0407b548-c4f0-488f-bdcf-8ba35dcb03eb 311 Alessandro 4 5 false false 1 1 decimal

&[DATE] &[TIME] - &[FILENAME]

WIND ON MAST IN ACCORDANCE WITH UNI EN 1991 - 1 - 4 : 2010

VA CONTROLLATO SE NELL'ULTIMA VERSIONE C'è QUALCOSA DI NUOVO

The wind velocity and the velocity pressure are composed of a mean and a fluctuating component.
The mean velocity vm, should be determined from the basic wind velocity vb, which depends on:
- Climate;
- Height variation of the wind (determined from terrain roughness and relative orography)
The fluctuating component of the wind is represented by the turbolence intensity

Flare description

ST_T Unlined welded steel stucks without ext. thermal insulation 0.012 Unlined welded steel stucks with ext. thermal insulation 0.020 Steel stack with internal brick liner 0.070 Steel stack with internal gunite 0.030 Coupled stacks without liner 0.015 Guyes steel stack without liner 0.04 62 mat :

Sur_T Glass 0.0015 mm * Polished metal 0.002 mm * Fine paint 0.006 mm * Spray paint 0.02 mm * Bright steel 0.05 mm * Cast iron 0.2 mm * Galvanized steel 0.2 mm * Smooth concrete 0.2 mm * Planed wood 0.5 mm * Rought concrete 1 mm * Raunght sawn wood 2 mm * Rust 2 mm * Brickwork 3 mm * 132 mat :

IC Ice-Free condition Iced condition 21 mat :

Structural type definition in accordance with Table F.2 EC 1 - 4 (Annex F)

"Unlined\0020\welded\0020\steel\0020\stucks\0020\without\0020\ext\002E\\0020\thermal\0020\insulation"
"Unlined\0020\welded\0020\steel\0020\stucks\0020\with\0020\ext\002E\\0020\thermal\0020\insulation"
"Steel\0020\stack\0020\with\0020\internal\0020\brick\0020\liner"
"Steel\0020\stack\0020\with\0020\internal\0020\gunite"
"Coupled\0020\stacks\0020\without\0020\liner"
"Guyes\0020\steel\0020\stack\0020\without\0020\liner"

"Glass"
"Polished\0020\metal"
"Fine\0020\paint"
"Spray\0020\paint"
"Bright\0020\steel"
"Cast\0020\iron"
"Galvanized\0020\steel"
"Smooth\0020\concrete"
"Planed\0020\wood"
"Rought\0020\concrete"
"Raunght\0020\sawn\0020\wood"
"Rust"
"Brickwork"

Ty_S

Type of treatment surface

"Ice\002D\Free\0020\condition"
"Iced\0020\condition"

ICE

Definition of ice presence

Geometrical and design data of mast

Material design parameter of mast

h 29 m * :

Maximum height of structure

f.y 230 MPa * :

Yelding stress of mast

W 395.3 kN * :

Total weight of structure

E.m 200000 MPa * :

Elastic modulus of elasticity

ρ.s 78.5 kN m 3^/ * :

Shell density of the element

ν.s 0.3 :

Poisson ratio

V.N 20 :

Nominal life of the structure (in years)

n.p 23 :

Total number of point in which mast is discretized

i 1 n.p range :

Discretized point considered

b.i b t 2 / - :

Inner diameter of the mast

b.o b.i t + :

Outer diameter of the structure

I i el π 64 / b.o i el 4^b.i i el 4^- (* :

Bending inertia of the mast

δ.s ST 2 el : 0.012

Thickness

Diameter

Inner diameter at each levels

mm 112 range :

nn 1323 range :

22007 - 2193

2200 4.5 - 2195.5

z 0 1.7 3.4 5.2 5.7 6.9 8.6 10 11.3 12.7 14 15.4 16.8 16.9 18.1 19.5 20.8 22.2 23.6 24.9 26.3 27.6 29231 mat m * :

t 14141414141499999 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 231 mat mm * :

b 219321932193219321932193 2195.5 2195.5 2195.5 2195.5 2195.5 2196.75 2196.75 2196.75 2196.75 2196.75 2196.75 2196.75 2196.75 2196.75 2196.75 2196.75 2196.75 231 mat mm * :

2200 3.25 - 2196.75

Table1 z b t bi bo Ii [m] [cm] [mm] [cm] [] [dm^4] z mm el m / b mm el cm / t mm el mm / b.i mm el cm / b.o mm el cm / I i el mm el dm 4^/ 36 mat :

Tablexx z b t bi bo Ii [m] [cm] [mm] [cm] [] [dm^4] z nn el m / b nn el cm / t nn el mm / b.i nn el cm / b.o nn el cm / I i el nn el dm 4^/ 36 mat :

Tablexx Table1 δ.s b.o b.i ν.s ρ.s E.m W f.y h Ty_S ST Tablexx Table1 b t z nn mm δ.s I b.o b.i i ν.s ρ.s E.m W f.y h IC δ.s b.o b.i ν.s ρ.s E.m W f.y h

Table1
z	b	t	bi	bo	Ii
[m]	[cm]	[mm]	[cm]	[]	[dm^4]
mat(0,1.7,3.4,5.2,5.7,6.9,8.6,10,11.3,12.7,14,15.4,12,1)	mat(219.3,219.3,219.3,219.3,219.3,219.3,219.55,219.55,219.55,219.55,219.55,219.68,12,1)	mat(14,14,14,14,14,14,9,9,9,9,9,6.5,12,1)	mat(218.6,218.6,218.6,218.6,218.6,218.6,219.1,219.1,219.1,219.1,219.1,219.35,12,1)	mat(220,220,220,220,220,220,220,220,220,220,220,220,12,1)	mat(289.92,289.92,289.92,289.92,289.92,289.92,187.01,187.01,187.01,187.01,187.01,135.3,12,1)
#
#
#
#
#

Tablexx Table1 δ.s b.o b.i ν.s ρ.s E.m W f.y h Ty_S ST Tablexx Table1 b t z nn mm δ.s I b.o b.i i ν.s ρ.s E.m W f.y h IC δ.s b.o b.i ν.s ρ.s E.m W f.y h

Tablexx
z	b	t	bi	bo	Ii
[m]	[cm]	[mm]	[cm]	[]	[dm^4]
mat(16.8,16.9,18.1,19.5,20.8,22.2,23.6,24.9,26.3,27.6,29,11,1)	mat(219.68,219.68,219.68,219.68,219.68,219.68,219.68,219.68,219.68,219.68,219.68,11,1)	mat(6.5,6.5,6.5,6.5,6.5,6.5,6.5,6.5,6.5,6.5,6.5,11,1)	mat(219.35,219.35,219.35,219.35,219.35,219.35,219.35,219.35,219.35,219.35,219.35,11,1)	mat(220,220,220,220,220,220,220,220,220,220,220,11,1)	mat(135.3,135.3,135.3,135.3,135.3,135.3,135.3,135.3,135.3,135.3,135.3,11,1)
#
#
#
#
#

INSERIRE TUTTI I DATI RELATIVI IN QUESTA FINESTRA

Where, with reference to each point of discretized structure
- z: Reference heigth - t : Thickness; - b0: Outer diameter of mast;
- b: Average diameter - bi: Inner diameter of mast; - Ii: Flexural inertia

WIND VELOCITY AND WIND PRESSURE DEFINITION

Basic values for wind action

Terrain categories and relative terrain parameters for studied location: ALABAMA (Lat. 32.766543 Long. -86.840287 )

115 mph * m s / 51.4096

TCat 0 Sea or coastal area exposed to the open sea 0.003 m * 1 m * I Lakes or flat and horizontal area with negligeble vegetation and without obstacles 0.01 m * 1 m * II Area with low vegetation such as grass and isolated obstacles (trees, buildings) with separation of at least 20 obstacle height 0.05 m * 2 m * III Area with regular cover of vegetation or buildings or with isolated obstacles with separation of maximum 20 obstacle heights (such as villages, suburban terrain, permanent forest 0.3 m * 5 m * IV Area in which at least 15 % of the surface is covered with buildings and their average height exceeds 15 m 1 m * 10 m * 54 mat :

"0"
"I"
"II"
"III"
"IV"

T_C

Terrain category (in accordance with Annex A of EC1)

Description T_C 2 el : Area with regular cover of vegetation or buildings or with isolated obstacles with separation of maximum 20 obstacle heights (such as villages, suburban terrain, permanent forest

z.0 T_C 3 el :

z.min T_C 4 el :

z.0 0.3 m *

Roughness length defined in table 4.1 depending on representative terrain categories

z.min 5 m *

Minimum height defined in table 4.1 depending on representative terrain categories

z.max 200 m * :

Maximum height for which wind analisys with EC1 is admissible

z.0_II 0.05 m * :

Reference roughness lenght for terrain of category II (Table 4.1)

ρ 1.25 kg m 3^/ * :

Reccomended value for air density. More accurate value could be defined by National Annex

ν 15106 -^* m 2^s / * :

Cinematic viscosity of the air

Definition of basic wind velocity and pressure

v.b0 66.96 m s / * :

Reference site for find this value:
https://www.dlubal.com/en/load-zones-for-snow-wind-earthquake/wind-lst-en-1991-1-4.html?#&center=55.39763106897136,25.221955874999992&zoom=7&marker=54.8985207,23.9035965

Foundamental value of the basic wind velocity (obtained by national Annex and related to the investigated Region)

c.dir 1 :

Recommended value for directional factor (different values could be set in relation of National Annex)

c.season 1 :

Recommended value for season factor (different values could be set in relation of National Annex)

Return Period and probabiliy factor

Dur ≤3 days 2 ≤3 months (but >3 days) 5 ≤1 years (but >3 months 10 >1 years 5042 mat :

"\2264\3\0020\days"
"\2264\3\0020\months\0020\\0028\but\0020\\003E\3\0020\days\0029\"
"\2264\1\0020\years\0020\\0028\but\0020\\003E\3\0020\months"
"\003E\1\0020\years"

Duration

Duration of permanent utilization of structure (as per Table 1 EN 1991-1- 6)

T.r Duration 2 el : 50

T.r 50

Return period of wind action

K.pr 0.2 :

Reccomended value of shape parameter depending on the coefficient of variation of the extreme value distribution

n.pr 0.5 :

Reccomended value of exponent

T.r c.prob 1 K.pr 11 T.r / - ln - ln * - 1 K.pr 0.98 ln - ln * - / (n.pr^: 1

Probability factor

T.r v.b c.dir c.season * T.r c.prob * v.b0 * : 67 m s / *

Basic wind velocity, defined as a function of wind direction and time of year at 10 m above ground of terrain category II

q.b 12 / ρ * T.r v.b 2^* : kPa 2.8

Basic velocity pressure

Mean Wind

Definition of mean wind characteristics is reported below.

Terrain roughness

The roughness factor accounts the variability of the mean wind velocity at the site of the structure due to:
- Height above ground level;
- Ground roughness of the terrein upwind of the structure in the wind direction considered.

k.r 0.19 z.0 z.0_II / (0.07^* : 0.2154

Terrain factor depending on the roughness lenght z0

z c.r z z.min ≤ k.r z.min z.0 / ln * k.r z z.0 / ln * if :

Roughness factor

Terrain roughness to be used for a given wind direction depends on the ground roughsess and the dstance with uniform terrain roughness in the angular sector around the wind direction.
Small areas with deviating roughness may be ignored.

Terrain orography and related coefficients

Orography coefficient take into account both topography and orography characteristics of site where the structure is locate.
Generally it is assumed equal to 1, but where the orograpy ( hills, cliffts etc.) increased wind velocies by more than 5%, the effects should be taken into account.

TS Valley or Terrain with slope less than 5% Upwind section for all orography Downwind section for cliff and escarpments Downwind section for hills and ridges 41 mat :

"Valley\0020\or\0020\Terrain\0020\with\0020\slope\0020\less\0020\than\0020\5\0025\"
"Upwind\0020\section\0020\for\0020\all\0020\orography"
"Downwind\0020\section\0020\for\0020\cliff\0020\and\0020\escarpments"
"Downwind\0020\section\0020\for\0020\hills\0020\and\0020\ridges"

Orography configuration investigate

x 1500 (m * :

Horizontal distance of the site from the top of the crest

L.u 100 m * :

Actual length of the downwind slope in the wind direction

z rappresenta l'altezza sul suolo della costruzione con riferimento al punto più alto

Vertical distance from the ground level of the site

z.top z max :

L.d 1000 m * :

Actual length of the upwind slope in the wind direction

NOTA : è importante utilizzare il segno per x al fine di identificare in che condizione mi trovo
- Per la condizione sopravento a) valida per tutte le fondigurazioni orografiche la x sarà SEMPRE NEGATIVA;
- Per la condizione sottovento b) valida solo per la figura A.2 x deve essere per forza POSITIVO ossia sopra il rilievo nella parte di destra;
- Anche per la condizione sottovento c) che fa riferimento alla figura A.3 x deve essere POSITIVO

H 0 m * :

Effective height of the feature

Φ H L.u abs / : % 0

Up wind slope in the wind direction

Slope_Type Φ 5 % * > (Φ 30 % * < (& Shallow Steep if : Steep

Slope_Type Steep

L.e Φ 5 % * > (Φ 30 % * < (& L.u H 0.3 / if : 0

Effective length of the upwind slope

A TS 1 ≡ - TS 2 ≡ (x L.u / 1.5 - ≥ (x L.u / 0 ≤ (& (z.top L.e / 0 ≥ (& (z.top L.e / 2 ≤ (& 0.1552 z.top L.e / (4^* 0.8575 z.top L.e / (3^* - 1.8133 z.top L.e / (2^* + 1.9115 z.top L.e / (* - 1.0124 + 0 if TS 3 ≡ x L.e / 0.1 ≥ (x L.e / 3.5 ≤ (& (z.top L.e / 0.1 ≥ (& (z.top L.e / 2 ≤ (& 1.342 z.top L.e / 10 log (3^* - 0.82222 z.top L.e / 10 log (2^* - 0.4609 z.top L.e / 10 log (* + 0.0791 - x L.e / 0 ≥ (x L.e / 0.1 ≤ (& (1.342 z.top L.e / 10 log (3^* - 0.82222 z.top L.e / 10 log (2^* - 0.4609 z.top L.e / 10 log (* + 0.0791 - z.top L.e / 1 < 1.342 0.1 10 log (3^* - 0.82222 0.1 10 log (2^* - 0.4609 0.1 10 log (* + 0.0791 - - if if if TS 4 ≡ 0.1552 z.top L.e / (4^* 0.8575 z.top L.e / (3^* - 1.8133 z.top L.e / (2^* + 1.9115 z.top L.e / (* - 1.0124 + 0 if if if if : -

B TS 1 ≡ - TS 2 ≡ (x L.u / 1.5 - ≥ (x L.u / 0 ≤ (& (z.top L.e / 0 ≥ (& (z.top L.e / 2 ≤ (& 0.3542 z.top L.e / (2^* 1.0577 z.top L.e / (* - 2.6456 + 0 if TS 3 ≡ x L.e / 0.1 ≥ (x L.e / 3.5 ≤ (& (z.top L.e / 0.1 ≥ (& (z.top L.e / 2 ≤ (& 1.0196 z.top L.e / 10 log (3^* - 0.8910 z.top L.e / 10 log (2^* - 0.5343 z.top L.e / 10 log (* + 0.1156 - x L.e / 0 ≥ (x L.e / 0.1 ≤ (& (1.0196 z.top L.e / 10 log (3^* - 0.8910 z.top L.e / 10 log (2^* - 0.5343 z.top L.e / 10 log (* + 0.1156 - z.top L.e / 1 < 1.0196 0.1 10 log (3^* - 0.8910 0.1 10 log (2^* - 0.5343 0.1 10 log (* + 0.1156 - - if if if TS 4 ≡ 0.3056 z.top L.e / (2^* - 1.0212 z.top L.e / * + 1.7637 - 0 if if if if : -

C TS 1 ≡ (TS 2 ≡ (TS 4 ≡ (& (| - TS 3 ≡ (x L.e / 0.1 ≥ (x L.e / 3.5 ≤ (& (z.top L.e / 0.1 ≥ (& (z.top L.e / 2 ≤ (& 0.8030 z.top L.e / 10 log (3^* 0.4236 z.top L.e / 10 log (2^* + 0.5738 z.top L.e / 10 log (* - 0.1606 + x L.e / 0 ≥ (x L.e / 0.1 ≤ (& (0.8030 z.top L.e / 10 log (3^* 0.4236 z.top L.e / 10 log (2^* + 0.5738 z.top L.e / 10 log (* - 0.1606 + z.top L.e / 0.1 < (0.8030 0.1 10 log (3^* 0.4236 0.1 10 log (2^* + 0.5738 0.1 10 log (* - 0.1606 + - if if if - if if : -

s TS 1 ≡ - TS 2 ≡ x L.u / 1.5 - ≥ (x L.u / 0 ≤ (& (z.top L.e / 0 ≥ (& (z.top L.e / 2 ≤ (& A e B x L.u / * (^* x L.u / 1.5 - < ((z.top L.e / 2 > (| 0 Check if if TS 3 ≡ x L.e / 0.1 ≥ (x L.e / 3.5 ≤ (& (z.top L.e / 0.1 ≥ (& (z.top L.e / 2 ≤ (& A x L.e / 10 log (2^* B x L.e / 10 log * + C + x L.e / 0 ≥ (x L.e / 0.1 ≤ (& (0 0.1 21 mat 0.1552 z.top L.e / (4^* 0.8575 z.top L.e / (3^* - 1.8133 z.top L.e / (2^* + 1.9115 z.top L.e / (* - 1.0124 + A 0.1 10 log (2^* B 0.1 10 log * + C + 21 mat x L.e / linterp z.top L.e / 1 < A x L.e / 10 log (2^* B x L.e / 10 log * + C + x L.e / 3.5 > (z.top L.e / 2 > (| 0 Check if if if if TS 4 ≡ x L.d / 0 ≥ (x L.d / 2 ≤ (& (z.top L.e / 0 ≥ (& (z.top L.e / 2 ≤ (& A e B x L.d / * (^* x L.d / 2 > (z.top L.e / 2 > (| 0 Check if if Check if if if if : -

Follow pictures highlight the different condition for evaluate orography location factor condirering the design configuration setted

s -

Orographic location factor, to be obtained from Figura A.2 or Figura A.3 scaled to the length of the effective upwind slope length, depends from the considered condition analized for the studied case (if no value is assessed means the terrain is flat)

z.top c.o Φ 5 % * < 1 Φ 5 % * ≥ (Φ 30 % * < (& 12 s * Φ * + 1 0.6 s * + if if : 1

Orography factor

z v.m z c.r z.top c.o * T.r v.b * :

Mean wind velocity at a height z above the terrain

Wind turbolance intensity

The wind turbolance intensity at height z is defined as the standard deviation of the turbolance devided by the mean wind velocity

k.l 1 :

Recommended value in accordance with EC1 (refer to National Annex for more accurated evaluation)

σ.v k.r T.r v.b * k.l * : 14.4225 m s / *

Standard deviation of the turbolance

z I.v z z.min < σ.v z.min v.m / σ.v z v.m / if :

Turbolance intensity at z height

Peak velocity pressure

The peack velocity pressure in evaluated including mean and short-term velocity fluctuation

z q.p 17 z I.v * + (12 / * ρ * z v.m (2^* :

Peak velocity pressure at different height

z c.e z q.p q.b / :

Exposure factor highlighted in figure 4.2 of EC1 for different terrain category

Table2 z vm Iv Ce qp [m] [m/s] - - [kPa] z mm el m / z v.m vectorize mm el m s / / z I.v vectorize mm el z c.e vectorize mm el z q.p vectorize mm el kPa / 35 mat :

Table_x1 z vm Iv Ce qp [m] [m/s] - - [kPa] z nn el m / z v.m vectorize nn el m s / / z I.v vectorize nn el z c.e vectorize nn el z q.p vectorize nn el kPa / 35 mat :

v._m z v.m vectorize m s / / z m / augment :

I._v z I.v vectorize z m / augment :

q._p z q.p vectorize kPa / z m / augment :

c._e z c.e vectorize z m / augment :

Definition of main wind design parameters

Table_x1 Table2 c.e q.p I.v σ.v v.m L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration v.b0 ν ρ z.0_II z.max z.min z.0 T_C Tablexx Table1 δ.s b.o b.i ν.s ρ.s E.m W f.y h Ty_S ST Table_x1 Table2 c.e q.p I.v σ.v v.m L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 b t z nn mm δ.s I b.o b.i i ν.s ρ.s E.m W f.y h IC c.e q.p σ.v v.m L.d L.u x k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 δ.s b.o b.i ν.s ρ.s E.m W f.y h

Table2
z	vm	Iv	Ce	qp
[m]	[m/s]	-	-	[kPa]
mat(0,1.7,3.4,5.2,5.7,6.9,8.6,10,11.3,12.7,14,15.4,12,1)	mat(40.58,40.58,40.58,41.14,42.47,45.22,48.4,50.57,52.34,54.02,55.43,56.8,12,1)	mat(0.36,0.36,0.36,0.35,0.34,0.32,0.3,0.29,0.28,0.27,0.26,0.25,12,1)	mat(1.28,1.28,1.28,1.3,1.36,1.47,1.61,1.71,1.79,1.87,1.93,2,12,1)	mat(3.59,3.59,3.59,3.65,3.81,4.13,4.52,4.79,5.01,5.23,5.42,5.6,12,1)
#
#
#
#
#

Table_x1
z	vm	Iv	Ce	qp
[m]	[m/s]	-	-	[kPa]
mat(16.8,16.9,18.1,19.5,20.8,22.2,23.6,24.9,26.3,27.6,29,11,1)	mat(58.06,58.14,59.13,60.2,61.14,62.08,62.96,63.73,64.52,65.22,65.93,11,1)	mat(0.25,0.25,0.24,0.24,0.24,0.23,0.23,0.23,0.22,0.22,0.22,11,1)	mat(2.06,2.06,2.11,2.16,2.21,2.26,2.3,2.34,2.38,2.42,2.45,11,1)	mat(5.77,5.78,5.92,6.06,6.19,6.33,6.45,6.56,6.67,6.77,6.88,11,1)
#
#
#
#
#

Where, with reference to each point of discretized structure :
- z: reference heigth; - Iv : turbolence intensities; - qp: Peack pressur.
- vm: average wind velocity - Ce: exposure factor;

Distribution of main parameters with height

Mean wind velocity profile

v._m

Wind turbolence profile

I._v

Exposure factor profile

c._e

Peack velocity pressure profile

q._p

DEFINITION OF WIND ACTIONS

Wind ctions on structures and structural elements shall be determined taking into account the definition of structural factors, force coefficient, peack pressure and the reference area.

Wind Force coefficients for circular cylinder

Force coefficient of cylinder without free-end flow definition

k Ty_S 2 el : mm 0.006

k mm 0.006

Equivalent surface coefficient as per Table 7.13

Reynolds Number definition

z v 2 z q.p * ρ / sqrt :

Peack wind velocity at reference height h

z R.e b z v * ν / :

Reynolds number for circular cylinder

Choosen curve A or B depends from diagram distribution of figure 7.28 that define the force coefficient for circular cylinder with - out free - end flow and for different equivalent roughness k / b

b.mean b i 1 + el z i 1 + el z i el - (* i 1 n.p 1 - sum z i 1 + el z i el - (i 1 n.p 1 - sum / : 2.2 m *

Eq.r i el k b i el / :

Equivalent roughness

c.f0 i el Eq.r i el 105 -^≤ (z R.e vectorize i el 5.5 105^* ≤ (& 0.11 z R.e vectorize i el 106^/ (1.4^/ 1.2 12 mat min Eq.r i el 105 -^≤ (z R.e vectorize i el 5.5 105^* > (& 1.2 0.18 10 Eq.r i el * log10 * 1 0.4 z R.e vectorize i el 106^/ log10 * + / + 0.4 12 mat max Eq.r i el 105 -^> (Eq.r i el 104 -^≤ (& z R.e vectorize i el 3.5 105^* ≥ 1.2 0.18 10 Eq.r i el * log10 * 1 0.4 z R.e vectorize i el 106^/ log10 * + / + 0.4 12 mat max 0.11 z R.e vectorize i el 106^/ (1.4^/ 1.2 12 mat min if Eq.r i el 104 -^> (Eq.r i el 103 -^≤ (& z R.e vectorize i el 2.8 105^* ≥ 1.2 0.18 10 Eq.r i el * log10 * 1 0.4 z R.e vectorize i el 106^/ log10 * + / + 0.4 12 mat max 0.11 z R.e vectorize i el 106^/ (1.4^/ 1.2 12 mat min if Eq.r i el 103 -^> (Eq.r i el 102 -^≤ (& z R.e vectorize i el 2.2 105^* ≥ 1.2 0.18 10 Eq.r i el * log10 * 1 0.4 z R.e vectorize i el 106^/ log10 * + / + 0.4 12 mat max 0.11 z R.e vectorize i el 106^/ (1.4^/ 1.2 12 mat min if Eq.r i el 102 -^> (z R.e vectorize i el 2.2 105^* ≥ (& 1.2 0.18 10 Eq.r i el * log10 * 1 0.4 z R.e vectorize i el 106^/ log10 * + / + 0.4 12 mat max 0.11 z R.e vectorize i el 106^/ (1.4^/ 1.2 12 mat min if if if if if if :

SCHEMA DI DEI PUNTI DI PASSAGGIO TRA LE DIVERSE CURVE

Table_1 Force coefficient for circular cylinder Equivalent surface coeff. k k mm / [mm] Reynolds number for circ. cylinder Re R.e - Equivalent roughness k/b k b / - Force coeff. for circ. cylinder without free-end flow cf0 c.f0 - Effective slenderness for circular cylinder λ λ - Solidity Ratio φ φ - End-effect factor in accordance ψλ ψ.λ - Force coefficient for a finite circular cylinder cf c.f - 84 mat 21 mat :

M.y h b.mean / 70 Max 0.7 h b.mean / * 70 Max 21 mat : 707021 mat

M.x 155021 mat :

End - effect factor need to be evaluated, where relevant, as a function of slenderness λ.
Reccomended values of λ for cylinder are provide in Table 7.16

λ h 15 m * < h b.mean / 70 Max h 15 m * ≥ (h 50 m * < (& M.x M.y h m / linterp 0.7 h b.mean / * 70 Max if if : 70

Effective slenderness for circular cylinder

Definition of the solidity ratio

A b i 1 + el z i 1 + el z i el - (* i 1 n.p 1 - sum :

Sum of the projected areas of the members

A.c b i 1 + el z i 1 + el z i el - (* i 1 n.p 1 - sum :

Overall envelope area

φ A A.c / : 1

Solidity Ratio for the mast

λ.rif 11020307010020071 mat :

φ.1 0.6 0.7 0.77 0.82 0.92 0.95 171 mat :

φ.0.95 0.73 0.8 0.85 0.87 0.94 0.96 171 mat :

φ.0.9 0.83 0.87 0.91 0.93 0.96 0.98 171 mat :

φ.0.5 0.88 0.91 0.93 0.94 0.97 0.99 171 mat :

φ.0.1 0.97 0.98 0.99 0.993 0.995 0.998 171 mat :

φ.rif 1 0.95 0.90 0.50 0.1 51 mat :

ψ.λ φ φ.rif 1 el ≡ (λ.rif φ.1 λ linterp φ φ.rif 1 el < (φ φ.rif 2 el ≥ (& λ.rif φ.1 λ linterp λ.rif φ.0.95 λ linterp λ.rif φ.1 λ linterp - (φ.rif 1 el φ - φ.rif 1 el φ.rif 2 el - / * (+ φ φ.rif 2 el ≡ (λ.rif φ.0.95 λ linterp φ φ.rif 2 el < (φ φ.rif 3 el ≥ (& λ.rif φ.0.95 λ linterp λ.rif φ.0.9 λ linterp λ.rif φ.0.95 λ linterp - (φ.rif 2 el φ - φ.rif 2 el φ.rif 3 el - / * (+ φ φ.rif 3 el ≡ λ.rif φ.0.9 λ linterp φ φ.rif 3 el < (φ φ.rif 4 el ≥ (& λ.rif φ.0.9 λ linterp λ.rif φ.0.5 λ linterp λ.rif φ.0.9 λ linterp - (φ.rif 3 el φ - φ.rif 3 el φ.rif 4 el - / * (+ φ φ.rif 4 el ≡ λ.rif φ.0.5 λ linterp φ φ.rif 4 el < (φ φ.rif 5 el ≥ (& λ.rif φ.0.5 λ linterp λ.rif φ.0.1 λ linterp λ.rif φ.0.5 λ linterp - (φ.rif 4 el φ - φ.rif 4 el φ.rif 5 el - / * (+ φ φ.rif 5 el ≡ λ.rif φ.0.1 λ linterp Check if if if if if if if if if : 0.92

End - effect factor takes into account the reduced resistence of the structure due to the wind flow around the end.

ψ.λ 0.92

End-effect factor in accordance with Figure 7.36 of EC1

c.f c.f0 ψ.λ * :

Force coefficient for a finite circular cylinder for each heigth

Table3 z Re k/b cf0 cf [m] - 10^-6 - - z mm el m / z R.e vectorize mm el Eq.r i el mm el 106^* c.f0 mm el c.f mm el 35 mat :

Table3x z Re k/b cf0 cf [m] - 10^-6 - - z nn el m / z R.e vectorize nn el Eq.r i el nn el 106^* c.f0 nn el c.f nn el 35 mat :

Table3x Table3 A.c b.mean R.e v k Table_x1 Table2 c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration v.b0 ν ρ z.0_II z.max z.min z.0 T_C Tablexx Table1 δ.s b.o b.i ν.s ρ.s E.m W f.y h Ty_S ST Table3x Table3 c.f φ.rif φ.0.1 φ.0.5 φ.0.9 φ.0.95 φ.1 λ.rif A.c M.x M.y Table_1 c.f0 Eq.r b.mean R.e v k Table_x1 Table2 c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 b t z nn mm δ.s I b.o b.i i ν.s ρ.s E.m W f.y h IC A.c b.mean R.e k c.e q.p σ.v v.m A L.d L.u x k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 δ.s b.o b.i ν.s ρ.s E.m W f.y h

Table3
z	Re	k/b	cf0	cf
[m]	-	10^-6	-	-
mat(0,1.7,3.4,5.2,5.7,6.9,8.6,10,11.3,12.7,14,15.4,12,1)	mat(11079331.6,11079331.6,11079331.6,11178562.4,11409810.4,11886732.1,12444175,12813050.9,13110024,13392321.9,13626805.1,13863131.6,12,1)	mat(2.7,2.7,2.7,2.7,2.7,2.7,2.7,2.7,2.7,2.7,2.7,2.7,12,1)	mat(0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,12,1)	mat(0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,12,1)
#
#
#
#
#

Table3x
z	Re	k/b	cf0	cf
[m]	-	10^-6	-	-
mat(16.8,16.9,18.1,19.5,20.8,22.2,23.6,24.9,26.3,27.6,29,11,1)	mat(14071070.9,14085229.2,14248661.5,14425711.3,14578717.4,14732815.5,14877192,15003555.3,15132250,15245584.8,15361650.1,11,1)	mat(2.7,2.7,2.7,2.7,2.7,2.7,2.7,2.7,2.7,2.7,2.7,11,1)	mat(0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,11,1)	mat(0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,0.6,11,1)
#
#
#
#
#

Where, with reference to each point of discretized structure :
- z: reference heigth; - cf0: standard force coefficient for circular cilinder;
- Re: Reynolds Number; - cf: force coefficient for circular cilinder.

Wind Force coefficients of for vertical cylinders in a row arrangement (Pipe -line EC1 Part. 4 - 1 Par. 7.9.3)

For any linear ancillary item such as pipe line, the total wind force coefficient in the direction of wind should be taken as follow.

c.f_P_0 2 :

Overall normal drag coefficient of cylinder with - out free end flow

ψ.λP 1 :

End effect factor

a.P 10 cm * :

Distance between the centerline of the pipeline

b.P 1 in * :

Diameter of element of pipe line close to flare stack

κ a.P b.P / 3.5 < 1.15 a.P b.P / 3.5 ≥ (a.P b.P / 30 < (& 210 a.P b.P / (- 180 / 1 if if : 1.1

Factor for the most unfavorable wind direction of the pipeline

c.f_P c.f_P_0 ψ.λP * κ * : 2.3

Force coefficient linear pipeline

Wind Force coefficients of discrete ancilleries

For any discrete ancillary item such as platform, the total wind force coefficient in the direction of wind should be taken as follow.

c.f_a0 2 :

Force coefficient for the item appropriate to the wind direction obteined from manufacturerer. In this case the reccomended value is associated to reference sharp profile that form the platform (Ec1 Part. 1 - 4 Par. 7.7)

K.a 1 :

Reduction factor to take into account the shielding of the component by the structure itself. It is assume 1 on safety side

c.f_a c.f_a0 K.a * : 2

Force coefficient for discrete ancilleries

Spectral analysis method

When the responce is calculate by spectral analysis, this should be used for the resonance contribution to the responce only.
The non - resonant responce may be determined using the general static procedures reported before, using scaling factor ks = 3.5.
Below is reported an alternative methos for assessment of along cross - section wind forces in accordance with Par. 5.3 of EC 1 Part. 4 -1. It return an equivalent static action of wind for the whole structure to be compared with the profile analysed before.
Only the worst scenario can be implemented in the FEM model.

Main parameter of structure

N.m 2 :

Number of natural mode considered. General rule is that responce should be calculated for all modes of wibration having natural frequencies less than 2Hz (EC 3 Part 3 - 1 Par. B 4. 4)

n 2.25 11.12 21 mat Hz * :

First and second Natural frequencies of the structure (for symmetrical structure it is equal to first and second natural frequencies).

n.m 1 N.m range :

j n.m :

S Vertical structures 0.6 h * z.min 12 mat max Parallel oscillator h.1 h 2 / + z.min 12 mat max Pointlike structures h.1 h 2 / + z.min 12 mat max None of figure shape h 42 mat :

Definition of structural categories for applicability of procedure in accordance with Figure 6.1

"Vertical\0020\structures"
"Parallel\0020\oscillator"
"Pointlike\0020\structures"
"None\0020\of\0020\figure\0020\shape"

F_6_1

z.s F_6_1 2 el : 29 m *

z.s 29 m *

Reference height for determining the structural factor (Figure 6.1 EC1)

Estimated logarithmic decrement of damping δ

Logarithmic decrement of structural damping

δ.s 0.012

Approximate values of logarithmic decrement of structural damping in the foundamental mode (Table F.2 EC1)

ATTENZIONE CHE C'è COMPLETA DIFFORMITà TRA EC E CNR. è molto importante in quanto da qui ne deriva ilo numeri di scruton!!!!

Logarithmic decrement of aerodynamic damping

QUA VA EDITATA EVENTUALE VARIAZIONE NELLA TABELLA DI RIFERIMENTO MODALE IN TEMRINI DI RIFERIMENTI DI CELLE

Definition of foundamental flexural vertical mode shape From FEM model

Below there are the parameter assesed with axcell

Path 413-Camino Alabama-Modale_R1 :

Sheet.1 m_eq :

Parameter Path Sheet.1 C39 L62 importData.XLSX :

Parameter S1 01 1.72 1938.1896 2253.735 0000 S2 1.72 2 1.72 3488.7412 2028.3621 0.0083 0.0593 - 0.1385 7.138 S3 3.44 3 1.72 3101.1033 1802.9885 0.0248 0.161 - 1.1083 46.7451 S4 5.1599 4 0.52 2019.3231 1802.9888 0.0496 0.2797 - 4.4333 141.0123 S51 5.6799 5 1.2 1874.8429 2180.0765 0.0579 0.3136 - 7.2962 214.3439 S5 6.8799 6 1.72 2632.3319 1802.988 0.0826 0.3898 - 12.3147 273.9962 S6 8.5999 7 1.36 2328.6013 1512.0964 0.124 0.4831 - 23.2376 352.8297 S7 9.9599 8 1.34 1544.6573 1144.2042 0.157 0.5424 - 28.2124 336.5886 S52 11.2999 9 0.02 1102.3409 1621.1098 0.1983 0.5847 - 63.777 554.3022 S8 11.3199 10 1.36 789.4915 1144.2047 0.1983 0.5847 - 45.0148 391.2352 S9 12.6798 11 1.36 1556.0993 1144.2041 0.2479 0.5932 - 70.3356 402.6573 1110 mat

M.k Parameter i 5 el kg * :

z ϕ Parameter i 7 el Parameter i 8 el 21 mat :

z m.avg Parameter i 6 el kg m / * :

z m Parameter i 9 el Parameter i 10 el 21 mat kg m / * :

Table8 Node z Δs Mk m(s) ϕ1 ϕ2 m1 m2 k [m] [m] [kg] [kg/m] - - kg/m kg/m Parameter i 1 el Parameter i 2 el Parameter i 4 el Parameter i 5 el Parameter i 6 el Parameter i 7 el Parameter i 8 el Parameter i 9 el Parameter i 10 el 39 mat :

Table8 z.s F_6_1 c.f_P κ b.P a.P Table3x Table3 A.c b.mean R.e v k Table_x1 Table2 c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration v.b0 ν ρ z.0_II z.max z.min z.0 T_C Tablexx Table1 δ.s b.o b.i ν.s ρ.s E.m W f.y h Ty_S ST Table8 m m.avg ϕ M.k z.s j n.m n c.f_P κ b.P a.P c.f φ.rif φ.0.1 φ.0.5 φ.0.9 φ.0.95 φ.1 λ.rif A.c M.x M.y Table_1 c.f0 Eq.r b.mean R.e v k c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 b t z nn mm δ.s I b.o b.i i ν.s ρ.s E.m W f.y h IC z.s c.f_P κ b.P a.P A.c b.mean R.e k c.e q.p σ.v v.m A L.d L.u x k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 δ.s b.o b.i ν.s ρ.s E.m W f.y h

Table8
Node	z	Δs	Mk	m(s)	ϕ1	ϕ2	m1	m2
k	[m]	[m]	[kg]	[kg/m]	-	-	kg/m	kg/m
mat("S1","S2","S3","S4","S51","S5","S6","S7","S52","S8","S9","S10","S11","S12","S53","S13","S14","S15","S16","S17","S18","S19","S20",23,1)	mat(0,1.72,3.44,5.16,5.68,6.88,8.6,9.96,11.3,11.32,12.68,14.04,15.4,16.76,16.92,18.12,19.48,20.84,22.2,23.56,24.92,26.28,27.64,23,1)	mat(1.72,1.72,1.72,0.52,1.2,1.72,1.36,1.34,0.02,1.36,1.36,1.36,1.36,0.16,1.2,1.36,1.36,1.36,1.36,1.36,1.36,1.36,1.36,23,1)	mat(1938.19,3488.74,3101.1,2019.32,1874.84,2632.33,2328.6,1544.66,1102.34,789.49,1556.1,1556.1,1453.47,754.88,3810.23,1271.38,1350.84,1350.84,1248.21,1145.58,1145.58,1145.58,1137.26,23,1)	mat(2253.74,2028.36,1802.99,1802.99,2180.08,1802.99,1512.1,1144.2,1621.11,1144.2,1144.2,1144.2,1068.74,993.28,5603.35,993.28,993.28,993.28,917.81,842.35,842.35,842.35,836.23,23,1)	mat(0,0.01,0.02,0.05,0.06,0.08,0.12,0.16,0.2,0.2,0.25,0.3,0.36,0.41,0.42,0.47,0.54,0.6,0.66,0.73,0.79,0.86,0.93,23,1)	mat(0,-0.06,-0.16,-0.28,-0.31,-0.39,-0.48,-0.54,-0.58,-0.58,-0.59,-0.58,-0.53,-0.46,-0.45,-0.36,-0.23,-0.08,0.08,0.25,0.44,0.63,0.81,23,1)	mat(0,0.14,1.11,4.43,7.3,12.31,23.24,28.21,63.78,45.01,70.34,101.28,134.97,169.61,995.45,220.42,286.63,351.69,401.2,445.54,530.23,622.28,716.46,23,1)	mat(0,7.14,46.75,141.01,214.34,274,352.83,336.59,554.3,391.24,402.66,379.98,304.64,208.01,1130.41,125.84,52,7.13,6.59,54.45,163.58,331.28,553.48,23,1)
#
#
#
#
#

Where, with reference to each point of discretized structure :
- k: Identification node
- z: reference heigth;
- m( s): Vibrating mass of structure;
-Δs: Span between point;
- Mk: Lumped masses;
- ϕ : Modal shape of mode 1 and 2;
- m: generalized mass of structure associated to 1 and second mode.

ATTENZIONE: se la forma modale presenta un picco con spostamento negativo, meglio renderlo positivo perchè non so se la funzione di ricerca dell'indice dell'altezza associata è capace di valutarlo

ϕ.1 z ϕ 1 el Parameter i 2 el augment :

ϕ.2 z ϕ 2 el Parameter i 2 el augment :

W.0 1 :

C.0 1 :

w 1 n.p range pt w el W.0 : Parameter W.0 7 el 1 ≡ (Parameter W.0 7 el 1 - ≡ (| break W.0 W.0 1 + : if 21 line for

r 1 n.p (range pt r el C.0 : Parameter C.0 8 el 1 ≡ (Parameter C.0 8 el 1 - ≡ (| break C.0 C.0 1 + : if 21 line for

pt w el 23

pt r el 23

Definition of main point where vortex shedding can occur (sections where modal shape response is higher )

Modal shape of structure normalized to
maximum modal displament

ϕ.1 ϕ.2 2 1 sys

z.vs z pt w el el z pt r el el 21 mat : m 292921 mat

Reference heigth of structure where is maximum the value of dispacement for each mode

b.vs b pt w el el b pt r el el 21 mat : m 2.2 2.2 21 mat

Reference width of structure where is maximum the value of dispacement for each mode

Definition of the equivalent mass of the structure.
For cantilevered structures with a varyng mass distribution, me may be approximate by the average value of m over the upper third of the structure.

m.e n.m el z m n.m el i el i 1 n.p sum z ϕ n.m el i el (2^i 1 n.p sum / :

Formulation proposed by Part. F.4 EC1 Part. 4 - 1

m.e kg m / 1087135521 mat

Equivelent mass per unit length related respectively to frist and second mode of structure

δ.a c.f n.p el ρ * b.mean * z.s v.m * 4 π * n 1 el * m.e 1 el * / : 0.003

IL primo modo avrà un unico antinodo, il secondo 2! e quindi due massimi relativi

Logarithmic decrement of aerodynamic damping for the foundamental mode.
In no case is possible generalize the expression in order to evaluate the aerodynamic coefficient related transversal or torsional vibration. In these situation, if necessary, specialized evaluation need to be provide for assessment of δa.

Logarithmic decrement of damping for special devices

δ.d 0 :

Logarithmic decrement of damping due to special devices (taken equal to 0 in case that no special devices are added to the structure)

Total logarithmic decrement of damping definition

δ δ.s δ.a + δ.d + : 0.02

Total logarithmic decrement of damping

Definition of structural factors (in accordance with procedure 1 of Annex B)

Wind turbolance length scale

The turbolent lenght scale L(z) represents the average gust size for natural wind. It can be avaluated with the follow expression for structures with eight below 200 m.

z.t z.max :

Reference height

L.t 300 m * :

Reference legth scale

α 0.67 0.05 z.0 m / ln * + : 0.61

Exponential parameter

z L z z.min < L.t z.min z.t / (α^* L.t z z.t / (α^* if :

Turbolent lenght scale expression

L_z z L vectorize m / z m / augment :

Turbolent lenght scale profile

L_z

The wind distribution over frequencies is expressed by the non-dimensional power spectral density function

n.1 n 1 el :

z n.1 f.L n.1 z L * z v.m / :

Non-dimensional frequency referred to first natural frequency period

z n.1 S.L 6.8 z n.1 f.L * 1 10.2 z n.1 f.L * + (53 /^/ :

Non-dimensional power spectral density function

Structural factor main parameters

Structural factors cs x cd should take into account the effect on wind actions from the non simultaneus occurrence of peak wind pressure on the surface (cs) together with the effect of the vibrations of the structure due to the turbolence (cd).
The structural factor, may be separated into the two component:
- cs : size factor;
- cd : dynamic factor.

B 11 0.9 b.mean h + z.s L / (0.63^* + / sqrt : 0.83

Squadre root of Background factor allowing for the lack of full correlation of the pressure on the structure surface

B 2^0.69

Background factor

Definition of the aerodynamic admittance functions

η.b 4.6 b.mean * z.s L / z.s n 1 el f.L * : 0.34

η.h 4.6 h * z.s L / z.s n 1 el f.L * : 4.55

η.h R.h η.h 0 ≡ 11 η.h / 12 η.h 2^* / 1 e 2 η.h * -^- (* - if : 0.2

η.b R.b η.b 0 ≡ 11 η.b / 12 η.b 2^* / 1 e 2 η.b * -^- (* - if : 0.8

Definition of the resonance responce factor for turbolence in resonance related to the vibration mode of structure

R π 2^2 δ * / z.s n 1 el S.L * η.h R.h * η.b R.b * sqrt :

NOTE : for mode shapes with internal node points more detailed calculations should be used

R 2^3.14

Resonance Responce factor

ν.f n 1 el R 2^B 2^R 2^+ / sqrt * 0.08 Hz * 12 mat max : 2.04 Hz *

Up - crossing frequency. The setted limit to 0.08 Hz corresponds to a peack factor of 3.0

Definition of peak factor, defined as the ratio of the maximum value of the fluctuating par of the responce to its standard deviation.

T 600 s * :

Average time for the wind velocity

k.p 2 ν.f T * ln * sqrt 0.6 2 ν.f T * ln * sqrt / + 312 mat max : 3.93

Peak factor

Structural factors

Definition of size factor that take into account the reduction effect on the wind action due to the non simultaneity of occurrence of the peack wind pressure on the surface.

c.s 17 z.s I.v * B 2^sqrt * + 17 z.s I.v * + / : 0.9

Size factor

Definition of dynamic factor that take into account the increasing effect from vibrations due to turbolence in resonance with the structure

c.d 12 k.p * z.s I.v B 2^R 2^+ sqrt * * + 17 z.s I.v * B 2^sqrt * + / : 1.92

Dynamic factor

Table9 Structural factors parameters Reference height zs z.s m / [m] Reference legth scale Lt L.t m / [m] Exponential parameter α α - Turbolent lenght scale L(zs) z.s L m / [m] First natural frequency of structure n n Hz / [Hz] Non-dimensional frequency fL(zs,n) z.s n 1 el f.L - Non-dimensional power spectral density function SL(zs,n) z.s n 1 el S.L - Equivalent mass per unit length of the first foundamental mode m1e m.e 1 el kg / m * [kg/m] Approximate values of log. decrement of structural damping δs δ.s - Logarithmic decrement of aerodynamic damping for the I mode δa δ.a - Logarithmic decrement of damping due to special devices δd δ.d - Total logarithmic decrement of damping δ δ - Background factor B^2 B 2^- Resonance Responce factor R^2 R 2^- Up - crossing frequency νf ν.f Hz / [Hz] Average time for the wind velocity T T s / [s] Peak factor kp k.p - Size factor cs c.s - Dynamic factor cd c.d - 194 mat 21 mat :

Table9 c.d c.s k.p T ν.f R.b R.h η.h η.b S.L f.L n.1 L α L.t z.t δ δ.a m.e b.vs z.vs m ϕ z.s F_6_1 j n.m n c.f_P κ b.P a.P A.c M.x M.y Table_1 b.mean R.e v k c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration v.b0 ν ρ z.0_II z.max z.min z.0 T_C δ.s b.o b.i ν.s ρ.s E.m W f.y h Ty_S ST IC Table9 c.d c.s k.p T ν.f R.b R.h η.h η.b S.L f.L n.1 L α L.t z.t δ δ.a m.e b.vs z.vs pt m m.avg ϕ M.k z.s j n.m n c.f_P κ b.P a.P c.f φ.rif φ.0.1 φ.0.5 φ.0.9 φ.0.95 φ.1 λ.rif A.c M.x M.y Table_1 c.f0 Eq.r b.mean R.e v k c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 b t z nn mm δ.s I b.o b.i i ν.s ρ.s E.m W f.y h IC c.d c.s k.p T ν.f η.h η.b S.L f.L n.1 α L.t z.t δ δ.a m.e b.vs z.vs m ϕ z.s j n.m n c.f_P κ b.P a.P A.c M.x M.y b.mean R.e k c.e q.p σ.v v.m A L.d L.u x k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 δ.s b.o b.i ν.s ρ.s E.m W f.y h

Table9
Structural factors parameters
mat("Reference height","zs",29,"[m]","Reference legth scale","Lt",300,"[m]","Exponential parameter","α",0.61,"-","Turbolent lenght scale","L(zs)",92.41,"[m]","First natural frequency of structure","n",mat(2.25,11.12,2,1),"[Hz]","Non-dimensional frequency","fL(zs,n)",3.15,"-","Non-dimensional power spectral density function","SL(zs,n)",0.06,"-","Equivalent mass per unit length of the first foundamental mode","m1e",1087.13,"[kg/m]","Approximate values of log. decrement of structural damping","δs",0.01,"-","Logarithmic decrement of aerodynamic damping for the I mode","δa",0,"-","Logarithmic decrement of damping due to special devices","δd",0,"-","Total logarithmic decrement of damping","δ",0.02,"-","Background factor","B^2",0.69,"-","Resonance Responce factor","R^2",3.14,"-","Up - crossing frequency","νf",2.04,"[Hz]","Average time for the wind velocity","T",600,"[s]","Peak factor","kp",3.93,"-","Size factor","cs",0.9,"-","Dynamic factor","cd",1.92,"-",19,4)
#
#
#
#
#

NOTE : zs is the reference heigth for guy rope calculation and not the total height of flare.
In order to determine the structural factor (cs cd as mentioned in Eurocode terminology), in accordance with par. 6.3 “Detailed procedure” (EC3 Part. 1-4), reference height is provided strictly for cantilever structures. In order to extend this approach to a guy ropes stabilised structure, it is necessary to calculate in new reference height. 40.8 is 0.6 times h=68m as per figure 6.1 (Par. 6.3.1 EC 3 Part 1-4).

Wind forces on mast for different quote

z F.w c.s c.d * c.f * z q.p * b * :

Wind force acting along pipe-line, referred to one line only

Table10 z Fw [m] [kN/m] z mm el m / z F.w vectorize mm el m kN / * 32 mat :

Table10x z Fw [m] [kN/m] z nn el m / z F.w vectorize nn el m kN / * 32 mat :

f.w1 z F.w vectorize kN m / / z m / augment :

Table10x Table10 F.w c.d c.s k.p T ν.f R.b R.h η.h η.b S.L f.L n.1 L α L.t z.t δ δ.a Table8 z.s F_6_1 c.f_P κ b.P a.P Table3x Table3 A.c b.mean R.e v k Table_x1 Table2 c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration v.b0 ν ρ z.0_II z.max z.min z.0 T_C Tablexx Table1 δ.s b.o b.i ν.s ρ.s E.m W f.y h Ty_S ST f.w1 Table10x Table10 F.w Table9 c.d c.s k.p T ν.f R.b R.h η.h η.b S.L f.L n.1 L_z L α L.t z.t δ δ.a m.e b.vs z.vs pt ϕ.2 ϕ.1 m m.avg ϕ M.k z.s F_6_1 S j n.m n c.f_P κ b.P a.P c.f φ.rif φ.0.1 φ.0.5 φ.0.9 φ.0.95 φ.1 λ.rif A.c M.x M.y Table_1 c.f0 Eq.r b.mean R.e v k c._e q._p I._v v._m c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration Dur v.b0 ν ρ z.0_II z.max z.min z.0 b t z nn mm δ.s I b.o b.i i ν.s ρ.s E.m W f.y h Ty_S ST IC Sur_T ST_T F.w c.d c.s k.p T ν.f η.h η.b S.L f.L n.1 α L.t z.t δ δ.a z.s c.f_P κ b.P a.P A.c b.mean R.e k c.e q.p σ.v v.m A L.d L.u x k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 δ.s b.o b.i ν.s ρ.s E.m W f.y h

Table10
z	Fw
[m]	[kN/m]
mat(0,1.7,3.4,5.2,5.7,6.9,8.6,10,11.3,12.7,14,15.4,12,1)	mat(7.75,7.75,7.75,7.9,8.25,8.99,9.89,10.52,11.04,11.55,11.98,12.41,12,1)
#
#
#
#
#

Table10x
z	Fw
[m]	[kN/m]
mat(16.8,16.9,18.1,19.5,20.8,22.2,23.6,24.9,26.3,27.6,29,11,1)	mat(12.81,12.84,13.15,13.5,13.8,14.11,14.4,14.66,14.93,15.16,15.41,11,1)
#
#
#
#
#

Where, with reference to each point of discretized structure :
- z: reference heigth;
-Fw: Wind loading on mast considering dynamic actions

Wind force profile profile on flare stack

f.w1

Wind forces on pipe-line

z F.w_P c.s c.d * c.f_P * z q.p * b.P * :

Wind force acting on a structure or a structural component

Table10.bis z Fw [m] [kN/m] z mm el m / z F.w_P vectorize mm el m kN / * 32 mat :

f.wP z F.w_P vectorize kN m / / z m / augment :

Table10.bisx z Fw [m] [kN/m] z nn el m / z F.w_P vectorize nn el m kN / * 32 mat :

Table10.bisx Table10.bis F.w_P Table10x Table10 F.w c.d c.s k.p T ν.f R.b R.h η.h η.b S.L f.L n.1 L α L.t z.t δ δ.a Table8 z.s F_6_1 c.f_P κ b.P a.P Table3x Table3 A.c b.mean R.e v k Table_x1 Table2 c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration v.b0 ν ρ z.0_II z.max z.min z.0 T_C Tablexx Table1 δ.s b.o b.i ν.s ρ.s E.m W f.y h Ty_S ST Table10.bisx f.wP Table10.bis F.w_P f.w1 Table10x Table10 F.w Table9 c.d c.s k.p T ν.f R.b R.h η.h η.b S.L f.L n.1 L_z L α L.t z.t δ δ.a m.e b.vs z.vs pt ϕ.2 ϕ.1 m m.avg ϕ M.k z.s F_6_1 S j n.m n c.f_P κ b.P a.P c.f φ.rif φ.0.1 φ.0.5 φ.0.9 φ.0.95 φ.1 λ.rif A.c M.x M.y Table_1 c.f0 Eq.r b.mean R.e v k c._e q._p I._v v._m c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration Dur v.b0 ν ρ z.0_II z.max z.min z.0 b t z nn mm δ.s I b.o b.i i ν.s ρ.s E.m W f.y h Ty_S ST IC Sur_T ST_T F.w_P F.w c.d c.s k.p T ν.f η.h η.b S.L f.L n.1 α L.t z.t δ δ.a z.s c.f_P κ b.P a.P A.c b.mean R.e k c.e q.p σ.v v.m A L.d L.u x k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 δ.s b.o b.i ν.s ρ.s E.m W f.y h

Table10.bis
z	Fw
[m]	[kN/m]
mat(0,1.7,3.4,5.2,5.7,6.9,8.6,10,11.3,12.7,14,15.4,12,1)	mat(0.36,0.36,0.36,0.37,0.38,0.41,0.45,0.48,0.5,0.52,0.54,0.56,12,1)
#
#
#
#
#

Table10.bisx
z	Fw
[m]	[kN/m]
mat(16.8,16.9,18.1,19.5,20.8,22.2,23.6,24.9,26.3,27.6,29,11,1)	mat(0.58,0.58,0.59,0.61,0.62,0.63,0.65,0.66,0.67,0.68,0.69,11,1)
#
#
#
#
#

Where, with reference to each point of discretized structure :
- z: reference heigth;
-Fwp: Wind loading on pipeline considering dynamic actions

Wind force profile profile of pipe-line

f.wP

Wind forces on discrete ancillary

z.anc 6 11.5 20.125 28.1 36.1 44.1 52.1 56.5 6591 mat m * :

z.anc F.w_a c.s c.d * c.f_a * z.anc q.p * :

Wind force acting on a structure or a structural component

Table10.tris z Fw [m] [kPa] z.anc m / z.anc F.w_a vectorize m 2^kN / * 32 mat :

f.w_an z.anc F.w_a vectorize kPa / z.anc m / augment :

Table10.tris F.w_a Table10.bisx Table10.bis F.w_P Table10x Table10 F.w c.d c.s k.p T ν.f R.b R.h η.h η.b S.L f.L n.1 L α L.t z.t δ δ.a Table8 z.s F_6_1 c.f_P κ b.P a.P Table3x Table3 A.c b.mean R.e v k Table_x1 Table2 c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration v.b0 ν ρ z.0_II z.max z.min z.0 T_C Tablexx Table1 δ.s b.o b.i ν.s ρ.s E.m W f.y h Ty_S ST f.w_an Table10.tris F.w_a z.anc Table10.bisx f.wP Table10.bis F.w_P f.w1 Table10x Table10 F.w Table9 c.d c.s k.p T ν.f R.b R.h η.h η.b S.L f.L n.1 L_z L α L.t z.t δ δ.a m.e b.vs z.vs pt ϕ.2 ϕ.1 m m.avg ϕ M.k z.s F_6_1 S j n.m n c.f_P κ b.P a.P c.f φ.rif φ.0.1 φ.0.5 φ.0.9 φ.0.95 φ.1 λ.rif A.c M.x M.y Table_1 c.f0 Eq.r b.mean R.e v k c._e q._p I._v v._m c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration Dur v.b0 ν ρ z.0_II z.max z.min z.0 b t z nn mm δ.s I b.o b.i i ν.s ρ.s E.m W f.y h Ty_S ST IC Sur_T ST_T F.w_a F.w_P F.w c.d c.s k.p T ν.f η.h η.b S.L f.L n.1 α L.t z.t δ δ.a z.s c.f_P κ b.P a.P A.c b.mean R.e k c.e q.p σ.v v.m A L.d L.u x k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 δ.s b.o b.i ν.s ρ.s E.m W f.y h

Table10.tris
z	Fw
[m]	[kPa]
mat(6,11.5,20.12,28.1,36.1,44.1,52.1,56.5,65,9,1)	mat(13.43,17.4,21.13,23.49,25.32,26.83,28.11,28.74,29.85,9,1)
#
#
#
#
#

Where, with reference to each point of discretized structure :
- z: reference heigth of ancilleries;
-Fw_a: Wind loading on ancillaries considering dynamic actions

DEFINIRE QUA DOVE SI TROVANO LE QUOTE DELLE PIATTAFORME

VORTEX SHEDDING AND AREOELASTIC INSTABILITIES

Vortex shedding occurs when vortices are shed alternately from opposite sides of the structure.
This give rise to a fluctuating load perpendicular to the wind direction. Structural vibration may occur if the frequency of vortex - shedding is the same as natural frequency of the structure. This condition occurs when the wind velocity is equal to the critical wind velocity. Typically, the critical wind velocity is a frequent wind velocity indicating the fatigue, and the thereby the number of load cycles, may become relevant.

Criteria for vortex shedding

Effect of vortex shedding should be investigated when the ratio of the largest to the smallest crosswind dimension of the structure, both taken in the plane perpendicular to the wind, exceed 6.

First criteria

b max b min / 6 ≤

Second criteria

v.crit i el 1.25 z.vs v.m * >

Wind critical velocity related to the i vibrational mode

z.vs v.m vectorize m s / 65.9 65.9 21 mat

Average wind velocity referred to the heigth vs where the modal shape assume greatest values

Definition of Strouhal number St

S.t 0.18 :

Value from Table E.1 adapt for cyrcular section and for each Reynolds number

Vortex shedding condition

v.crit j el b.vs j el n j el * S.t / : m s / 27.5 135.7 21 mat

Critical wind velocity respectively for first and second bending vibration mode

Vortex_Shedding v.crit 1 el 1.25 z.vs 1 el v.m * / 1 > (v.crit 2 el 1.25 z.vs 2 el v.m * / 1 > (& ((b max b min / 6 ≤ (& Need NO to be investigate v.crit 1 el 1.25 z.vs 1 el v.m * / 1 ≤ (v.crit 2 el 1.25 z.vs 2 el v.m * / 1 > (& Only First vibration mode need to be investigate v.crit 1 el 1.25 z.vs 1 el v.m * / 1 > (v.crit 2 el 1.25 z.vs 2 el v.m * / 1 ≤ (& Only Second vibration mode need to be investigate v.crit 1 el 1.25 z.vs 1 el v.m * / 1 ≤ (v.crit 2 el 1.25 z.vs 2 el v.m * / 1 ≤ (& Both vibration modes need to be investigate Check if if if if :

Vortex_Shedding Only First vibration mode need to be investigate

Basic parameters for vortex shedding

Definition of Scruton number Sc

Scruton Number express the supsceptibility of vibration depends on the structural damping and the ratio of the structural mass to fluid mass.

S.c j el 2 δ.s * m.e j el * ρ b.vs j el 2^* / : 4.33 5.39 21 mat

Scruton number for the 2 main foundamental mode of vibration

In resonance conditions, smaller Scruton Numbers (ligth or no dumper structures) means bigger resonance responce.

For circular cross section structures, follow parameter can be take into consideration in order to evaluate the supsceptibily to resonance vibration

Study_Condition S.c min 30 ≥ Reduced risk of vortex shedding S.c min 30 < (S.c min 5 ≥ (& Analyze vortex shedding and fatigue tension S.c min 5 < Analyze vortex shedding with caution - if if if :

Study_Condition Analyze vortex shedding with caution

Reynolds number Re at critical wind velocity

The vortex shedding action on a citcular cylinder depends on the Reynolds number Re at the critical wind velocity.

R.e_vcr j el b.vs j el v.crit j el * ν / : 4.02 106^* 1.99 107^* 21 mat

Reynolds number for critical wind velocity

Vortex shedding action

The effect of vibrations induced by vortex shedding should be calculated from the effect of the inertia force per unit length Fw, acting perpendicular to the wind direction at locations on the structure

Calculation of largest displacement for the first shape mode

Approach 1 - Armonic Method (interactive method)

This approach can be used for various kind of structures and mode shape. It include turbolence and roughness effects and may be used for normal climatic conditions.
Explination below is set considering just the first foundamental mode of vibration,

c.lat_0 j el R.e_vcr j el 3105^* ≤ 0.7 R.e_vcr j el 3105^* > (R.e_vcr j el 5105^* ≤ (& 3105^* 5105^* 21 mat 0.7 0.2 21 mat R.e_vcr 1 el linterp R.e_vcr j el 5105^* > (R.e_vcr j el 5106^* ≤ (& 0.2 R.e_vcr j el 5106^* > (R.e_vcr j el 1107^* ≤ (& 5106^* 1107^* 21 mat 0.2 0.3 21 mat R.e_vcr j el linterp R.e_vcr j el 1107^* > 0.3 Check if if if if if : 0.2 0.3 21 mat

S.c 1 el 4.3254

S.t 0.18

K z ϕ 1 el i el i 1 n.p sum 4 π * z ϕ 1 el i el (2^i 1 n.p sum * / : 0.1357

PROBLEM EXPLINATION

My problem is assess the value of Yf after the iteration process.
The varius steps are following described:
- Set first value of L = 6b;
- Define reference quote z = h-L/2;
- Evaluate vm at set height z;
- Evaluate varios parameter;
- Assess Yf (that is my target);
- If Yf/ b <0.1 stop the cicle else continue setting L equal 4.8 + 12 y /b;
- estimate the new y until convergence

Below an example of the procedure

I'M NOT ABLE TO WRITE IN THE CORRECT WAY THE CICLE :(

L.0 6 b.vs 1 el * : 13.2 m *

L.0 c.lat 1 el v.crit 1 el h L.0 2 / - v.m / 0.83 ≤ c.lat_0 1 el v.crit 1 el h L.0 2 / - v.m / 0.83 ≥ (v.crit 1 el h L.0 2 / - v.m / 1.25 < (& 3 2.4 v.crit 1 el h L.0 2 / - v.m / * - (c.lat_0 1 el * v.crit 1 el h L.0 2 / - v.m / 1.25 ≥ 0 Check if if if : K.W 3 L.0 b.vs 1 el / h b.vs 1 el / / * 1 L.0 b.vs 1 el / h b.vs 1 el / / - 13 / L.0 b.vs 1 el / h b.vs 1 el / / (2^* + (* 0.6 12 mat min : y.0 1 S.t 2^/ 1 S.c 1 el / * K * K.W * c.lat 1 el * (b.vs 1 el * : y.0 b.vs 1 el / 0.1 < (y.max y.0 ≡ j 1 mm * (6 b.vs 1 el * (range L.0 4.8 12 y.0 b.vs 1 el / (* + (b.vs 1 el * : c.lat 1 el v.crit 1 el h L.0 2 / - v.m / 0.83 ≤ c.lat_0 1 el v.crit 1 el h L.0 2 / - v.m / 0.83 ≥ (v.crit 1 el h L.0 2 / - v.m / 1.25 < (& 3 2.4 v.crit 1 el h L.0 2 / - v.m / * - (c.lat_0 1 el * v.crit 1 el h L.0 2 / - v.m / 1.25 ≥ 0 Check if if if : K.W 3 L.0 b.vs 1 el / h b.vs 1 el / / * 1 L.0 b.vs 1 el / h b.vs 1 el / / - 13 / L.0 b.vs 1 el / h b.vs 1 el / / (2^* + (* 0.6 12 mat min : y.max 1 S.t 2^/ 1 S.c 1 el / * K * K.W * c.lat 1 el * (b.vs 1 el * : L.1 4.8 12 y.max b.vs 1 el / (* + (b.vs 1 el * : y.max b.vs 1 el / 0.1 ≥ (y.max b.vs 1 el / 0.6 < (& y.max y.0 - abs 1 mm * < (L.1 L.0 - abs 1 mm * < (& break L.0 L.1 j + : if if 61 line for if y.max 61 line

y.max y.0 - abs 0

6 b.vs 1 el * 13.1805 m *

12 b.vs 1 el * 26.361 m *

y.0 0.2553 m *

y.0 b.vs 1 el / 0.1162

y.max 0.2553 m *

y.max b.vs 1 el / 0.1162

L.0 13.6076 m *

4.8 12 y.0 b.vs 1 el / (* + (b.vs 1 el * 13.6076 m *

L.1 13.6076 m *

y.max b.vs 1 el / 0.1162

Table11 Vortex shedding parameters (1° Method) Stroual Number St S.t - Scruton Number Sc S.c 1 el - Eff. correlation length factor Kw K.W - Correlation length Ld L d el m / [m] Mode shape factor K K - Lateral force coefficient clat c.lat 1 el - 64 mat 21 mat :

Basic value of the lateral force coefficient for different value of Re(vcrit) number. Figure E.2 is related for circular cylinder shape only.

c.lat_0 0.2 0.3 21 mat

c.lat 1 el 0.2

Lateral force coefficient in accordance with table E.3 for first mode

L #

Effective correlation length in accordance with Table E.4 and in function of vibration amplitude for the first vibration mode

Table11 L.1 y.max y.0 K.W c.lat L.0 K c.lat_0 R.e_vcr S.c v.crit S.t F.w_a F.w_P F.w Table9 c.d c.s k.p T ν.f R.b R.h η.h η.b S.L f.L n.1 L α L.t z.t δ δ.a m.e b.vs z.vs m ϕ z.s F_6_1 j n.m n c.f_P κ b.P a.P A.c M.x M.y Table_1 b.mean R.e v k c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr Duration v.b0 ν ρ z.0_II z.max z.min z.0 T_C δ.s b.o b.i ν.s ρ.s E.m W f.y h Ty_S ST IC Table11 L.1 y.max y.0 K.W c.lat L.0 K c.lat_0 R.e_vcr S.c v.crit S.t F.w_a z.anc F.w_P F.w Table9 c.d c.s k.p T ν.f R.b R.h η.h η.b S.L f.L n.1 L α L.t z.t δ δ.a m.e b.vs z.vs pt m m.avg ϕ M.k z.s j n.m n c.f_P κ b.P a.P c.f φ.rif φ.0.1 φ.0.5 φ.0.9 φ.0.95 φ.1 λ.rif A.c M.x M.y Table_1 c.f0 Eq.r b.mean R.e v k c.e q.p I.v σ.v v.m A L.d z.top L.u x c.r k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 b t z nn mm δ.s I b.o b.i i ν.s ρ.s E.m W f.y h IC L.1 y.max y.0 K.W c.lat L.0 K c.lat_0 R.e_vcr S.c v.crit S.t F.w_a F.w_P F.w c.d c.s k.p T ν.f η.h η.b S.L f.L n.1 α L.t z.t δ δ.a m.e b.vs z.vs m ϕ z.s j n.m n c.f_P κ b.P a.P A.c M.x M.y b.mean R.e k c.e q.p σ.v v.m A L.d L.u x k.r q.b v.b c.prob n.pr K.pr v.b0 ν ρ z.0_II z.max z.min z.0 δ.s b.o b.i ν.s ρ.s E.m W f.y h

Table11
Vortex shedding parameters (1° Method)
mat("Stroual Number","St",9/50,"-","Scruton Number","Sc",570014688212735/131784114310661,"-","Eff. correlation length factor","Kw",3/5,"-","Correlation length","Ld",el(L,d)/'m,"[m]","Mode shape factor","K",5789721004934/42663553839995,"-","Lateral force coefficient","clat",1/5,"-",6,4)
#
#
#
#
#

Largest displacement for Approach1 relative to the first mode of vibration

y.max 1 S.t 2^/ 1 S.c 1 el / * K * k.W * c.lat 1 el * (b.vs 1 el * :

The results of this formulation came from an interactive process

y.max_1 cm #

Maximum displacement for approach 1 on the tip on structure, Mode 1 approssimate to cantiliver element.