**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:

$$\begin{bmatrix} \sigma^T e(\delta)dV - \int\_V \delta^T f dV - \int\_S \left[ \begin{matrix} \left( n\_x \sigma\_{\mathbf{x}} + n\_y \tau\_{\mathbf{xy}} + n\_x \tau\_{\mathbf{xz}} \right) \delta\_{\mathbf{x}} \\ + \left( n\_x \tau\_{\mathbf{xy}} + n\_y \sigma\_{\mathbf{y}} + n\_x \tau\_{\mathbf{yz}} \right) \delta\_{\mathbf{y}} \\ + \left( n\_x \tau\_{\mathbf{xx}} + n\_y \tau\_{\mathbf{yz}} + n\_x \sigma\_{\mathbf{z}} \right) \delta\_{\mathbf{z}} \end{matrix} \right] dS \tag{1}$$

*Perspective Chapter: Dynamic Analysis of High-Rise Buildings Using Simplified Numerical… DOI: http://dx.doi.org/10.5772/intechopen.108556*

Here, *<sup>σ</sup>* represents six independent component of stress *<sup>σ</sup>* <sup>¼</sup> *<sup>σ</sup>x*, *<sup>σ</sup>y*, *<sup>σ</sup>z*, *<sup>τ</sup>yz*, *<sup>τ</sup>xz*, *<sup>τ</sup>xy* � �*<sup>T</sup>* normal stresses and shear stresses; *ε* represents six strains corresponding to stresses *<sup>ε</sup>* <sup>¼</sup> *<sup>ε</sup>x*, *<sup>ε</sup>y*, *<sup>ε</sup>z*, *<sup>γ</sup>yz*, *<sup>γ</sup>xz*, *<sup>γ</sup>xy* h i*<sup>T</sup>* ; *dV* is volume integration *dV* = *dx dy dz;* distributed force per unit volume *f* ¼ *f <sup>x</sup>*, *f <sup>y</sup>*, *f <sup>z</sup>* h i*<sup>T</sup>* ; displacement vector is *δ* ¼ *ux*, *uy*, *uz*, *θx*, *θy*, *θ<sup>y</sup>* � �*<sup>T</sup>* ; the unit normal to surface *dA* is *n* ¼ *nx*, *ny*, *nz* � �*<sup>T</sup>* .

#### **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. Oneto-one relationship (based on generalized Hooke's law) between the state of stress and state of strain will be written as:

$$\begin{Bmatrix} \sigma\_{11} \\ \sigma\_{22} \\ \sigma\_{33} \\ \sigma\_{23} \\ \sigma\_{13} \\ \sigma\_{12} \\ \sigma\_{12} \end{Bmatrix} = \begin{bmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} & \mathbf{C}\_{13} & \mathbf{C}\_{14} & \mathbf{C}\_{15} & \mathbf{C}\_{16} \\ \mathbf{C}\_{21} & \mathbf{C}\_{22} & \mathbf{C}\_{23} & \mathbf{C}\_{24} & \mathbf{C}\_{25} & \mathbf{C}\_{26} \\ \mathbf{C}\_{31} & \mathbf{C}\_{32} & \mathbf{C}\_{33} & \mathbf{C}\_{34} & \mathbf{C}\_{35} & \mathbf{C}\_{36} \\ \mathbf{C}\_{41} & \mathbf{C}\_{42} & \mathbf{C}\_{43} & \mathbf{C}\_{44} & \mathbf{C}\_{45} & \mathbf{C}\_{46} \\ \mathbf{C}\_{51} & \mathbf{C}\_{52} & \mathbf{C}\_{53} & \mathbf{C}\_{54} & \mathbf{C}\_{55} & \mathbf{C}\_{56} \\ \mathbf{C}\_{61} & \mathbf{C}\_{62} & \mathbf{C}\_{63} & \mathbf{C}\_{64} & \mathbf{C}\_{65} & \mathbf{C}\_{66} \end{bmatrix} \begin{Bmatrix} \varepsilon\_{11} \\ \varepsilon\_{21} \\ \varepsilon\_{31} \\ \varepsilon\_{33} \\ \varepsilon\_{32} \\ \varepsilon\_{13} \\ \varepsilon\_{12} \end{Bmatrix} \tag{2}$$

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:

$$\begin{Bmatrix} \sigma\_{11} \\ \sigma\_{22} \\ \sigma\_{33} \\ \sigma\_{23} \\ \sigma\_{13} \\ \sigma\_{12} \\ \sigma\_{12} \end{Bmatrix} = \begin{bmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} & \mathbf{C}\_{13} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{C}\_{21} & \mathbf{C}\_{22} & \mathbf{C}\_{23} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{C}\_{31} & \mathbf{C}\_{32} & \mathbf{C}\_{33} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C}\_{44} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C}\_{55} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C}\_{66} \end{bmatrix} \begin{Bmatrix} \boldsymbol{e}\_{11} \\ \boldsymbol{e}\_{22} \\ \boldsymbol{e}\_{33} \\ \boldsymbol{e}\_{23} \\ \boldsymbol{e}\_{13} \\ \boldsymbol{e}\_{12} \end{Bmatrix} \tag{3}$$

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

*ε*11 *ε*22 *ε*33 *ε*23 *ε*13 *ε*12 8 >>>>>>>>< >>>>>>>>: 9 >>>>>>>>= >>>>>>>>; ¼ *S*<sup>11</sup> *S*<sup>12</sup> *S*<sup>13</sup> 000 *S*<sup>21</sup> *S*<sup>22</sup> *S*<sup>23</sup> 000 *S*<sup>31</sup> *S*<sup>32</sup> *S*<sup>33</sup> 000 000 *S*<sup>44</sup> 0 0 0000 *S*<sup>55</sup> 0 00000 *S*<sup>66</sup> 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 *σ*<sup>11</sup> *σ*<sup>22</sup> *σ*<sup>33</sup> *σ*<sup>23</sup> *σ*<sup>13</sup> *σ*<sup>12</sup> 8 >>>>>>>>< >>>>>>>>: 9 >>>>>>>>= >>>>>>>>; (4) *ε*11 *ε*22 *ε*33 *ε*23 *ε*13 *ε*12 8 >>>>>>>>< >>>>>>>>: 9 >>>>>>>>= >>>>>>>>; ¼ 1 *E*1 � *<sup>ν</sup>*<sup>21</sup> *E*2 � *<sup>ν</sup>*<sup>31</sup> *E*3 000 � *<sup>ν</sup>*<sup>12</sup> *E*1 1 *E*2 � *<sup>ν</sup>*<sup>32</sup> *E*3 000 � *<sup>ν</sup>*<sup>13</sup> *E*1 � *<sup>ν</sup>*<sup>23</sup> *E*2 1 *E*3 000 <sup>000</sup> <sup>1</sup> *G*<sup>23</sup> 0 0 0 0 00 <sup>1</sup> *G*<sup>13</sup> 0 0 0 0 00 <sup>1</sup> *G*<sup>12</sup> 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 *σ*<sup>11</sup> *σ*<sup>22</sup> *σ*<sup>33</sup> *σ*<sup>23</sup> *σ*<sup>13</sup> *σ*<sup>12</sup> 8 >>>>>>>>< >>>>>>>>: 9 >>>>>>>>= >>>>>>>>; (5)

Here, [S]6 � <sup>6</sup> denotes the compliance coefficients; [S] = [C]�<sup>1</sup> ; *E*1, *E*2, *E*<sup>3</sup> 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 *j*th direction to the axial strain in *i*th direction when load is applied along longitudinal direction or stressed in *i*th direction; *G*12, *G*13, and *G*<sup>23</sup> 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: *<sup>ν</sup>*<sup>21</sup> *<sup>E</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ν</sup>*<sup>12</sup> *<sup>E</sup>*<sup>1</sup> ; *<sup>ν</sup>*<sup>31</sup> *<sup>E</sup>*<sup>3</sup> <sup>¼</sup> *<sup>ν</sup>*<sup>13</sup> *<sup>E</sup>*<sup>1</sup> ; *<sup>ν</sup>*<sup>32</sup> *<sup>E</sup>*<sup>3</sup> <sup>¼</sup> *<sup>ν</sup>*<sup>23</sup> *<sup>E</sup>*<sup>2</sup> *or <sup>ν</sup>ij Ei* <sup>¼</sup> *<sup>ν</sup>ji Ej* (no sum on *i*, *j*).

Here *i*, *j* = 1,2,3. Hence, there are only nine independent material coefficients (*E*1, *E*2, *E*3, *G*12, *G*13, *G*23, *ν*12, *ν*13, *ν*23) for an orthotropic material. For an isotropic material (material having infinite number of planes of material symmetry) independent elastic coefficients are reduced to two (*E*<sup>1</sup> = *E*<sup>2</sup> = *E*<sup>3</sup> = *E*, *ν*<sup>12</sup> = *ν*<sup>13</sup> = *ν*<sup>23</sup> = *ν*, *G*<sup>12</sup> = *G*<sup>13</sup> = *G*<sup>23</sup> =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:

$$\begin{aligned} \left\{ \begin{array}{cccc} \varepsilon\_{1} \\ & \varepsilon\_{2} \\ & \varepsilon\_{3} \\ \varepsilon\_{12} = \varepsilon\_{6} \end{array} \right\} &= \begin{bmatrix} \mathbf{S}\_{11} & \mathbf{S}\_{12} & \mathbf{0} \\ \mathbf{S}\_{12} & \mathbf{S}\_{22} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{S}\_{66} \end{bmatrix} \begin{Bmatrix} \sigma\_{1} \\ \sigma\_{2} \\ \sigma\_{12} \\ \sigma\_{12} = \sigma\_{6} \end{Bmatrix} \\ &= \begin{bmatrix} \mathbf{1}/E\_{1} & -\nu\_{21}/E\_{2} & \mathbf{0} \\ -\nu\_{12}/E\_{1} & \mathbf{1}/E\_{2} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{1}/G\_{12} \end{bmatrix} \begin{Bmatrix} \sigma\_{1} \\ \sigma\_{2} \\ \sigma\_{2} \\ \sigma\_{12} = \sigma\_{6} \end{Bmatrix} \end{aligned} \tag{6}$$

*Perspective Chapter: Dynamic Analysis of High-Rise Buildings Using Simplified Numerical… DOI: http://dx.doi.org/10.5772/intechopen.108556*

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

$$\begin{Bmatrix} \sigma\_1 \\ \sigma\_2 \\ \sigma\_{12} = \sigma\_6 \end{Bmatrix} = \begin{bmatrix} Q\_{11} & Q\_{12} & \mathbf{0} \\ Q\_{12} & Q\_{22} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & Q\_{66} \end{bmatrix} \begin{Bmatrix} \varepsilon\_1 \\ \varepsilon\_2 \\ \varepsilon\_{12} = \varepsilon\_6 \end{Bmatrix} \tag{7}$$

Here, *Q*ij is known as plane stress-reduced stiffness, are expressed by:

$$Q\_{11} = \frac{S\_{22}}{S\_{11}S\_{22} - S\_{12}^2} = \frac{E\_1}{1 - \nu\_{12}\nu\_{21}}$$

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

$$Q\_{22} = \frac{S\_{11}}{S\_{11}S\_{22} - S\_{12}^2} = \frac{E\_2}{1 - \nu\_{12}\nu\_{21}}\tag{8}$$

$$Q\_{66} = \frac{1}{S\_{66}} = G\_{12}$$

Thus, reduced stiffness involved four independent material constants *E*1, *E*2, *ν*12, *G*12.

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

$$\begin{Bmatrix} \sigma\_{23} = \sigma\_4\\ \sigma\_{13} = \sigma\_5 \end{Bmatrix} = \begin{bmatrix} Q\_{44} & \mathbf{0} \\ \mathbf{0} & Q\_{55} \end{bmatrix} \begin{Bmatrix} \varepsilon\_{23} = \varepsilon\_4\\ \varepsilon\_{13} = \varepsilon\_5 \end{Bmatrix} \tag{9}$$

$$\text{Here, } Q\_{44} = C\_{44} = G\_{23} \text{ and } Q\_{55} = C\_{55} = G\_{13}.$$

#### **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 (*x*1, *x*2, *x*3) and unbarred (*x*1, *x*2, *x*3) coordinate systems are related as shown in **Figure 2** and written in Eqs. (10) and (11).

$$\begin{aligned} \mathfrak{x}\_1 &= \mathfrak{x}\_1(\overline{\mathfrak{x}}\_1, \overline{\mathfrak{x}}\_2, \overline{\mathfrak{x}}\_3) \\ \mathfrak{x}\_2 &= \mathfrak{x}\_2(\overline{\mathfrak{x}}\_1, \overline{\mathfrak{x}}\_2, \overline{\mathfrak{x}}\_3) \\ \mathfrak{x}\_3 &= \mathfrak{x}\_3(\overline{\mathfrak{x}}\_1, \overline{\mathfrak{x}}\_2, \overline{\mathfrak{x}}\_3) \end{aligned} \tag{10}$$

Inverse relations are written as:

$$\begin{aligned} \overline{\boldsymbol{\varpi}}\_1 &= \overline{\boldsymbol{\varpi}}\_1(\boldsymbol{\varpi}\_1, \boldsymbol{\varpi}\_2, \boldsymbol{\varpi}\_3) \\ \overline{\boldsymbol{\varpi}}\_2 &= \overline{\boldsymbol{\varpi}}\_2(\boldsymbol{\varpi}\_1, \boldsymbol{\varpi}\_2, \boldsymbol{\varpi}\_3) \\ \overline{\boldsymbol{\varpi}}\_3 &= \overline{\boldsymbol{\varpi}}\_3(\boldsymbol{\varpi}\_1, \boldsymbol{\varpi}\_2, \boldsymbol{\varpi}\_3) \end{aligned} \tag{11}$$

**Figure 2.** *Unbarred and barred rectangular coordinate system.*

#### **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).

$$\mathbf{C}\_{ijkl} = a\_{im} a\_{jn} a\_{kp} a\_{lq} \mathbf{C}\_{mnpq} \tag{12}$$

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 *Qij* in the reference coordinate system is written as:

$$\overline{\Omega}\_{11} = Q\_{11}\cos^4\theta + 2(Q\_{12} + 2Q\_{d6})\sin^2\theta\cos^2\theta + Q\_{22}\sin^4\theta$$

$$\overline{\Omega}\_{12} = (Q\_{11} + Q\_{22} - 4Q\_{66})\sin^2\theta\cos^2\theta + Q\_{32}(\sin^4\theta + \cos^4\theta)$$

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

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

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

$$\overline{\Omega}\_{44} = (Q\_{41} + Q\_{22} - 2Q\_{42} - 2Q\_{66})\sin^2\theta\cos^2\theta + Q\_{66}(\sin^4\theta + \cos^4\theta)$$

$$\overline{\Omega}\_{46} = Q\_{44}\cos^2\theta + Q\_{55}\sin^2\theta$$

$$\overline{\Omega}\_{48} = (Q\_{55} - Q\_{44})\cos\theta\sin\theta$$

$$z = x\_{\overline{x}}$$

$$z = z\_{\overline{x}}$$

$$z = z\_{\overline{x}}$$

**Figure 3.** *Rectangular plate with global and material coordinate systems.*

*Perspective Chapter: Dynamic Analysis of High-Rise Buildings Using Simplified Numerical… DOI: http://dx.doi.org/10.5772/intechopen.108556*

#### **2.4 Governing equations**

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

$$\prod = \frac{1}{2} \int\_{V} \sigma^{T} \epsilon dV - \int\_{V} \delta^{T} f dV - \int\_{S} \delta^{T} T dS - \sum\_{i} \delta\_{i}^{T} P\_{i} \tag{14}$$

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:

$$[K\_L + K\_{NL1}(\delta) + K\_{NL2}(\delta, \ \delta)] \{\delta\} = \{F\_P\} + \{F\_{wind}\} \tag{15}$$

Here, [*K*L] is linear stiffness matrix; [*KN*L] is nonlinear stiffness matrix; {*δ*} is displacement vector represents six degree of freedom three displacements *u*x, *u*y, *u*<sup>z</sup> and three rotations *θ*x, *θ*y, *θ*<sup>z</sup> 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:

$$[\mathbf{M}]\{\ddot{\delta}\} + [\mathbf{K}\_L + \mathbf{K}\_{\text{NL1}}(\delta) + \mathbf{K}\_{\text{NL2}}(\delta, \ \delta)]\{\delta\} = \{F\_p\} + \{F\_{wind}\} \tag{16}$$

Here, [*M*] is mass matrix; €*δ* � � 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:

$$[\mathbf{M}]\{\ddot{\delta}\} + [\mathbf{K}\_L + \mathbf{K}\_{\text{NL1}}(\delta) + \mathbf{K}\_{\text{NL2}}(\delta, \ \delta)]\{\delta\} = \mathbf{0} \tag{17}$$

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

$$\left\{-\left[M\right]\alpha^{2} + \left[K\_{L} + K\_{\text{NL1}}(\delta) + K\_{\text{NL2}}(\delta, \ \delta)\right]\right\}\{\delta\} = \mathbf{0} \tag{18}$$

Here, *ω* is the natural frequency of high-rise building represented as an Eigenvalues; {*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 *δ* ¼ *δ* max sin *θt*, the weighted residual of Eq. (16) with {FP + Fwind = 0} along the path *t* = 0 to *T*/4 (*δ* = 0 to *δ max*) may be expressed as:

$$\int\_{0}^{T/4} R\sin\theta t dt = \{0\}\tag{19}$$

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

$$\{R\} = \left[K\_L + K\_{NL1}(\delta\_{\max})\sin\theta t + K\_{NL2}(\delta\_{\max}, \ \delta\_{\max})\sin^2\theta t - \theta^2 M\right] \{\delta\_{\max}\} \sin\theta t\tag{20}$$

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

$$\left[K\_L + \frac{4}{3\pi}K\_{\rm NL1}(\delta\_{\rm max}) + \frac{3}{4}K\_{\rm NL2}(\delta\_{\rm max}, \ \delta\_{\rm max}) - \theta^2\mathcal{M}\right] \{\delta\_{\rm max}\} = \{0\} \tag{21}$$

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**

*Perspective Chapter: Dynamic Analysis of High-Rise Buildings Using Simplified Numerical… DOI: http://dx.doi.org/10.5772/intechopen.108556*
