STOCHASTIC DYNAMICS OF A GUYED MAST UNDERWIND LOAD

Ballaben, J.S.1, Rosales, M.B.1 Sampaio, R.2

1 Department of Engineering - Universidad Nacional del SurAlem 1253, 8000, Bahía Blanca, Argentina

CONICET, Bahía Blanca, Argentinajorgeballaben@gmail.com

2Department of Mechanical Engineering, Pontificia Universidade CatólicaRua Marquês de São Vicente, 225, Rio de Janeiro, Brazil

SUMMARY: Detailed studies of guyed structures under natural actions as wind or earthquake, are not frequentdespite the potential of adverse impact, e.g. fatigue of structural members or affected quality of the transmission incommunication guyed towers. In this work, the stochastic dynamics of a guyed mast is analyzed under wind loadaction. The simplified structural model consists of a beam-column model accounting for the second order effectdue to axial loads and one guy which is represented by a non-linear extensible cable. The resulting governingsystem is discretized using finite elements and a reduced order model is afterwards constructed using vibrationmodes as a basis. The model is stochastic due to the load and some structural properties uncertainties. The windload is calculated from the wind velocity field by means of the Spectral Representation Method. Since nonlinearstructures show special sensitivity to dynamic loads, the reference nominal wind velocity (suggested in the stan-dards) is also considered as a stochastic parameter. As the guy pretension is a significant parameter in the behaviorof guyed structures, its variation is a relevant issue. Furthermore, the bending stiffness can vary due structure re-inforcements. Since it is not easy to foresee their behavior, an uncertainty quantification study appears necessary.After performing Monte Carlo simulations, the probability distribution functions (PDF) of the results are observedaccounting for each parameter (guy tension, bending stiffness and wind nominal velocity) separately. Also, param-eters combinations are considered. The studied variables exhibit very different types of influence on the statisticsof the dynamic response.

KEYWORDS: Guyed Mast, Wind Loads, Non-Linear Dynamics, Uncertainty Quantification

1. INTRODUCTION

Within guyed structures, the guyed masts are extensively employed, e.g. to support communication devices suchas antennas (Figure 1 a). In this work, the stochastic dynamics of a guyed mast is analyzed under wind load action.The simplified structural model consists of a beam-column model accounting for the second order effect due toaxial loads and one guy (Figure 1 b). Since the wind load contains energy that interacts with flexible structures,the dynamic response becomes important in the analysis of guyed masts. The mast acts strongly in a non-linearfashion when the guys vary between slack and taut states. The wind load is variable in space and time (during anevent) and it changes from one event to the other. Hence, a study considering the wind load as a stochastic event,and the study of the response of the structure to a variety of events with different nominal velocities (that can beconsidered stochastic as well), is desirable to understand the behavior of the structural system. Usually, the guypretension tension varies, during the structure service life w.r.t. the design value, affecting the system performance.As the guy pretension is a significant parameter in the behavior the structure, its variation is a relevant issueand the introduction of uncertainty is adequate. Also, the column-beam stiffness can be a variable value. Forexample, various companies can share the use of one structure to install their antennas; this situation requires, inmost of the cases, the retrofit of the mast and one strategy consists in the partial reinforcement of the legs of thelattice mast. However, the stiffness of the reinforced mast is uncertain due to the construction procedures. Thus,a stochastic treatment of the mast bending stiffness seems indicated. In the present study, the stochastic dynamic

(a) (b)

Figure 1 – (a) Typical guyed tower for mobile signal transmission. (b) Reduced model under study

response of a reduced and simplified model of a guyed mast is analyzed under random wind loads. The fluctuatingwind component is obtained through the Spectral Representation Method (SRM) [1] starting from a given PowerSpectral Density function (psdf ). The temporal and spatial correlations are taken into account by finding the cross-spectrum and introducing a coherence function. The method yields a temporal record of the fluctuating windvelocity. Combined with the standard recommendations, the fluctuating wind pressure are applied on the mast.Using the classical extended Hamilton’s principle, the equations of motion that govern the transverse vibration ofthe system are obtained. The nonlinear model of the cable are similar to the one reported by Gattulli and Lepidi [2].Then, the equations are discretized with finite elements. In particular, the linerized equations are used to performa modal analysis of the structure. Finally, by means of a Galerkin projection, the main nonlinear PDE system isreduced to a 2-DOF model, using two selected normal modes extracted from the previous modal analysis. With thisreduced order model (ROM), the nonlinear response of the structure excited by a stochastic wind load is evaluated,considering uncertainties on the guy tension, the mast bending stiffness and the nominal wind velocity.

2. MODEL DESCRIPTION

2.1. Problem formulationThe plane configuration of a guyed column (Fig 2 a,b) is described, referenced to the static equilibrium state, bythe cable displacement components uc(xc, t) and vc(xc, t) along the abscissa xc, and by the beam-column trans-verse displacements vb(xb, t) and ub(xb, t) along xb. The following assumptions are made: a) both the cable andthe beam-column are considered as homogeneous one-dimensional elastic continua obeying a linear stress-strainrelationship; b) the equilibrium configuration for the inclined cable is described through a quadratic parabola underthe assumptions of small sag to length ratio; c) axial extensions of the cable are described by the Lagrangian strainof the centerline; d) the bending, torsional and shear stiffness of the cable are assumed negligible; e) the torsionaland shear strain of the beam are neglected; f) the nonlinearity of the problem arises from the cable formulation;g) a second order effect due of the axial load (assumed constant) is accounted for in the column equation. Thecable formulation follows the approach reported by [2]. Under these assumptions and using the classical extendedHamilton’s principle, the equations of motion governing the transverse vibration of the guyed column are obtained.

Vb : mbvb + cbvb +EIv′′′′b +PHv′′b = Fvb(t,xb)

Ub : mbub + cbub +EAbu′′b = Fub(t,xb)

Vc : mcvc + ccvc− [Hv′c +EAc(Y ′c + v′c)εc]′ = Fvc(t,xc)

Uc : mcuc +[EAcεc]′ = Fuc(t,xc)

(1)

(a) (b)

(c) (d) (e)

Figure 2 – Cable-stayed tower configurations: a) static; b) dynamic and FEM modal analysis. c); d) and e).

with the following set of geometric and mechanical boundary conditions:

uc(lc) = vc(lc) = 0vb(lb) = v′b(lb) = ub(lb) = 0EIv′′b(0) = 0vb(0) =−uc(0)cosθ + vc(0)sinθ

ub(0) = uc(0)sinθ + vc(0)cosθ

EIv′′′b (0)+(EAcεc +H)cosθ +[EAcεc(y′c(0)+ v′c(0))+Hv′c(0)]sinθ = 0

(2)

Here (∗)′ = d(∗)/dxc and ˙(∗) = d(∗)/dt, mb and mc denote the beam and cable mass per unit length, respec-tively, EI = EbIb the beam bending stiffness, PH is the component of the initial pretension in the direction of themast, EAc = EcAc and EAb = EbAb the cable and beam axial stiffness, respectively, H is the mean static ten-sion in the cable, Yc is the initial configuration of the cable and, due the hypothesis of small sag to span ratio,Yc(xc) = 4D(Xc/lc− (Xc/lc)2), finally εc = uc + y′cv′c + 1/2v′2c is the expression of the elongation for the cable.Since in cable-stayed structures the beam and the cable materials can generally have different viscous behavior,the model accounts separately for beam and cable transverse damping per unit length, Cb and Cc, respectively;here an estimate of system damping is assumed to give the conventional values ci = 8.5e−3(2miωi), with i = c,band ωi is a natural frequency of the system. Here ωi is chosen by observation of the modal shapes used in theapproximation, i.e. if the modal shape selected is a cable-like mode, then the corresponding frequency is used tothe damping term of the cable; the value of ωi were found with Ib = 8.33e−4m4. The values of the constants forthe problem are detailed in Tab. 1.

Table 1 – Values of the constants for Eqs. (1-2). ∗ Other values are considered in particular examples.

lb Eb Ib Ab mb lc Ec Ac mc H30 2.1e11 4.16e−4 - 13e−4 0.04 314 32.31 1.5e11 7.854e−5 0.61 10500∗

m N/m2 m4 m2 kg m N/m2 m2 kg N

2.2. Finite Element discretization and Reduced Order Model (ROM).After stating the weak formulation for the guyed column, the governing system of equations 1 is discretized bymeans of an ad-hoc finite element formulation. The column is modeled using 6-DOF beam elements and the cableusing 6-DOF (three node) cable elements. Figure 2c-e depicts three selected modal shapes found with a modalanalysis.Then, by means of a Galerkin Method, the main nonlinear equations system is further reduced into a 2-DOFmodel, using two selected normal modes extracted from the previous modal analysis. With this ROM the nonlinearresponse of the structure, excited by a stochastic wind load is evaluated, considering the uncertainty of the stiffnessof the mast. The 2-DOF ROM was verified with an analogous finite element model from a commercial softwareand reproduces correctly the amplitude, shape and frequency.The displacements have been expressed in the modal space as v = ΦΦΦq where the displacement vector is or-dered as v = {vb(xb, t),ub(xb, t),vc(xc, t),uc(xc, t)}T , the modal matrix ΦΦΦ = [φφφ 1| φφφ 2] of the eigenfunctions φφφ i ={φbvi(xb),φbui(xb),φvci(xc),φuci(xc)}T and the modal amplitude vector is qqq = {q1(t),q2(t)}T . Imposing the sta-tionarity of the associated Hamiltonian, the following nonlinear ordinary differential equations in terms of thegeneralized coordinates q1 and q2 are obtained:{

m1q1 +a1q1 + c1q1 + c2q2 + c12q1q2 + c11q21 + c22q2

2 + c211q21q2 + c122q1q2

2 + c111q31 + c222q3

2 = p1(t)m2q2 +a2q2 +d1q1 +d2q2 +d12q1q2 +d11q2

1 +d22q22 +d211q2

1q2 +d122q1q22 +d111q3

1 +d222q32 = p2(t)

(3)

2.3. Wind loadA wind load with a mean varying with the height (y coordinate, see Fig. 2) in a potential law fashion (see Fig 1 bgiven by the standards, and with a temporal variation generated by means of the Spectral Representation Method(SRM) [1] is applied on the structure. The wind load takes into account both temporal and spatial correlations. Fordetails, the reader is referred to [3].

3. UNCERTAINTIES QUANTIFICATION

The uncertainty quantification is performed considering different stochastic sources: the guy tension (H), the bend-ing stiffness of the mast (EI) and the wind nominal velocity (Vn). In all cases, the wind load is also random sinceit is derived from a stochastic process. Monte Carlo simulations are carried out by taking the three parameters asrandom variable separately. Additionally, some combinations of these parameters are evaluated. The Principle ofMaximum Entropy (PME) [4] allows to determine the best PDF that suits with the imposed constraints, and intro-duces no unwarranted information. The application of the PME under the constraints of positiveness and boundedsecond moment leads to a gamma PDF. Both the guy tension H and the bending stiffness EI fullfill these condi-tions. For the Vn parameter, both a Gumbel and a Frechet PDFs are used, since that are common for extreme winds.Monte Carlo simulations are performed in order to find the influence of the stochastic parameters on the responseof the structure. To achieve significant statistical results, a convergence study on the standard deviation was firstperformed in each problem to determine the minimum number of realizations of the Monte Carlo simulations.

4. RESULTS

In this section, all the results correspond to the displacements at the top of the mast. Various cases are consideredin the following subsections.

4.1. Guy tension as stochastic variablesThis case is briefly reported (more details in [5]). A Gamma probability density function (PDF) to represent theguy tension H is used for the Montecarlo simulations.. The wind load is stochastic while the nominal velocity Vn isdeterminstic. A convergence study was carried out and it was found that at least 2000 realizations were necessary.The PDF graphs of the results were constructed using the ksdensity function of MATLAB. In this example, themean values of the guy tension vary in the range 7000-14000 N.Figure 3 shows the evolution of the PDF with Hm for the peak displacements (PD, local maxima), when H isconsidered stochastic, and for a fixed value of Vn = 70 m/s. Fig. 3 a contains the (interpolated) confiability bands.The bandwidth remains almost constant. though it has a trend to decrease, specially for the lowest and highestvalues of Hm. On the other hand, the PDF modes present high variability: in some cases they correspond to thelower values and in others, to the highest ones, without a clear trend. Other interesting aspect is that the supportof the PDFs also presents a high variability (Fig. 3b). Figure 4 illustrates the evolution of the displacements PDFwith Vn for the peaks (local maxima), when H is considered stochastic, and for a fixed value of E(H) = 7500 N. It

(a) (b)

Figure 3 – Evolution of the displacements PDF with E(H) (H is considered stochastic, with a Gamma PDF) of the peak values(local maximua) of the transverse displacement of the beam-column at the tip of the mast for Vn = 70 m/s. a) top view ofdisplacement PDFs; b) some arbitrary PDFs.

can be seen from Fig. 4 a that the width of the bands gets higher with the value of Vn . The displacements becomemore disperse and less predictable (the mode, shape and support of the resulting PDF result different for each Vn)for the higher values of Vn. In Fig. 4 b, three PDFs and different shapes and widths are depicted.

(a) (b)

Figure 4 – Evolution of the displacements PDF with Vn (H is considered stochastic, with a Gamma PDF and E(H) = 7500N) of the peak values (local maxima) of the transverse displacement of the beam-column at the tip of the mast. a) top view ofdisplacement PDFs; b) some arbitrary PDFs.

4.2. Stiffness of the mast as stochastic variableIn this section, the EI parameter is considered stochastic. Parameters H and Vn are deterministic although thewind load is stochastic. A uniform PDF, a truncated exponential PDF and a gamma PDF are respectively usedto represent EI and to observe the propagation of uncertainty of this parameter over the local PD statistics. Theinfluence of the Vn is also studied, since the statistics using EI as stochastic parameter are behold for each individualcase of Vn. Figure 5 depicts a plan view of the PDF of the PD (in m) at the top of the guyed mast. Each verticalstrip corresponds to a case of Vn (in m/s). The warmer colors indicate higher probability and the location of themodes while the colder colors, lower. All the resulting distributions are gamma-like PDFs and the mean and themode mostly coincide. No differences can be seen when using either a uniform PDF or a truncated exponentialPDF to represent the random variable EI in the PDF of the results. The use of a gamma PDF instead, leads asmaller dispersion in the results. Nevertheless, the differences are small and it seems that the choice of the PDF forEI does not affect widely the statistics of the peak displacements. The Vn parameter seems to influence the modeand dispersion in a linear fashion. More results can be found in [6].

(a) (b) (c)

Figure 5 – EI as random variable. Plan view of the peak displacements PDFs for each value of Vn. EI PDF: a) uniform; b)truncated exponential; c) gamma.

4.3. Wind nominal velocity as stochastic variableIn this section, the Vn parameter is considered stochastic. Gumbel and Frechet PDFs are respectively used toobserve the propagation of uncertainty of this parameter over statistics of the PD. The influence of the EI variationis also studied, since the statistics using Vn as stochastic parameter are considered for each individual case of EI.Figure 6 depicts the PDF of the PD at the top of the guyed mast in which each vertical strip corresponds to a

(a) (b)

Figure 6 – Plan view of the PDFs of peak displacements for each value of EI, using Vn as stochastic variable with distributions:(a) Gumbel and (b) Frechet.

case of EI. The differences using the different PDFs are evident. The PDF of the results using a Gumbel PDF aremore disperse and often multimodalities can be observed (particularly in the third portion corresponding to lowervalues of EI). On the other hand, using a Frechet distribution for the input random variable, the PDF of the PDresults more compact and Gamma shaped; the modes and the PDFs are located in a zone of higher PD values,in agreement with the Frechet PDF, which yields higher probabilities on the zone of higher Vn while the Gumbeldistribution leads to a more distributed probability among all the values of Vn. Yet, the mode corresponding to thehigher value of PD in the Gumbel case matches the mode of the Frechet case, then, the prediction of the mostdangerous displacements is accomplished similarly in both situations. The difference lies in that the use of theGumbel PDF results in wider dispersion, since lower values also have high probabilities of occurrence. The EIparameter affects the mean and the dispersion, strongly in the first third region of the values of EI and then, itsinfluence is reduced, but still present.

4.4. Wind nominal velocity and stiffness of the mast as joint stochastic variablesNext, both EI and Vn are considered stochastic together. The resulting PDFs of PD for the three EI PDFs and thetwo Vn PDFs above employed, are depicted in Fig. 7. Again, the distribution of EI almost does not affect the PDFsof peak displacements. For the case of Frechet PDF, the result is more compact than the case using a Gumbel PDF.In this study, no multimodalities are observed and the modes do not match, being the more conservative results theones corresponding to the Frechet PDF.

(a) (b) (c)

Figure 7 – PDF of peak displacements using EI and Vn as joint stochastic variables. Each subfigure plots both the cases ofFrechet and Gumbel and: (a) uniform; (b) truncated expontential; c) gamma.

4.5. Probability of exceeding and conditional probability of exceeding a limit valueIn this section, the probability that the displacements (D) (all the displacements of each experiment, not exclusivelythe peaks) exceed a limit value, arbitrarily fixed on Dlim = 0.20m, is evaluated considering EI and Vn stochastic.The probability of exceeding a given limit value in some parameter combination is assessed, and furthermore, ifthat parameter combination is probable. The results are plotted in Fig. 8, for all the combinations of PDF studiedin section 4.4. As before, the warmer colors indicate a higher concentration of probabilities and the colder, theopposite. The results show different patterns when the cases of PDF used for EI are compared. When the uniform

(a) (b) (c)

(d) (e) (f)

Figure 8 – Probability of exceeding of a limit value of 0.20 m given a pair of values of EI and Vn.

PDF or the truncated exponential PDF are used, the highest probabilities are concentrated in the zone of lesservalues of EI (more dispersion is observed using the uniform PDF, as expected). Nevertheless, the results withboth cases of PDF are very disperse. When the gamma PDF is chosen, the probabilities are focused in the zoneof reference Vn (55 m/s) and the mean value of EI. The use of a Gumbel PDF (Fig. 8a-c) is characterized for thepresence of a larger dispersion in the direction of the Vn axis, when compared with the results using a Frechet PDF(Fig. 8d-f). This study reveals that, for this system, the lesser dispersion in the input PDFs, the lesser dispersion isobserved in the conditional probabilities of exceedance. The fact that the different choices for the parameter PDF

give qualitatively different results, indicates that a better knowing of the statistics of the parameters leads to a bettercharacterization of the response.

5. FINAL REMARKS

In this work, a nonlinear formulation of a cable-stayed mast structure under stochastic wind load was presentedand the model was used to perform a stochastic dynamic analysis. First, a deterministic model was stated and thegoverning system solved via a finite elements discretization. A 2-DOF reduced order model (ROM) of the systemwas then constructed using the eigendata obtained from the linearized problem. Once calibrated, the ROM wasemployed to perform an uncertainty propagation study using the guy tension, the mast bending stiffness and thenominal wind velocity as stochastic parameters, separately and in combinations, always with a stochastic load tosimulate the wind actuating on the mast. The wind load accounts for both spatial and temporal correlations. ThePDFs of the guy tension and the stiffness were found by means of the Principle of Maximum Entropy. For the Vnparameter, two commonly used distributions were selected (Frechet and Gumbel). Monte Carlo simulations wereemployed for the numerical realizations. Regarding the case of stochastic guy tension, the following features wereobserved. A sudy of the local maxima (peaks) showed that, for a fixed value of Vn and considering H stochastic ( fora wide range of values of Hm), the width of the bands of confiability is not modified with the value of Hm. Also, themodes varies from the zone of higher to lower displacements without following any trend. In turn, when the valueof E(Hm) is fixed (treating H as a stochastic variable) and the results are plotted for all the values of Vn, the absenceof multimodalities is observed. The case of random stiffness was studied with three different PDFs. It was foundthat the PDF of the local peak displacements present similar gamma-like shapes for all the three cases, suggestingthat the statistic of EI have a small importance in the statistic analysis of the response. On the other hand, the Vnparameter affects the mode and the dispersion of the results in a linear fashion. When Vn is considered stochastic,multimodalities and a larger dispersion are found when using a Gumbel PDF, while when using a Frechet PDF theresults are more concentrated but also conservative; the influence of EI in the shape and dispersion of the resultingPDFs is remarkable in the first third portion of values of EI. When both EI and Vn are considered stochastic, and thePDF of the peak displacements is studied, similar conclusions as before can be stated, but no multimodalities areobserved. The influence of the PDF used for EI becomes noticeable when the probability for the displacements ofexceeding a limit value is studied. Further studies in order to statistically characterize the EI are proved necessary.At present, the authors are analyzing the effect of considering combination of the three stochastic parameters withjoin distribution functions.

ACKNOWLEDGEMENTS

The authors acknowledge the financial support from CONICET, MINCyT and UNS (Argentina); CAPES, CNPq,and FAPERJ (Brazil).

REFERENCES

[1] Shinozuka, M., Jan, C., “Digital simulation of random processes and its applications”, Journal of Sound andVibration, 25(1), 1972, pp. 111–128.

[2] Gattulli, V., Lepidi, M., “Localization and veering in the dynamics of cable-stayed bridges”, Computers &Structures, 85, 2007, pp. 1661–1678.

[3] Ballaben, J., Rosales, M., “Parametric study of the dynamic along-wind response of a guyed tower”, Proceed-ings of Segundo Congreso Latinoamericano de Ingeniería del Viento (CLIV2), La Plata, Argentina, 2012.

[4] Shannon, C., “A matemathical theory of communication.”, The Bell Technical Journal, 27, 1948, pp. 379–423.

[5] Ballaben, J., Rosales, M., Sampaio, R., “Dynamic analysis of a guyed mast with uncertainties on the guyinitial tension and the wind load”, Proceedings of ENIEF 2014, Mecánica Computacional, vol. XXXIII,Bariloche, Argentina, 2014, pp. 1751–1765, http://www.cimec.org.ar/ojs/index.php/mc/article/view/4774/4706.

[6] Ballaben, J., Rosales, M., Sampaio, R., “Dynamic analysis of a guyed mast with uncertainties on the stiffnessand the nominal wind velocity”, Proceedings of 1st. Pan-American Congress on Computational Mechanics(PANACM 2015), Buenos Aires, Argentina, 2015.