NONLINEAR DYNAMIC ANALYSIS BASED ON EXPERIMENTAL DATA OF RC TELECOMMUNICATION TOWERS SUBJECTED TO WIND LOADING

The goal of this paper is to propose a nonlinear dynamic model based on experimental data and NBR-6123-87 to accomplish a nonlinear dynamic analysis of slender structures subjected to wind loading. At first we compute the static answer given by the mean wind speed. In this part of the problem we consider the concept of effective stiffness to represent the physical nonlinearity of material and a P-Delta method to represent the geometrical nonlinearity. Considering the final stiffness obtained in that P-Delta method, we compute the dynamic answer given by the floating wind speed, according to the discrete dynamic model given by NBR-6123-87. A 40 m RC telecommunication tower was analyzed, and the results obtained were compared with those given by linear static and dynamic models.


Introduction
The models proposed by the Brazilian code NBR-6123-87 [2] to accomplish a dynamic analysis of structures-subjected wind loading are based on linear dynamic models.In RC structures where the effective stiffness changes continuously due to nonlinear material behavior and the level of strength, linear models could not describe precisely the structure behavior.Computation of cross-sections properties, and consequently the displacements and internal loads, in slender RC structures subjected to wind loading is a very difficult task because as the loads change along time, cross-sections properties change too.Which stiffness do we consider?Wind speed is defined by two components, one is the mean wind speed and the other is the floating wind speed.Mean wind speed applies on the structures static loads, while floating wind speed applies on dynamic loading.The models given by NBR-6123-87 [2] are based on linear dynamic models, in other words, they consider a constant stiffness along time, what does not happen in practice.In this work the authors analyze a prefabricated 40 m RC telecommunication tower (Figure 2.1) similar to others erected at Minas Gerais and Espírito Santo states of Brazil.For the effects of the mean wind speed on structure, the authors consider a nonlinear behavior.In this phase, a P-Delta effect will be considered on the structure.In each iteration, the effective stiffness is given by Brasil and Silva [1].After this method of converging, we initiate the computation of the dynamic effects of wind given by the floating wind speed.The authors consider that the structure vibrates around an equilibrium position.This position is that one given by the last iteration of the P-Delta method.Then, the natural modes and frequencies of vibration are computed considering the effective stiffness given by the last iteration of P-Delta method.Once the natural shapes and frequencies are known, the dynamic analysis can be done according to NBR-6123-87 [2].The sum of the static, given by P-Delta method with Brasil and Silva [1] curves, and dynamic components, provided by the discrete dynamic model of NBR-6123-87 [2], gives the structure behavior.

Linear static analysis (LSA).
According to NBR-6123-87 [2], V 0 (meters per second) is the mean wind speed computed on 3 seconds, at 10 meters above ground, at a plain terrain with no roughness, and with recurrence of 50 years.The topographic factor is S 1 , while the terrain roughness is given by factor S 2 , which is a function given by where b, p, and F r are factors which depend on the terrain characteristics, and z is the height above the ground in meters.The statistic factor is S 3 .Both S 1 , S 2 , and S 3 are given by tables in Brazilian code NBR-6123-87 [2].The characteristic wind speed (meters per second) and the wind pressure (Pascal) are, respectively, 2) The wind load (Newton) on an area A (projection on a vertical plane of a given object area in square meters) is computed as where C a is the aerodynamical coefficient.The Brazilian code NBR-6123-87 [2] presents tables for C a values.

Linear dynamic analysis (LDA).
According to NBR-6123-87 [2], for the jth degree of freedom, the total load X j due to direct along wind is the sum of the mean and floating load given by where the mean load X j is given  20 of NBR-6123-87]; z r is the level of reference, equal to 10 meters in this work; and V p is the design wind speed corresponding to the mean speed during 10 minutes at 10 meters above the ground level, for a terrain roughness (S 2 ) category II.The floating component X j is given by where m i , m 0 , A i , A 0 , ξ, and C ai being, respectively, the lumped mass at the ith degree of freedom, a reference mass, the equivalent area at the ith degree of freedom, a reference area, the dynamic amplification coefficient [2, Figure 17 of NBR-6123-87], and the area A i aerodynamical coefficient.Note that ϕ = [ϕ i ] is a given mode of vibration.To compute ϕ i and ξ, it is necessary to consider the structure mass and stiffness.The lumped mass can be easily calculated by summing the mass around an influence region of the node.The total homogenized moment of inertia of the cross-section is given by (2.9) E s , E c sec , I s , I shom , I c , and f ck being, respectively, the elasticity modulus of steel, the secant elasticity modulus of concrete (NBR-6118-78 [3]), the moment of inertia related to the structure axis of the total longitudinal steel area, the homogenized moment of inertia of the longitudinal steel area, the moment of inertia of the total cross-section area, and the characteristic compressive resistance in MPa at 28 days concrete.Since this model is based on linear dynamic models, we consider the cross-section moment of inertia as the total stiffness, such as of each section to compute stiffness matrix of the structure.This assumption may be justified because if this is a linear elastic model, any cross-section damage can be considered in this analysis, so the stiffness to be considered must be the total stiffness.When r modes are considered in the analysis, the combination of these modes, for a given dynamic variable Q, is computed as and is a transversal dynamic load.

Nonlinear dynamic analysis (NDA).
As stated before, the loads due to the wind speed present two components: the static loads due to mean wind speed and the dynamic loads due to the floating wind speed.The static loads are computed as given in (2.5) and (2.6).We call the first results obtained using these equations as the first-order static internal loads.At this point, we consider that the structure under those static loads is subjected to the P-Delta effect.The static displacements (δ i( j) ) at the ith node and the jth iteration of the P-Delta method are computed considering the effective stiffness.Differently of what occurs in Section 2.2, we consider the following expressions to compute the moment of inertia (Brasil and Silva [1]) at the ith node and the jth iteration of the P-Delta method: I EF , w, x, M k , and M u being, respectively, the effective moment of inertia, the parameter of effective stiffness, the level of strength, the working bending moment due to mean wind speed, and the ultimate code-based moment of a given cross-section.In (2.13) we consider the damage that occurred in the cross-sections is represented by the effective stiffness concept.
Finally, the P-Delta effect is computed, at the ith node and the jth iteration of the P-Delta method, as (2.14) We call the final results obtained using these equations as the second order static internal loads.Considering the stiffness obtained in the final iteration of P-Delta method we compute the modes and frequencies of the vibration of the structure and so accomplish the dynamic analysis, as described by (2.7) and (2.8).We considered that the structure displaces around the equilibrium position given by the P-Delta method.

Structure analyzed and numerical results
The structure analyzed here is an RC telecommunication tower 40 m long and with a diameter of 60 cm.The structure is cylindrical with cross-section in circular ring.Properties change along the structure axis, because the thickness and steel area vary along the axis.The concrete used in the fabrication of the structure presents characteristic resistance ( f ck ) at 28 days equal to 45 MPa, which represents, according to (2.9), E c sec = 41.4GPa.We consider the elasticity modulus of the structure E = E c sec .The concrete covering is 25 mm.The concrete design resistance is f cd = 45/1.3MPa.The steel used in confection of the structure presents f yd = 500/1.15MPa(steel design stress) and E s = 210GPa.The structure is discretized into 40 elements of one meter long each one.The properties are shown in Table 3.1.
In Table 3.1 we used the following notations: Node is the node number in the finite elements method (FEM) program; Height is the level related to the ground level; Øext is the external diameter of the cross-section; Thick is the thickness of the cross-section; M is the nodal mass (lumped mass); A total is the cross-section area; Ic is the moment of inertia of the circular ring; nb is the number of longitudinal bars of the reinforced concrete section; øb is the diameter of longitudinal bars; As is the total longitudinal steel area; Rb is the radius of the circle that passes along the longitudinal bars axis; Is is the total moment of inertia of the steel area; I total is the total homogenized moment of inertia of the reinforced concrete cross-section; and Is/I total = ws is the lower boundary value for w in each section.According to NBR-6123-87 [2], we consider the basic wind speed of V 0 = 35m/s, the topographic factor is S 1 = 1, terrain roughness category IV, class B, which gives S 2 ≡ (b; p; F r ), and the statistic factor is S 3 = 1.1.As we stated before, the wind load on an area A is F = q • C a • A, where C a is the aerodynamical coefficient.Several equipments are installed on the structure, they are stairway with anti-falls cable, platform with antennas supports, night signer lights, protection against atmospheric discharges system, and installed antennas.The values of A and C a are tower, 0 ≤ z ≤ 40m, A = 0.6 m 2 /m, and C a = 0.6; stairways, 0 ≤ z ≤ 40m, A = 0.05 m 2 /m and C a = 2; cables, 0 ≤ z ≤ 40m, A = 0.15 m 2 /m, and C a = 1.2; platform and antennas supports, z = 40m, A = 1 m 2 , and C a = 2; antennas, z = 40m, A = 3 m 2 , and C a = 1.Table 3.1 shows the nodal mass (M) and area for the structure analyzed.
Based on the results obtained by Brasil and Silva [1], in this section we adopt the following equation (Figure 3.1) for the effective stiffness parameters: Note that the upper value is equal to 1.0 and lower values vary.Because of the safety coefficients adopted materials, and design process, usually in tests structures present values of x = M k /M u ≥ 1.0.For a 30 m structure tested by Brasil and Silva [1], the maximum value assumed by x was 1.33 and for other similar 40 m structure the maximum was x = 1.53.
Considering the lumped mass (M) given in Table 3.1, the total homogenized moment of inertia for the LDA model, and the effective moment of inertia of the final iteration of P-Delta method for the NDA, we compute the natural modes and frequencies of vibration (Figure 3  The values obtained for bending moment in both models analyzed are shown in Figure 3.3.In this figure we can see the following bending moments obtained: M lsa (LSA), M lda (LDA), M nda (NDA), M d (design bending moment), and M t (bending moment applied in tests).The LDA presented values of bending moment 1.3 times those given by the LSA, while the NDA presented values 1.5 times those from LSA.The design moments are 1.4 times those given by LSA and 1.1 times those given by LDA.Comparing the results we conclude that the design bending moment is satisfactory to the LDA, but is not
.6) b and p are indicated [2, Table
.2).Note that in nonlinear model the frequencies are smaller than in LSA.The coefficient of amplification ξ presented values until 2.35 for LDA and 2.65 for NDA.