Keywords: continuum, shear lag, tube structure, potential energy, tube-tube interaction, additional bending.

Framed tube structure with multiple internal tubes can be widely used because of their high stiffness in resisting lateral loads and the availability of the internal tubes in supporting vertical loads. The structure, which include smaller size of internal tubes, reduce the effect of shear lag in internal tubes and induce more effective participation of internal columns in resisting lateral forces. The internal tubes are often designed to provide lateral stiffness to the structure. As a result, the framed tube structure with multiple internal tubes can be assessed as a system able to maximise the structural efficiency of the system. However, the existing methods of modelling a framed tube structure as equivalent orthotropic plate panels with shear and bending rigidities is not sufficient to capture the true behaviour of the framed tube structure with multiple internal tubes.

A typical framed-tube structure under lateral loading behaves differently from that predicted by the primary bending theory, in that the stress distribution in the flange wall panels is not uniform, and that in the web wall panels is nonlinear. This (nonlinear) phenomenon is referred to as shear lag. Positive shear lag refers to the case where the stresses in the corner columns of the flange frame panels exceed those in the centre columns. This leads to the warping of the floor slabs which, in turn, causes the deformation of the interior partitions and other secondary components. In the case of negative shear lag, where the stresses in the centre columns exceed those in the corner columns, local buckling on the compression side and cracking on the tension side of the flange frame may occur. In addition, the tube-tube interactive stresses, referred to as the additional bending stresses, would further complicate the shear lag prediction.

The negative shear lags along with the existence of the tube-tube interaction in the tubes makes it difficult to estimate the structural performance and the accurate analysis of such system. Furthermore, existing models [1,2] for approximate analysis not only ignore the contribution to lateral stiffness provided by the internal tubes but also neglect the negative shear lag effects and the tube-tube interaction in the tubes.

In the proposed method, Reissner's function [3] is modified to account for the independent distribution of the vertical displacement in the flange frame panels of each tube, thereby taking the net shear lag into consideration. By simplifying the assumptions in relation to the patterns of the vertical displacement distributions in the tubes, the complex three-dimensional structural analysis is reduced to the mere solution of a single second-order linear differential equation.

The numerical analysis for stress distribution is made based on the principle of minimum potential energy in conjunction with the variational approach. The total potential energy of a framed tube structure with multiple internal tubes is obtained by summing up the total strain energy of external and internal tubes and the potential energy of the applied loading as applied to the external and internal tubes. This total potential energy must be minimised which can be achieved by using the governing differential equation and the required set of boundary conditions based on the variational approach. The governing differential equations describe the overall behaviour of the framed tube structures with multiple internal tubes subjected to a lateral load. The proposed method is applicable to the analysis of framed-tube structures with any number of internal tubes. This is an advancement over the existing methods which are restricted to the single tube structures.

A series of framed tube structure with multiple internal tubes subjected to lateral loading is analysed to verify the accuracy and reliability of the proposed method. Three 40-storey reinforced concrete framed structures (of tube, tube-in-tube and 2 tubes-in-tube construction) are analysed using the proposed method and a 3-D frame analysis program [4]. The results are compared to estimate the stress distribution in columns of each tube.

In view of its simplicity, efficiency and accuracy, the proposed method is considered to be a suitable design tool for framed-tube structures, particularly at the preliminary stages where numerous analysis iterations need to be carried out.

1

Coull, A., Bose, B., "Simplified analysis of framed-tube structures", Journal of Structural Engineering, ASCE, 101(11), 2223-2240, 1975.

2

Kwan, A.K.H., "Simple method for approximate analysis of framed tube structures", Journal of Structural Engineering, ASCE, 120(4), 1221-1239. 1994. doi:10.1061/(ASCE)0733-9445(1994)120:4(1221)

3

Reisnner, E., "Analysis of shear lag in Box Beam by the Principle of Minimum Potential Energy", Quarterly of Applied Mathematics, 4(3), 268-278. 1945.

4

A. Habibullah, "ETABS Version 6.2 User's Manual", Computers and Structures Inc., California, USA, 1997.

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