Static analysis of tall buildings based on Timoshenko beam theory
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Abstract
In this paper, the continuum model, which is known as Kwan model, has been presented for the analysis of tall buildings that have been as an appropriate approximation of the overall behavior of the structure. Tall building was modeled as a cantilever beam and analyzed with the assumption of flexural behavior based on Euler–Bernoulli Beam Theory, then the displacement of floors was calculated. o consider the shear lag effects in the overall displacement of the structure, Timoshenko’s beam model has been considered and related relations were extracted. The lateral displacement formulas obtained and calculated for the framed tube system modeled by Kwan’s method. To verify the results, numerical models were created in software (ETABS) and statically were analyzed for lateral loading. Finally the results were compared with those obtained by computer analysis and the corresponding diagrams were presented. At the end, the shape factor formula has been developed to improve the results of the Timoshenko’s theory.
Keywords
Framed tube system Equivalent continuous method Euler–Bernoulli beam theory Timoshenko’s beam theoryIntroduction
Tall buildings are a logical and economical solution for the settlement of the population, jobs, departments, etc. in a small area of land, which can help the town planners, besides its appearance and glory. However, for many cosmopolitans, highrise buildings are the only response to the continuous growth of population concentration. The purpose of constructing highrise buildings is to use more area to meet the needs of different citizens. This is a way to save on land use and share prices to users, thus avoiding the unnecessary land use, coping with its scarcity, and expanding urban levels. The conventional analysis of tall structures is often timeconsuming and costly due to having high degrees of freedom. The designers of this kind of buildings require a proper and reasonable design for their design and it is necessary to be able to analyze the static and dynamic characteristics of the structure quickly and accurately. Additionally, the effect of each variable in assumed structural system should be determined and finally, the first appropriate design should be recommended. Considering the importance of framed tube structures in highrise buildings as a system resistant to lateral loads, it is necessary to have a simple but precise method for analyzing these structures.
There are different forms of tall buildings and numerous researches have been carried out about the approximate and exact methods of investigating the behaviour, deflection, vibration, optimal design and control of such buildings. Coull and Bose (1975) presented a method based on the theory of elasticity. In this method, the structure is equivalent to orthotopic planes, and the equations and equilibrium relations are satisfied in the equivalent structure. Coull and Ahmed (1978) presented a method in calculation of stories displacement in the framed tube. Kwan (1994) presented equations for determining the stress in columns and obtaining the lateral displacement of the framed tube structure using equivalence orthotropic planes, energy relations and the theory of elasticity. Connor and Pouangare (1991) recommended the fivestringer model, in which the structure is equalled with beams and vertical planes and by calculating the shear and moment rigidity of members, the formulas for stresses in the column are proceeded. One of the other methods for improving the behavior of the framed tube is adding internal tubed frames to the original structure. In this case, the stress distribution and displacement will be appreciably reduced. Other methods for analyzing framedtube are presented by researchers such as Paulino (2010), Mahjub et al. (2011), Kamgar and Rahgozar (2013), Rahgozar et al. (2014), Malekinejad and Rahgozar (2014), Mohammadnejad (2015), Rahgozar et al. (2015), Khajuai Rad et al. (2017). A parametric dimensionless formula has been developed for flexural stiffness of highrise buildings with the objective of optimization problem by Alavi et al. (2017). Kamgar and Rahgozar (2019) introduced a method to reduce the static roof displacement and axial forces in the columns of tall buildings. Tavakoli et al. (2019) considered a direct method for highrise structures with elastic and inelastic analysis with fixedbase system. Davari and Rahgozar (2019) conducted their research on the static analysis of highrise buildings without shear lag effect, too. Davari et al. (2019) explained the Euler–Bernoulli and the Timoshenk’s beam theory to analyse the tall buildings, which it concluded that the Timoshenko theory is more accurate than the Euler–Bernoulli. In this paper, a simple approximation method has been proposed for static analysis of a tall structure with a framed tube system with symmetrical plan and fixed profile in height using a Timoshenko beam model with regard to shear and flexural hardening and regardless of the shear lag effects (Herrmann 1955). Finally, a revised formula for the shape factor of the highrise buildings is proposed to minimize the error between ETABS modeling and the Timoshenko’s theory.
Method
In this study, the framed tube was equalized with four orthotropic orthogonal planes with a hollow box section, the structure of the framed tube was subjected under a uniform load and triangular distributed loading, and static analysis was performed. Davari et al. presented the Euler–Bernoulli’s theory and the Timoshenko’s beam theory to calculate the displacement of the stories.
Uniform distributed loading
Triangular distributed loading
In the following, two examples are presented to examine the proposed method and compare the results of the numerical investigation with two others theories. Each of these structures has been analyzed and studied with ETABS modeling, proposed method and Euler–Bernoulli’s theory under a uniform static load pattern.
Concrete framedtube with symmetric plan
Parameters of equivalent structure
\(t = \frac{{A_{\text{c}} }}{S}\)  \(I = 4a^{2} bt + \frac{4}{3}a^{3} t\)  \(k\)  \(\frac{{\Delta_{\text{b}} }}{Q} = \frac{{\left( {h  d_{\text{b}} } \right)^{3} }}{{12E_{\text{m}} I_{\text{c}} }} + \left( {\frac{h}{S}} \right)^{2} \frac{{\left( {S  d_{\text{c}} } \right)^{2} }}{{12E_{\text{m}} I_{\text{b}} }}\)  \(\frac{{\Delta_{\text{s}} }}{Q} = \frac{{\left( {h  d_{\text{b}} } \right)}}{{G_{\text{m}} A_{\text{Sc}} }} + \left( {\frac{h}{S}} \right)^{2} \frac{{\left( {S  d_{\text{c}} } \right)}}{{G_{\text{m}} A_{\text{Sb}} }}\)  \(G = \frac{{\frac{h}{St}}}{{\frac{{\Delta_{\text{b}} }}{Q} + \frac{{\Delta_{\text{s}} }}{Q}}}\) 

0.26 m  5184 m^{4}  \(\frac{5}{6}\)  2.1634 × 10^{−9}  0.9078 × 10^{−9}  1.526 GPa 
Results
Lateral displacements calculated in different area of concrete structure
Area of concrete structure (m^{2})  ETABS modeling (mm)  Timoshenko (mm)  Error (%) 

750  496.6  451.2  9.14 
1050  354  292.1  17.49 
1400  299.5  201.8  32.62 
1800  260.7  146.4  43.84 
Lateral displacements calculated in concrete structure with different heights of the stories
Height of concrete structure’s stories (m)  ETABS modeling (mm)  Timoshenko (mm)  Error (%) 

3  354  292.1  17.49 
3.5  584  534.2  8.53 
4  908  839.7  7.52 
4.5  1347.5  1266.2  6.03 
5  1925.8  1844  4.25 
Lateral displacements calculated in different area of steel structure
Area of steel structure (m^{2})  ETABS modeling (mm)  Timoshenko (mm)  Error (%) 

750  265.4  258.3  2.68 
1050  214  176.8  17.38 
1400  185.5  128.2  30.89 
1800  165.8  116.4  29.79 
Lateral displacements calculated in different areas of steel structure
Height of steel structure’s stories (m)  ETABS modeling (mm)  Timoshenko (mm)  Error (%) 

3  214  212.2  0.84 
3.5  332.5  317.6  4.48 
4  493.8  458.2  7.21 
4.5  706.3  642.2  9.08 
5  1052  878.7  16.47 
Comparison of results for shape factor and displacement
Analysis type  Shape factor  Displacement (mm) 

ETABS  –  567.3 
Old proposed method  0.83  642.6 
New proposed method  1.7  571.4 
Conclusion

Proposed model for the analysis of tall structures with symmetrical plan produced acceptable results, particularly in the initial design stage.

Although numerical methods and commercial software were more accurately able to model tall structures, the analytical approach presented in the proposed method provided the ability to determine the parameters affecting structure behavior and their sensitivity analysis in the structure response.

Proposed method was less costly compared to the numerical methods such as finite element, and required less computing procedures. This issue becomes more important in the case of highrise structures with a large number of elements.

New coefficients for the shape factor equation induced better accuracy for the displacement of the framedtube structures.

Besides the accuracy of the shape factor equation for the framedtube structures, the timeconsuming analysis could be reduced significantly.

The results of the analysis with shear effect is considerable because Timoshenko beam theory adapted vigorously on the ETABS modeling results.
Notes
References
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