4. Column as a structure for transmitting system

### 4.1. Formulation of the undamped vibration problem

Assuming the well-known trigonometric function

Figure 6. Frequency of the beam with time with viscoelasticity. (a) Frequency with time for capacity of the section, (b)

Safety factor γ—Collapse at 90th day.

358 Numerical Simulations in Engineering and Science

$$\phi(\mathbf{x}) = 1 - \cos\left(\frac{\pi\mathbf{x}}{2L}\right),\tag{14}$$

which includes the self-weight of the column on considered part, and the lumped forces from

Analytical and Mathematical Analysis of the Vibration of Structural Systems Considering Geometric Stiffness…

with m0, ms are defined below, and g is the acceleration of gravity. The generalized mass is then given by M = m0 + m, where m0 is the lumped mass on the top and m is the generalized mass

where ms is the mass to each segment s, found by multiplying the cross-sectional area As to the density r of the material at the respective intervals, that is, ms, mass per unit length, and m, generalized mass of the system due the density of the material, with n as defined previously.

ð Þ rd=<sup>s</sup> <sup>∴</sup>f tðÞ¼ <sup>ω</sup>ð Þ<sup>t</sup>

taking into consideration that, for a compressive force being positive, the temporal stiffness is:

It is important to mention that Eq. (14) has been evaluated by [19] as a valid shape for the first mode of vibration with geometric nonlinear characteristics, applied for actual structures, even those with variable geometry, being a function valid throughout the entire domain of the

A 40-m-high reinforced concrete pole structure with an external 60-cm hollow circular crosssectional diameter, with variable thickness (Figure 9) and a slenderness ratio of 472 was used for analysis. The properties of the sections change along the length due to the changes in

The concrete used in the manufacture of the structure had the compression characteristic

The concrete cover c' specified for the reinforcing steel was 25 mm and the steel used in the construction of the pole was CA-50, with yield strength of 500 MPa and modulus of elasticity Es of 205 GPa. The secant modulus of elasticity of the concrete Ec is 31931.05 MPa. The numerical simulation was performed considering that all elastic parameters in Eq. (6) are equal

dx, withms <sup>¼</sup> As<sup>r</sup> and <sup>m</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>

N xð Þ¼ f g m<sup>0</sup> þ ms½ � ð Þ� Ls � Ls�<sup>1</sup> x g, (17)

s¼1

K tðÞ¼ K0ð Þ� t Kg: (20)

<sup>2</sup><sup>π</sup> ð Þ Hertz , (19)

s, and a density r of 2600 kg/m3

.

ms, (18)

http://dx.doi.org/10.5772/intechopen.75615

361

upper segments, or better

ms ¼

L ðs

ms <sup>ϕ</sup>ð Þ<sup>x</sup> � �<sup>2</sup>

Ls�<sup>1</sup>

The first natural frequency of the structure is calculated:

ωðÞ¼ t

ffiffiffiffiffiffiffiffiffi K tð Þ M

r

found with

structure.

4.2. Numerical simulation 2

thickness and variation of the steel area.

strength fck of 45 MPa, viscosity η<sup>1</sup> of 51089681149.92 MPa.

to the modulus of elasticity of the concrete, E0 = E1 = Ec.

where x is an independent variable of the problem originating in the base, in the cantilever position, and L is length of the column, q(t) is the generalized coordinate, and e(t) is the vertical displacement of the top due the vibratory movement, as shown in Figure 8. By using the Rayleigh method, in a similar way as described in the previous Section 3.2, the conventional stiffness is found by

$$k\_{0s}(t) = \int\_{L\_{s-1}}^{L} E(t) I\_s F\_s \left(\phi''\right)^2 d\mathbf{x}, \text{ with } K\_0(t) = \sum\_{s=1}^n k\_{0s}(t), \tag{15}$$

where kos(t) and Fs are the parcel of the stiffness and the homogenizing factor of the concrete cross section due to the reinforcement steel at the segment s. K0 is the final conventional stiffness, where n is the total number of intervals given by the structural geometry. E(t) represents the variable modulus of elasticity on time, according Eq. (6), and Is is the moment of inertia of each section.

The geometric stiffness is obtained by the following equation:

$$k\_{\mathfrak{g}s} = \int\_{L\_{s-1}}^{L\_s} N(\mathfrak{x}) \left(\phi''\right)^2 d\mathfrak{x}, \text{ with} \\ K\_{\mathfrak{g}} = \sum\_{s=1}^n k\_{\mathfrak{g}s\prime} \tag{16}$$

where kgs is the geometric stiffness at the interval s; Kg is the total geometric stiffness of the structure, with n as defined before; N(x) is a normal effort function at the respective interval,

Figure 8. Mathematical model of vibration of a column.

which includes the self-weight of the column on considered part, and the lumped forces from upper segments, or better

$$N(\mathbf{x}) = \{m\_0 + \overline{m}\_s[(L\_s - L\_{s-1}) - \mathbf{x}]\} \mathbf{g}\_{\prime} \tag{17}$$

with m0, ms are defined below, and g is the acceleration of gravity. The generalized mass is then given by M = m0 + m, where m0 is the lumped mass on the top and m is the generalized mass found with

$$m\_s = \int\_{L\_{s-1}}^{L\_s} \overline{m}\_s \left(\phi(\mathbf{x})\right)^2 d\mathbf{x}, \text{with} \,\overline{m}\_s = A\_s \rho \quad \text{and} \quad m = \sum\_{s=1}^n m\_{s\nu} \tag{18}$$

where ms is the mass to each segment s, found by multiplying the cross-sectional area As to the density r of the material at the respective intervals, that is, ms, mass per unit length, and m, generalized mass of the system due the density of the material, with n as defined previously. The first natural frequency of the structure is calculated:

$$
\omega(t) = \sqrt{\frac{\mathcal{K}(t)}{M}} (rd/s) . \mathbf{:} f(t) = \frac{\omega(t)}{2\pi} (Hert \mathbf{z}) . \tag{19}
$$

taking into consideration that, for a compressive force being positive, the temporal stiffness is:

$$K(t) = K\_0(t) - K\_\%. \tag{20}$$

It is important to mention that Eq. (14) has been evaluated by [19] as a valid shape for the first mode of vibration with geometric nonlinear characteristics, applied for actual structures, even those with variable geometry, being a function valid throughout the entire domain of the structure.

### 4.2. Numerical simulation 2

ϕð Þ¼ x 1 � cos

E tð ÞIsFs <sup>ϕ</sup><sup>00</sup> � �<sup>2</sup>

stiffness is found by

360 Numerical Simulations in Engineering and Science

of inertia of each section.

k0sðÞ¼ t

L ðs

Ls�<sup>1</sup>

The geometric stiffness is obtained by the following equation:

L ðs

N xð Þ <sup>ϕ</sup><sup>00</sup> � �<sup>2</sup>

Ls�<sup>1</sup>

kgs ¼

Figure 8. Mathematical model of vibration of a column.

where x is an independent variable of the problem originating in the base, in the cantilever position, and L is length of the column, q(t) is the generalized coordinate, and e(t) is the vertical displacement of the top due the vibratory movement, as shown in Figure 8. By using the Rayleigh method, in a similar way as described in the previous Section 3.2, the conventional

where kos(t) and Fs are the parcel of the stiffness and the homogenizing factor of the concrete cross section due to the reinforcement steel at the segment s. K0 is the final conventional stiffness, where n is the total number of intervals given by the structural geometry. E(t) represents the variable modulus of elasticity on time, according Eq. (6), and Is is the moment

where kgs is the geometric stiffness at the interval s; Kg is the total geometric stiffness of the structure, with n as defined before; N(x) is a normal effort function at the respective interval,

πx 2L � �

dx, withK0ðÞ¼ <sup>t</sup> <sup>X</sup><sup>n</sup>

dx, withKg <sup>¼</sup> <sup>X</sup><sup>n</sup>

s¼1

s¼1

, (14)

k0sð Þt , (15)

kgs, (16)

A 40-m-high reinforced concrete pole structure with an external 60-cm hollow circular crosssectional diameter, with variable thickness (Figure 9) and a slenderness ratio of 472 was used for analysis. The properties of the sections change along the length due to the changes in thickness and variation of the steel area.

The concrete used in the manufacture of the structure had the compression characteristic strength fck of 45 MPa, viscosity η<sup>1</sup> of 51089681149.92 MPa. s, and a density r of 2600 kg/m3 . The concrete cover c' specified for the reinforcing steel was 25 mm and the steel used in the construction of the pole was CA-50, with yield strength of 500 MPa and modulus of elasticity Es of 205 GPa. The secant modulus of elasticity of the concrete Ec is 31931.05 MPa. The numerical simulation was performed considering that all elastic parameters in Eq. (6) are equal to the modulus of elasticity of the concrete, E0 = E1 = Ec.

Figure 9. A column as a mast transition system.

The structure also has an array of antennas and accessories, such as a platform, stairs, cables, and mats (characteristics shown in Table 1), which exert compressive forces. It is important to mention that the viscous parameter was adjusted so that the deformations converged at 90 days (Figure 10(a)), as observed by [8]. With this, it was possible to obtain the variable modulus of elasticity E(t) (Figure 10(b)). The gravitational (g) acceleration was assumed to be 9.806650 m/s2 .

Since this is a cylindrical concrete reinforcement structure, it is necessary to take into account the presence of reinforcement bars at the moment of inertia of the cross-sectional area, which must be done by homogenizing the concrete area. Considering a circular ring cross section with external diameter D; thickness of the wall ts; s relative to the considerate segment of the structure; a reinforcement bar bi any occupies a position i in the section defined by Rbi and θi,

as shown in Figure 11. Rbi determines the center position of each bar in relation to the section

Analytical and Mathematical Analysis of the Vibration of Structural Systems Considering Geometric Stiffness…

<sup>2</sup> � <sup>c</sup> 0 � dbi

As θ<sup>i</sup> is the independent variable, the distance between the center of each bar relative to the

The spacing between the center of each bar section was obtained for sp. = 2πRbi/nbi, where nbi is the number of bars of the reinforcement steel. The angular phase shift between them is Δθ = sp/Rbi.

<sup>2</sup> : (21)

http://dx.doi.org/10.5772/intechopen.75615

363

yð Þ¼ θ<sup>i</sup> senð Þ θ<sup>i</sup> Rbi: (22)

center. c' is the concrete cover of the reinforcement and dbi is the diameter of the i bar.

Figure 10. Deformation and modulus of elasticity over time. (a) Deformation, (b) Elasticity.

Rbi <sup>¼</sup> <sup>D</sup>

axis center of inertia of the section is


Table 1. Structure's characteristics and devices.

Analytical and Mathematical Analysis of the Vibration of Structural Systems Considering Geometric Stiffness… http://dx.doi.org/10.5772/intechopen.75615 363

Figure 10. Deformation and modulus of elasticity over time. (a) Deformation, (b) Elasticity.

The structure also has an array of antennas and accessories, such as a platform, stairs, cables, and mats (characteristics shown in Table 1), which exert compressive forces. It is important to mention that the viscous parameter was adjusted so that the deformations converged at 90 days (Figure 10(a)), as observed by [8]. With this, it was possible to obtain the variable modulus of elasticity E(t) (Figure 10(b)). The gravitational (g) acceleration was assumed to be 9.806650 m/s2

Figure 9. A column as a mast transition system.

362 Numerical Simulations in Engineering and Science

Table 1. Structure's characteristics and devices.

Since this is a cylindrical concrete reinforcement structure, it is necessary to take into account the presence of reinforcement bars at the moment of inertia of the cross-sectional area, which must be done by homogenizing the concrete area. Considering a circular ring cross section with external diameter D; thickness of the wall ts; s relative to the considerate segment of the structure; a reinforcement bar bi any occupies a position i in the section defined by Rbi and θi,

Dispositive Height Weight Pole from 0 to 40 m 25.48 kN/m3 Stair from 0 to 40 m 0.15 kN/m Cables from 0 to 40 m 0.25 kN/m Platform and supports 40 m 4.90 kN Antennas 40 m 1.88 kN

.

as shown in Figure 11. Rbi determines the center position of each bar in relation to the section center. c' is the concrete cover of the reinforcement and dbi is the diameter of the i bar.

$$R\_{bi} = \frac{D}{2} - \mathbf{c'} - \frac{d\_{bi}}{2} \,. \tag{21}$$

As θ<sup>i</sup> is the independent variable, the distance between the center of each bar relative to the axis center of inertia of the section is

$$y(\theta\_i) = \text{sen}(\theta\_i) R\_{\text{bi}}.\tag{22}$$

The spacing between the center of each bar section was obtained for sp. = 2πRbi/nbi, where nbi is the number of bars of the reinforcement steel. The angular phase shift between them is Δθ = sp/Rbi.

Figure 11. Parameters for homogenizing concrete section.

Since the θ<sup>i</sup> varies from 0 to 2π at intervals defined by Δθ, the total inertia of the steel bars in relation to the section of the structure could be obtained by the theorem of parallel axes with the expression (14).

$$I\_s = \sum\_{\theta}^{2n} \left( \frac{\pi d\_{bi}^4}{64} + y(\theta\_i)^2 \frac{\pi d\_{bi}}{4} \right). \tag{23}$$

Height Ls (m) External diameter D (cm) Thickness ts (cm) Number of bar (nbi) Bar diameter dbi (mm) Factors of homogenizing Fs

Analytical and Mathematical Analysis of the Vibration of Structural Systems Considering Geometric Stiffness…

http://dx.doi.org/10.5772/intechopen.75615

365

40 60 10 20 13 1.0963

31 60 13 20 13 1.0869 30 60 12 15 16 1.0995 29 60 11 15 16 1.1029

 60 11 16 16 1.1091 60 11 17 16 1.1153 60 11 18 16 1.1214 60 11 19 16 1.1274 60 11 20 16 1.1334 60 14 20 16 1.1230 60 15 15 20 1.1374 60 16 15 20 1.1354 60 13 16 20 1.1512

15 60 13 17 20 1.1594 14 60 13 18 20 1.1675 13 60 13 19 20 1.1755

 60 13 20 20 1.1833 60 13 22 20 1.1987 60 16 22 20 1.1889 60 16 15 25 1.1961 60 17 15 25 1.194 60 14 16 25 1.2132

4 60 14 17 25 1.2241

1 60 18 17 25 1.2136

 60 10 20 13 60 10 20 13 60 10 20 13 60 10 20 13 60 10 20 13 60 10 20 13 60 10 20 13 60 10 20 13

28 60 11 15 16 27 60 11 15 16 26 60 11 15 16

16 60 13 16 20

12 60 13 19 20

5 60 14 16 25

3 60 14 17 25 2 60 14 17 25

0 60 18 17 25

Table 2. Structural properties and homogenizing factors of sections.

The homogenized moment of inertia of the steel bars will thus be:

$$I\_{\rm schom} = \sum\_{\theta}^{2\pi} I(\theta\_i) \left(\frac{E\_s}{E\_c} - 1\right). \tag{24}$$

The total homogenized inertia of the section will be obtained by Itot =I+Ishom, with I being the inertia of the circular section, I = π/64 [D<sup>4</sup> - (D - 2ts) 4 ]. Thus, to find a factor Fs, which multiplies the nominal inertia of the section in terms of total steel inertia, the homogenized section is made by Fs =1+(Ishom/Itot). Factors of homogenizing, the structural properties and the geometry of the structure are shown in Table 2.

Considering that the actual structure has variable proprieties along the height, the expressions (16), (17) and (18) must be resolved for each interval defined by structural geometry. The frequency was calculated for the 90th day according to Eq. (10) (see Figure 12).

Figure 13(a) shows the structural frequency over time, for a height limit of losing stability, calculated for the 90th day, considering viscoelasticity (L = 50.6975 m). To a height of 57.0000 m, for example, the behavior of Figure 13(b) is found, with the structural collapse occurring on the 60th day. The height limit without viscoelasticity (instant 0) is 71.29 m, a frequency of 0.0000 Hz.

Similar simulations for evaluation of the viscoelasticity can be found in [20, 21].

Analytical and Mathematical Analysis of the Vibration of Structural Systems Considering Geometric Stiffness… http://dx.doi.org/10.5772/intechopen.75615 365


Table 2. Structural properties and homogenizing factors of sections.

Since the θ<sup>i</sup> varies from 0 to 2π at intervals defined by Δθ, the total inertia of the steel bars in relation to the section of the structure could be obtained by the theorem of parallel axes with the

> <sup>2</sup> πdbi 2 4

: (23)

: (24)

]. Thus, to find a factor Fs, which multiplies

� �

Es Ec � 1 � �

πdbi 4 <sup>64</sup> <sup>þ</sup> <sup>y</sup>ð Þ <sup>θ</sup><sup>i</sup>

θ

Ið Þ θ<sup>i</sup>

The total homogenized inertia of the section will be obtained by Itot =I+Ishom, with I being the

the nominal inertia of the section in terms of total steel inertia, the homogenized section is made by Fs =1+(Ishom/Itot). Factors of homogenizing, the structural properties and the geom-

Considering that the actual structure has variable proprieties along the height, the expressions (16), (17) and (18) must be resolved for each interval defined by structural geometry. The

Figure 13(a) shows the structural frequency over time, for a height limit of losing stability, calculated for the 90th day, considering viscoelasticity (L = 50.6975 m). To a height of 57.0000 m, for example, the behavior of Figure 13(b) is found, with the structural collapse occurring on the 60th day. The height limit without viscoelasticity (instant 0) is 71.29 m, a

4

Is <sup>¼</sup> <sup>X</sup> 2π

The homogenized moment of inertia of the steel bars will thus be:

inertia of the circular section, I = π/64 [D<sup>4</sup> - (D - 2ts)

Figure 11. Parameters for homogenizing concrete section.

364 Numerical Simulations in Engineering and Science

etry of the structure are shown in Table 2.

frequency of 0.0000 Hz.

θ

Ishom <sup>¼</sup> <sup>X</sup> 2π

frequency was calculated for the 90th day according to Eq. (10) (see Figure 12).

Similar simulations for evaluation of the viscoelasticity can be found in [20, 21].

expression (14).

• The effect of geometric stiffness produced by the horizontal loading and the corresponding possibility of introducing resonant regimes in the structural support system were demon-

Analytical and Mathematical Analysis of the Vibration of Structural Systems Considering Geometric Stiffness…

http://dx.doi.org/10.5772/intechopen.75615

367

• It can be concluded, therefore, that due to the increase of the axial compressive force, resonance conditions can occur, as represented by the intersection of the curves in Figure 7. In the present study, resonance occurs when the axial compression force reaches 210 kN at

• Since the force of post-tension decreases the stiffness of the beam, this can lead to the resonant regime if it has not been previously evaluated in the structural analysis.

• The technique studied in this chapter offers an efficient tool to provide the removal of the support structure of that unwanted regime, avoiding the production of harmful effects on

• In further work, it is necessary to introduce normative criteria, perform experimental activity, and evaluate the influence of the prestressing bar stiffness on the structural response.

• The modulus of elasticity calculated by Eq. (6) on the ninth day was 16027.64 MPa, which represents a decrease of 49.81% in relation to the initial value of 31931.05 MPa.

• The frequency of the structure calculated at the initial moment was 0.215715 Hz, and on

• The simulated structure finds its limit of stability when reaching 50.6975 m, collapsing at 90 days. If the viscoelastic effect were not considered, the height limit would be 71.29 m, 20.47% above the first one. The obtained result was taken for an exactitude of five decimal

• The previous aspect is relevant because if the height were considered between the limit established without the viscoelasticity and that defined with it, the structure would collapse before the end of 90 days in service. For a height of 57 m, for example, the

• Others rheological models for representing viscoelastic behavior can be tested in order to evaluate the frequency of a column of reinforced concrete as well as criteria from regula-

• The critical load of buckling can be obtained by using the same process present in this work and comparing it to other tools for calculations as, for example, finite element

This work was supported by the National Council for Scientific and Technological Development (CNPq) from Brazil by process 443,044/2014–7 in Call MCTI/CNPQ/Universal 14/2014.

the 90th day, of 0.135021 Hz, representing a reduction of 37.41%.

the equipment, fabricated products, and work environment of the operators.

strated by calculating their frequencies.

significant algorisms (f = 0.00000).

collapse would occur 60 days after being loaded.

Simulation 2

tory codes.

method (FEM).

Acknowledgements

10 days. Other instants might also be considered.

Figure 12. Frequency variation on structure at 90 days.

Figure 13. Structural frequency on height of losing stability. (a) L = 50.6975 m—Collapse on the 90th day, (b) L = 57.0000 m —Collapse on the 60th day.
