Design Wind Speeds for Mexico:An optimum approach of wind design of structures

Celso J. Muñoz Black1, Jorge Sánchez Sesma2, Alberto López López3 and Luis E. Pérez Rocha4

1Researcher, Civil Engineering Department, Instituto de Investigaciones Eléctricas, Cuernavaca,Morelos, México, cjmb@iie.org.mx

2Researcher, Instituto Mexicano de Tecnología del Agua, Cuernavaca, Morelos, México,jsanchez@tlaloc.imta.mx

3Researcher, Civil Engineering Department, Instituto de Investigaciones Eléctricas, Cuernavaca,Morelos, México, alopezl@iie.org.mx

4Researcher, Civil Engineering Department, Instituto de Investigaciones Eléctricas, Cuernavaca,Morelos, México, lepr@iie.org.mx

ABSTRACT

Traditional structural wind design that has been a common practice in Mexico, is based onregional maximum gust wind velocities of specific return periods associated with differentstructural importance or accepted risks. However, nowadays it is a matter of controversy tospecify the optimum return periods to design different type of structures against wind effects thatconduct to minimum cost of losses in case of a structural failure. Although, wind hazarddistribution in Mexico has been well studied, transmission line failures are still occurring,mainly due to hurricane winds. Taking into account the important direct and indirect losses dueto lifeline structural failures occurred recently in Mexico, a research study focused on thebalance between the cost of losses and the structural reliability level has been developed. For awind design practice based on optimal criterion a well established optimum design model, whichis usual for seismic design in Mexico, is now being implemented. Described in this paper is anew optimum wind design model, as well as a sensitivity analysis of the involved variables.Finally, conventional and optimal gust wind velocity maps were developed and are presentedand discussed in this work for practical applications.

INTRODUCTION

During the last twenty three years, a unified design criteria for the determination of windpressures on high voltage transmission systems and structures, has been developed for Mexico[1,2,3]. The design wind determination procedure was based on annual maximum gusts arisingfrom a detailed terrain and topography homogenization followed by an extreme probabilisticanalysis of wind speeds registered in different meteorological stations located all over thecountry. Final maximum wind speed estimations were made also considering specificallyhurricane winds. Following this procedure, different isotach maps for predefined return periodsof 10, 50, 100 and 200 years, considering the importance of the structural system, have beenupdated[4].

On the other hand, although wind hazard distribution in Mexico has been well studied,transmission line failures are still occurring mainly due to hurricane winds like occurred inMexico during 2005 and 2007 (e.g. Wilma in 2005, Lane in 2006 and Dean in 2007), whichcaused important damages and losses to transmission structures lines as shown in Figures 1 and2.

Taking into account all these facts mentioned above, the Mexican Electrical Utility (CFE,as per Spanish abbreviation) focused its interest on developing different research studies in orderto improve their wind design process both for wind reliability and also for risk analysisformulation. The first of these studies was devoted to update the distribution of wind hazards forMexico [4], taking advantage of the improved methodologies for the estimation of hurricane andorography effects and completing the wind speeds database up to 2006. Figure 3 shows theisotach map for return period of 50 years, which is normally associated to structures of mediumimportance level. The second study was focused to the balance between the cost and thestructural reliability level, allowing for the relevance of the optimal criterion based design. Forthis, a well recognized optimum design criterion, initially applied for seismic regionalization ofthe Mexican territory [5, 6], was implemented in order to define the best return period for whichimportant facilities and life line structures must be designed against wind actions with minimumtotal costs.

Figure 1: Damage caused by hurricane Wilma in Cancun (2005)

Figure 2: Damages caused by hurricane Lane in 2006

Figure 3: Isotach map for return period of 50 years updated in 2006

OPTIMAL DESIGN OF WIND RESISTANCE STRUCTURES

In the case of optimal design of wind resistance structures, it is recognized that the cost ofdiscriminating between feasible design alternatives, is the total cost involved during the structureservice life which integrates not only its initial construction and operating costs, but the futureexpected costs, the last ones being related to upkeep expenses, for example, repairs, maintenance

and replacement of contents. Utility losses caused by services interruption, fatalities and injuriesand other consequences of failure or damage, are commonly added to indirect economicexpected costs. In many facilities, for example in electrical transmissions systems, the mostimportant expected cost involved is due to electrical service interruption when severe failuredamage is presented in one or more times in its service life.

In the optimum design formulation it is assumed that all these costs are dependent on asingle parameter: the nominal resistance design, which is expressed in terms of wind pressureover an exposed area. Then, a parameter of design is optimal if it minimizes the amount ofpresent value of the total costs of a structure or a system. As a result, optimal values are notassociated with constant return periods. Indeed, the optimization leads to a situation that isintuitively a design rule: in areas of low wind danger, where a structure design against windactions is relatively cheap, the design is an optimum for return periods higher than those thatwould be used in areas of greater danger.

In the present study, the methodology proposed in the pioneering work by Esteva [6],initially applied for seismic regionalization of the Mexican territory, has been followed to findoptimal wind velocities for design. According to this document, it is considered that a variable ofdesign is optimal if it minimizes the amount of expected costs from the decision of usingprecisely that value design.

In the following paragraphs, the two components of the total cost of a structure during itsservice life are discussed.

INITIALCOST

The following variation for the initial construction cost, )(vCI , is adopted,

000

00

if)(

if)(

vvvvCCvvC

vCIR

(1)

Where:

v Wind speed for structural design,0v Wind speed that the structure should resist if it has not been designed against

wind actions,0C Structure cost even if this is not designed to withstand lateral wind loads, and

RC , Constants of the initial cost function.

If the Equation 1 is normalized with respect to 0C , we have:

00

0

0 if)(1

if1)(vvvvKvv

CvCI

(2)

Where, evidently, 0CCK /R .

EXPECTED LOSSES COSTAs a proposed wind loss model, it is assumed that each time the design speed, v , is

exceeded, it will provide a total loss cost of the structure. This model is obviously too simple.The real resistance of a structure is, in general terms, uncertain but has an average resistancehigher than the nominal resistance that arises when adopting a value of speed design. Thus, whenthe nominal design wind speed is exceeded, not necessarily a total loss is presented, and can onlybe given a probabilistic estimation on the value of the loss. Moreover, it is conceivable that evenif the demand does not exceed the nominal wind speed design, partial failures are presented. Thiswould require the development of functions of vulnerability and its formal inclusion in thecalculation of the losses.

However, the optimization process was carried out only to determine relative levels ofexpected costs between structures in different parts of the country. Because of this, it wasconsidered that the use of a more refined model would not achieve substantial improvements.

According to Rosenblueth [7], if it is assumed that wind hazard follows a Poissonprocess, and if the updated value of money is properly described by an exponential function, thepresent value of the expected losses, )(vEVP , when a structure is designed to resist the windspeed “ v”, is:

)()()( vvCPvEVP (3)

Where:

)(vCP Cost of the losses generated by wind actions,)(v Rate of the exceeding demand that produces a structural failure when it has

been designed for a wind speed “ v”, and Annual net rate of discount of the value of money.

As indicated by Ordaz [8], the cost of loss includes more than that of just buildingdamage; since the loss of buildings affects the economy performance in a general way, the totallosses are greater than just the material losses. Taking this into account, it is proposed that:

)1()()( QvCIvCP (4)

Where )(vCI is the initial cost, given in the Equation 1, and Q is a proportional factorof the initial cost that measures the importance of losses in buildings.

Substituting the value given for )(vCP in Equation 4 in Equation 3 yields

)()1()()( vQvCIvEVP (5)

TOTALCOST

The objective cost to be minimized is the total cost, )(vCT , which is given by

)()1(1)()()()( vQvCIvEVPvCIvCT (6)

Or,

)()1(1)()(

00

vQC

vCIC

vCT (7)

Where 0/)( CvCI is given by Equation 2. In Figure 4 a qualitative drawn of Equation 7is shown.

Figure 4: Optimization of the total cost of a structure

In a similar manner to the Equation 7, the total cost can be written also as function of thebase shear as follows:

)()1(1)()(

00

SQC

SCIC

SCT (8)

Where S is the base shear of the structure. In Figure 5 it is shown the relation betweenthe initial cost and the design base shear for a steel pole supporting structure. This figure wasobtained designing the pole following the commonly used allowable stress design criteria fordifferent intensities of wind velocities and varying the thickness with constant external diameter.Then, from Equation 2 and the slope of the straight part, values for K and can be deduced. Itis evident that in this case 1 , but it will necessary to evaluate these parameters depending ofthe type of structure.

In the next paragraph it will be explained how the parameters K and are used toelaborate an optimum base shear map focused to combine different hazards as will be the seismiceffect in Mexico.

Figure 5: Initial cost normalized versus base shear

CRITERIA FOR DETERMINING WIND VELOCITIES FOR OPTIMAL DESIGN

In the following paragraphs, calculations that were performed to determine optimal windvelocities for design are described in a general way.

It will be supposed that common structures are those whose loss is not particularlyundesirable and, furthermore, that their cost is not excessive compared to the value of theconstruction itself. Also it will be supposed that the isotach maps in edition for the new WindDesign Manual [9], are applicable to the type of terrain of interest and for common structuresassociated to a return period of 50 year (see Figure 3). The procedure starts with an acceptedvalue of base shear as optimum value in a site subjected to a strong wind velocities. The selectedsite was Cancun placed in the Caribbean coast of the Peninsula of Yucatan. For our steel polesupporting structure, the base shear recommended as optimum was 120 ton. The procedurefollows an iteration scheme in which the minimum error between the recommended base shearand the one obtained with the Equation 8, varying the value of Q, is found. The values ofK =0.05 and =1 (see Equation. 2) were obtained from Figure 5,.The procedure yielded a valueQ=10 and a base shear S =115 ton. For the selected site, this result is accepted.

In a practical point of view the Q value means that the cost of loss, additional to theinitial cost, is 10 times that of the initial cost. In this cost of loss may be considered namely thecost of the losses due to the structural failure, including the possible business interruptions, aswell as the cost of repairs and the cost of losses of human life. As noted, these values are difficultto quantify and its accuracy will not bring substantial improvements.

In this description of optimal design applied to wind danger, the most importantstructures are essential structures, whose loss is particularly undesirable. In general, these maynot be particularly costly structures, but their loss is undesirable because they may becomeunusable or reparation may be extremely high. Itis considered that initial costs remain the sameand there only are differences between the costs of future losses.

Using the values of K , , and Q, the map of base shear distribution in Mexico for theparticular case of a steel pole supporting structure was computed and is shown in Figure 6. Thecorresponding wind speed was computed by using the relation between the wind velocity and thebase shear as follows:

exp5.0 ACSv

a (9)

Where:

v Wind speed for structural design,S Shear base of the structure, Air density at sea level,

aC Drag coefficient of the structure, andAexp exposed area of the structure, and

The respective map of wind velocity is shown in Figure 7. It can be observed that forQ=10 it is obtained a good fitness for shear values and corresponding wind speeds. This kind ofpresentation becomes relevant as it can be used to combine different types of hazards, as is thecase of seismic effect in Mexico, to provide practical information for rational use of structuresexposed to this type of hazards. This type of combination of hazards is in development to beapplied to latticed framed electrical substations structures.

-115 -110 -105 -100 -95 -90

15

20

25

30

35

Figures 6: Optimum base shears for Q = 10

-115 -110 -105 -100 -95 -90

15

20

25

30

35

Figures 7: Wind speeds corresponding to optimum base shears for Q = 10

CONCLUSIONS

An overview of developments related to optimal wind design of structures has been presented.The procedure leading to the adoption of intensities that result in minimum total costs has beendescribed, including start-up costs adding the future losses updated to the present value. Theseintensities, known as optimums are the starting point for the probabilistic approach followedthroughout this investigation. The first result is an optimal base shear map related to polesupporting structure for wind design purposes, which takes into account that the parameter thatcontrols the safety level are related to the cost of losses, or in other words, the importance of thestructure. Also this map can be applied for a combination of different hazards in terms of theshear force. Finally the corresponding wind speed map to optimum base shear distribution waspresented.

ACKNOWLEDGEMENTS

The authors would like to thank the financial support of Mexican Electrical Utility (CFE) andDrs. Luis Esteva and Mario Ordaz for the technical comments on the optimisation procedures.Also, grateful to Eng. Rosa Ma. Soberanes and Eng. Ismael E. Arzola for preparing the isotachand optimum maps.

REFERENCES

[1] CFE (Comisión Federal de Electricidad), Manual de Diseño de Obras Civiles – Diseño por Viento,México, 1993.

[2] A. López, C. J. Muñoz, J. I. Vilar and E. R. Neri, Unificación de criterios en el cálculo de presiones,flechas y tensiones para el diseño mecánico de líneas de transmisión, Informe Final IIE/42/18/I 02/F/95,México, 1995.

[3] J. Sánchez-Sesma, A. López, J. Aguirre and C. J. Muñoz, Wind design for Mexico: A review of theperiod 1964-2003, 11th International Conference on Wind Engineering (Lubbock, Texas, 2003), 2003.

[4] A. López and C. J. Muñoz, Actualización de isotacas en la república mexicana para fines de diseñocontra viento de líneas de transmisón y subtransmisión y subestaciones eléctricas, Informe FinalIIE/42/13083/I01/F/DC, México, 2007.

[5] L. Esteva, Seismic risk and seismic design decisions, Seminar on Seismic Design for Nuclear PowerPlants, Massachusetts Institute of Technology, Cambridge, Mass., 1969.

[6] L. Esteva, Regionalización sísmica de México para fines de ingeniería, Serie Azul de Instituto deIngeniería, 1970, pp.-246.

[7] E. Rosenblueth, Optimum design for infrequent disturbances, J. Structural Div., ASCE, 102, 1976,pp. 1807-1825.

[8] M. Ordaz, J. M. Jara and S. K. Singh, Riesgo sísmico y espectros de diseño en el estado de Guerrero.Informe conjunto del II-UNAM y el Centro de Investigación Sísmica AC de la Fundación Javier BarrosSierra al Gobierno del estado de Guerrero, Instituto de Ingeniería, UNAM, proyectos 8782 y 9745,México, 1989.

[9] CFE (Comisión Federal de Electricidad), Manual de Diseño de Obras Civiles – Diseño por Viento,México, updated version in edition.