2.2.1 Bending stiffness and capacity

Figure 4 shows the comparative results of elastic bearing capacity, ultimate bearing capacity, and bending stiffness for all six specimens. When the longitudinal length of the specimen increases from 1.5 to 2.0 m, the elastic and ultimate bearing capacities of STS composite slabs decreases by 13.7%, and the bending stiffness decreases by 41.2%. When the transverse length of the specimen increases from 1.25 to 1.75 m, the elastic bearing capacity of STS composite slabs increases by 10.3%, the ultimate bearing capacity increases by 14.7%, and the bending stiffness decreases by 24.2%. When the centre-to-centre distance of adjacent steel tubes increases from 250 to 300 mm, the elastic bearing capacity of STS composite slabs decreases by 6.5%, the ultimate bearing capacity decreases by 2.0%, and the bending stiffness decreases by 19.6%. This investigation aims to increase the transverse length of STS composite slabs by increasing the number of concrete-filled steel tubes while keeping other parameters constant. The results demonstrate that the bearing capacity of STS composite slabs increases with the increase of transverse length and decreases slightly with the increase of tube spacing. Moreover, reduction in bending stiffness of STS composite slabs is the most significant, when the longitudinal length of the specimen increases. This indicates that the longitudinal flexural resistance is more significant in the STS composite slab design than its transverse flexural resistance.

Figure 4全屏查看

Comparison of bending stiffness and capacity

2.2.2 Deformation characteristics

Figure 5 shows the deformation contours of STS composite slabs at different load stages. The STS composite slab demonstrates good resistance to deformation before reaching the elastic limit, with a maximum deformation not exceeding 6 mm. Before reaching the yield limit, the deformation growth rate of STS composite slab is relatively small with the increase of loading. However, the deformation growth rate increases significantly after reaching the yield limit. In addition, although the deformation rate of STS structures increases rapidly after reaching the yield limit, the deformation pattern remains unchanged. This indicates that STS composite slabs have good coordinated deformation ability among their components due to good overall and continuity properties.

Figure 5全屏查看

Deformation contours of STS composite slab specimens at different loads: (a) DCS-1; (b) DCS-2; (c) DCS-3; (d) DCS-4; (e) DCS-5; (f) DCS-6

Taking the DCS-1 specimen as an example, the transverse and longitudinal deformation patterns of STS composite slab are plotted in Figure 6. One can see that the deformation modes of STS composite slabs in the two directions are almost identical, being symmetric about the centreline in a half-sine wave form. In addition, although the transverse section of the STS structure can be divided into intra-tube and inter-tube parts, which are discontinuous, its deformation shows good continuity through different components. It can be concluded that the inter-tube connection (flange plates, bolts, and concrete) can effectively coordinate the deformation between steel tubes. This STS structure can then be considered as an integral unit, and its deformation is similar to that of a homogeneous slab.

Figure 6全屏查看

Deformation curves of the specimen DCS-1: (a) Transversal; (b) Longitudinal

Figure 7 compares the deformation values at the centre point of the STS composite slab at different loading stages. When the longitudinal length of the specimen increases from 1.5 to 2.0 m, the measured deformations at the centre point for elastic, yield, and ultimate loads increase by 35.3%, 55.9% and 57.2%, respectively. When the transverse length of the specimen increases from 1.25 to 1.75 m, the deformations of STS composite slab at its centre point upon the application of elastic, yield, and ultimate loads increase by 35.0%, 48.9% and 64.7%, respectively. When the centre-to-centre distance between adjacent steel tubes increases from 250 to 300 mm, the corresponding deflection at the centre point for elastic, yield, and ultimate loads increase by 86.1%, 80.5% and 116.5%, respectively. Thus, the STS composite slab deformation increases with the increase of transverse and longitudinal lengths. Moreover, the specimen deformation increases more significantly with the increase of transverse length under the ultimate load. This is because the transverse stiffness of the STS composite slab is less than its longitudinal stiffness. Before the specimen is yielded, the magnitude of deformation increases when the transverse length is smaller than the longitudinal length. This is because increasing the transverse length can increase the number of concrete-filled steel tubes simultaneously. However, the specimen deformation increases sharply after the transverse length is increased by enlarging the spacing between adjacent tubes. Therefore, an STS composite slab primarily bears the load in the longitudinal direction.

Figure 7全屏查看

Comparison of deformation at the centre point of the specimen at different loads

3.1.1 Assumptions

The test results show that the deformation of the STS composite slab has good continuity and uniformity, which is similar to the deformation characteristics of a homogeneous plate. Hence, the bidirectional deformation behaviour of STS composite slabs is estimated analytically based on the theory of elastic plates and the basic assumptions of elastic mechanics, including continuity, homogeneity, elasticity, isotropy and small deformations. Besides, the following assumptions should be supplemented to calculate the pipe-roof deformation:

1) The positive strain perpendicular to the mid-plane is zero, and the displacement component at any point in the plate is simply a function of x and y, and the same normal has the same displacement at the mid-plane;

2) The normal of the mid-plane does not stretch with the deformation of the pipe-roof and is always straight, and the shear strain caused by the horizontal shear stress can be disregarded;

3) No strain occurs in the mid-plane of the pipe-roof, i.e., there is no displacement parallel to the midplane at any point in the midplane.

3.1.2 Calculation model

The coordinate system is established according to the assumption of small deflection bending for elastic plates, as shown in Figure 8. The x- and y-directions represent the transverse and longitudinal lengths of STS composite slab, respectively. Furthermore, the basic equation can be obtained as follows:

(1)

Figure 8全屏查看

Coordinate system of STS composite slab under the “four-edge simply supported” condition

The STS composite slab is considered as a four-edge simply supported system, and the boundary conditions should satisfy the following form:

(2)

The following double-series solution is used:

(3)

Equation (3) satisfies the boundary conditions of Eq. (2). The coefficient Amn is determined based on a given load q(x, y). Expanding q(x, y) into the eigenfunction set, it yields:

(4)

Multiplying both sides of Eq. (4) by sin(iπx/a)sin(jπy/b) and integrating over the rectangular region, it yields:

(5)

Exchanging the order of summation and integration, it yields:

(6)

The following forms are then obtained from Eq. (3):

(7)

(8)

(9)

Substituting Eqs. (4), (7), (8) and (9) into Eq. (1), it yields:

(10)

Substituting Eq. (6) into Eq. (10), and considering that the coefficients on both sides of the equation are equal, it yields:

(11)

Substituting Eq. (11) into Eq. (3), the deflection calculation formula for STS structure subjected to a uniformly distributed load under the four-edge simply supported condition can be obtained.

(12)

where m and n = 1, 2, 3, ….

3.2.1 Assumptions

This study considers the transverse and longitudinal stress characteristics of the STS structure, and simulates the STS structure as a bidirectional slab during the deformation calculation. The bending stiffness D is a core parameter that determines the deformation capacity of pipe-roof structure. The experimental results indicate that the bending resistance of STS composite slab differs in the transverse and longitudinal directions. Moreover, the bending resistance along the longitudinal direction is higher than that in the transverse direction. Furthermore, the calculation method for bidirectional plate stiffness can be determined by multiplying the unidirectional plate stiffness with an enlargement coefficient [38]. Thus, this study calculates the bending stiffness of STS composite slab based on the bending stiffness along its longitudinal direction, which fully considers the concrete-filled steel tubular part and the connecting part between tubes of the pipe-roof structure. The cross-sectional area of STS composite slab is defined following the idea of equivalent section conversion for steel-concrete composite structures. Moreover, the equivalent cross-sectional moment of inertia is calculated according to the assumption of homogenous isotropic material to evaluate the bending stiffness of STS composite slab.

3.2.2 Equivalent section conversion

First, steel material is equivalently converted into concrete material.

(13)

Based on the equivalence, the radius of steel tube is expressed as:

(14)

Similarly, the flange plate length is expressed as:

(15)

3.2.3 Cross-sectional moment of inertia of concrete during the uncracked stage

The concrete and steel tube can work as a system during the uncracked concrete stage. In addition to compressive forces, concrete can bear tensile forces to a certain extent. Figure 9 shows a simplified calculation schematic of the section moment of inertia for an STS composite slab.

Figure 9全屏查看

Calculation schematic of the section moment of inertia

The cross-sectional moment of inertia of steel tube is expressed as:

(16)

The cross-sectional moment of inertia of concrete in steel tube is expressed as:

(17)

The cross-sectional moment of inertia of flange plate is expressed as:

(18)

The cross-sectional moment of inertia of concrete between steel tubes is expressed as:

(19)

where

(20)

Therefore, during the concrete uncracked stage, the cross-sectional moment of inertia of STS structure can be expressed as:

(21)

3.2.4 Cross-sectional moment of inertia of concrete during the cracked stage

As the load gradually increases, concrete in the bottom of STS structure initiates cracks under tensile stresses. The bottom concrete gradually losses its mechanical capacity with the crack development. At this point, the section bending stiffness can be significantly decreased compared with that under the uncracked state. Figure 9 shows the calculation schematic of the section moment of inertia for an STS composite slab.

The distance from the centroid to the neutral axis of the uncracked concrete region inside the steel tube and the associated cross-sectional area are expressed as:

(22)

(23)

The distance from the centroid to the neutral axis of the uncracked concrete region between steel tubes and the associated cross-sectional area are expressed as:

(24)

(25)

The distance from the centroid to the neutral axis of the steel tube in the uncracked region and the associated cross-sectional area are expressed as:

(26)

, (yr)(27)

, (r<yr_e)(28)

The distance from the centroid to the neutral axis of the steel tube in the cracked region and the associated cross-sectional area are expressed as:

(29)

,, (-r<yr_e-d_n)(30)

, (-r_ey-r)(31)

The height of the uncracked concrete zone and the cross-sectional moment of inertia of different member (including the uncracked concrete inside and between steel tubes, steel tube in the uncracked region, steel tube in the cracked region, and flange plate) can be obtained using Eqs. (32) and (33).

(32)

(33)

During the concrete cracked stage, the cross-sectional moment of inertia of STS structure can be obtained based on Eqs. (32) and (33) under the translation axis theorem.

(34)

3.2.5 Bending stiffness of STS composite slab

Based on the design provisions specified in the American Society of Civil Engineers (ASCE) [39], the arithmetic mean of the moments of inertia of the uncracked (I_u) and cracked (I_c) sections is taken as the effective cross-sectional moment of inertia of STS composite slab.

(35)

Therefore, the longitudinal bending stiffness of STS composite slab is computed as follows:

(36)

According to the theory of two-way slabs, the bending stiffness for STS composite slab can be obtained by multiplying the longitudinal stiffness with an enlargement coefficient φ.

(37)

This study determines the stiffness enlargement coefficient φ using numerical simulations. Finite element (FE) models are established using the software ABAQUS (version 6.12), and more information (including cell simulation, material simulation, boundary constraints, and so on) of the FE models could be referred in Ref. [37], which has also defined the relevant parameters used for this study, and presents that the average errors of load capacity and mid-span displacement between the FE models and test results in the elastic stage are 4.1% and 9.1%, respectively. Based on the calibrated FE models, five analyses (named STS-1 to STS-5) are carried out by keeping the transverse length of STS structure constant while changing its longitudinal length. The load-displacement curves for STS composite slabs under uniaxial and biaxial loading are calculated, as shown in Figure 10. The stiffness enlargement coefficient is then determined through fitting analysis.

Figure 10全屏查看

Load versus midspan displacement under different length-to-width ratio: (a) Uniaxial loading; (b) Biaxial loading

Based on Figure 10, the bending stiffnesses of STS structures under different boundary conditions are obtained, and the results are summarized in Table 2. The stiffness enlargement coefficient varied between 1.4 and 1.7 with changes in the length-to-width ratio of the composite slab.