**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* = 0.75*b* 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/m<sup>2</sup> ), wind (triangular load q *y*/b; here *q* = 1400 N/m2 ) load and combined uniformly distributed and wind load (q/2 + (q/2) *<sup>y</sup>*/b) of isotropic (*<sup>v</sup>* = 0.3, *<sup>E</sup>* = 210 GPa, *<sup>ρ</sup>* = 7800 kg/m<sup>3</sup> ) and RCC (*v* = 0.2, *E* = 30 GPa, *ρ* = 2500 kg/m<sup>3</sup> ) are obtained and presented in **Tables 1** and **2**, respectively.

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.

#### **Figure 4.**

*Analysis of model structure under uniformly distributed load.*


**Table 1.**

*Deformations* **δ** *(*qb*<sup>4</sup> /*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.*


**Table 2.**

*Deformations* δ *(*Wb*<sup>4</sup> /*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.*

#### **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 (*ε*2) 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 (*ε*2) 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 (*q*<sup>0</sup> = 1400 N/m<sup>2</sup> ).

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 (*q*<sup>0</sup> � *y*/*b*, *q*<sup>0</sup> = 1400 N/m2 ). It is observed that the distribution of longitudinal strain (*ε*xx) and second principle strain (*ε*2) 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)**.

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.

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/m<sup>3</sup> ) 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 (*ϖ<sup>i</sup>* <sup>¼</sup> *<sup>ω</sup>ib*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffi *ρh=D* p ) 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,

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

#### **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 σxy and (e) Second principle strain* ε*<sup>2</sup> 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/m<sup>3</sup> ) under uniformly distributed load (*q*<sup>0</sup> = 1400 N/m<sup>2</sup> ).*

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.

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

#### **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

#### **Figure 6.**

*Distribution of displacement and in-plane strains (εxx,* ε*xy) and principle strains (*ε*22,* ε*33) under wind load (*q*<sup>0</sup>* <sup>x</sup>*/*b; q0 *= 1400 N/m<sup>2</sup> ) on isotropic (*ν *= 0.3) cantilever plate.*

**Figure 7.**

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

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.

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

#### **Figure 8.**

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


#### **Table 3.**

*Non-dimensional fundamental frequency (ϖ<sup>i</sup>* <sup>¼</sup> *<sup>ω</sup>ib*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffi *ρh=D* p *) 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.*


#### **Table 4.**

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

**Figure 9.**

*Mode shapes of Isotropic and RCC cantilever plates.*

### **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 (1*B*, 2*B* and 3*B*). It is observed that nondimensional 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.

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

**Figure 10.** *Schematic representation of deformation verses y-axis for Isotropic (*ν *= 0.3) and RCC (*ν *= 0.2) cantilever plates.*

**Figure 11.** *The frequency (ωi,* i *= 1,9) verses non-dimensional deformation (*w*/*h*) of Isotropic cantilever plate.*

**Figure 12.** *The frequency (ωi,* i *= 1, 9) verses non-dimensional deformation (*w*/*h*) of RCC structure.*
