Lifetime-Oriented Structural Design Concepts- P3 pps

The failure risk tremes of gust wind speeds at a building location at building height of 35 m above ground d comparison of the load factors of the sectors; the largest load factor is val

7x10 7

3.1x10 11

N 0.85

Fig 2.4 Distribution of absolute frequencies of normalized gust responses into

subsequent classes of different levels of effect

con-struction component, W is the elastic section modulus, A is the loaded area.

the statistical mean

i (N i /N ci) is formed in order to assess resistance of the considered component with respect

per-to fatigue Figure 2.5 shows an example taken from a fatigue analysis of the

S - N c u r v e ( W ö h l e r c u r v e ) o f

s t r e s s c o n c e n t r a t i o n c a t e g o r y 3 6 *

Fig 2.5 Comparison of the distribution of cyclic stress amplitudes with the S-N

curve (W¨ohler curve) of stress concentration category 36* after [30]

gust responses of steel archs of a road bridge The considered cerb is cient to resist the repeated gust impacts The application of the Equations2.12 or 2.17 permits a detailed and safe method for the fatigue analysis ofgust-induced effects at building structures.

suffi-2.1.2 Influence of Wind Direction on Cycles of Gust Responses

Meteorological observations document that the intensity of a storm isstrongly related to its wind direction Figure 2.6(a) shows the wind rosette ofthe airport Hannover, Germany, as an example The probability of the firstpassage of the same threshold value can strongly vary for different sectors ofwind direction That means that the risk of a high wind induced stressing of astructural component is different between the wind directions The failure risk

tremes of gust wind speeds at a building location at building height of 35 m above

ground (d) comparison of the load factors of the sectors; the largest load factor is

valid for the design of the fa¸cade element after Figure 2.8

of the structure or structural components is determined by the superposition

of all probability fractions originating from the sectors of wind direction.Usually, codes follow the conservative approach to assume the same prob-ability of an extreme wind speed for all wind directions In general, more re-alistic and very often also more economic results can be achieved if the effect

of wind direction is considered This can be done by employing wind speedsfor the structural loading which are adjusted in each sector with a directionalfactor Such procedure is in principle permitted by the Eurocode [32] It isleft to the national application documents to regulate the procedures.The wind load is a non-permanent load; within statical proofs of the loadbearing capacity it is employed using a characteristic value, which is defined

as a 98% fractile, and an associated safety factor of 1.5 A load level is requiredwhich is exceeded not more than 0.02 times a year in a statistical sense Suchvalue is statistically evaluated from the collective of yearly extremes of thewind speeds The intensity of the wind load is deduced from the level of thewind speed, or more exact, from its dynamic pressure The related statisticalparameters are used to determine the characteristic value of the load.The wind load depends on the wind direction as the wind speed is differentlydistributed regarding their compass, and as the aerodynamic coefficients varieswith respect to the angle of flow attack Taking this into account the mostunfavourable load can originate from combining a lower characteristic value

of the wind speed, which might be associated to a directional sector, and therelated aerodynamic coefficient for this sector In order to evaluate completelythe effect of the influence of the wind direction it is required to take thestructural response into account, e.g after [227] In such procedure a responsequantity, which is a representative value of the wind action, is evaluated withthe restriction to limit its exceedance probability of its yearly extremes to avalue lower than 0.02 instead of focussing on loads Using this requirement thecharacteristic wind velocities related to the different sectors can be deduced

2.1.2.1 Wind Data in the Sectors of the Wind Rosette

The maximum wind load effect on a structural component is resulting fromthe most unfavourable superposition of the function of the aerodynamic coeffi-cient and the dynamic pressure Both variables are independent and functions

of the direction of mean wind The usual zoning in statistical meteorology into

distri-bution effects The prediction of the risk requires an analysis of the extremewind velocities for each sector at the building location If available a completeset of data is taken from a local station for meteorological observations nearthe considered building location The wind statistics of a considered buildinglocation in the city of Hannover in Germany is shown in Figure 2.6(a) as anexample The wind rosette is evaluated from data collected at the observationstation at the airport of Hannover The terrain in the environment of the sta-tion is plain with a relatively homogeneous surface represented by a roughness

Table 2.1 Conversion of the wind data of the observation station at the airport of

Hannover into data for the building location

Sectors of wind directions

in a standard height of 10 m above ground level, cf J Christoffer and M

are ranked in each sector F , and respective probability distributions are

iden-tified In the presented example distributions of Gumbel-type were adapted.The occurrence probability of an extreme value in a year, which is lower than

In Equation 2.19 U is the modal parameter, and the parameter a describes the

diffusion The wind velocities with return periods of 50 years for all sectorsare listed in Table 2.1, line 1 In opposite to the conditions at the observa-tion station, the building location is surrounded by a terrain with stronglynon-homogeneous surface roughnesses The effect of the varying roughnessessuperpose the undisturbed conditions evaluated for the location of the obser-vation station

These additional effects influence the wind velocity in reference height,its profile and the profile of gustiness over height, which vary between thedirections according to the respective roughness conditions of a sector.The surface roughnesses for each sector are required The local roughness

a radius of 50 to 100 times the height of the considered building, e.g ca 5 km

in case of the considered stadium, Figure 2.7 Mixed profiles are evaluated forthose sectors with significantly changing surface roughnesses; for approxima-tion an equivalent roughness length is adapted The results are shown in line 2

of Table 2.1; the conditions within each sector are described by conversion tors related to the undisturbed wind rosette The factor in line 3 of Table 2.1relates the mean wind speeds with a return period of 50 years at the building

Fig 2.7 Roughness lengths of the

ter-rain in the farther vicinity of the building

location [771]

Fig 2.8 Sketch of a building contour

(top view) with b < 2 h and fa¸cade

el-ement exposed to a pressure coefficient

cp=−1.4 [32] at the eastern fa¸cade in

the case of winds from 0◦

location at a building height of 35 m of the stadium and the reference windspeed of the same return period at the location of the observation station inreference height of 10 m The logarithmic law for the profile of the mean wind

an empirical relation (Equation 2.21)

The gust velocity in the last row of Table 2.1 is calculated from Equation 2.23,

called background response factor after [32]

v =

is also affecting the turbulence intensity, as shown in line 5 of Table 2.1.The statistical evaluation for all sectors leads to a mean wind of 50 years

return period of 23.8 m/s at the building location.

Figure 2.6(b) represents the rosette of mean wind speeds at the buildinglocation In comparison of both wind rosettes, representing the building lo-cation and the location of the observation station, it can be concluded thatthe main character of the local wind climate is preserved but relevant changesdue to the terrain roughness are introduced

2.1.2.2 Structural Safety Considering the Occurrence Probability

of the Wind Loading

The wind load effect on a structure can be expressed in terms of a response

quantity Y For a linear, stiff structure without dynamic amplification, Y is

the surface of the structure for a given wind direction Φ; ρ - mass density of air; A - pressure exposed influence area.

A certain response force Y forms the basis for the determination of a

2ρ · v2

effect by use of the gust velocity v The wind effect admittance depending on

2.24 It covers the distribution and the value of the aerodynamic coefficientwithin the influence area of the load as well as the mechanical admittance,which is the transfer from the dynamic pressure into the response quantity

the complete risk is evaluated as the exceedance probability of the response

quantity Y , which adds up from the contributions from each sector The safety

requirements are met if the total risk has a value smaller than 0.02

iteration until a value smaller 0.02 is achieved In an analogeous manner a

decreased value of v i,lim is introduced aiming on an economical optimization

if the first iteration yields a value much smaller than 0.02

proved within the following steps The main idea of the procedure is to make

2ρ · v2

2ρ · v2 Aprobability of non-exceedance of 0.98 of the applied force must be guaranteedfor both in the sectors and in total

The effect of the direction of the wind on the wind effect is expressed through

a directional wind effect factor:

be calculated from the probability distribution of the mean wind velocity inthe sector as given by Equation 2.19 The probability of the non-exceedance

probabilities under the condition that the yearly extremes in the differentsectors are statistically independent

2.1.2.3 Advanced Directional Factors

The responses of a structure must be taken into consideration for the mination of the relevant wind speeds and wind loads for each sector This

deter-Table 2.2 Determination of a reduced characteristic suction force on the fa¸cadeelement after Figure 2.8 through the consideration of the effect of wind direction

on loading line 1: extreme gust speed at a building location at Hannover at

build-ing height of 35 m; line 2: c p,10-values at the considered fa¸cade element for wind

flow from the respective directions; line 3: directional wind effect factor after tion 2.8; line 4: iterative determination of applicable wind speeds in sectors and associated non-exceedance probabilities in sectors; line 5: applicable fraction of

Equa-codified standard load after the proposed method

Sectors of wind directions

northern, eastern, southern and western directions The question is if reducedvalues of the suction forces at the cladding elements at the edge of the easternfa¸cade can be adopted as the wind rosettes clearly indicate different wind ex-tremes when comparing the sectors, cf line 1 in Table 2.2 Wind from easterndirections generate pressure forces at the element, whereas suction forces at thesame element are generated through winds from all other sectors Suction co-efficients from [26], Table 3, are used to describe the aerodynamic admittance

pres-sure minimum — or maximum suction — occurs for northern directions and is

in-serted for western wind directions (cf line 2 in Table 2.2)

The directional wind effect factor a(φ) in line 3 after Equation 2.28 is

calculated refering the sectorial pressure coefficents to the minimum pressure

the probability of non-exceedance of the values given in line 1, or it is 1 in

application of Equation 2.29 leads to P = 0.8171 < 0.98 In a second iteration

the extreme wind speeds are increased in such a way that the total probability

of non-exceedance after Equation 2.29 results to be larger or equal to 0.98.The third and fourth row in line 4 of Table 2.2 represent a valid solution for

which P = 0.9866 and results larger than the required value of P = 0.98.

The codified standard design procedure requires a reference wind speed of

after the wind profile for midlands ([26], Table B.3) leads to a characteristic

gust speed of v = 41.3 m/s at building height of 35 m The standard suction

force for the considered element — without any consideration of the

quotient is listed in line 5 of Table 2.2, and it is represented in Figure 2.6,(d) The largest factor in line 5 must be applied The respective characteris-

tic velocity is ca 59 m/s but the associated characteristic suction force after

Equation 2.26 is lower than the standard suction force after the code Thereason is in the application of the much higher pressure coefficent — or lower

The procedure can also be adopted for a fatigue analysis after Equation 2.9

2.1.3 Vortex Excitation Including Lock-In

Vortex excitations represent an aerodynamic load type which can causevibrations leading to fatigue, especially for slender bluff cylindrical structures

— bridge hangers, towers or chimneys

The nature of air flow around the structure depends strongly on the windvelocity and on the dimensions of the structure Accordingly, different windvelocity ranges can be defined, depending on the value of a non-dimensionalparameter called the Reynolds-number

body in the across-wind direction — for cylindrical structures, the diameter

formed and alternately shed in the wake of the cylinder creating the von

alter-nating force which acts on the structure in the across-wind direction.The nature of the vortex shedding and of the lift force is considerablyinfluenced by the wind turbulence

¯

Fig 2.9 Von K´arm´anvortex trail formed by vortex shedding

the across-wind force is a harmonic function of the time t:

F l (t) = ρ¯ u

2

is the frequency of the vortex shedding, also called the Strouhal-frequency

The non-dimensional coefficient S in (2.33) is the Strouhal-number which

turbulent flow, the excitation frequencies are distributed in an interval aroundthe mean frequency, the width of the interval depending on the turbulence.When the Strouhal-frequency approaches one of the natural frequencies

ampli-tudes because the resonance, an aeroelastic phenomenon, the so-called lock-ineffect occurs This results in the synchronization of the vortex shedding pro-cess to the motion of the excited structure (Figure 2.10), acting as a negativeaerodynamic damping, and can lead to very large oscillation amplitudes Con-sequently, the lock-in effect can play an essential role in the evolution of thefatigue processes in the damage-sensitive parts of the structure

The width of the lock-in range is zero for a fixed system and increaseswith increasing oscillation amplitude As the amplitude depends on mass anddamping, these system-parameters have a large influence on the lock-in effect.This influence can be numerically catched by introducing the dimensionless

where μ denotes the mass of the structure per unit length, and δ is the

structural logarithmic damping decrement The width of the lock-in range is

1 Generally only the first natural frequency is of practical importance

Fig 2.10 Dependence of the vortex shedding frequencyfvon the wind velocity ¯u.

fn is the natural frequency of the structure

reduced with increasing Scruton-number, and for very large values of Sc, no

lock-in effect occurs at all

In the case of a uniform smooth flow, the lift force per unit span acting on

a circular cylinder fixed in both the along-wind and across-wind directions isgiven by (2.32) However, the force is not fully correlated along the cylinderspan If the cylinder is allowed to oscillate, the magnitude of the lift force andalso the correlation increases The equation of motion of the cylinder is givenby

where y denotes the across-wind displacement, m, c and k are the mass, the

damping coefficient and the stiffness of the cylinder per unit span As the lift

diameter and on time, but also on the displacement, on the velocity and on the

expression Furthermore, the wind velocity u(t) is a stochastic variable which

generally describes a turbulent wind process, and consequently a suitable windload model must also correctly describe the oscillations in turbulent flow.Much effort has been done in order to find an expression for the across-windforce which fits the experimentally observed facts However, all of the wind-load models developed up to the present can only describe the experimentallyobserved oscillations correctly if some limiting conditions are fulfilled

2.1.3.1 Relevant Wind Load Models

The Ruscheweyh-model [695], which is implemented in the German CodesDIN 4131 (Steel radio towers and masts) and DIN 4133 (Steel stacks), de-scribes the across-wind oscillations in the time domain The lift force per unit

2 The lift force also depends on the roughness of the cylinder surface, which is herenot explicitely shown

span is given by (2.32) The lift coefficient is C l = 0.7 for Re ≤ 3 × 105,

synchro-nized vortex shedding along the cylinder span The increase of correlation

This model predicts the oscillation amplitudes of slender cylindrical tures in a smooth wind flow for constant mean wind velocities within andoutside of the lock-in range with a remarkable accuracy Large estimationerrors occur, however, in the case of high turbulence, or if the mean windvelocity considerably varies in time — especially in the case of entering orexiting the lock-in range

struc-The Vickery-model [811] uses a frequency-domain-approach to describethe across-wind vibrations Assuming a Gaussian distribution for the spectraldensity of the lift force, the standard deviation (rms-value or effective value)

of the across-wind deflection is obtained as

effective mass and damping ratio of the structure, h is the height of the der and B is a dimensionless parameter which describes the relative width

depends on the wind turbulence

The model is suitable for predicting the oscillation amplitudes, both insmooth and turbulent flow, but it is limited to the case of stationary flow, i e

to constant mean wind velocities, and it doesn’t take the lock-in effect intoconsideration

The model of Vickery and Basu [810] describes the across-wind tions in smooth or turbulent flow, with mean wind velocities outside or withinthe lock-in range The lift force is written as the sum of two forces: a narrow-band stochastic term with a normal distribution of the spectral density and amotion dependent term — negative aerodynamic damping — which describesthe lock-in effect For the lock-in range, the rms-value of the displacement isobtained as

oscilla-σ y = 2.5 C l ρD

3L c 16π2S2

aerodynamic damping ratio The aerodynamic damping is negative in the

is defined in such a way that it limits the amplitude to a predefined value.The most exhaustive model of vortex-induced across-wind vibrations hasbeen developed by ESDU [262], mainly based on the work of Vickery and

oscilla-tion amplitude and incorporate the influences of turbulence and of the lock-ineffect The system response is obtained from the superposition of a broad-and of a narrow-band term A very large variety of parameters, such as thesurface roughness or the integral length of the turbulent wind, is included inthe calculation Also, the dependence of the lock-in range width on the oscil-lation amplitude is taken into consideration Because of their complexity, theresponse equations will not be presented here Like all the models presentedabove, also this model is only suitable to describe the across-wind vibrations

in a stationary or quasi-stationary flow, i e if the mean wind velocity doesn’tchange too rapidly and if there is no transition into or from the lock-in range

sug-gested by Vickery and Clark [811], Lou has developed a convolution model[507] which describes the lift force in the time-domain, for a stationary tur-bulent flow, outside of the lock-in range:

2.1.3.2 Wind Load Model for the Fatigue Analysis of Bridge

Fig 2.11 Wind velocity, measured and simulated deflection vs time for the bridge

hanger 1 (left) and 2 (right) The horizontal lines in the upper panels show the meanwidth of the lock-in range

their extremely damage-sensitive welded connections Therefor, the vibrations

of two hangers have been filmed by digital cameras, and the time histories

of the deflections have been extracted from the videos by means of a Javaprogram Simultaneously, the fluctuating wind velocity has been recorded with

an ultrasonic 3D-anemometer The mean wind velocity varied with time insuch a way, that one of the hangers entered and exited the lock-in range severaltimes during the measurement, while the other one stayed outside of the lock-

in range, see Figure 2.11 Because of the low oscillation amplitude, the lock-inrange of the second hanger was very narrow; it lies within the width of thehorizontal line in the upper right panel

In order to check the validity of the previously presented wind load els for bridge hangers, the amplitudes measured on hanger 1 in the lock-inrange, in the time-interval between 8.5–12.5 min have been compared to thepredictions of the Ruscheweyh- [695] and ESDU-models [262] The experi-ment shows a peak amplitude of ca 9 mm and an rms-amplitude of ca 6 mm,while the Ruscheweyh-model predicts peak amplitude of about 5 mm andthe ESDU-model an rms-amplitude of ca 30 mm

mod-As both models show a substantial discrepancy compared to the sured values, a new wind load model for the across-wind vibrations of bridgetie-rods in non-stationary, turbulent flow, including the lock-in effect, hasbeen developed [296], based on the model by Lou [507] For this purpose, a

mea-power-function dependence of the parameter β in (2.40) on the fluctuating wind velocity u has been supposed:

with the fit-parameters K and n Furthermore, in order to describe the

non-stationary wind process, the mean values in (2.40) have been replaced by

supposed to be constant The lift force per unit span obtained this way is:

in the lock-in range, it is set in phase with the rod motion:

natural frequency of the rod The increase of the force amplitude caused by thephase-synchronization is compensated by the reduction of the multiplicative

parameter K in equation (2.43) for the lock in range.

It has been assumed that the lock-in range is symmetric with respect to the

critical wind velocity (2.37) with a half width Δu depending on the oscillation

experimental data

The fit of the model parameters to the experimental data has been formed by simulating the vortex-induced vibrations in the time domain, on

per-a finite-element model of the hper-anger, which hper-as been excited by the force

calculated using equation (2.43) applied to the experimental wind data u(τ ).

The time dependent deflections have been calculated using the

n = 3, are shown in the lower panels of Figure 2.11 For the lock-in range, the multiplicative parameter K has been reduced by a factor 4.