Mechanical Performance of Built-Up Columns Composed of Four Cold-Formed Square Steel Tubes

: This study presents an experimental investigation into the mechanical performance of built-up columns composed of four cold-formed square steel tubes under axial load. The four tubes were assembled together with several C-shaped connectors through two self-tapping screws in each junction. The inﬂuence of parameters including spacing between tubes, type of connectors and transverse diaphragm were analyzed based on the failure modes, ultimate loads, load-displacement relationships and load-strain relationships measured in the tests. Moreover, a further numerical analysis was carried out to study the effect of the number of connectors, web height of connectors and installing connectors at column ends by means of the veriﬁed ﬁnite element models. Finally, the numerical results were compared with the strengths predicted by the AISI-S100-2012 code. Results show that the performance of built-up columns can be inﬂuenced by the change in the number of connectors and ratio of web height of connectors to spacing between tubes as well as the installation of connectors at column ends. In addition, the current AISI-S100-12 speciﬁcations do not provide a good prediction of the built-up columns composed of four cold-formed square steel tubes.


Introduction
Galvanized lipped channel sections are mainly used as the bearing components in cold-formed steel (CFS) wall-stud systems, which are confined to low-rise buildings. To apply this system to taller buildings [1][2][3][4], several lipped channel sections are usually combined to form built-up sections. However, the inherent open and thin-walled characteristics of lipped channel sections cause them to suffer from local buckling and distortional buckling. Further, these buckling modes increase the complexity of bearing capacity calculation and greatly cause the decrease of bearing capacity. Therefore, recent research has primarily focused on eliminating the occurrence of such buckling modes and improving the bearing capacity of cold-formed thin-walled built-up sections.
The most common built-up section is the combination of two channels placed back-to-back. Either they were welded together and researched by Dabaon, M. [5,6] and Hosseini Hashemi, B. [7] or connected through self-tapping screws and tested by Stone, T. A. [8] and Lu, Y. [9] Later the channel section is improved at the web stiffened lipped channel section by Anbarasu, M. [10]. In this way, the local buckling of web could be avoided. Maia and Vieira [11] combined two angle steel through battened plates, which are connected to the angles by bolt or welding. EI Aghoury, M. A. [12][13][14]

Test Setup and Procedure
The specimens were loaded horizontally by a testing equipment as shown in Figure 2. The specimens were fixed together with the loading frame through the stay bolts and fixed together with the loading plate through bolts. The horizontality of specimens was calibrated by a levelling rod. This loading equipment ensured the axial loading to the maximum extent possible. This was mainly The nomenclature and dimension of each specimen are listed in Table 1. In this experiment, two sizes of spacing between tubes were designed to investigate their influence on the stability of built-up columns, which were 110 mm and 200 mm respectively. The influence parameter of the connector's type is studied in such a way that the C-shaped connectors around the column were replaced with a loop formed by the bending of the steel strip. Furthermore, several X-type steel plates (transverse diaphragm) were welded with the four steel tubes to investigate their influence on the performance of the novel built-up section.

Test Setup and Procedure
The specimens were loaded horizontally by a testing equipment as shown in Figure 2. The specimens were fixed together with the loading frame through the stay bolts and fixed together with the loading plate through bolts. The horizontality of specimens was calibrated by a levelling rod. This loading equipment ensured the axial loading to the maximum extent possible. This was mainly because, the non-axial displacements of cylindrical steel block used to transfer the load were limited by two steel plates with openings. In addition, the cylindrical steel block had a slot along the generatrix, and the bulge at the edge of the openings of two steel plates could just fit into the slot. Therefore, the rotational displacement of the cylindrical steel block was also limited.

Test Setup and Procedure
The specimens were loaded horizontally by a testing equipment as shown in Figure 2. The specimens were fixed together with the loading frame through the stay bolts and fixed together with the loading plate through bolts. The horizontality of specimens was calibrated by a levelling rod. This loading equipment ensured the axial loading to the maximum extent possible. This was mainly because, the non-axial displacements of cylindrical steel block used to transfer the load were limited by two steel plates with openings. In addition, the cylindrical steel block had a slot along the generatrix, and the bulge at the edge of the openings of two steel plates could just fit into the slot. Therefore, the rotational displacement of the cylindrical steel block was also limited. As shown in Figure 2, the load was applied by a hydraulic jack of 500 kN capacity and measured by a load cell of 500 kN capacity. The measure devices were shown in Figure 3. Two Linear Variable Displacement Transducer (LVDT) (D1 and D2) were used to measure the axial shortening of the specimens along y-axis. Eight (LVDT) transducers (Dv1-Dv4 and Dh1-Dh4) were placed at the flange and web of four tubes at half the height of the column to measure the lateral displacement. In addition, the lateral displacement along z-axis at quarter and three quarter of the length of the column was measured by the LVDT Dv1/4 and Dv3/4. Sixteen strain gauges (T1-1-T4-4) were pasted on the four sides of the four tubes at mid-height and four strain gauges (B1-B4) were glued at the web of the connectors. A data acquisition system was adopted to record the applied load and readings of the LVDTs and strain gauges. The load increment was 5 kN/5 min until the column buckled. In the postbuckling range, the compression was increased with displacement control in a rate of 0.03 mm/s. Prior to testing, a preload was applied to specimens to make full contact with the loading frame.

Material Properties
The material properties of tubes, connectors and tracks were measured in this paper. The material grade of tubes, connectors and tracks were Q345 stipulated in GB50018-2002 [28]. All the connectors and U-shaped tracks were fabricated from the same batch of steel sheets. The dimensions of the tensile coupon were conformed to the GB/T 228.1-2010 standard [29]. An electronic universal material testing machine was used to apply load and record readings. An extensometer having a gauge length of 50 mm was attached to each coupon and the corresponding readings were used to determinate the elastic modulus. The relationships between stress and strain of tubes and steel sheets are shown in Figure 4. The measured yield and ultimate stresses are listed in Table 2. As shown in Figure 2, the load was applied by a hydraulic jack of 500 kN capacity and measured by a load cell of 500 kN capacity. The measure devices were shown in Figure 3. Two Linear Variable Displacement Transducer (LVDT) (D1 and D2) were used to measure the axial shortening of the specimens along y-axis. Eight (LVDT) transducers (Dv1-Dv4 and Dh1-Dh4) were placed at the flange and web of four tubes at half the height of the column to measure the lateral displacement. In addition, the lateral displacement along z-axis at quarter and three quarter of the length of the column was measured by the LVDT Dv1/4 and Dv3/4. Sixteen strain gauges (T1-1-T4-4) were pasted on the four sides of the four tubes at mid-height and four strain gauges (B1-B4) were glued at the web of the connectors. A data acquisition system was adopted to record the applied load and readings of the LVDTs and strain gauges. The load increment was 5 kN/5 min until the column buckled. In the post-buckling range, the compression was increased with displacement control in a rate of 0.03 mm/s. Prior to testing, a preload was applied to specimens to make full contact with the loading frame.

Material Properties
The material properties of tubes, connectors and tracks were measured in this paper. The material grade of tubes, connectors and tracks were Q345 stipulated in GB50018-2002 [28]. All the connectors and U-shaped tracks were fabricated from the same batch of steel sheets. The dimensions of the tensile coupon were conformed to the GB/T 228.1-2010 standard [29]. An electronic universal material testing machine was used to apply load and record readings. An extensometer having a gauge length of 50 mm was attached to each coupon and the corresponding readings were used to determinate the elastic modulus. The relationships between stress and strain of tubes and steel sheets are shown in Figure 4. The measured yield and ultimate stresses are listed in Table 2.

Failure Mode
The ultimate loads of the four specimens are listed in Table 3. The deformed shapes at failure of the four specimens were shown in Figure 5. Only one mode of failure was observed in the tests, which was flexural buckling. For the four specimens, lateral deformations at the middle length of the column were visible when the loads were close to the peak. After the loads crossed the peak, the bending deformations increased rapidly. It is worth noting that all the specimens generated lateral displacement on both the x-axis and z-axis. However, the displacement on the x-axis was more pronounced. In addition, after the tubes bended, on the one hand, the end of the tubes was detached from the web of track (Figure 6a), and on the other hand, the angle between C-shaped connector and tube was no longer perpendicular ( Figure 6b). The connectors rotated as the tubes bended, which led to the joint of the connector and tube being similar to a hinge point. The rotation of connectors was due to the non-negligible extrusion deformation of the tube plate caused by screws. This indicates that the bending deformation of tubes cannot effectively be avoided by the small web height of connectors and only two screws adopted for connecting connector and tube. Maia and Vieira [11] once concluded that the type of connection is more important than the size of connectors. Therefore, the influence of screws connection should be investigated in future research.

Failure Mode
The ultimate loads of the four specimens are listed in Table 3. The deformed shapes at failure of the four specimens were shown in Figure 5. Only one mode of failure was observed in the tests, which was flexural buckling. For the four specimens, lateral deformations at the middle length of the column were visible when the loads were close to the peak. After the loads crossed the peak, the bending deformations increased rapidly. It is worth noting that all the specimens generated lateral displacement on both the x-axis and z-axis. However, the displacement on the x-axis was more pronounced. In addition, after the tubes bended, on the one hand, the end of the tubes was detached from the web of track (Figure 6a), and on the other hand, the angle between C-shaped connector and tube was no longer perpendicular ( Figure 6b). The connectors rotated as the tubes bended, which led to the joint of the connector and tube being similar to a hinge point. The rotation of connectors was due to the non-negligible extrusion deformation of the tube plate caused by screws. This indicates that the bending deformation of tubes cannot effectively be avoided by the small web height of connectors and only two screws adopted for connecting connector and tube. Maia and Vieira [11] once concluded that the type of connection is more important than the size of connectors. Therefore, the influence of screws connection should be investigated in future research.     Figure 7 shows the load-strain curves of connectors. It can be seen from Figure 7 that the strain of connectors for S1, S2 and S4 are small and most of them are lower than 25 µε. This finding shows that the deformation of connectors are mainly in the form of rigid motion, which indirectly validates the hinge points observed in Figure 6b. As shown in Figure 6c, it can be seen that a fracture occurred at the weld, which connected the X-type transverse diaphragm and tubes (S4). This shows that the X-type transverse diaphragm does play a role in resisting the relative deformation between tubes. However, the transverse diaphragm does not play a role in improving the carrying capacity, for the bearing capacity of S4 is 12.3% lower than that of S1. Figure 6d shows the distortion of the steel strip and pullout of the screws. The steel strip has undergone a large deformation, which can be verified by the large strain values as shown in Figure 7(S3). Although the loop formed by steel strips does not have rigid motion like C-shaped connectors, the steel strip cannot restrain the flexural deformation of tubes well because of the poor bending resistance. Therefore, the bearing capacity of specimen S3 is 15.85% lower than that of S1. The spacing between tubes of S1 and S2 are 200 mm and 110 mm, respectively, with a difference of 2.65% in ultimate bearing capacity, which illustrates that the factor of spacing between tubes has no obvious effect on improving the ultimate load of novel built-up columns.   Figure 7 shows the load-strain curves of connectors. It can be seen from Figure 7 that the strain of connectors for S1, S2 and S4 are small and most of them are lower than 25 με. This finding shows that the deformation of connectors are mainly in the form of rigid motion, which indirectly validates the hinge points observed in Figure 6b. As shown in Figure 6c, it can be seen that a fracture occurred at the weld, which connected the X-type transverse diaphragm and tubes (S4). This shows that the Xtype transverse diaphragm does play a role in resisting the relative deformation between tubes. However, the transverse diaphragm does not play a role in improving the carrying capacity, for the bearing capacity of S4 is 12.3% lower than that of S1. Figure 6d shows the distortion of the steel strip and pullout of the screws. The steel strip has undergone a large deformation, which can be verified by the large strain values as shown in Figure 7(S3). Although the loop formed by steel strips does not have rigid motion like C-shaped connectors, the steel strip cannot restrain the flexural deformation of tubes well because of the poor bending resistance. Therefore, the bearing capacity of specimen S3 is 15.85% lower than that of S1. The spacing between tubes of S1 and S2 are 200 mm and 110 mm, respectively, with a difference of 2.65% in ultimate bearing capacity, which illustrates that the factor of spacing between tubes has no obvious effect on improving the ultimate load of novel built-up columns.

Load-Lateral Displacement Curves
The load-lateral displacement relationships for specimen S1 are plotted in Figure 8. Dv1/4 and Dv3/4 represent the displacement at the column quarter-length point along the z-axis (see Figure 3). As can be seen from Figure 8, the deformation of a quarter point near the loading end is different from that of a quarter point far away from the loading end, which means that the deformation of the column is not perfectly symmetric. In addition, Dv1 and Dh3 represent the deformation at the column mid-length point along the z-axis and the x-axis respectively. To further investigate the effect of the transverse diaphragm, the mid-length lateral deformation of S4 and S1 are compared. The displacement of S4 at the ultimate load in the X and Z directions is 20.67 mm and 7.15 mm respectively, and the ratio of them is 2.89, but the displacements of S1 is 23.74 mm and 14.02 mm respectively, and the ratio of them is 1.69. The comparison indicates that the deformation of S4 is closer to one-way bending than that of S1. It is possible that the transverse diaphragms of S4 enhance the torsional resistance of the built-up section. To sum up, the transverse diaphragm could improve the torsional resistance of the built-up section. But it does not work on improving the bearing capacity of the built-up column.

Load-Lateral Displacement Curves
The load-lateral displacement relationships for specimen S1 are plotted in Figure 8. Dv1/4 and Dv3/4 represent the displacement at the column quarter-length point along the z-axis (see Figure 3). As can be seen from Figure 8, the deformation of a quarter point near the loading end is different from that of a quarter point far away from the loading end, which means that the deformation of the column is not perfectly symmetric. In addition, Dv1 and Dh3 represent the deformation at the column mid-length point along the z-axis and the x-axis respectively. To further investigate the effect of the transverse diaphragm, the mid-length lateral deformation of S4 and S1 are compared. The displacement of S4 at the ultimate load in the X and Z directions is 20.67 mm and 7.15 mm respectively, and the ratio of them is 2.89, but the displacements of S1 is 23.74 mm and 14.02 mm respectively, and the ratio of them is 1.69. The comparison indicates that the deformation of S4 is closer to one-way bending than that of S1. It is possible that the transverse diaphragms of S4 enhance the torsional resistance of the built-up section. To sum up, the transverse diaphragm could improve the torsional resistance of the built-up section. But it does not work on improving the bearing capacity of the built-up column. Lateral displacement at half-length point of S1 and S4.

Load-Strain Curves
For ease of description, the four tubes (T1, T2, T3 and T4) belonging to the same built-up column are numbered. Figure 9 plots the strain of tubes of S1 versus the load. It is specified that the compression strain is negative and the tensile strain is positive. It can be seen that the curves kept growing linearly at the initial stage and the load dropped especially fast after the curve reached the peak point. The tube T1 and T3 kept in compression zone throughout the loading process. The tubes T2 and T4 went into tensile zone in the post-buckling stage. In addition, it can be found that not all the four tubes generated strain at the initial stage of loading. When the load didn't exceed 20 kN, the value of strain for tubes (except tube T1) was small, which means that the four tubes are not subjected Load (kN) S1-Dv1 S1-Dh3 S4-Dv1 S4-Dh3 Lateral displacement (mm) 23.74mm Figure 8. Load-lateral displacement curves: (a) Lateral displacement at quarter-length point of S1; (b) Lateral displacement at half-length point of S1 and S4.

Load-Strain Curves
For ease of description, the four tubes (T1, T2, T3 and T4) belonging to the same built-up column are numbered. Figure 9 plots the strain of tubes of S1 versus the load. It is specified that the compression strain is negative and the tensile strain is positive. It can be seen that the curves kept growing linearly at the initial stage and the load dropped especially fast after the curve reached the peak point. The tube T1 and T3 kept in compression zone throughout the loading process. The tubes T2 and T4 went into tensile zone in the post-buckling stage. In addition, it can be found that not all the four tubes generated strain at the initial stage of loading. When the load didn't exceed 20 kN, the value of strain for tubes (except tube T1) was small, which means that the four tubes are not subjected to load at the same time. According to Table 2, the yield stress of the tube is 375 MPa and the Young's modulus is 204 GPa. Therefore, the yield strain is 1850 µε derived from 375 MPa divided by 204 GPa. It can be known from Figure 9 that most of the strain of four tubes are smaller than the yield strain during the whole loading process. Moreover, only a portion of the strain exceeds 1850 µε in the post-buckling phase. That is to say, there are very few cross-section areas that went into the plastic zone. The above shows that the four tubes lost stability before the strain of the tubes reached the yield value. The bearing capacity of the steel tube has not been fully utilized during the loading process. Therefore, it is necessary to improve the structure so as to maximize the bearing capacity of the four tubes. diaphragm have no effect on improving the bearing capacity of built-up columns. Therefore, it is necessary to further study the influence of more factors on the bearing capacity performance of builtup columns by means of the finite element method.

Numerical Models
The tubes, connectors and tracks of built-up columns were modeled by commercial ANSYS software with SHELL181 element, which has four nodes and six degrees of freedom at each node. Since the two ends of the column was not rigidly connected to the web of the tracks, the nodes on the end of tubes and the nodes on the web of the tracks in the vicinity were modelled as contact pairs. The contact segment between tubes and tracks was simulated by element of TARGE 170 and CONTA 175. In addition, the transverse diaphragm was modeled with BEAM188 element. For the screws used in tests, since no damage was found in the test, the self-tapping screw was simulated by coupling the displacement in three directions. The elastic modulus and yield strength of the tubes, connectors and tracks are referred in Table 2. Elastic perfect plastic stress-strain curve complying with isotropic hardening and Von Mises yield criterion was adopted for the material model. In addition, the loading end of the column was restrained against translational and rotational degrees of freedom in both X and Z directions. The other end was restrained against all three translational degrees of freedom and two rotational degrees of freedom in both X and Z directions. The linear load was applied on the node of tubes directly in the finite element models. An initial imperfection in the form of half of a sine wave was introduced into finite element models with an amplitude of 1/1000 of column length From the above test results and analysis, it can be concluded that changing the spacing between tubes, replacing the C-shaped connectors with loops formed by steel strips and setting the transverse diaphragm have no effect on improving the bearing capacity of built-up columns. Therefore, it is necessary to further study the influence of more factors on the bearing capacity performance of built-up columns by means of the finite element method.

Numerical Models
The tubes, connectors and tracks of built-up columns were modeled by commercial ANSYS software with SHELL181 element, which has four nodes and six degrees of freedom at each node. Since the two ends of the column was not rigidly connected to the web of the tracks, the nodes on the end of tubes and the nodes on the web of the tracks in the vicinity were modelled as contact pairs. The contact segment between tubes and tracks was simulated by element of TARGE 170 and CONTA 175. In addition, the transverse diaphragm was modeled with BEAM188 element. For the screws used in tests, since no damage was found in the test, the self-tapping screw was simulated by coupling the displacement in three directions. The elastic modulus and yield strength of the tubes, connectors and tracks are referred in Table 2. Elastic perfect plastic stress-strain curve complying with isotropic hardening and Von Mises yield criterion was adopted for the material model. In addition, the loading end of the column was restrained against translational and rotational degrees of freedom in both X and Z directions. The other end was restrained against all three translational degrees of freedom and two rotational degrees of freedom in both X and Z directions. The linear load was applied on the node of tubes directly in the finite element models. An initial imperfection in the form of half of a sine wave was introduced into finite element models with an amplitude of 1/1000 of column length [30][31][32]. Residual stresses were not included in the model for it had little effect on the performance of built-up column [33,34]. The arc-length method was used in solving the nonlinear system of equations.

Validation of Finite Element Model
The numerical models were verified by the test results. The comparison results including ultimate loads, failure shapes and load-axial shortening curve (take S1 as an example) are given in Table 3, Figures 10 and 11. From the Table 3, it can be seen that, the difference between numerical results and test results are within 10%. Moreover, as you can see in Figure 10, both the simulations and experiments exhibit similar failure modes. The ultimate loads and the failure shapes obtained from numerical analysis are in good agreement with those from tests. As for the comparison of load-axial shortening curves, there are some differences in the axial displacement corresponding to the peak value of the curves (see Pu-FEA (couple) in Figure 11a. The difference may be due to the fact that there is a small slip at the screw connection in the tested component, while the slippage of each connection adds up to a considerable axial displacement. Therefore, another numerical model, in which the screw was simulated by the nonlinear spring element Combin39 [35] (Slippage can be considered. The spring stiffness was shown in Figure 11b), was established and compared with the experimental results, as shown in Figure 11a. It can be seen that the curve of Pu-FEA (COMBIN 39) meets the tested curve better. The new established finite element model is used to the in-depth study of the influence of the web height of connector and number of connectors on the bearing character of built-up columns.

Effect of Web Height of Connector
According to the experiments, the connectors having web height of 40 mm cannot effectively restrain the deformation of tubes, so the web height of the connector becomes a factor to be considered in the numerical analysis. The influence of web height of the connector on the mechanical performance of built-up column was analyzed by means of a dimensionless parameter named as web height/spacing ratio, that is, the ratio of the web height of connector to the spacing between tubes. The models for comparison were established based on S1 and the web height of the connectors was 50 mm, 60 mm, 80 mm and 100 mm respectively, and the corresponding web height/spacing ratios were 0.25, 0.3, 0.4 and 0.5 respectively. The number of screws in each junction were kept at two.
The change of bearing capacity with the web height/spacing ratio is shown in Figure 12. From the figure, we can see that the bearing capacity of the built-up column increased obviously with the increase of web height/spacing ratio. When the web height/spacing ratio is 0.5, the local plastic deformation of tubes was observed during the post-buckling period (see Figure 13). This indicates that with the increase of the web height/spacing ratio, the ability of the connector to restrain the deformation of tubes is improved, and the composed action between tubes is also strengthened. However, when the parameter increases from 0.4 mm to 0.5 mm, the bearing capacity of the built-up column increases only 2.6 %. The increasing range decreases significantly after the parameter is greater than 0.4. This shows that the increase in the web height/spacing ratio does not result in a sustained significant increase in the bearing capacity of the built-up column. Therefore, for the built-up column with length of 3000 mm, the ratio of the web height of connector to the spacing between the tubes is suggested to be 0.4. from numerical analysis are in good agreement with those from tests. As for the comparison of loadaxial shortening curves, there are some differences in the axial displacement corresponding to the peak value of the curves (see Pu-FEA (couple) in Figure 11a. The difference may be due to the fact that there is a small slip at the screw connection in the tested component, while the slippage of each connection adds up to a considerable axial displacement. Therefore, another numerical model, in which the screw was simulated by the nonlinear spring element Combin39 [35] (Slippage can be considered. The spring stiffness was shown in Figure 11b), was established and compared with the experimental results, as shown in Figure 11a. It can be seen that the curve of Pu-FEA (COMBIN 39) meets the tested curve better. The new established finite element model is used to the in-depth study of the influence of the web height of connector and number of connectors on the bearing character of built-up columns.

Effect of Web Height of Connector
According to the experiments, the connectors having web height of 40 mm cannot effectively restrain the deformation of tubes, so the web height of the connector becomes a factor to be considered in the numerical analysis. The influence of web height of the connector on the mechanical performance of built-up column was analyzed by means of a dimensionless parameter named as web height/spacing ratio, that is, the ratio of the web height of connector to the spacing between tubes. The models for comparison were established based on S1 and the web height of the connectors was 50 mm, 60 mm, 80 mm and 100 mm respectively, and the corresponding web height/spacing ratios were 0.25, 0.3, 0.4 and 0.5 respectively. The number of screws in each junction were kept at two.
The change of bearing capacity with the web height/spacing ratio is shown in Figure 12. From the figure, we can see that the bearing capacity of the built-up column increased obviously with the increase of web height/spacing ratio. When the web height/spacing ratio is 0.5, the local plastic deformation of tubes was observed during the post-buckling period (see Figure 13). This indicates that with the increase of the web height/spacing ratio, the ability of the connector to restrain the deformation of tubes is improved, and the composed action between tubes is also strengthened. However, when the parameter increases from 0.4 mm to 0.5 mm, the bearing capacity of the built-up column increases only 2.6 %. The increasing range decreases significantly after the parameter is greater than 0.4. This shows that the increase in the web height/spacing ratio does not result in a

Effect of Web Height of Connector
According to the experiments, the connectors having web height of 40 mm cannot effectively restrain the deformation of tubes, so the web height of the connector becomes a factor to be considered in the numerical analysis. The influence of web height of the connector on the mechanical performance of built-up column was analyzed by means of a dimensionless parameter named as web height/spacing ratio, that is, the ratio of the web height of connector to the spacing between tubes. The models for comparison were established based on S1 and the web height of the connectors was 50 mm, 60 mm, 80 mm and 100 mm respectively, and the corresponding web height/spacing ratios were 0.25, 0.3, 0.4 and 0.5 respectively. The number of screws in each junction were kept at two.
The change of bearing capacity with the web height/spacing ratio is shown in Figure 12. From the figure, we can see that the bearing capacity of the built-up column increased obviously with the increase of web height/spacing ratio. When the web height/spacing ratio is 0.5, the local plastic deformation of tubes was observed during the post-buckling period (see Figure 13). This indicates that with the increase of the web height/spacing ratio, the ability of the connector to restrain the deformation of tubes is improved, and the composed action between tubes is also strengthened. However, when the parameter increases from 0.4 mm to 0.5 mm, the bearing capacity of the built-up column increases only 2.6 %. The increasing range decreases significantly after the parameter is greater than 0.4. This shows that the increase in the web height/spacing ratio does not result in a sustained significant increase in the bearing capacity of the built-up column. Therefore, for the builtup column with length of 3000 mm, the ratio of the web height of connector to the spacing between the tubes is suggested to be 0.4.

Effect of Number of Connector
To investigate the effect of the number of connectors on the performance of the built-up column, the numerical models with column length ranged from 600-3000 mm were established and each column length had a variable number of connectors, varying from 0-7. An increase in the number of connectors means a decrease in the spacing between connectors. The remaining dimension of these models were consistent with that of the specimen S1, and the details were listed in Table 4. The loadaxial shortening curves of models with different number of connectors were compared, as shown in Figure 14. Note that, the number of connectors described here refers to the number of connectors in only one side of the built-up column. From Figure 14, it can be observed that with the increase of the number of connectors, the peak of the curves increases and the ductility of the columns also improves, which shows that increasing the number of connectors can improve the mechanical performance of the built-up column.

Effect of Number of Connector
To investigate the effect of the number of connectors on the performance of the built-up column, the numerical models with column length ranged from 600-3000 mm were established and each column length had a variable number of connectors, varying from 0-7. An increase in the number of connectors means a decrease in the spacing between connectors. The remaining dimension of these models were consistent with that of the specimen S1, and the details were listed in Table 4. The load-axial shortening curves of models with different number of connectors were compared, as shown in Figure 14. Note that, the number of connectors described here refers to the number of connectors in only one side of the built-up column. From Figure 14, it can be observed that with the increase of the number of connectors, the peak of the curves increases and the ductility of the columns also improves, which shows that increasing the number of connectors can improve the mechanical performance of the built-up column. Table 4. Strength-to-weight (STW) ratio of built-up columns and comparison between numerical results and design strength according to AISI-S100-12.

Effect of Number of Connector
To investigate the effect of the number of connectors on the performance of the built-up column, the numerical models with column length ranged from 600-3000 mm were established and each column length had a variable number of connectors, varying from 0-7. An increase in the number of connectors means a decrease in the spacing between connectors. The remaining dimension of these models were consistent with that of the specimen S1, and the details were listed in Table 4. The loadaxial shortening curves of models with different number of connectors were compared, as shown in Figure 14. Note that, the number of connectors described here refers to the number of connectors in only one side of the built-up column. From Figure 14, it can be observed that with the increase of the number of connectors, the peak of the curves increases and the ductility of the columns also improves, which shows that increasing the number of connectors can improve the mechanical performance of the built-up column. Furthermore, the effect of the number of connectors was also analyzed from the aspect of the strength-to-weight (STW) ratio, as listed in Table 4. Note, the strength-to-weight ratio is derived from dividing the ultimate capacity obtained from the numerical calculation by the total weight of the models. As listed in Table 4, for the columns having length of 2400 mm and 3000 mm, when the column length is the same, the STW ratio increases with the increase in the number of connectors. For example, for the column with length of 3000 mm, the ultimate load of the model with 7 connectors is Furthermore, the effect of the number of connectors was also analyzed from the aspect of the strength-to-weight (STW) ratio, as listed in Table 4. Note, the strength-to-weight ratio is derived from dividing the ultimate capacity obtained from the numerical calculation by the total weight of the models. As listed in Table 4, for the columns having length of 2400 mm and 3000 mm, when the column length is the same, the STW ratio increases with the increase in the number of connectors. For example, for the column with length of 3000 mm, the ultimate load of the model with 7 connectors is increased by 54% compared with the model without connectors, and the STW ratio is improved by 28%. However, for the columns with length of 600 mm, 1200 mm and 1800 mm, the STW ratio decreases as the number of connectors increases. For example, for the column with length of 1800 mm, the ultimate load of the model with 4 connectors is only increased by 8% compared with the model without connectors, but the STW ratio is reduced by 9%. This shows that the increase in the number of connectors can improve the economic performance of long columns but reduce the economic performance of middle and short columns.
Finally, in order to better improve the mechanical performance of the built-up column from the structure, two additional connectors were mounted at both ends of S1. The results of the model with end connectors were listed in Table 4 as L3000S700-5 (end). It can be observed that the bearing capacity of L3000S700-5 (end) is increased by 14.2% compared to that of L3000S700-3 (S1, same spacing between connectors), but decreased by 3.4% compared to that of L3000S500-5 (same number of connectors). This shows that the performance of the built-up column can be improved by mounting connectors at the column ends without changing the spacing between connectors. However, on the premise of the same number of connectors, it has a more obvious effect on improving the mechanical performance to reduce spacing between connectors than to install connectors at column ends.

Comparisons with AISI-S100-12 Code
The strengths of the novel built-up section are evaluated according to the current AISI-S100-12 code [36] of practice. The comparison between numerical results and predictions are shown in Table 4. It is clear that the AISI-S100-12 is conservative for the built-up columns without connectors, and the conservatism increased with the increase of column length. For the columns having connectors, it can be seen that the predictions are unconservative for the built-up columns with length of 3000 mm, 2400 mm and 1800 mm. The ratio of P-FEA/Pu-AISI decreased with the increase of column length. Especially for the columns with a length of 3000 mm, the ratios of P-FEA/Pu-AISI were between 0.58 and 0.84, which were far less than 1. This may be due to the fact that the slenderer the column, the weaker the restraint effect of the connectors, which leads to the poor combination effect between tubes. However, the predictions are conservative for the columns with length of 1200 mm and 600 mm. In addition, the comparisons show that the code does not provide a good prediction, with a mean of 1.11 and a coefficient of variation (COV) of 0.556.

Conclusions
The performance of built-up columns composed of four cold-formed steel tubes assembled by several connectors were experimentally and numerically investigated. The conclusions are as follows: (1) The experimental results showed that the built-up columns composed of four cold-formed thin-walled steel square tubes failed in flexural buckling. (2) It has no effect on improving the mechanical performance of built-up columns to change spacing between tubes and to replace the C-shaped connectors with loops formed by steel strip as well as to install a transverse diaphragm. However, the performance can obviously be improved by the change of number of connectors and ratio of web height of connectors to spacing between tubes as well as the installation of connectors at column ends. For the built-up column with length of 3000 mm, it is suggested that the web height/spacing ratio should be 0.4.
(3) The increase in the number of connectors can improve the economic performance of longer columns but reduce the economic performance of shorter columns. In addition, the performance of the built-up column can be improved by mounting connectors at the column ends without changing the spacing between connectors. However, on the premise of the same number of connectors, it has a more obvious effect on improving the mechanical performance to reduce spacing between connectors than to install connectors at column ends. (4) The current AISI-S100-12 code does not provide a good prediction to the novel built-up columns composed of four cold-formed steel tubes.
In further study, the influence of the number of screws connecting the tubes and connectors on the combination action between tubes will be investigated.