Torsional effect analysis of high-rise reinforced concrete space grid cassette multi-tube structure system

3.1. Model of space grid cassette multi-tube structure

The model of space grid cassette multi-tube structure is established in accordance with the latest standards of “Code for Seismic Design of Buildings” (GB50011-2010) [24] and “Technical Specification for Concrete Structures of Tall Buildings” (JGJ3-2010) [25]. The design parameters of each component of the structural model are shown in Table 1. Standard floor plan, standard floor axonometric view, and three-dimensional view, as shown in Fig. 4.

The floor is composed of four grid shaped hollow plates without beams, as well as ordinary floor plates. The grid size of hollow plates is 2.5 m×2.5 m. The length of the short span of the floor is 10.0 m, and the thickness of plate is calculated as 1/30 of the short span length. So thickness of plate $h=$ 10000 / 30 = 333.33 mm, and the thickness of plate is taken as 380 mm. The thickness of the upper and lower layers of the grid shaped hollow plate is 60 mm, and 4 phosphogypsum mold boxes are placed in each grid, and each phosphogypsum mold box is weighed 25 kg. In order to facilitate the software design and calculation, the method of equal generation of well-shaped beam is adopted for the hollow plate [26]. The grid shaped hollow plate is dispersed into the cross-beam system roof, and the flange of the cross beams is taken 6 times the thickness of the plates to form an I-shaped beam [27]. According to the principle of equal bending stiffness, the I-beam is replaced by a rectangular beam, $I=$ 302844 cm⁴, the width of the hollow plate rib beam is 180 mm, $h=\sqrt[3]{12I/B}$ = 586.6 mm, then after conversion, the height of beam is 580 mm (Fig. 5).

Fig. 4Three-dimensional model of cassette multi-tube structure

a) Standard floor plan

b) Standard layer axonometry

c) 3D Finite element model

Fig. 5Convert I-beam to rectangular beam (unit: mm)

3.2. Model of frame-core tube structures

In order to compare with the space grid cassette multi-tube structure effectively, the size of the core tube of the conventional frame-core tube structure is the same as that of the space grid cassette multi-tube structure, the minimum column spacing is 4 m and the maximum column spacing is 7.5 m. Based on the principle that the axial compression ratio of the column of the frame-core tube structure is the same as that of the column of the space grid cassette multi-tube structure, the column size of the frame-core tube structure is also divided into three changes, The design parameters of each component of the structural model are shown in Table 2. Standard floor plan, standard floor axonometric view, and three-dimensional view, as shown in Fig. 6.

Fig. 6Three-dimensional model of conventional frame-core tube structure

a) Standard floor plan

b) Standard layer axonometry

c) 3D Finite element model

3.3. Software applicability analysis

In this article, dimensions of each component in the two structural models are calculated by the calculation and analysis software “PKPM” to obtain data that is met the actual engineering requirements. Then “MIDAS-Gen” software is used to analyze the basic mechanical properties of the two structures. By conducting static analysis on the models in the two types of software, and models of first 15 order cycles and structural mass are compared (Fig. 7, 8), it can be concluded that the model in “MIDAS Gen” software is accurate and can be used for the next step of analysis.

It can be seen that the cycle of the cassette multi-tube structure model calculated by the two software is relatively close, and the maximum relative error of the cycle is 4.76 %, both of which are less than 5 %. And it can be seen that the cycle of the frame-core tube structure model calculated by the two software is also relatively close, and the maximum relative error of the cycle is 4.40 %, both of which are less than 5 %. It is indicated that the calculation results are accurate, and the modeling in “MIDAS Gen” can be directly used the component dimensions which are obtained by “PKPM” software.

In Table 3, M1 is the total mass generated by dead load, M2 is the total mass generated by live load, and M3 is the total mass of the structure. It can be seen that the quality of the cassette multi-tube structure model calculated by the two software is relatively close, and the quality error is less than 4 %. And it can be seen that the quality of the frame-core tube structure model calculated by the two software is also relatively close, and the quality error is less than 3 %. It is shown that the model is more accurate in “MIDAS-Gen” software.

Fig. 7Two kinds of software calculate the first 15 order cycles of cassette multi-tube structure and the relative error of cycles

a) The first 15 order cycles which are calculated by two kinds of software

b) The relative error of cycles which are calculated by two kinds of software

Fig. 8Two kinds of software calculate the first 15 order cycles of frame-core tube structure and the relative error of cycles

a) The first 15 order cycles which are calculated by two kinds of software

b) The relative error of cycles which are calculated by two kinds of software

Based on the comparison of cycle and structure quality of cassette multi-tube structure and frame-core structure model in “PKPM” software and “MIDAS-Gen” software respectively, it can be seen that the modeling of cassette multi-tube structure and frame-core structure in “MIDAS-Gen” software is accurate, and can be analyzed in the next step.

3.4. Design parameters

In this project, it is assumed that the embedded end of the structure is at the top of the basement, so the model is only included the part above the ground. The fortification earthquake group is the first group, the fortification intensity is 6 degrees, the site category is class II, the basic wind pressure is 0.35 KN/m², the characteristic period $T_g=$ 0.35 s, and the structural damping ratio is 0.05, the periodic reduction coefficient is 0.8, the accidental eccentric and bidirectional earthquake is considered, the formation combination method is CQC, and the number of formations is calculated as 26, the influence of higher order modes is considered. The constant load on the floor is 2.5 KN/m² (excluding the dead-weight of the grid shaped hollow plate), the dead-weight of the external wall is equivalent to the line load of 6.5 KN/m, the partition wall is calculated according to the equivalent uniform load on the floor, and the total live load on the floor is 3.5 KN/m².

In this paper, the software “MIDAS-Gen” is used to perform static elastic-plastic analysis of the new structure under rare earthquakes to be evaluated the torsional resistance of the structure. In the model, beam elements are used for beams and columns, and wall elements are used for shear walls. Steel bars are automatically imported and generated by the program after calculating the internal forces of each component of the structure through modal decomposition response spectrum analysis and reinforcement. The beam hinge is defined as the bending moment-rotation angle, the correlation type is “none”, components are My and Mz, the position distribution of the hinge is “I end” and “J end”, the skeleton curve is of “FEMA” type, the material type is “concrete”, and the unit type is “beam”. The column hinge is defined as bending moment-rotation angle, the correlation type is “P-M-M” type, components are My and Mz, the position distribution of hinge is “I end” and “J end”, the skeleton curve is of “FEMA” type, the material type is “concrete”, and the element type is “column”. The wall hinge is defined as the bending moment-rotation angle, the related type is “P-M-M”, components are Fz, My and Mz, the position distribution of the hinge is “I end” and “J end”, the skeleton curve is of “FEMA” type, the material type is “concrete”, and the unit type is “wall”, the unit model of the wall is adopted the plate unit model. Component skeleton curve in Fig. 9.

Fig. 9Component skeleton curves

4.1. The problem of torsion control in Chinese standard

In the latest standards of “Code for Seismic Design of Buildings” (GB50011-2010) and “Technical Specification for Concrete Structures of Tall Buildings” (JGJ3-2010), Both of them are put forward some requirements and specific judgment indicators for the torsion control of structures. But there are some differences between the two specifications in some regulations, mainly in the judgment indicators of irregular torsion and the consideration of accidental eccentricity (Table 4).

Due to different regulations of China on torsion control, the same structure may be classified as a regular structure when it is calculated according to the “Code for Seismic Design of Buildings”, while it may be classified as an irregular structure when it is calculated according to the “Technical Specification for Concrete Structures of Tall Buildings”. This situation is due to the fact that the regular structure may be classified into the irregular structure because of the torsional rigidity is not enough to meet the requirements of the torsional displacement ratio, so that the division is unreasonable from the principle. And the accidental eccentricity is objective, and it will be affected the torsional effect of the structure, it will not be changed due to the structural form.

The torsional period ratio is referred to the ratio of the first natural vibration period $T_t$ which is based on torsion to the first natural vibration period $T_1$ which is based on translation, that is $T_t$/$T_1$. The control index of torsional period ratio will be given the designer an illusion that the structure is safe if the ratio is within the requirements of the specification. When the ratio is failed to meet the requirements of the standard, the structural designer will simply to reduce the value of $T_t$ or to expand the value of $T_1$, without considering the adverse effects of changing the value of $T_t$ and $T_1$ on other aspects of the structure. It is unreasonable and unsafe to be caused the torsional rigidity of the structure to be too large, the size of the external component of the structure to be too large and the internal component to be too small, or to be made the size of the component smaller in the first translational direction to be met the requirements of the specification [28]. For example, for a frame structure that is met the requirements of torsional period ratio, if other conditions are consistent, shear walls be arranging in the middle of the structure should be more safely in general, but the torsional period ratio will be became difficult to be met the requirements of the specification, which is indicated that the torsional period ratio is unreasonable in certain situations. Therefore, it can be seen that there are some problems in the indicators of torsion control under the standard. It is suggested to seek a new index to control the torsion of the structure.

4.2. Inter-layer torsion angle

The torsional displacement ratio of the floor is stipulated in both “Code for Seismic Design of Buildings” and “Technical Specification for Concrete Structures of Tall Buildings”, and the torsional displacement ratio of the floor can be defined as the following formula [29]:

where ${\mu }^{\mathrm{*}}$ is torsional displacement ratio, ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is maximum Inter-layer displacement, ${\mathrm{\Delta }}_a$ is average Inter-layer displacement, ${\mathrm{\Delta }}_{t\mathrm{m}\mathrm{a}\mathrm{x}}=\theta \mathrm{*}{X}_m$, and ${\mu }^{\mathrm{*}}=1+\mu$, the formula can be obtained:

From the Eq. (2), it can be concluded:

where $\theta$ is Inter-layer torsion angle, $X_m$ is the horizontal distance between the maximum Inter-layer displacement and the average Inter-layer displacement in the calculation direction. Because of $\theta$ is small, $X_m$ can be approximately equal to half of the length of the calculated direction. When $\mu$, ${\mathrm{\Delta }}_a$, $X_m$ are determined, the value of $\theta$ can be controlled to be avoided exceeding the limit value. $\theta$ as a new torsional control index which is derived from the formula of torsional displacement ratio, it is also needs to be studied for its applicability in structural analysis.

5.1. Seismic action of $\mathit{X}$-direction

In the case of accidental eccentricity and $X$-direction seismic action, the inter-layer displacement ratio and the layer displacement ratio of the high-rise cassette multi-tube structure and high-rise frame-core structure are linearly decreasing in general trend. However, a sharp point is appeared in the curve at the 17th floor, which is indicated that the displacement ratio of the 17th floor is became larger, that is caused by the smaller size of each component when the structural model is in the 17th floor. With the decrease of the number of layers, the inter-layer displacement ratio and layer displacement ratio is increased. The inter-layer displacement ratio of the cassette multi-tube structure is ranged from 1.055 to 1.426, and the layer displacement ratio is ranged from 1.125 to 1.430. And the inter-layer displacement ratio of the frame-core structure is ranged from 1.075 to 1.497, and the layer displacement ratio is ranged from 1.180 to 1.493. Both two structures are met the requirements of the code, and the inter-layer displacement ratio curve and layer displacement ratio curve of the frame-core structure are above the curves of the cassette multi-tube structure, which is indicated that the displacement ratio of the frame-core structure is greater than that of the cassette multi-tube structure, and the torsional effect of the cassette multi-tube structure is less than that of the frame-core structure under the $X$-direction seismic action.

Fig. 10Inter-layer displacement ratio

Fig. 11Layer displacement ratio

In the case of accidental eccentricity and $X$-direction seismic action, the maximum inter-layer displacement and the average inter-layer displacement of the high-rise cassette multi-tube structure and high-rise frame-core structure are upward convex curves. The maximum inter-layer displacement of the cassette multi-tube structure is reached the maximum value at the 10th and 17th floors, which are 0.813 cm and 0.812 cm respectively, and the average inter-layer displacement is reached the maximum value at the 18th floor, which is 0.688 cm. The maximum and average inter-layer displacement of the frame-core structure is reached the maximum value at the 17th floor, which are 0.869 cm and 0.780 cm respectively. The maximum inter-layer displacement curve and average inter-layer displacement curve of the frame-core structure are above the curves of the cassette multi-tube structure, which is indicated that the maximum inter-layer displacement and the average inter-layer displacement of the frame-core structure is greater than that of the cassette multi-tube structure. According to the curves of maximum inter-layer displacement and the average inter-layer displacement, it can be seen that the minimum inter-layer displacement of frame-core structure is smaller than that of cassette multi-tube structure below the 7th floor, and that the minimum inter-layer displacement of frame-core structure is greater than that of cassette multi-tube structure above the 7th floor.

Fig. 12Maximum inter-layer displacement

Fig. 13Average inter-layer displacement

Fig. 14Maximum layer displacement

Fig. 15Average layer displacement

In the case of accidental eccentricity and $X$-direction seismic action, the maximum layer displacement and the average layer displacement of the high-rise cassette multi-tube structure and high-rise frame-core structure are linearly increasing in general trend, and as the number of floors is increased, the displacement is increase. The layer displacement of each layer is superimposed by the inter-layer displacement of the layer and the below layers. At any layer, the layer displacement of the frame-core structure is greater than that of the cassette multi-tube structure, which is indicated that the lateral stiffness of the cassette multi-tube structure is better than that of the frame-core structure under the combined action of grid walls and grid hollow plates.

In the case of accidental eccentricity and $X$-direction seismic action, the inter-layer displacement angle of the high-rise cassette multi-tube structure and high-rise frame-core structure are upward convex curves. The inter-layer displacement angle of the cassette multi-tube structure is reached the maximum value at the 10th and 17th floors, which are 2.624×10^-4 and 2.619×10^-4 respectively, the inter-layer displacement angle of the frame-core structure is reached the maximum value at the 17th floor, which is 2.987×10^-4. The inter-layer displacement angle curve of the frame-core structure is only close to the curve of the cassette multi-tube structure below the 7th floor, while the distance between the inter-layer displacement angle curve of the two structures is increasing continuously above the 8th floor. which is indicated that the difference between the inter-layer displacement angle of the cassette multi-tube structure and the frame-core structure is very small at low floors, it is because of the frame-core structure is ensured strong stiffness by using large-sized components at the lower floors, which is resisted the earthquake action, so the inter-layer displacement angle of the frame-core structure is close to that of the cassette multi-tube structure. However, with the increase of the number of layers, the size of the frame-core structural components is constantly decreasing, and the stiffness is also decreasing. But the stiffness reduction of the cassette multi-tube structure is smaller than that of the frame-core structure due to the existence of grid walls and grid hollow plates, so the gap between the inter-layer displacement angle is constantly increasing.

Fig. 16Inter-layer displacement angle

Fig. 17Inter-layer torsion angle

The inter-layer torsional angle is a more intuitive control index for the torsional performance of the structure. In the case of accidental eccentricity and $X$-direction seismic action, the inter-layer torsion angle of the high-rise cassette multi-tube structure and high-rise frame-core structure are upward convex curves. The inter-layer torsion angle of the frame-core structure and the cassette multi-tube structure are reached the maximum value at the 7th floor, which are 452×10^-4and 1.701×10^-4 respectively. The torsional displacement angle of the frame-core structure at each layer is larger than that of the cassette multi-tube structure. It can be seen that the $X$-direction torsional rigidity of the cassette multi-tube structure is greater than that of the frame-core structure, and the $X$-direction torsional performance of the cassette multi-tube structure is better than that of the frame-core structure.

5.2. Seismic action of $\mathit{Y}$-direction

In the case of accidental eccentricity and $Y$-direction seismic action, the inter-layer displacement ratio and the layer displacement ratio of the high-rise cassette multi-tube structure and high-rise frame-core structure are linearly decreasing in general trend. The minimum value of inter-layer displacement ratio and the minimum value of layer displacement ratio in the $Y$-direction are smaller than those in the $X$-direction. The inter-layer displacement ratio of the frame-core structure is ranged from 1.051 to 1.483, and the layer displacement ratio is ranged from 1.170 to 1.493. And the inter-layer displacement ratio of the cassette multi-tube structure is ranged from 1.016 to 1.435, and the layer displacement ratio is ranged from 1.106 to 1.432. In the displacement ratio curves of the two directions, the difference of layer displacement ratio between the cassette multi-tube structure and the frame-core structure in the $Y$-direction is larger than that between the two structures in the $X$-direction. It is because of the grid walls of the cassette multi-tube structure are more distributed in the $Y$-direction, while the columns of the frame-core structure are not distributed enough in the $Y$-direction. So lateral stiffness of the two structures in the $Y$-direction is more different than that in the $X$-direction.

Fig. 18Inter-layer displacement ratio

Fig. 19Layer displacement ratio

Fig. 20Maximum inter-layer displacement

Fig. 21Average inter-layer displacement

In the case of accidental eccentricity and $Y$-direction seismic action, the maximum inter-layer displacement and the average inter-layer displacement of the high-rise cassette multi-tube structure and high-rise frame-core structure are upward convex curves. The slope of the inter-layer displacement curve in the $Y$-direction and the magnitude of increase of the inter-layer displacement value in the $Y$-direction are similar to that of the $X$-direction in the 0-10 layers. However maximum inter-layer displacement curve $Y$-direction is tended to be gentle in the 10-16 layers, and the average inter-layer displacement curve is increased slightly in the 10-16 layers. The curve in the $Y$-direction is also reached maximum value at the 17th floor, but after the 17th floor, the magnitude of decrease of the inter-layer displacement value in the $Y$-direction is much smaller than that in the $X$-direction. Therefore, it can be seen that the inter-layer displacement in the $Y$-direction is larger than that in the $X$-direction. The maximum inter-layer displacement and the average inter-layer displacement of the cassette multi-tube structure are 0.986 cm and 0.840 cm respectively, and the maximum inter-layer displacement and the average inter-layer displacement of the frame-core structure are 1.104 cm and 0.897 cm respectively. The maximum inter-layer displacement curve and average inter-layer displacement curve of the frame-core structure are above the curves of the cassette multi-tube structure in the $Y$-direction, which is indicated that the lateral stiffness of the cassette multi-tube structure is better than that of the frame-core structure in the $Y$-direction.

In the case of accidental eccentricity and $Y$-direction seismic action, the maximum layer displacement and the average layer displacement of the high-rise cassette multi-tube structure and high-rise frame-core structure are linearly increasing in general trend. But the maximum value of the layer displacement curve in the $Y$-direction is greater than the maximum value the layer displacement curve in the $X$-direction. The maximum values of maximum layer displacement and average layer displacement of the cassette multi-tube structure are 21.340 cm and 17.809 cm respectively, and the maximum values of maximum layer displacement and average layer displacement of the frame-core structure are 23.424 cm and 20.099 cm respectively. The maximum value of two structures in the $Y$-direction are greater than that in the $X$-direction. Which is indicated that the lateral stiffness of two structures in the $X$-direction is better than that in the $Y$-direction, and the lateral stiffness of the cassette multi-tube structure is also better than that of the frame-core structure in the $Y$-direction. The inter-layer displacement of the two structures is uniformly increased, it means that the structure is uniformly deformed, and the stiffness of the structure does not mutate in the vertical direction, that is met the requirements of the code.

Fig. 22Maximum layer displacement

Fig. 23Average layer displacement

Fig. 24Inter-layer displacement angle

Fig. 25Inter-layer torsion angle

In the case of accidental eccentricity and $Y$-direction seismic action, the inter-layer displacement angle of the high-rise cassette multi-tube structure and high-rise frame-core structure are upward convex curves. The inter-layer displacement angle of the cassette multi-tube structure is reached the maximum value at the 17th floor, which is 3.182×10^-4, and the inter-layer displacement angle of the frame-core structure is also reached the maximum value at the 17th floor, which is 3.679×10^-4. The inter-layer displacement angle curve of the frame-core structure are above the curve of the cassette multi-tube structure. It can be seen that the inter-layer displacement angle of the two structures in the $Y$-direction is greater than the inter-layer displacement angle in the $X$-direction, it is because of the length of the two structures in the $Y$-direction is greater than the length in the $X$-direction, and part of the seismic force in the $X$-direction is acted on the shear walls that is reduced the seismic force which is acted on the frame in the $X$-direction, and the overall inter-layer displacement angle in the $X$-direction is smaller than the inter-layer displacement angle in the $Y$-direction.

In the case of accidental eccentricity and $Y$-direction seismic action, the inter-layer torsion angle of the high-rise cassette multi-tube structure and high-rise frame-core structure are upward convex curves. The inter-layer torsion angles of the cassette multi-tube structure and the frame-core structure are reached the maximum value at the 7th floor, which are 1.506×10^-4 and 1.898×10^-4 respectively. The torsional displacement angle of the frame-core structure at each layer is also larger than that of the cassette multi-tube structure. It can be seen that the $Y$-direction torsional rigidity of the cassette multi-tube structure is greater than that of the frame-core structure, and the $Y$-direction torsional performance of the cassette multi-tube structure is better than that of the frame-core structure.

By comparing the $X$-direction and $Y$-direction inter-layer torsion angles, it can be seen that The torsional angle of the cassette multi-tube structure in the $Y$ direction is larger than that in the $X$ direction, and the difference between the two directions is small, and the maximum difference between the inter-layer torsion angle is only 3.72 %, which is indicated that the torsional properties of the cassette multi-tube structure are similar in the two directions, and the torsional stiffness distribution is relatively average. The torsional angle of the frame-core structure in the $Y$ direction is larger than that in the $X$ direction, and the maximum difference is 11.58 %, which is indicated that the distribution of torsional stiffness in the $X$ direction is larger than that in the $Y$ direction. Comparing the data in Table 5, the values corresponding to the frame-core structure are greater than those corresponding to the cassette multi-tube structure. It can be seen that the torsional rigidity of the cassette multi-tube structure is greater than that of the frame-core structure, and the torsional performance of the cassette multi-tube structure is better than that of the frame-core structure.