According to the above statement, it indicates that IBRBs in the condition of reasonable design can have good energy dissipation ability. Because of the limitation of equipment, reduced scale test was done by us. Geometrical parameters of IBRBs are shown in Table 2 and Fig. (3).

Fig. (3). Figure of constructional detail of different IBRBs.

After development of the deformation of inner tube, it will be constrained by outer tube. Assume outer tube is in elastic stage, it provides inner tube with some equivalent force (see Fig. 4). The force depends on the deformation of outer tube and geometric parameters. When the number of the spring approaches to infinite, it equals to the double-tube without clearance. When the number of the spring is finite, it refers to the IBRBs. However, some more assumptions should be considered. 1) Both ends of the brace are articulated and the brace is considered as axially loaded member; 2) the outer tube is considered as elastic model and inner tube is considered as elastoplastic model; 3) the contact rings and outer tube are considered as springs, with the stiffness of k_i.

Assume that each contact ring provides inner tube with stiffness k_i and the stability strength capacity can be obtained according to the energy theory. The derivation can be expressed as Eq. (2.8):

(2.8)

In the formula, E_i is elastic modulus, when σ_x < f_y, E_i equals E, when σ_x ≥ f_y, E_iequals 0;

k_i is the ith stiffness coefficient of the spring

v_i is y axial displacement of the ith spring

u is x axial displacement of inner tube

v is y axial displacement of inner tube

x is the external load

In local system, Eq. (2.8) can be expressed as Eq. (2.9),

(2.9)

Due to the axial displacement has little effect on the whole system, it can be omitted. The formula can be expressed as:

(2.10)

The incremental displacement of the element in local coordinate can be expressed as:

(2.11)

The node incremental displacement and shape function matrix in local coordinate are:

(2.12)

(2.13)

(2.14)

The relationship of the incremental displacement of the element between local coordinate and global coordinate is:

(2.15)

In which,

Discrete Eq. (2.10), Eq. (2.16) can be obtained.

(2.16)

In which,

Consider the relationship between local system and whole system, set

Then, the whole system can be expressed as Eq. (2.17)

(2.17)

For δΠ^k=0, the differential calculus of Eq. (2.17) is Eq. (2.18)

(2.18)

In Eq. (2.18), the last item is the part of spring. This item can also be expressed as a column vector where the nodes k_i with springs has non zero value and the others without springs have zero values. Thus,

(2.19)

The characteristics is that V ≠ 0, Thus | K-pM + k_i^k | = 0, the minimum value of p is the value of stability strength capacity.

Fig. (4). Brief figure of the inner tube of IBRBs.

According to the above process, a program can be applied by using MATLAB; this program can calculate the values of stability strength capacity of different IBRBs. The values are listed in Table 1.

The model is established by using the ANSYS finite element analysis software. Solid45 is applied to the inner tube, outer tube and contact rings of IBRBs. There is a large room of contact area between inner tube and outer tube. Conta174 which has high order quadrilateral element of 8 nodes is adopted. For the displacement of structure will cause nonlinear structure. The comprehensive yield strength of inner tube is 235Mpa. The yield strength of outer tube is 235Mpa, employing steel Q235 with 0.3 elastic modulus. Mapped meshing method and displacement load method are applied, both ends are articulated. The figures of different IBRBs are as Fig. (5).

Fig. (5). Figure of finite element model.

Generally, the strength capacity of the member mainly depends on its geometry and material. As for the IBRBs, inner tube is under constraint of outer tube and the strength capacity has been significantly improved. The main factors are: 1) The number of contact rings; 2) The restraining stiffness; 3) Geometric imperfection.

3.2.1. The Effect of the Number of Contact Rings

Through tracking one-way loading of IBRBs-1, IBRBs-2 and IBRBs-3, the load displacement curves (see Fig. 6) and the calculation results (see Table 3) can be obtained.

Fig. (6). Axial load displacement curves.

(1) The curves can be divided into two stages. Namely, elastic stage, which is from the beginning to the first turning point, and plastic stage, which is from the first turning point to the end. In the plastic stage, the curve is smooth, assuming it as a flat line. Therefore, the model can be simplified as an ideal elastoplastic model.

(2) The yield strengths of the members are respectively: 124.8kN, 126.4kN and 128.6kN with the increase of contact rings. It shows that the effect of contact ring is good and can improve the yield strength.

(3) After the development of the four curves reaching to the first turning point, there is a small fluctuation on the curves with the increase of the number of contact rings. The member of IBRBs-1 has the greatest curve fluctuation and the load has a downward trend. However, the curves of IBRBs-2 and IBRBs-3 have a remarkable rising trend. The stability of IBRBs-3 is good and fully meets the standard of ideal plasticity.

Table 3.

Table of calculation results.

Number	Yield strength Pu/kN	Ultimate strength Pu/kN	Clearance/mm	Number of contact ring
IBRBs-1	124.8	131.6	10	2
IBRBs-2	126.4	135.3	10	3
IBRBs-3	128.6	142.5	10	5

Through the calculation of restraining stiffness of the three members, three curves (see Fig. 6 and Fig. 7) could be obtained. In general; the curve of IBRBs-5 has a poor stability. In contrast, the curves of IBRBs-3 and IBRBs-4 are stable.

Fig. (7). The displacement curves of one way loading.

(1) The development tendency of IBRBs-3, IBRBs-4 and IBRBs-5 agreed with each other. They are all in the ideal linear period. The initial stiffness of the three members has a little difference, for it mainly reflects the stiffness of inner tube at the initial stage.

(2) The curves of the three members from the yield point to the displacement reaching to 20mm are relatively stable. It means they are all in the plastic stage with mainly axial deformation.

(3) After the displacement of 20mm, the strength capacity of IBRBs-3 and IBRBs-4 have been enhanced. They are all in the strengthening stage. However, the strength capacity of IBRBs-5 is unstable. It means that the member has been affected by the outer tube. For the comprehensive restraining stiffness coefficient η is large and the clearance is small, with deformation of inner tube well constrained by outer tube. Thus, the continuing development is stable. The comprehensive restraining stiffness coefficient η of IBRBs-5 is small and the clearance is large, so the constraint can hardly meet the deformation of inner tube. As a result, the constraint effect is declined and the curve is fluctuated.

(4) After the displacement of 60mm the curve is a horizontal line, IBRBs-3 and IBRBs-4 reach to their ultimate strength.

The loading device consists of a rigid base and loading portal frame (see Fig. 8). The test adopts one way loading and cyclic loading. In order to check the experimental apparatus and data acquisition system, 1.5mm preload should be applied for the specimen before loading. After then, 3 specimens are under one way loading and cyclic loading, the load should be terminated under these following circumstance:

(1) Overall buckling of the brace.

(2) Insufficient rigidity of constraint leads to higher order buckling instability.

(3) Crack of weld seam between flanges and brace.

(4) Fracture of the end of inner tube.

Considering the strength capacity of the loading device and the load program of exiting test, displacement load method is adapted for the whole loading process. Before the value of 100kN, the incremental displacement is 1.5mm, when the value exceeds 100kN; the incremental displacement is 1mm, each stage of load cycling once.

Fig. (8). Test loading device.

There is a load sensor arranged between the upper flanges plate and the load beam (see Fig. 9). The arrangements of displacement meter are:

(1) One is arranged at the upper flanges, mainly to determine the out of plane lateral displacement.

(2) Three of them are distributed along the trisection of circumference of outer tube.

(3) Two of them are arranged at the brace of bottom flanges, mainly to measure axial displacement. Strain gauges (see Fig. 9) are distributed along the position of the trisection of circumference of the steel tube; with a strain gauge set in each position. Thus, three strain gauges are arranged in each section.

Fig. (9). The arrangement of strain gauge.

Table 5 lists the main information of the test and Fig. (10) is the load displacement curve. Following conclusions can be obtained.

Fig. (10). The load displacement curve.

(1) The yield strengths of IBRBs-3, IBRBs-4 and IBRBs-5 are close to each other. On the whole, the yield strength increases with the increase of restraining stiffness. Because at the initial stage, the main deformation is the axial deformation and the contact ring is closely combined with outer tube. The grater the constraint of outer tube is, the better the strength capacity is. The strength capacity is proportional to the constraining stiffness. Compared with the finite element numerical calculation results, the test results are relatively large, because the value of yield strength in reality is larger than that in theory. However the main principles between them are the same.

(2) In the process of the test, the three members of lateral displacement are very small and the maximum lateral displacement is 2.1mm. The deformation of the member is 1.75‰ of its length. It means overall buckling doesn't occur. The test value of strength capacity has exceeded the calculated value of yield strength (see Table 5). In view of the results, the inner tube is on yield form. That means it reaches to the goal that the buckling restrained brace of the tube is on yield form without buckling.

The hysteretic loops of the specimens under cyclic loading are shown in Fig. (11). The weld in IBRBs-5 was cracked due to the poor quality in the test and the core of the member wasn’t in plastic stage. Thus, the figure of IBRBs-5 hasn’t been presented below. The load and displacement loop is plump and fusiform shaped so that specimen energy dissipation is good. Hysteresis loops show:

Fig. (11). The hysteretic curves.

(1) The shapes of the hysteresis loops of both IBRBs-3 and IBRBs-4 are similar with each other, they have basically the same envelope area and the same energy dissipation ability.

(2) The hysteresis loops without big fluctuations and strength degradation are plump. The reason is that, with the increment of contact rings, the outer tube plays an important role in constraining the inner tube so that it will yield without buckling, reaching to the effect of same performance in compression and tension.

The strength back-bone curves of IBRBs-3 and IBRBs-4 are shown in Fig. (12). As can be seen from the figure:

Fig. (12). The skeleton Curves.

(1) The maximum load in tension and compression of IBRBs-3 and IBRBs-4 are similar with each other.

(2) The compression and tension curves of IBRBs-3 and IBRBs-4 are relatively smooth.

(3) The shape of the strength back-bone curves by cyclic loading is similar with the curve of the one way loading, however, the maximum load is slightly lower.