Perspective Chapter: Dynamic Analysis of High-Rise Buildings Using Simplified Numerical Method

Abstract

This chapter emphasizes on the static and dynamic characteristics of multi-story building subjected to uniformly distributed and wind load. First-order shear deformation theory is used to formulate governing equations based on the finite element method. The multi-story building is considered as a vertical cantilever beam/plate and modeled using nine-node degenerated shell element. Fictitious membrane and shear stresses are eliminated by considering Mixed Interpolation Tonsorial Component (MITC) technique. Here, the static and dynamic characteristics of multi-story buildings have been investigated take into account as a vertical cantilever plate subjected to UDL, triangular load (wind load) and combination of both. In this chapter authors demonstrated the deformation shapes, longitudinal stress and in-plane shear stress and principle strains in various loading conditions of vertical cantilever flat panel. Moreover, investigated the dynamic characteristics of multi-story buildings considering as a vertical cantilever plates and governing equations of motion are derived by employing Hamilton’s principle. Moreover, nonlinear transient response of high-rise structures has been studied here by employing the energy and momentum conservation implicit time integration scheme. The structural analysis of tall buildings has been carried out here through commercial software ANSYS. Matrix amplitude method is employed to investigate the large-amplitude flexural vibration responses of flat panels. Also, plotted the fast Fourier transform and phase portraits for first three bending modes.

1. Introduction

In the recent times, dynamically increase in population and limitation of land acquisition high-rise structures or multi-story building become major concern of researchers. The critical issue in high-rise buildings is selecting proper structural form to resist the lateral and flexural loads. Therefore, the static and dynamic responses of high-rise building is essential to design safe and reliable structure for engineers. These high-rise buildings may be transformed into multi-story buildings in order to make more floors space but occupy less lands space [1]. These multistory buildings are used for residential flats, offices, shopping complex, Malls, Hotels and public centers. Lateral loads such as wind load and seismic load play predominant role in high-rise structures.

1.1 Wind loading

There are several instances where structures have failed owing to an instability that needs second-order analysis (P-Delta). One of the problems resulted from wind loading. The wind induces outward and inward as a triangular load act on the surfaces of multi-story buildings. Structural instability issues arises when structures could not bear certain loads and buckling of structures +occurs due to dead load, wind load and seismic load like earthquake [2]. Ankireddi and Yang [3] considered gradually increased load along building height (wind load) as shown in Figure 1. To simplify the analysis [4, 5] load is reduced by 50 percent at the center of building as consideration of BS EN 1991-1-4:2005.

Figure 1.

Wind loads act on the surface of the building [3, 5].

The direction of wind is very important throughout the life of structure. Wen [6] found that the wind direction analysis is not conducted seriously. Analysis of wind direction becomes necessary for the case of high-rise buildings and suspension bridges. Moreover, wind speed is also plays crucial role on construction sites, structures will collapse without warning due to speed of wind. For example, building, power lines and trees are collapsed in Hurricane Frederic east of Pascagoula, Mississippi on September 12, 1979 in coastal areas. Mehta et al. [7] estimated the effect of wind speed based on an indirect approach. Authors collected comprehensive data to introduce an alternate approach and convenient equipment. It is noticed that wind speed and direction significantly contribute in deflection of multi-story structures.

1.2 Urban context and sustainability

For energy saving and sustainable development high-rise building requires sunlight access, considering optimum utilization of limited conventional resources. Therefore, new multi-story buildings are optimized on its shape, height and orientation considering environmental factors such as harmony between buildings and their urban and environmental contexts.

Environment friendly structures are classified into three basic sets of strategies. First one is the minimization of operation cost and material consumption; second one minimization of energy consumption or maximum utilization of solar energy and sun light; third one is consideration of whether conditions according to different climate zones.

To design building for different climatic zones, then consider the hot seasons and the cold seasons to deal with, and you have the two mid- seasons. So, you have to design the enclosure, the skin as a responsive environmental filter for energy efficiency.

For the analysis purpose, a cantilever beam (l = 60 m, b = 12 m and h = 1 m) with fixed support at bottom and free at remaining all three ends is modeled as high-rise building. The analysis is performed considering the building experiencing different loading conditions viz., uniformly distributed load, uniformly varying load (wind load) with zero magnitude at bottom and the combination of the two said loads.

The static behavior of hybrid high rise buildings have been studied here considering a vertical cantilever flat panel under uniformly distributed load, wind load and combination of both. Linear bending response of vertical cantilever plates has been analyzed under triangular load. The structural analysis of tall buildings has been carried out here through commercial software ANSYS.

1.3 High-rise building considered as cantilever beams

An approximate method for the static and dynamic analysis of high-rise buildings is continuum method in which these structures are substituted by a continuum beam/plate, adopting Euler-Bernoulli or Timoshenko beam theory/Classical or Mindlin or higher-order plate theory. The bending analysis of tall building have been performed by employing continuum-based Kwan model, here, tall buildings were considered as a cantilever beam and studied bending behavior of tall building by employing Euler-Bernoulli beam theory [8] and also compared the analytical results with numerical results obtained by ETABS software. Framed tube structures primarily act like cantilever hollow box beams. These beams could resist more moments of the lateral loads; the beam bending action of the framed tube structures were complicated due to shear lag in the web and flange panel; [9] introduced a simple hand-calculation method considering shear lag effect for approximate static analysis of framed tube structures. Alavi et al. [10, 11] proposed a simple mathematical technique to design minimum cost tall and slender structures that might be used as conceptual/early-stage design. Authors considered the tall structures as cantilever beam and studied the flexural vibration behavior [10] and peak lateral deflection response [11] of high-rise buildings. They also performed the parametric study considering 42-story and 60-story buildings.

Most of the researchers modeled the tall building as cantilever beam to study the free vibration behavior [12, 13, 14, 15, 16]. The discrepancy of empirical formulas given by various researchers for the fundamental frequency of tall buildings was examined by Dym et al. [12] and used the Euler-Bernoulli and Timoshenko beam model to estimate the natural frequencies of these structures. The natural frequency of multistory building under seismic load was determined by Kaviani et al. [14] using Timoshenko beam theory. They assumed each lateral load carrying subcomponent (floor) of tall building is considered continuous cantilever beam with variable cross-section.

The natural frequencies of framed tube and shear walls structures have been calculated by using energy method and Hamilton’s principle [17]. Authors considered the framed tube and shear walls structures as a cantilever beam based on continuum approach. They also compared the analytical results with numerical results obtained by SAP2000. Piccardo et al. [18] introduced an equivalent three-dimensional space continuous Timoshenko beam model to study the static and dynamic behavior of tower building.

Hallebrand and Jakobsson [19] investigated the high-rise building under static and dynamic loads to investigate the effect of deflections, resonance frequencies, accelerations and stability. They discussed the modeling techniques and issues to model high-rise building using finite element method. Authors also compared the vertical loading (such as self-weight, imposed loads, snow loads and live loads) and horizontal loading (for example wind load or design load and unintended inclinations) considering different modeling techniques.

Miranda [20] introduced an approximate method based on an equivalent continuum beam model to determine lateral displacement and maximum inter-story drift of tall buildings under seismic load. Multistory structures were considered as an equivalent continuum model taking in account the combination of shear cantilever beam and flexural cantilever beam. Author examined the effect of lateral force along the height of tall structures, shear deformations, flexural deformations on the multistory buildings that was considered as a beam structure. An approximate method was further generalized [21] to consider the non-uniform lateral stiffness of tall structures. The non-uniform Timoshenko beam was modeled [22] to describe the spectrum analysis of it. They assessed the first four vibration modes of non-uniform Timoshenko beam model under seismic loading conditions. Attention of researchers is also focused on the eigenvalue solutions of beams (Rafezy and Howson 2008) that would be considered as a high-rise building.

Authors considered the step change of properties along the height of the structures. Recently, the nonlinear static and dynamic response of thin-wall composite structures were investigated [23, 24, 25, 26] by employing finite element method with first-order shear deformation theory. Here, author considered a high-rise building as a cantilever plat to study the static and dynamic behavior of these thin-wall structures.

2. Finite element formulation

The static and dynamic analysis of high-rise buildings under uniform pressure and wind load are performed by using finite element method based commercial software ANSYS. To discretize the high-rise building type cantilever beam/plate considered the eight-node 281 shell element. To solve the multi-dimensional problem using Green-Gauss theorem will expressed as:

$$\underset{V}{\int }{\sigma }^T\epsilon \left(\delta \right)\mathit{dV}-\underset{V}{\int }{\delta }^T\mathit{fdV}-\underset{S}{\int }\left[\begin{array}{l}\left(n_x{\sigma }_x+n_y{\tau }_{\mathit{xy}}+n_z{\tau }_{\mathit{xz}}\right){\delta }_x\\ +\left(n_x{\tau }_{\mathit{xy}}+n_y{\sigma }_y+n_z{\tau }_{\mathit{yz}}\right){\delta }_y\\ +\left(n_x{\tau }_{\mathit{xz}}+n_y{\tau }_{\mathit{yz}}+n_z{\sigma }_z\right){\delta }_z\end{array}\right]\mathit{dS}$$E1

Here, σ represents six independent component of stress $\sigma ={\left[{\sigma }_x,{\sigma }_y,{\sigma }_z,{\tau }_{\mathit{yz}},{\tau }_{\mathit{xz}},{\tau }_{\mathit{xy}}\right]}^T$ normal stresses and shear stresses; ε represents six strains corresponding to stresses $\epsilon ={\left[{\epsilon }_x,{\epsilon }_y,{\epsilon }_z,{\gamma }_{\mathit{yz}},{\gamma }_{\mathit{xz}},{\gamma }_{\mathit{xy}}\right]}^T$; dV is volume integration dV = dx dy dz; distributed force per unit volume $f={\left[f_x,f_y,f_z\right]}^T$; displacement vector is $\delta ={\left[u_x,u_y,u_z,{\theta }_x,{\theta }_y,{\theta }_y\right]}^T$; the unit normal to surface dA is $n={\left[n_x,n_y,n_z\right]}^T$.

2.1 Constitutive equations

The kinematic correlations and the mechanical and thermodynamic concepts are applicable at all continuum irrespective of its physical constitutions. Here, considered the equations characterizing the individual material and its reaction to apply loads. These equations are known as constitutive eqs.

A material body said to be isotropic/homogeneous if the properties of material are same throughout the body. In an anisotropic/heterogeneous body, the properties of material are function of position.

A material body supposed to be ideally elastic under isothermal conditions, the body will recover its original form with removal of forces causing deformation. One-to-one relationship (based on generalized Hooke’s law) between the state of stress and state of strain will be written as:

$$\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{23}\\ {\sigma }_{13}\\ {\sigma }_{12}\end{array}\right\}=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& C_{14}& C_{15}& C_{16}\\ C_{21}& C_{22}& C_{23}& C_{24}& C_{25}& C_{26}\\ C_{31}& C_{32}& C_{33}& C_{34}& C_{35}& C_{36}\\ C_{41}& C_{42}& C_{43}& C_{44}& C_{45}& C_{46}\\ C_{51}& C_{52}& C_{53}& C_{54}& C_{55}& C_{56}\\ C_{61}& C_{62}& C_{63}& C_{64}& C_{65}& C_{66}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ {\epsilon }_{23}\\ {\epsilon }_{13}\\ {\epsilon }_{12}\end{array}\right\}$$E2

Where C_ij is the elastic coefficient.

The elastic coefficient matrix C_ij is a symmetric (C_ij = C_ji) therefore, there is 21 independent coefficients of the matric [C].

For three perpendicular planes (x-y, x-z and y-z) to each other known orthogonal planes due to symmetry the number of elastic coefficients are reduced to nine, and these materials are known as orthotropic. The stress-strain relations for an orthotropic material will be expressed as:

$$\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{23}\\ {\sigma }_{13}\\ {\sigma }_{12}\end{array}\right\}=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{21}& C_{22}& C_{23}& 0& 0& 0\\ C_{31}& C_{32}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{55}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ {\epsilon }_{23}\\ {\epsilon }_{13}\\ {\epsilon }_{12}\end{array}\right\}$$E3

The inverse relations, strain–stress relations may be written as:

$$\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ {\epsilon }_{23}\\ {\epsilon }_{13}\\ {\epsilon }_{12}\end{array}\right\}=\left[\begin{array}{cccccc}{S}_{11}& S_{12}& S_{13}& 0& 0& 0\\ S_{21}& S_{22}& S_{23}& 0& 0& 0\\ S_{31}& S_{32}& S_{33}& 0& 0& 0\\ 0& 0& 0& S_{44}& 0& 0\\ 0& 0& 0& 0& S_{55}& 0\\ 0& 0& 0& 0& 0& S_{66}\end{array}\right]\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{23}\\ {\sigma }_{13}\\ {\sigma }_{12}\end{array}\right\}$$E4

$$\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ {\epsilon }_{23}\\ {\epsilon }_{13}\\ {\epsilon }_{12}\end{array}\right\}=\left[\begin{array}{cccccc}\frac{1}{E_1}& -\frac{{\nu }_{21}}{E_2}& -\frac{{\nu }_{31}}{E_3}& 0& 0& 0\\ -\frac{{\nu }_{12}}{E_1}& \frac{1}{E_2}& -\frac{{\nu }_{32}}{E_3}& 0& 0& 0\\ -\frac{{\nu }_{13}}{E_1}& -\frac{{\nu }_{23}}{E_2}& \frac{1}{E_3}& 0& 0& 0\\ 0& 0& 0& \frac{1}{G_{23}}& 0& 0\\ 0& 0& 0& 0& \frac{1}{G_{13}}& 0\\ 0& 0& 0& 0& 0& \frac{1}{G_{12}}\end{array}\right]\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{23}\\ {\sigma }_{13}\\ {\sigma }_{12}\end{array}\right\}$$E5

Here, [S]_6 × 6 denotes the compliance coefficients; [S] = [C]⁻¹; E₁, E₂, E₃ are Young’s modulus in 1 (longitudinal), 2 (transverse) and 3 (normal) material directions, respectively; Poisson’s ratio ν_ij is the ratio of transverse strain in jth direction to the axial strain in ith direction when load is applied along longitudinal direction or stressed in ith direction; G₁₂, G₁₃, and G₂₃ are shear moduli in the x-y, x-z and y-z planes, respectively. The compliance matrix [S] is symmetric matrix, because compliance matrix is the inverse of stiffness matrix. Symmetric matrix inverse is also symmetric.

Therefore: $\frac{{\nu }_{21}}{E_2}=\frac{{\nu }_{12}}{E_1};\frac{{\nu }_{31}}{E_3}=\frac{{\nu }_{13}}{E_1};\frac{{\nu }_{32}}{E_3}=\frac{{\nu }_{23}}{E_2}\mathit{\text{or}}\frac{{\nu }_{\mathit{ij}}}{E_i}=\frac{{\nu }_{\mathit{ji}}}{E_j}$ (no sum on i, j).

Here i, j = 1,2,3. Hence, there are only nine independent material coefficients (E₁, E₂, E_3,G₁₂, G₁₃, G_23,ν₁₂, ν₁₃, ν₂₃) for an orthotropic material. For an isotropic material (material having infinite number of planes of material symmetry) independent elastic coefficients are reduced to two (E₁ = E₂ = E₃ = E, ν₁₂ = ν₁₃ = ν₂₃ = ν, G₁₂ = G₁₃ = G₂₃ = G = E/2(1 + ν).

The state of plane stress is expressed to be one in which transverse stresses are neglected. Then, for the orthotropic material the strain-stress relations to describe the state of plane stress is:

$$\begin{array}{rcl}\left\{\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_{12}={\epsilon }_6\end{array}\right\}& =& \left[\begin{array}{ccc}{S}_{11}& S_{12}& 0\\ S_{12}& S_{22}& 0\\ 0& 0& S_{66}\end{array}\right]\left\{\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_{12}={\sigma }_6\end{array}\right\}\\ & =& \left[\begin{array}{ccc}1/E_1& -{\nu }_{21}/E_2& 0\\ -{\nu }_{12}/E_1& 1/E_2& 0\\ 0& 0& 1/G_{12}\end{array}\right]\left\{\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_{12}={\sigma }_6\end{array}\right\}\end{array}$$E6

The strain-stress relation expressed in Eq. (6) are inverted to obtain the stress-strain relations:

$$\left\{\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_{12}={\sigma }_6\end{array}\right\}=\left[\begin{array}{ccc}{Q}_{11}& Q_{12}& 0\\ Q_{12}& Q_{22}& 0\\ 0& 0& Q_{66}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_{12}={\epsilon }_6\end{array}\right\}$$E7

Here, Q_ij is known as plane stress-reduced stiffness, are expressed by:

$$Q_{11}=\frac{S_{22}}{S_{11}{S}_{22}-S_{12}^2}=\frac{E_1}{1-{\nu }_{12}{\nu }_{21}}$$

$$Q_{12}=\frac{S_{12}}{S_{11}{S}_{22}-S_{12}^2}=\frac{{\nu }_{12}{E}_2}{1-{\nu }_{12}{\nu }_{21}}$$

$$Q_{22}=\frac{S_{11}}{S_{11}{S}_{22}-S_{12}^2}=\frac{E_2}{1-{\nu }_{12}{\nu }_{21}}$$E8

$$Q_{66}=\frac{1}{S_{66}}=G_{12}$$

Thus, reduced stiffness involved four independent material constants E₁, E₂, ν₁₂, G₁₂.

The transverse shear stresses and shear strain relations for orthotropic materials are defined as:

$$\left\{\begin{array}{c}{\sigma }_{23}={\sigma }_4\\ {\sigma }_{13}={\sigma }_5\end{array}\right\}=\left[\begin{array}{cc}{Q}_{44}& 0\\ 0& Q_{55}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_{23}={\epsilon }_4\\ {\epsilon }_{13}={\epsilon }_5\end{array}\right\}$$E9

Here, Q₄₄ = C₄₄ = G₂₃ and Q₅₅ = C₅₅ = G₁₃.

2.2 Transformation of components

In structural analysis, it is required to consider all the quantities for common structural coordinate system. Scalars are independent of any coordinate system, whereas vectors and tensors are independent of a particular coordinate system, and their components are not. The same vectors and tensors have different components in different coordinate systems, but any two sets of components of a vectors and tensor will be related by writing one set of components in terms of the other. Transformation of vectors component considering barred (${\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3$) and unbarred (x₁, x₂, x₃) coordinate systems are related as shown in Figure 2 and written in Eqs. (10) and (11).

Figure 2.

Unbarred and barred rectangular coordinate system.

$$x_1=x_1\left({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3\right)$$

$$x_2=x_2\left({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3\right)$$E10

$$x_3=x_3\left({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3\right)$$

Inverse relations are written as:

$${\overline{x}}_1={\overline{x}}_1\left(x_1,x_2,x_3\right)$$

$${\overline{x}}_2={\overline{x}}_2\left(x_1,x_2,x_3\right)$$E11

$${\overline{x}}_3={\overline{x}}_3\left(x_1,x_2,x_3\right)$$

2.3 Transformation of material stiffness

The material stiffness C_ijkl is the fourth order tensor. Thus, considering the in general law of the fourth order tensor transforms as given in Eq. (12).

$${\overline{C}}_{\mathit{\text{ijkl}}}=a_{\mathit{im}}{a}_{\mathit{jn}}{a}_{\mathit{kp}}{a}_{\mathit{lq}}{C}_{\mathit{\text{mnpq}}}$$E12

Schematic representation of rectangular plate with global and material coordinates is shown in Figure 3. For the plane stress case, the elastic stiffness Q_ij in the principal material system are related to ${\overline{Q}}_{\mathit{ij}}$ in the reference coordinate system is written as:

Figure 3.

Rectangular plate with global and material coordinate systems.

$${\overline{Q}}_{11}=Q_{11}{cos}^4\theta +2\left(Q_{12}+2Q_{66}\right){sin}^2\theta {cos}^2\theta +Q_{22}{sin}^4\theta$$

$${\overline{Q}}_{12}=\left(Q_{11}+Q_{22}-4Q_{66}\right){sin}^2\theta {cos}^2\theta +Q_{12}\left({sin}^4\theta +{cos}^4\theta \right)$$

$${\overline{Q}}_{22}=Q_{11}{sin}^4\theta +2\left(Q_{12}+2Q_{66}\right){sin}^2\theta {cos}^2\theta +Q_{22}{cos}^4\theta$$

$${\overline{Q}}_{16}=\left(Q_{11}-Q_{12}-2Q_{66}\right)sin\theta {cos}^3\theta +\left(Q_{12}-Q_{22}+2Q_{66}\right){sin}^3\theta cos\theta$$

$${\overline{Q}}_{26}=\left(Q_{11}-Q_{12}-2Q_{66}\right){sin}^3\theta cos\theta +\left(Q_{12}-Q_{22}+2Q_{66}\right)sin\theta {cos}^3\theta$$

$${\overline{Q}}_{66}=\left(Q_{11}+Q_{22}-2Q_{12}-2Q_{66}\right){sin}^2\theta {cos}^2\theta +Q_{66}\left({sin}^4\theta +{cos}^4\theta \right)$$

$${\overline{Q}}_{44}=Q_{44}{cos}^2\theta +Q_{55}{sin}^2\theta$$

$${\overline{Q}}_{45}=\left(Q_{55}-Q_{44}\right)cos\theta sin\theta$$

$${\overline{Q}}_{55}=Q_{55}{cos}^2\theta +Q_{44}{sin}^2\theta$$E13

2.4 Governing equations

The total potential energy of the general elastic body is written as:

$$\prod =\frac{1}{2}\underset{V}{\int }{\sigma }^T\mathit{\epsilon dV}-\underset{V}{\int }{\delta }^T\mathit{fdV}-\underset{S}{\int }{\delta }^T\mathit{TdS}-\sum _i{\delta }_i^T{P}_i$$E14

Here, σ = [D][B]{δ} and ε = [B]{δ}; [D] is the flexural rigidity matrix and [B] is the strain–displacement matrix.

2.5 Static analysis

Firstly, for the static analysis of multi-story building are performed by solving this governing equation that may be expressed as:

$$\left[K_L+K_{\mathit{NL}1}\left(\delta \right)+K_{\mathit{NL}2}\left(\delta ,\delta \right)\right]\left\{\delta \right\}=\left\{F_P\right\}+\left\{F_{\mathit{\text{wind}}}\right\}$$E15

Here, [K_L] is linear stiffness matrix; [K_NL] is nonlinear stiffness matrix; {δ} is displacement vector represents six degree of freedom three displacements u_x, u_y, u_z and three rotations θ_x, θ_y, θ_z along x-axis, y-axis and z-axis, respectively; {F_p} and {F_wind} are uniformly distributed load and wind load vector.

2.6 Dynamic analysis

Thereafter, for the dynamic analysis of multi-degree of freedom systems the governing equation of motion may be written as:

$$\left[M\right]\left\{\ddot{\delta }\right\}+\left[K_L+K_{\mathit{NL}1}\left(\delta \right)+K_{\mathit{NL}2}\left(\delta ,\delta \right)\right]\left\{\delta \right\}=\left\{F_p\right\}+\left\{F_{\mathit{\text{wind}}}\right\}$$E16

Here, [M] is mass matrix; $\left\{\ddot{\delta }\right\}$ is acceleration vector.

For the dynamic forced vibration analysis of large story building Eq. (16) used here, whereas for the free vibration analysis of high-rise building governing equation may be written as:

$$\left[M\right]\left\{\ddot{\delta }\right\}+\left[K_L+K_{\mathit{NL}1}\left(\delta \right)+K_{\mathit{NL}2}\left(\delta ,\delta \right)\right]\left\{\delta \right\}=0$$E17

The natural frequency and deformations of high-rise building are expressed by Eigen-values and Eigen-vector solutions of Eq. (17).

$$\left\{-\left[M\right]{\omega }^2+\left[K_L+K_{\mathit{NL}1}\left(\delta \right)+K_{\mathit{NL}2}\left(\delta ,\delta \right)\right]\right\}\left\{\delta \right\}=0$$E18

Here, ω is the natural frequency of high-rise building represented as an Eigen-values; {δ} is the deformation of structures along six degree of freedom displacements (u_x, u_y, u_z) and rotations (θ_x, θ_y, θ_z) is represented by Eigen-vector of Eq. (17).

2.7 Solution Procedure for large amplitude flexural vibration analysis

Assume a harmonic solution of the displacement vector $\delta ={\delta }_{\max }sin\mathit{\theta t}$, the weighted residual of Eq. (16) with {F_P + F_wind = 0} along the path t = 0 to T/4 (δ = 0 to δ _max) may be expressed as:

$$\underset{0}{\overset{T/4}{\int }}Rsin\mathit{\text{θtdt}}=\left\{0\right\}$$E19

Where, the residual of Eq. (19) is:

$$\left\{R\right\}=\left[K_L+K_{\mathit{NL}1}\left({\delta }_{\max }\right)sin\mathit{\theta t}+K_{\mathit{NL}2}\left({\delta }_{\max },{\delta }_{\max }\right){sin}^2\mathit{\theta t}-{\theta }^{2}M\right]\left\{{\delta }_{\max }\right\}sin\mathit{\theta t}$$E20

Evaluating the integral of Eq. (20); the matrix amplitude equation may be written as [27]:

$$\left[K_L+\frac{4}{3\pi }{K}_{\mathit{NL}1}\left({\delta }_{\max }\right)+\frac{3}{4}{K}_{\mathit{NL}2}\left({\delta }_{\max },{\delta }_{\max }\right)-{\theta }^{2}M\right]\left\{{\delta }_{\max }\right\}=\left\{0\right\}$$E21

The matrix amplitude Eq. (21) is solved iteratively (Naghsh and Azhari 2015) to find the frequency verses amplitude relationship of cantilever plates.

2.8 Flow chart for static analysis through ANSYS

3. Results and discussion

Schematic representation of high-rise buildings considered as a vertical cantilever plate under uniformly distributed load and triangular load (wind load) is shown in Figure 4.

Figure 4.

Analysis of model structure under uniformly distributed load.

3.1 Validation study

First authors performed the validation study of developed model and then compared the present ANSYS results with available published results [28]. For comparison the deformations at specific locations (y = 0, y = 0.25b, y = 0.5b, y = 0.75b and y = b) for a rectangular cantilever (CFFF) thin plate (a = 12 m, b = 60 m and h = 1 m) under uniformly distributed load (q = 1400 N/m²), wind (triangular load q × y/b; here q = 1400 N/m²) load and combined uniformly distributed and wind load (q/2 + (q/2) × y/b) of isotropic (v = 0.3, E = 210 GPa, ρ = 7800 kg/m³) and RCC (v = 0.2, E = 30 GPa, ρ = 2500 kg/m³) are obtained and presented in Tables 1 and 2, respectively.

δ	y = 0		y = 0.25b		Y = 0.5b		y = 0.75b		x = b
	Ref [28]	Present	Ref [28]	Present	Ref [28]	Present	Ref [28]	Present	Ref [28]	Present
y = 0	0.000	0.000	0.0117	0.01394	0.0433	0.04181	0.0841	0.08361	0.127	0.1254
y = b	0.000	0.000	0.0117	0.01394	0.0433	0.04181	0.0841	0.08361	0.127	0.1254

Table 1.

Deformations δ (qb⁴/D) [28] of the free edges x = 0 and x = a for a rectangular plate (CFFF) under uniformly distributed loading for mild steel material with v = 0.3.

δ	y = 0		y = 0.25b		y = 0.5b		y = 0.75b		y = b
	Ref [28]	Present	Ref [28]	Present	Ref [28]	Present	Ref [28]	Present	Ref [28]	Present
y = 0	0.000	0.000	0.0117	0.01368	0.0433	0.04103	0.0841	0.0821	0.127	0.1231
y = b	0.000	0.000	0.0117	0.01368	0.0433	0.04103	0.0841	0.0821	0.127	0.1231

Table 2.

Deformations δ (Wb⁴/D) [28] of the free edges x = 0 and x = a for a rectangular plate (CFFF) under uniformly distributed loading for RCC material with v = 0.2.

The agreement between the present results and those from the literature is satisfactory. The method developed in this article is suitable for the problems of rectangular cantilever thin plates under uniformly distributed load, a wind load and combined uniformly distributed load and wind loads.

3.2 Distribution of displacement, stresses and strains

Next, schematic distribution of deformation (with deformed and un-deformed shape and contour plot), membrane stresses (σ_xx,σ_xy) and second principle strain (ε₂) is shown in Figure 5. It is noticed that by changing the material /material properties (in terms of Young’s modulus, Poisson’s ratio and density) qualitatively distribution of displacement, stresses and strains are similar for cantilever towers made by isotropic material/reinforced concrete composites. In-plane minimum and maximum stresses makes a strip/band as shown in Figure 5(d) having magnitude 2.13 MPa and 1.53 MPa for isotropic and composite cantilever plates, respectively. Moreover, distribution of second principle strains (ε₂) is shown in Figure 5(e), maximum principle strains predicts at 10 m from the base represented by red color; whereas distribution of second principle strains is similar for both the cantilever plates (ν = 0.3 and ν = 0.2) under uniformly distributed load (q₀ = 1400 N/m²).

Figure 5.

Schematic representation of (a) displacement with deformed and un-deformed shape (b) Contour Plot of total displacement (c) Distribution of longitudinal stress σ_xx (d) In-plane shear stress σ_xy and (e) Second principle strain ε₂ for high rise structure considered as vertical cantilever plate (a = 12 m, b = 60 m, h = 1 m; E = 210 GPa, ν = 0.3 and ρ = 7800 kg/m³) under uniformly distributed load (q₀ = 1400 N/m²).

Thereafter, the change in total displacement, in-plane strains and principle strains is shown in Figure 6(a–e) with schematic scales under wind load/triangular load (q₀ × y/b, q₀ = 1400 N/m²). It is observed that the distribution of longitudinal strain (ε_xx) and second principle strain (ε₂) is qualitatively similar as shown in Figure 6(b and d); whereas in-plane shear strain (ε_xy) makes a banded strip as given in Figure 6(c).

Figure 6.

Distribution of displacement and in-plane strains (ε_xx,ε_xy) and principle strains (ε_22,ε₃₃) under wind load (q₀ × x/b; q₀ = 1400 N/m²) on isotropic (ν = 0.3) cantilever plate.

For the vibration analysis, a high-rise building was modeled as a cantilever (CFFF) plate with 12 m × 60 m with unit thickness (h = 1 m). The plate is analyzed in linear bending for displacement and vibrations using commercial software ANSYS considering two different materials- isotropic material (mild-steel young’s modulus 210 GPa, poison’s ratio 0.3) and RCC (young’s modulus 30 GPa, poison’s ratio 0.2) under three different loading conditions i.e. uniformly distributed loading, wind load and wind load with uniformly distributed loading. The results are presented in Figures 7 and 8 for isotropic and RCC cantilever panels under various loading conditions.

Figure 7.

Non-dimensional deformation of isotropic (ν = 0.3) thin cantilever plate.

Figure 8.

Non-dimensional deformation of RCC (ν = 0.2) thin cantilever Plate.

Next, dynamic analysis has been performed for high-rise buildings considered as a cantilever (CFFF) plates made of isotropic (E = 210 GPA, ν = 0.3 and ρ = 7800 kg/m³) material and RCC (E = 30 GPA, ν = 0.2 and ρ = 2500 kg/m³) material and presented the vibration frequencies and mode shapes in Tables 3 and 4 and Figure 9, respectively. The non-dimensional fundamental vibration frequencies (${\varpi }_i={\omega }_i{b}^2\sqrt{\mathit{\rho h}/D}$) are given in Table 3, compared the present numerical results with commercial software ANSYS. Mesh convergence study is also performed to get the converge results considering mesh size 1 × 5 and 2 × 10. Non-dimensional fundamental frequencies should not change with change in material properties as given in Table 3. Thereafter, natural frequencies of isotropic (0.3) and composite (0.2) cantilever plates is presented in Table 4. Moreover, vibration mode shapes are shown in Figure 9 representing in-plane and out-off plane bending, bending 1B, 2B, 3B; and torsion 1 T, 2 T, 3 T modes for mild steel and RCC cantilever plates. It is noticed that vibration mode shapes are same for isotropic and RCC structures.

Modes	Isotropic			RCC
	Present		ANSYS	Present		ANSYS
	1 × 5	2 × 10		1 × 5	2 × 10
1	3.3954	3.3930	3.3906	3.3954	3.393	3.3691
2	21.2275	21.2151	21.2000	21.2275	21.2151	21.0788
3	33.3339	33.1739	32.8655	33.3339	33.1739	34.0947
4	39.1755	39.1446	39.1403	39.1755	39.1446	39.1478
5	59.4395	59.392	59.3469	59.4395	59.3920	58.9589
6	101.9307	101.3963	100.4799	101.9307	101.3963	103.9966
7	116.6200	116.3912	116.3008	116.6200	116.3912	115.3795
8	176.0709	175.0056	173.4994	176.0709	175.0056	178.8652

Table 3.

Non-dimensional fundamental frequency (${\varpi }_i={\omega }_i{b}^2\sqrt{\mathit{\rho h}/D}$) of an isotropic (ν = 0.3) and RCC ((ν = 0.2) cantilever plate (a = 12 m, b = 60 m and h = 1 m) considered as a ten story building.

Mode	1	2	3	4	5	6	7	8
Mild Steel	0.23673	1.4806	2.2944	2.7205	4.1456	7.0159	8.1275	12.119
RCC	0.16292	1.0197	1.6664	1.8929	2.8524	5.0816	5.5827	8.7343

Table 4.

Natural vibration frequencies of high rise building (considered as cantilever plate).

* all values are in Hz.

Figure 9.

Mode shapes of Isotropic and RCC cantilever plates.

From Figures 9 and 10 it is noticed that mode shapes of Isotropic and RCC flat panel structures is same.

Figure 10.

Schematic representation of deformation verses y-axis for Isotropic (ν = 0.3) and RCC (ν = 0.2) cantilever plates.

3.3 Large amplitude flexural vibration analysis

Next, Nonlinear vibration responses of cantilever isotropic (ν = 0.3) and RCC (ν = 0.2) structures is presented in Figures 11 and 12. It is observed that bending modes (1st B, 2nd B, 1st Out-off plane B, 2nd Torsion, 3rd Bending, 3rd Torsion, 4th Bending, 4th Torsion) gives hardening response whereas 1st Torsion mode gives softening effect. Qualitatively large-amplitude flexural vibration response of isotropic and RCC structures is similar.

Figure 11.

The frequency (ω_i, i = 1,9) verses non-dimensional deformation (w/h) of Isotropic cantilever plate.

Figure 12.

The frequency (ω_i, i = 1, 9) verses non-dimensional deformation (w/h) of RCC structure.

3.4 Fast Fourier Transform (FFT) analysis

Next, the fast Fourier transform (FFT) and phase portraits are shown in Figure 13 considering first three bending modes (1B, 2B and 3B). It is observed that non-dimensional transverse deflection (w/h) at tip of plate is maximum for first bending mode and amplitude (w/h) is reduces for higher order bending modes. Therefore, to design high-rise building first bending mode amplitude (w/h) should be minimum.

Figure 13.

Fast Fourier Transform and phase portrait for first thre`e bending modes of cantilever plate.

4. Summary

In this chapter, first Authors elaborated the finite element method based mathematical formulation considering first-order shear deformation theory. Also explained the stiffness matric and compliance matrix and relationship between three dimensional stresses and strains. Authors also explained the state of in-plane stresses and in-plane stains with mathematical formulation of isotropic and anisotropic materials. Here, authors also elaborated the transformation of stiffness matrixes and explained how scaler components are independent on coordinate system and vector as well as tensors are independent, but components of vectors and tensors are dependent on coordinate system.

The static and dynamic analysis of high rise building subjected to uniform pressure (q) and wind load (considered as triangular load as W = q × y/L) and combination of both (q/2 + (q/2) × y/L) has been performed eight node shell 281 element through commercial finite element-based software ANSYS.

The large amplitude flexural vibration responses of isotropic and RCC flat panels is also investigated considering first 9 modes. It is found that 1st Torsion mode gives the softening responses whereas remaining bending, out-off plane bending and higher order torsion mode fives hardening effect.

Fast Fourier transform shows the amplitude verses time response and phase portrait gives displacement verses velocity response, represents effect mode shapes to design safe structures.

Acknowledgments

I like thank to editors of Intech Open to provide opportunity to write a book chapter to Vibration Monitoring and Analysis - Recent Advances. I also like to thank to Department of Mechanical Engineering, MBM University to provide computational facility in computer lab to work finite element method based commercial software ANSYS. I am so much thankful to editors for reviewing this chapter and help me to improve it.

Conflict of interest

The author declare no conflict of interest.