Dynamics of Structures - Appunti

Classification of buildings

Depending on wind analysis buildings can be classified as:

• Low-rise: wind effect acts as static load and is often overshadowed, on design grounds, by seismic actions. Roof loading and internal pressures are important for the structural design.
• Normal: moderate dynamic effects, single-mode dynamic response; we can use equivalent static forces
• Tall: sensitive to dynamic effects, multi-mode response, low-frequency vortex shedding.
• Supertall: aeroelastic behavior; aeroelastic testing is required

Tall buildings

Tall buildings are sensitive to dynamic effects due to fluctuation of fluid particles velocity. This is caused by:

• turbulence in the earth boundary layer
• vortex shedding at low frequency (i.e. lower than the structural natural frequencies)

be also prone to more complex phenomena (galloping). The typical problem for tall buildings is comfort: the check is done with comparison in terms of acceleration values, which are much more difficult to estimate than other quantities such as displacements.

5.5 Wind forces in EC1

In the Eurocode 1 the wind forces can be applied as equivalent static loads without any dynamic analysis (except for special cases).

F = c c c q (z ) A q = ρ v (z) [1 + 7 I (z)]w s p s p vd f ref m2

This approach can be adopted for structures having a "simple" dynamic behaviour, i.e. characterized by a single "dominant" modal response. The coefficients are:

• A is the reference area (to obtain a force we need an area)
• q (z ) is the peak pressure
• v (z) is the average velocity at elevation z, depends on site "windyness" and terrain
• I (z) represents the intensity of turbulence, depends on the roughness of terrain
• c is the aerodynamic force

coefficient (it is a drag coefficient)f• c c are two different coefficients but we will derive them as a single coefficient, thes dstructural coefficientDynamics of Structures 455. Wind engineering 5.5. Wind forces in EC1

The average velocity is given as: v (z) = c (z) c (z) vm r o b

Where c (z) is the roughness factor and c (z) is the orography factor. The rough-r oness factor is needed since the mean velocity is intended to be computed at elevation of10 meters and, since we have a logaritmic profile, we need a coefficient that allow us tocompute the average velocity at different elevations.

The turbulence intensity I (z) is equal to the standard deviation of the turbulence dividedvby the mean wind velocity.

How the peak velocity is derived?

1 1 12 2′2 q (z) = ρ v = ρ v (z) + v (z) ρ [v (z) + g σ (z)] ==p m m v vp p2 2 211 h i2 22 2 ≃ρ v (z) [1 + g I (z)] = ρ v (z) 1 + (g I (z)) + 2 g I (z)= v v v v v vm m 2 2 1 2≃ ρ v (z) [1 + 7 I (z)]vm2

′Where g = 3.5 is the peak factor of wind fluctuation, v (z) is the fluctuation of the windv p ′≫velocity. The squared term is negligible under the hypothesis of v (z) v (z).m p5.5.1 Spectrum of the longitudinal turbulenceAnother important quantity defined by the Eurocode is the turbulent length scaleL(z) which is a measure of the width of vortexes. The turbulent length scale grows withelevation and with growing L(z) also the probability of reaching the peak pressure grows.The non dimensional power spectrum of longitudinal fluctuation is:

·n S (n) 6.28 f (z, n)v LS (z, n) = =L 2 5/3σ ·(1 + 10.2 f (z, n))v LWhere n is the frequency and: n L (z )i ef (z, n) =L v (z )m eIn defining the spectral model we must take in account coherency, which is not explicitelymentioned in Eurocode. We call point coherency the coherency in one point betweenthe longitudinal average velocity, the orthogonal component and the vertical one. Thecorrelation is small. Then we

Define the space coherency, which is the coherency between two points for the same component. It decays with distance and frequency.

Correlation of wind components means correlation in wind loads and so we ask ourselves how this correlation reflects itself on the response of structures. We look at this topic from the modal superposition perspective:

(j),T2 Qÿ + 2 ξ ω ẏ + ω y = φ = Q̄ (t)j j j j j jj

Where:

n n(j) (j)X XQ̄ (t) = φ ρ d l W C w (t) = φ g w (t)j k k k k d 1,k k 1,kk kk=1 k=1

Dynamics of Structures 465. Wind engineering 5.5. Wind forces in EC1

In order to explicit coherency we look at the variance of the modal forcing:

n n" #(j)X X2 (j) E Q̄ (t) = E φ φ g g w (t) w (t) =r 1,rk 1,kj rkr=1k=1

n n (j)X X (j)= E [w (t) w (t)] φ φ g g1,r r1,k krkr=1k=1

Where E [w (t) w (t)] is the covariance between the two time histories of wind velocity 1,r1,k fluctuation, which can be expresses as the integral of the cross spectral density

function: 
n +∞Z (j)X X (j)2  (f ) df φ φ g g =Q̄ (t) = SE rw w krj krk0r=1k=1 n n+∞Z q (j)X X (j)|z − |) gS (f ) S (f ) φ φ g df= exp (−α(z , z ) f z w rwr r kk k rkrk0 r=1k=1
But the variance is the integral of PSD, therefore must hold:
n n q (j)X X (j)|z − |)S (f ) = exp (−α(z , z ) f z S (f ) S (f ) φ φ g gr r w w rk k krQ̄ krkj r=1k=1
We found the bond between the spectral density of wind velocity and that of the force Q̄ .j
Now we will analyze two different cases:
• perfect correlation of time histories regarless of the distance (unitary coherency).
2n n n #"q q (j)(j)X X X(j)S SS (f ) = (f ) S (f ) φ g(f ) φ φ g g =w ww r kkrQ̄ kkrk kj r=1k=1 k=1
The dominating term here is the first mode.
• delta-correlated time histories, with zero coherency even at infinitesimal distance.
n n n 2q h i(j) (j)X X X(j)S (f ) = δ S (f ) S (f ) φ φ g g = S (f ) φ gw w r wkr k krQ̄ k

krk kj r=1k=1 k=1All the normal modes contribute.5.5.2 Structural factorThe definition is based on the following hypotheses and criteria:

• the structure is subjected only to drag forces
• turbulence fluctuations are modelled as a stationary (zero mean) and gaussian random process
• only longitudinal (alongwind) turbulence is considered structural response is "dominated" by a single normal mode
• it is possible to define a single time history of alongwind velocity fluctuation which is equivalent, in terms of forcing of the relevant mode, to the effect of the multi-correlated velocity fluctuation acting on the structure

Dynamics of Structures 475. Wind engineering 5.5. Wind forces in EC1The factor is equal to: √ 2 21 + 2 k I (z ) B + Rp v sc c =s d 1 + 7 I (z )v sWhere: z = 0.6 h is the height; k is the peak factor of the response; B is the backgrounds presponse factor; R is the resonant response factor.We can see the generic effect as the sum of the

static one, due to the average wind velocity,and the fluctuating one: Z (t) = Z + Z(t)tot m

On the other hand, the code definies this effect as a product:Z = c c Ztot,max s Qs,maxd

If we substitute statics concepts to the relation we get:|Z(t)|Z (t) = Z + max = Z + g σtot,max m m z z

Comparing the result with the code formula we obtain:Z g σm z zc c = +s d Z ZQs,max Qs,max

The first term can be analyzed observing that Z has a functional dependency on themsquare of average velocity while Z on the square of the peak valued. Their quotientQs,maxis: 2|w Zmax (z, t)| Qs,maxt = = 1 + 7 I (z)w2w (z) Z mm

The equation becomes: 1 g σ /Zz z mc c = +s d 1 + 7 I (z ) 1 + 7 I (z )w s w s

In the second term comes into play the structural dynamics:n ! w (t) w (t)j j(j)X2 e eÿ + 2 ξ ω ẏ + ω y = φ ρ d l W C = αj j j j j jk dj k M Mj jk=1

When we want to compute a generic effect we do:T T (j)q(t) φ y (t)Z(t) = S = S jT (j)σ = S φ σz y j

σσ yzT T (j) j−→Z = S q = S φ y =m j,mm Z ym j,m

To compute y is sufficient to go in statics:j,m n !2W 1 1 W(j)X f2 kω y = φ ρ d l C = αj jdj k 2 M 2 Mj jk=1

Then, to compute dynamic modal response to random stationary loading w (t):je+∞ +∞Z Z2 2 2|σ = S (f ) df = α H (f )| S (f ) dfy j wey jjj e0 0

Dynamics of Structures 485. Wind engineering 5.6. SDOF model for galloping

This integral can be simplified by considering the fact that the response is divided in abackground response, in which we have the long period fluctuations that excite thestructure essentially as a static load, and a resonant response, in which we have a largeamplification even though the wind power is small.

(Z )−δf S (f )S (f )j jww2 2≃σ α df + eey j 2 3j K 8 ξ M ωj j0 j j

Making all substitutions, rearranging terms and introducing the non dimensional spectrumand logarithmic decay damping we get the following: