3.1. Basic considerations prestressing in reinforced concrete

A piece can be considered as prestressed reinforced concrete when it is subjected to the action of the so-called prestressing forces and of permanent and variable loads, so that the concrete is not subjected to tension or it occurs below the limit of its resistance. As an example, take the normal stresses beam diagrams of the prestressed beam of Figure 2, where P is the prestressing force, MP the bending moment due to eccentricity of the load P, Mp is the bending moment due to uniformly distributed load p and R is the resultant, each one of these with their corresponding normal stresses. Under the conditions presented, the lower fibers of the beam, under positive bending moment, will have the tension stresses overturned by the superposition of those produced by the normal stress of the applied stress eccentrically.

Prestressed concrete was developed scientifically from the beginning of the last century. Prestressing can be defined as the artifice of introducing, in a structure or a part, a previous state of stresses, in order to improve its resistance or its behavior in service, under the action of

Figure 2. Normal stresses in a prestressed beam.

several effects. Due to the characteristics of the concrete as a structural material, the use of prestressing can bring a great advantage from the economic point of view. When comparing the cost of a prestressed structure with a similar one of conventional reinforced concrete, there is a reduction in the final cost of the structure due to the reduction of steel reinforcement [11]. In addition, the prestressing allows the part to overcome large spans, improves the control and reduction of deformations and fissures. It can also be used for structural recovery and reinforcement, as well as for slender systems and prefabricated or precast parts. There are three types of prestressing systems: (1) prestressing with initial adherence, (2) prestressing with posterior adherence, and (3) prestressing without adherence. The latter type is composed of a post-tensioning system characterized by the slipping freedom of the steel reinforcement in relation to the concrete, along the whole extension of the cable, except for the anchorages.

the engine axis and the part is initially ignored. The vertical displacement of the central joint is

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

By using the Rayleigh method [14], the undamped vibration frequency in its first mode considering viscoelasticity is obtained. It is worth mentioning that Rayleigh assumed that a system containing infinite degrees of freedom could be associated with another with a single degree of freedom (SDOF) to approximate its frequency. It is important to note that the technique developed by Rayleigh aimed at calculating the fundamental vibration frequency of elastic systems, as

The basic concept of the method is the principle of energy conservation, and can, therefore, be applied to linear and nonlinear structures. [15] applies the Rayleigh technique to determine the fundamental period of vibration to verify the stability of mechanical systems. The process is described in relation to the principle of virtual works and as the appropriate choice of the generalized coordinate describing the first mode of vibration. At the end of the process, the generalized properties of the system are obtained as stiffness and mass, necessary for the

Consider that the vertical displacement of a generic section of the beam in Figure 4 is given by:

in which ϕ(x) is a shape function that attempts to define the boundary conditions in the supports and value 1 in the central section of the beam, whose displacement with time is q(t). In this case, one adopts the shape function ϕ(x) = sin (πx/L), which is the exact solution of the problem without the P load. A prime mark will denote a derivative of the function in relation

Applying the Rayleigh method, one has the conventional bending stiffness, K0, as a function of

E tð Þ<sup>I</sup> <sup>ϕ</sup><sup>00</sup> � �<sup>2</sup>

where E(t)I is the known flexural bending with viscoelasticity, represented by multiplication of the temporal material modulus of elasticity with the inertia of the section in relation to the considered movement, the vertical vibration mode (1st mode). In turn, the geometric stiffness,

ϕ<sup>0</sup> � �<sup>2</sup>

The total generalized mass of the system is found by calculating M=MC + MV where MC is the concentrated mass at the middle span and MV is the mass coming from the beam self-weight

dx <sup>¼</sup> <sup>P</sup>π<sup>2</sup>

KG, as a function of the normal force of compression (or even tension), is equivalent to:

ð L

0

dx <sup>¼</sup> <sup>π</sup><sup>4</sup>E tð Þ<sup>I</sup>

the material behavior and the geometry of the cross, which is equivalent to:

ð L

0

KG ¼ P

K0ðÞ¼ t

v xð Þ¼ ; t ϕð Þx q tð Þ, (7)

<sup>2</sup>L<sup>3</sup> (8)

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<sup>2</sup><sup>L</sup> (9)

its precision is dependent on the function chosen to represent this mode of vibration.

the generalized coordinate of the system.

calculation of the frequency.

to x (Lagrange's notation).

given by:

In a non-adherent prestressing, the cables or chutes are wrapped in two or three layers of resistant paper. The wires and paper are painted with bituminous paint in order to tension them after the concrete has hardened. The bitumen avoids the penetration of the cement cream inside the cable and, in this way, it eliminates adhesion between the concrete and the reinforcement [12]. The prestressed concrete is a composite material of the aggregate mixture and a cement paste associated with prestressing cables and/or passive reinforcing bars. Because of the combination of several materials, these structures develop a highly complex behavior, presenting a non-linear response, which is due, among other factors, to time-dependent effects, such as the creep of the concrete [13].

### 3.2. Mathematical model for the nonlinear vibration problem

Consider a rotary machine mounted on a beam subjected to a pre-tensioning force, without adhesion. It is known that such forces affect the geometric stiffness and, consequently, the values of the undamped free-vibration frequencies. If the structure is designed, as is usually the case, to have frequencies farther from the machine's service speed rotation, the changes in the frequency due to geometric stiffness may lead to the appearance of potentially dangerous resonance conditions.

Take a beam model of Bernoulli-Euler applied to a simply supported beam AB of length L and inertia I, intended to function as the base of an engine Eg, composed of viscoelastic material, represented by the temporal modulus of elasticity E(t) as shown in Figure 3. A normal force of compression P reproduces the post-tensioning force, which changes the stiffness, and consequently, the natural frequency of vibration of the structure with time. The eccentricity between

Figure 3. Beam model.

the engine axis and the part is initially ignored. The vertical displacement of the central joint is the generalized coordinate of the system.

several effects. Due to the characteristics of the concrete as a structural material, the use of prestressing can bring a great advantage from the economic point of view. When comparing the cost of a prestressed structure with a similar one of conventional reinforced concrete, there is a reduction in the final cost of the structure due to the reduction of steel reinforcement [11]. In addition, the prestressing allows the part to overcome large spans, improves the control and reduction of deformations and fissures. It can also be used for structural recovery and reinforcement, as well as for slender systems and prefabricated or precast parts. There are three types of prestressing systems: (1) prestressing with initial adherence, (2) prestressing with posterior adherence, and (3) prestressing without adherence. The latter type is composed of a post-tensioning system characterized by the slipping freedom of the steel reinforcement in relation to the concrete, along the whole extension of the cable, except for the anchorages.

In a non-adherent prestressing, the cables or chutes are wrapped in two or three layers of resistant paper. The wires and paper are painted with bituminous paint in order to tension them after the concrete has hardened. The bitumen avoids the penetration of the cement cream inside the cable and, in this way, it eliminates adhesion between the concrete and the reinforcement [12]. The prestressed concrete is a composite material of the aggregate mixture and a cement paste associated with prestressing cables and/or passive reinforcing bars. Because of the combination of several materials, these structures develop a highly complex behavior, presenting a non-linear response, which is due, among other factors, to time-dependent effects,

Consider a rotary machine mounted on a beam subjected to a pre-tensioning force, without adhesion. It is known that such forces affect the geometric stiffness and, consequently, the values of the undamped free-vibration frequencies. If the structure is designed, as is usually the case, to have frequencies farther from the machine's service speed rotation, the changes in the frequency due to geometric stiffness may lead to the appearance of potentially dangerous

Take a beam model of Bernoulli-Euler applied to a simply supported beam AB of length L and inertia I, intended to function as the base of an engine Eg, composed of viscoelastic material, represented by the temporal modulus of elasticity E(t) as shown in Figure 3. A normal force of compression P reproduces the post-tensioning force, which changes the stiffness, and consequently, the natural frequency of vibration of the structure with time. The eccentricity between

such as the creep of the concrete [13].

354 Numerical Simulations in Engineering and Science

resonance conditions.

Figure 3. Beam model.

3.2. Mathematical model for the nonlinear vibration problem

By using the Rayleigh method [14], the undamped vibration frequency in its first mode considering viscoelasticity is obtained. It is worth mentioning that Rayleigh assumed that a system containing infinite degrees of freedom could be associated with another with a single degree of freedom (SDOF) to approximate its frequency. It is important to note that the technique developed by Rayleigh aimed at calculating the fundamental vibration frequency of elastic systems, as its precision is dependent on the function chosen to represent this mode of vibration.

The basic concept of the method is the principle of energy conservation, and can, therefore, be applied to linear and nonlinear structures. [15] applies the Rayleigh technique to determine the fundamental period of vibration to verify the stability of mechanical systems. The process is described in relation to the principle of virtual works and as the appropriate choice of the generalized coordinate describing the first mode of vibration. At the end of the process, the generalized properties of the system are obtained as stiffness and mass, necessary for the calculation of the frequency.

Consider that the vertical displacement of a generic section of the beam in Figure 4 is given by:

$$
\sigma(\mathbf{x}, t) = \phi(\mathbf{x}) \cdot \eta(t), \tag{7}
$$

in which ϕ(x) is a shape function that attempts to define the boundary conditions in the supports and value 1 in the central section of the beam, whose displacement with time is q(t). In this case, one adopts the shape function ϕ(x) = sin (πx/L), which is the exact solution of the problem without the P load. A prime mark will denote a derivative of the function in relation to x (Lagrange's notation).

Applying the Rayleigh method, one has the conventional bending stiffness, K0, as a function of the material behavior and the geometry of the cross, which is equivalent to:

$$K\_0(t) = \int\_0^L E(t) I \left(\phi''\right)^2 d\mathbf{x} = \frac{\pi^4 E(t) I}{2L^3} \tag{8}$$

where E(t)I is the known flexural bending with viscoelasticity, represented by multiplication of the temporal material modulus of elasticity with the inertia of the section in relation to the considered movement, the vertical vibration mode (1st mode). In turn, the geometric stiffness, KG, as a function of the normal force of compression (or even tension), is equivalent to:

$$K\_G = P \int\_0^L \left(\phi'\right)^2 d\mathbf{x} = \frac{P\pi^2}{2L} \tag{9}$$

The total generalized mass of the system is found by calculating M=MC + MV where MC is the concentrated mass at the middle span and MV is the mass coming from the beam self-weight given by:

Figure 4. Rayleigh method.

$$M\_v = \int\_0^L m\_V \left(\phi(\mathbf{x})\right)^2 d\mathbf{x} = \frac{m\_V L}{2} \tag{10}$$

Section data:

• External height: H ¼ 16 cm

• Wall thickness: t ¼ 5 cm

• Total inertia: <sup>I</sup> <sup>¼</sup> <sup>H</sup>4�h<sup>4</sup>

Figure 5. Beam section characteristics.

• Concrete cover: c<sup>0</sup> ¼ 25 mm • Reference beam span: L ¼ 3m

• Internal height: h ¼ H � 2 � t ¼ 6 cm

• Reinforced bar diameters: db ¼ 8 mm • Number of reinforcement bars: nb ¼ 4

• Gross-sectional area:<sup>A</sup> <sup>¼</sup> <sup>H</sup><sup>2</sup> � <sup>h</sup><sup>2</sup> <sup>¼</sup> 220 cm2

<sup>12</sup> <sup>¼</sup> 5353 cm<sup>4</sup>

The concrete section was homogenized by the transformation of the steel bars of the reinforcement, which led to a homogenization factor of 1.061 to be considered in the material and geometric properties of the beam section. The homogenizing technic is presented in Section 4.2. For the simulation, all the elements that constitute physical parts to be added to the system, such as the bar used in prestressing systems and an electric induction motor that

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represents periodic excitation, were considered as lumped or distributed masses.

in which mV represents the total mass per length unit. Finally, the frequency of undamped free vibration (in rad/s) is found by way of Eq. (11):

$$
\omega(t) = \sqrt{\frac{\mathcal{K}(t)}{M}}\tag{11}
$$

Considering the total beam stiffness as K(t) = K0(t) � KG, the free undamped frequency of vibration of the 1st mode is found, in Hertz, admitting the compressive force as positive, by:

$$f(t) \ = \ \frac{\omega(t)}{2\pi} = \ \frac{1}{2} \left[ \frac{\pi^2 E(t)I - PL^2}{L^3 (L\,m\_V + 2\,Mc)} \right]^{\frac{1}{2}} \tag{12}$$

For a better understanding of the Rayleigh method and the importance of the geometric stiffness to the structural analysis, the work [16, 17] should be consulted.

### 3.3 Numerical simulation 1

The beam gross cross section was estimated with a passive reinforcement arrangement capable of resisting the predicted load in the simulation, being treated by the homogenized section method, with geometry as indicated in Figure 5. The modulus of elasticity of the concrete was calculated according to NBR 6118/2014 [18] recommendations, following Eq. (13), for a concrete characteristic compressive strength, fck, equal to 30 MPa.

$$\begin{aligned} E\_0 &= a\_i \cdot 5600 \sqrt{f\_{ck}} \quad = \text{ 26838.405 MPa} \\ a\_i &= \quad 0.8 + 0.2 \cdot \frac{f\_{ck}}{80 MPa} \quad = \text{ 0.875} \end{aligned} \tag{13}$$

The reinforced concrete specific weight γ<sup>c</sup> was obtained for a material density r of 2500 kg/m<sup>3</sup> and a gravitational acceleration g of 9.8061 m/s<sup>2</sup> , therefore γ<sup>c</sup> = 24.52 kN/m3 .

Section data:

Mv ¼ ð L

f tðÞ ¼ <sup>ω</sup>ð Þ<sup>t</sup>

stiffness to the structural analysis, the work [16, 17] should be consulted.

<sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>α</sup><sup>i</sup> � <sup>5600</sup> ffiffiffiffiffi

<sup>α</sup><sup>i</sup> <sup>¼</sup> <sup>0</sup>:<sup>8</sup> <sup>þ</sup> <sup>0</sup>:<sup>2</sup> � <sup>f</sup> ck

crete characteristic compressive strength, fck, equal to 30 MPa.

and a gravitational acceleration g of 9.8061 m/s<sup>2</sup>

vibration (in rad/s) is found by way of Eq. (11):

3.3 Numerical simulation 1

Figure 4. Rayleigh method.

356 Numerical Simulations in Engineering and Science

0

mV ϕð Þx

in which mV represents the total mass per length unit. Finally, the frequency of undamped free

Considering the total beam stiffness as K(t) = K0(t) � KG, the free undamped frequency of vibration of the 1st mode is found, in Hertz, admitting the compressive force as positive, by:

For a better understanding of the Rayleigh method and the importance of the geometric

The beam gross cross section was estimated with a passive reinforcement arrangement capable of resisting the predicted load in the simulation, being treated by the homogenized section method, with geometry as indicated in Figure 5. The modulus of elasticity of the concrete was calculated according to NBR 6118/2014 [18] recommendations, following Eq. (13), for a con-

f ck

The reinforced concrete specific weight γ<sup>c</sup> was obtained for a material density r of 2500 kg/m<sup>3</sup>

<sup>p</sup> <sup>¼</sup> <sup>26838</sup>:<sup>405</sup> MPa;

<sup>80</sup>MPa <sup>¼</sup> <sup>0</sup>:<sup>875</sup>

, therefore γ<sup>c</sup> = 24.52 kN/m3

.

r

ωðÞ¼ t

<sup>2</sup><sup>π</sup> <sup>¼</sup> <sup>1</sup> 2 2

ffiffiffiffiffiffiffiffiffi K tð Þ M

dx <sup>¼</sup> mVL

<sup>π</sup><sup>2</sup>E tð Þ<sup>I</sup> � P L<sup>2</sup> <sup>L</sup><sup>3</sup> ð Þ L mV <sup>þ</sup> <sup>2</sup>Mc � �<sup>1</sup>

2

<sup>2</sup> (10)

(11)

(12)

(13)


The concrete section was homogenized by the transformation of the steel bars of the reinforcement, which led to a homogenization factor of 1.061 to be considered in the material and geometric properties of the beam section. The homogenizing technic is presented in Section 4.2. For the simulation, all the elements that constitute physical parts to be added to the system, such as the bar used in prestressing systems and an electric induction motor that represents periodic excitation, were considered as lumped or distributed masses.

Figure 5. Beam section characteristics.

By fixing the force on the section-resistant capacity, one can observe the variation of the natural frequency of the beam with time when considering viscoelasticity, as shown in Figure 6(a). A safety factor of 1.170426 to be applied to the loading can be found, which defines the beam collapse at the 90th day, as can be seen in Figure 6(b). By varying force P, which represents a non-adherent post-tension force, from zero to the resistant capacity of the section, the variation of the natural frequency of the beam is obtained, given in the graph in Figure 7. There, it is possible to see that, with the increase of the axial compression force, the beam frequency decreases, since the geometric portion (KG) stiffness is changed, consequently decreasing the total stiffness (K) of the beam.

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Figure 7. Resonant and non-resonant frequencies as a function of axial compression force P.

Since the motor rotation is set at 1200 rpm (20 Hz), represented by the dotted horizontal line, there is no resonance without consideration of viscoelasticity, but the resonance appears when the natural frequency of the beam is calculated with the introduction of the viscoelastic behavior of the material. For 10 days after application of load, for example, the resonant regime can be observed by the intersection of two curves, dotted (horizontal) and dashed (sloped), defining exactly for which prestressing force the phenomenon occurs at that instant (210 kN).

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.

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Figure 7. Resonant and non-resonant frequencies as a function of axial compression force P.

By fixing the force on the section-resistant capacity, one can observe the variation of the natural frequency of the beam with time when considering viscoelasticity, as shown in Figure 6(a). A safety factor of 1.170426 to be applied to the loading can be found, which defines the beam collapse at the 90th day, as can be seen in Figure 6(b). By varying force P, which represents a non-adherent post-tension force, from zero to the resistant capacity of the section, the variation of the natural frequency of the beam is obtained, given in the graph in Figure 7. There, it is possible to see that, with the increase of the axial compression force, the beam frequency decreases, since the geometric portion (KG) stiffness is changed, consequently decreasing the total stiffness (K) of the beam.

Since the motor rotation is set at 1200 rpm (20 Hz), represented by the dotted horizontal line, there is no resonance without consideration of viscoelasticity, but the resonance appears when the natural frequency of the beam is calculated with the introduction of the viscoelastic behavior of the material. For 10 days after application of load, for example, the resonant regime can be observed by the intersection of two curves, dotted (horizontal) and dashed (sloped), defining exactly for which prestressing force the phenomenon occurs at that instant (210 kN).
