**1. Introduction**

The framed tube idea is an effective framing system for high-rise buildings. This type of structural system is mainly comprises closely spaced circumferential columns, which are connected by deep spandrel beams. The whole system works as a giant vertical cantilever, and its high efficiency is due to the large distance between windward and leeward columns. Among the most important specifications of tubular systems is their high economic efficiency. A case in point is that the material consumed in this kind of system is reduced by half in comparison with other systems [1]. In a rigid frame the "strong" bending direction of columns is aligned perpendicular to the face, while this factor is typically aligned along the face of the building in a framed tube system. In a framed tube system, the tube form resists overturning produced by lateral load—a leading cause of compression and tension in columns. Bending in columns and beams or rotation of the beam-column joint in the web section resists the shear force produced by lateral load. Gravity loads are resisted partly by exterior frames and partly by interior columns [2].

In an ideal tubular structure, circumferential columns and beams are assumed to be completely rigid, so both web and flange panels act separately and bend against lateral loads like a true cantilever. While the above system has a tubular form, it also has a more complicated behavior than a solid tube. To be more specific, the components in a framed tube cannot be completely rigid due to technical and economic constraints [3]. This will be a leading cause of nonuniform distribution of loads in columns.

Consequently, in a framed-tube structure under lateral load, the stress distribution in the flange wall panels is nonuniform and is nonlinear in the web wall panels. This anomaly which reduces the efficiency of the structures is referred to as "shear lag" [4]. To obtain a better understanding of this phenomenon, a factor called the shear lag factor is defined. This factor is a ratio of the corner column axial force to the middle column axial force. When the stresses in the corner columns of the flange frame panels exceed those in the middle columns, the shear lag is positive. Nevertheless, in some cases, it is vice-versa where the stress in the middle columns exceeds those in the corner columns, and this is referred to as the negative shear lag.

This paper studied the effect of lateral load type on the shear lag phenomenon in framed-tube reinforced concrete tall buildings with different plan geometries.

## **2. Review of Literature**

With regard to the great importance of framed tube systems in the construction of high-rise buildings, it is no wonder that multifarious research has been carried out on this structural system and its shortcomings in order to make framed tube systems more effective. However, previous studies did not pay enough attention to the subjected load type and its relation to the shear lag factor and the shape of the structures, which play an important role in the amount of this factor.

In 1969, Fazlur Khan proposed a chart named "structural systems for height" which has classified the different types of tubular systems with regard to their efficiency for high-rise buildings of different heights [5]. In the same year, Chang and Zheng tried to find out more about negative shear lag and its influential factors on a cantilever box girder. In this research, they found that negative shear lag will change with the different boundary conditions of displacement and external force applied to the girder [6]. In 1988, Shiraishi et al. studied the aerodynamic stability effects on rectangular cylinders by altering their shapes subjected to the lateral loads such as wind and water flow. They cut some squares with different sizes in each edge in a rectangular cross section and observed that it has a controlling effect on the separated shear layer generating from the leading edge. These sections with various sizes of corner-cuts had totally different behaviors against wind force and water flume [7]. In 1990, Hayashida and Iwasa investigated the effects of the geometry of structures on aerodynamic forces and displacement response for tall buildings. They tested four different plan shapes, with and without corner-cuts in a wind tunnel and identified the aerodynamic damping effects produced by changing some parts of the basic cross section and also the aerodynamic character of basic shapes [8]. In 1991, Connor and Pouangare presented a simple model for the design of framed tube structures. They modeled the structures as a series of stringers which resisted axial forces without bending rigidity and shear panels which resisted shear forces without bending or axial rigidity. They proposed a model that gives accurate results for the preliminary design and analysis of tubular structures with different geometry and material properties [9]. Kwan, in 1994, proposed a hand calculation method for approximate analysis in framed tube structures by considering the shear lag factor. This method could be useful for preliminary design and quick evaluation and could provide a better perception of the effect of multifarious parameters on the structure's behavior [10]. In 2000, Han et al. investigated the shear lag factor in the web panels of shear-core walls [11]. Lee et al., in 2002, looked into the behavior of the shear lag of framed tube structures with and without internal tube(s) for the behavioral characteristics of the structures and also the relation between their performance and various structural parameters. They also proposed a simple numerical method for the prediction of the shear lag effect in framed tube structures. It has been found that the stiffness factor has an effective role in producing shear lag in tubular structures [12]. Haji-Kazemi and Company proposed a new method to analyze

shear lag in framed tube structures using an analogy between the shear lag behavior of a cantilever box which represents a uniform framed tube building. This method is able to accurately analyze positive as well as negative shear lag effects in tubular structures accurately [13]. Furthermore, Moghadasi and Keramati, in 2009, studied the effects of internal tubes in shear lag reduction. They reduced the lateral displacement and shear lag amount in high-rise buildings by adding internal tubes to the framed tube structures [14].

In 2012, Shin et al. investigated different parameters such as depth and width of beams and columns on the behavior of shear lag in a frame-wall tube building. The results showed that the effect of column depth on the shear lag behavior of framed tube was more outstanding than other parameters [15]. In 2014, Mazinani et al. compared the shear lag amount of pure tube structural systems with braced tube systems and different types of X-diagonal bracing. It was observed that these braces started from corner-to-corner, increased the stiffness of the structure and consequently reduced the story drift and shear lag factor in the tubular system [16]. In addition, Nagvekar and Hampali studied the shear lag phenomenon in both the web and wing panel of a hollow structure and measured it in various heights of a structure [17]. The plan geometry, building's body form, the ratio of height to width and three-dimensional stiffness for the transfer of wind and seismic loads are the most important structural system properties affect the behavior of tall buildings, as reported by Szolomicki and Golasz-Szolomicka [18]. Alaghmandan et al. in 2016 inquired about the architectural strategies on wind effects in tall buildings. These tactics included altering the geometry of the whole building scale such as tapering and setbacks, and attenuated the wind effects in some models by architectural strategies [19]. In 2019, Shi and Zhang proposed a simplified method for calculation of shear lag in diagrid framed tube structures. In their study, the diagrid tube structure is assumed to be equivalent to continuous orthogonal elastic membrane. They tried to solve two key problems of finding the optimized angle of the diagonal column and the shear lag assessment in the preliminary design of the above structures [20].

In terms of microstructural view of concrete subjected to dynamic loading at high strain rates, Hentz et al. used a 3D discrete element method and verified it [21]. Furthermore, the evaluation of concrete cracks occurring in complex states of stress was studied by Golewski and Sadowski [22]. In their study, crack development at shear was investigated through experimental tests using two types of aggregates.

As seen from the review above of previous studies, the possible relation between shear lag and the type of lateral load subjected to these systems is not yet considered.

## **3. Structural Models and Analyses Specifications**

## *3.1. Characteristics and Final Dimensions*

In this paper, 12 reinforced concrete framed tube buildings with different heights and different shapes are modeled and analyzed. From a shape point of view, they were divided into three various groups of structures with (a) rectangular, (b) triangular and (c) hexagonal plan shapes (See Figure 1). Each structural plan shape consists of four different heights: 20-story, 40-story, 60-story and 80-story buildings. Figure 2 shows example pictures of each type of buildings illustrated in Figure 1. Table 1 shows the terminology of the models used in this paper. Figure 3 displays the 40-story models in ETABS software version 18.1.1 as examples. This software is chosen because it is developed specifically for the analysis and design of building structures. Furthermore, it has a high ability for static and dynamic analyses regardless of the number of nodes and stories.

**Figure 1.** Three different plan shapes: (**a**) rectangular, (**b**) triangular and (**c**) hexagonal (all dimensions are in meters).

**Figure 2.** Three example pictures of: (**a**) rectangular plan building (WTC buildings, New York, USA) [23], (**b**) triangular plan building (Flatiron Building, New York, USA) [24] and (**c**) hexagonal plan (Hoxton Press towers, London, UK) [25].


**Table 1.** Terminology of models.

The spaces between columns are normal (10 m in rectangular and triangular plans and 5 m in hexagonal plans) on the first floor, due to the existence of entrances, and the circumference of the second floor is also isolated by deep beams. As a result, the tubular behavior of framed tube structures begins from the third floor. To obtain an accurate analogy, the equivalent length of 30 m for each side of the rectangular and triangular plans is assumed which is equal to the hexagonal plan diameters. Specifically, each side of the hexagonal plan is obtained as 15 m. The distances between columns in the first floor are 10 m for rectangular and triangular plans and five meters for hexagonal plans. The spaces between peripheral columns, which create a tubular form in the structures, are 2.5 m in all stories. The dimensions of all circumferential and gravitational beams and columns are lessened in each 10 stories from the bottom to top of the structures.

**Figure 3.** Three-dimensional view of 40-story models in ETABS Software: (**a**) rectangular, (**b**) triangular and (**c**) hexagonal.

#### *3.2. Loading, Structural Analyses and Final Design Specifications*

The national Iranian building code was used in order to calculate the loading of structures [26]. The total dead load for each model was 308 kg/m<sup>2</sup> with regard to this point that ETABS software automatically calculates result loads from beams, columns and slabs weights. The dead load consisted of two parts—63 kg/m2 for floor weight and 245 kg/m<sup>2</sup> for partition weight. Moreover, 250 kg/m2 was assumed as a live load.

The structures were analyzed by both static and pseudo-spectral dynamic methods, and to achieve a real and accurate structural analysis and design, earthquake and wind load were also applied separately on every 12 framed tube structures. The earthquake load was applied based on the Iranian seismic code [27], and the wind load specifications were according to the ASCE 7-16 code [28].

#### 3.2.1. Earthquake Load

In this study, all the models were analyzed against the earthquake load in two manners—static equivalent and pseudo-spectral dynamic analyses—and the Iranian seismic code was used for both methods.

#### Static Equivalent Earthquake Load

Like every static equivalent analysis, the equivalent static base shear in this research was determined in accordance with the following equations [27]:

$$\mathbf{V} = \mathbf{C}\mathbf{W} \tag{1}$$

where V is base shear, C is the seismic response coefficient and W is the effective weight of building (all dead load + a percentage of live load)

And:

$$\mathcal{C} = \frac{\text{ABI}}{\text{R}}\tag{2}$$

where A is the design basis acceleration over the bedrock dependent on the seismic zone of the building's location; B is the reflection coefficient of the building related to seismic zone, soil type and vibration period of the structure (T) (See Figure 4); I is the importance factor; and R is the response modification factor which determines the nonlinear performance of the building during earthquakes and affected directly from the type of the structural system. It should be noticed that the structural system of the modeled structures is reinforced concrete special moment frames. A, I and R were considered as 0.3, 1.0 and 10, respectively, due to the Iranian seismic code. The minimum value of V was *Vmin* = 0.12AIW [27].

**Figure 4.** Response spectra intended for dynamic analysis (Iranian seismic code, 2015).

#### Dynamic Earthquake Load

In the dynamic analysis procedure, the lateral seismic load is determined from the dynamic response of a building subjected to an appropriate ground motion, and the pseudo-spectral dynamic analysis method was used in this paper according to the Iranian seismic code. The dynamic analysis spectrum was also obtained from the same code reflecting the effects of ground motion for design earthquake level. The spectrum shape is based on the type of soil, seismic zone and period of the structures. The information about the spectrum used in the dynamic analysis is shown in Figure 4 [27].

#### Wind Load

Wind load was applied to the aforementioned structures to obtain more accurate results and a comparison of framed tube behavior against different load types. Table 2 shows the specifications of the wind load obtained from the ASCE7-16 code [28].


**Table 2.** Specification of the wind load.
