The height and span of the frame are 1.2m and 2.4m, respectively. The beam section is H150×150×6×6, and the column section is H150×150×6×8. The plane stiffness at the bottom of the frame structure with special central support is infinite. The reduced integral element C3D8R was used in the model, and the plastic deformation analysis of the casting steel connector in the stress process is more accurate by using a smaller mesh size. The material of the inner tube and outer sleeve was Q345B, its constitutive model adopted the Von Mises multi-linear isotropic hardening trilinear model. The material of casting steel connector was Q235B, its constitutive model adopted the ideal elastic-plastic model, Young's modulus E = 2.06 × 10 ⁵Mpa, Poisson ' s ratio = 0.3. The material stress-strain curve of steel is shown in Figs. ( 3 and 4). The material property curve of the casting steel connector is shown in Fig. ( 3). The material property curve of the inner tube, contact ring and outer sleeve is shown in Fig. ( 4). The special central support frame with casting steel connectors was numbered KF1 ~ KF5, and the finite element model of the frame is shown in Fig. ( 5).

Fig. (3). Material property curve of casting steel connectors.

Fig. (4). Material property curve of inner-tube.

Fig. (5). Finite element model of the special concentrically braced single-story frame with casting steel connector.

Fig. (6). Concentrically braced steel frames.

Fig. ( 6) shows the schematic diagram of the concentrically braced frame. The initial configuration of the structure is the same as the current configuration when the buckling-restrained brace frame has a small deformation, so it can be assumed that the 0  in the structure is unchanged. And then, the lateral stiffness of the single-layer structure of the buckling-restrained brace was calculated as follows:

The axial displacement of single support, ∆ L, is:

(1)

The horizontal component of support, F _n, is:

(2)

The lateral stiffness of single diagonal brace, K _D, is:

(3)

The herringbone support, as shown in Fig. ( 6), can be equivalent to the joint action of two monoclines, so the axial displacement, ∆ L, is:

(4)

The lateral stiffness of herringbone braces, K _D, is:

(5)

Assuming that the cross-sectional area of the inner unit is A, when the support enters the yield, the yield strength of the inner unit reaches ƒ _y. Then, it can be concluded that the yield force of the support in the horizontal direction is:

The yield stress of the single diagonal brace, F _Y, is:

(6)

The yield stress of chevron buckling restrained braces, F _Y, is:

(7)

So the yield displacement of the buckling-restrained brace, u _y, is:

(8)

Fig. (7). Monoclinic braced steel frame.

According to the above Equation ( 8), it can be seen that the yield displacement u _y of the buckling-restrained brace is only related to the material properties, support height and tilt angle of the supporting member. It is independent of the section size of the frame and the layout of the support (chevron braces, one-way diagonal support) and the section area of the supporting core.

In this paper, Q345 steel is used for the inner tube and outer tube, and both ends of the support need to be articulated with the frame. The yield displacement angle of the buckling-restrained brace, 0  _DY, is:

(9)

In this paper, the angle of support is 30° <0 <60°,so the yield displacement angle range of buckling-restrained brace can be obtained as: 1/298 < 0  _DY< 1/259

It can be seen from above that the yield displacement angle of the buckling restrained brace is less than the specification requirements of the elastic story drift ratio limit under a small earthquake. This shows that the condition of support yielding prior to the frame is easy to meet, and the support may have yielded under small earthquakes and entered the plastic stage.

The China’s code stipulates that the limit of elastic story drift ratio of steel structure is 1 / 250, and the limit of elastic-plastic story drift ratio is 1 / 50. Buckling restrained brace as energy dissipation components should remain in the elastic stage. In the elastic-plastic stage, the buckling-restrained brace should enter the plastic state to dissipate energy. From the above derivation, it can be seen that the yield displacement of a buckling restrained brace does not depend on the layout form of support. So, in order to simplify the calculation, the monocline support is taken to indicate that the support has entered the plastic stage under frequent earthquakes. Monoclinic braced steel frame is shown in Fig. ( 7). When a small deformation occurs, the relationship between the relevant parameters can be obtained.

(10)

The total deformation of buckling-restrained braces can be obtained from a geometric relationship.

(11)

Under the limit of elastic inter-story drift ratio, the axial displacement of the support is:

(12)

Under the limit of elastic-plastic inter-story drift ratio, the axial displacement of the support is:

(13)

The buckling restrained brace with casting steel connectors is generally divided into monoclinal and herring-shaped. The main connection parts of casting steel connectors exist in the beam-column joint and column-foot joints, and the local design depends on the design of the overall structure. When the cross-sectional area of the core element increases, the lateral stiffness of BRB and the deformation resistance of the structure become stronger, so the lateral displacement of the structure decreases under horizontal load. However, when the area of BRB core unit continues to increase, the lateral stiffness of the structure will be too large, and the absorbed seismic energy is more, but the overall lateral displacement of the structure shows an increasing trend. In order to give full play to the performance of BRB in buckling restrained brace frame, its lateral stiffness ratio is one of the important indexes. Therefore, the lateral stiffness ratio should be determined before the design. Jian-Bin Liu [ 11] of Tsinghua University carried out a finite element simulation of a three-story frame. The simulation showed that the displacement and acceleration of the top floor decreased when the lateral stiffness ratio was greater than 2, but the influence on the structure was not obvious when the lateral stiffness ratio was greater than 3. It is suggested that the lateral stiffness ratio should be between 2 and 5. The relationship is shown in Fig. ( 8).

The elastic-plastic time history analysis of four three-span frames was carried out by Ying Zhao of Tsinghua University [ 16]. The results showed that the story drift ratio decreases with the increase of lateral stiffness and tends to a constant value. Therefore, it is suggested that the lateral stiffness ratio should be between 1 and 3.

The relationship between BRB and steel frame is deduced theoretically by taking BRBF arranged in the shape of a chevron as an example. As shown in Fig. 9, it is assumed that the structure has a slight lateral displacement along the horizontal direction, and the axial deformation of the beam is ignored, it can be considered that the angle does not change, then the lateral stiffness K _B of the chevron-shaped support can be written as:

(14)

Where, A _b is the sectional area of the inner element, E _b is the elastic modulus of the core element of support, and h is the height of the frame column.

Assuming that the lateral displacements of the same column are equal, the horizontal lateral stiffness K _a of the frame structure can be obtained according to the D-value method [ 17]:

(15)

Where Σ(EaIi) is the sum of the bending stiffness of the i-story column, E _a is the elastic modulus of the material used in the column, I _i is the moment of inertia of the column, α is the correction coefficient of the line stiffness of the beam-column, which is related to the line stiffness of the beam-column.

Fig. (8). The relationship between displacement, acceleration, interstory displacement angle and lateral stiffness ratio of frames.

Fig. (9). BRBF arranged in the shape of a chevron.

The lateral stiffness ratio is the ratio of BRB horizontal lateral stiffness K _b to frame horizontal lateral stiffness K _a, then the structural lateral stiffness ratio K _i can be expressed as:

(16)

According to Equation (16), the influencing factors of lateral stiffness ratio are only section size and the angle of support arrangement, which have little relation with the form and quantity of support arrangement. The overall lateral stiffness can be regarded as a linear superposition of the horizontal lateral stiffness of the frame and support [ 18]. In design, it is generally considered that the horizontal lateral stiffness K _a of the frame is fixed, and the overall lateral stiffness ratio is adjusted by changing the value of K _b.

The stiffness of BRB can be regarded as the stiffness of the inner element, which is composed of the connecting segment, the transition segment and the energy dissipation segment. The steel performance and section area of each part are different, so it is necessary to make an equivalent treatment for its stiffness. According to the stiffness series principle [ 19], the equivalent stiffness K _b of the supporting member in the elastic stage is:

(17)

Where K _ed^B is the stiffness of the BRB energy dissipation segment, K _tr^B is the stiffness of the BRB transition segment, and K _con^B is the stiffness of the BRB connection segment. K can be obtained by Equation K _{i =} EiAi/Li, where E _i is the elastic modulus, A _i is the sectional area of the corresponding position, and _Li is the length of the corresponding position.

It is assumed that L₁, L_2,L_3,A_1,A₂ and A₃ are the length and sectional area of the core segment, the connecting segment and the transition segment, respectively, and E _B is the stiffness of the core element. In order to simplify the calculation, sectional changes of L₂ and L₃ are usually ignored. And assuming that the sectional area of the core unit is constant A ₁, then Equation ( 17) can be simplified as follows:

(18)

After the support enters the yield stage, both the connection segment and the transition segment should be in the elastic stage. Only the core segment becomes the plastic state, and the stiffness of the core segment needs to be adjusted. According to the assumption of the double broken line model, by introducing the stiffness ratio α, the formula can be expressed as:

(19)

Where the stiffness ratio coefficient of buckling restrained brace is generally 0.02 ~ 0.05 [ 20].

The overall stiffness of the buckling-restrained brace with casting steel connector can take the stiffness series principle to obtain. The equivalent stiffness K _C of the casting in the elastic stage is:

(20)

K ^C_ed is the stiffness of the energy dissipation segment of the casting part, K ^C_tr is the stiffness of the transition segment of the casting part, K ^C_con is the stiffness of the connecting segment of the casting part. Then the line stiffness of the buckling restrained brace with casting steel connector under the elastic stage is:

(21)

Then, by using the Equation K _e = F _e / ∆ _e , the load on the elastic limit of the whole specimen in the elastic stage can be obtained.

After the support enters the yield stage, the BRB transition segment is still in the elastic stage, and only the ductile casting part and energy dissipation segment of BRB become the plastic state, and the stiffness of the energy dissipation segment needs to be adjusted. The equivalent stiffness can be changed into:

(22)

The bearing capacity is determined by formula F _p = ∆ _LK according to the buckling restrained brace design principle of ductile casting. The energy dissipation segment of ductile casting is the first to yield, and the stress is the largest, so the elastic ultimate F _p that the energy dissipation segment can bear should be

(23)

When the axial displacement of the support reaches 1/50, the ultimate load of the support in the elastic-plastic stage can be calculated by Equation ( 24).

(24)

The following conclusions can be drawn from Fig. ( 10) and Table 3:

(1) According to Fig. ( 10), the development of KF-1 ~ KF-12 under horizontal loading went through three stages: elastic stage, yield stage and plastic strengthening stage. The load and displacement increased linearly in the elastic stage, and the curve rose rapidly, indicating that it has good stiffness. At this time, a buckling restrained brace acts as the first line of defense of the structure. According to the stress nephogram of the loading process, with the continuous loading, the stress of the casting steel connector and the inner tube increased rapidly, and the casting steel connector reached the yield, but the inner tube had not yet yielded. As the load continued, the buckling restrained brace inner tube reached yield stress, and this stage is the yield stage. With the continuous loading, the casting steel connector is completely out of work, and the buckling-restrained brace relies on the plastic deformation of the inner tube to dissipate energy. As this time, as the last line of defense, the pure frame bore all the horizontal loads and the curve began to rise slowly to provide the final safety. This stage is the strengthening stage.

Fig. (10). Load-displacement curve of the frame.

(2) According to the comparison of KF-1~ KF-5 in Table 3, it can be seen that the overstrength coefficient of the energy dissipation segment and the inner tube is the same, but the width-thickness ratio of the energy dissipation plate is 2.31, 2.22, 1.90, 1.64 and 1.42, respectively, and the width-thickness ratio of the stiffener is 1.44, 1.64,1.90,2.22 and 2.63 respectively. As the width-thickness ratio of the energy dissipation plate and stiffener changes, yield load and ultimate load are almost the same. Therefore, it can be concluded that the width-thickness ratio of stiffeners and energy dissipation plates will affect the ultimate load and yield load of the frame, but this factor is not the main influence.

(3) By taking the same overstrength coefficient but different energy dissipation lengths, KF-2, KF-6 and KF-7 are obtained. It can be seen that the yield load of KF-7 is longer than that of KF-2, and the ultimate load is reduced by 13.1% and 3.99%, respectively. It can be seen that the length of the energy dissipation segment is the main factor affecting the bearing capacity of the specimen.

(4) The buckling restraint ratio is defined as:

E _tI _t — The bending stiffness of the outer casing;

µ — The calculation length coefficient of two ends constraint;

ƒ _y — The yield stress of the inner tube;

A — The sectional area of the inner tube;

When the constraint ratio of the outer tube is inconsistent, the frames KF-2, KF-11 and KF-12 are taken to know that the yield load is almost consistent, and the ultimate load increases by 12.15%. It can be seen that the constraint ratio of the outer constraints has an obvious influence on the bearing capacity of the component.

The initial stiffness reflects the bearing capacity and stability of the frame. The following analysis can be obtained from Table 3: according to the comparison of frames KF1~KF-5, when the overstrength coefficient was the same, the initial stiffness of the model frame was basically the same. Compared with KF-2 and KF-8~KF-10, the initial stiffness had no obvious change when the overstrength coefficient increased. When the length of the energy dissipation section increased, the initial stiffness decreased significantly. Compared with frame KF-2, the initial stiffness of KF-7 decreased by 18.6%. It can be seen that the length of the energy dissipation segment of the casting steel connector has a significant impact on the initial stiffness of the frame.

The results in Table 3 show that the ductility coefficients of each frame are 4.57 and 5.81. It can be seen that the special central support steel frame with casting steel connectors has good ductility and deformation ability.

(1) By comparing frame KF-1~KF-5, it can be seen that the ductility of the frame increased by 2.28% when the ratio of the width-thickness ratio of the stiffener and the width-thickness ratio of the energy dissipation plate changed from 0.62 to 1.00; when the ratio changed from 1.00 to 1.85, the ductility of the frame decreased by 4.06%. Therefore, it can be seen that only when the width-thickness ratio of the energy dissipation plate and stiffener of the casting steel connector is controlled in a reasonable range, the structure has good ductility. It is recommended that the width-thickness ratio of the energy dissipation plate and stiffener be set at 0.74~1.73.

(2) Compared with frames KF-2,KF-6 and KF-7, when the overstrength coefficient of casting steel connector and the inner tube was consistent, the ductility of the frame was significantly improved by increasing the length of the energy dissipation segment. When the length of the energy dissipation segment increased from 90mm to 120mm, the ductility of the frame was improved by 27.1%. It can be seen that the length of the energy dissipation segment has an obvious influence on the ductility of the frame.

The stress distribution of each frame is basically similar, as shown in Fig. ( 11), the maximum stress occurred at the column foot and the end of the loaded beam. The stress of the rest of the frame was very small. The energy dissipation segment of the casting steel connector located in the column foot and the middle of the beam reached the yield stress. The stress of the transition segment of the casting steel connector is very small, and did not reach the yield stress. The stress of the outer casing as a constraint unit did not change significantly from the beginning to the end of loading, and the stress was small. The maximum stress of the frame reached 460.3MPa, which was mainly distributed in the area composed of column foot and beam end stiffening plate. As shown in Fig. ( 11), the maximum stress of the casting steel connector was 415 MPa, which was mainly distributed in the energy dissipation segment of the casting steel connectors. It fully shows that the protective effect of ductile casting steel on the structure is obvious.

Fig. (11). Finite element simulation results of each frame under unidirectional loading.