*4.1. The E*ff*ect of Lateral Load Type on Shear Lag of Framed Tube Structures*

There was ample evidence from analysis results, which indicated that wind load could distribute forces among the columns more unequally. In most cases, shear lag phenomenon observed in structures subjected to the wind load were more severe than those for structures subjected to the dynamic or static earthquake load. Wind load in comparison with dynamic earthquake load caused a greater positive shear lag factor in almost all models. The positive Shear Lag factor obtained from the wind load for the 20R model was almost 1.5 times greater than the same factor for the same model against the dynamic earthquake load. Negative shear lag intensity of the 20R model against wind load was 1.22 times more than the same model subjected to the dynamic earthquake load. Although positive shear lag factors calculated from structures subjected to the wind load and dynamic earthquake load for 20T and 20H models were almost equal, negative shear lag factors for these structures subjected to the wind load were 39.08% and 21.18% more than the same models subjected to the dynamic earthquake load. The intensity of positive and negative shear lag factors observed from the 40R model subjected to the wind load was higher than the factors calculated from this model against dynamic earthquake load by 19.59% and 33.85%, respectively. The above percentages for 40T model were 3.3% and 25.71% and, for the 40H model, were 0.0% and 5.59%, respectively. This trend was observed for 60- and 80-story structures as well, and shear lag factors in structures subjected to wind load had greater amounts. Table 3 shows the percentage differences between shear lag obtained from wind load and dynamic

earthquake load. The nature of wind load and its distribution and application on the surface of the structure in comparison with the seismic load applied on the center mas of the rigid diagram floors by ETABS could be the reason for the above differences illustrated in Table 3.


**Table 3.** The differences between shear lag obtained from the wind load and dynamic earthquake load.

Moreover, although the dynamic analysis shows a more accurate result in comparison with equal static analysis, the shear lag factors for structures subjected to the static earthquake load were also less than those factors for structures subjected to the wind load in more than 80% of the models.

#### *4.2. The E*ff*ect of Geometry of Plan on Shear Lag of Framed Tube Structures*

Analysis results indicated that the amount of shear lag in framed tube structures could be highly dependent on the plan geometry of the structure. Shear lag diagrams for three types of 80-story framed tube structures subjected to earthquake loads are shown in Figures 5 and 6. Although these diagrams are more similar to a straight line in a particular story, the framed tube structure has less shear lag in that story. Diagrams with a minimum in the middle represent the positive shear lag, and those with a maximum in the middle illustrate negative shear lag.

**Figure 5.** Shear lag diagram of 20th story of 80 story models subjected to the dynamic earthquake load.

**Figure 6.** Shear lag diagram of 60th story of 80 story models subjected to the dynamic earthquake load.

According to these diagrams, it is observed that hexagon shaped plan structures have a better performance in terms of shear lag in comparison with the other two shapes. This structural behavior is in line with a previous study conducted by Awida, who stated that the octagon shape as a plan geometry can be the best in the structural response against wind load in comparison with other possible plan geometries [30]. Triangular-shaped structures act much better against lateral loads than rectangular ones which have the most inequality of load distribution in their flange columns. In addition, shear lag factors for every 10 stories in all structures were obtained, and the average of positive and negative shear lag factors for each structure was calculated as shown in Table 4. This table shows a similar trend in the case of shear lag in all structures. For instance, in terms of positive shear lag of a dynamic earthquake load, the 80H model acted 8.2% better than the 80R model. Moreover, the 80T model had 3.6% less shear lag in comparison with 80R. Furthermore, the negative shear lag factor of the 80H and 80T models—23% and 10.2%, respectively—performed better than the 80R model.


**Table 4.** The average of positive and negative shear lag factor in all models analyzed with three different load types.

To obtain a better understanding of shear lag fluctuation in the models, shear lag factors for odd stories were investigated, and Figures 7–9 shows the shear lag factor diagram for 80-story structures for three types of loads. In regard to these Figures, the shear lag factor diagrams for hexagonal-shaped plan structures in most of the stories are close to one, which means shear lag is at a minimum in these types. The shear lag phenomenon that showed up in rectangular-shaped plan structures was the maximum, especially in the first and last 10 stories. Furthermore, triangular-shaped plan structures exhibited a better behavior, in general, in terms of shear lag in comparison with rectangular ones mostly in the top half of the buildings, but still, shear lag in these structures was far higher than framed tube structures with hexagonal plan shape.

**Figure 7.** Shear lag factor diagram for 80 story structures subjected to the dynamic earthquake load.

**Figure 8.** Shear lag factor diagram for 80 story structures subjected to the static earthquake load.

**Figure 9.** Shear lag factor diagram for 80 story structures subjected to the wind load.

Shear lag factors for static earthquake load are also available in Table 4. A lower positive shear lag up to 13% in the 80H model in comparison with 80R could be observed, and the 80T model had 5.21% less positive shear lag in comparison with 80R. This amount of reduction in negative shear lag was 21.3% for 80H model and 5.4% for 80T model as compared to the 80R model. This tendency could be observed in all the other models.

In addition to the earthquake load, lateral wind load was also applied to the structures in the static analysis method. The behavior in shear lag reduction in this part of analysis is also like previous sections (See Table 4). For positive shear lag, 80H and 80T specimens—13% and 6%, respectively—behaved better than the 80R model. Likewise, for negative shear lag, the 80H model responded at 24.7%, and the 80T model responded almost 8.2% better than 80R structures.

It is evident that from Figure 7, the shear lag effect is more intense in the top half of the structures, and this trend was observed in all other models. The shear lag switch-level from positive to negative in 20-story structures is between the 10th to 15th floors, for 40- story structures is between the 25th to 30th floors, for 60-story structures is between the 35th to 40th floors, and for 80-story structures is between the 45th to 50th floors.

#### *4.3. The E*ff*ect of Structural Height on Shear Lag of Framed Tube Structures*

Structural height in tubular buildings has a direct effect on shear lag phenomenon. As the height of a framed tube structure increases, the positive shear lag in each story will decrease as well.

Table 5 indicates that 20 story structures had the highest positive shear lag factor in all types of plan geometry. This factor decreased as the number of stories increased almost in all the subsequent models. For example, the positive shear lag factor for 20R model against the dynamic earthquake load was 1.99, and this number was 1.19, 1.09 and 1.09 for 40R, 60R and 80R models, respectively. Furthermore, the average of positive shear lag factors for all models subjected to various lateral loads, without considering their plan geometry, is shown in Table 5. It is observed that positive shear lag factors in taller structures are fewer than this factor in shorter models and has nothing to do with the factor of geometry. In addition, the average of positive shear lag factors without considering any other factor such as load type and plan geometry was calculated. The positive shear lag factor in 20-story structures was 31.91% fewer than in 40-story structures, 37.02% fewer than in 60-story structures and 37.92% fewer than in 80-story structures. It is concluded that the positive shear lag phenomenon has a negative correlation with the height in framed tube structures subjected to any type of lateral load and with any plan geometry.


**Table 5.** Average of positive shear lag factor among three types of plan geometry for each height.

#### *4.4. Comparison and Verification of the Results*

In this study, a 40-story reinforced concrete framed tube building was chosen to compare the Matrix method [31] and Haji-Kazemi and Company [13] analyses results. Beams and columns dimensions in this example are 0.8 × 0.8 m. Each story is 3 m in height, and center to center spacing between columns is 2.5 m. The modulus of elasticity and shear modulus of concrete are 20 and 8.0 GPa, respectively. The external load is 120 kN⁄m and uniformly distributed along the height of the structure. Figures 10 and 11 show the axial forces in columns of the web and flange of the structure at the base and 10th floor of the framed tube, respectively.

**Figure 10.** Axial force distribution in the flange and web columns at the base of the building.

**Figure 11.** Axial force distribution in the flange and web columns at the 10th floor of the building.

Due to the symmetry of the structure, only half of the web and flange was considered. These diagrams illustrate that the axial forces in corner columns obtained from the proposed model have 9% and 10% difference in comparison with Haji-Kazemi and Company [13] and the Matrix method [31], respectively. In all models and analyses, the shear lag phenomenon was positive at lower heights and negative in the upper stories. Specifically, this anomaly was at a minimum in the middle of each structure.

This factor will decrease gradually in upper stories so that shear lag factor in the middle of the structures decreases to 1 which is considered no shear lag. This trend had no change, and the shear lag factor reduced till the last story. In sharp contrast with lower floors, minimum and negative shear lag factors were obvious in the upper stories. The above results are consistent with the previous studies [6].

#### **5. Conclusions**

This study investigated the effect of lateral load type, plan geometry and height on shear lag behavior of framed tube structures. For this goal, 12 models in four different heights and three different plan geometry against three different load types were considered. From the structural analyses performed, it could be concluded that:

(1) Type of the lateral load could affect the distribution of forces in peripheral columns in tubular structures. Wind load caused a greater amount of positive shear lag in comparison with the dynamic earthquake load and the static earthquake load by 9% and 7.5%, respectively. These numbers for negative shear lag were 14% and 1.5%, respectively. In regard to the importance of wind load in the design of high-rise structures and the severity of shear lag in framed tube structures designed based on it, the above results should be seriously considered by structural designers.

(2) Shear lag phenomenon could be affected significantly by the geometry plan in framed tube structures. Hexagon shaped plan structures had a reasonable behavior against lateral loads. Specifically, the average of positive and negative shear lag factors in the three types of analyses were 28.76%, and 25% less in hexagon shaped plan structures, respectively, in comparison with the control model (rectangular-shaped plan). This superiority may lead the structures towards being more laterally load resistant, of lighter weight and more economical due to its equal load distribution in the whole frame.

(3) Rectangular-shaped plan structures had the most inequality of axial force distribution in the flange frame columns. In the above mentioned structures, the average of positive shear lag rose to near two and even more and the average of negative shear lag fell down to below 0.5 in some cases. These amounts of shear lag are not evident in any other shaped plans.

(4) The structures with triangular-shaped plan had almost the same amount of shear lag with rectangular-shaped plan structures in shorter buildings, but the triangular plan had a better behavior in terms of shear lag than the rectangular plan in taller structures. The triangular performed almost 5% better, in the case of positive, and 8.4% in the case of negative shear lag on average in comparison with the rectangular-shaped plan.

(5) Shear lag of framed tube structures is highly affected by the height of the structure. Column axial forces were distributed more unequally in shorter structures, and taller buildings had smaller amount of shear lag factors. It can be concluded that in taller buildings, the structural behavior of the box-shaped cantilever beam that represents the whole building is more similar to Euler–Bernoulli beam than the shorter building and, as it is known from theory of structures, the effect of shear lag in Euler–Bernoulli cantilever beams (taller buildings) is lower than the shorter ones.

**Author Contributions:** Formal analysis, S.T.; Funding acquisition, S.S.R.K. and M.P.; Investigation, S.T.; Project administration, M.M., S.T., S.S.R.K. and M.P.; Resources, M.M. and S.S.R.K.; Software, S.T.; Supervision, M.M.; Writing—review & editing, M.M. and S.T.; and All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the Ministry of Education, Youth, and Sports of the Czech Republic and the European Union (European Structural and Investment Funds Operational Program Research, Development, and Education) in the framework of the project "Modular platform for autonomous chassis of specialized electric vehicles for freight and equipment transportation", Reg. No. CZ.02.1.01/0.0/0.0/16\_025/0007293, as well as financial support from internal grants in the Institute for Nanomaterials, Advanced Technologies and Innovations (CXI), Technical University of Liberec (TUL).

**Acknowledgments:** The authors would like to acknowledge the financial support by Ministry of Education, Youth and Sports of the Czech Republic and the European Union (European Structural and Investment Funds—Operational Program Research, Development and Education), Reg. No. CZ.02.1.01/0.0/0.0/16\_025/0007293, as well as the financial support from internal grants in the Institute for Nanomaterials, Advanced Technologies and Innovations (CXI), Technical University of Liberec (TUL).

**Conflicts of Interest:** The authors declare no conflict of interest.
