# Parametric Analysis of the Shear Lag Effect in Tube Structural Systems of Tall Buildings
## 2 Shear Lag Effect in Tube Structural Systems  
As mentioned previously, this type of structural system is derived from the rigid frame system, which implies that the exterior columns, which form the tube, are closely spaced and connected with high beams on every floor. So, it can be concluded that the tube system consists of multiple very rigid and connected frame panels which, while bending, act as webs and flanges of a tube cross section. During a horizontal load, due to their flexibility, beams in the web and flange frames deform in bending. This results in the increase of axial forces in the columns at the end of the frame panels and the decrease of axial forces in the columns towards the middle of the panel. Consequently, deformations occur in thefloor slabs and the interior secondary elements which leads to the Euler-Bernoulli beam theory bending of cross sections being no longer valid. Hence, the distribution of stress becomes nonlinear and very complex [5]. The nonlinearity is found in the flange frames (frame panels perpendicular to the direction of the horizontal load) as well as in the web frames (frame panels parallel to the direction of the horizontal load). This phenomenon can be seen in Figure 1 and is known as the positive shear lag [6]. On the other hand, the negative shear lag effect is characterized with increased axial forces in the columns in the middle of the panels (Figure 2). To explain this phenomenon, a simple box section 5000 mm high cantilever model is used, loaded at mid-height by point load (Figure 3a) and uniform continuous load (Figure 3b). This model is introduced as an illustrative example, and as such it comprises a rectangular thin-walled steel section consisting of 1000 mm wide and 100 mm thick panels. These panels are modelled each with 10 shell elements along the panel width allowing an in-plane shear deformation which is responsible for shear lag effect. The stresses of the flange opposite to the load are shown in the graphic, ranging from zero (dark blue) to maximum compression (red). Iso-areas of the same color show matching stress levels on the panel surface, thus allowing sections with positive or negative shear lag to be recognized. Since the value of the bending moment and the shear force at the free end of the cantilever equal zero, the flange stays undeformed in this position. On the fixed end of the cantilever, positive shear lag occurs due to the longitudinal shear in the flange. Due to the compatibility of deformations for all adjacent sections between the beginning and the end of the cantilever, stresses of the opposite distribution occur along the height as we move away from the fixed edge (negative shear lag). On the quarter of the cantilever length, closer to the fixed end, the axial stresses are almost equal along the whole flange width for the point load, while for continuous load some negative shear lag occurs at that section. So, in conclusion, the negative shear lag effect is a result of the compatibility of displacements (the cross section at the end of the cantilever must be straight and in-plane) [7]. Figure 1: Shear lag effect in a tube structural system (positive shear lag effect). Figure 3: _Cont_. Figure 2: Shear lag effect in a tube structural system (negative shear lag effect). The first researches of the shear lag and the effect it has on different structural systems date back to the 1980s [8]. In the very beginning this phenomenon was examined in steel and concrete girders, mostly bridges. Researches involving steel box girders and bridges were the most prominent ones [8; 9; 10; 11]. In [9; 10] theoretical foundations for the shear lag effect are given with a special emphasis on steel box girders. At about the same time [8; 10; 11] basic mathematical models and formulas for the calculation of the shear lag effect in box girders were given.