Expressions for deriving seismic base shear for free-standing TL guyed towers are suggested based on the outcomes of the numerical simulations performed for the two guyed towers studied herein. These expressions are derived for each one of the orthogonal directions (transversal, longitudinal and vertical). In the present study, these expressions are presented as a percentage of structure’s weight. The free-standing guyed towers studied herein were subject to detailed nonlinear transient simulations. In this analysis, 20 scaled ground motion records given in Chapter 3 were considered separately in each orthogonal direction (transversal, longitudinal and vertical directions). Then, the time-history series of base reactions were extracted and compiled. The type of seismic input considered in these simulations was synchronous ground motion applied at the tower base (tower mast base and guy-anchors). These ground motion were taken from the PEER Ground Motion Database (2016). A uniform damping of 2% of the critical viscous damping was considered for the tower’s latticed structure. The choice for this damping value was based on the findings of Kotsubo at al. (1985) and El Attar et al. (1995). For the guy- cables, a damping ratio of 0.1% was considered. Based on the set of outcomes from these simulations, expressions were derived based on regression analyses.
5.4.1. Assessment of Horizontal Base Shear under Earthquake Excitations
A first attempt was made to correlate the base shear responses from an equivalent SDOF system with those obtained from nonlinear transient simulations for the MDOF system. The base shear response for the equivalent SDOF system was obtained by solving the equation of motions of a SDOF system for all scaled seismic accelerograms considered in the present study for the horizontal direction. Fig. 5.3 shows a dispersion diagram of the results of both SDOF and MDOF systems. The dashed lines in this diagram represent an absolute error of 0.2. As shown in this diagram, some of the responses from SDOF and MDOF systems have a good correlation but the dispersion is considered high and, overall, the responses obtained by solving the equation motion for an equivalent SDOF system are not considered satisfactory for estimating the base reaction of these towers. This poor correlation indicates that the use of SDOF response acceleration spectra values for deriving seismic inertia forces is not considered suitable for these types of structures. Then, regression analyses were performed on the results of nonlinear transient simulations in an attempt to derive expressions for estimating base shear response as a function of the tower’s natural flexural period and ground motion parameters. It was observed that the magnitude of base shear not only had a direct proportional correlation with peak ground motion acceleration but also at resonant frequencies (i.e. ground motion predominant periods close to or coinciding with the natural period of the structure) the magnitude of the base shear response was amplified. A second order polynomial function with two independent variables was chosen and regression analyses were carried out to determine the values of the coefficients of this function that resulted in a best fit to the base shear responses obtained by the nonlinear transient simulations. This function is given in Eq. (5.1a) where the ratio of maximum base shear response to structure’s weight is expressed in terms of the peak ground motion acceleration , expressed in units of g (gravity), and the variable , herein named frequency term and defined by Eq. (5.1b). From Eq. (5.1b) it can be seen that the frequency term introduces a relationship between natural period of vibration of the tower structure *$ and the predominant period of the ground motion record * . This variable attains its maximum value when *$ = * , thus introducing the effect of dynamic resonance into the base shear response of Eq. (5.1a). All the other coefficients given in Eqs. (5.1a) and (5.1b) are given in Table 5.5 for each tower structure analyzed.
Figure 5.3. Dispersion diagram for the base shear response from SDOF and MDOF systems. Delta guyed tower.
/ 4 = × H+ × H + r × × + × + s × + m =C1 − * 1 $⁄ * HEH+ C¤ × *$⁄ E* H (5.1.a) (5.1.b)
Table 5.5. Coefficients for Eqs. (5.1a) and (5.1b).
Coefficient Tower Delta guyed tower Mast guyed tower
Transversal Longitudinal Transversal Longitudinal
a -0.79 -0.92 -0.50 -0.55 b -1.03 -3.03 -3.39 -4.60 c 4.32 4.55 4.32 4.02 d 1.75 2.04 1.81 1.92 e 0.55 0.21 0.64 0.64 f 1.30 0.52 0.26 0.29 g 0.67 0.91 0.73 1.05 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 B a se S h ea r to We ig h t R a ti o ( S D OF S y st em )
These fitted functions are depicted in the surface graphs of Figs. 5.4 to 5.7 for the guyed towers. The dispersion diagrams in these figures presents the absolute error of the fitted functions in regards to the base shear responses obtained with nonlinear transient simulations. The dashed lines in these diagrams represents the maximum absolute errors found between the base shear responses predicted with Eq. (5.1a) which included Eq. (5.1b) and those obtained with numerical simulations.
Figure 5.4. Mast guyed tower. Surface representation of base shear function (left). Dispersion diagram (right). Transversal calculation direction.
Figure 5.5. Mast guyed tower. Surface representation of base shear function (left). Dispersion diagram (right). Longitudinal calculation direction.
Figure 5.6. Delta guyed tower. Surface representation of base shear function (left). Dispersion diagram (right). Transversal calculation direction.
Figure 5.7. Delta guyed tower. Surface representation of base shear function (left). Dispersion diagram (right). Longitudinal calculation direction.
5.4.2. Vertical Earthquake Excitation
The towers studied herein were also analyzed considering the 20 scaled earthquake ground motion records acting in the vertical direction. Similar to the horizontal component of earthquake excitation, an attempt was also made to derive an expression for the vertical base reaction based on regression analyses. By analyzing the results of the numerical simulations and comparing them with the parameters of each seismic case, it was observed that the vertical base reaction only had a reasonable correlation with peak ground motion acceleration. Therefore, a polynomial quadratic function with only one independent variable (in this case the vertical peak ground motion acceleration) was used for regression analyses. The adjusted coefficients of this function for both towers are presented in Table 5.6. In Fig. 5.8 is given the dispersions diagrams depicting the absolute errors of Eq. (5.2) in regards to the results of nonlinear transient simulations.
V¦ §
W = a × AH+ b × A + c (5.2)
Table 5.6. Coefficients for equation 5.2.
Coefficient Tower Delta guyed tower Mast guyed tower a -1.88 -3.89 b 3.16 6.47 c 1.45 1.77
Figure 5.8. Dispersion diagrams for Mast Guyed tower (left) and Delta Guyed tower (right) for earthquake base reaction in the vertical direction.