Critical load and application of core-separated buckling-restrained braces

Journal of Constructional Steel Research 106 (2015) 1–10

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Journal of Constructional Steel Research

Critical load and application of core-separated buckling-restrained braces Yan-lin Guo, Bo-hao Zhang ⁎, Zi-qin Jiang, Hang Chen Department of Civil Engineering, Tsinghua University, Beijing, 100084, PR China

a r t i c l e

i n f o

Article history: Received 12 April 2014 Accepted 28 November 2014 Available online xxxx Keywords: Core-separated Buckling-restrained brace Critical load Equilibrium method Restrain ratio

a b s t r a c t A new type of buckling-restrained braces (BRBs), called core-separated buckling-restrained braces (CSBRBs) is presented and investigated in this study. The CSBRBs are composed of two chord members and one or several continuous web plates that connect those members longitudinally. This core-separated section improves bending stiffness, thus resulting in lightness and convenience in fabrication and erection compared with conventional BRBs. The elastic buckling performance of CSBRBs is initially investigated because such performance depends directly on the restrain ratios, which should be specified in the design of all types of BRBs. The critical loads of pin-ended and fix-ended CSBRBs are then obtained based on the equilibrium method and verified by finite element (FE) elastic buckling analysis, respectively. Subsequently, the relationship between axial compression and core compressive strain of CSBRBs with different restrain ratios is studied by using FE elasto-plastic analysis. These research achievements provide the foundation for further investigations and applications for CSBRBs. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Buckling-restrained braces (BRBs) have been widely used in frame structures, long-span structures and bridges due to their significant energy dissipation capacity and ductility. A typical BRB is usually composed of an internal structural steel core plate and an external restraining member. The steel core plate resists the axial force and it does not globally buckle because of external restraining member. Therefore such plate can undergo considerable yielding under both axially tensile and compressive forces, thus supplying stable hysteresis characteristics. The restraining member resists forces transverse to the axis of the BRB while resisting little or no force in the axis of the BRB, thereby restraining core plate buckling [1]. In conventional BRBs, the typical restraining members encasing core plates are concrete-filled steel tubes or reinforced concrete members [2–4]. However, the axial forces of braces increase in practice, such that the cross sections of conventional BRBs inevitably become largescale, thus resulting to heavier dead weight and more difficulties in fabrication, transportation, and erection. Problems concerning quality control in the manufacturing process, as well as flexibility in the detailed design of both ends of the core plate, also have to be solved. Therefore, several new types of light-weight BRBs have been developed recently. Tsai et al. [5] proposed a double-tube BRB, in which two steel tubes acting as restraining members were combined by a series of batten plates. Usami et al. [6] studied the overall buckling of a lightened BRB composed of only steel members without filling concrete. Chou and Chen [7] proposed a sandwiched BRB, in which the core plate was ⁎ Corresponding author. Tel.: +86 010 62788124. E-mail address: [email protected] (B. Zhang).

http://dx.doi.org/10.1016/j.jcsr.2014.11.011 0143-974X/© 2014 Elsevier Ltd. All rights reserved.

restrained by two identical restraining members, and each restraining member was formed by welding a steel channel with a flat plate. Genna and Gelfi [8,9] proposed another type of BRB, in which restraining members were steel channels strengthened by stiffeners. Wang et al. [10] recently proposed a four-channel assembled BRB, in which the core was restrained by four steel channels combined by longitudinally distributed high-strength bolts. After an earthquake, the damaged core plate can be replaced independently by disassembling the restraining member. A potential disadvantage of all-steel BRBs is that such BRBs have more parts (i.e. bolts) and are thus more expensive than traditional BRBs. Quasi-static and dynamic loading tests were conducted [11–13] to investigate the performance and dynamic response of BRBs under earthquakes. The system-level performance of BRB frames was studied by numerical and large-scale experimental simulations [14–16]. The design method of isolated conventional BRBs was first proposed by Takahashi and Mochizuki [3] and then developed by Fujimoto et al. [17]. The core plate shall be designed to resist the axial force in the BRB. Thus, the net cross-sectional area of the core can be determined by the brace design axial force obtained from the frame structure. The restraining member shall be designed to ensure that it does not buckle or lose strength at force levels corresponding to the yielding of the core. This demand can be expressed by a performance parameter called restraining ratio, which is defined as the ratio between the overall elastic buckling load Pcr and the axial yield strength of the core Py,c, which can be expressed as

ζ¼

P cr : P y;c

ð1Þ

2

Y. Guo et al. / Journal of Constructional Steel Research 106 (2015) 1–10

a) Section of single-web type

b) Section of double-web type

c) Section of all-steel type

d) 3D diagram of single-web type

Fig. 1. Schematics of core-separated buckling-restrained braces: (a) section of single-web type, (b) section of double-web type, (c) section of all-steel type, and (d) 3D diagram of single-web type.

In the ideal elastic situation of materials, if the restrain ratio of a BRB is larger than 1.0, the restraining member is considered strong enough to prevent buckling prior to the yielding of the core. However, considering the influence of initial geometric imperfection and the clearance between the core and the restraining member, Fujimoto et al. [17] recommended that the restrain ratio should be larger than 1.5. Therefore, the investigation of elastic buckling loads of BRBs plays an important role in the design of all types of BRBs.

a) Actual model

This paper presents a new type of light-weight BRB called coreseparated buckling-restrained brace (CSBRB). CSBRBs are composed of two chord members which are conventional single-core BRBs, and the chord members are connected together by one or several continuous web plates. Compared with conventional BRBs, CSBRBs have the potential to be widely used as large-tonnage BRBs in civil engineering because of their lightness and convenience in fabrication and erection. Subsequently the elastic buckling behavior of CSBRBs is investigated,

(b) Simplified analytical model

c) Buckling deformation of the simplified analytical model

Fig. 2. Simplified analytical model of pin-ended CSBRBs: (a) actual model, (b) simplified analytical model, and (c) buckling deformation of the simplified analytical model.

Y. Guo et al. / Journal of Constructional Steel Research 106 (2015) 1–10

a) Half of the model

b) Half of the simplified analytical model

3

c) Interaction among two cores, a restraining member and an end plate

Fig. 3. Analysis of the elastic buckling load about the y axis: (a) half of the model, (b) half of the simplified analytical model, and (c) interaction among two cores, a restraining member and an end plate.

and formulas to predict the critical loads of pin-ended and fix-ended CSBRBs are deduced by using the equilibrium method and verified by finite element (FE) elastic buckling analysis, respectively. Finally, the employment of elasto-plastic analysis of several CSBRB specimens, demonstrates that the compressive behavior of CSBRBs is closely related to the restrain ratio. Thus studying the critical loads of CSBRBs is meaningful for the further development of the design procedure. 2. Components of core-separated buckling-restrained braces The components of several types of CSBRBs and connection configurations at the ends are introduced in this section. As shown in Fig. 1(a) and (d), single-web CSBRBs consist of two chord members, which are conventional single-core concrete-filled steel tube restrained BRBs. The two chord members are connected by a web plate, to form a biaxially symmetrical I-shaped section. The steel tube can be thin-walled cold-formed steel or welded tube formed by four steel plates. A thin layer of unbonded material is placed between the steel core plate and the surrounding concrete to eliminate axial force transfer and enable the lateral expansion of the steel core plate under compression. If a brace is subjected to a large axial force, such brace can be designed as a double-web CSBRB to guarantee the strength and stability of webs [Fig. 1(b)]. For convenient fabrication, the chord members can be welded by four steel plates. Two of the steel plates are restraining plates and the others are web plates. The two chord members are connected to form an all-steel CSBRB by the continuous fillet welding of two web plates with each other along their entire length [Fig. 1(c)]. The proposed CSBRB has a few practical advantages over conventional BRBs. First is the capability to deliver two chord members separately for on-site erection. Second, the core-separated section improves bending stiffness, and the performance of this section can be adjusted by changing the distance between two chord members, which significantly improves the material utilization efficiency of braces. Furthermore, the web plates provide continuous lateral support for chord members and improve their stability. Finally, the core in conventional BRBs contains projections beyond the restraining member to realize a connection with adjoining elements. Thus, the unstrained zone of the core can easily collapse, which is an undesirable failure mode that does not permit the full utilization of brace ductility capacity. At the end of CSBRBs, two core plates and one end plate are welded into an H-shaped section [Fig. 1(d)]. As a result, flexural rigidity is enhanced remarkably compared with that of conventional BRBs, which results in a reliable brace-end connection design.

3. Critical loads of pin-ended CSBRBs CSBRBs connected to frames by gusset plates and bolts can be supposed to have a fix-ended structural design. Meanwhile CSBRBs connected to frames by ear plates and pins can be considered to have a pin-ended structural design. The critical loads of pin-ended CSBRBs are investigated in this section based on two assumptions. Firstly, the two restraining tubes and the web plate connecting them are considered as a unified restraining member as a whole under bending, where the plane-section assumption is satisfied between them. Secondly, the local buckling of steel plates is not considered. Therefore, the actual model of a pin-ended CSBRB [Fig. 2(a)] can be simplified to the analytical model shown in Fig. 2(b). In the simplified analytical model, the notations EbIb and b represent the flexural rigidity and the length of the end plate, respectively. EeIe, Ec1Ic1, and Ec2Ic2 represent the flexural rigidity of restraining member and that of the two cores respectively. Ac1 and Ac2 represent the cross-sectional area of cores. l represents the length of the brace. The buckling deformation of the simplified analytical model is illustrated in Fig. 2(c). Also, assuming that the small gap between the cores and the restraining member is neglected, the lateral deflection of cores, denoted by wc1 or wc2 is completely compatible with that of the

Fig. 4. Comparison between exact results and approximate results by Eq. (12).

4

a) γ → ∞

Y. Guo et al. / Journal of Constructional Steel Research 106 (2015) 1–10

b) γ → ∞ and β → ∞

c) γ → ∞ and β → 0

Fig. 5. Various limit cases of CSBRBs (a) γ → ∞, (b) γ → ∞ and β → ∞, and (c) γ → ∞ and β → 0.

restraining member we, i.e., wc1(z) = wc2(z) = we(z), where z stands for the axial coordinate measured from the lower end. This compatible relation is reflected in the simplified analytical model by setting several virtual absolutely rigid bars, as shown in Fig. 2(b). In particular, the lateral deflection of cores and the restraining member at mid height is denoted by δ. It is obvious that according to the geometrical and deformation symmetry of the CSBRBs, the upper half of the simplified model [Fig. 2(c)] could be only considered as an analytical model where its bottom is fixed and its top is free, and it behaves as a sway frame [Fig. 3(b)]. In the frame, the middle column represents a restraining member which is simulated as a whole by using a beam element in FE analysis, and other two side columns are respectively the two cores. The beam of the frame, i.e., the end plate, is rigid-jointed with the side columns, and is independent at the ends of the middle column. The coordinate system for half of the simplified model is shown in Fig. 3. The end plate is rigid-jointed with the cores at points C and D, and no interaction occurs between the end plate and the restraining member. The restraining member is also fixed at the lower end. As discussed above, the lateral deflection of cores wc1(z1) or wc2 (z1) is compatible with that of the restraining member we(z1) at any height of the frame, with a same lateral deflection δ of the cores and restraining member at their upper ends. The lateral deflections of the cores and

restraining member at their upper ends are equal to the lateral deflection at mid height of the simplified model shown in Fig. 2(c). It is obvious that the elastic buckling analysis of the pin-end CSBRB [Fig. 2(c)] is equal to that of the sway frame [Fig. 3(b)]. The axial force P subjected to the pin-end CSBRB is divided into two axial forces P/2 applied at two cores, thus introducing only a slight error in the calculation of critical loads [18]. Assuming that the two cores are identical, then Ec1Ic1 = Ec2Ic2, and Ac1 = Ac2. The interaction among two cores, a restraining member and an end plate, which form a sway frame as discussed above, is shown in Fig. 3(b). Q i (i = 1,2) is the concentrated shearing force at the top, whereas qi(z1) (i = 1,2) is the intensity of the lateral contact force. Moreover, ΔP is the increment of the axial force acting on the cores because of the secondorder effect, and M is the moment at the rigid joints. Thus, the differential equations governing the deflections of two cores and the restraining member are as follows:

 Z l=2 d2 wc1 P l ¼ q1 ðt Þðt−zÞdt−Q 1 −ΔP ðδ−wc1 Þ−M− −z 2 2 2 dz

 Zz l=2 2 d wc2 P l ¼ ð Þ−M− q ð t Þ ð t−z Þdt−Q þ ΔP δ−w −z Ec2 Ic2 c2 2 2 2 2 dz2 z
 Z l=2 d2 we l ¼ ½q1 ðt Þ þ q2 ðt Þðt−zÞdt þ ðQ 1 þ Q 2 Þ −z : Ee Ie 2 dz2 z ð2Þ Ec1 Ic1

Substituting wc1(z) = wc2(z) = we(z) and Ec1Ic1 = Ec2Ic2 into the sum of three above formulas yields ð2Ec1 Ic1 þ Ee Ie Þ d2 wc1 P ¼ ðδ−wc1 Þ−M: dz 2 2

ð3Þ

For simplicity, the notation k2 = P/(2Ec1Ic1 + EeIe) is introduced. The general solution of this equation is then obtained as wc1 ðzÞ ¼ A cos kz þ B sin kz þ δ−2M=P

ð4Þ

where A and B are constants of integration. These constants can be determined from the conditions of the fixed end. Substituting wc1(0) = wc1′(0) = 0 into Eq. (4), we obtain A = 2M/P − δ and B = 0. Therefore the deflection of the core can be expressed as wc1 ðzÞ ¼ ðδ−2M=P Þð1− cos kzÞ:

Fig. 6. Calculation of critical loads of fix-ended CSBRBs.

ð5Þ

Y. Guo et al. / Journal of Constructional Steel Research 106 (2015) 1–10

a) Core element

5

b) Restraining member element

Fig. 7. Equilibrium of an element of length dz: (a) core element and (b) restraining member element.

As shown in Fig. 3, the lateral deflection of the core at the top is equal to δ, namely wc1 ðl=2Þ ¼ δ:

ð6Þ

The rotation of the end plate consists of two portions. One is deduced by the moment M at the end of the end plate, and is equal to Mb/6EbIb. The other is the overall rotation of the end plate attributed to the axial deformation of cores. The left core is subjected to a tension force 2M/b, whereas the right core is subjected to an equal compressive force, such that the corresponding axial deformations are both lM/bEc1Ac1. This condition results in a rotation of the end plate equal to 2lM/b2Ec1Ac1. Therefore, the rotation of the end plate at the rigid joints C and D is equal to the sum of the above two parts, namely Mb/6EbIb + 2lM/b2Ec1Ac1. Based on the continuity conditions between the cores and the end plate at the rigid joints, the rotation of cores at the rigid joints C and D can be expressed as dwc1 Mb 2Ml ðl=2Þ ¼ þ : dz 6Eb Ib Ec1 Ac1 b2

ð7Þ

Then, Eq. (9) is simplified to

kl=2 12γ  : ¼ − tan ðkl=2Þ Ac1 b2 =Ic1 þ 12γ β

As a transcendental equation, Eq. (11) can be solved by using numerical methods, but the solution is not in a closed-form and is inapplicable to practical design. The approximate solution of kl/2 can be derived by using the least square method [19] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi u u12γ þ 0:798β Ac1 b2 =Ic1 þ 12γ kl t  : ¼π 2 12γ þ 3:192β Ac1 b2 =Ic1 þ 12γ

δ cos ðkl=2Þ þ

ð8Þ

To obtain a nontrivial solution, the coefficient matrix about δ and M must be singular. In other words, the determinant of the matrix should be eliminated. Therefore, the transcendental equation about k for determining the critical value of the load P must be satisfied kl=2 6lEb Ib 1 : ¼− bð2Ec1 Ic1 þ Ee Ie Þ 1 þ 12lEb Ib =Ec1 Ac1 b3 tan ðkl=2Þ

ð9Þ

The linear stiffness ratio between the end plate and the core γ = lEbIb/bEc1Ic1 is introduced, and parameter β is defined as β¼

2ð2Ec1 Ic1 þ Ee Ie Þ Ec1 Ac1 b2

ð10Þ

the sum of flexural rigidity of cores and that of the restraining member is 2Ec1Ic1 + EeIe, and the flexural rigidity of the compound section formed by two cores with a distance b is equal to Ec1Ac1b2/2. Thus, β represents the ratio of the two types of flexural rigidity.

ð12Þ

The approximate solution to kl/2 by Eq. (12) and the exact solution by Eq. (11) are compared, as shown in Fig. 4. The approximate results obtained by Eq. (12) show good agreement with the exact results, and the maximum error is 1.4%.

Substituting Eq. (5) into Eqs. (6) and (7) for wc1 yields 2M ½1− cos ðkl=2Þ ¼ 0 P " # 2M bP lP ¼ 0: k δ sinðkl=2Þ− þ k sin ðkl=2Þ þ P 12Eb Ib Ec1 Ac1 b2

ð11Þ

Fig. 8. Sketch for finite element model of pin-ended CSBRB.

6

Y. Guo et al. / Journal of Constructional Steel Research 106 (2015) 1–10

(2) If the distance between two cores b is considerably small, β approaches infinity, then tan(kl/2) approaches infinity, and one obtains kl/2 = π/2, such that

Table 1 Dimensions of numerical specimens with infinitely rigid end plates. Group no.

Ie (mm4)

Ic (mm4)

L (m)

b (m)

β

1 2 3 4 5 6 7

6.67 × 104–1.15 × 108 6.67 × 104–1.15 × 108 5.33 × 105 1.15 × 108 1.15 × 108 1.44 × 107 6.67 × 104

6.67 × 104 6.67 × 104 6.67 × 104 6.67 × 104 6.67 × 104 6.67 × 104 6.67 × 104

4 14 4–14 4–14 4 4 4

0.2 0.2 0.2 0.2 0.2–2 0.2–2 0.2–2

0.01–2.88 0.01–2.88 0.02 2.88 2.88–0.03 0.36–0.01 0.005–0.0001

The critical load about y axis can be deduced by substituting k2 = P/(2Ec1Ic1 + EeIe) into Eq. (12), such that   2 48γ þ 3:192β Ac1 b =Ic1 þ 12γ π 2 ð2E I þ E I Þ c1 c1 e e   : P cr ¼ 12γ þ 3:192β Ac1 b2 =Ic1 þ 12γ l2

ð13Þ

As discussed above, the load Pcr expressed in Eq. (13) is the elastic buckling load of the sway frame shown in Fig. 3(a), and is equal to the elastic buckling load of a pin-ended CSBRB shown in Fig. 2. As illustrated in Fig. 1(d), the cores and end plates of CSBRBs are all steel plates. The flexural rigidity of end plates about the y axis EbIb is considerably larger than that of cores Ec1Ic1. If the linear stiffness ratio between the end plate and the core γ = lEbIb/bEc1Ic1 is considered as infinite, namely the end plates are assumed as infinitely rigid [Fig. 5(a)], then Eq. (13) yields 2

P cr ¼

4 þ 3:192β π ð2Ec1 Ic1 þ Ee Ie Þ  1 þ 3:192β l2

ð14Þ

Eq. (13) indicates that the critical load of a CSBRB with infinitely rigid end plates is a function of parameter β. The above formula will correspond to various limit cases for particular parameter β, which is discussed below. (1) If the distance between two cores b is considerably large, β approaches zero based on Eq. (10), then tan(kl/2) approaches zero according to Eq. (14), and one obtains kl/2 = π, such that

P cr ¼

π2 ð2Ec1 Ic1 þ Ee Ie Þ : l2

ð16Þ

In this limit case, the two cores and the restraining member buckle together as a pin-ended superposed column, as shown in Fig. 5(c). In fact, the buckled configurations are different for CSBRBs with different β, and the critical loads change accordingly. The influence of parameter β on the buckled configuration of CSBRBs will be further discussed below. 4. Critical loads of fix-ended CSBRBs When CSBRBs are fixed at both ends, cores and restraining members at both ends will not rotate when they buckle (Fig. 6). Similar to the analysis of pin-ended CSBRBs, the axial force P is divided into two axial forces equal to P/2 applied to each core. The equilibrium of a core element and a restraining member element of length dz is illustrated in Fig. 7. The shearing force and internal moment acting on the elements are assumed positive in the direction shown [20]. Fig. 7(a) shows that the core element is subjected to an axial force P/2 and the lateral contact force applied by the restraining member qi(z). According to the equilibrium of forces in the x direction

dQ ci ðzÞ dz þ qi ðzÞdz ¼ 0 Q ci ðzÞ− Q ci ðzÞ þ dz

ð17Þ

where i = 1, 2 represents the element of each core. Taking the moments of the core element about point A yields

dMci ðzÞ P 1 2 dz þ dwci þ Q ci ðzÞdz þ qi ðzÞðdzÞ ¼ 0: Mci ðzÞ− Mci ðzÞ þ dz 2 2 ð18Þ

2

P cr ¼

4π ð2Ec1 Ic1 þ Ee Ie Þ : l2

ð15Þ

In this limit case, the infinitely rigid end plates will not rotate. The two cores and the restraining member buckle in the same way as fix-ended columns, as shown in Fig. 5(b).

a) Different moments of inertia

Based on the small strain theory, the curvature of the axis of the core is approximately the second derivative of deflection. Thus, the internal moment of the core is Mci(z) = − EciIciw″ci. Combining Eqs. (17) and (18) as well as eliminating the terms of the second order, yields P ″ ″″ Eci Ici wci þ wci ¼ −qi ðzÞ: 2

b) Different lengths of braces

ð19Þ

c) Different distances between cores

of restraining member Fig. 9. Critical loads of pin-ended CSBRBs with infinitely rigid end plates: (a) different moments of inertia of restraining member, (b) different lengths of braces, and (c) different distances between cores.

Y. Guo et al. / Journal of Constructional Steel Research 106 (2015) 1–10 Table 2 Dimensions of numerical specimens considering the deformations of end plates. Group no.

Ie (mm4)

Ic (mm4)

l (m)

b (m)

β

tb (mm)

1 2 3 4 5

6.67 × 104–1.15 × 108 5.33 × 105 1.15 × 108 1.44 × 107 6.67 × 104

6.67 × 104 6.67 × 104 6.67 × 104 6.67 × 104 6.67 × 104

4 4–14 4–14 4 4

0.2 0.2 0.2 0.2–2 0.2–2

0.01–2.88 0.02 2.88 0.36–0.01 0.005–0.0001

20–50 30–50 30–50 30–50 30–50

Similarly, for the restraining member element of length dz as illustrated in Fig. 7(b) E e I e we

0000

¼ q1 ðzÞ þ q2 ðzÞ:

ð20Þ

Substituting wc1 = wc2 = we into the sum of Eqs. (19) and (20), the differential equation of a fix-ended CSBRB can be expressed as ð2Ec1 Ic1 þ Ee Ie Þwc1

0000

00

þ Pwc1 ¼ 0:

ð21Þ

Solving Eq. (21) and based on the end conditions of the core wc1(0) = wc1(l) = 0 and wc1′(0) = wc1′(l) = 0, the critical load of a fix-ended CSBRB about the y axis can be derived as P cr ¼

4π 2 ð2Ec1 I c1 þ Ee I e Þ ¼ 2P cr;c1 þ P cr;e : l2

ð22Þ

The findings indicate that a fix-ended CSBRB buckles as a superposed beam, and its critical load Pcr is the combination of the critical load of fixended cores 2Pcr,c1 and the critical load of fix-ended restraining member Pcr,e. 5. FE elastic buckling analysis 5.1. FE model building In this section, critical loads of CSBRBs are calculated by using the FE package ANSYS [21]. The beam element, rather than the shell element, in ANSYS is used because the local buckling of plates is not considered in this study. The buckling about the x axis and the influence of shear deformation are not considered. Thus planar beam element beam 3, which ignores the effects of shear deformation, is used to simulate the cores, restraining member and end plates (Fig. 8). To simulate the lateral contact between cores and the restraining member in the FE model, translations of cores and the restraining member are coupled in the x direction. No friction occurs between cores and the restraining member. Therefore, the cores and restraining member

a) Different moments of inertia of

7

can deform freely in the z direction. However, translations at the center of the restraining member and the core are coupled together in the z direction to prevent the rigid body displacement of the restraining member. The connection between the end plate and the core is a rigid joint, and no interaction occurs between the end plates and the restraining member. When CSBRBs are hinged at both ends, the translation at the center of the upper end plate in the x direction is fully restrained, and translations at the center of the lower end plate in the x and z directions are fully restrained. When CSBRBs are fixed at both ends, the translation at the center of the upper end plate in the x direction is fully restrained, translations at the center of the lower end plate in the x and z directions are fully restrained, and rotations at the center of the lower and upper end plates are both fully restrained. A concentrated force P is applied downwards to the center of the upper end plate. 5.2. Results of pin-ended CSBRBs 5.2.1. Results of pin-ended CSBRBs with infinitely rigid end plates Seven groups of all-steel type numerical specimens are analyzed (Table 1), with different moments of inertia of restraining members Ie, lengths of braces l and distances between cores b. The braces are hinged at both ends, and the rigidity of the end plates is considerably large. The steel of cores and restraining members is assumed to be perfectly elastic without considering the material nonlinearity. And Young's modulus is Ec1 = Ec2 = Ee = 2.06 × 105 MPa. The parameter β obtained by Eq. (10) is also summarized in Table 1. A comparison between critical loads Pcr obtained by Eq. (14) and those via FE analysis is shown in Fig. 9. The vertical coordinate stands for the ratio Pcr/Pcr,0. Pcr,0 = π2(2Ec1Ic1 + EeIe)/l2 is the sum of the critical loads of two isolated pin-ended cores and that of an isolated pin-ended restraining member. As shown in Fig. 9(a), with the increase of the moment of inertia of the restraining member Ie, the critical load reduces from 4Pcr,0 to Pcr,0. The buckled configuration also changes significantly. In particular, a CSBRB with very small Ie has a small ratio β, and end plates will not rotate when the CSBRB buckles. Thus, the buckled configuration is similar to that of a sway frame, and the critical load can be calculated by using Eq. (15). For a CSBRB with very large Ie, the ratio β is large, end plates rotate, and remarkable axial deformations exist in cores when the CSBRB buckles. Thus, the buckled configuration is similar to that of a superposed beam. The critical load can be obtained by using Eq. (16). Fig. 9(b) indicates that the critical load and buckled configuration are both independent of length l. Fig. 9(c) shows that the critical load approaches 4Pcr,0 with the increasing distance between cores b. The parameter β obtained by Eq. (10) deceases with increasing b. As a result, the rotation of end plates and the axial deformation of cores are

b) Different lengths of braces

c) Different distances between cores

restraining member Fig. 10. Critical loads of pin-ended CSBRBs considering deformations of end plates: (a) different moments of inertia of restraining member, (b) different lengths of braces, and (c) different distances between cores.

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Y. Guo et al. / Journal of Constructional Steel Research 106 (2015) 1–10

a) Different moments of inertia

b) Different length of braces

c) Different distances between cores

of restraining member Fig. 11. Critical loads of fix-ended CSBRBs: (a) different moments of inertia of restraining member, (b) different length of braces, and (c) different distances between cores.

prevented from occurring, and the buckled configuration of a CSBRB approaches that of a sway frame. This phenomenon is more significant for CSBRBs with small 2Ec1Ic1 + EeIe, the critical load of which is equal to approximately 4Pcr,0 with a smaller b. According to Fig. 9, if a CSBRB is hinged at both ends and its end plates are infinitely rigid, the critical load obtained by Eq. (14) coincides exactly with that obtained via FE elastic buckling analysis. Thus, Eq. (14) deduced by equilibrium method is a precise method for calculating the critical load.

5.2.2. Results of pin-ended CSBRBs considering the deformations of end plates Table 2 summarizes the five groups of numerical specimens analyzed to evaluate the quality of Eq. (13), considering the deformations of end plates. The braces are hinged at both ends. The steel of cores and the restraining member, as well as that of the end plates, is assumed to be perfectly elastic. And Young's modulus is Ec1 = Ec2 = Ee = Eb = 2.06 × 105 MPa. The influence of the end plate moment of inertia on the critical load is studied in this section by changing the thickness of end plates tb. A comparison between critical loads Pcr obtained by Eq. (13) and those via FE analysis is shown in Fig. 10. As illustrated in Fig. 10(a), with the increase of Ie, the critical load reduces from 4Pcr,0 to Pcr,0. For CSBRBs with identical Ie, the constraints of cores applied by the end plates are enhanced with increasing end plate thickness tb. As a result, the critical load increases, and the buckled configuration approaches that of a sway frame. Fig. 10(b) shows that the critical load and buckled configuration are both independent of the length of brace l. Fig. 10(c) shows that the influence of the distance between cores b on the critical load is complicated. With the increase in b, the moment of inertia of the compound section formed by two cores increases, thus preventing the rotation of end plates and axial deformations of cores, the buckled configuration approaches that of a sway frame, and critical load increases. This influence is similar to that of CSBRBs with infinitely rigid end plates [Fig. 9(c)]. Meanwhile, the constraints of cores applied by end plates become weaker, such that the critical load decreases. Taking numerical specimens in groups 4 and 5 for example, as b in group 4 increases with tb = 0.03 m, Pcr increases first but then reduces. For specimens in group 5, Ie is very small, such that the buckled configuration is similar to that of frames. Therefore, the second kind of influence of b discussed above is more significant than the first. Thus, Pcr of specimens in group 5 always decreases with increasing b. According to Fig. 10, when a CSBRB is hinged at both ends and deformations of end plates are considered, the critical load obtained by Eq. (13) fits well with that via FE elastic buckling analysis, and the maximum error is 4.8%.

5.3. Results of fix-ended CSBRBs When a fix-ended CSBRB buckles, its end plates cannot rotate, and no axial deformation occurs in cores. Consequently, the buckled configuration of fix-ended CSBRBs is similar to that of a sway frame. Critical loads of numerical specimens in Table 1 are obtained by FE analysis, with both ends fixed. These results are compared with those obtained by using Eq. (22) (Fig. 11). As shown in Fig. 11, the critical loads of fix-ended CSBRBs obtained by Eq. (22) fit well with those obtained via FE elastic buckling analysis, and the maximum error is 3.6%. Critical loads of CSBRBs with different dimensions are always approximately equal to 4Pcr,0, independent of moment of inertia of the restraining member Ie, lengths of braces l or distances between cores b. The buckled configuration of fix-ended CSBRBs with different dimensions is similar to that of sway frames. According to the comparison between Figs. 9 and 11, when a pinended CSBRB with small β buckles, its end plates will not rotate, and no axial deformation occurs in the cores. Therefore, the buckled configuration and critical load of a pin-ended CSBRB with small β are the same as those of a fix-ended CSBRB. Taking the numerical specimens of groups 3 and 7 in Table 1 for example (β ≤ 0.02), the buckled configurations of those cases under two end conditions both resemble that of sway frames, and the difference of critical loads under two end conditions is less than 1%. In fact, when β approaches zero, the critical load of pin-ended CSBRBs expressed by Eq. (13) corresponds to that of fix-ended CSBRBs obtained by using Eq. (22).

6. Verification of general design method In this section, ANSYS [21] is used to perform elastic-plastic large deformation analysis of six CSBRB specimens to evaluate their ductility under axial monotonic compression, as well as to investigate the influence of the restrain ratios on the behavior of CSBRBs. According to the study of Iwata and Murai [22], the core compressive strain is approximately identical to the drift angle that occurs in the frame under the influence of horizontal forces. Hence, the maximum Table 3 Dimensions of numerical specimens for elasto-plastic analysis. Designation

l (m)

d (mm)

he (mm)

be (mm)

tf (mm)

tw (mm)

Pcr (kN)

ζy

AS-4 AS-5 AS-6 AS-7 AS-8 AS-9.5

4.0 5.0 6.0 7.0 8.0 9.5

120 120 120 120 120 120

140 140 140 140 140 140

40 40 40 40 40 40

4 4 4 4 4 4

4 4 4 4 4 4

5919.5 3859.2 2707.6 2001.7 1538.8 1105.7

2.87 1.87 1.31 0.97 0.75 0.53

Y. Guo et al. / Journal of Constructional Steel Research 106 (2015) 1–10

Fig. 12. Dimensions of the restraining member.

core compressive strain, equivalent in magnitude to an allowable drift angle of 1/50 degree specified in the Chinese seismic design code [23], is set to be 2% in this study. Notably, in the case of core steel with yield stress fy = 235 MPa and Young's modulus E = 206GPa, the ratio of BRB maximum deformation Δmax and yield deformation Δby is approximately 20 at core compressive strain of 2%, which satisfies the requirement for ductility of BRBs prescribed in the Seismic Provisions [24]. The dimensions of six CSBRB specimens are illustrated in Table 3, with different values of length l. As shown in Fig. 12, tf and tw are respectively the wall thickness of the restraining plate and web plate, he and be are respectively the height and width of steel tube, and d is the net distance between two steel tubes. The cross-sectional areas of two core plates are both 137 × 22 mm, such that the axial yield strength Py,c is equal to 2060.5 kN. The restrain ratios ζy of different CSBRBs, computed by Eq. (1), are also shown in Table 3. The core plates and restraining member are modeled using beam 188 elements with bilinear kinematic hardening material. The material grade of the core plate is Q235 in GB50017 [25] with a yield stress fy = 235 MPa, and that of the restraining member is Q345 with fy = 345 MPa. Young's modulus E = 206GPa, and the tangent modulus of steel after yielding Et is taken as 0.005E. The end plates are assumed to be infinitely rigid. The cross section of restraining member is uncommon. Thus, a custom cross section is created with a user-defined mesh method to obtain the exact stress distributions of restraining members.

9

The residual stresses are not assigned in the FE model to simplify the numerical analysis. An initial geometrical imperfection proportional to the overall buckling shape of CSBRB is usually assigned, with the magnitude of imperfections at the center assumed to be l/1000 [25]. CSBRB is composed of several structural members. Thus, imperfections can be easily induced in the fabrication, and residual stresses are unassigned. Therefore, a larger magnitude equal to l/750 is adopted in this study. The CSBRBs are hinged at both ends, and only the in-plane behavior of CSBRBs in the x direction is studied. The out-of-plane deformation in the y direction is constrained in the analysis. An axial displacement equal to 0.02 l is applied to the center of the upper edge plate. Fig. 13 shows the relationship between the ratio of axial force to the axial yield strength (P/Py) and the ratio of core compressive strain to the core yield strain (ε/ε y) for different CSBRB specimens. The core yield strain ε y is computed by fy,c/E. According to Fig. 13, CSBRBs with different restrain ratios ζy show three kinds of behavior under axial compressive forces. For specimens AS-4 and AS-5 with restrain ratios ζy no less than 1.87, the axial force versus core compressive strain relationships show almost a stable increasing trend, and overall buckling is not observed. No strength degradation is noted for specimens AS-4 and AS-5. The ratio P/Py reaches 1.0 when ε/εy = 1.5, indicating that the entire core area yields. The ratio P/Py increases slightly because of the strain hardening of steel material. For specimen AS-6 with restrain ratio ζy = 1.31, the entire core area also yields when ε/εy = 1.5. However, overall buckling occurs when ε/εy = 5.7, and strength degradation is noted afterwards. According to the numerical simulation, the occurrence of plastic spread at the center of the restraining member results in the overall buckling of CSBRB. The restrain ratios ζy of specimens AS-7, AS-8 and AS-9.5 are relatively low (ζy ≤ 0.97). As a result, overall buckling occurs prior to the yielding of the entire core area, and the strength deceases rapidly at a very low core compressive strain. The maximum P/P y ratio of the three specimens are less than 1.0 (0.58–0.88), and the maximum P/Py ratio decreases with decreasing restrain ratio ζy. The above numerical simulation results show that the restrain ratio is a reliable parameter for predicting the performance and ductility of CSBRBs. The results obtained through monotonic-loading numerical analysis indicate that restrain ratios above 1.8 can guarantee that the CSBRB will work without overall buckling. The validity of this minimum proposed value of restrain ratio relative to a CSBRB that is subjected to cyclic loading requires further verification. This study is being conducted by the authors.

Fig. 13. Axial force versus core compressive strain relationships.

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Y. Guo et al. / Journal of Constructional Steel Research 106 (2015) 1–10

7. Conclusion This paper presented a new type of BRB called CSRBR. The elastic critical load of braces is remarkably improved by the core-separated section. As a result, CSBRBs can be used as a type of large-tonnage BRBs in practical applications. The unidirectional elastic buckling performance of CSBRBs was investigated, and the design procedure of CSBRBs was discussed in this study. These research achievements provide a sound foundation for further investigations and application of CSBRBs. The elastic buckled configurations are different for pin-ended CSBRBs with different dimensions. In particular cases, pin-ended CSBRBs may buckle as superposed beams or sway frames. An important flexural ratio parameter β was presented, and formulas to predict elastic critical loads of pin-ended CSBRBs were proposed by using the equilibrium method. The end plates of fix-ended CSBRBs cannot rotate. Thus, the buckled configuration of fix-ended CSBRBs is always similar to that of sway frames. The critical loads of fix-ended CSBRBs were also deduced by the equilibrium method. Numerical simulations were performed, demonstrating the accuracy and efficiency of the proposed formulas. A parametric study, conducted on six CSBRB specimens with different restrain ratios and loaded monotonically in axial compression, showed that the restrain ratio is a reliable indicator of the axial compressive behavior of CSBRBs. For CSBRBs with restrain ratio greater than about 1.8, no strength degradation was noted at a core compressive strain of 2%. For CSBRBs with restrain ratio less than 0.97, overall buckling occurred obviously prior to the yielding of entire core cross section, and substantial strength deterioration was noted. Acknowledgments This study has been supported by research grants from the National Natural Science Foundation of China (No. 51178243) and Tsinghua University of China under grant (No. 2012Z10134) awarded to the first author. References [1] FEMA450. NEHRP recommended provisions for seismic regulations for new buildings and other structures. Washington: Federal Emergency Management Agency; 2004.

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