on the web. This indicated a larger friction force on the flange than on the web, which could result in axial stress in the flange larger than the average axial stress while that in the web smaller than the average.

The proposed half-wavelength calculation method can also be applied to other section steel, for example, the angle steel. Zhao et al. (2014) conducted cyclic tests on a type of all angle steel BRBs and investigated their local buckling properties. The data of six BRBs in Zhao et al. (2014) are used herein to predict half wavelength with Lundquist and Bleich’s theory, and the results are listed in Table 10. It can be found that the theory proposed by Lundquist is proved to be more consistent with the experimental data of the BRBs with steel core of different width–thickness ratios of flanges and different clearance between the core and restraining member.

Table 10.

Comparison of theoretical values and test results of half wavelength for angle steel BRBs.

Specimen no.	b (mm)	t (mm)	b/t	c (mm)	ε _c,max (%)	P _c,max (kN)	K _t (kN/mm)	η (%)	$a'_{Test}$ (mm)	$a'_{Lundquist}$ (mm)	$a'_{Bleich}$ (mm)
ABRB1	39.4	4.6	8.6	2.2	1.06	700.3	10.5	1.70	41	39.6	17.2
ABRB2	37.9	4.6	8.2	3.5	0.53	572.5	11.6	1.94	47	48.4	22.6
ABRB3	39.4	4.6	8.6	1.8	1.38	743.9	7.0	1.13	31	32.9	12.7
ABRB4	37.9	4.6	8.2	3.1	1.40	700.9	8.7	1.45	32	37.1	15.5
ABRB7	33.4	7.6	4.4	1.5	1.87	1267.3	6.1	0.66	57	60.2	25.4
ABRB8	32.0	7.5	4.3	2.8	1.31	1072.3	12.5	1.40	66	81.2	–

b: calculation width of the flange of angle steel core; t: measured thickness of the flange of angle steel core; c: thickness of the clearance between inner core and restraining member on one side; ε_c,max and P_c,max: the ultimate strain and reaction force of each specimen under compression before the occurrence of stiffness degradation; K_t: the tangent stiffness corresponding to ε_c,max and P_c,max; η: modulus degradation factor; –: no buckling; BRB: buckling-restrained brace.

It should be noted that values of the modulus degradation factor η in Table 10 are different from the results in Zhao et al. (2014). The values of η in this article are calculated based on the ultimate strain of each specimen, while in Zhao et al. (2014), the intersection between the theoretical P_cr–ε curve and the experimental hysteretic loop at the final cycle was chosen as the calculation point to obtain the values of η. Because the buckling mode of the core member will evolve with the increase of axial load, the method in this article appears more appropriate than Zhao et al.’s.

Conclusion

The quasi-static cyclic tests on two HBRBs and a BRB with flat steel core are described first in this article to further investigate the effect of configuration details on seismic performance of HBRBs. Then, two methods for evaluating the half wavelength of higher-order local buckling of H-section steel core based on elastoplastic buckling theory of plates proposed by Bleich and Lundquist, respectively, are reviewed and compared with the test results from the four HBRBs (including the two previously tested).

It is found from the test results that due to H-section steel core’s higher self-stability, the compression force fluctuation was not observed on the hysteretic loops of HBRB with even larger clearance but on the BRB with flat core; the HBRB was also advantageous over the BRB with flat core in terms of having lower compression strength adjustment factor β. Test results indicated that a stopper in the middle of the core member and the gradual change of cross section of the core plate around the end of stiffeners could help to improve the fatigue performance of BRBs. Compared with the measured half wavelength, Lundquist’s theory was proved to be more reliable for evaluating the half wavelength of higher-order local buckling for H-section steel core. In addition, the half-wavelength calculation method proposed in this article can also be applied to other section steel, and the accuracy was validated based on the test results of angle steel cores of BRBs proposed by Zhao et al.

To further extend the fatigue life of BRBs, more investigations on the fabrication details are still needed, such as improving the welding process, changing the cross section of the core plate around the end of stiffeners with a style of arc chamfering, and so on. To make the half-wavelength calculation method more precise for H-section steel core, further study can be conducted on the uneven distribution of stress on flanges and web caused by the different friction forces from restraining member onto the flange and web.

Footnotes

Acknowledgements

The authors thank Mr Yunfei Ma for his kind assistance on the operation of MTS test machine.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the National Science Foundation of China (grant no. 51161120360, no. 91315301-09, and no. 51308169).

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