Download pdf - MecaStack Software Validation Manual · 2018. 10. 16. · C BS 6399 Wind + BS 4076 D BS CP 3 Ch V Wind + BS 4076 E Indian Standards F Canadian (NBCC 2010) Wind & Seismic G Eurocodes

Revision Date Description Engineer

0 07-Jul-16 Initial Issue with CICIND 2010 and ASME STS-1

validation

CR

1 23-Apr-17 Add new codes to validation, and update existing

validation to latest version of MecaStack

CR

2 01-Aug-17 Add new codes to validation for latest version of

MecaStack

CR

MecaStack Software Validation Manual

This manual contains worked examples that are intended to validate the accuracy of the MecaStack

software analysis and output. It is not possible to provide validation for every single aspect of the

software, because the software is so broad in its scope and application; however, we attempt to utilze

some representative designs which can be verified by an independent method (Code example,

Mathcad calculations, etc..) and compare those results to MecaStack to verify that they are in

reasonable agreement.

MecaStack_Validation_Rev2 Do Not Reproduce or Distribute without Permission.Copyrighted (c) 2017 by Meca Enterprises, Inc.

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Section Description

A ASME STS-1-2011

A-1 NonMan App Ex E-1 & E-4 (Along & Across Wind)

A-2 NonMan App E Table E-1.1-1 Examples (Along Wind)

A-3 NonMan App E-5 (Response Spectrum)

B CICIND 2010

B-1 Appendix 5 Example - CICIND Calc's

B-2 Mathcad Calc's to replicate CICIND Example

B-3 StaadPro Model to Verify Mode Shape

B-4 Mathcad Calculation with Corrected Values

B-5 MecaStack (Reduced Formulations 7.2.4.2 / 8.5.2)

B-6 MecaStack (Full Formulations per Commentary 3/4)

B-7 MecaStack (Ovalling Ring Verification)

C BS 6399 Wind + BS 4076

D BS CP 3 Ch V Wind + BS 4076

E Indian Standards

F Canadian (NBCC 2010) Wind & Seismic

G Eurocodes

Table of Contents


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Section A - ASME STS-1-2011


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Section A-1 - NonMan App Ex E-1 & E-4 (Along & Across Wind)


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Chris Rosencutter

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Matches 0.956

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1615 kip-ft


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Chris Rosencutter

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187.01 / 2 = 93.5 ksi

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18,617 ft-kip

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17,453 ft-kip

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4.815 ft

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4.514 ft

MecaStack v5314 Software Developer: Meca Enterprises Inc., www.meca.biz, Copyright © 2017

Calculations Prepared by: Calculations Prepared For: Meca Enterprises Client: Meca 816 W. Elgin St Project #: Ex_1_and_4 Broken Arrow, OK, 74012 Location: Broken Arrow Date: Apr 22, 2017 Description: Designer: C. Rosencutter ASME STS-1 Example 1 & 4

File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\STS2011\ Along_Across_Wind\v5314_STS_Ex1_and_4.Stk

INPUT PARAMETERS:

* Linear Static Analysis * Stress analysis based upon Limit States Design

Total Stack Height = 140.0 ft Top of Stack Elevation = 140.0 ft Grade Elevation = 0.0 ft Bottom of Stack = 0.0 ft

All elevations are based upon the bottom of stack being at 0 ft

Stack Geometry:

Elevation Outer Diameter | Elevation Thickness | Elevation Corrosion Allow ft ft ft in ft in

--------- -------------- - --------- --------- - --------- ---------------140.0 8.0 | 140.0 0.3125 | 140.0 0.0000

Materials:

Elev Material Temp Fyld Elas Mod Alpha Allow Strs Density ft ksi ksi in/in/F ksi lb/ft^3

----- -------- ---- ----- -------- -------- ---------- ------- 140.0 A-36 70.0 36.00 28,950 6.80E-06 16.60 490.0 70.0 36.00 28,950 6.80E-06 16.60 490.0

Design Codes

Comprehensive Design Standard: ASME STS-1-2011 'Steel Stack Design'

Stress Criteria: ASME STS-1-2011 'Steel Stack Design'

Wind Load Criteria: ASCE 7-05 'Minimum Design Loads for Buildings and Other Structures'

V = Design Wind Speed = 100.0 mph ASCE 7-10 = Is the wind speed based upon ASCE 7-10? = False Exposure = Terrain Exposure = C Category = Structural Category = II DisableSTS= Disable application of wind loads per ASME STS-1 = False

Vortex Shedding Criteria: ASME STS-1-2011 'Steel Stack Design'

Arrange = Stack Arrangement = Single Spacing = Center to center stack spacing (Only required for Group) = 10.0 ft S = Manually entered Strouhal Number (Only used for Custom) = 0.2000 Const_Dia = Force the analysis to use Constant Diameter Equations. = False Life = Fatigue life of the stack = 20.0 Years Cycles = Override Number of Cycles (0 to calculate automatically) = 0

Fatigue Criteria: American Institute of Steel Construction 360-10

Elev Cat Cf Fth 2a w tp ft ksi in in in

------ --- -------- ----- ------ ------ ------ 140.00 C 4.40E+09 10.00 0.0000 0.0000 0.0000


Page 11 of 180

Seismic Criteria: None

Deflection Criteria:

Deflection Criteria for Static Loading: MaxDefl = No Deflection Limit = 0.0000 in

Deflection Criteria for Vortex Loading: DeflLimit = No Deflection Limit = 0.0000 in

Section Properties

Properties calculated based upon Corroded stack

Top Bot Outer Thick Rad of Area Plastic Elastic Mom of Elev Elev Diameter Gyration Sec Modulus Sec Modulus Inertia ft ft in in in sq in in^3 in^3 in^4

------ ------ -------- ------ -------- ----- ----------- ----------- ---------- 140.00 130.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 130.00 120.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 120.00 110.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 110.00 100.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 100.00 90.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 90.00 80.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 80.00 70.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 70.00 60.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 60.00 50.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 50.00 40.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 40.00 30.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 30.00 20.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 20.00 10.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75 10.00 0.00 96.0000 0.3125 33.8308 93.94 2,861.29 2,239.95 107,517.75

Weight Detail

Breakdown of Weight by Component

Cylindrical Shells: (Wt = 44752 lb, El COG = 70.0 ft)

Elev Elev Elev OD Thk Density Wt Lin Wt Wt Top Bot COG Corroded UnCorroded UnCorroded ft ft ft ft in lb/ft^3 lb lb/ft lb

------ ---- ----- ----- ------ ------- -------- ---------- ---------- 140.00 0.00 70.00 8.000 0.3125 490.0 44,752 319.66 44,752

------ ---- ----- ----- ------ ------- -------- ---------- ---------- Total 70.00 44,752 44,752

Weight Summary

Component Elev COG Weight ft lb

---------------------------------- -------- ------ Cylinders (Corroded Wt = 44752 lb) 70.000 44,752

---------------------------------- -------- ------ Total 70.000 44,752

Wind Areas

Wind Area Summary

Elev Shape Fac Riser Ladder Platform Piping Tot Uni Total Riser Area Area Area Area Area Area ft ft^2/ft ft^2/ft ft^2/ft ft^2/ft ft^2/ft sq ft

------ --------- ------- ------- -------- ------- ------- ------ 140.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67 130.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67


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120.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

110.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

100.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

90.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

80.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

70.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

60.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

50.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

40.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

30.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

20.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

10.00 0.658 5.27 0.00 0.00 0.00 5.27 52.67

------ --------- ------- ------- -------- ------- ------- ------

Total 737.33

Frequency Summary

Description Direction Mode Mode Mode

# 1 # 2 # 3

Deg Hz Hz Hz

----------------- --------- ----- ----- ------

Cold & Uncorroded 0.00 1.331 8.295 23.021

Wind Loads per ASME STS-1-2011

Pressures calculated based upon lowest natural frequency

V: Wind Speed = 100.0 mph f: Natural Frequency = 1.331 Hz

C: Structure Category = II Exp: Exposure Category = C

I: Importance Factor = 1.00 H: Structure Height = 140.0 ft

D: Structure Depth = 8.0 ft b: Structure Width = 8.0 ft

Rho = Air Density = 0.0765 lb/ft^3

Dm = Average Diameter = 8.0 ft

Cf = Shape factor of Cylinder = 0.658

ma = Average mass of top 1/3 of stack = 319.66 lb/ft

Ba = Aerodynamic Damping (Eqn 5-1): (Cf*Rho*Dm*Vzmean) / (4*PI*ma*f) = 0.0083

Bs = Structural Damping of Mode # 1 = 0.0040

B: Damping Coeff. (beta) = 0.012 Zmin: Const from Table 6-2 = 15.000

ATri: Const from Table 6-2 = 0.105 Btri: Const from Table 6-2 = 1.000

Amean: Const from Table 6-2 = 0.154 Bmean: Const from Table 6-2 = 0.650

C: Const from Table 6-2 = 0.200 Epsilon: Const from Table 6-2 = 0.200

ElAdj: Stack Elev Above Grade = 0.0 ft BDist: Stack Enclosed Elev = 0.0 ft

Zmean = 0.6 * H = 84.000

Izm = c * (33 / Zmean) ^ (1 / 6) = 0.171

Lzmean = l * (Zmean / 33) ^ eps = 602.730

Q = Sqrt(1/(1+.63*((b+h)/Lzmean)^.63)) = 0.891

Kd = Directionality factor = 1.000

Kzt = Terrain factor = 1.000

Since ASME STS criteria being following, always calculate flexible gust factor

Gust factor (Flexible Or Dynamically sensitive Structure) :

VZmean = Bmean * (Zmean/33)Âmean * V = 110.1 ft/s

N1 = f * Lzmean / VZmean = 7.288 Hz

Rn = 7.465 * N1 / (1 + 10.302 * N1)^(5/3) = 0.040

nh = 4.6 * f * h / VZmean = 7.787

Rh = (1 / nh) - (1 / (2 * nh^2)) *(1 - Exp(-2*nh)) = 0.120

nb = 4.6 * f * b / VZmean = 0.445

Rb = (1 / nb) - (1 / (2 * nb^2)) * (1 - Exp(-2 * nb)) = 0.759

nd = 15.4 * f * d / VZmean = 1.490

Rd = (1 / nd) - (1 / (2 * nd^2)) * (1 - Exp(-2 * nd)) = 0.457

R = Sqrt((1 / Beta) * Rn * Rh * Rb * (.53 + .47 * Rd)) = 0.469

g = (2 * Ln(3600 * NF1))^0.5 + 0.577 / (2 * Ln(3600 * NF1))^0.5 = 4.257

G = .925*((1+1.7*Izm*(3.4^2 * Q^2 +g^2 * R^2)^.5)/(1+1.7*3.4*Izm)) = 0.956

qz = 0.00256 * Kzt * Kd * V^2 * I = 25.60 psf


Page 13 of 180

z Kz Qz wm wd w Total

Force

ft psf lb/ft lb/ft lb/ft lb

------ ----- ----- ----- ----- ----- ------

140.00 1.359 34.78 84.6 124.3 209.0 2,090

130.00 1.337 34.24 83.3 115.5 198.8 1,988

120.00 1.315 33.67 81.9 106.6 188.5 1,885

110.00 1.291 33.06 80.5 97.7 178.2 1,782

100.00 1.266 32.40 78.9 88.8 167.7 1,677

90.00 1.238 31.69 77.1 79.9 157.1 1,571

80.00 1.208 30.91 75.2 71.1 146.3 1,463

70.00 1.174 30.06 73.2 62.2 135.3 1,353

60.00 1.137 29.10 70.8 53.3 124.1 1,241

50.00 1.094 28.00 68.2 44.4 112.6 1,126

40.00 1.044 26.72 65.0 35.5 100.5 1,005

30.00 0.982 25.15 61.2 26.6 87.8 878

20.00 0.902 23.09 56.2 17.8 74.0 740

10.00 0.849 21.73 52.9 8.9 61.8 618

------ ----- ----- ----- ----- ----- ------

Total 19,416

DQ = D*Qz^0.5

Qz = qz * Kz * Kzt (Kzt = 1.0)

wm = Mean Along wind load on stack: Cf*qz*Kz*D / (1+6.8*Izbar)

Mo = Moment due to Mean Along Wind Load = 760.36 k-ft

wd = Fluctuating Along Wind Load: (3*Z*Mo/h^3)*((Gf*(1+6.8*Izbar))-1)

w = wm + wd

Wind Loads Applied to Model

Top Bot Uniform Total

Elev Elev Load Load

ft ft lb/ft lb

------ ------ ------- ------

140.00 130.00 209.0 2,090

130.00 120.00 198.8 1,988

120.00 110.00 188.5 1,885

110.00 100.00 178.2 1,782

100.00 90.00 167.7 1,677

90.00 80.00 157.1 1,571

80.00 70.00 146.3 1,463

70.00 60.00 135.3 1,353

60.00 50.00 124.1 1,241

50.00 40.00 112.6 1,126

40.00 30.00 100.5 1,005

30.00 20.00 87.8 878

20.00 10.00 74.0 740

10.00 0.00 61.8 618

------ ------ ------- ------

Total 19,416

Vortex Shedding Summary

Mode Configuration Frequency Crit Wind Reduced Defl Comment

Num Speed Mass Max

Hz mph lb ft

---- ----------------- --------- --------- ------- ----- -----------

1 Cold & Uncorroded 1.331 36.3 11,172 3.010

2 Cold & Uncorroded 8.295 226.2 10,940 0.000 Vc>1.2*Vzcr

3 Cold & Uncorroded 23.021 627.8 10,508 0.000 Vc>1.2*Vzcr

Vortex Shedding Analysis per ASME STS-1-2011

H = Stack Height = 140.0 ft

V = Design Wind Speed based upon 50 yr Return, 3 sec gust @ 10 m = 100.0 mph

I = Importance Factor per ASCE 7-05 = 1.000

Zcr = Elevation Of 5/6 of Stack Height: (5/6)*(H + BaseDist) = 116.667 ft

Zb = Elevation to 6/10 of Stack Height = 84.0 ft


Page 14 of 180

n = Natural Frequency For Mode # 1 = 1.331 Hz Bs = Structural Damping of Mode # 1 = 0.0040 Del = Log Decrement Damping: 2 * PI * Bs = 0.0251 D = Average Outer Diameter Of top 1/3 Of Stack = 8.0 ft S = Strouhal Number for a Single Stack is 0.2 = 0.2000

Wind Velocity Imperial Imperial Metric ft/s mph m/s

--------------------------------------------------- -------- -------- ------ V: Design Wind Speed: 50 yr, 3 sec Gust & 10 m 146.7 100.0 44.70 Vr: Reference Design Speed: V*(I)^0.5 146.7 100.0 44.70 Vzcr: Mean Hourly Speed at Zcr: b*(Zcr/33)Âlpha*Vr 115.8 78.9 35.29 Vupr: Upr Limit for Vortex Shedding: 1.2*Vzcr 138.9 94.7 42.35 Vc: Critical Wind Speed: n*D/S 53.2 36.3 16.23

me = Equivalent Uniform Mass per Unit Length = 319.66 lb/ft Mred = Reduced Mass at Top of Stack = 11172 lb Rv = Vc * I^0.5 / Vzcr = 0.46 N = # Cycles: n*(T/50)*(10^10)*(Ratio^2)*Exp(-15*Rv^2) {App E-3.3} = 4.72E+07 Sc = Scruton Number: 4*PI*me*Bs / (Rho * D^2) = 3.2818

Factor = Since Vc < Vzcr use 1.0 = 1.00 R = Constant Of 1.0 For Parallel Stack = 1.00

Phi(zm) = Maximum value of Phi(z) at Maximum deflection = 1.000 Int2 = Integral From 0 to H of [Phi^2 dz] = 34.951 Cm = Mode Shape Const: Phi(zm)*[(1/H)*Int2]^0.5 / [(1/H)*Int2] = 2.001 C1 = Constant for grouped/isolated stacks = 0.120 C2 = Constant: 0.6 * r = 0.600 Mr = Dimensionless Mass: Me / (0.00238 * D ^ 2 * 32.2) = 65.17 Lambda = Aspect Ratio: H / D = 17.500

A1 = Constant: C1 * Cm / [mr * (Bs*Lambda)^0.5] = 0.014 A2 = Constant: C2 / (mr * Bs) = 2.302 Accel = Acceleration at the top of the stack: Ã¢m*(2 * PI * n)^2 = 210.5 ft/s^2

mr*Bs = Product of mr * Bs = 0.261 0.8*r = 0.8 * R = 0.800 0.4*r = 0.4 * R = 0.400 gh = mr*Bs [0.261] < 0.4*r [0.400] --> 1.6 = 1.6 ft/s^2 gs = Since Mr*Bs [0.261] < 0.4*r [0.400] --> 1.5 = 1.5 ft/s^2

am_num = Numerator: -(1-A2) + ((1-A2)^2 + 16*A1^2 * A2)^0.5 = 2.606 am = Para E-3.2, rms dynamic displ: D* [am_num / (8*A2)]^0.5 = 3.01 ft ah = Max value amplitude for static equiv design loads: gh * am = 4.815 ft as = Max value amplitude for static equiv fatigue loads: gs * am = 4.514 ft

Vortex Shedding Deflections: No deflection limit has been specified

Vortex Shedding Static Loads Applied to Model

Top Bot Uniform Total Elev Elev Load Load ft ft lb/ft lb

------ ------ -------- --------140.00 130.00 -3,178.8 -31,788

130.00 120.00 -2,850.4 -28,504 120.00 110.00 -2,523.3 -25,233 110.00 100.00 -2,199.5 -21,995 100.00 90.00 -1,881.7 -18,817 90.00 80.00 -1,573.3 -15,733 80.00 70.00 -1,278.4 -12,784 70.00 60.00 -1,001.5 -10,015 60.00 50.00 -747.6 -7,476 50.00 40.00 -521.9 -5,219 40.00 30.00 -330.0 -3,300 30.00 20.00 -177.6 -1,776 20.00 10.00 -70.5 -705


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10.00 0.00 -14.5 -145

------ ------ -------- --------

Total -183,490

Vortex Shedding Fatigue Loads Applied to Model


Elev Elev Load Load

ft ft lb/ft lb

------ ------ -------- --------

140.00 130.00 -2,980.2 -29,802

130.00 120.00 -2,672.2 -26,722

120.00 110.00 -2,365.6 -23,656

110.00 100.00 -2,062.0 -20,620

100.00 90.00 -1,764.1 -17,641

90.00 80.00 -1,474.9 -14,749

80.00 70.00 -1,198.5 -11,985

70.00 60.00 -938.9 -9,389

60.00 50.00 -700.8 -7,008

50.00 40.00 -489.3 -4,893

40.00 30.00 -309.4 -3,094

30.00 20.00 -166.5 -1,665

20.00 10.00 -66.1 -661

10.00 0.00 -13.6 -136

------ ------ -------- --------

Total -172,022

Vortex Shedding Fatigue Analysis for Load # 7 - Vortex Fatigue

Fatigue Analysis per AISC 360-10

Top El Bot El Cat Cf Fth 2a w tp Moment Srange Fsr Unity

ft ft ksi in in in k-ft ksi ksi

------ ------ --- -------- ----- --- --- --- --------- ------ ----- -----

140.00 130.00 C 4.40E+09 10.00 N/A N/A N/A 149.01 1.60 10.00 0.16

130.00 120.00 C 4.40E+09 10.00 N/A N/A N/A 313.41 3.36 10.00 0.34

120.00 110.00 C 4.40E+09 10.00 N/A N/A N/A 1,027.60 11.01 10.00 1.10

110.00 100.00 C 4.40E+09 10.00 N/A N/A N/A 1,962.86 21.03 10.00 2.10

100.00 90.00 C 4.40E+09 10.00 N/A N/A N/A 3,088.86 33.10 10.00 3.31

90.00 80.00 C 4.40E+09 10.00 N/A N/A N/A 4,375.93 46.89 10.00 4.69

80.00 70.00 C 4.40E+09 10.00 N/A N/A N/A 5,795.41 62.09 10.00 6.21

70.00 60.00 C 4.40E+09 10.00 N/A N/A N/A 7,320.06 78.43 10.00 7.84

60.00 50.00 C 4.40E+09 10.00 N/A N/A N/A 8,924.54 95.62 10.00 9.56

50.00 40.00 C 4.40E+09 10.00 N/A N/A N/A 10,585.89 113.42 10.00 11.34

40.00 30.00 C 4.40E+09 10.00 N/A N/A N/A 12,284.00 131.62 10.00 13.16

30.00 20.00 C 4.40E+09 10.00 N/A N/A N/A 14,002.21 150.03 10.00 15.00

20.00 10.00 C 4.40E+09 10.00 N/A N/A N/A 15,727.81 168.52 10.00 16.85

10.00 0.00 C 4.40E+09 10.00 N/A N/A N/A 17,453.96 187.01 10.00 18.70

Sr = 2 x Bending Stress

Fsr= Design Stress Range per Section 3.3

N = Num of Cycles Calculated from Vortex Shedding Analysis = 4.72E07

2a = Length of the nonwelded root face in dir of the thk of the tension-loaded plate

w = Leg size of reinf or contouring fillet, in dir of thk of tension-loaded plate.

tp = Thickness of the tension loaded plate

N/A= Not Applicable, the terms 2a, w and tp are only necessary if Category is C' or C''

Stack Deflection Load # 8

Load Combination D+W

Elev Dx Dy Dz Dres Limit Ratio Rresult

ft in in in in in

------ ------- ------- ------ ------ -------- ----- --------

140.00 -4.6086 -0.0138 0.0000 4.6086 No Limit 0.00 No Limit

130.00 -4.1675 -0.0138 0.0000 4.1675 No Limit 0.00 No Limit

120.00 -3.7260 -0.0135 0.0000 3.7260 No Limit 0.00 No Limit

110.00 -3.2859 -0.0132 0.0000 3.2859 No Limit 0.00 No Limit

100.00 -2.8500 -0.0127 0.0000 2.8500 No Limit 0.00 No Limit

90.00 -2.4222 -0.0121 0.0000 2.4222 No Limit 0.00 No Limit


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80.00 -2.0073 -0.0113 0.0000 2.0073 No Limit 0.00 No Limit

70.00 -1.6111 -0.0104 0.0000 1.6111 No Limit 0.00 No Limit

60.00 -1.2399 -0.0093 0.0000 1.2399 No Limit 0.00 No Limit

50.00 -0.9014 -0.0081 0.0000 0.9014 No Limit 0.00 No Limit

40.00 -0.6034 -0.0068 0.0000 0.6034 No Limit 0.00 No Limit

30.00 -0.3546 -0.0053 0.0000 0.3546 No Limit 0.00 No Limit

20.00 -0.1645 -0.0037 0.0000 0.1645 No Limit 0.00 No Limit

10.00 -0.0429 -0.0019 0.0000 0.0429 No Limit 0.00 No Limit

0.00 0.0000 0.0000 0.0000 0.0000 0.0000 0.00 Pass

Dres = Max Lateral Displacement: (Dx^2 + Dz^2)

Ratio = Dres / Limit

Stack Deflection Load # 9

Load Combination D+L+W

Elev Dx Dy Dz Dres Limit Ratio Rresult

ft in in in in in

------ ------- ------- ------ ------ -------- ----- --------

140.00 -4.6086 -0.0138 0.0000 4.6086 No Limit 0.00 No Limit

130.00 -4.1675 -0.0138 0.0000 4.1675 No Limit 0.00 No Limit

120.00 -3.7260 -0.0135 0.0000 3.7260 No Limit 0.00 No Limit

110.00 -3.2859 -0.0132 0.0000 3.2859 No Limit 0.00 No Limit

100.00 -2.8500 -0.0127 0.0000 2.8500 No Limit 0.00 No Limit

90.00 -2.4222 -0.0121 0.0000 2.4222 No Limit 0.00 No Limit

80.00 -2.0073 -0.0113 0.0000 2.0073 No Limit 0.00 No Limit

70.00 -1.6111 -0.0104 0.0000 1.6111 No Limit 0.00 No Limit

60.00 -1.2399 -0.0093 0.0000 1.2399 No Limit 0.00 No Limit

50.00 -0.9014 -0.0081 0.0000 0.9014 No Limit 0.00 No Limit

40.00 -0.6034 -0.0068 0.0000 0.6034 No Limit 0.00 No Limit

30.00 -0.3546 -0.0053 0.0000 0.3546 No Limit 0.00 No Limit

20.00 -0.1645 -0.0037 0.0000 0.1645 No Limit 0.00 No Limit

10.00 -0.0429 -0.0019 0.0000 0.0429 No Limit 0.00 No Limit

0.00 0.0000 0.0000 0.0000 0.0000 0.0000 0.00 Pass

Dres = Max Lateral Displacement: (Dx^2 + Dz^2)

Ratio = Dres / Limit

Static Summation of Forces

Total sum of all loads acting on system, Shear = (Fx^2+Fz^2)^0.5

Ld Load Case Dir Fx Fy Fz Shear

(P)=Primary, (C)=Combination

Deg Kip Kip Kip Kip

-- ---------------------------- --- ----- ----- ------- ------

1 (P)Dead 0 0.00 44.75 0.00 0.00

2 (P)Live 0 0.00 0.00 0.00 0.00

3 (P)Operating 0 0.00 0.00 0.00 0.00

4 (P)Thermal Hot 0 0.00 0.00 0.00 0.00

5 (P)Wind 0 19.42 0.00 0.00 19.42

6 (P)Vortex Static 0 0.00 0.00 -183.49 183.49

7 (P)Vortex Fatigue 0 0.00 0.00 -172.02 172.02

8 (C)D+W 0 19.42 44.75 0.00 19.42

9 (C)D+L+W 0 19.42 44.75 0.00 19.42

10 (C)D+O+P 0 0.00 44.75 0.00 0.00

11 (C)D+V 0 0.00 44.75 -183.49 183.49

Detailed Support Loads for Stack Base

Loads acting on the support points (+Y is Vertical and Upward)

Ld Load Case Dir Fx Fy Fz Mx My Mz

(P)=Primary,(C)=Combo

Deg Kip Kip Kip k-ft k-ft k-ft

-- --------------------- --- ----- ----- ------- ---------- ---- ---------

1 (P)Dead 0 0.00 44.75 0.00 0.00 0.00 0.00

2 (P)Live 0 0.00 0.00 0.00 0.00 0.00 0.00

3 (P)Operating 0 0.00 0.00 0.00 0.00 0.00 0.00

4 (P)Thermal Hot 0 0.00 0.00 0.00 0.00 0.00 0.00

5 (P)Wind 0 19.42 0.00 0.00 0.00 0.00 -1,615.21


Page 17 of 180

6 (P)Vortex Static 0 0.00 0.00 -183.49 -18,617.56 0.00 0.00

7 (P)Vortex Fatigue 0 0.00 0.00 -172.02 -17,453.96 0.00 0.00

8 (C)D+W 0 19.42 44.75 0.00 0.00 0.00 -1,615.21

9 (C)D+L+W 0 19.42 44.75 0.00 0.00 0.00 -1,615.21

10 (C)D+O+P 0 0.00 44.75 0.00 0.00 0.00 0.00

11 (C)D+V 0 0.00 44.75 -183.49 -18,617.56 0.00 0.00

Load Load Case Vertical Shear Moment


Kip Kip k-ft

---- --------------------- -------- ------ ---------

1 (P)Dead 44.75 0.00 0.00

2 (P)Live 0.00 0.00 0.00

3 (P)Operating 0.00 0.00 0.00

4 (P)Thermal Hot 0.00 0.00 0.00

5 (P)Wind 0.00 19.42 1,615.21

6 (P)Vortex Static 0.00 183.49 18,617.56

7 (P)Vortex Fatigue 0.00 172.02 17,453.96

8 (C)D+W 44.75 19.42 1,615.21

9 (C)D+L+W 44.75 19.42 1,615.21

10 (C)D+O+P 44.75 0.00 0.00

11 (C)D+V 44.75 183.49 18,617.56

These are the resultant forces and the shear and moment can occur in any direction

Shear = Resultant Shear: (Fx^2 + Fz^2)^0.5

Moment = Resultant Mom: (Mx^2 + Mz^2)^0.5

Vertical load is the Fy force.

Maximum Support Loads for All Load Combinations

Maximum loading for each restraint, but loads may not necessarily occur at the same time

Restraint Description Vertical Uplift Shear Moment Torsion

Kip Kip Kip k-ft k-ft

--------------------- -------- ------ ------ --------- -------

Stack Base 0.00 44.75 183.49 18,617.56 0.00

Vertical = Max downward acting Fy (Largest negative Fy value)

Uplift = Max upward acting Fy (Largest postive Fy value)


Moment = Resultant Overturning Moment: (Mx^2 + Mz^2)^0.5

Torsion = Maximum magnitude of My


Page 18 of 180

Section A-2 - NonMan App E Table E-1.1-1 Examples (Along Wind)


Page 19 of 180


Page 20 of 180

Chris Rosencutter

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Chris Rosencutter

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Chris Rosencutter

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0.988 if we set aerodnamic damping to zero.

Chris Rosencutter

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Chris Rosencutter

Callout


Chris Rosencutter

Text Box

we are permitted to use aerodynamic damping for along wind loads, which would reduce the value of wind. These examples are ignoring the aerodynamic damping, and so we manually set the aero damping to zero in order to match the results.


Date: Apr 22, 2017

File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\STS2011\ Along_Gust_Stack1\v5314_STS_TableE111_Stack1.stk

INPUT PARAMETERS:

*Non-Linear P-Delta Analysis Loads applied in 10 steps, with the forces And deflections calculated after each step. * Stress analysis based upon Limit States Design



Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------80.0 5.0 | 80.0 0.2500 | 80.0 0.0000

Materials:


---- -------- ---- ----- -------- -------- ---------- -------80.0 A-36 70.0 36.00 29,300 6.80E-06 16.60 490.0

70.0 36.00 29,300 6.80E-06 16.60 490.0



V: Wind Speed = 110.0 mph f: Natural Frequency = 2.6 Hz C: Structure Category = II Exp: Exposure Category = C I: Importance Factor = 1.00 H: Structure Height = 80.0 ft D: Structure Depth = 5.0 ft b: Structure Width = 5.0 ft


B: Damping Coeff. (beta) = 0.006 Zmin: Const from Table 6-2 = 15.000 ATri: Const from Table 6-2 = 0.105 Btri: Const from Table 6-2 = 1.000 Amean: Const from Table 6-2 = 0.154 Bmean: Const from Table 6-2 = 0.650 C: Const from Table 6-2 = 0.200 Epsilon: Const from Table 6-2 = 0.200 ElAdj: Stack Elev Above Grade = 0.0 ft BDist: Stack Enclosed Elev = 0.0 ft

Zmean = 0.6 * H = 48.000 Izm = c * (33 / Zmean) ^ (1 / 6) = 0.188 Lzmean = l * (Zmean / 33) ^ eps = 538.909 Q = Sqrt(1/(1+.63*((b+h)/Lzmean)^.63)) = 0.914 Kd = Directionality factor = 1.000 Kzt = Terrain factor = 1.000


Gust factor (Flexible Or Dynamically sensitive Structure) : VZmean = Bmean * (Zmean/33)Âmean * V = 111.1 ft/s N1 = f * Lzmean / VZmean = 12.613 Hz Rn = 7.465 * N1 / (1 + 10.302 * N1)^(5/3) = 0.028 nh = 4.6 * f * h / VZmean = 8.613 Rh = (1 / nh) - (1 / (2 * nh^2)) *(1 - Exp(-2*nh)) = 0.109


Page 21 of 180

nb = 4.6 * f * b / VZmean = 0.538 Rb = (1 / nb) - (1 / (2 * nb^2)) * (1 - Exp(-2 * nb)) = 0.720 nd = 15.4 * f * d / VZmean = 1.802 Rd = (1 / nd) - (1 / (2 * nd^2)) * (1 - Exp(-2 * nd)) = 0.405 R = Sqrt((1 / Beta) * Rn * Rh * Rb * (.53 + .47 * Rd)) = 0.514 g = (2 * Ln(3600 * NF1))^0.5 + 0.577 / (2 * Ln(3600 * NF1))^0.5 = 4.411 G = .925*((1+1.7*Izm*(3.4^2 * Q^2 +g^2 * R^2)^.5)/(1+1.7*3.4*Izm)) = 0.988 qz = 0.00256 * Kzt * Kd * V^2 * I = 30.98 psf

z Kz Qz wm wd w Total Force ft psf lb/ft lb/ft lb/ft lb

----- ----- ----- ----- ----- ----- ----- 80.00 1.208 37.40 53.4 92.6 146.0 1,460 70.00 1.174 36.37 51.9 81.1 132.9 1,329 60.00 1.137 35.21 50.2 69.5 119.7 1,197 50.00 1.094 33.88 48.3 57.9 106.2 1,062 40.00 1.044 32.33 46.1 46.3 92.4 924 30.00 0.982 30.43 43.4 34.7 78.2 782 20.00 0.902 27.94 39.9 23.2 63.0 630 10.00 0.849 26.30 37.5 11.6 49.1 491

----- ----- ----- ----- ----- ----- ----- Total 7,876 DQ = D*Qz^0.5 Qz = qz * Kz * Kzt (Kzt = 1.0) wm = Mean Along wind load on stack: Cf*qz*Kz*D / (1+6.8*Izbar) Mo = Moment due to Mean Along Wind Load = 158.00 k-ft wd = Fluctuating Along Wind Load: (3*Z*Mo/h^3)*((Gf*(1+6.8*Izbar))-1) w = wm + wd


Page 22 of 180


Date: Apr 22, 2017

File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\STS2011\ Along_Gust_Stack2\v5314_STS_TableE111_Stack2.Stk

INPUT PARAMETERS:

*Non-Linear P-Delta Analysis Loads applied in 10 steps, with the forces And deflections calculated after each step. * Stress analysis based upon Limit States Design



Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------160.0 10.0 | 160.0 0.3125 | 160.0 0.0000

Materials:


----- -------- ----- ----- -------- -------- ---------- -------160.0 A-36 100.0 36.00 28,950 6.80E-06 16.60 490.0

100.0 36.00 28,950 6.80E-06 16.60 490.0



V: Wind Speed = 100.0 mph f: Natural Frequency = 1.3 Hz C: Structure Category = III Exp: Exposure Category = C I: Importance Factor = 1.15 H: Structure Height = 160.0 ft D: Structure Depth = 10.0 ft b: Structure Width = 10.0 ft





Gust factor (Flexible Or Dynamically sensitive Structure) : VZmean = Bmean * (Zmean/33)Âmean * V = 112.4 ft/s N1 = f * Lzmean / VZmean = 7.163 Hz Rn = 7.465 * N1 / (1 + 10.302 * N1)^(5/3) = 0.040 nh = 4.6 * f * h / VZmean = 8.516 Rh = (1 / nh) - (1 / (2 * nh^2)) *(1 - Exp(-2*nh)) = 0.111


Page 23 of 180

nb = 4.6 * f * b / VZmean = 0.532 Rb = (1 / nb) - (1 / (2 * nb^2)) * (1 - Exp(-2 * nb)) = 0.723 nd = 15.4 * f * d / VZmean = 1.782 Rd = (1 / nd) - (1 / (2 * nd^2)) * (1 - Exp(-2 * nd)) = 0.408 R = Sqrt((1 / Beta) * Rn * Rh * Rb * (.53 + .47 * Rd)) = 0.622 g = (2 * Ln(3600 * NF1))^0.5 + 0.577 / (2 * Ln(3600 * NF1))^0.5 = 4.252 G = .925*((1+1.7*Izm*(3.4^2 * Q^2 +g^2 * R^2)^.5)/(1+1.7*3.4*Izm)) = 1.006 qz = 0.00256 * Kzt * Kd * V^2 * I = 29.44 psf


------ ----- ----- ----- ----- ----- ------ 160.00 1.397 41.14 125.0 197.5 322.6 3,226 150.00 1.378 40.58 123.4 185.2 308.5 3,085 140.00 1.359 39.99 121.6 172.8 294.4 2,944 130.00 1.337 39.38 119.7 160.5 280.2 2,802 120.00 1.315 38.72 117.7 148.1 265.8 2,658 110.00 1.291 38.01 115.6 135.8 251.4 2,514 100.00 1.266 37.26 113.3 123.5 236.7 2,367 90.00 1.238 36.44 110.8 111.1 221.9 2,219 80.00 1.208 35.55 108.1 98.8 206.8 2,068 70.00 1.174 34.56 105.1 86.4 191.5 1,915 60.00 1.137 33.46 101.7 74.1 175.8 1,758 50.00 1.094 32.20 97.9 61.7 159.6 1,596 40.00 1.044 30.72 93.4 49.4 142.8 1,428 30.00 0.982 28.92 87.9 37.0 124.9 1,249 20.00 0.902 26.55 80.7 24.7 105.4 1,054 10.00 0.849 24.99 76.0 12.3 88.3 883

------ ----- ----- ----- ----- ----- ------ Total 33,767 DQ = D*Qz^0.5 Qz = qz * Kz * Kzt (Kzt = 1.0) wm = Mean Along wind load on stack: Cf*qz*Kz*D / (1+6.8*Izbar) Mo = Moment due to Mean Along Wind Load = 1464.79 k-ft wd = Fluctuating Along Wind Load: (3*Z*Mo/h^3)*((Gf*(1+6.8*Izbar))-1) w = wm + wd


Page 24 of 180

MecaStack Version 5.3.0.2

Reference No.: 0 Copyright © 2013 Date : September 14, 2016 Meca Enterprises Inc. Designed by : 0 www.mecaenterprises.com FileLocation : C:\Users\Chris Rosencutter\Docu ments\Software\MecaStack\Validation\STS2011\Along_Gust_Stack3\v5302\STS_TableE111_Stack3_v5302 .Stk

ANALYSIS TYPE: * Linear Static Analysis * Stress analysis based upon Allowable Stress D esign

STACK GEOMETRY:

Total Stack Height = 240.000 ft Top of Stack Elevation = 240.000 ft Grade Elevation = 0.000 ft Bottom of Sta ck Elevation = 0.000 ft

All elevations are based upon the bottom of sta ck being at 0 ft

OUTER DIAMETERS STACK THICKNESS CORROSION ALLOWANCE Elevation Diameter Elevation Thickness Elevation Cor. Allow (ft) (ft) (ft) (in) (ft) (in) --------- -------- --------- --------- --------- ---------- 240.00 15.00 240.00 0.3750 240.00 0.0000 MATERIAL:

Elev Material Temp Fyld Elas M od Alpha Allow Strs Density ft ksi ksi in/in/F ksi lb/ft^3 ------ -------- ---- ----- ------ -- ------- ---------- ------- 240.00 A-36 70.0 36.00 29,3 00 6.80E-06 16.60 490.0 70.0 36.00 29,3 00 6.80E-06 16.60 490.0

DAMPING CALCULATIONS: Structural Damping Criteria: Manually Ent ered by User Mode # 1: Bs : 0.0060

Aerodynamic Damping Criteria: Manually Entere d by User Ba : 0

DESIGN CODES: OVERALL DESIGN CODE: ASME STS-1-2011 'Ste el Stack Design' ALLOWABLE STRESS CODE: ASME STS-1-2011 'Ste el Stack Design' DEFLECTION LIMITS: No Deflection Limit was Specified

FATIGUE: Design Code : American Institute of Steel Construction ASD 9th Ed. Fatigue Life : 20 Years Number of Cycles: Calculated per ASME STS -1-2011 Appendix E-5 Stress Category : C

VORTEX SHEDDING CODE: ASME STS-1-2011 Stack Arrangement : SINGLE

ALONG WIND CODE: ASCE 7-05 Wind Speed : 110.0 mph Exposure : C Structural Category : 2 Importance Factor : 1 Shape Factor (Piping D*qz^0.5 > 2.5) : 0.700 Shape Factor (Piping D*qz^0.5 < 2.5) : 0.000 Shape Factor (L&P's) : 2.000 Stack Enclosed to Elev : 0. 0 ft (No wind below this elev) Stack Elev above Grade : 0. 0 ft (used for gust calculations)

SEISMIC CODE: NONE Seismic Zone : 0

ASME STS-1-2011 Wind LoadsPressures calculated based upon lowest natural frequency

Wind Speed = 110.00 mph Natural Fre quency = 0.90 Hz Structure Category = II Exposure Categ ory = C


Page 25 of 180

Importance Factor = 1.00 Structure Heig ht = 240.00 ft Structure Depth = 15.00 ft Structure Wi dth = 15.00 ft Stack Elev Above Grade = 0.00 ft Stack Enclos ed to Elev = 0.00 ft Damping Coeff. (beta) = 0.0060

TABLE 6-2 VALUES: ATri = 0.105 Btri = 1.000 Amean = 0.154 Bmean = 0.650 C = 0.200 Epsilon = 0.200 Zmin = 15.000

Zmean : .6 * h = 144.0 Izm : c * (33 / Zmean) ^ (1 / 6) = 0.156 Lzmean: l * (Zmean / 33) ^ eps = 671.3 Q : Sqr(1/(1+.63*((b+h)/Lzmean)^.63)) = 0.863

Gust factor (Flexible or Dynamically sensitive structure) : VZmean: Bmean*(Zmean/33)Âmean*Wspeed (ft/sec ) = 131.5 ft/s N1 : NatFreq * Lzmean / VZmean = 4.593 Hz Rn : 7.465*N1/(1 + 10.302 * N1)^(5/3) = 0.053 nh : 4.6 * NatFreq * h / VZmean = 7.553 Rh : (1/nh)-(1/(2*nh^2))*(1-Exp(-2*nh)) = 0.124 nb : 4.6 * NatFreq * b / VZmean = 0.472 Rb : (1/nb)-(1/(2*nb^2))*(1-Exp(-2*nb)) = 0.747 nd : 15.4 * NatFreq * d / VZmean = 1.580 Rd : (1/nd)-(1/(2*nd^2))*(1-Exp(-2*nd)) = 0.441 R : Sqr((1/Beta)*Rn*Rh*Rb*(.53+.47*Rd)) = 0.779 g : (2*Ln(3600*NF1))^0.5 + 0.577/(2*Ln(36 00*NF1))^0.5 = 4.164 G :.925*((1+1.7*Izm*(3.4^2*Q^2+g^2*R^2)^. 5)/(1+1.7*3.4*Izm)) = 1.051

qz : 0.00256*Kzt*Kd*WindSpeed^2*I (Kd=Kz t=1) = 30.98 psf

z G Kz Qz wm wd w Cf Riser Cf Pipe1 Cf Pipe2 ft lb/ft lb/ft lb/ft D*Qz^.5>2.5 D*Qz^.5<=2.5 ------- ----- ----- -------- -------- -------- -------- ---------- ----------- ------------ 240.00 1.05 1.52 46.93 221.70 353.39 575.09 0.65 0.70 1.20 230.00 1.05 1.50 46.50 219.68 338.66 558.34 0.65 0.70 1.20 220.00 1.05 1.49 46.06 217.59 323.94 541.52 0.65 0.70 1.20 210.00 1.05 1.47 45.60 215.42 309.21 524.63 0.65 0.70 1.20 200.00 1.05 1.46 45.12 213.16 294.49 507.65 0.65 0.70 1.20 190.00 1.05 1.44 44.62 210.81 279.76 490.57 0.65 0.70 1.20 180.00 1.05 1.42 44.11 208.36 265.04 473.40 0.65 0.70 1.20 170.00 1.05 1.41 43.56 205.79 250.31 456.11 0.65 0.70 1.20 160.00 1.05 1.39 42.99 203.10 235.59 438.69 0.65 0.70 1.20 150.00 1.05 1.37 42.39 200.27 220.87 421.14 0.65 0.70 1.20 140.00 1.05 1.35 41.76 197.28 206.14 403.42 0.65 0.70 1.20 130.00 1.05 1.33 41.09 194.11 191.42 385.53 0.65 0.70 1.20 120.00 1.05 1.30 40.37 190.73 176.69 367.43 0.65 0.70 1.20 110.00 1.05 1.28 39.61 187.11 161.97 349.08 0.65 0.70 1.20 100.00 1.05 1.25 38.78 183.21 147.24 330.46 0.65 0.70 1.20 90.00 1.05 1.22 37.89 178.97 132.52 311.49 0.65 0.70 0.70 80.00 1.05 1.19 36.90 174.32 117.80 292.11 0.65 0.70 0.70 70.00 1.05 1.16 35.80 169.15 103.07 272.22 0.65 0.70 0.70 60.00 1.05 1.12 34.57 163.30 88.35 251.65 0.65 0.70 0.70 50.00 1.05 1.07 33.14 156.54 73.62 230.17 0.65 0.70 0.70 40.00 1.05 1.01 31.43 148.48 58.90 207.37 0.65 0.70 0.70 30.00 1.05 0.95 29.28 138.32 44.17 182.50 0.65 0.70 0.70 20.00 1.05 0.85 26.30 124.22 29.45 153.67 0.65 0.70 0.70 10.00 1.05 0.85 26.30 124.22 14.72 138.94 0.65 0.70 0.70

Definitions: DQ = D*Qz^0.5 Qz = qz * Kz * Kzt (Kzt = 1.0) wm = Mean Along wind load on stack: Cf*qz* Kz*D / (1+6.8*Izbar) Mo = Moment due to Mean Along Wind Load = 5,803 K-ft wd = Fluctuating Along Wind Load: (3*Z*Mo/ h^3)*((Gf*(1+6.8*Izbar))-1) w = wm + wd


Page 26 of 180


Date: Apr 22, 2017

File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\STS2011\ Along_Gust_Stack3\v5314_STS_TableE111_Stack3.Stk

INPUT PARAMETERS:




Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------240.0 15.0 | 240.0 0.3750 | 240.0 0.0000

Materials:


----- -------- ---- ----- -------- -------- ---------- -------240.0 A-36 70.0 36.00 29,300 6.80E-06 16.60 490.0

70.0 36.00 29,300 6.80E-06 16.60 490.0



V: Wind Speed = 110.0 mph f: Natural Frequency = 0.9 Hz C: Structure Category = II Exp: Exposure Category = C I: Importance Factor = 1.00 H: Structure Height = 240.0 ft D: Structure Depth = 15.0 ft b: Structure Width = 15.0 ft





Gust factor (Flexible Or Dynamically sensitive Structure) : VZmean = Bmean * (Zmean/33)Âmean * V = 131.5 ft/s N1 = f * Lzmean / VZmean = 4.593 Hz Rn = 7.465 * N1 / (1 + 10.302 * N1)^(5/3) = 0.053 nh = 4.6 * f * h / VZmean = 7.553 Rh = (1 / nh) - (1 / (2 * nh^2)) *(1 - Exp(-2*nh)) = 0.124 nb = 4.6 * f * b / VZmean = 0.472


Page 27 of 180

Rb = (1 / nb) - (1 / (2 * nb^2)) * (1 - Exp(-2 * nb)) = 0.747 nd = 15.4 * f * d / VZmean = 1.580 Rd = (1 / nd) - (1 / (2 * nd^2)) * (1 - Exp(-2 * nd)) = 0.441 R = Sqrt((1 / Beta) * Rn * Rh * Rb * (.53 + .47 * Rd)) = 0.779 g = (2 * Ln(3600 * NF1))^0.5 + 0.577 / (2 * Ln(3600 * NF1))^0.5 = 4.164 G = .925*((1+1.7*Izm*(3.4^2 * Q^2 +g^2 * R^2)^.5)/(1+1.7*3.4*Izm)) = 1.051 qz = 0.00256 * Kzt * Kd * V^2 * I = 30.98 psf


------ ----- ----- ----- ----- ----- ------ 240.00 1.522 47.14 222.7 356.1 578.8 5,788 230.00 1.508 46.72 220.7 341.3 562.0 5,620 220.00 1.494 46.28 218.6 326.4 545.1 5,451 210.00 1.480 45.83 216.5 311.6 528.1 5,281 200.00 1.464 45.36 214.3 296.7 511.0 5,110 190.00 1.449 44.88 212.0 281.9 493.9 4,939 180.00 1.432 44.37 209.6 267.1 476.7 4,767 170.00 1.415 43.84 207.1 252.2 459.3 4,593 160.00 1.397 43.28 204.5 237.4 441.9 4,419 150.00 1.378 42.70 201.7 222.6 424.3 4,243 140.00 1.359 42.08 198.8 207.7 406.5 4,065 130.00 1.337 41.43 195.7 192.9 388.6 3,886 120.00 1.315 40.74 192.4 178.0 370.5 3,705 110.00 1.291 40.00 189.0 163.2 352.2 3,522 100.00 1.266 39.20 185.2 148.4 333.6 3,336 90.00 1.238 38.34 181.1 133.5 314.7 3,147 80.00 1.208 37.40 176.7 118.7 295.4 2,954 70.00 1.174 36.37 171.8 103.9 275.7 2,757 60.00 1.137 35.21 166.3 89.0 255.3 2,553 50.00 1.094 33.88 160.1 74.2 234.2 2,342 40.00 1.044 32.33 152.7 59.3 212.1 2,121 30.00 0.982 30.43 143.7 44.5 188.2 1,882 20.00 0.902 27.94 132.0 29.7 161.7 1,617 10.00 0.849 26.30 124.2 14.8 139.1 1,391

------ ----- ----- ----- ----- ----- ------ Total 89,487 DQ = D*Qz^0.5 Qz = qz * Kz * Kzt (Kzt = 1.0) wm = Mean Along wind load on stack: Cf*qz*Kz*D / (1+6.8*Izbar) Mo = Moment due to Mean Along Wind Load = 5847.53 k-ft wd = Fluctuating Along Wind Load: (3*Z*Mo/h^3)*((Gf*(1+6.8*Izbar))-1) w = wm + wd


Page 28 of 180

Section A-3 - NonMan App E E-5 (Response Spectrum)


Page 29 of 180


Page 30 of 180

Chris Rosencutter

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Chris Rosencutter

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Due to complexity of response spectrum, the results do have some variance but match within reason. We do no know all of the parameters used in the STS example either.

Chris Rosencutter

Callout

4,452 ft-kip


Date: Apr 22, 2017

File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\STS2011\ Response Spectrum\v5314_Response_Spectrum_Example.Stk

INPUT PARAMETERS:




Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------210.0 12.0 | 210.0 0.5000 | 210.0 0.0000

Materials:


----- -------- ----- ----- -------- -------- ---------- -------210.0 A-36 100.0 36.00 28,950 6.80E-06 16.60 490.0

100.0 36.00 28,950 6.80E-06 16.60 490.0

Design Codes

Comprehensive Design Standard: ASME STS-1-2011 'Steel Stack Design'


Wind Load Criteria: ASCE 7-05 'Minimum Design Loads for Buildings and Other Structures'

V = Design Wind Speed = 120.0 mph ASCE 7-10 = Is the wind speed based upon ASCE 7-10? = False Exposure = Terrain Exposure = C Category = Structural Category = III DisableSTS= Disable application of wind loads per ASME STS-1 = False


Arrange = Stack Arrangement = Single Spacing = Center to center stack spacing (Only required for Group) = 10.0 ft S = Manually entered Strouhal Number (Only used for Custom) = 0.2000 Const_Dia = Force the analysis to use Constant Diameter Equations. = False Life = Fatigue life of the stack = 50.0 Years Cycles = Override Number of Cycles (0 to calculate automatically) = 0


Elev Cat Cf Fth 2a w tp ft ksi in in in

------ --- -------- ----- ------ ------ ------ 210.00 C 4.40E+09 10.00 0.0000 0.0000 0.0000


Page 31 of 180

Seismic Criteria: Response Spectrum Analysis

Period Acceleration

s g's

--------- ------------

0.000 0.30

99999.000 0.30

Mode Shape (Hot & UnCorroded)

Elevation Mode # 1 Mode # 2 Mode # 3 Mode # 4 Mode # 5

(0.887 Hz) (5.529 Hz) (15.343 Hz) (29.687 Hz) (48.298 Hz)

ft Normalized Normalized Normalized Normalized Normalized

--------- ---------- ---------- ----------- ----------- -----------

210.00 1.000 1.000 -1.000 -1.000 1.000

200.00 0.934 0.772 -0.623 -0.469 0.313

190.00 0.869 0.545 -0.257 0.024 -0.275

180.00 0.804 0.322 0.078 0.412 -0.619

170.00 0.738 0.108 0.356 0.631 -0.630

160.00 0.674 -0.091 0.553 0.646 -0.334

150.00 0.610 -0.271 0.651 0.468 0.125

140.00 0.547 -0.426 0.643 0.152 0.539

130.00 0.485 -0.551 0.536 -0.212 0.724

120.00 0.425 -0.643 0.346 -0.523 0.597

110.00 0.367 -0.700 0.102 -0.697 0.213

100.00 0.312 -0.722 -0.162 -0.686 -0.263

90.00 0.260 -0.708 -0.410 -0.491 -0.624

80.00 0.211 -0.663 -0.608 -0.163 -0.712

70.00 0.165 -0.590 -0.730 0.213 -0.489

60.00 0.125 -0.497 -0.763 0.540 -0.048

50.00 0.089 -0.390 -0.707 0.735 0.423

40.00 0.058 -0.279 -0.576 0.755 0.728

30.00 0.034 -0.174 -0.397 0.610 0.751

20.00 0.015 -0.085 -0.210 0.360 0.514

10.00 0.004 -0.023 -0.061 0.113 0.177

0.00 0.000 0.000 0.000 0.000 0.000

Mode Shape (Cold & Uncorroded)


(0.887 Hz) (5.529 Hz) (15.343 Hz) (29.687 Hz) (48.298 Hz)

ft Normalized Normalized Normalized Normalized Normalized

--------- ---------- ---------- ----------- ----------- -----------

210.00 1.000 1.000 -1.000 -1.000 1.000

200.00 0.934 0.772 -0.623 -0.469 0.313

190.00 0.869 0.545 -0.257 0.024 -0.275

180.00 0.804 0.322 0.078 0.412 -0.619

170.00 0.738 0.108 0.356 0.631 -0.630

160.00 0.674 -0.091 0.553 0.646 -0.334

150.00 0.610 -0.271 0.651 0.468 0.125

140.00 0.547 -0.426 0.643 0.152 0.539

130.00 0.485 -0.551 0.536 -0.212 0.724

120.00 0.425 -0.643 0.346 -0.523 0.597

110.00 0.367 -0.700 0.102 -0.697 0.213

100.00 0.312 -0.722 -0.162 -0.686 -0.263

90.00 0.260 -0.708 -0.410 -0.491 -0.624

80.00 0.211 -0.663 -0.608 -0.163 -0.712

70.00 0.165 -0.590 -0.730 0.213 -0.489

60.00 0.125 -0.497 -0.763 0.540 -0.048

50.00 0.089 -0.390 -0.707 0.735 0.423

40.00 0.058 -0.279 -0.576 0.755 0.728

30.00 0.034 -0.174 -0.397 0.610 0.751

20.00 0.015 -0.085 -0.210 0.360 0.514

10.00 0.004 -0.023 -0.061 0.113 0.177

0.00 0.000 0.000 0.000 0.000 0.000


Page 32 of 180

Response Spectrum Analysis

Mode Freq Circ Period Participation Acceleration Modal Modal Mass

Num Freq Factor Mass Ht Participation

Hz rad/sec s g's lb ft

---- ------ ------- ------ ------------- ------------ ------ ------- -------------

1 0.887 5.574 1.127 1.564 0.30 98,668 152.704 0.613

2 5.529 34.737 0.181 -0.861 0.30 30,150 43.664 0.187

3 15.343 96.404 0.065 -0.511 0.30 10,741 28.887 0.067

4 29.687 186.532 0.034 0.352 0.30 5,173 15.679 0.032

5 48.298 303.463 0.021 0.289 0.30 3,572 21.930 0.022

1. Total Mass Participation for All Modes Considered is 83.3%

Response Spectrum - Normalized Mode Shapes

Elev Mass Mode # Mode # Mode # Mode # Mode #

1 2 3 4 5

ft lb

------ ----- ------ ------ ------ ------ ------

210.00 3,835 1.000 1.000 -1.000 -1.000 1.000

200.00 7,670 0.934 0.772 -0.623 -0.469 0.313

190.00 7,670 0.869 0.545 -0.257 0.024 -0.275

180.00 7,670 0.804 0.322 0.078 0.412 -0.619

170.00 7,670 0.738 0.108 0.356 0.631 -0.630

160.00 7,670 0.674 -0.091 0.553 0.646 -0.334

150.00 7,670 0.610 -0.271 0.651 0.468 0.125

140.00 7,670 0.547 -0.426 0.643 0.152 0.539

130.00 7,670 0.485 -0.551 0.536 -0.212 0.724

120.00 7,670 0.425 -0.643 0.346 -0.523 0.597

110.00 7,670 0.367 -0.700 0.102 -0.697 0.213

100.00 7,670 0.312 -0.722 -0.162 -0.686 -0.263

90.00 7,670 0.260 -0.708 -0.410 -0.491 -0.624

80.00 7,670 0.211 -0.663 -0.608 -0.163 -0.712

70.00 7,670 0.165 -0.590 -0.730 0.213 -0.489

60.00 7,670 0.125 -0.497 -0.763 0.540 -0.048

50.00 7,670 0.089 -0.390 -0.707 0.735 0.423

40.00 7,670 0.058 -0.279 -0.576 0.755 0.728

30.00 7,670 0.034 -0.174 -0.397 0.610 0.751

20.00 7,670 0.015 -0.085 -0.210 0.360 0.514

10.00 7,670 0.004 -0.023 -0.061 0.113 0.177

0.00 3,835 0.000 0.000 0.000 0.000 0.000

Response Spectrum - Static Shear Loads

Elev Mass Mode Mode Mode Mode Mode SRSS SRSS

# 1 # 2 # 3 # 4 # 5 Shear Cumulative

ft lb lb lb lb lb lb lb lb

------ ------- ------ ------ ------ ------ ----- ------ ----------

210.00 3,835 1,799 -991 588 -405 333 2,200 2,200

200.00 7,670 5,162 -2,519 1,321 -784 541 3,770 5,970

190.00 7,670 8,289 -3,598 1,623 -765 358 3,249 9,219

180.00 7,670 11,180 -4,236 1,531 -431 -54 2,842 12,061

170.00 7,670 13,837 -4,451 1,113 79 -473 2,525 14,586

160.00 7,670 16,262 -4,270 463 602 -695 2,259 16,844

150.00 7,670 18,456 -3,734 -302 981 -611 2,023 18,868

140.00 7,670 20,424 -2,890 -1,058 1,104 -253 1,818 20,685

130.00 7,670 22,170 -1,798 -1,688 933 229 1,642 22,327

120.00 7,670 23,700 -524 -2,095 509 626 1,485 23,812

110.00 7,670 25,023 864 -2,215 -55 768 1,335 25,147

100.00 7,670 26,146 2,294 -2,025 -611 593 1,191 26,338

90.00 7,670 27,080 3,697 -1,543 -1,008 178 1,056 27,394

80.00 7,670 27,838 5,010 -829 -1,140 -296 928 28,322

70.00 7,670 28,434 6,179 30 -967 -621 798 29,120

60.00 7,670 28,882 7,163 927 -531 -653 664 29,783

50.00 7,670 29,202 7,935 1,759 64 -372 530 30,314


Page 33 of 180

40.00 7,670 29,411 8,488 2,436 675 113 402 30,716

30.00 7,670 29,532 8,832 2,903 1,169 612 273 30,988

20.00 7,670 29,586 8,999 3,150 1,460 954 145 31,134

10.00 7,670 29,600 9,045 3,222 1,552 1,072 42 31,176

0.00 3,835 0 0 0 0 0 0 31,176

------ ------- ------ ------ ------ ------ ----- ------ ----------

Total 161,074 31,176

Seismic Loads Applied to Model


Elev Elev Load Load

ft ft lb/ft lb

------ ------ ------- ------

210.00 200.00 220.0 2,200

200.00 190.00 377.0 3,770

190.00 180.00 324.9 3,249

180.00 170.00 284.2 2,842

170.00 160.00 252.5 2,525

160.00 150.00 225.9 2,259

150.00 140.00 202.3 2,023

140.00 130.00 181.8 1,818

130.00 120.00 164.2 1,642

120.00 110.00 148.5 1,485

110.00 100.00 133.5 1,335

100.00 90.00 119.1 1,191

90.00 80.00 105.6 1,056

80.00 70.00 92.8 928

70.00 60.00 79.8 798

60.00 50.00 66.4 664

50.00 40.00 53.0 530

40.00 30.00 40.2 402

30.00 20.00 27.3 273

20.00 10.00 14.5 145

10.00 0.00 4.2 42

------ ------ ------- ------

Total 31,176



Ld Load Case Dir Fx Fy Fz Mx My Mz


Deg Kip Kip Kip k-ft k-ft k-ft

-- ---------------------- --- ----- ------ ------ --------- ---- ----------

1 (P)Dead 0 0.00 161.07 0.00 0.00 0.00 0.00

2 (P)Live 0 0.00 0.00 0.00 0.00 0.00 0.00

3 (P)Operating 0 0.00 0.00 0.00 0.00 0.00 0.00

4 (P)Thermal Hot 0 0.00 0.00 0.00 0.00 0.00 0.00

5 (P)Wind 0 81.87 0.00 0.00 0.00 0.00 -10,285.82

6 (P)Seismic 0 31.18 0.00 0.00 0.00 0.00 -4,514.16

7 (P)Vortex Static 0 0.00 0.00 512.41 78,092.13 0.00 0.00

8 (P)Vortex Fatigue 0 0.00 0.00 480.39 73,211.37 0.00 0.00

9 (C)D+W 0 81.87 161.07 0.00 0.00 0.00 -10,285.82

10 (C)D+L+W 0 81.87 161.07 0.00 0.00 0.00 -10,285.82

11 (C)D + 0.7*E 0 21.82 161.07 0.00 0.00 0.00 -3,159.91

12 (C)D+0.75*L+0.75*0.7*E 0 16.37 161.07 0.00 0.00 0.00 -2,369.94

13 (C)0.6*D+0.7*E 0 21.82 96.64 0.00 0.00 0.00 -3,159.91

14 (C)D+O+P 0 0.00 161.07 0.00 0.00 0.00 0.00

15 (C)D+V 0 0.00 161.07 512.41 78,092.13 0.00 0.00

Load Load Case Vertical Shear Moment


Kip Kip k-ft

---- ---------------------- -------- ------ ---------


Page 34 of 180

1 (P)Dead 161.07 0.00 0.00

2 (P)Live 0.00 0.00 0.00

3 (P)Operating 0.00 0.00 0.00

4 (P)Thermal Hot 0.00 0.00 0.00

5 (P)Wind 0.00 81.87 10,285.82

6 (P)Seismic 0.00 31.18 4,514.16

7 (P)Vortex Static 0.00 512.41 78,092.13

8 (P)Vortex Fatigue 0.00 480.39 73,211.37

9 (C)D+W 161.07 81.87 10,285.82

10 (C)D+L+W 161.07 81.87 10,285.82

11 (C)D + 0.7*E 161.07 21.82 3,159.91

12 (C)D+0.75*L+0.75*0.7*E 161.07 16.37 2,369.94

13 (C)0.6*D+0.7*E 96.64 21.82 3,159.91

14 (C)D+O+P 161.07 0.00 0.00

15 (C)D+V 161.07 512.41 78,092.13






Page 35 of 180

Section B - CICIND 2010


Page 36 of 180

CICIND 2010 Validation

In order to validate the CICIND 2010 analysis in MecaStack, the example provided in the CICIND

2010 Steel Chimneys Manual – Commentaries and Appendices Appendix No 5 (A.5.1) was

utilized. This worked example is for a 710 mm OD chimney, that is 25 m tall with a constant

wall thickness of 6.3 mm. The example demonstrates the vortex shedding (cross wind) analysis

as well as the fatigue analysis per CICIND 2010.

In the process of working through the example, many problems were encountered with our

attempt to replicate the calculations in the example. CICIND was contacted on several

occasions with little success in getting any meaningful explanations on how their numbers were

calculated. In order to validate the calculations, we had to make our own assumptions

regarding discrepancies and proceed accordingly.

As a first step, a Mathcad calculation was created in an attempt to replicate the calculations in

the example. On the following pages we present our Mathcad calculations, and then show the

correlation with the CICIND example A.5.1. It matches reasonably well, and we have attempted

to explain why we believe there to be some discrepancies.


Page 37 of 180

Section B-1 - Appendix 5 Example


Page 38 of 180

page 66 CICIND Steel Chimneys Manual ─ Commentaries and Appendices

APPENDIX No. 5 ─ EXAMPLE COMPUTATIONS

A.5.1 Vortex shedding

In this section an example calculation is given of a vortex shedding design as per Section 7.2.4.2.

The chimney is 25m tall with a constant diameter of 0.71m and a constant wall thickness of 6.3mm. The design life time of the chimney is 50 years. No insulation, ladders or other details are applied to keep the example simple. The support of the chimney is assumed to be rigid, the welding detail category is 71N/mm2 and the yield strength is 355N/mm2. The wind velocity at the top of the stack is 29.6m/s. The computation is executed for the first natural frequency only. The analysis for the higher modes is straightforward given the simple geometry of the chimney.

The chimney is divided in 17 height levels as indicated in Table A5.1.

Level z (m) d(z) (m) t(z) (mm) u1(z)

17 25.000 0.710 6.3 1.000

16 23.000 0.710 6.3 0.814

15 21.000 0.710 6.3 0.678

14 19.000 0.710 6.3 0.555

13 17.000 0.710 6.3 0.444

12 15.000 0.710 6.3 0.346

11 13.000 0.710 6.3 0.260

10 11.000 0.710 6.3 0.186

9 9.000 0.710 6.3 0.125

8 8.000 0.710 6.3 0.098

7 6.000 0.710 6.3 0.055

6 5.000 0.710 6.3 0.038

5 4.000 0.710 6.3 0.025

4 3.000 0.710 6.3 0.014

3 2.000 0.710 6.3 0.006

2 1.000 0.710 6.3 0.002

1 0.000 0.710 6.3 0.000

Table A5.1 ─ Geometrical properties.

The mode shape is approximated by a parabolic profile:

2

1z

u (z)h

=

where h is the height of the stack.

Top 1/3 mean diameter:

h

12 h/3

3d d(z) dz 0.710

h⋅

= ⋅ ⋅ =∫ (m)

First natural frequency, see ref. [1]:


Page 39 of 180

CICIND Steel Chimneys Manual ─ Commentaries and Appendices page 67

11 2

1

d 2 tf 1023 1 1.15

dh

⋅≈ ⋅ ⋅ − = (Hz)

This analytic expression is only relevant for chimneys with a constant diameter and wall thickness, a rigid support and without additional masses fitted to the chimney. For more realistic situations the natural frequency should be computed with a finite element or other computer program.

Strouhal number:

St 0.20=

Critical wind speed:

1 1cr,1

f dV 4.08

St

⋅= = (m/s)

Reynolds number:

4 51 cr,1 1Re 6.67 10 V d 1.83 10= ⋅ ⋅ ⋅ = ⋅

Site surface roughness (terrain category III, Table 7.2):

0z 0.30= (m)

Turbulence intensity:

0.22h

I (h) 0.28 0.2310

−

ν

= ⋅ =

Structural damping ratio, see Table 7.4:

s 0.002ζ =

Mass per unit length:

m(z) 7850 d(z) t(z) 110.3= ⋅ π ⋅ ⋅ = (kg/m), for all z

Equivalent mass per unit length:

( ) ( )

( ) ( )

h N 12 2 21 i 1,i i 1 1,i 1 i 1 i

0 i 1o,1 h N 1

2 221,i 1,i 1 i 1 i1

i 10

m(z) u (z) dz 0.5 m u m u h h

m

0.5 u u h hu (z) dz

110.3 (kg/m)

−

+ + +=

−

+ +=

⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ −

= =

⋅ + ⋅ −⋅

=

∫ ∑

∑∫

Scruton number:

s o,11 2

a 1

4 mSc 4.45

d

⋅ π ⋅ζ ⋅= =

ρ ⋅

Aerodynamic damping parameter:

a,maxK 2.8= , see expression (7.17)

a,1 a,maxK K 1.0 3 I

0.87ν= ⋅ − ⋅

=

The ratio 1

a,1

Sc0.4

4 K≈

⋅ π ⋅, i.e. smaller than one. So, referring to expression (7.8 ) we may expect a large

cross-wind deflection amplitude. To find this amplitude we continue with the remaining parameters.

Limiting deflection amplitude as a fraction of d1:


Page 40 of 180

Chris Rosencutter

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We obtain 1.93*10^5. We suspect maybe the 1.83 is a simple typo.

Chris Rosencutter

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This is obtained using the approximate formula, but using the exact formula we obtain 109.3 kg/m

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7.37

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4.4

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0.877


1L,1

1

fa 0.4 0.4

f= ⋅ =

Coefficient 1,1c :

2 211,1 ,1

,1

0.5 1- 4.7 104

−

= ⋅ ⋅ = ⋅ ⋅ ⋅ L

a

Scc a

Kπ

Width of lift spectrum:

B 0.1 I 0.33 0.35ν= + = ≤

RMS lift coefficient:

LC 0.7= , see expression (7.24)

Coefficient a,1C :

( ) ( ){ }

4 5 2 4 5 21 L 1 L

a,1 h N 14 2 24 2

o,1 i i 1 i 1 io,n 1i 10

5

1.78 10 d C 1.78 10 d CC

St m B 0.5 u u z zSt m B u (z) dz

5.40 10

− −

−

+ +=

−

⋅ ⋅ λ ⋅ ⋅ ⋅ ⋅ ⋅= =

⋅ ⋅ ⋅ ⋅ + ⋅ −⋅ ⋅ ⋅ ⋅

= ⋅

∑∫

Coefficient 2,1c :

1

2L,1 a,1 5

2,1 2a

a Cc = 2.02 10

K d−⋅

= ⋅⋅

Peak factor:

4

1p,1

a,1

Sck 2 1 1.2 arctan 0.75

4 K

1.45

= ⋅ + ⋅ ⋅ ⋅ π ⋅

=

First mode standard deviation of deflection:

2y,1 1 1,1 1,1 2,1 d c c c

0.22

σ = ⋅ + +

=

First mode cross-wind deflection amplitude:

1 p,1 y,1 1y (h) k u (h)

0.320 (m)

= ⋅σ ⋅

=

or

1

1

y (h)0.45

d=

The resulting cross-wind deflection amplitude is indeed a substantial fraction of the diameter as was expected by the ratio ( )aSc / 4 K⋅ π ⋅ being much smaller than one.

With the full implementation of the Model Code the deflection is:

1

1

y (h)0.47

d=

The main reason for this difference is that the maximum of the response curve occurs at a velocity slightly larger than crV .


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Per Eqn 7.39 this should be 3, not 5. This is confirmed by the units, which will end up being m^2 if we use d^5, and all other indications are that the value of Ca1 should be dimensionless.

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Eqn: 7.39

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If we use d^5 then we get 5.7*10^5 which is close to this value, but the units are m^2 which we think must be mistaken. If we use d^3 then the value is dimensionless and we get a value of 1.13*10^-4, which we will use for the remainder of calculations.

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Should have units of meters.

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We get C2n = 2.06*10-5, which is fairly close, so we think the Ca1 used is 1.13*10^-4

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1.45

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0.33

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4.81*10^-2

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0.320 m

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0.45

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We are unable to replicate this value, and CICIND has offered no supporting calculations to demonstrate how the value was obtained. We get a value of 0.45, which matches exactly the value obtained above using the reduced equations. Therefore we are assuming that our calculations.

Chris Rosencutter

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This is consistent with our findings, the maximum displacement occurs at approximately 1.1*Vcr.

CICIND Steel Chimneys Manual ─ Commentaries and Appendices page 69

To estimate the expected life time of the chimney we calculate the Palmgren-miner sum for the first mode.

Moment of inertia:

( )

( )

2

4 4

I(z) d (z) t(z) d(z) 3 t(z)8

8.6 10 m for all values of z−

π= ⋅ ⋅ ⋅ − ⋅

= ⋅

Stress standard deviation:

( )

( ) ( ) ( ){ }( )

h2

1 1a,1 z

y,1

N 1i j 1 j2a,1 i

1 j 1, j j i j 1 1, j 1 j 1 iy,1 ij i

d(z) m(s) 2 f u (s) (s - z) ds(z)

2 I(z)

d z z(z )2 f 0.5 m u z z m u z z

2 I

−+

+ + +=

⋅ ⋅ ⋅ π ⋅ ⋅ ⋅ ⋅σ

=σ ⋅

⋅ −σ= ⋅π ⋅ ⋅ ⋅ ⋅ ⋅ − + ⋅ ⋅ − ⋅

σ ⋅

∫

∑

Standard deviation of wind distribution function:

oV 0.20 V(h) 5.92= ⋅ = (m/s)

Number of load cycles in 50years:

{ }7 2 2 2 21 1 1 o 2 oN = 3.15 10 T f exp(- V / V ) - exp(- V / V )⋅ ⋅ ⋅ ⋅

V1 and V2 follow from the crossings of the horizontal line

a,0a,max

ScK 0.126

4 K= =

⋅ π ⋅

with the line for I 0.23ν = in Figure C3.12:

1 cr 2 crV 0.85 V and V 1.5 V≈ ⋅ ≈ ⋅

and

81N 6.65 10≈ ⋅

Because the number of load cycles is larger than 81.10 the stresses must be smaller than the cut-off-limit for the Palmgren-Miner sum to be smaller than one.

Palmgren-Miner sum (with 1γ = ):

{ } ( )

y,T

min

kf 2

1 1 2 2W,0 W,0 a,1 a,1a

I 1

1 i i 1 i 1 ii 1

1 a a aM (z) N exp - da

N a (z) 2 (z)

N 0.5 f f a a−

+ +=

= ⋅ ⋅ ⋅ ⋅ ⋅ σ ⋅σ

= ⋅ ⋅ + ⋅ −

∫

∑

with:

mina 14.4= (N/mm2)

6W,0 W,0N 5 10 and a 26.2= ⋅ = (N/mm2)

k 2i i i

i 2 2W,0 W,0 a,1 i a,1 i

a a a1f exp -

N a (z ) 2 (z )

= ⋅ ⋅ ⋅ σ ⋅σ

I is the number of intervals used. When using equidistant intervals I = 100 is recommended.


Page 42 of 180

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Using CICIND tables we get slightly different values of 0.86 and 1.49.

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Slightly different: 6.46*10^8, but we forced our value to match example for comparison purposes.

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0.125

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8.6*10^-4 m^4

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5.92 m/s

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Used same I = 100

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Used same values.


The resulting values are given in Table A5.2.

z (m) ,1( )a izσσσσ M(z)_red. M(z)_full

25.000 0 0 0

23.000 9.91E+05 4.02E-44 4.31E-26

21.000 3.76E+06 8.81E-02 2.04E-02

19.000 8.00E+06 5.51E+00 5.99E+00

17.000 1.35E+07 5.62E+01 5.57E+01

15.000 1.99E+07 2.10E+02 2.12E+02

13.000 2.71E+07 5.47E+02 5.68E+02

11.000 3.48E+07 1.17E+03 1.26E+03

9.000 4.29E+07 2.21E+03 2.43E+03

8.000 4.71E+07 2.92E+03 3.26E+03

6.000 5.57E+07 4.82E+03 5.49E+03

5.000 6.00E+07 6.02E+03 6.94E+03

4.000 6.43E+07 7.43E+03 8.63E+03

3.000 6.86E+07 9.04E+03 1.06E+04

2.000 7.30E+07 1.09E+04 1.28E+04

1.000 7.74E+07 1.29E+04 1.52E+04

0.000 8.17E+07 1.52E+04 1.79E+04

Table A5.2 ─ Standard deviation of the bending moment, and

the Palmgren-Miner sum for the reduced and full

formulations (partial safety factor 1=γγγγ ).

The Palmgren-Miner sum is much larger than one, so the expected lifetime is much smaller than the design life time. This was to be expected because the stresses are much larger than the stress cut-off limit of 14.4N/mm2 for detail category 71.

Compared with the full detail computation:

• the deflections are very similar as might have been expected for this straight chimney;

• the Palmgren-Miner sum is about 18% smaller than computed with the full expression and so the expected operational life time is about 18% longer. The results are so close due to the way the velocity range is determined in which cross-wind vibrations can occur.

References

[1] Szabo, I -- Höhere Technische Mechanik, 6 Auflage. Springer –Verlag Berlin, 2001.


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Values match these within 5%.

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Ratio of Mathcad/Example values, 100% is a perfect match.

Chris Rosencutter

Callout

If we force our Mathcad to use same values as in column 2 (Sigma a,1, then we get the same values with the exception of the value in row 3 *

Chris Rosencutter

Image

Chris Rosencutter

Highlight

Chris Rosencutter

Image

Chris Rosencutter

Highlight

Chris Rosencutter

Callout

*We think maybe there is a typo -2 vs -3 in the table, since all other values match well.

Chris Rosencutter

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Chris Rosencutter

Image

Chris Rosencutter

Arrow

Chris Rosencutter

Callout

Using commentary 4 we get slightly different values for the Stress Std Deviation, but if we force the values to match the example we get these values for Mn which match reasonably close.

Section B-2 - Mathcad Calculation to replicate App 5


Page 44 of 180

CICIND 2010 (App 5 Example)

Include << T:\Mathcad\Meca_Functions.mcdx

≔Cat 3 ≔f1 1.15 ≔h 25 ≔kt 1 ≔Vb 31.42 ― ≔ξs 0.002

≔ρa 1.25 ――3

≔ki 1

≔t 6.3

≔z

2523211917151311986543210

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔dz

0.710.710.710.710.710.710.710.710.710.710.710.710.710.710.710.710.71

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔tz

6.36.36.36.36.36.36.36.36.36.36.36.36.36.36.36.36.3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔d1 0.71 (Avg of top 1/3, or 6*d for region of vortex shedding)

≔a =_a ((Cat)) 0.77 ≔c =_c ((Cat)) 0.28 ≔Zmin =_Zmin ((Cat)) 8

≔b =_b ((Cat)) 0.22 ≔Zo =_Zo ((Cat)) 0.3 ≔β =_β ((Cat)) 0.37

≔Zs =⋅0.6 h 15

≔V (( ,z Cat)) ⋅⋅_k (( ,z Cat)) kt Vb

=V (( ,h Cat)) 29.6 ―

≔Lzs =L ⎛⎝ ,Zs Cat⎞⎠ 99.02 ≔Vzs =V ⎛⎝ ,Zs Cat⎞⎠ 26.45 ― ≔Ivs =_Iv ⎛⎝ ,Zs Cat⎞⎠ 0.26

≔fL (( ,Z Cat)) ⋅f1 ――――L (( ,Z Cat))

V (( ,Z Cat))=fL ⎛⎝ ,Zs Cat⎞⎠ 4.31

≔ηx (( ,,x Zs Cat)) ―――――――⋅⋅4.6 fL (( ,Zs Cat)) x

L (( ,Zs Cat))≔ηd =ηx ⎛⎝ ,,d1 Zs Cat⎞⎠ 0.14 ≔ηh =ηx ⎛⎝ ,,h Zs Cat⎞⎠ 5

≔_SL(( ,Z Cat)) ―――――――――

⋅6.8 fL (( ,Z Cat))

⎛⎝ +1 ⋅10.2 fL (( ,Z Cat))⎞⎠

⎛⎜⎝―5

3

⎞⎟⎠

≔Rd =−―1

ηd⋅―――

1

⋅2 ηd2

⎛⎝ −1 ⋅−2 ηd⎞⎠ 0.91

≔SL =_SL⎛⎝ ,Zs Cat⎞⎠ 0.05 ≔Rh =−―

1

ηh⋅―――

1

⋅2 ηh2

⎛⎝ −1 ⋅−2 ηh⎞⎠ 0.18

≔ξa =―――――――⋅⋅2.7 10−6 V (( ,h Cat))

⋅f1 t0.01103 ≔ξ =+ξs ξa 0.01303 =V (( ,h Cat)) 29.6 ―

≔R =‾‾‾‾‾‾‾‾‾‾‾‾‾

⋅⋅⋅――⋅4 ξ

SL Rd Rh 0.71 ≔B =‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾――――――――

1

+1 ⋅0.9⎛⎜⎝――――

+⋅1 b h

Lzs

⎞⎟⎠

0.630.85

=R2 0.51

=B2 0.72

≔vT =

‖‖‖‖‖‖‖‖‖

|||||||

|

←vt ⋅⋅⋅600 f1

‾‾‾‾‾‾‾‾―――

R2

+B2

R2

⎛⎜⎝――

1

1

⎞⎟⎠

|||

if <vt 48‖‖ ←vt 48

((vt))

443.57 ≔g =+‾‾‾‾‾‾‾‾⋅2 ln ((vT)) ――――0.577

‾‾‾‾‾‾‾‾⋅2 ln ((vT))3.66

4‾‾‾‾‾‾‾2 2


Page 45 of 180

≔G =+1 ⋅⋅⋅2 g Ivs‾‾‾‾‾‾‾+B2 R2 3.08 ≔_Re (( ,V d)) ⋅⋅⋅6.9 104 ――

2V d

≔Re =_Re ⎛⎝ ,V (( ,h Cat)) d1⎞⎠ ⋅1.45 106

=T

CD0 ⎛⎝ ,,z Cat dz⎞⎠ 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 …[[ ]]

≔ka =_ka ⎛⎝ ,h d1⎞⎠ 1 =ki 1

≔CD =_CD ⎛⎝ ,,z Cat dz⎞⎠

0.70.70.7⋮

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

≔Vz =_Vz (( ,z Cat))

29.629.128.527.927.226.525.624.7⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―

≔_wg ((zz)) ⋅⋅――――⋅3 (( −G 1))

h2

―zz

h

⌠⌡ d0

h

⋅_wm1((z)) z z

≔wg((z))

‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((z)) 1‖‖‖

←M,j 0

_wg ⎛⎝z

,j 0⎞⎠

((M))

≔wmz =wm((z))

272262252241230217204190174165165165

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― ≔wgz =wg((z))

706649593536480423367311254226169141

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― ≔wz =+wmz wgz

978912845778710641571500428391334306

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―

Across Wind Vibration (Using Para 7.2.4.2)


Page 46 of 180


≔fn =f1 1.15 ≔d1 0.71 ≔t1 6.3

≔St 0.2 ≔m =⋅⋅―4

⎛⎝ −d1

2 ⎛⎝ −d1 ⋅2 t1⎞⎠2 ⎞⎠ 490 ――

3109.32 ――

Examples uses approximate formula for mass, so override to match:

≔m 110.3 ――

≔dz =Vfill ⎛⎝ ,18 d1⎞⎠

0.710.710.710.71⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

≔u

1.00.8140.6780.5550.4440.3460.2600.1860.1250.0980.0550.0380.0250.0140.0060.002

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔tz =Vfill ⎛⎝ ,18 t1⎞⎠

0.010.010.010.01⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

≔mz =Vfill (( ,18 m))

110.3110.3110.3110.3

⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

――

≔Nrows =−rows ((u)) 1 16 ≔Vb =V (( ,h Cat)) 29.6 ― ≔Vcr =―――⋅fn d1

St4.08 ―

≔Vlimit =⋅1.25 V (( ,h Cat)) 37 ― Can be ignored if Vcr > Vlimit≔Ren =⋅⋅⋅6.67 104 ――2

Vcr d1 ⋅1.93 105

≔Iv =_Iv (( ,h Cat)) 0.229

=Bv 0.33≔CLn =_CLn⎛⎝Ren⎞⎠ 0.7

≔Integral_un2 =∑=j 0

−Nrows 1

⎛⎜⎝

⋅0.5 ⎛⎜⎝

⋅⎛⎜⎝

+uj

2 u+j 1

2 ⎞⎟⎠

⎛⎝

−zj

z+j 1

⎞⎠⎞⎟⎠⎞⎟⎠

4.74 ≔Integral_mun2 =⋅Integral_un2 m 523.2

≔mon =―――――Integral_mun

Integral_un110.3 ――

≔λ 1

=ρa 1.25 ――3

=⋅⋅⋅1.42 10−4 ρa λ ⎛⎝ ⋅1.78 10−4⎞⎠ ――3

≔Can =―⋅⋅⋅⋅⋅1.42 10−4 ρa λ d1

5CLn

2

⋅⋅⋅St4

mon Bv Integral_un2⎝ ⋅5.7 10−5⎠ 2 <--(Using the wrong d^5) ≔Can =―

Can

1 2⋅5.7 10−5

≔Can =―――――――――⋅⋅⋅⋅⋅1.42 10−4 ρa λ d1

3CLn

2

⋅⋅⋅St4

mon Bv Integral_un2⋅1.13 10−4 <--(Using the right d^3)


Page 47 of 180

≔Ka_max_n =_Ka_max_n ⎝Ren⎠ 2.8=Iv 0.23

≔Kan =_Kan 0.877 ≔Scn =―――――⋅⋅⋅4 mon ξs

⋅ρa d12

4.4 =ξs 0.002 ≔Ratio =――――Scn

⋅⋅4 Kan

0.4

≔aln =―――⋅0.4 f1

fn0.4

≔c1n =⋅⋅0.5 aln2

⎛⎜⎜⎝

−1 ――――Scn

⋅⋅4 Kan

⎞⎟⎟⎠

⋅4.81 10−2 ≔c2n =――――⋅aln

2Can

Kan

⋅2.06 10−5

≔kpn =⋅‾‾2⎛⎜⎜⎝

+1 ⋅1.2 atan⎛⎜⎜⎝

⋅0.75⎛⎜⎜⎝――――

Scn

⋅⋅4 Kan

⎞⎟⎟⎠

4 ⎞⎟⎟⎠

⎞⎟⎟⎠

1.45

≔σyn =⋅d1‾‾‾‾‾‾‾‾‾‾‾‾‾‾

+c1n‾‾‾‾‾‾‾‾+c1n

2 c2n 0.22

≔_yn ((j)) ⋅⋅kpn σyn uj

=―――_yn ((0))

d10.45≔_Fn

((j)) ⋅⋅⎛⎝ ⋅⋅2 fn⎞⎠2

m _yn ((j))

≔yn =‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((z)) 1‖‖‖

←yj

_yn ((j))

((y))

0.320.260.220.180.140.110.080.060.040.030.020.01⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Fn =‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((z)) 1‖‖‖

←Fj

_Fn((j))

((F))

183614951245101981563547734123018010170⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―

Fatigue Strength Due to Cross Wind Vibration (8.5.2)


Page 48 of 180


≔Nwo ⋅5 106 ≔FatCat 71

≔fyt 355

≔I =⋅――64

⎛⎝ −dz

4 ⎛⎝ −dz ⋅2 tz⎞⎠4 ⎞⎠

⋅8.6 10−4

⋅8.6 10−4

⋅8.6 10−4

⋮

⎡⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦

4

≔Tyrs 50

≔aint 100 ≔γ 1

Values taken from example.

≔σan =‖‖‖‖‖‖‖‖‖‖

||||||||

|

←N −rows ((z)) 1←i 0

for ∊ |||||

j , ‥0 1 N‖‖‖‖‖

←σi

_σan((j))

←i +i 1

((σ))

0⋅1.05 106

⋅3.79 106

⋅7.96 106

⋅1.33 107

⋅1.95 107

⋅2.65 107

⋅3.4 107

⋅4.19 107

⋅4.6 107

⋅5.42 107

⋅5.84 107

⋅6.26 107

⋅6.68 107

⋅7.1 107

⋅7.52 107

⋅7.95 107

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σapp5

⋅1 10−99

⋅9.91 105

⋅3.76 106

⋅8 106

⋅1.35 107

⋅1.99 107

⋅2.71 107

⋅3.48 107

⋅4.29 107

⋅4.71 107

⋅5.57 107

⋅6 107

⋅6.43 107

⋅6.86 107

⋅7.3 107

⋅7.74 107

⋅8.17 107

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=――σan

σapp5

0%105%101%99%98%98%98%98%98%98%97%97%97%97%97%97%97

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σan σapp5

≔Vh =V (( ,h Cat)) 29.6 ― (Provided in Example) ≔Vo =⋅Vh 0.2 5.92 ―

≔Ka_max =_Ka_max_n⎛⎝Ren⎞⎠ 2.8 Force to match example.

≔Kao =―――――Scn

⋅⋅4 Ka_max

0.125 ≔Vzero =_Vzero⎛⎝Kao

⎞⎠0.861.49

⎡⎢⎣

⎤⎥⎦

≔Vzero0.851.5

⎡⎢⎣

⎤⎥⎦

≔V1 =⋅Vzero0Vcr 3.47 ― =_Kao

⎛⎜⎝

,Vzero0Iv⎞⎟⎠

0.11 ≔V2 =⋅Vzero1Vcr 6.12 ―

Force Nn to match example.

≔Nn =

‖‖‖‖‖‖‖‖‖‖

||||||||

|

←N ⋅⋅⋅⋅3.15 107 Tyrs

⎛⎜⎝――fn

1

⎞⎟⎠

⎛⎜⎝ −

⎛⎜⎜⎝

―――−V1

2

Vo2

⎞⎟⎟⎠

⎛⎜⎜⎝

―――−V2

2

Vo2

⎞⎟⎟⎠

⎞⎟⎠

||

|

if <N ⋅200 Tyrs

‖‖‖

←N ⋅200 Tyrs

((N))

⋅6.63 108 ≔Nn ⋅6.65 108

≔awo =⋅0.5 σsr⎛⎝ ,Nwo FatCat⎞⎠ 26.2

≔amin =⋅0.5 σsr⎛⎝ ,⋅1 108 FatCat⎞⎠ 14.4

=σsr⎛⎝ ,⋅1 108 FatCat⎞⎠ 28.73


Page 49 of 180

=Ta 14.37 17.77 21.18 24.59 27.99 31.4 34.81 38.21 41.62 45.02 48.43 51.84 55.24 …[[ ]]

≔_M ⎛⎝zi⎞⎠ ⋅Nn ∑=i 0

−aint 1

⋅0.5 ⎛⎝

⋅⎛⎝ +_f ⎛⎝ ,i zi⎞⎠ _f ⎛⎝ ,+i 1 zi⎞⎠⎞⎠ ⎛⎝

−a+i 1

ai⎞⎠⎞⎠

Values taken from example.

≔Mz_red =‖‖‖‖‖‖‖

for ∊j , ‥1 2 −rows ((z)) 1‖‖‖

←Mj

⋅γ _M ((j))

((M))

0

⋅3.78 10−44

⋅8.96 10−3

5.52⋅5.7 10⋅2.11 102

⋅5.5 102

⋅1.17 103

⋅2.2 103

⋅2.92 103

⋅4.83 103

⋅6.03 103

⋅7.43 103

⋅9.02 103

⋅1.09 104

⋅1.29 104

⋅1.52 104

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Mz_red_app5

0.0001⋅4.02 10−44

⋅8.81 10−2

5.51

⋅5.62 101

⋅2.1 102

⋅5.47 102

⋅1.17 103

⋅2.21 103

⋅2.92 103

⋅4.82 103

⋅6.02 103

⋅7.43 103

⋅9.04 103

⋅1.09 104

⋅1.29 104

⋅1.52 104

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=――――Mz_red

Mz_red_app5

0%94%10%100%101%101%101%100%100%100%100%100%100%100%100%100%100

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Vortex Shedding (Using Commentary 3)


Page 50 of 180


≔Vcr =―――⋅fn d1

St4.08 ― ≔zcr =⋅―

5

6h 20.83 ≔λ 1 ≔d1 0.71 ≔t 6.3

=Cat 3 ≔m =⋅⋅―4

⎛⎝ −d1

2 ⎛⎝ −d1 ⋅2 t⎞⎠2 ⎞⎠ 490 ――

3109.32 ――

Examples uses approximate formula for mass, so override to match:

≔m 110.3 ――

Avg Diameter d1Mode 1: Avg of top 1/3:> Mode 1: where Vortex occurs over 6 diameter length

≔row =rows ((z)) 17

≔dz =Vfill ⎛⎝ ,row d1⎞⎠

0.710.710.710.71⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

=d1 0.71

=z

2523211917151311986543210

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=u

10.8140.6780.5550.4440.3460.260.1860.1250.0980.0550.0380.0250.0140.0060.0020

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔tz =Vfill (( ,row 6.3 ))

6.36.36.36.3⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

≔mz =Vfill (( ,row m))

110.3110.3110.3110.3

⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

――

We find the Vb that will give us Vcr that we want at the top of the stack:

≔_Vb((V)) ――――――

V

⋅_k ⎛⎝ ,zcr Cat⎞⎠ kt≔_V (( ,z Cat)) ⋅⋅_k (( ,z Cat)) kt Vb

=V

3.19 3.4 3.82 4.25 4.46 4.67 4.89 5.1 6.37 7.22 8.07 8.53.13 3.34 3.76 4.17 4.38 4.59 4.8 5.01 6.26 7.09 7.93 8.343.07 3.27 3.68 4.09 4.29 4.5 4.7 4.91 6.13 6.95 7.77 8.183 3.2 3.6 4 4.2 4.4 4.6 4.8 6 6.8 7.6 82.93 3.12 3.51 3.9 4.1 4.29 4.49 4.68 5.86 6.64 7.42 7.812.85 3.04 3.42 3.8 3.99 4.18 4.37 4.56 5.7 6.46 7.22 7.62.76 2.94 3.31 3.68 3.86 4.05 4.23 4.42 5.52 6.26 6.99 7.362.66 2.84 3.19 3.55 3.72 3.9 4.08 4.26 5.32 6.03 6.74 7.092.55 2.72 3.05 3.39 3.56 3.73 3.9 4.07 5.09 5.77 6.45 6.79

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―

=Vcr 4.08 ― ≔Ren =_Ren ⎛⎝ ,Vcr d1⎞⎠ ⋅1.942 105 ≔CLn =_CLn⎛⎝Ren⎞⎠ 0.7

=TBz 0.33 0.33 0.34 0.34 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35[[ ]]

=TIv 0.23 0.23 0.24 0.24 0.25 0.26 0.26 0.27 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 …[[ ]]

≔Kao_zero =―――――Scn

⋅⋅4 Ka_max

0.125 ≔Vzero =_Vzero⎛⎝Kao_zero

⎞⎠0.861.49

⎡⎢⎣

⎤⎥⎦

≔Vlwr =Vzero00.86

≔Vupr =Vzero11.49

0.78 0.83 0.94 1.04 1.09 1.15 1.2 1.25 1.56 1.77 1.98 2.08⎡ ⎤


Page 51 of 180

=Vr

0.78 0.83 0.94 1.04 1.09 1.15 1.2 1.25 1.56 1.77 1.98 2.080.77 0.82 0.92 1.02 1.07 1.12 1.18 1.23 1.53 1.74 1.94 2.040.75 0.8 0.9 1 1.05 1.1 1.15 1.2 1.5 1.7 1.9 20.73 0.78 0.88 0.98 1.03 1.08 1.13 1.18 1.47 1.67 1.86 1.960.72 0.77 0.86 0.96 1 1.05 1.1 1.15 1.43 1.63 1.82 1.910.7 0.74 0.84 0.93 0.98 1.02 1.07 1.12 1.4 1.58 1.77 1.860.68 0.72 0.81 0.9 0.95 0.99 1.04 1.08 1.35 1.53 1.71 1.80.65 0.7 0.78 0.87 0.91 0.96 1 1.04 1.3 1.48 1.65 1.740.62 0.67 0.75 0.83 0.87 0.91 0.96 1 1.25 1.41 1.58 1.660.61 0.65 0.73 0.81 0.85 0.89 0.93 0.97 1.22 1.38 1.54 1.620.61 0.65 0.73 0.81 0.85 0.89 0.93 0.97 1.22 1.38 1.54 1.620.61 0.65 0.73 0.81 0.85 0.89 0.93 0.97 1.22 1.38 1.54 1.62

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Kao

0 0 0.218 0.301 0.316 0.315 0.302 0.279 0 0 0 00 0 0.198 0.288 0.308 0.312 0.304 0.286 0 0 0 00 0 0.177 0.272 0.297 0.307 0.304 0.29 0 0 0 00 0 0.155 0.253 0.283 0.299 0.301 0.293 0.134 0 0 00 0 0 0.232 0.266 0.287 0.295 0.292 0.153 0 0 00 0 0 0.207 0.245 0.271 0.285 0.288 0.172 0 0 00 0 0 0.181 0.22 0.25 0.269 0.278 0.191 0 0 00 0 0 0.152 0.191 0.224 0.248 0.262 0.208 0.115 0 00 0 0 0 0.16 0.193 0.22 0.239 0.22 0.142 0 00 0 0 0 0 0.177 0.204 0.225 0.224 0.155 0 00 0 0 0 0 0.177 0.204 0.225 0.224 0.155 0 00 0 0 0 0 0.177 0.204 0.225 0.224 0.155 0 0

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ka_max_n =_Ka_max_n⎛⎝Ren⎞⎠ 2.8

≔Karn =―――Karn_num

Karn_den

0 0 0.436 0.746 0.816 0.845 0.837 0.8 0.138 0.009 0 0[[ ]]

≔aLn =―――⋅0.4 f1

fn0.4

=fs

0.9 0.96 1.08 1.2 1.26 1.32 1.38 1.44 1.8 2.04 2.27 2.390.88 0.94 1.06 1.18 1.23 1.29 1.35 1.41 1.76 2 2.23 2.350.86 0.92 1.04 1.15 1.21 1.27 1.32 1.38 1.73 1.96 2.19 2.30.85 0.9 1.01 1.13 1.18 1.24 1.3 1.35 1.69 1.92 2.14 2.250.82 0.88 0.99 1.1 1.15 1.21 1.26 1.32 1.65 1.87 2.09 2.20.8 0.86 0.96 1.07 1.12 1.18 1.23 1.28 1.6 1.82 2.03 2.140.78 0.83 0.93 1.04 1.09 1.14 1.19 1.24 1.55 1.76 1.97 2.070.75 0.8 0.9 1 1.05 1.1 1.15 1.2 1.5 1.7 1.9 20.72 0.76 0.86 0.96 1 1.05 1.1 1.15 1.43 1.63 1.82 1.910.7 0.75 0.84 0.93 0.98 1.02 1.07 1.12 1.4 1.58 1.77 1.860.7 0.75 0.84 0.93 0.98 1.02 1.07 1.12 1.4 1.58 1.77 1.860.7 0.75 0.84 0.93 0.98 1.02 1.07 1.12 1.4 1.58 1.77 1.86

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―1

≔_gn (( ,j k)) ⋅―――――――

⋅⋅⋅⋅ρa V,j k

2 dzjCLn u

j

⋅2‾‾‾‾‾‾‾‾‾‾‾‾

⋅⋅‾‾ Bzjfs ,j k

⎛⎜⎜⎜⎜⎜⎜⎝

⋅――−1

2

⎛⎜⎜⎜⎜⎜⎜⎝

――――

−1 ―――fn

fs,j k

Bzj

⎞⎟⎟⎟⎟⎟⎟⎠

2⎞⎟⎟⎟⎟⎟⎟⎠


Page 52 of 180

=Integral_gn 30.81 56.15 121.23 183.27 207.41 225.88 238.85 246.88 233.09 203.96 176.58 164.71[[ ]] ―――⋅3

≔ωn =⋅⋅2 fn 7.23 ――

≔Can ――――――――――――⋅λ Integral_gn

⋅⋅⋅⋅⋅4 ωn3

ρa d1 mon((Integral_un2))2

=Can ⋅9.27 10−6 ⋅1.69 10−5 ⋅3.65 10−5 ⋅5.51 10−5 ⋅6.24 10−5 ⋅6.8 10−5 ⋅7.19 10−5 ⋅7.43 10−5 ⋅7.01 10−5 …⎡⎣ ⎤⎦ 2

≔Scn =―――――⋅⋅⋅4 mon ξs

⋅ρa d12

4.4

=c1n 0 0 0.02 0.04 0.05 0.05 0.05 0.04 −0.12 −3.17 0 0⎡⎣ ⎤⎦

=c2n 0 0 ⋅2.66 10−5 ⋅2.35 10−5 ⋅2.43 10−5 ⋅2.55 10−5 ⋅2.73 10−5 ⋅2.95 10−5 ⋅1.61 10−4 ⋅2.26 10−3 0 0⎡⎣ ⎤⎦

=kpn 0 0 1.93 1.48 1.46 1.45 1.45 1.46 4.02 4.08 0 0[[ ]]

≔_yn (( ,j k)) ⋅⋅⋅ujkpn ,0 k

d1‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾

+c1n ,0 k

‾‾‾‾‾‾‾‾‾‾‾‾+c1n ,0 k

2 c2n ,0 k

=yn

0 0 0.246 0.306 0.313 0.316 0.315 0.312 0.073 0.055 0 00 0 0.2 0.249 0.255 0.257 0.257 0.254 0.06 0.045 0 00 0 0.167 0.207 0.212 0.214 0.214 0.211 0.05 0.037 0 00 0 0.137 0.17 0.174 0.175 0.175 0.173 0.041 0.03 0 00 0 0.109 0.136 0.139 0.14 0.14 0.138 0.032 0.024 0 00 0 0.085 0.106 0.108 0.109 0.109 0.108 0.025 0.019 0 00 0 0.064 0.08 0.081 0.082 0.082 0.081 0.019 0.014 0 00 0 0.046 0.057 0.058 0.059 0.059 0.058 0.014 0.01 0 0

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=max ⎛⎝yn⎞⎠ 0.316 =――――max ⎛⎝yn⎞⎠

d10.45

=σyn 0 0 0.13 0.21 0.21 0.22 0.22 0.21 0.02 0.01 0 0[[ ]] =Fn

1820148112341010808630473338227178100

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―

Fatigue Strength Due to Cross Wind Vibration (Commentary 4)


Page 53 of 180


≔Nwo ⋅5 106 ≔Tyrs 50 ≔aint 100

≔I =⋅――64

⎛⎝ −dz

4 ⎛⎝ −dz ⋅2 tz⎞⎠4 ⎞⎠

⋅8.6 10−4

⋅8.6 10−4

⋅8.6 10−4

⋮

⎡⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦

4

≔Category 71 ≔fyt 355 ≔γ 1

=σan_full

0 0 0 0 0 0 0 0 0 0 0 00 0 0.61 0.98 1.02 1.03 1.03 1.01 0.09 0.06 0 00 0 2.2 3.57 3.7 3.74 3.73 3.67 0.31 0.23 0 00 0 4.61 7.48 7.76 7.86 7.83 7.7 0.66 0.48 0 00 0 7.69 12.49 12.95 13.11 13.07 12.86 1.09 0.81 0 00 0 11.31 18.37 19.04 19.29 19.22 18.91 1.61 1.19 0 00 0 15.35 24.93 25.84 26.18 26.09 25.66 2.19 1.61 0 00 0 19.7 32 33.18 33.6 33.49 32.94 2.81 2.07 0 00 0 24.28 39.43 40.89 41.41 41.27 40.59 3.46 2.55 0 00 0 26.62 43.24 44.84 45.41 45.26 44.52 3.79 2.8 0 00 0 31.4 51.01 52.89 53.57 53.39 52.51 4.47 3.3 0 00 0 33.82 54.93 56.95 57.69 57.49 56.55 4.82 3.55 0 00 0 36.24 58.87 61.04 61.83 61.62 60.61 5.16 3.81 0 00 0 38.68 62.82 65.14 65.98 65.76 64.68 5.51 4.06 0 00 0 41.12 66.78 69.25 70.14 69.9 68.75 5.86 4.32 0 00 0 43.56 70.75 73.36 74.3 74.05 72.83 6.2 4.58 0 00 0 46 74.71 77.47 78.47 78.2 76.91 6.55 4.83 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔maxcol 5 Values taken from example.

=σan_full⟨⟨maxcol⟩⟩

0⋅1.03 106

⋅3.74 106

⋅7.86 106

⋅1.31 107

⋅1.93 107

⋅2.62 107

⋅3.36 107

⋅4.14 107

⋅4.54 107

⋅5.36 107

⋅5.77 107

⋅6.18 107

⋅6.6 107

⋅7.01 107

⋅7.43 107

⋅7.85 107

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σapp5

⋅1 10−99

⋅9.91 105

⋅3.76 106

⋅8 106

⋅1.35 107

⋅1.99 107

⋅2.71 107

⋅3.48 107

⋅4.29 107

⋅4.71 107

⋅5.57 107

⋅6 107

⋅6.43 107

⋅6.86 107

⋅7.3 107

⋅7.74 107

⋅8.17 107

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=―――――σan_full

⟨⟨maxcol⟩⟩

σapp5

0%104%100%98%97%97%97%97%97%96%96%96%96%96%96%96%96

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(Force values to match example)

≔σan_full⟨⟨maxcol⟩⟩ σapp5

=Vh 29.6 ― ≔Vo =⋅Vh 0.2 5.92 ― ≔Ka_max =_Ka_max_n⎛⎝Ren⎞⎠ 2.8


⋅⋅4 Ka_max


⎞⎠0.861.49

⎡⎢⎣

⎤⎥⎦

≔Vzero0.851.5

⎡⎢⎣

⎤⎥⎦


Page 54 of 180


⎛⎜⎝

,Vzero0Iv ,0 0

⎞⎟⎠

0.11 ≔V2 =⋅Vzero1Vcr 6.12 ―

=Nn ⋅6.65 108 =Vo 5.92 ―

≔awo =⋅0.5 σsr⎛⎝ ,Nwo Category⎞⎠ 26.2 ≔amin =⋅0.5 σsr

⎛⎝ ,⋅1 108 Category⎞⎠ 14.4

=Ta 14.37 17.77 21.18 24.59 27.99 31.4 34.81 38.21 41.62 45.02 48.43 51.84 …[[ ]]

≔_Mfull_sub⎛⎝ ,zi k⎞⎠ ∑

=i 0

−aint 1

⋅0.5 ⎛⎝

⋅⎛⎝ +_f ⎛⎝ ,,i zi k⎞⎠ _f ⎛⎝ ,,+i 1 zi k⎞⎠⎞⎠ ⎛⎝

−a+i 1

ai⎞⎠⎞⎠

=TVrange 0 1 2 3 4 5[[ ]] ―

≔_Mfull⎛⎝ ,zi k⎞⎠ ⋅⋅⋅⋅3.15 107 Tyrs

⎛⎜⎝――fn

1

⎞⎟⎠

∑=j 0

−Nrange 2

⋅⋅⋅――――

⋅2 Vrangej

Vo2

exp

⎛⎜⎜⎜⎝−―――

Vrangej

2

Vo2

⎞⎟⎟⎟⎠

_Mfull_sub⎛⎝ ,zi k⎞⎠ ⎛

⎜⎝−Vrange +j 1Vrange

j⎞⎟⎠

=Mz_full

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0.1 4.5 5.5 6.5 5.8 5.3 0 0 0 00 0 5.2 50.2 57.7 67.4 59.8 56.1 0 0 0 00 0 33.8 193.4 217.3 250 224 212.4 0 0 0 00 0 106.3 504.1 563.3 651 579.7 551 0 0 0 00 0 242.1 1077.6 1202.1 1388.7 1236.9 1176.4 0 0 0 00 0 464.8 2023.2 2256 2607.1 2321 2208 0 0 0 00 0 616.6 2670 2976.9 3451.8 3062.5 2913.5 0 0 0 00 0 1018.2 4384.8 4888.4 5711.3 5028.8 4784.4 0.1 0 0 00 0 1273.7 5477.2 6106.1 7139.5 6281.4 5976.2 0.2 0 0 00 0 1569.7 6743.6 7517.7 8787.7 7733.6 7357.9 0.3 0 0 00 0 1909 8196.2 9136.8 10671.2 9399.2 8942.7 0.6 0 0 00 0 2294.5 9846.3 10975.9 12857 11290.8 10742.8 0.9 0.1 0 00 0 2728.7 11704.8 13046.2 15316.7 13419.9 12769.4 1.4 0.1 0 00 0 3214.5 13781.4 15356.8 17992.1 15795.3 15031.9 2 0.2 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Mz_full_max =Mz_full⟨⟨Mz_max_col

⟩⟩

0

⋅4.48 10−44

⋅1.06 10−2

6.53⋅6.74 10

⋅2.5 102

⋅6.51 102

⋅1.39 103

⋅2.61 103

⋅3.45 103

⋅5.71 103

⋅7.14 103

⋅8.79 103

⋅1.07 104

⋅1.29 104

⋅1.53 104

⋅1.8 104

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Mz_full_app5

0.000001⋅4.31 10−26

⋅2.04 10−2

5.99

⋅5.57 101

⋅2.12 102

⋅5.68 102

⋅1.26 103

⋅2.43 103

⋅3.26 103

⋅5.49 103

⋅6.94 103

⋅8.63 103

⋅1.06 104

⋅1.28 104

⋅1.52 104

⋅1.79 104

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=――――Mz_full_max

Mz_full_app5

00%52%109%121%118%115%110%107%106%104%103%102%101%100%101%101

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 55 of 180

Section B-3 - StaadPro Model to verify Mode shape


Page 56 of 180

CICIND 2010 Validation

Now that the example has been replicated reasonably well, we attempt to validate the

calculations for the same example using MecaStack. Here we ran into more discrepancies, with

the largest discrepancy being due to the difference in mode shape determined in MecaStack

compared to that provided in the example. In attempt to determine the correct mode shape,

we also constructed a model in StaadPro. The results are shown below and it was verified that

the MecaStack mode shape was correct. So using the MecaStack calculated Mode shape we

utilized our Mathcad calculation again to determine the new calculated results, and these were

what was used to compare to the MecaStack output.

Comparison of Mode Shapes Calculated

Elev (m) Appendix 5 Example StaadPro Model MecaStack

25 1.000 1.000 1.000

23 0.814 0.890 0.890

21 0.678 0.780 0.780

19 0.555 0.671 0.671

17 0.444 0.565 0.564

15 0.346 0.461 0.461

13 0.260 0.363 0.363

11 0.186 0.272 0.272

9 0.125 0.191 0.190

8 0.098 0.154 0.154

7 0.055 0.120 0.120

6 0.038 0.090 0.090

5 0.025 0.064 0.064

4 0.014 0.042 0.042

3 0.006 0.024 0.024

2 0.002 0.011 0.011

1 0.000 0.003 0.003

0 0.000 0.000 0.000

In the following example, we present the same Mathcad solution; however, we use the values

we believe to be correct rather than forcing them to match the example as we did in the

previous calculation. This Mathcad calculation is then compared to the MecaStack output in

order to validate the MecaStack calculations.


Page 57 of 180

Software licensed to

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Ref

By Date Chd

File Date/Time

1

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18-Apr-2016 14:24App5.std

Print Time/Date: 19/04/2016 08:42 Print Run 1 of 9STAAD.Pro V8i (SELECTseries 4) 20.07.09.31

Job Information Engineer Checked Approved

Name:

Date: 18-Apr-16

Structure Type SPACE FRAME

Number of Nodes 18 Highest Node 18

Number of Elements 17 Highest Beam 17

Number of Basic Load Cases 1

Number of Combination Load Cases 0

Included in this printout are data for:

All The Whole Structure

Included in this printout are results for load cases:

Type L/C Name

Primary 1 DEAD


Page 58 of 180


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2

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Mode Shape 1Load 1 :

XY

Z

Whole Structure


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3

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18

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1


XY

Z

Node #'s


Page 60 of 180


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4

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17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1


XY

Z

Beam #'s

NodesNode X

(m)

Y

(m)

Z

(m)

1 0.000 25.000 0.000

2 0.000 23.000 0.000

3 0.000 21.000 0.000

4 0.000 19.000 0.000

5 0.000 17.000 0.000

6 0.000 15.000 0.000

7 0.000 13.000 0.000

8 0.000 11.000 0.000

9 0.000 9.000 0.000

10 0.000 8.000 0.000

11 0.000 7.000 0.000

12 0.000 6.000 0.000


Page 61 of 180


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Nodes Cont...Node X

(m)

Y

(m)

Z

(m)

13 0.000 5.000 0.000

14 0.000 4.000 0.000

15 0.000 3.000 0.000

16 0.000 2.000 0.000

17 0.000 1.000 0.000

18 0.000 0.000 0.000

BeamsBeam Node A Node B Length

(m)

Property β(degrees)

1 1 2 2.000 1 0

2 2 3 2.000 1 0

3 3 4 2.000 1 0

4 4 5 2.000 1 0

5 5 6 2.000 1 0

6 6 7 2.000 1 0

7 7 8 2.000 1 0

8 8 9 2.000 1 0

9 9 10 1.000 1 0

10 10 11 1.000 1 0

11 11 12 1.000 1 0

12 12 13 1.000 1 0

13 13 14 1.000 1 0

14 14 15 1.000 1 0

15 15 16 1.000 1 0

16 16 17 1.000 1 0

17 17 18 1.000 1 0

Section PropertiesProp Section Area

(in2)

Iyy

(in4)

Izz

(in4)

J

(in4)

Material

1 PIPE 21.757 2.09 E +3 2.09 E +3 4.17 E +3 STEEL

MaterialsMat Name E

(kip/in2)

ν Density

(kip/in3)

α(/°F)

1 STEEL 30.5 E +3 0.300 0.000 6 E -6

2 STAINLESSSTEEL 28 E +3 0.300 0.000 10 E -6

3 ALUMINUM 10 E +3 0.330 0.000 13 E -6

4 CONCRETE 3.15 E +3 0.170 0.000 5 E -6


Page 62 of 180


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SupportsNode X

(kip/in)

Y

(kip/in)

Z

(kip/in)

rX

(kip-ft/deg)

rY

(kip-ft/deg)

rZ

(kip-ft/deg)

18 Fixed Fixed Fixed Fixed Fixed Fixed

Basic Load CasesNumber Name

1 DEAD

Combination Load CasesThere is no data of this type.

Reactions

Horizontal Vertical Horizontal Moment

Node L/C FX

(kg)

FY

(kg)

FZ

(kg)

MX

(kip-ft)

MY

(kip-ft)

MZ

(kip-ft)

18 1:DEAD -2.75 E +3 0.000 0.000 0.0 0.0 248.5

Calculated Modal Frequencies & Mass ParticipationsMode Frequency

(Hz)

Period

(sec)

Participation X

(%)

Participation Y

(%)

Participation Z

(%)

1 1.149 0.870 62.371 0.000 0.000

2 7.108 0.141 19.414 0.000 0.000

3 19.471 0.051 6.775 0.000 0.000

4 37.408 0.027 3.347 0.000 0.000

5 60.413 0.017 2.187 0.000 0.000

6 86.319 0.012 1.434 0.000 0.000


Page 63 of 180


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Mode ShapesNode Mode X Y Z rX rY rZ

1 1 1.000 0.000 0.000 0.000 0.000 -0.001

2 -1.000 0.000 0.000 0.000 0.000 0.005

3 1.000 0.000 0.000 0.000 0.000 -0.008

4 1.000 0.000 0.000 0.000 0.000 -0.012

5 1.000 0.000 0.000 0.000 0.000 -0.016

6 1.000 0.000 0.000 0.000 0.000 -0.021

2 1 0.890 0.000 0.000 0.000 0.000 -0.001

2 -0.613 0.000 0.000 0.000 0.000 0.005

3 0.356 0.000 0.000 0.000 0.000 -0.008

4 0.077 0.000 0.000 0.000 0.000 -0.010

5 -0.215 0.000 0.000 0.000 0.000 -0.012

6 -0.500 0.000 0.000 0.000 0.000 -0.012

3 1 0.780 0.000 0.000 0.000 0.000 -0.001

2 -0.237 0.000 0.000 0.000 0.000 0.005

3 -0.211 0.000 0.000 0.000 0.000 -0.006

4 -0.576 0.000 0.000 0.000 0.000 -0.005

5 -0.760 0.000 0.000 0.000 0.000 -0.000

6 -0.703 0.000 0.000 0.000 0.000 0.006

4 1 0.671 0.000 0.000 0.000 0.000 -0.001

2 0.107 0.000 0.000 0.000 0.000 0.004

3 -0.585 0.000 0.000 0.000 0.000 -0.003

4 -0.682 0.000 0.000 0.000 0.000 0.002

5 -0.306 0.000 0.000 0.000 0.000 0.009

6 0.340 0.000 0.000 0.000 0.000 0.012

5 1 0.565 0.000 0.000 0.000 0.000 -0.001

2 0.395 0.000 0.000 0.000 0.000 0.003

3 -0.678 0.000 0.000 0.000 0.000 0.001

4 -0.246 0.000 0.000 0.000 0.000 0.007

5 0.542 0.000 0.000 0.000 0.000 0.008

6 0.892 0.000 0.000 0.000 0.000 -0.002

6 1 0.461 0.000 0.000 0.000 0.000 -0.001

2 0.603 0.000 0.000 0.000 0.000 0.002

3 -0.482 0.000 0.000 0.000 0.000 0.004

4 0.390 0.000 0.000 0.000 0.000 0.007

5 0.800 0.000 0.000 0.000 0.000 -0.003

6 0.059 0.000 0.000 0.000 0.000 -0.013

7 1 0.363 0.000 0.000 0.000 0.000 -0.001

2 0.714 0.000 0.000 0.000 0.000 0.001

3 -0.083 0.000 0.000 0.000 0.000 0.006

4 0.761 0.000 0.000 0.000 0.000 0.001

5 0.167 0.000 0.000 0.000 0.000 -0.010

6 -0.863 0.000 0.000 0.000 0.000 -0.004

8 1 0.272 0.000 0.000 0.000 0.000 -0.001

2 0.725 0.000 0.000 0.000 0.000 -0.000

3 0.364 0.000 0.000 0.000 0.000 0.005


Page 64 of 180


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Mode Shapes Cont...Node Mode X Y Z rX rY rZ

4 0.598 0.000 0.000 0.000 0.000 -0.005

5 -0.655 0.000 0.000 0.000 0.000 -0.006

6 -0.431 0.000 0.000 0.000 0.000 0.011

9 1 0.191 0.000 0.000 0.000 0.000 -0.001

2 0.644 0.000 0.000 0.000 0.000 -0.001

3 0.695 0.000 0.000 0.000 0.000 0.003

4 0.012 0.000 0.000 0.000 0.000 -0.008

5 -0.755 0.000 0.000 0.000 0.000 0.004

6 0.680 0.000 0.000 0.000 0.000 0.009

10 1 0.154 0.000 0.000 0.000 0.000 -0.001

2 0.576 0.000 0.000 0.000 0.000 -0.002

3 0.777 0.000 0.000 0.000 0.000 0.001

4 -0.314 0.000 0.000 0.000 0.000 -0.007

5 -0.458 0.000 0.000 0.000 0.000 0.009

6 0.876 0.000 0.000 0.000 0.000 -0.000

11 1 0.120 0.000 0.000 0.000 0.000 -0.001

2 0.494 0.000 0.000 0.000 0.000 -0.002

3 0.792 0.000 0.000 0.000 0.000 -0.000

4 -0.585 0.000 0.000 0.000 0.000 -0.005

5 -0.015 0.000 0.000 0.000 0.000 0.011

6 0.675 0.000 0.000 0.000 0.000 -0.008

12 1 0.090 0.000 0.000 0.000 0.000 -0.001

2 0.403 0.000 0.000 0.000 0.000 -0.002

3 0.743 0.000 0.000 0.000 0.000 -0.002

4 -0.757 0.000 0.000 0.000 0.000 -0.003

5 0.442 0.000 0.000 0.000 0.000 0.010

6 0.166 0.000 0.000 0.000 0.000 -0.013

13 1 0.064 0.000 0.000 0.000 0.000 -0.001

2 0.309 0.000 0.000 0.000 0.000 -0.002

3 0.638 0.000 0.000 0.000 0.000 -0.003

4 -0.802 0.000 0.000 0.000 0.000 0.000

5 0.781 0.000 0.000 0.000 0.000 0.006

6 -0.429 0.000 0.000 0.000 0.000 -0.012

14 1 0.042 0.000 0.000 0.000 0.000 -0.001

2 0.217 0.000 0.000 0.000 0.000 -0.002

3 0.493 0.000 0.000 0.000 0.000 -0.004

4 -0.721 0.000 0.000 0.000 0.000 0.003

5 0.907 0.000 0.000 0.000 0.000 0.000

6 -0.861 0.000 0.000 0.000 0.000 -0.006

15 1 0.024 0.000 0.000 0.000 0.000 -0.000

2 0.134 0.000 0.000 0.000 0.000 -0.002

3 0.330 0.000 0.000 0.000 0.000 -0.004

4 -0.541 0.000 0.000 0.000 0.000 0.005

5 0.800 0.000 0.000 0.000 0.000 -0.005

6 -0.958 0.000 0.000 0.000 0.000 0.002


Page 65 of 180


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Mode Shapes Cont...Node Mode X Y Z rX rY rZ

16 1 0.011 0.000 0.000 0.000 0.000 -0.000

2 0.065 0.000 0.000 0.000 0.000 -0.001

3 0.173 0.000 0.000 0.000 0.000 -0.003

4 -0.312 0.000 0.000 0.000 0.000 0.005

5 0.517 0.000 0.000 0.000 0.000 -0.007

6 -0.711 0.000 0.000 0.000 0.000 0.008

17 1 0.003 0.000 0.000 0.000 0.000 -0.000

2 0.019 0.000 0.000 0.000 0.000 -0.001

3 0.053 0.000 0.000 0.000 0.000 -0.002

4 -0.105 0.000 0.000 0.000 0.000 0.004

5 0.192 0.000 0.000 0.000 0.000 -0.006

6 -0.292 0.000 0.000 0.000 0.000 0.008

18 1 0.000 0.000 0.000 0.000 0.000 0.000

2 0.000 0.000 0.000 0.000 0.000 0.000

3 0.000 0.000 0.000 0.000 0.000 0.000

4 0.000 0.000 0.000 0.000 0.000 0.000

5 0.000 0.000 0.000 0.000 0.000 0.000

6 0.000 0.000 0.000 0.000 0.000 0.000


Page 66 of 180

Section B-4 - Mathcad Calculation with Corrected Values


Page 67 of 180

CICIND 2010 (App 5 Example)


≔Cat 3 ≔f1 1.15 ≔h 25 ≔kt 1 ≔Vb 31.42 ― ≔ξs 0.002

≔ρa 1.25 ――3

≔ki 1

≔t 6.3

≔z

2523211917151311986543210

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔dz

0.710.710.710.710.710.710.710.710.710.710.710.710.710.710.710.710.71

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔tz

6.36.36.36.36.36.36.36.36.36.36.36.36.36.36.36.36.3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔d1 0.71

(Avg of top 1/3, or 6*d for region of vortex shedding)

≔a =_a ((Cat)) 1 ≔c =_c ((Cat)) 0 ≔Zmin =_Zmin ((Cat)) 8 ≔Zs =⋅0.6 h 15

≔b =_b ((Cat)) 0 ≔Zo =_Zo ((Cat)) 0 ≔β =_β ((Cat)) 0 ≔Lzs =L ⎛⎝ ,Zs Cat⎞⎠ 99

≔V (( ,z Cat)) ⋅⋅_k (( ,z Cat)) kt Vb =V (( ,h Cat)) 30 ― ≔Vzs =V ⎛⎝ ,Zs Cat⎞⎠ 26 ―

≔fL (( ,Z Cat)) ⋅f1 ――――L (( ,Z Cat))

V (( ,Z Cat))=fL ⎛⎝ ,Zs Cat⎞⎠ 4 ≔Ivs =_Iv ⎛⎝ ,Zs Cat⎞⎠ 0

≔ηx (( ,,x Zs Cat)) ―――――――⋅⋅4.6 fL (( ,Zs Cat)) x

L (( ,Zs Cat))≔ηd =ηx ⎛⎝ ,,d1 Zs Cat⎞⎠ 0 ≔ηh =ηx ⎛⎝ ,,h Zs Cat⎞⎠ 5

≔_SL(( ,Z Cat)) ―――――――――

⋅6.8 fL (( ,Z Cat))

⎛⎝ +1 ⋅10.2 fL (( ,Z Cat))⎞⎠

⎛⎜⎝―5

3

⎞⎟⎠

≔Rd =−―1

ηd⋅―――

1

⋅2 ηd2

⎛⎝ −1 ⋅−2 ηd⎞⎠ 1 ≔SL =_SL⎛⎝ ,Zs Cat⎞⎠ 0

≔Rh =−―1

ηh⋅―――

1

⋅2 ηh2

⎛⎝ −1 ⋅−2 ηh⎞⎠ 0 =V (( ,h Cat)) 30 ―

≔ξa =―――――――⋅⋅2.7 10−6 V (( ,h Cat))

⋅f1 t0.01103 ≔ξ =+ξs ξa 0.01303

≔R =‾‾‾‾‾‾‾‾‾‾‾‾‾

⋅⋅⋅――⋅4 ξ

SL Rd Rh 1 =R2 1 ≔B =‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾――――――――

1

+1 ⋅0.9⎛⎜⎝――――

+⋅1 b h

Lzs

⎞⎟⎠

0.631 =B2 1


Page 68 of 180

≔vT =

‖‖‖‖‖‖‖‖‖

|||||||

|

←vt ⋅⋅⋅600 f1

‾‾‾‾‾‾‾‾―――

R2

+B2

R2

⎛⎜⎝――

1

1

⎞⎟⎠

|||

if <vt 48‖‖ ←vt 48

((vt))

444 ≔g =+‾‾‾‾‾‾‾‾⋅2 ln ((vT)) ――――0.577

‾‾‾‾‾‾‾‾⋅2 ln ((vT))4

≔_Vz (( ,z Cat))‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((z)) 1‖‖‖

←M,j 0

V ⎛⎝

,z,j 0

Cat⎞⎠

((M))≔G =+1 ⋅⋅⋅2 g Ivs‾‾‾‾‾‾‾+B2 R2 3.08

≔Re =_Re ⎛⎝ ,V (( ,h Cat)) d1⎞⎠ ⋅1.45 106 ≔ka =_ka ⎛⎝ ,h d1⎞⎠ 1 =ki 1

≔_CD1 (( ,,z Cat d)) ⋅⋅ki ka _CD0 (( ,,z Cat d))

=CD0 ⎛⎝ ,,z Cat dz⎞⎠

111⋮

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

≔_CD ⎛⎝ ,,z Cat dz⎞⎠ ⋅⋅ki ka CD0 ⎛⎝ ,,z Cat dz⎞⎠

≔CD =_CD ⎛⎝ ,,z Cat dz⎞⎠

1111⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

≔Vz =_Vz (( ,z Cat))

29.629.128.527.927.226.525.624.723.6⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―

≔_wm1((zz)) ⋅⋅⋅⋅0.5 ρa V (( ,zz Cat))

2

_CD1 ⎛⎝ ,,zz Cat d1⎞⎠ d1

≔wm((z))

‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((z)) 1‖‖‖

←M,j 0

_wm1⎛⎝z

,j 0⎞⎠

((M))≔_wg ((zz)) ⋅⋅――――⋅3 (( −G 1))

h2

―zz

h

⌠⌡ d0

h

⋅_wm1((z)) z z

≔wg((z))

‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((z)) 1‖‖‖

←M,j 0

_wg ⎛⎝z

,j 0⎞⎠

((M))

=⌠⌡ d0

h

⋅_wm1((z)) z z 70648

≔wmz =wm((z))

272262252241230217204190174165165165

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― ≔wgz =wg((z))

706649593536480423367311254226169141

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― ≔wz =+wmz wgz

978912845778710641571500428391334306

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―



Page 69 of 180


≔fn =f1 1 ≔d1 0.71 ≔t1 6.3

≔St 0.2 ≔m =⋅⋅―4

⎛⎝ −d1

2 ⎛⎝ −d1 ⋅2 t1⎞⎠2 ⎞⎠ 490 ――

3109.32 ――

Can be ignored if Vcr > Vlimit

≔dz =Vfill ⎛⎝ ,18 d1⎞⎠

1111⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

≔Vlimit =⋅1.25 V (( ,h Cat)) 37 ―


St4.08 ―

≔Vb =V (( ,h Cat)) 30 ―

≔tz =Vfill ⎛⎝ ,18 t1⎞⎠

0000⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

≔u

1.00.8900.7800.6710.5640.4610.3630.2720.1900.1540.1200.0900.0640.0420.0240.011

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Nrows =−rows ((u)) 1 16

≔Ren =⋅⋅⋅6.67 104 ――2

Vcr d1 ⋅1.93 105

≔mz =Vfill (( ,18 m))

109.32109.32109.32109.32

⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

―― ≔Iv =_Iv (( ,h Cat)) 0.229

≔CLn =_CLn⎛⎝Ren⎞⎠ 1 =Bv 0.33

≔Integral_un2 =∑=j 0

−Nrows 1

⎛⎜⎝

⋅0.5 ⎛⎜⎝

⋅⎛⎜⎝

+uj

2 u+j 1

2 ⎞⎟⎠

⎛⎝

−zj

z+j 1

⎞⎠⎞⎟⎠⎞⎟⎠

6 ≔Integral_mun2 =⋅Integral_un2 m 689

≔mon =―――――Integral_mun

Integral_un109.3 ――

≔λ 1 =ρa 1 ――3

=⋅⋅⋅1.42 10−4 ρa λ ⎛⎝ ⋅1.78 10−4⎞⎠ ――3

≔Can =―――――――――⋅⋅⋅⋅⋅1.42 10−4 ρa λ d1

3CLn

2

⋅⋅⋅St4

mon Bv Integral_un2⋅8.59 10−5 <--(Using the right d^3)

≔Ka_max_n =_Ka_max_n⎛⎝Ren⎞⎠ 3

≔Kan =_Kan 0.877 ≔Scn =―――――⋅⋅⋅4 mon ξs

⋅ρa d12

4.36 =ξs 0.002 ≔Ratio =――――Scn

⋅⋅4 Kan

0.4

≔aln =―――⋅0.4 f1

fn0

≔c1n =⋅⋅0.5 aln2

⎛⎜⎜⎝

−1 ――――Scn

⋅⋅4 Kan

⎞⎟⎟⎠

⋅4.84 10−2 ≔c2n =――――⋅aln

2Can

Kan

⋅1.57 10−5

‾‾⎛ ⎛ ⎛ Scn ⎞4 ⎞⎞


Page 70 of 180

≔kpn =⋅‾‾2⎛⎜⎜⎝

+1 ⋅1.2 atan⎛⎜⎜⎝

⋅0.75⎛⎜⎜⎝――――

Scn

⋅⋅4 Kan

⎞⎟⎟⎠

4 ⎞⎟⎟⎠

⎞⎟⎟⎠

1.445

≔σyn =⋅d1‾‾‾‾‾‾‾‾‾‾‾‾‾‾

+c1n‾‾‾‾‾‾‾‾+c1n

2 c2n 0

≔_yn ((j)) ⋅⋅kpn σyn uj

=―――_yn ((0))

d10.45

≔_Fn((j)) ⋅⋅⎛⎝ ⋅⋅2 fn⎞⎠

2

m _yn ((j))

≔yn =‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((z)) 1‖‖‖

←yj

_yn ((j))

((y))

0.3190.2840.2490.2140.180.1470.1160.0870.0610.0490.0380.0290.020.0130.0080.0040

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Fn =‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((z)) 1‖‖‖

←Fj

_Fn((j))

((F))

182316231422122310288406624963462812191641177744200

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―



Page 71 of 180


≔Nwo ⋅5 106 ≔FatCat 71 ≔fyt 355 ≔I =⋅――64

⎛⎝ −dz

4 ⎛⎝ −dz ⋅2 tz⎞⎠4 ⎞⎠

⋅8.6 10−4

⋅8.6 10−4

⋅8.6 10−4

⋮

⎡⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦

4

≔aint 100 ≔γ 1 ≔Tyrs 50

=TVr_table 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 …[[ ]]

≔Vh =V (( ,h Cat)) 30 ― ≔Vo =⋅Vh 0.2 5.92 ― ≔Ka_max =_Ka_max_n⎛⎝Ren⎞⎠ 3


⋅⋅4 Ka_max


⎞⎠0.861.5

⎡⎢⎣

⎤⎥⎦


⎛⎜⎝

,Vzero0Iv⎞⎟⎠

0 ≔V2 =⋅Vzero1Vcr 6.11 ―

≔Nn =

‖‖‖‖‖‖‖‖‖‖

||||||||

|

←N ⋅⋅⋅⋅3.15 107 Tyrs

⎛⎜⎝――fn

1

⎞⎟⎠

⎛⎜⎝ −

⎛⎜⎜⎝

―――−V1

2

Vo2

⎞⎟⎟⎠

⎛⎜⎜⎝

―――−V2

2

Vo2

⎞⎟⎟⎠

⎞⎟⎠

||

|

if <N ⋅200 Tyrs

‖‖‖

←N ⋅200 Tyrs

((N))

⋅6.48 108

≔awo =⋅0.5 σsr⎛⎝ ,Nwo FatCat⎞⎠ 26.2

≔amin =⋅0.5 σsr⎛⎝ ,⋅1 108 FatCat⎞⎠ 14.4 =σsr

⎛⎝ ,⋅1 108 FatCat⎞⎠ 28.73

=Ta 14 18 21 25 28 31 35 38 42 45 48 52 …[[ ]]

=Mz_red

000.027.14

69.02259.42690.17

1518.522931.723917.916581.948311.49

10338.612681.515349.3918337.121621.28

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



Page 72 of 180



St4 ― ≔zcr =⋅―

5

6h 21 ≔λ 1 ≔d1 0.71 ≔t 6.3

=Cat 3 ≔m =⋅⋅―4

⎛⎝ −d1

2 ⎛⎝ −d1 ⋅2 t⎞⎠2 ⎞⎠ 490 ――

3109 ――

Avg Diameter d1Mode 1: Avg of top 1/3:> Mode 1: where Vortex occurs over 6 diameter length

≔row =rows ((z)) 17

≔dz =Vfill ⎛⎝ ,row d1⎞⎠

1111⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

=d1 1

=z

2523211917151311986543210

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=u

10.890.780.6710.5640.4610.3630.2720.190.1540.120.090.0640.0420.0240.0110

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔tz =Vfill (( ,row 6.3 ))

6666⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

≔mz =Vfill (( ,row m))

109109109109

⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

――

=V

3.19 3.61 4.04 4.25 4.46 4.67 4.89 5.1 5.95 6.8 7.65 8.53.13 3.55 3.96 4.17 4.38 4.59 4.8 5.01 5.84 6.68 7.51 8.343.07 3.48 3.89 4.09 4.29 4.5 4.7 4.91 5.73 6.54 7.36 8.183 3.4 3.8 4 4.2 4.4 4.6 4.8 5.6 6.4 7.2 82.93 3.32 3.71 3.9 4.1 4.29 4.49 4.68 5.47 6.25 7.03 7.812.85 3.23 3.61 3.8 3.99 4.18 4.37 4.56 5.32 6.08 6.84 7.62.76 3.13 3.5 3.68 3.86 4.05 4.23 4.42 5.15 5.89 6.62 7.362.66 3.02 3.37 3.55 3.72 3.9 4.08 4.26 4.97 5.68 6.39 7.092.55 2.89 3.22 3.39 3.56 3.73 3.9 4.07 4.75 5.43 6.11 6.792.48 2.81 3.14 3.31 3.47 3.64 3.8 3.97 4.63 5.29 5.95 6.61

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―


St4 ― ≔Ren =_Ren ⎛⎝ ,Vcr d1⎞⎠ ⋅1.942 105 ≔CLn =_CLn

⎛⎝Ren⎞⎠ 1

=TBz 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0[[ ]]

=TIv 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0[[ ]]

≔Kao_zero =―――――Scn

⋅⋅4 Ka_max

0.124 ≔Vzero =_Vzero⎛⎝Kao_zero

⎞⎠0.861.5

⎡⎢⎣

⎤⎥⎦

≔Vlwr =Vzero01

≔Vupr =Vzero11

1 1 1 1 1 1 1 1 1 2 2 2⎡ ⎤


Page 73 of 180

=Vr

1 1 1 1 1 1 1 1 1 2 2 21 1 1 1 1 1 1 1 1 2 2 21 1 1 1 1 1 1 1 1 2 2 21 1 1 1 1 1 1 1 1 2 2 21 1 1 1 1 1 1 1 1 2 2 21 1 1 1 1 1 1 1 1 1 2 21 1 1 1 1 1 1 1 1 1 2 21 1 1 1 1 1 1 1 1 1 2 21 1 1 1 1 1 1 1 1 1 1 21 1 1 1 1 1 1 1 1 1 1 21 1 1 1 1 1 1 1 1 1 1 21 1 1 1 1 1 1 1 1 1 1 2

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Kao

0 0.155 0.268 0.301 0.316 0.315 0.302 0.279 0.149 0 0 00 0.136 0.25 0.288 0.308 0.312 0.304 0.286 0.164 0 0 00 0 0.231 0.272 0.297 0.307 0.304 0.29 0.18 0 0 00 0 0.209 0.253 0.283 0.299 0.301 0.293 0.196 0 0 00 0 0.186 0.232 0.266 0.287 0.295 0.292 0.211 0 0 00 0 0.161 0.207 0.245 0.271 0.285 0.288 0.225 0.117 0 00 0 0 0.181 0.22 0.25 0.269 0.278 0.237 0.14 0 00 0 0 0.152 0.191 0.224 0.248 0.262 0.245 0.163 0 00 0 0 0 0.16 0.193 0.22 0.239 0.247 0.184 0 00 0 0 0 0 0.177 0.204 0.225 0.244 0.193 0.115 00 0 0 0 0 0.177 0.204 0.225 0.244 0.193 0.115 00 0 0 0 0 0.177 0.204 0.225 0.244 0.193 0.115 0

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ka_max_n =_Ka_max_n⎛⎝Ren⎞⎠ 3

=Kan

0 0 1 1 1 1 1 1 0 0 0 00 0 1 1 1 1 1 1 0 0 0 00 0 1 1 1 1 1 1 1 0 0 00 0 1 1 1 1 1 1 1 0 0 00 0 1 1 1 1 1 1 1 0 0 00 0 0 1 1 1 1 1 1 0 0 00 0 0 1 1 1 1 1 1 0 0 00 0 0 0 1 1 1 1 1 0 0 00 0 0 0 0 1 1 1 1 1 0 00 0 0 0 0 0 1 1 1 1 0 00 0 0 0 0 0 1 1 1 1 0 00 0 0 0 0 0 1 1 1 1 0 0

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Karn =―――Karn_num

Karn_den

0 0.164 0.587 0.72 0.795 0.832 0.83 0.799 0.517 0.06 0.004 0[[ ]] ≔aLn =―――⋅0.4 f1

fn0

=fs

1 1 1 1 1 1 1 1 2 2 2 21 1 1 1 1 1 1 1 2 2 2 21 1 1 1 1 1 1 1 2 2 2 21 1 1 1 1 1 1 1 2 2 2 21 1 1 1 1 1 1 1 2 2 2 21 1 1 1 1 1 1 1 1 2 2 21 1 1 1 1 1 1 1 1 2 2 21 1 1 1 1 1 1 1 1 2 2 21 1 1 1 1 1 1 1 1 2 2 21 1 1 1 1 1 1 1 1 1 2 21 1 1 1 1 1 1 1 1 1 2 21 1 1 1 1 1 1 1 1 1 2 2

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 74 of 180

≔_gn (( ,j k)) ⋅―――――――

⋅⋅⋅⋅ρa V,j k

2 dzjCLn u

j

⋅2‾‾‾‾‾‾‾‾‾‾‾‾

⋅⋅‾‾ Bzjfs ,j k

⎛⎜⎜⎜⎜⎜⎜⎝

⋅――−1

2

⎛⎜⎜⎜⎜⎜⎜⎝

――――

−1 ―――fn

fs,j k

Bzj

⎞⎟⎟⎟⎟⎟⎟⎠

2⎞⎟⎟⎟⎟⎟⎟⎠

=Integral_gn 37.91 109.63 196.23 235.13 267.82 293.41 311.94 324.02 326.62 294.82 257.59 224.8[[ ]] ―――⋅2

3

≔ωn =⋅⋅2 fn 7 ―― ≔Scn =―――――⋅⋅⋅4 mon ξs

⋅ρa d12

4

≔Can ――――――――――――⋅λ Integral_gn

⋅⋅⋅⋅⋅4 ωn3

ρa d1 mon((Integral_un2))2

=Can ⋅6.52 10−6 ⋅1.89 10−5 ⋅3.38 10−5 ⋅4.05 10−5 ⋅4.61 10−5 ⋅5.05 10−5 ⋅5.37 10−5 ⋅5.58 10−5 ⋅5.62 10−5 …⎡⎣ ⎤⎦ 2

=c1n 0 0 0 0 0 0 0 0 0 0 −7 0⎡⎣ ⎤⎦

=c2n 0 ⋅3.64 10−5 ⋅1.83 10−5 ⋅1.78 10−5 ⋅1.84 10−5 ⋅1.93 10−5 ⋅2.05 10−5 ⋅2.22 10−5 ⋅3.45 10−5 ⋅2.7 10−4 ⋅3.83 10−3 0⎡⎣ ⎤⎦

=kpn 0 4 2 1 1 1 1 1 2 4 4 0[[ ]]

≔_yn (( ,j k)) ⋅⋅⋅ujkpn ,0 k

d1‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾

+c1n ,0 k

‾‾‾‾‾‾‾‾‾‾‾‾+c1n ,0 k

2 c2n ,0 k

=yn

0 0.04 0.286 0.304 0.312 0.315 0.315 0.312 0.274 0.054 0.046 00 0.036 0.254 0.27 0.277 0.281 0.281 0.278 0.244 0.048 0.041 00 0.031 0.223 0.237 0.243 0.246 0.246 0.243 0.214 0.042 0.036 00 0.027 0.192 0.204 0.209 0.212 0.211 0.209 0.184 0.036 0.031 00 0.023 0.161 0.171 0.176 0.178 0.178 0.176 0.154 0.031 0.026 00 0.019 0.132 0.14 0.144 0.145 0.145 0.144 0.126 0.025 0.021 00 0.015 0.104 0.11 0.113 0.114 0.114 0.113 0.099 0.02 0.017 0

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

=max ⎛⎝yn⎞⎠ 0.315 =――――max ⎛⎝yn⎞⎠

d10

=σyn 0 0 0 0 0 0 0 0 0 0 0 0[[ ]] =Fn

18001602140412081015830653490342277216162

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―



Page 75 of 180


≔Nwo ⋅5 106 ≔Tyrs 50 ≔aint 100

≔I =⋅――64

⎛⎝ −dz

4 ⎛⎝ −dz ⋅2 tz⎞⎠4 ⎞⎠

⋅8.6 10−4

⋅8.6 10−4

⋅8.6 10−4

⋮

⎡⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦

4

≔Category 71 ≔fyt 355 ≔γ 1

=σyn 0 0 0 0 0 0 0 0 0 0 0 0[[ ]]

=σan_full

0 0 0 0 0 0 0 0 0 0 0 00 0 1 1 1 1 1 1 1 0 0 00 0 3 4 4 4 4 4 3 0 0 00 0 7 8 8 8 8 8 6 1 0 00 1 12 13 14 14 14 14 11 1 1 00 1 18 20 21 21 21 21 16 1 1 00 1 24 27 28 29 29 28 22 2 2 00 2 31 35 37 38 38 37 28 2 2 00 2 39 44 46 47 47 46 35 3 2 00 2 43 49 51 51 51 51 39 3 3 00 3 51 58 60 61 61 60 46 4 3 00 3 55 62 65 66 66 65 50 4 3 0

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Vh 30 ― ≔Vo =⋅Vh 0.2 5.92 ― ≔Ka_max =_Ka_max_n⎛⎝Ren⎞⎠ 3


⋅⋅4 Ka_max

0 ≔Vzero =_Vzero⎛⎝Kao

⎞⎠0.861.5

⎡⎢⎣

⎤⎥⎦

=Vo 6 ―


⎛⎜⎝

,Vzero0Iv ,0 0

⎞⎟⎠

0 ≔V2 =⋅Vzero1Vcr 6.11 ―

≔awo =⋅0.5 σsr⎛⎝ ,Nwo Category⎞⎠ 26.2 ≔amin =⋅0.5 σsr

⎛⎝ ,⋅1 108 Category⎞⎠ 14.4

=Ta 14 18 21 25 28 31 35 38 42 45 48 52 …[[ ]]

≔_Mfull_sub⎛⎝ ,zi k⎞⎠ ∑

=i 0

−aint 1

⋅0.5 ⎛⎝

⋅⎛⎝ +_f ⎛⎝ ,,i zi k⎞⎠ _f ⎛⎝ ,,+i 1 zi k⎞⎠⎞⎠ ⎛⎝

−a+i 1

ai⎞⎠⎞⎠

=TVrange 0 1 2 3 4 5[[ ]] ―

≔_Mfull⎛⎝ ,zi k⎞⎠ ⋅⋅⋅⋅3.15 107 Tyrs

⎛⎜⎝――fn

1

⎞⎟⎠

∑=j 0

−Nrange 2

⋅⋅⋅――――

⋅2 Vrangej

Vo2

exp

⎛⎜⎜⎜⎝――――

−Vrangej

2

Vo2

⎞⎟⎟⎟⎠

_Mfull_sub⎛⎝ ,zi k⎞⎠ ⎛

⎜⎝−Vrange +j 1Vrange

j⎞⎟⎠

=Mz_full

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0.01 0.01 0.01 0.01 0.01 0 0 0 00 0 2.86 5.73 7.2 7.89 7.85 7.29 1.44 0 0 00 0 40.3 63.45 73.9 78.61 78.34 74.48 26.16 0 0 00 0 169.31 247.47 282.21 297.82 296.93 284.13 119.73 0 0 00 0 462.93 663.99 753.5 793.71 791.42 758.44 335.48 0 0 00 0 1025.73 1464.01 1659.3 1747.07 1742.07 1670.08 748.41 0 0 00 0 1984.38 2828.17 3204.32 3373.38 3363.75 3225.08 1450.88 0 0 00 0 2653.02 3779.99 4282.42 4508.24 4495.38 4310.14 1940.61 0 0 00 0 4458.84 6351.03 7194.68 7573.88 7552.29 7241.24 3262.93 0.01 0 00 0 5631.26 8020.37 9085.54 9564.25 9536.99 9144.31 4121.33 0.03 0 00 0 7006.36 9978.1 11302.56 11897.59 11863.72 11375.62 5128.08 0.06 0.01 00 0 8599.66 12245.16 13867.91 14596.21 14554.76 13957.37 6294.6 0.13 0.02 00 0.01 10425.6 14838.39 16796.66 17673.58 17623.71 16904.44 7631.66 0.23 0.05 00 0.03 12497.01 17767.07 20091.63 21128.56 21069.67 20219.22 9149.46 0.39 0.09 00 0.05 14824.19 21029.05 23738.77 24940.59 24872.45 23886.9 10857.65 0.63 0.16 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 76 of 180

Check Ovalling Vibration per Para 7.2.5.2


Assume a stack with stiffeners at the top, midspan and bottom that are 50 mm deep x 12 mm thk.

≔do =dz01 ≔t 6.3 ≔E 209947 ≔ρs 7849 ――

3≔z 25

≔tstiff 12 ≔Dstiff 50

≔br =Dstiff 50

Check Static Ovaling: =z0

25

≔V5sec =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((z)) 1‖‖‖

←Vj

⋅1.4 Vzj

((V))

41.4440.6839.8839.0138.0637.0335.8834.5933.0932.2532.2532.2532.2532.2532.2532.2532.25

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― ≔w5sec =⋅⋅0.5 ρa V5sec2

10731034994951906857805748685650650650650650650650650

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Astiff =⋅tstiff Dstiff 6 2

≔Shell_Width =‖‖‖‖‖‖‖‖‖

|||||||

|

←br ‾‾‾‾⋅do t

|||||

if >⋅br t Astiff

‖‖‖‖

←br ――Astiff

t

((br))

0.067

≔Ir_min =―――――――――⋅⋅⋅0.06 w5sec do

4 ⎛⎝ +do ⋅6 br⎞⎠

⋅E t

1.251.21.161.111.0510.940.870.80.760.760.760.760.760.760.760.76

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

4

≔Mstiff

“Shell” 1 Shell_Width t 0 ―t

2

“Stiffener” 1 tstiff Dstiff 0 +t ――Dstiff

2

⎡⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦

≔Iact =Rect_Prop_Ix ⎛⎝Mstiff⎞⎠ 32 4 =Iact 1 4


Page 77 of 180

≔UnityIr =‖‖‖‖‖‖‖‖‖‖

||||||||

|

for ∊ ||||||

|

j , ‥0 1 −rows ((z)) 1‖‖‖‖‖‖‖

←Un,j 0

Unity ⎛⎜⎝

,Ir_minjIact⎞

⎟⎠0

←Un,j 1

Unity ⎛⎜⎝

,Ir_minjIact⎞

⎟⎠1

((Un))

0.04 “Pass”0.04 “Pass”0.04 “Pass”0.03 “Pass”0.03 “Pass”0.03 “Pass”0.03 “Pass”0.03 “Pass”0.02 “Pass”0.02 “Pass”0.02 “Pass”0.02 “Pass”0.02 “Pass”0.02 “Pass”0.02 “Pass”0.02 “Pass”0.02 “Pass”

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ls 6.35 ≔Ls_max =⋅⋅do 0.56‾‾‾―do

t4 =Unity ⎛⎝ ,Ls Ls_max

⎞⎠2

“FAIL”⎡⎢⎣

⎤⎥⎦

Check the stress in the stiffening ring due to static wind (Eqn C3.5.10):

≔Mr =⋅⋅⋅⋅0.027 w5_top dz02 ⎛

⎜⎝+dz0

⋅6 br⎞⎟⎠

‾‾‾‾

――

dz0

tz0

156.6 ⋅≔w5_top =w5sec01073

≔Sact =Rect_Prop_Sx ⎛⎝Mstiff⎞⎠ 8.81 3

≔fb_ring =――Mr

Sact

17.78 ≔fk =――fyt

1.1322.73

≔γ 1.4 =Unity ⎛⎝ ,⋅γ fb_ring fk⎞⎠0.08

“Pass”⎡⎢⎣

⎤⎥⎦

Since stack doesn't meet all criteria to be considered "Stiffened" we check the static ovalling stress in the shell per Para 7.2.5.1:

≔V5sec((i)) ⋅Vz

i1.4 ≔Ishell =――

t3

1220.837 ――

4

≔Sshell =――Ishell

⎛⎜⎝―t

2

⎞⎟⎠

6.62 ――3

≔σb_allow =⋅0.9 fyt 320

≔w5sec =‖‖‖‖‖‖‖

‖

|||||||

|

for ∊ |||||

|

j , ‥0 1 −rows (z) 1‖‖‖‖

‖

←wj

―

⋅1.42 wzj

⋅dzj

CDj

(w)

3856.453595.123331.873066.382798.292527.092252.121972.421686.561540.651317.981206.641095.31983.97872.64761.3649.96

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔M =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((z)) 1‖‖‖‖

←wj

⋅⋅0.08 w5secjdz

j

2

((w))

43.2741.7240.0838.3536.5234.5632.4530.1527.6126.2126.2126.2126.2126.2126.2126.2126.21

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 78 of 180

≔σoval =――M

Sshell

6.546.316.065.85.525.234.914.564.173.963.963.963.963.963.963.963.96

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Unityσ =‖‖‖‖‖‖‖‖‖‖

||||||||

|

for ∊ ||||||

|

j , ‥0 1 −rows ((z)) 1‖‖‖‖‖‖‖

←Un,j 0

Unity ⎛⎜⎝

,σovaljfk⎞

⎟⎠0

←Un,j 1

Unity ⎛⎜⎝

,σovaljfk⎞

⎟⎠1

((Un))


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Dynamic Ovaling:

Determine if Ovalling vibrations are even possible based upon wind speed.

If Vr < Vt then ovaling can occur, need stiff rings

≔Vt =Vz030 ―

≔fo1 =⋅―――⋅0.5 tz

dz2

‾‾‾―E

ρs

32.3232.3232.3232.3232.3232.3232.3232.3232.3232.3232.3232.3232.3232.3232.3232.3232.32

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Vr =‖‖‖‖‖‖‖‖

||||||

|

for ∊ ||||

|

j , ‥0 1 −rows ((z)) 1‖‖‖‖‖

←wj

―――

⋅fo1jdz

j

⋅2 St

((w))

57.3657.3657.3657.3657.3657.3657.3657.3657.3657.3657.3657.3657.3657.3657.3657.3657.36

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― =Vz

3029282827262625242323232323232323

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―


Page 79 of 180

≔Unityσ =‖‖‖‖‖‖‖‖‖‖

||||||||

|

for ∊ ||||||

|

j , ‥0 1 −rows ((z)) 1‖‖‖‖‖‖‖

←Un,j 0

Unity ⎛⎜⎝

,VzjVr

j⎞⎟⎠0

←Un,j 1

Unity ⎛⎜⎝

,VzjVr

j⎞⎟⎠1

((Un))


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Determine the Maximum Ring Spacing:

=E 209947 =St 0 ≔ρsteel 7850 ――3

=h 25

=Ls_max 4.22 <-- Taken from Static analysis

≔Lmax2 =⋅――⋅dz

4

‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾−⋅

‾‾‾‾‾――E

ρsteel――――

1

⋅⋅⋅2 St Vt

1

6.556.556.556.556.556.556.556.556.556.556.556.556.556.556.556.556.55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Lmax2_c3_5_24 =――――⋅10.9 dz

⎛⎜⎝――h

1

⎞⎟⎠

0.06

6.386.386.386.386.386.386.386.386.386.386.386.386.386.386.386.386.38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Lmax =min ⎛⎝ ,Ls_max Lmax2⎞⎠ 4.22 =Ls 6.35 =Unity ⎛⎝ ,Ls Lmax

⎞⎠2

“FAIL”⎡⎢⎣

⎤⎥⎦

Determine if the Ring Moment of Inertia is Adeqaute:

=Vt 30 ― =br 0 =E 210 =t 6

≔Ir_min =――――――――⋅⋅⋅0.06 po dz

4 ⎛⎝ +dz ⋅6 br⎞⎠

⋅E tz

10.8410.8410.8410.8410.8410.8410.8410.84⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

4

≔po =⋅⋅0.5 ρa Vt2 547

=Iact 32 4


Page 80 of 180

≔Unityσ =‖‖‖‖‖‖‖‖‖‖

||||||||

|

for ∊ ||||||

|

j , ‥0 1 −rows ((z)) 1‖‖‖‖‖‖‖

←Un,j 0

Unity ⎛⎜⎝

,Ir_minjIact⎞

⎟⎠0

←Un,j 1

Unity ⎛⎜⎝

,Ir_minjIact⎞

⎟⎠1

((Un))

0.34 “Pass”0.34 “Pass”0.34 “Pass”0.34 “Pass”

⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

Along Wind Loads on Stack: (Unfactored)

≔Swind =‖‖‖‖‖‖‖‖‖

||||||||

for ∊ ||||||

j , ‥0 1 −rows ((z)) 2‖‖‖‖‖

←L −zj

z+j 1

←Sj

⋅wzjL

((S))

1955.771823.241689.731555.091419.131281.61142.151000.3427.66781.33334.2305.97277.74249.51221.28193.04

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Mwind =‖‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||||

for ∊ ||||||||||

j , ‥0 1 −rows ((z)) 2‖‖‖‖‖‖‖‖‖‖

←Mj

⋅0

for ∊ ||||||

k , ‥0 1 j‖‖‖‖‖

←L −zk

zj

←Mj

+Mj

⋅SwindkL

((M))

03912

114702240736455533417279094523

118257130552156704170114183830197824212068226532

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅

≔Stot =∑=j 0

−rows ((z)) 2

Swindj

14658

Along Wind Loads on Stack: (factored)

≔γwind 1.4 ≔γmatl 1.1 ≔γpermanent 1.1 ≔Stot =⋅γwind Stot 20521

≔Mw_fac =⋅γwind Mwind

05476

16057313705103774677

101906132333165560182773219385238160257362276954296895317145

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅ ≔S =――I

⎛⎜⎝―dz

2

⎞⎟⎠

2428.682428.682428.682428.682428.682428.682428.682428.682428.682428.682428.682428.682428.682428.682428.682428.682428.68

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

3 ≔A =⋅―4

⎛⎝ −dz

2 ⎛⎝ −dz ⋅2 tz⎞⎠2 ⎞⎠

139.28139.28139.28139.28139.28139.28139.28139.28139.28139.28139.28139.28139.28139.28139.28139.28139.28

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

2

2144.36⎡ ⎤ 2359⎡ ⎤


Page 81 of 180

≔V =‖‖‖‖‖‖‖‖‖

||||||||

for ∊ ||||||

j , ‥0 1 −rows ((z)) 2‖‖‖‖‖

←L −zj

z+j 1

←Vj

⋅⋅⋅ρsteel AjL

((V))

2144.362144.362144.362144.362144.362144.362144.362144.361072.182144.361072.181072.181072.181072.181072.181072.18

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Vcum =‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||||||

for ∊ ||||||||||||

j , ‥0 1 −rows ((z)) 2‖‖‖‖‖‖‖‖‖‖‖

←Vcj

0

for ∊ ||||||

k , ‥0 1 j‖‖‖‖‖

←L −zk

zj

←Vcj

+Vcj

Vk

←Vcj

⋅Vcjγpermanent

((Vc))

2359471870769435

117941415316512188702005022409235882476725947271262830629485

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σN =‖‖‖‖‖‖‖‖‖

|||||||

|

for ∊ ||||||

j , ‥0 1 −rows ((z)) 2‖‖‖‖‖‖

←Nj

――

Vcumj

Aj

((N))

0.170.340.510.680.851.021.191.351.441.611.691.781.861.952.032.12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σB =‖‖‖‖‖‖‖‖‖

|||||||

|

for ∊ ||||||

j , ‥0 1 −rows ((z)) 2‖‖‖‖‖‖

←Nj

―――

Mw_facj

Sj

((N))

02.256.61

12.9221.0130.7541.9654.4968.1775.2690.3398.06

105.97114.03122.25130.58

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Biaxial Stress: ≔τ 0 ≔fk =――fyt

γmatl

322.73

≔σx1 =+σN σB

0.172.597.12

13.5921.8631.7643.1455.8469.6176.8792.0299.84

107.83115.98124.28132.7

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σx2 =−σN σB

0.17−1.92−6.1

−12.24−20.17−29.73−40.77−53.13−66.73−73.65−88.64−96.28

−104.11−112.09−120.21−128.47

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σy =σoval

6.546.316.065.85.525.234.914.564.173.963.963.963.963.963.963.963.96

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔_σbiaxial⎛⎝ ,,σx σy τ⎞⎠

‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾+−+σx2 σy

2 ⋅σx σy ⋅3 τ2

6.46⎡ ⎤ 6.46⎡ ⎤


Page 82 of 180

≔σbiaxial1 =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((z)) 2‖‖‖

←Bj

_σbiaxial⎛⎜⎝

,,σx1jσy

jτ⎞⎟⎠

((B))

6.465.496.65

11.8119.6929.540.9153.7167.6274.9690.1197.92

105.91114.05122.34130.76

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σbiaxial2 =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((z)) 2‖‖‖

←Bj

_σbiaxial⎛⎜⎝

,,σx2jσy

jτ⎞⎟⎠

((B))

6.467.45

10.5315.9523.4232.6643.4455.5568.9175.7190.6898.32

106.14114.12122.24130.49

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔_αN(( ,d t))

‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||||||||||

←r −⋅0.5 d t|||||||||||||

|

if

else

≤―r

t212

‖‖‖‖‖‖

←an ―――――0.83

‾‾‾‾‾‾‾‾‾+1 ―――

r

⋅100 t

‖‖‖‖‖‖

←an ―――――0.7

‾‾‾‾‾‾‾‾‾‾+0.1 ―――

r

⋅100 t

((an))

≔_αB(( ,d t)) +0.189 ⋅0.811 _αN

(( ,d t))

=E 209947

=tz06.3 =dz0

710

≔_σk‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||||||||||||||||||||||||||||||||||||

for ∊ ||||||||||||||||||||||||||||||||||||||||

|

j , ‥0 1 −rows ((z)) 2‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖

←r −⋅0.5 dzj

tzj

←σcr ―――――

⋅⋅0.605 E tzj

r

←αN _αN⎛⎜⎝

,dzjtz

j⎞⎟⎠

←αB _αB⎛⎜⎝

,dzjtz

j⎞⎟⎠

←α ――――――

+⋅αN σNj

⋅αB σBj

+σNj

σBj

←λ‾‾‾‾‾‾―――fyt

⋅α σcr

|||||||

|

if

else

≤λ ‾‾2‖‖‖

←σk ⋅⎛⎝ −1 ⋅0.412 λ1.2⎞⎠ fyt

‖‖‖‖

←σk ⋅0.75 ――fyt

λ2

←M,j 0

σcr

←M,j 1

αN

←M,j 2

αB

←M,j 3

α

←M,j 4

λ

←M,j 5

σk

((M))

≔σcr =_σk⟨⟨0⟩⟩

2294.852294.852294.852294.852294.852294.852294.852294.852294.852294.852294.852294.852294.852294.852294.852294.85

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔αN =_σk⟨⟨1⟩⟩

0.6660.6660.6660.6660.6660.6660.6660.6660.6660.6660.6660.6660.6660.6660.6660.666

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔αB =_σk⟨⟨2⟩⟩

0.7290.7290.7290.7290.7290.729⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

≔α =_σk⟨⟨3⟩⟩

0.6660.7210.7250.7260.7270.7270.727⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔λ =_σk⟨⟨4⟩⟩

0.4820.4630.4620.4620.4610.4610.461⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 83 of 180

≔σk =_σk⟨⟨5⟩⟩

294.08296.91297.09297.15297.19297.21297.22297.23297.24297.24297.25297.25297.25297.25297.25297.26

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Check1 =‖‖‖‖‖‖‖‖

||||||

|

for ∊ ||||

|

j , ‥0 1 −rows ((z)) 2‖‖‖‖‖

←Cj

――――

⎛⎜⎝

+σNj

σBj⎞⎟⎠

fk

((C))

00.010.020.040.070.10.130.170.220.240.290.310.330.360.390.41

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Check2 =‖‖‖‖‖‖‖‖‖‖‖

||||||||||

for ∊ ||||||||

j , ‥0 1 −rows ((z)) 2‖‖‖‖‖‖‖‖

←Cj

――――

⎛⎜⎝

+σNj

σBj⎞⎟⎠

⎛⎜⎜⎝――

σkj

γmatl

⎞⎟⎟⎠

((C))

00.010.030.050.080.120.160.210.260.280.340.370.40.430.460.49

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Check3 =‖‖‖‖‖‖‖‖

||||||

|

for ∊ ||||

|

j , ‥0 1 −rows ((z)) 2‖‖‖‖‖

←Cj

―――

σbiaxial2j

fk

((C))

0.020.020.030.050.070.10.130.170.210.230.280.30.330.350.380.4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 84 of 180

Section B-5 - MecaStack (Reduced Formulations 7.2.4.2 & 8.5.2)


Page 85 of 180


Calculations Prepared by: Meca Enterprises 816 W. Elgin St Broken Arrow, OK, 74012 Date: Apr 22, 2017

File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\CICIND2010\ Reduced\v5314_App5_Reduced.Stk

INPUT PARAMETERS:


Total Stack Height = 25.0 m Top of Stack Elevation = 25.0 m Grade Elevation = 0.0 m Bottom of Stack = 0.0 m

All elevations are based upon the bottom of stack being at 0 m

Stack Geometry:

Elevation Outer Diameter | Elevation Thickness | Elevation Corrosion Allow m m m mm m mm

--------- -------------- - --------- --------- - --------- ---------------25.0 0.71 | 25.0 6.3 | 25.0 0.0

| 8.0 6.3 | | 7.0 6.3 | | 6.0 6.3 | | 5.0 6.3 | | 4.0 6.3 | | 3.0 6.3 | | 2.0 6.3 | | 1.0 6.3 |

Materials:

Elev Material Temp Fyld Elas Mod Alpha Allow Strs Density m MPa MPa mm/mm/C MPa Kg/m^3

---- -------------- ---- ------ -------- -------- ---------- ------- 25.0 S355 (FE 510A) 20.0 354.94 209,947 1.15E-05 122.73 7,849 20.0 354.94 209,947 1.15E-05 122.73 7,849

Design Codes

Comprehensive Design Standard:

Stress Criteria: CICIND 2010 'Model Code for Steel Chimneys'

Chimney = Chimney is considered to be Critical = False Zone = Chimney is located in a Tropical Zone = False

Wind Load Criteria: CICIND 2010 'Model Code for Steel Chimneys'

Vb = Wind Speed = 31.40 m/s Kt = Topographical Factor = 1.000 Category = Terrain Exposure = III

Vortex Shedding Criteria: CICIND 2010 'Model Code for Steel Chimneys'

Group = Stack Groups = Single Spacing = Center to center stack spacing (Only required for Group) = 0.0 m S = Manually entered Strouhal Number (Only used for Custom) = 0.2000 Full = Use full formulations per Commentary 3 = False


Page 86 of 180

Fatigue Criteria:


Section Properties


Top Bot Outer Thick Rad of Area Plastic Elastic Mom of Elev Elev Diameter Gyration Sec Modulus Sec Modulus Inertia m m mm mm mm sq cm cm^3 cm^3 cm^4

----- ----- -------- ----- -------- ------ ----------- ----------- -------- 25.00 23.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 23.00 21.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 21.00 19.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 19.00 17.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 17.00 15.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 15.00 13.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 13.00 11.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 11.00 9.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 9.00 8.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 8.00 7.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 7.00 6.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 6.00 5.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 5.00 4.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 4.00 3.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 3.00 2.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 2.00 1.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 1.00 0.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0

Weight Detail


Cylindrical Shells: (Wt = 2733.0 kg, El COG = 12.5 m)

Elev Elev Elev OD Thk Density Wt Lin Wt Wt Top Bot COG Corroded UnCorroded UnCorroded m m m m mm Kg/m^3 kg kg/m kg

----- ---- ----- ----- ---- ------- -------- ---------- ---------- 25.00 8.00 16.50 0.710 6.30 7,849 1,858.4 109.32 1,858.4 8.00 7.00 7.50 0.710 6.30 7,849 109.3 109.32 109.3 7.00 6.00 6.50 0.710 6.30 7,849 109.3 109.32 109.3 6.00 5.00 5.50 0.710 6.30 7,849 109.3 109.32 109.3 5.00 4.00 4.50 0.710 6.30 7,849 109.3 109.32 109.3 4.00 3.00 3.50 0.710 6.30 7,849 109.3 109.32 109.3 3.00 2.00 2.50 0.710 6.30 7,849 109.3 109.32 109.3 2.00 1.00 1.50 0.710 6.30 7,849 109.3 109.32 109.3 1.00 0.00 0.50 0.710 6.30 7,849 109.3 109.32 109.3

----- ---- ----- ----- ---- ------- -------- ---------- ---------- Total 12.50 2,733.0 2,733.0

Weight Summary

Component Elev COG Weight m kg

----------------------------------- -------- ------- Cylinders (Corroded Wt = 2733.0 kg) 12.500 2,733.0

----------------------------------- -------- ------- Total 12.500 2,733.0

Wind Areas

Wind Area Summary

Elev Shape Fac Riser Ladder Platform Piping Tot Uni Total Riser Area Area Area Area Area Area m m^2/m m^2/m m^2/m m^2/m m^2/m sq m


Page 87 of 180

----- --------- ----- ------ -------- ------ ------- ------

25.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

23.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

21.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

19.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

17.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

15.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

13.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

11.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

9.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

8.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

7.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

6.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

5.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

4.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

3.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

2.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

1.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

----- --------- ----- ------ -------- ------ ------- ------

Total 12.425

Frequency Summary


# 1 # 2 # 3

Deg Hz Hz Hz

----------------- --------- ----- ----- ------

Cold & Uncorroded 0.00 1.152 7.212 20.149


Elevation Mode # 1 Mode # 2 Mode # 3

(1.152 Hz) (7.212 Hz) (20.149 Hz)

m Normalized Normalized Normalized

--------- ---------- ---------- -----------

25.00 -1.000 1.000 1.000

23.00 -0.890 0.618 0.376

21.00 -0.780 0.246 -0.180

19.00 -0.671 -0.096 -0.554

17.00 -0.564 -0.383 -0.655

15.00 -0.461 -0.590 -0.473

13.00 -0.363 -0.702 -0.088

11.00 -0.272 -0.713 0.345

9.00 -0.190 -0.633 0.665

8.00 -0.154 -0.565 0.743

7.00 -0.120 -0.484 0.757

6.00 -0.090 -0.394 0.708

5.00 -0.064 -0.301 0.605

4.00 -0.042 -0.210 0.464

3.00 -0.024 -0.128 0.306

2.00 -0.011 -0.062 0.156

1.00 -0.003 -0.016 0.044

0.00 0.000 0.000 0.000

CICIND 2010 Wind Loadings


Vb = Basic Wind Speed, 10 min mean wind speed @ 10 m above grade = 31.40 m/s

h = Height of Chimney = 25.0 m

d = Average Diameter = 0.71 m

t = Average Thickness = 6.3 mm

f1 = Minimum Natural Frequency = 1.152 Hz

Vtop = Velocity at top of Chimney/Stack = 29.58 m/s

Bs = Structural Damping Ratio (Table 7.5) = 0.0020

Ba = Aerodynamic Damping Ratio: 2.7e-6*Vtop/(f1*t) (Eqn 7.51) = 0.0110


Page 88 of 180

B = Total Along Wind Damping Ratio (Ba + Bs) = 0.0130

zb = Stack Base Elev above Grade (for Kz adjustment) = 0.0 m

ze = Stack Enclosed below this Elev (no wind <= He) = 0.0 m

Kt = Topographical Factor = 1.000

Cat = Terrain Category (Table 7.1) = III

a = Terrain Category Factor (Table 7.2) = 0.770

b = Terrain Category Factor (Table 7.2) = 0.220

c = Terrain Category Factor (Table 7.2) = 0.280

Beta = Terrain Category Factor (Table 7.4) = 0.370

Zo = Terrain Cateogry Value (Table 7.1) = 0.300

Zmin = Height below which the wind speed is the same (Table 7.1) = 8.00 m

Zs = 0.6 * h (Para 7.2.3.3.2) = 15.00 m

L(Zs) = 300*(Z/300)^Beta if Z less than Zmin then Z=Zmin (Eqn 7.21) = 99.02 m

Nd = 4.6*fl*d / L(Zs) (Eqn 7.23) = 0.142

Nh = 4.6*fl*h / L(Zs) (Eqn 7.23) = 5.013

Rd = 1/Nd - (1/(2*Nd))*(1-exp(-2*Nd)) (Eqn 7.22) = 0.911

Rh = 1/Nh - (1/(2*Nh))*(1-exp(-2*Nh)) (Eqn 7.22) = 0.180

Sl = Power Spectral Density: 6.8*fl/(1+10.2*fl)^(5/3) (Eqn 7.19) = 0.052

B^2 = Background Factor 1/(1+0.9*(b+h / L(Zs))^0.63) (Eqn 7.17) = 0.725

R^2 = Resonance Response Factor: (PI/(4*B))*Sl*Rh*Rd (Eqn 7.18) = 0.509

fl = f1*L(zs) / V(zs) (Eqn 7.20) = 4.317

pa = Air Density = 1.250 Kg/m^3

vT = 600*f1 * (R^2 / (B^2 + R^2))^0.5 & vT >= 48 (Eqn 7.16) = 444.196

g = Peak Factor (Eqn 7.15) = 3.657

G = Gust Factor (Eqn 7.14) = 3.081

Ki = Interference Factor (Taken as 1 per Para 7.2.6) = 1.000

Ka = End Effect Factor (Eqn 7.12) = 1.000

Re = Reynolds Number: 6.9*10^4 * Vtop * d (Eqn 7.11) = 1.45E6

Wind Loading Summary

z1 z1 K(z1) V(z1) Aw Wm Wg W Pe Force Force

m ft m/s m^2/m N/m N/m N/m kPa lb N

----- ----- ----- ----- ----- --- --- --- ----- ----- ------

25.00 82.02 0.942 29.58 0.497 272 703 975 1.962 438 1,950

23.00 75.46 0.925 29.04 0.497 262 647 909 1.829 409 1,818

21.00 68.90 0.907 28.46 0.497 252 591 842 1.695 379 1,685

19.00 62.34 0.887 27.84 0.497 241 535 775 1.560 349 1,551

17.00 55.77 0.865 27.17 0.497 229 478 708 1.424 318 1,415

15.00 49.21 0.842 26.43 0.497 217 422 639 1.286 287 1,278

13.00 42.65 0.816 25.61 0.497 204 366 570 1.146 256 1,139

11.00 36.09 0.786 24.69 0.497 189 309 499 1.004 224 998

9.00 29.53 0.752 23.62 0.497 173 253 427 0.858 96 427

8.00 26.25 0.733 23.02 0.497 165 225 390 0.784 88 390

7.00 22.97 0.733 23.02 0.497 165 197 362 0.727 81 362

6.00 19.69 0.733 23.02 0.497 165 169 333 0.671 75 333

5.00 16.40 0.733 23.02 0.497 165 141 305 0.614 69 305

4.00 13.12 0.733 23.02 0.497 165 113 277 0.558 62 277

3.00 9.84 0.733 23.02 0.497 165 84 249 0.501 56 249

2.00 6.56 0.733 23.02 0.497 165 56 221 0.444 50 221

1.00 3.28 0.733 23.02 0.497 165 28 193 0.388 43 193

----- ----- ----- ----- ----- --- --- --- ----- ----- ------

Total 3,280 14,590

z1 = Elevation above grade: z + zb

K(z1) = Height Factor: a*(z1/10)^b & if z1

V(z1) = 10 Min mean design wind speed @ elev z1: K(z1)*Kt*Vb

Aw = Total effective wind area, refer to 'Wind Area Summary' for details

Area includes shape factors. Units are Area per unit length.

Wm = 10 Min mean wind load per unit height: 0.5* pa * V(Z1)^2 * Aw (Eqn 7.6)

Wg = Wind Gust: (3*(G-1)/h)*(z/h)*{Integral of [wm*z dz] from 0 to h} (Eqn 7.13)

W = Design wind load per unit height: Wm + Wg (Eqn 7.5)

Pe = Equiv Press (Used elsewhere in analysis): W / Aw

Wind Loads Applied to the Model



Page 89 of 180

Chris Rosencutter

Image

Chris Rosencutter

Arrow

Chris Rosencutter

Text Box

Pressures agree well with MathCad

Elev Elev Load Load

m m N/m N

----- ----- ------- ------

25.00 23.00 975 1,950

23.00 21.00 909 1,818

21.00 19.00 842 1,685

19.00 17.00 775 1,551

17.00 15.00 708 1,415

15.00 13.00 639 1,278

13.00 11.00 570 1,139

11.00 9.00 499 998

9.00 8.00 427 427

8.00 7.00 390 390

7.00 6.00 362 362

6.00 5.00 333 333

5.00 4.00 305 305

4.00 3.00 277 277

3.00 2.00 249 249

2.00 1.00 221 221

1.00 0.00 193 193

----- ----- ------- ------

Total 14,590


Mode Configuration Frequency Crit Wind Reduced Defl Defl Defl

Num Speed Mass Max Limit Ratio

Hz m/s kg m m

---- ----------------- --------- --------- ------- ----- ----- -----

1 Cold & Uncorroded 1.152 4.09 682 0.027 0.003 9.00

2 Cold & Uncorroded 7.212 19.21 664 0.001 0.002 0.24

3 Cold & Uncorroded 20.149 53.66 637 0.000 0.002 0.00

1. Defl Ratio = Defl Max / Defl Limit

Vortex Shedding Loads per CICIND 2010

Ref CICIND 2010 'Model Code for Steel Chimneys' Reduced Formulations per Para 7.2.4

h = Overall Height of Stack = 25.00 m

fn = Natural Frequency of Mode under Consderation = 1.152 Hz

f1 = Natural Frequency of first Mode = 1.152 Hz

d1 = Average outside diameter of top one third of stack = 0.71 m

Wind Velocity Imperial Imperial Metric

ft/s mph m/s

------------------------------------------------------------- -------- -------- ------

Vb: Basic Wind speed (10 min mean @ 10 m above grade) 103.0 70.2 31.40

Vtop: Wind Speed at Top of Stack: k(z)*kt*Vb {Eqn 7.1} 97.0 66.2 29.58

Vlimit: Limit to Ignore Vortex Shedding: 1.25*Vtop {Eqn 7.27} 121.3 82.7 36.97

Vc: Critical Wind Speed: fn*d1/St {Eqn 7.25} 13.4 9.2 4.09

St = Strouhal Number for a Single Stack (Eqn 7.24) = 0.200

Re = Reynolds Number: 6.67*10^4 * Vcr * d1 = 1.95E5

Iv(h) = Turbulence Intensity: c*(z/10)^-b if z z=zmin (Eqn 7.3) = 0.229

Cln = RMS Lift Coefficient (Eqn 7.41) = 0.700

Bv = Width of Lift Spectrum: 0.1+Iv (but <= 0.35) = 0.329

Mon = Equiv mass per unit length (Eqn 7.35) = 109.32 kg/m


Int1 = Integral of un^2 dz from 0 to h = 6.28

Can = Factor (Eqn 7.39) = 8.61E-05

Kamax = Factor (Eqn 7.37) = 2.800

Kan = Aerodynamic damping parameter (Eqn 7.36) = 0.877

Bs = Structural Damping Ratio = 0.0020

Scn = Scruton Number: 4*PI*Mon*Bs / (pa * d1^2) (Eqn 7.34) = 4.360

aln = 0.4*f1 / fn (Eqn 7.33) = 0.400

c1n = 0.5 * aln^2 * (1-Scn / (4*PI*Kan)) (Eqn 7.31) = 4.84E-02

c2n = aln^2 * Can / Kan (Eqn 7.32) = 1.57E-05

Kpn = Peak: (2)^0.5*(1+1.2*arctan(0.75*(Sc/(4*PI*Kan))^4) (Eqn 7.30) = 1.445

Syn = Std Deviation of Defl: d1*(c1n+(c1n^2+c2n)^0.5)^0.5 (Eqn 7.29) = 0.221 m

Mred = Reduced Mass at top of Stack = 682.2 kg

Mred = Reduced Mass at top of Stack = 1504 lb

Acc = Acceleration at Top of Stack = 11.59 m/s^2

Ymax = Max Value of: Kpn * Syn * un(z) = 0.319 m

Ymax = Max Value of: Kpn * Syn * un(z) = 1.048 ft


Page 90 of 180

Vortex Shedding Loading Calculations

z z un(z) yn(z) m(z) Fn(z) Ftot(z) Ftot(z)

m ft m kg/m N/m lb N

----- ----- ------ ------ ------ ------ ------- -------

25.00 82.02 -1.000 -0.319 109.32 -1,830 -823 -3,661

23.00 75.46 -0.890 -0.284 109.32 -1,629 -732 -3,258

21.00 68.90 -0.780 -0.249 109.32 -1,428 -642 -2,856

19.00 62.34 -0.671 -0.214 109.32 -1,229 -552 -2,457

17.00 55.77 -0.564 -0.180 109.32 -1,033 -465 -2,066

15.00 49.21 -0.461 -0.147 109.32 -844 -380 -1,688

13.00 42.65 -0.363 -0.116 109.32 -664 -299 -1,329

11.00 36.09 -0.272 -0.087 109.32 -498 -224 -996

9.00 29.53 -0.190 -0.061 109.32 -348 -78 -348

8.00 26.25 -0.154 -0.049 109.32 -281 -63 -281

7.00 22.97 -0.120 -0.038 109.32 -220 -49 -220

6.00 19.69 -0.090 -0.029 109.32 -165 -37 -165

5.00 16.40 -0.064 -0.020 109.32 -117 -26 -117

4.00 13.12 -0.042 -0.013 109.32 -76 -17 -76

3.00 9.84 -0.024 -0.008 109.32 -44 -10 -44

2.00 6.56 -0.011 -0.003 109.32 -20 -4 -20

1.00 3.28 -0.003 -0.001 109.32 -5 -1 -5

----- ----- ------ ------ ------ ------ ------- -------

Total -4,403 -19,588

z = Elevation above base of stack

un(z) = Mode shape for stack

yn(z) = Estimated Deflection of Stack: Kpn * Syn * un(z) (Eqn 7.28)

m(z) = Average mass per unit length

Fn(z) = Inertial force per unit length: (2*PI*fn)^2 * m(z) * yn(z)

Ftot(z)= Total Force For Beam Element: Fn(z)*(z1-z2), z1 current elev, z2 next elev

Vortex Shedding Loads Applied to Model


Elev Elev Load Load

m m N/m N

----- ----- ------- -------

25.00 23.00 -1,830 -3,661

23.00 21.00 -1,629 -3,258

21.00 19.00 -1,428 -2,856

19.00 17.00 -1,229 -2,457

17.00 15.00 -1,033 -2,066

15.00 13.00 -844 -1,688

13.00 11.00 -664 -1,329

11.00 9.00 -498 -996

9.00 8.00 -348 -348

8.00 7.00 -281 -281

7.00 6.00 -220 -220

6.00 5.00 -165 -165

5.00 4.00 -117 -117

4.00 3.00 -76 -76

3.00 2.00 -44 -44

2.00 1.00 -20 -20

1.00 0.00 -5 -5

----- ----- ------- -------

Total -19,588

Fatigue Analysis

Ref CICIND 2010 'Model Code for Steel Chimneys' (Para 8.5.2)


Vh = Wind Speed at top of Stack = 29.58 m/s

T = Fatigue Life of Chimney (Entered by User) = 50.0 Years

fn = Frequency of Structure = 1.152 Hz

d1 = Average diameter = 0.71 m

St = Strouhal Number = 0.200

Vcr = Critical Wind Speed: fn * d / St = 4.09 m/s

Mon = Equiv mass per unit length (Eqn 7.35) = 109.32 kg/m


Bs = Structural Damping Ratio = 0.0020


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Displacement & Forces agree well with MathCad

Scn = Scruton Number: 4*PI*Mon*Bs / (pa * d1^2) (Eqn 7.34) = 4.36

= Kao for Upr/Lwr limits for cross-wind vib: Scn / (4*PI*Kan) = 0.12

V0 = Std Dev of Wind Distribution Function: 0.2*Vh (Eqn 8.14) = 5.92 m/s

V1 = Lower Limit for cross wind vib: Use & Fig C3.3.4 = 3.52 m/s

V1 = Upper Limit for cross wind vib: Use & Fig C3.3.4 = 6.12 m/s

Nn = Cycles: 3.15*10^7*T*fn*[exp(-V1^2/V0^2)-Exp(-V2^2/V0^2)] (8.13) = 6.51E+08

Fatigue Analysis Summary

Partial Safety Factor = 1

z FatCat Fyt un d m I San Mn

m^4

m MPa m kg/m MPa

----- ------ ------ ------ ----- ------ -------- ------- --------

25.00 71 354.94 -1.000 0.710 109.32 8.62E-04 0.00 0.00

23.00 71 354.94 -0.890 0.710 109.32 8.62E-04 1.04 0.00

21.00 71 354.94 -0.780 0.710 109.32 8.62E-04 3.94 0.02

19.00 71 354.94 -0.671 0.710 109.32 8.62E-04 8.47 7.32

17.00 71 354.94 -0.564 0.710 109.32 8.62E-04 14.39 70.28

15.00 71 354.94 -0.461 0.710 109.32 8.62E-04 21.50 263.76

13.00 71 354.94 -0.363 0.710 109.32 8.62E-04 29.56 701.52

11.00 71 354.94 -0.272 0.710 109.32 8.62E-04 38.39 1543.42

9.00 71 354.94 -0.190 0.710 109.32 8.62E-04 47.78 2979.80

8.00 71 354.94 -0.154 0.710 109.32 8.62E-04 52.62 3982.29

7.00 71 354.94 -0.120 0.710 109.32 8.62E-04 57.54 5208.96

6.00 71 354.94 -0.090 0.710 109.32 8.62E-04 62.53 6684.50

5.00 71 354.94 -0.064 0.710 109.32 8.62E-04 67.56 8432.33

4.00 71 354.94 -0.042 0.710 109.32 8.62E-04 72.63 10473.34

3.00 71 354.94 -0.024 0.710 109.32 8.62E-04 77.72 12823.27

2.00 71 354.94 -0.011 0.710 109.32 8.62E-04 82.82 15488.28

1.00 71 354.94 -0.003 0.710 109.32 8.62E-04 87.93 18460.11

0.00 71 354.94 0.000 0.710 109.32 8.62E-04 93.04 21712.79

z = Elevation above base of stack

FatCat = Fatigue Detail Category per Figure 8.4

Fyt = Yield Stress of Material

un = Mode shape at elevation under consideration

d = Diameter of stack an elevation under consideration

m = Mass per unit length

I = Moment of Inertia of Stack and elevation under consderation.

San = Standard deviation of the cross-wind stress per Eqn 8.16

Mn = Palmgren-Miner sum for the mode (>1 is a failure) (Eqn 8.15)


Page 92 of 180

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Section B-6 - MecaStack (Full Formulations Commentary 3&4)


Page 93 of 180



File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\CICIND2010\Full\ v5314_App5_Full.Stk

INPUT PARAMETERS:




Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------25.0 0.71 | 25.0 6.3 | 25.0 0.0

| 8.0 6.3 | | 7.0 6.3 | | 6.0 6.3 | | 5.0 6.3 | | 4.0 6.3 | | 3.0 6.3 | | 2.0 6.3 | | 1.0 6.3 |

Materials:


---- -------------- ---- ------ -------- -------- ---------- ------- 25.0 S355 (FE 510A) 20.0 354.94 209,947 1.15E-05 122.73 7,849 20.0 354.94 209,947 1.15E-05 122.73 7,849

Design Codes

Comprehensive Design Standard: CICIND 2010 'Model Code for Steel Chimneys'






Group = Stack Groups = Single Spacing = Center to center stack spacing (Only required for Group) = 0.0 m S = Manually entered Strouhal Number (Only used for Custom) = 0.2000 Full = Use full formulations per Commentary 3 = True


Page 94 of 180

Fatigue Criteria: CICIND 2010 'Model Code for Steel Chimneys'


Section Properties



----- ----- -------- ----- -------- ------ ----------- ----------- --------25.00 23.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0

23.00 21.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 21.00 19.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 19.00 17.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 17.00 15.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 15.00 13.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 13.00 11.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 11.00 9.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 9.00 8.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 8.00 7.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 7.00 6.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 6.00 5.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 5.00 4.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 4.00 3.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 3.00 2.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 2.00 1.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 1.00 0.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0

Weight Detail




----- ---- ----- ----- ---- ------- -------- ---------- ---------- 25.00 8.00 16.50 0.710 6.30 7,849 1,858.4 109.32 1,858.4 8.00 7.00 7.50 0.710 6.30 7,849 109.3 109.32 109.3 7.00 6.00 6.50 0.710 6.30 7,849 109.3 109.32 109.3 6.00 5.00 5.50 0.710 6.30 7,849 109.3 109.32 109.3 5.00 4.00 4.50 0.710 6.30 7,849 109.3 109.32 109.3 4.00 3.00 3.50 0.710 6.30 7,849 109.3 109.32 109.3 3.00 2.00 2.50 0.710 6.30 7,849 109.3 109.32 109.3 2.00 1.00 1.50 0.710 6.30 7,849 109.3 109.32 109.3 1.00 0.00 0.50 0.710 6.30 7,849 109.3 109.32 109.3

----- ---- ----- ----- ---- ------- -------- ---------- ---------- Total 12.50 2,733.0 2,733.0

Weight Summary



----------------------------------- -------- ------- Total 12.500 2,733.0

Wind Areas

Wind Area Summary



Page 95 of 180

----- --------- ----- ------ -------- ------ ------- ------

25.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

23.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

21.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

19.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

17.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

15.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

13.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

11.00 0.700 0.497 0.000 0.000 0.000 0.497 0.994

9.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

8.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

7.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

6.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

5.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

4.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

3.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

2.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

1.00 0.700 0.497 0.000 0.000 0.000 0.497 0.497

----- --------- ----- ------ -------- ------ ------- ------

Total 12.425

Frequency Summary


# 1 # 2 # 3

Deg Hz Hz Hz

----------------- --------- ----- ----- ------

Cold & Uncorroded 0.00 1.152 7.212 20.149



(1.152 Hz) (7.212 Hz) (20.149 Hz)


--------- ---------- ---------- -----------

25.00 -1.000 1.000 1.000

23.00 -0.890 0.618 0.376

21.00 -0.780 0.246 -0.180

19.00 -0.671 -0.096 -0.554

17.00 -0.564 -0.383 -0.655

15.00 -0.461 -0.590 -0.473

13.00 -0.363 -0.702 -0.088

11.00 -0.272 -0.713 0.345

9.00 -0.190 -0.633 0.665

8.00 -0.154 -0.565 0.743

7.00 -0.120 -0.484 0.757

6.00 -0.090 -0.394 0.708

5.00 -0.064 -0.301 0.605

4.00 -0.042 -0.210 0.464

3.00 -0.024 -0.128 0.306

2.00 -0.011 -0.062 0.156

1.00 -0.003 -0.016 0.044

0.00 0.000 0.000 0.000












Page 96 of 180


zb = Stack Base Elev above Grade (for Kz adjustment) = 0.0 m ze = Stack Enclosed below this Elev (no wind <= He) = 0.0 m Kt = Topographical Factor = 1.000

Cat = Terrain Category (Table 7.1) = III a = Terrain Category Factor (Table 7.2) = 0.770 b = Terrain Category Factor (Table 7.2) = 0.220 c = Terrain Category Factor (Table 7.2) = 0.280 Beta = Terrain Category Factor (Table 7.4) = 0.370 Zo = Terrain Cateogry Value (Table 7.1) = 0.300 Zmin = Height below which the wind speed is the same (Table 7.1) = 8.00 m

Zs = 0.6 * h (Para 7.2.3.3.2) = 15.00 m L(Zs) = 300*(Z/300)^Beta if Z less than Zmin then Z=Zmin (Eqn 7.21) = 99.02 m Nd = 4.6*fl*d / L(Zs) (Eqn 7.23) = 0.142 Nh = 4.6*fl*h / L(Zs) (Eqn 7.23) = 5.013 Rd = 1/Nd - (1/(2*Nd))*(1-exp(-2*Nd)) (Eqn 7.22) = 0.911 Rh = 1/Nh - (1/(2*Nh))*(1-exp(-2*Nh)) (Eqn 7.22) = 0.180 Sl = Power Spectral Density: 6.8*fl/(1+10.2*fl)^(5/3) (Eqn 7.19) = 0.052 B^2 = Background Factor 1/(1+0.9*(b+h / L(Zs))^0.63) (Eqn 7.17) = 0.725 R^2 = Resonance Response Factor: (PI/(4*B))*Sl*Rh*Rd (Eqn 7.18) = 0.509 fl = f1*L(zs) / V(zs) (Eqn 7.20) = 4.317 pa = Air Density = 1.250 Kg/m^3 vT = 600*f1 * (R^2 / (B^2 + R^2))^0.5 & vT >= 48 (Eqn 7.16) = 444.196 g = Peak Factor (Eqn 7.15) = 3.657 G = Gust Factor (Eqn 7.14) = 3.081

Ki = Interference Factor (Taken as 1 per Para 7.2.6) = 1.000 Ka = End Effect Factor (Eqn 7.12) = 1.000 Re = Reynolds Number: 6.9*10^4 * Vtop * d (Eqn 7.11) = 1.45E6


z1 z1 K(z1) V(z1) Aw Wm Wg W Pe Force Force m ft m/s m^2/m N/m N/m N/m kPa lb N

----- ----- ----- ----- ----- --- --- --- ----- ----- ------ 25.00 82.02 0.942 29.58 0.497 272 703 975 1.962 438 1,950 23.00 75.46 0.925 29.04 0.497 262 647 909 1.829 409 1,818 21.00 68.90 0.907 28.46 0.497 252 591 842 1.695 379 1,685 19.00 62.34 0.887 27.84 0.497 241 535 775 1.560 349 1,551 17.00 55.77 0.865 27.17 0.497 229 478 708 1.424 318 1,415 15.00 49.21 0.842 26.43 0.497 217 422 639 1.286 287 1,278 13.00 42.65 0.816 25.61 0.497 204 366 570 1.146 256 1,139 11.00 36.09 0.786 24.69 0.497 189 309 499 1.004 224 998 9.00 29.53 0.752 23.62 0.497 173 253 427 0.858 96 427 8.00 26.25 0.733 23.02 0.497 165 225 390 0.784 88 390 7.00 22.97 0.733 23.02 0.497 165 197 362 0.727 81 362 6.00 19.69 0.733 23.02 0.497 165 169 333 0.671 75 333 5.00 16.40 0.733 23.02 0.497 165 141 305 0.614 69 305 4.00 13.12 0.733 23.02 0.497 165 113 277 0.558 62 277 3.00 9.84 0.733 23.02 0.497 165 84 249 0.501 56 249 2.00 6.56 0.733 23.02 0.497 165 56 221 0.444 50 221 1.00 3.28 0.733 23.02 0.497 165 28 193 0.388 43 193

----- ----- ----- ----- ----- --- --- --- ----- ----- ------ Total 3,280 14,590 z1 = Elevation above grade: z + zb K(z1) = Height Factor: a*(z1/10)^b & if z1 V(z1) = 10 Min mean design wind speed @ elev z1: K(z1)*Kt*Vb Aw = Total effective wind area, refer to 'Wind Area Summary' for details Area includes shape factors. Units are Area per unit length. Wm = 10 Min mean wind load per unit height: 0.5* pa * V(Z1)^2 * Aw (Eqn 7.6) Wg = Wind Gust: (3*(G-1)/h)*(z/h)*{Integral of [wm*z dz] from 0 to h} (Eqn 7.13) W = Design wind load per unit height: Wm + Wg (Eqn 7.5) Pe = Equiv Press (Used elsewhere in analysis): W / Aw Wind Loads Applied to the Model



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Pressures agree well with MathCad

Elev Elev Load Load m m N/m N

----- ----- ------- ------ 25.00 23.00 975 1,950 23.00 21.00 909 1,818 21.00 19.00 842 1,685 19.00 17.00 775 1,551 17.00 15.00 708 1,415 15.00 13.00 639 1,278 13.00 11.00 570 1,139 11.00 9.00 499 998 9.00 8.00 427 427 8.00 7.00 390 390 7.00 6.00 362 362 6.00 5.00 333 333 5.00 4.00 305 305 4.00 3.00 277 277 3.00 2.00 249 249 2.00 1.00 221 221 1.00 0.00 193 193

----- ----- ------- ------ Total 14,590 Vortex Shedding Summary

Mode Configuration Frequency Crit Wind Reduced Defl Defl Defl Num Speed Mass Max Limit Ratio Hz m/s kg m m

---- ----------------- --------- --------- ------- ----- ----- ----- 1 Cold & Uncorroded 1.152 4.09 682 0.027 0.010 2.74 2 Cold & Uncorroded 7.212 19.21 664 0.001 0.007 0.07 3 Cold & Uncorroded 20.149 53.66 637 0.000 0.007 0.00 1. Defl Ratio = Defl Max / Defl Limit Vortex Shedding Loads per CICIND 2010 Commentary 3

Ref CICIND 2010 'Model Code for Steel Chimneys' Commentary 3

h = Overall Height of Stack = 25.00 m fn = Natural Frequency of Mode under Consderation = 1.152 Hz f1 = Natural Frequency of first Mode = 1.152 Hz d1 = Average outside diameter of top one third of stack = 0.71 m Wind Velocity Imperial Imperial Metric ft/s mph m/s

------------------------------------------------------------- -------- -------- ------ Vb: Basic Wind speed (10 min mean @ 10 m above grade) 103.0 70.2 31.40 Vct: Speed @ top to produce Vcr at Zcr: Vcr/k(Zcr) 14.8 10.1 4.52 Vtop: Wind Speed at Top of Stack: k(z)*kt*Vb {Eqn 7.1} 97.0 66.2 29.58 Vlimit: Limit to Ignore Vortex Shedding: 1.25*Vtop {Eqn 7.27} 121.3 82.7 36.97 Vc: Critical Wind Speed: fn*d1/St {Eqn 7.25} 13.4 9.2 4.09 St = Strouhal Number for a Single Stack (Eqn 7.24) = 0.200 Re = Reynolds Number: 6.67*10^4 * Vcr * d1 = 1.95E5 Cln = RMS Lift Coefficient (Eqn 7.41) = 0.700 Bv = Width of Lift Spectrum: 0.1+Iv (but <= 0.35) = 0.329 Mon = Equiv mass per unit length (Eqn 7.35) = 109.32 kg/m pa = Air Density = 1.250 Kg/m^3 Int1 = Integral of un^2 dz from 0 to h = 6.28 Kamax = Factor (Eqn 7.37) = 2.800 Bs = Structural Damping Ratio = 0.0020 Scn = Scruton Number: 4*PI*Mon*Bs / (pa * d1^2) (Eqn 7.34) = 4.360 aln = 0.4*f1 / fn (Eqn 7.33) = 0.400 Zcr = Elevation of mid-point where vortex shedding occurs = 20.833 m Vc = Wind speed @ Top to produce Vcr at Zcr: Vcr / k(Zcr) = 4.52 m/s Kao_z = Horiz Line on Fig C3.3.4: Scn / (4*PI*Ka_max) = 0.12 Vlwr = Lower Reduced Velocity which intersects Kao_z = 0.86 m/s Vupr = Upper Reduced Velocity which intersects Kao_z = 1.50 m/s Mred = Reduced Mass at top of Stack = 682.2 kg Mred = Reduced Mass at top of Stack = 1504 lb Acc = Acceleration at Top of Stack = 11.59 m/s^2 yn_max = Max standard deviation of the maximal deflection (Eqn C3.3.12) = 0.315 m yn_max = Max standard deviation of the maximal deflection (Eqn C3.3.12) = 1.035 ft Vc_max = Top Wind velocity that produces the largest defl: 1.10* Vc = 4.68 m/s


Page 98 of 180

z_max = Elevation that produces the largest deflection = 25.00 m Vortex Shedding Deflections

z z un(z) m(z) B(z) Iv(z) yn(z) Fn(z) Fntot Fntot m ft kg/m m N/m lb N

----- ----- ------ ------ ----- ----- ------ ------ ------ ------- 25.00 82.02 -1.000 109.32 0.329 0.229 -0.315 -1,808 -813 -3,616 23.00 75.46 -0.890 109.32 0.333 0.233 -0.281 -1,609 -723 -3,218 21.00 68.90 -0.780 109.32 0.338 0.238 -0.246 -1,410 -634 -2,821 19.00 62.34 -0.671 109.32 0.343 0.243 -0.212 -1,214 -546 -2,427 17.00 55.77 -0.564 109.32 0.349 0.249 -0.178 -1,020 -459 -2,041 15.00 49.21 -0.461 109.32 0.350 0.256 -0.145 -834 -375 -1,667 13.00 42.65 -0.363 109.32 0.350 0.264 -0.115 -656 -295 -1,313 11.00 36.09 -0.272 109.32 0.350 0.274 -0.086 -492 -221 -984 9.00 29.53 -0.190 109.32 0.350 0.287 -0.060 -344 -77 -344 8.00 26.25 -0.154 109.32 0.350 0.294 -0.048 -278 -62 -278 7.00 22.97 -0.120 109.32 0.350 0.294 -0.038 -217 -49 -217 6.00 19.69 -0.090 109.32 0.350 0.294 -0.028 -163 -37 -163 5.00 16.40 -0.064 109.32 0.350 0.294 -0.020 -115 -26 -115 4.00 13.12 -0.042 109.32 0.350 0.294 -0.013 -75 -17 -75 3.00 9.84 -0.024 109.32 0.350 0.294 -0.008 -43 -10 -43 2.00 6.56 -0.011 109.32 0.350 0.294 -0.003 -20 -4 -20 1.00 3.28 -0.003 109.32 0.350 0.294 -0.001 -5 -1 -5

----- ----- ------ ------ ----- ----- ------ ------ ------ ------- Total -4,349 -19,346 z = Elevation above base of stack un(z) = Mode shape for stack yn(z) = Estimated Deflection of Stack: Kpn * Syn * un(z) (Eqn 7.28) m(z) = Average mass per unit length B(z) = Width of the lift spectrum (Eqn C3.3.4) Iv(z) = Turbulence intensity (Eqn 7.3) V: Wind Velocities

z 0.75 0.85 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00 *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr m m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s

----- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- 25.00 3.19 3.62 4.05 4.26 4.47 4.68 4.90 5.11 5.96 6.81 7.66 8.52 23.00 3.14 3.55 3.97 4.18 4.39 4.60 4.81 5.02 5.85 6.69 7.53 8.36 21.00 3.07 3.48 3.89 4.10 4.30 4.51 4.71 4.92 5.74 6.56 7.38 8.20 19.00 3.01 3.41 3.81 4.01 4.21 4.41 4.61 4.81 5.61 6.41 7.22 8.02 17.00 2.93 3.33 3.72 3.91 4.11 4.30 4.50 4.69 5.48 6.26 7.04 7.82 15.00 2.85 3.23 3.62 3.81 4.00 4.19 4.38 4.57 5.33 6.09 6.85 7.61 13.00 2.77 3.13 3.50 3.69 3.87 4.06 4.24 4.43 5.16 5.90 6.64 7.38 11.00 2.67 3.02 3.38 3.55 3.73 3.91 4.09 4.27 4.98 5.69 6.40 7.11 9.00 2.55 2.89 3.23 3.40 3.57 3.74 3.91 4.08 4.76 5.44 6.12 6.80 8.00 2.49 2.82 3.15 3.31 3.48 3.65 3.81 3.98 4.64 5.30 5.97 6.63 7.00 2.49 2.82 3.15 3.31 3.48 3.65 3.81 3.98 4.64 5.30 5.97 6.63 6.00 2.49 2.82 3.15 3.31 3.48 3.65 3.81 3.98 4.64 5.30 5.97 6.63 5.00 2.49 2.82 3.15 3.31 3.48 3.65 3.81 3.98 4.64 5.30 5.97 6.63 4.00 2.49 2.82 3.15 3.31 3.48 3.65 3.81 3.98 4.64 5.30 5.97 6.63 3.00 2.49 2.82 3.15 3.31 3.48 3.65 3.81 3.98 4.64 5.30 5.97 6.63 2.00 2.49 2.82 3.15 3.31 3.48 3.65 3.81 3.98 4.64 5.30 5.97 6.63 1.00 2.49 2.82 3.15 3.31 3.48 3.65 3.81 3.98 4.64 5.30 5.97 6.63 Determine velocities (V) that will achieve 'Factor * Vcr' at the elevation Zcr. Vr: Reduced Velocities

z 0.75 0.85 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00 *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr m

----- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- 25.00 0.78 0.88 0.99 1.04 1.09 1.15 1.20 1.25 1.46 1.67 1.87 2.08 23.00 0.77 0.87 0.97 1.02 1.07 1.12 1.18 1.23 1.43 1.64 1.84 2.04 21.00 0.75 0.85 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00 19.00 0.73 0.83 0.93 0.98 1.03 1.08 1.13 1.18 1.37 1.57 1.76 1.96 17.00 0.72 0.81 0.91 0.96 1.00 1.05 1.10 1.15 1.34 1.53 1.72 1.91 15.00 0.70 0.79 0.88 0.93 0.98 1.02 1.07 1.12 1.30 1.49 1.67 1.86 13.00 0.68 0.77 0.86 0.90 0.95 0.99 1.04 1.08 1.26 1.44 1.62 1.80 11.00 0.65 0.74 0.83 0.87 0.91 0.96 1.00 1.04 1.22 1.39 1.56 1.74


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Agrees well with MathCad calculations

9.00 0.62 0.71 0.79 0.83 0.87 0.91 0.96 1.00 1.16 1.33 1.50 1.66

8.00 0.61 0.69 0.77 0.81 0.85 0.89 0.93 0.97 1.13 1.30 1.46 1.62

7.00 0.61 0.69 0.77 0.81 0.85 0.89 0.93 0.97 1.13 1.30 1.46 1.62

6.00 0.61 0.69 0.77 0.81 0.85 0.89 0.93 0.97 1.13 1.30 1.46 1.62

5.00 0.61 0.69 0.77 0.81 0.85 0.89 0.93 0.97 1.13 1.30 1.46 1.62

4.00 0.61 0.69 0.77 0.81 0.85 0.89 0.93 0.97 1.13 1.30 1.46 1.62

3.00 0.61 0.69 0.77 0.81 0.85 0.89 0.93 0.97 1.13 1.30 1.46 1.62

2.00 0.61 0.69 0.77 0.81 0.85 0.89 0.93 0.97 1.13 1.30 1.46 1.62

1.00 0.61 0.69 0.77 0.81 0.85 0.89 0.93 0.97 1.13 1.30 1.46 1.62

Vr = Reduced velocity: V / Vcr

Kao: Turbulence Mean Aerodynamic Damping Parameter per Figure C3.3.4

z 0.75 0.85 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00

*Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr

m

----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- -----

25.00 0.000 0.155 0.268 0.301 0.316 0.315 0.302 0.279 0.149 0.000 0.000 0.000

23.00 0.000 0.136 0.250 0.288 0.308 0.312 0.304 0.286 0.164 0.000 0.000 0.000

21.00 0.000 0.000 0.231 0.272 0.297 0.307 0.304 0.290 0.180 0.000 0.000 0.000

19.00 0.000 0.000 0.209 0.253 0.283 0.299 0.301 0.293 0.196 0.000 0.000 0.000

17.00 0.000 0.000 0.186 0.232 0.266 0.287 0.295 0.292 0.211 0.000 0.000 0.000

15.00 0.000 0.000 0.161 0.207 0.245 0.271 0.285 0.288 0.225 0.117 0.000 0.000

13.00 0.000 0.000 0.000 0.181 0.220 0.250 0.269 0.278 0.237 0.140 0.000 0.000

11.00 0.000 0.000 0.000 0.152 0.191 0.224 0.248 0.262 0.245 0.163 0.000 0.000

9.00 0.000 0.000 0.000 0.000 0.160 0.193 0.220 0.239 0.247 0.184 0.000 0.000

8.00 0.000 0.000 0.000 0.000 0.000 0.177 0.204 0.225 0.244 0.193 0.115 0.000

7.00 0.000 0.000 0.000 0.000 0.000 0.177 0.204 0.225 0.244 0.193 0.115 0.000

6.00 0.000 0.000 0.000 0.000 0.000 0.177 0.204 0.225 0.244 0.193 0.115 0.000

5.00 0.000 0.000 0.000 0.000 0.000 0.177 0.204 0.225 0.244 0.193 0.115 0.000

4.00 0.000 0.000 0.000 0.000 0.000 0.177 0.204 0.225 0.244 0.193 0.115 0.000

3.00 0.000 0.000 0.000 0.000 0.000 0.177 0.204 0.225 0.244 0.193 0.115 0.000

2.00 0.000 0.000 0.000 0.000 0.000 0.177 0.204 0.225 0.244 0.193 0.115 0.000

1.00 0.000 0.000 0.000 0.000 0.000 0.177 0.204 0.225 0.244 0.193 0.115 0.000

If VrVupr then Kao = 0, since Kao is negative in Fig C3.3.4

Kan: Aerodynamic Damping Parameter per Eqn C3.3.20

z 0.75 0.85 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00


m

----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- -----

25.00 0.000 0.433 0.750 0.842 0.884 0.882 0.845 0.782 0.418 0.000 0.000 0.000

23.00 0.000 0.380 0.701 0.806 0.862 0.875 0.851 0.800 0.461 0.000 0.000 0.000

21.00 0.000 0.000 0.647 0.761 0.832 0.860 0.851 0.812 0.504 0.000 0.000 0.000

19.00 0.000 0.000 0.586 0.709 0.793 0.837 0.844 0.819 0.548 0.000 0.000 0.000

17.00 0.000 0.000 0.520 0.649 0.745 0.803 0.826 0.818 0.590 0.000 0.000 0.000

15.00 0.000 0.000 0.451 0.581 0.685 0.758 0.797 0.805 0.630 0.327 0.000 0.000

13.00 0.000 0.000 0.000 0.506 0.615 0.699 0.753 0.779 0.663 0.391 0.000 0.000

11.00 0.000 0.000 0.000 0.426 0.535 0.626 0.693 0.734 0.686 0.455 0.000 0.000

9.00 0.000 0.000 0.000 0.000 0.449 0.541 0.616 0.670 0.691 0.515 0.000 0.000

8.00 0.000 0.000 0.000 0.000 0.000 0.496 0.572 0.630 0.683 0.539 0.322 0.000

7.00 0.000 0.000 0.000 0.000 0.000 0.496 0.572 0.630 0.683 0.539 0.322 0.000

6.00 0.000 0.000 0.000 0.000 0.000 0.496 0.572 0.630 0.683 0.539 0.322 0.000

5.00 0.000 0.000 0.000 0.000 0.000 0.496 0.572 0.630 0.683 0.539 0.322 0.000

4.00 0.000 0.000 0.000 0.000 0.000 0.496 0.572 0.630 0.683 0.539 0.322 0.000

3.00 0.000 0.000 0.000 0.000 0.000 0.496 0.572 0.630 0.683 0.539 0.322 0.000

2.00 0.000 0.000 0.000 0.000 0.000 0.496 0.572 0.630 0.683 0.539 0.322 0.000

1.00 0.000 0.000 0.000 0.000 0.000 0.496 0.572 0.630 0.683 0.539 0.322 0.000

fs: Shedding frequencies per Eqn 7.24: Vc * d / St

z 0.75 0.85 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00


m

----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- -----

25.00 0.900 1.020 1.140 1.199 1.259 1.319 1.379 1.439 1.679 1.919 2.159 2.399

23.00 0.883 1.001 1.119 1.178 1.237 1.295 1.354 1.413 1.649 1.884 2.120 2.355

21.00 0.866 0.981 1.097 1.154 1.212 1.270 1.327 1.385 1.616 1.847 2.078 2.309

19.00 0.847 0.960 1.073 1.129 1.186 1.242 1.299 1.355 1.581 1.807 2.033 2.258

17.00 0.826 0.937 1.047 1.102 1.157 1.212 1.267 1.322 1.543 1.763 1.983 2.204

15.00 0.804 0.911 1.018 1.072 1.126 1.179 1.233 1.286 1.501 1.715 1.930 2.144


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MathCad Calculation:

13.00 0.779 0.883 0.987 1.039 1.091 1.143 1.195 1.247 1.454 1.662 1.870 2.078

11.00 0.751 0.851 0.951 1.001 1.051 1.101 1.151 1.202 1.402 1.602 1.802 2.003

9.00 0.719 0.814 0.910 0.958 1.006 1.054 1.102 1.150 1.341 1.533 1.724 1.916

8.00 0.700 0.793 0.887 0.934 0.980 1.027 1.074 1.120 1.307 1.494 1.680 1.867

7.00 0.700 0.793 0.887 0.934 0.980 1.027 1.074 1.120 1.307 1.494 1.680 1.867

6.00 0.700 0.793 0.887 0.934 0.980 1.027 1.074 1.120 1.307 1.494 1.680 1.867

5.00 0.700 0.793 0.887 0.934 0.980 1.027 1.074 1.120 1.307 1.494 1.680 1.867

4.00 0.700 0.793 0.887 0.934 0.980 1.027 1.074 1.120 1.307 1.494 1.680 1.867

3.00 0.700 0.793 0.887 0.934 0.980 1.027 1.074 1.120 1.307 1.494 1.680 1.867

2.00 0.700 0.793 0.887 0.934 0.980 1.027 1.074 1.120 1.307 1.494 1.680 1.867

1.00 0.700 0.793 0.887 0.934 0.980 1.027 1.074 1.120 1.307 1.494 1.680 1.867

Parameters Calculated for Vortex Shedding Analysis

Velocity Karn Can C1n C2n Kpn

Eqn C3.3.5 Eqn C3.3.9 Eqn C3.3.13 Eqn C3.3.14 Eqn C3.3.11

---------- ---------- ---------- ----------- ----------- -----------

0.75 * Vcr 0.0000 6.56E-06 0.000 0.00E+00 0.000

0.85 * Vcr 0.1648 1.90E-05 -0.088 3.65E-05 3.965

0.95 * Vcr 0.5887 3.39E-05 0.033 1.83E-05 1.567

1.00 * Vcr 0.7226 4.07E-05 0.042 1.79E-05 1.482

1.05 * Vcr 0.7973 4.63E-05 0.045 1.84E-05 1.460

1.10 * Vcr 0.8334 5.07E-05 0.047 1.93E-05 1.452

1.15 * Vcr 0.8308 5.39E-05 0.047 2.06E-05 1.453

1.20 * Vcr 0.7994 5.60E-05 0.045 2.22E-05 1.459

1.40 * Vcr 0.5170 5.64E-05 0.026 3.46E-05 1.670

1.60 * Vcr 0.0582 5.09E-05 -0.397 2.77E-04 4.078

1.80 * Vcr 0.0027 4.44E-05 -10.188 5.21E-03 4.080

2.00 * Vcr 0.0000 3.88E-05 0.000 0.00E+00 0.000

All values are dimensionless except Can which has the units of sq meters.

yn: Standard deviation in meters of the maximal deflection per Eqn C3.3.12

z 0.75 0.85 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00

* Vcr * Vcr * Vcr * Vcr * Vcr * Vcr * Vcr * Vcr * Vcr * Vcr * Vcr * Vcr

m

----- ----- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ -----

25.00 0.000 -0.040 -0.286 -0.304 -0.312 -0.315 -0.315 -0.312 -0.274 -0.054 -0.046 0.000

23.00 0.000 -0.036 -0.254 -0.270 -0.278 -0.281 -0.281 -0.278 -0.244 -0.048 -0.041 0.000

21.00 0.000 -0.032 -0.223 -0.237 -0.243 -0.246 -0.246 -0.244 -0.214 -0.042 -0.036 0.000

19.00 0.000 -0.027 -0.192 -0.204 -0.209 -0.212 -0.212 -0.210 -0.184 -0.036 -0.031 0.000

17.00 0.000 -0.023 -0.161 -0.171 -0.176 -0.178 -0.178 -0.176 -0.155 -0.031 -0.026 0.000

15.00 0.000 -0.019 -0.132 -0.140 -0.144 -0.145 -0.145 -0.144 -0.126 -0.025 -0.021 0.000

13.00 0.000 -0.015 -0.104 -0.110 -0.113 -0.115 -0.114 -0.113 -0.099 -0.020 -0.017 0.000

11.00 0.000 -0.011 -0.078 -0.083 -0.085 -0.086 -0.086 -0.085 -0.074 -0.015 -0.013 0.000

9.00 0.000 -0.008 -0.054 -0.058 -0.059 -0.060 -0.060 -0.059 -0.052 -0.010 -0.009 0.000

8.00 0.000 -0.006 -0.044 -0.047 -0.048 -0.048 -0.048 -0.048 -0.042 -0.008 -0.007 0.000

7.00 0.000 -0.005 -0.034 -0.036 -0.037 -0.038 -0.038 -0.038 -0.033 -0.007 -0.006 0.000

6.00 0.000 -0.004 -0.026 -0.027 -0.028 -0.028 -0.028 -0.028 -0.025 -0.005 -0.004 0.000

5.00 0.000 -0.003 -0.018 -0.019 -0.020 -0.020 -0.020 -0.020 -0.017 -0.003 -0.003 0.000

4.00 0.000 -0.002 -0.012 -0.013 -0.013 -0.013 -0.013 -0.013 -0.011 -0.002 -0.002 0.000

3.00 0.000 -0.001 -0.007 -0.007 -0.007 -0.008 -0.008 -0.007 -0.007 -0.001 -0.001 0.000

2.00 0.000 0.000 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.001 -0.001 0.000

1.00 0.000 0.000 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 0.000 0.000 0.000

All values are in units of meters.



Elev Elev Load Load

m m N/m N

----- ----- ------- -------

25.00 23.00 -1,808 -3,616

23.00 21.00 -1,609 -3,218

21.00 19.00 -1,410 -2,821

19.00 17.00 -1,214 -2,427

17.00 15.00 -1,020 -2,041

15.00 13.00 -834 -1,667

13.00 11.00 -656 -1,313

11.00 9.00 -492 -984

9.00 8.00 -344 -344

8.00 7.00 -278 -278


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7.00 6.00 -217 -217 6.00 5.00 -163 -163 5.00 4.00 -115 -115 4.00 3.00 -75 -75 3.00 2.00 -43 -43 2.00 1.00 -20 -20 1.00 0.00 -5 -5

----- ----- ------- ------- Total -19,346 Fatigue Analysis

Ref CICIND 2010 'Model Code for Steel Chimneys' (Commentary 4)

Vb = Basic Wind Speed, 10 min mean wind speed @ 10 m above grade = 31.40 m/s T = Fatigue Life of Chimney (Entered by User) = 50.0 Years fn = Frequency of Structure = 1.152 Hz Mode = Mode Number = 1 d1 = Average diameter = 0.71 m St = Strouhal Number = 0.200 Vcr = Critical Wind Speed: fn * d / St = 4.09 m/s Mon = Equiv mass per unit length (Eqn 7.35) = 109 N/m pa = Air Density = 0.077 Kg/m^3 Bs = Structural Damping Ratio = 0.0020 Scn = Scruton Number: 4*PI*Mon*Bs / (pa * d1^2) (Eqn 7.34) = 4.36 Kao_z = Kao for Upr/Lwr limits for cross-wind vib: Scn / (4*PI*Kan) = 0.12 V0 = Std Dev of Wind Distribution Function: 0.2*Vh (Eqn 8.14) = 5.92 m/s V1 = Lower Limit for cross wind vib: Use Kao and Fig C3.3.4 = 3.52 m/s V2 = Upper Limit for cross wind vib: Use Kao and Fig C3.3.4 = 6.12 m/s Nn = Cycles: 3.15*10^7*T*fn*[exp(-V1^2/V0^2)-Exp(-V2^2/V0^2)] (8.13) = 6.51E+08 Fatigue Analysis Parameters

z z FatCat Fyt un d m I m^4 m ft MPa m kg/m

----- ----- ------ ------ ------ ----- ------ -------- 25.00 82.02 71 354.94 -1.000 0.710 109.32 8.62E-04 23.00 75.46 71 354.94 -0.890 0.710 109.32 8.62E-04 21.00 68.90 71 354.94 -0.780 0.710 109.32 8.62E-04 19.00 62.34 71 354.94 -0.671 0.710 109.32 8.62E-04 17.00 55.77 71 354.94 -0.564 0.710 109.32 8.62E-04 15.00 49.21 71 354.94 -0.461 0.710 109.32 8.62E-04 13.00 42.65 71 354.94 -0.363 0.710 109.32 8.62E-04 11.00 36.09 71 354.94 -0.272 0.710 109.32 8.62E-04 9.00 29.53 71 354.94 -0.190 0.710 109.32 8.62E-04 8.00 26.25 71 354.94 -0.154 0.710 109.32 8.62E-04 7.00 22.97 71 354.94 -0.120 0.710 109.32 8.62E-04 6.00 19.69 71 354.94 -0.090 0.710 109.32 8.62E-04 5.00 16.40 71 354.94 -0.064 0.710 109.32 8.62E-04 4.00 13.12 71 354.94 -0.042 0.710 109.32 8.62E-04 3.00 9.84 71 354.94 -0.024 0.710 109.32 8.62E-04 2.00 6.56 71 354.94 -0.011 0.710 109.32 8.62E-04 1.00 3.28 71 354.94 -0.003 0.710 109.32 8.62E-04 0.00 0.00 71 354.94 0.000 0.710 109.32 8.62E-04 z = Elevation above base of stack | Fyt = Yield Stress of Material FatCat = Fatigue Detail Category {Figure 8.4} | m = Mass per unit length un = Normalized Mode Shape | d = Outer Diameter of Stack I = Moment of Inertia of Stack Syn (m): Standard Deviation of Deflection in meters (Eqn C3.3.10)

0.75 0.85 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00 *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr *Vcr

----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 0.000 0.010 0.182 0.205 0.214 0.217 0.217 0.214 0.164 0.013 0.011 0.000 San (MPa): Standard deviation of stress (Eqn C4.8)


----- ---- ----- ------ ------ ------ ------ ------ ------ ------ ----- ----- ---- 25.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00


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23.00 0.00 -0.05 -0.86 -0.97 -1.01 -1.03 -1.02 -1.01 -0.77 -0.06 -0.05 0.00 21.00 0.00 -0.18 -3.25 -3.66 -3.81 -3.87 -3.87 -3.82 -2.92 -0.24 -0.20 0.00 19.00 0.00 -0.39 -6.99 -7.86 -8.19 -8.32 -8.31 -8.20 -6.28 -0.51 -0.44 0.00 17.00 0.00 -0.66 -11.88 -13.35 -13.92 -14.15 -14.13 -13.93 -10.67 -0.86 -0.74 0.00 15.00 0.00 -0.99 -17.74 -19.94 -20.79 -21.13 -21.11 -20.81 -15.94 -1.29 -1.10 0.00 13.00 0.00 -1.36 -24.40 -27.43 -28.58 -29.06 -29.03 -28.62 -21.92 -1.77 -1.52 0.00 11.00 0.00 -1.77 -31.68 -35.61 -37.11 -37.73 -37.69 -37.16 -28.46 -2.30 -1.97 0.00 9.00 0.00 -2.20 -39.43 -44.32 -46.19 -46.96 -46.91 -46.25 -35.42 -2.87 -2.46 0.00 8.00 0.00 -2.43 -43.42 -48.82 -50.88 -51.72 -51.66 -50.94 -39.02 -3.16 -2.70 0.00 7.00 0.00 -2.66 -47.49 -53.38 -55.64 -56.56 -56.50 -55.71 -42.67 -3.45 -2.96 0.00 6.00 0.00 -2.89 -51.60 -58.01 -60.46 -61.46 -61.39 -60.53 -46.36 -3.75 -3.21 0.00 5.00 0.00 -3.12 -55.76 -62.68 -65.33 -66.41 -66.34 -65.41 -50.10 -4.06 -3.47 0.00 4.00 0.00 -3.35 -59.94 -67.38 -70.22 -71.39 -71.31 -70.31 -53.85 -4.36 -3.73 0.00 3.00 0.00 -3.59 -64.14 -72.10 -75.14 -76.39 -76.31 -75.24 -57.63 -4.67 -3.99 0.00 2.00 0.00 -3.82 -68.35 -76.83 -80.08 -81.40 -81.31 -80.18 -61.41 -4.97 -4.26 0.00 1.00 0.00 -4.06 -72.56 -81.57 -85.01 -86.42 -86.33 -85.12 -65.20 -5.28 -4.52 0.00 0.00 0.00 -4.29 -76.77 -86.31 -89.95 -91.44 -91.34 -90.07 -68.98 -5.59 -4.78 0.00 Mn: Partial Palmgren-Miner Sum for Mode 1 (Eqn C4.7)

Partial Safety Factor = 1


----- ---- ---- ----- ----- ----- ----- ----- ----- ----- ---- ---- ---- 25.00 0 0 0 0 0 0 0 0 0 0 0 0 23.00 0 0 0 0 0 0 0 0 0 0 0 0 21.00 0 0 0 0 0 0 0 0 0 0 0 0 19.00 0 0 3 6 7 8 8 7 1 0 0 0 17.00 0 0 41 65 76 80 80 76 27 0 0 0 15.00 0 0 173 253 288 303 302 289 121 0 0 0 13.00 0 0 473 677 768 807 805 771 340 0 0 0 11.00 0 0 1048 1493 1691 1777 1771 1697 758 0 0 0 9.00 0 0 2028 2884 3266 3430 3420 3278 1470 0 0 0 8.00 0 0 2712 3855 4364 4585 4570 4381 1966 0 0 0 7.00 0 0 3548 5043 5709 5997 5978 5730 2572 0 0 0 6.00 0 0 4554 6471 7326 7696 7671 7354 3302 0 0 0 5.00 0 0 5745 8164 9242 9708 9678 9277 4167 0 0 0 4.00 0 0 7138 10142 11481 12059 12021 11523 5177 0 0 0 3.00 0 0 8746 12425 14061 14767 14721 14113 6344 0 0 0 2.00 0 0 10584 15027 16996 17844 17788 17059 7677 0 0 0 1.00 0 0 12663 17955 20285 21283 21218 20359 9187 0 0 0 0.00 0 0 14992 21205 23911 25063 24988 23997 10883 1 0 0


Page 103 of 180

Chris Rosencutter

Image

Chris Rosencutter

Arrow

Chris Rosencutter

Text Box

Agrees well with MathCad calculations.

Section B-6 - MecaStack (Ovalling Ring Verification)


Page 104 of 180



File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\CICIND2010\ Ovaling\v5314_App5_Ovaling.Stk

INPUT PARAMETERS:




Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------25.0 0.71 | 25.0 6.3 | 25.0 0.0

| 8.0 6.3 | | 7.0 6.3 | | 6.0 6.3 | | 5.0 6.3 | | 4.0 6.3 | | 3.0 6.3 | | 2.0 6.3 | | 1.0 6.3 |

Materials:


---- -------------- ---- ------ -------- -------- ---------- ------- 25.0 S355 (FE 510A) 20.0 355.00 209,947 6.80E-06 122.73 7,849 20.0 355.00 209,947 6.80E-06 122.73 7,849

Stiffening Rings:

ID Country Shape Size Config Material-- -------------- ----- ----------------------------- ------ --------

A Not Applicable FLAT Depth: 50.0 mm X Thk: 12.0 mm A-36

Stiff Elevations ID (m)

----- ---------- A 24.8, 12.5

Baseplate:

Ds = Outer Diameter of Stack at Base = 710.0 mm Di = Inside Diameter of Baseplate = 456.0 mm Dbc = Bolt Circle Diameter = 837.0 mm Do = Outside Diameter of Baseplate = 964.0 mm Tb = Thickness of Bottom Plate = 25.0 mm Tt = Thickness of Top Plate = 25.0 mm Pw = Plate Washer Width = 114.0 mm Pl = Plate Washer Length = 127.0 mm Pt = Plate Washer Thickness = 19.0 mm Pdh = Plate Washer Bolt Hole = 53.0 mm Fw = Fillet Welds = 6.0 mm


Page 105 of 180

Db = Nominal Bolt Diameter = 50.0 mm Dbh = Diameter of Bolt Hole = 62.0 mm Nb = Number of Bolts = 8 Fu = Ultimate Bolt Tensile Stress = 399900.9 KPa CA = Total Corrosion Allowance for Bolts and Baseplate = 0.0 mm Ec = Modulus of Elasticity of Concrete = 20684250.0 KPa Fc = Allowable Compressive Stress of Concrete = 6894.8 KPa Fy = Yield Stress of Baseplate Material = 355.0 KPa Es = Modulus of Elasticity of Baseplate = 209947.0 KPa Dens = Density of Baseplate = 490 Kg/m^3 u = Poisons Ratio for Baseplate Material = 0.300 Ngus = Number of Gussets per Bolt = 2 hg = Height of Chair = 305.0 mm tg = Thickness of Gusset = 12.0 mm wg = Spacing of Gussets = 102.0 mm

Design Codes

Comprehensive Design Standard: CICIND 2010 'Model Code for Steel Chimneys'






Group = Stack Groups = Single Spacing = Center to center stack spacing (Only required for Group) = 0.0 m S = Manually entered Strouhal Number (Only used for Custom) = 0.2000 Full = Use full formulations per Commentary 3 = True

Fatigue Criteria: CICIND 2010 'Model Code for Steel Chimneys'



Deflection Criteria for Static Loading: MaxDefl = Limitation based upon Stack Ht: Stack_Ht / 133 = 7.4 mm

Deflection Criteria for Vortex Loading: DeflLimit = Limitation based upon a % of Top OD: Stack_Top_OD x 0.05 = 0.036 m

Section Properties



----- ----- -------- ----- -------- ------ ----------- ----------- -------- 25.00 24.80 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 24.80 21.80 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 21.80 18.80 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 18.80 15.80 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 15.80 12.80 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 12.80 12.50 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 12.50 9.50 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 9.50 8.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 8.00 7.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 7.00 6.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 6.00 5.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0 5.00 4.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0


Page 106 of 180

4.00 3.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0

3.00 2.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0

2.00 1.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0

1.00 0.31 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0

0.31 0.00 710.00 6.30 248.81 139.28 3,119.8 2,428.7 86,218.0

Weight Detail



Elev Elev Elev OD Thk Density Wt Lin Wt Wt

Top Bot COG Corroded UnCorroded UnCorroded

m m m m mm Kg/m^3 kg kg/m kg

----- ---- ----- ----- ---- ------- -------- ---------- ----------

25.00 8.00 16.50 0.710 6.30 7,849 1,858.4 109.32 1,858.4

8.00 7.00 7.50 0.710 6.30 7,849 109.3 109.32 109.3

7.00 6.00 6.50 0.710 6.30 7,849 109.3 109.32 109.3

6.00 5.00 5.50 0.710 6.30 7,849 109.3 109.32 109.3

5.00 4.00 4.50 0.710 6.30 7,849 109.3 109.32 109.3

4.00 3.00 3.50 0.710 6.30 7,849 109.3 109.32 109.3

3.00 2.00 2.50 0.710 6.30 7,849 109.3 109.32 109.3

2.00 1.00 1.50 0.710 6.30 7,849 109.3 109.32 109.3

1.00 0.31 0.65 0.710 6.30 7,849 76.0 109.32 76.0

0.31 0.00 0.15 0.710 6.30 7,849 33.3 109.32 33.3

----- ---- ----- ----- ---- ------- -------- ---------- ----------

Total 12.50 2,733.0 2,733.0

Stiffening Rings: (Wt = 24.0 kg, El COG = 18.65 m)

Description Elevation Weight

m kg

----------------------------- --------- ------

A: Flat Bar 50.0 mm X 12.0 mm 24.800 12.0

A: Flat Bar 50.0 mm X 12.0 mm 12.500 12.0

----------------------------- --------- ------

Total 18.650 24.0

Baseplate: (Wt = 28.7 kg, El COG = 0.139 m)

Description Elevation Weight

COG

m kg

------------ --------- ------

Bottom Plate 0.013 6.9

Top Plate 0.293 4.1

Gussets 0.153 3.0

Washers 0.153 14.6

------------ --------- ------

Baseplate 0.139 28.7

Weight Summary

Component Elev COG Weight

m kg

----------------------------------- -------- -------

Cylinders (Corroded Wt = 2733.0 kg) 12.500 2,733.0

Stiffening Rings 18.650 24.0

Baseplate 0.139 28.7

----------------------------------- -------- -------

Total 12.426 2,785.7

Frequency Summary


# 1 # 2 # 3

Deg Hz Hz Hz

----------------- --------- ----- ----- ------

Cold & Uncorroded 0.00 1.142 7.136 20.035


Page 107 of 180



(1.142 Hz) (7.136 Hz) (20.035 Hz)


--------- ---------- ---------- -----------

25.00 1.000 -1.000 1.000

24.80 0.989 -0.961 0.936

21.80 0.824 -0.387 0.011

18.80 0.660 0.137 -0.598

15.80 0.502 0.528 -0.587

12.80 0.353 0.717 -0.040

12.50 0.339 0.724 0.028

9.50 0.210 0.668 0.619

8.00 0.153 0.571 0.759

7.00 0.120 0.489 0.772

6.00 0.090 0.398 0.721

5.00 0.064 0.303 0.616

4.00 0.042 0.212 0.472

3.00 0.024 0.129 0.311

2.00 0.011 0.062 0.159

1.00 0.003 0.017 0.045

0.31 0.000 0.002 0.005

0.00 0.000 0.000 0.000












zb = Stack Base Elev above Grade (for Kz adjustment) = 0.0 m

ze = Stack Enclosed below this Elev (no wind <= He) = 0.0 m

Kt = Topographical Factor = 1.000

Cat = Terrain Category (Table 7.1) = III

a = Terrain Category Factor (Table 7.2) = 0.770

b = Terrain Category Factor (Table 7.2) = 0.220

c = Terrain Category Factor (Table 7.2) = 0.280

Beta = Terrain Category Factor (Table 7.4) = 0.370

Zo = Terrain Cateogry Value (Table 7.1) = 0.300

Zmin = Height below which the wind speed is the same (Table 7.1) = 8.00 m

Zs = 0.6 * h (Para 7.2.3.3.2) = 15.00 m

L(Zs) = 300*(Z/300)^Beta if Z less than Zmin then Z=Zmin (Eqn 7.21) = 99.02 m

Nd = 4.6*fl*d / L(Zs) (Eqn 7.23) = 0.141

Nh = 4.6*fl*h / L(Zs) (Eqn 7.23) = 4.970

Rd = 1/Nd - (1/(2*Nd))*(1-exp(-2*Nd)) (Eqn 7.22) = 0.912

Rh = 1/Nh - (1/(2*Nh))*(1-exp(-2*Nh)) (Eqn 7.22) = 0.181

Sl = Power Spectral Density: 6.8*fl/(1+10.2*fl)^(5/3) (Eqn 7.19) = 0.052

B^2 = Background Factor 1/(1+0.9*(b+h / L(Zs))^0.63) (Eqn 7.17) = 0.725

R^2 = Resonance Response Factor: (PI/(4*B))*Sl*Rh*Rd (Eqn 7.18) = 0.513

fl = f1*L(zs) / V(zs) (Eqn 7.20) = 4.279


vT = 600*f1 * (R^2 / (B^2 + R^2))^0.5 & vT >= 48 (Eqn 7.16) = 441.195

g = Peak Factor (Eqn 7.15) = 3.655

G = Gust Factor (Eqn 7.14) = 3.082

Ki = Interference Factor (Taken as 1 per Para 7.2.6) = 1.000

Ka = End Effect Factor (Eqn 7.12) = 1.000


Page 108 of 180

Re = Reynolds Number: 6.9*10^4 * Vtop * d (Eqn 7.11) = 1.45E6


z1 z1 K(z1) V(z1) Aw Wm Wg W Pe Force Force

m ft m/s m^2/m N/m N/m N/m kPa lb N

----- ----- ----- ----- ----- --- --- --- ----- ----- ------

25.00 82.02 0.942 29.58 0.497 266 691 958 1.927 43 192

24.80 81.36 0.940 29.53 0.497 265 686 951 1.914 642 2,854

21.80 71.52 0.914 28.70 0.497 251 603 854 1.718 576 2,561

18.80 61.68 0.885 27.78 0.497 235 520 755 1.519 509 2,265

15.80 51.84 0.852 26.74 0.497 218 437 655 1.317 441 1,964

12.80 41.99 0.813 25.53 0.497 198 354 552 1.111 37 166

12.50 41.01 0.809 25.39 0.497 196 346 542 1.091 366 1,626

9.50 31.17 0.761 23.91 0.497 174 263 437 0.879 147 655

8.00 26.25 0.733 23.02 0.497 161 221 383 0.770 86 383

7.00 22.97 0.733 23.02 0.497 161 194 355 0.714 80 355

6.00 19.69 0.733 23.02 0.497 161 166 327 0.659 74 327

5.00 16.40 0.733 23.02 0.497 161 138 300 0.603 67 300

4.00 13.12 0.733 23.02 0.497 161 111 272 0.547 61 272

3.00 9.84 0.733 23.02 0.497 161 83 244 0.492 55 244

2.00 6.56 0.733 23.02 0.497 161 55 217 0.436 49 217

1.00 3.28 0.733 23.02 0.497 161 28 189 0.380 30 131

0.31 1.00 0.733 23.02 1.928 626 8 634 0.329 43 193

----- ----- ----- ----- ----- --- --- --- ----- ----- ------

Total 3,306 14,704

z1 = Elevation above grade: z + zb

K(z1) = Height Factor: a*(z1/10)^b & if z1

V(z1) = 10 Min mean design wind speed @ elev z1: K(z1)*Kt*Vb

Aw = Total effective wind area, refer to 'Wind Area Summary' for details

Area includes shape factors. Units are Area per unit length.

Wm = 10 Min mean wind load per unit height: 0.5* pa * V(Z1)^2 * Aw (Eqn 7.6)

Wg = Wind Gust: (3*(G-1)/h)*(z/h)*{Integral of [wm*z dz] from 0 to h} (Eqn 7.13)

W = Design wind load per unit height: Wm + Wg (Eqn 7.5)

Pe = Equiv Press (Used elsewhere in analysis): W / Aw



Elev Elev Load Load

m m N/m N

----- ----- ------- ------

25.00 24.80 958 192

24.80 21.80 951 2,854

21.80 18.80 854 2,561

18.80 15.80 755 2,265

15.80 12.80 655 1,964

12.80 12.50 552 166

12.50 9.50 542 1,626

9.50 8.00 437 655

8.00 7.00 383 383

7.00 6.00 355 355

6.00 5.00 327 327

5.00 4.00 300 300

4.00 3.00 272 272

3.00 2.00 244 244

2.00 1.00 217 217

1.00 0.31 189 131

0.31 0.00 634 193

----- ----- ------- ------

Total 14,704

Riser Stiffening Ring Property Summary

All stiffening rings as well as other components that behave as stiffening rings

Ring Elev Stiff Eff Shell Shell Ls Area Mom of Section Xc

Num Ring Width Thickness Inertia Modulus

m mm mm m sq cm cm^4 cm^3 mm

---- ----- ------------- --------- --------- ----- ------ --------- --------- ------

1 24.80 A 66.88 6.30 6.35 10.21 32.3 8.8 13.39

2 12.50 A 66.88 6.30 12.25 10.21 32.3 8.8 13.39

3 0.31 Baseplate Top 231.68 6.30 6.25 46.35 871.4 103.1 42.51

4 0.00 Baseplate Bot 115.84 6.30 0.15 70.80 4,523.0 322.1 113.58


Page 109 of 180

Properties are for the combined stiffener and effective shell Ls = Ring spacing, 1/2 of shell on each side of ring. Xc = Centroid of combined section from OD of stack shell

Stiffening Ring: [1, 2] A

El 24.8, 12.5 m

Description Dx Dy Xc Yc Area Iy_Loc Iy_Tot mm mm mm mm sq cm cm^4 cm^4

----------- ----- ----- ----- ------ ----- ------ ------ Flat_Bar 50.00 12.00 25.00 0.00 6.00 12.5 20.6 Shell Above 6.30 33.44 -3.15 16.72 2.11 0.1 5.8 Shell Below 6.30 33.44 -3.15 -16.72 2.11 0.1 5.8

----------- ----- ----- ----- ------ ----- ------ ------ Total 13.39 10.21 32.3 X & Y axis are local used for this calculation & may not correspond to global X & Y. Dx = Dimension of the rectangle in X Dir, Dy = Dimension of Rectangle in Y Dir. Xc = Centroid of Rect in X Dir, Yc = Centroid of Rect in Y Dir Ix_Tot = Ix_Loc + Area*(Yc-Yc_Tot)^2, Iy_Tot = Iy_Loc + Area*(Xc-Xc_Tot)^2 Sy_Tot = 8.8 cm^3 Stiffening Ring: [3] Baseplate Top

El 0.305 m


----------- ------ ------ ----- ------ ----- ------ ------ Flat_Bar 127.00 25.00 63.50 0.00 31.75 426.7 566.6 Shell Above 6.30 115.84 -3.15 57.92 7.30 0.2 152.4 Shell Below 6.30 115.84 -3.15 -57.92 7.30 0.2 152.4

----------- ------ ------ ----- ------ ----- ------ ------ Total 42.51 46.35 871.4 X & Y axis are local used for this calculation & may not correspond to global X & Y. Dx = Dimension of the rectangle in X Dir, Dy = Dimension of Rectangle in Y Dir. Xc = Centroid of Rect in X Dir, Yc = Centroid of Rect in Y Dir Ix_Tot = Ix_Loc + Area*(Yc-Yc_Tot)^2, Iy_Tot = Iy_Loc + Area*(Xc-Xc_Tot)^2 Sy_Tot = 103.1 cm^3 Stiffening Ring: [4] Baseplate Bot

El 0.0 m


----------- ------ ------ ------ ----- ----- ------- ------- Flat_Bar 254.00 25.00 127.00 0.00 63.50 3,414.0 3,528.3 Shell Above 6.30 115.84 -3.15 57.92 7.30 0.2 994.7 Shell Below 6.30 0.00 -3.15 0.00 0.00 0.0 0.0

----------- ------ ------ ------ ----- ----- ------- ------- Total 113.58 5.97 70.80 4,523.0 X & Y axis are local used for this calculation & may not correspond to global X & Y. Dx = Dimension of the rectangle in X Dir, Dy = Dimension of Rectangle in Y Dir. Xc = Centroid of Rect in X Dir, Yc = Centroid of Rect in Y Dir Ix_Tot = Ix_Loc + Area*(Yc-Yc_Tot)^2, Iy_Tot = Iy_Loc + Area*(Xc-Xc_Tot)^2 Sy_Tot = 322.1 cm^3 SUMMARY OF STATIC OVALING CHECK OF RING SPACING & Ir

Are Stiff Rings are Adequate to consider riser 'Stiffened' per CICIND 2010 Para 7.2.5.2

Ring Elev br d w5 Ls LsMax Iact Ireqd Unity Unity Num Ls/LsMax Ireqd/Iact m mm m kPa m m cm^4 cm^4

---- ------ ------ ----- ----- ------ ----- ------- ----- -------- ---------- 1 24.800 50.00 0.710 1.051 6.350 4.221 32.3 1.2 1.50 0.04 2 12.500 50.00 0.710 1.047 12.248 4.221 32.3 0.9 2.90 0.03 3 0.305 127.00 0.710 0.989 6.250 4.221 871.4 1.1 1.48 0.00 4 0.000 254.00 0.710 0.927 0.153 4.221 4,523.0 1.6 0.04 0.00 br = Width of StiffeningRing d = Outer Diameter of Stack Ls = 1/2 of dist each side of stiff LsMax = Max allowed stiff spacing 0.56*(d/t)^0.5 Iact = Act Mom of Inertia of Ring+Shell Ireqd = Req'd Mom of Inertia of Ring + Shell w5 = Wind Press: 0.5*Rho*[1.4*V(z)]^2 ** The stiffening and/or spacing is inadequate, please revise ** SUMMARY OF STATIC OVALING CHECK OF STIFFENING RING STRESS

Determine if Stiffening Ring Stress is Acceptable per CICIND 2010 Para 7.2.5.1


Page 110 of 180

Ring Elev br d w5 Sact Mr fb fk Unity

Num

m mm m kPa cm^3 N-m MPa MPa

---- ------ ------ ----- ----- ----- ----- ----- ------ -----

1 24.800 50.00 0.710 1.051 8.8 153.3 17.40 322.95 0.08

2 12.500 50.00 0.710 1.047 8.8 152.7 17.34 322.95 0.08

3 0.305 127.00 0.710 0.989 103.1 210.3 2.04 322.95 0.01

4 0.000 254.00 0.710 0.927 322.1 299.0 0.93 322.95 0.00

Mr = Bend Moment in ring per Eqn C3.5.10 fb = Bend Stress: Mr / Sact

fk = Fyld / 1.1 Gamma= Partial Safety Factor 1.4

Unity = Gamma*fb / fk For d, br and w5 see previous section

** Since one or more failures, this stack is not considered to be 'Stiffened' **

** The ovaling stresses 'fo' will be calculated and included in the stack analysis **

CALCULATION OF STATIC OVALLING STRESSES IN SHELLCheck Unstiffened Shell Ovalling Stresses per CICIND 2010 Para 7.2.5.1

Elev d t V5sec w5sec Iact Sact M fo fk Unity

fo/fk

m m mm m/s kPa mm^3 sq mm N MPa MPa

----- ----- ---- ------- ------- ------ ------ --- ---- ------ -----

25.00 0.710 6.30 41.41 1.051 20.8 6.62 42 6.41 322.73 0.02

24.80 0.710 6.30 41.34 1.047 20.8 6.62 42 6.39 322.73 0.02

21.80 0.710 6.30 40.18 0.989 20.8 6.62 40 6.03 322.73 0.02

18.80 0.710 6.30 38.89 0.927 20.8 6.62 37 5.65 322.73 0.02

15.80 0.710 6.30 37.43 0.859 20.8 6.62 35 5.24 322.73 0.02

12.80 0.710 6.30 35.74 0.783 20.8 6.62 32 4.77 322.73 0.01

12.50 0.710 6.30 35.55 0.774 20.8 6.62 31 4.72 322.73 0.01

9.50 0.710 6.30 33.47 0.686 20.8 6.62 28 4.19 322.73 0.01

8.00 0.710 6.30 32.23 0.636 20.8 6.62 26 3.88 322.73 0.01

7.00 0.710 6.30 32.23 0.636 20.8 6.62 26 3.88 322.73 0.01

6.00 0.710 6.30 32.23 0.636 20.8 6.62 26 3.88 322.73 0.01

5.00 0.710 6.30 32.23 0.636 20.8 6.62 26 3.88 322.73 0.01

4.00 0.710 6.30 32.23 0.636 20.8 6.62 26 3.88 322.73 0.01

3.00 0.710 6.30 32.23 0.636 20.8 6.62 26 3.88 322.73 0.01

2.00 0.710 6.30 32.23 0.636 20.8 6.62 26 3.88 322.73 0.01

1.00 0.710 6.30 32.23 0.636 20.8 6.62 26 3.88 322.73 0.01

0.31 0.710 6.30 32.23 0.636 20.8 6.62 26 3.88 322.73 0.01

Iact = Unit Shell Mom of Inertia: t^3/12 Sact = Unit Shell Section Moduls: t^2 / 6

V5sec = Velocity for 5 sec gust: 1.4*V(z) fk1 = Fyld / 1.1 (Eqn 6.1)

w5sec = Wind Press: 0.5*Rho*[V5sec]^2 M = Unit Moment on Shell: 0.08*w5sec*d^2

fo = Ovaling Stress: M / Sact, which will be used in the stack biaxial stress analysis

==> The stack was checked as an Unstiffened stack And found to be acceptable. <==

==> Any failures in the Ovalling Static Stiffening Ring check can be disregarded <==

Verify that Beam Theory can be Used per Para 8.2For unstiffened chimney need L/R > 50 to use Beam Theory

L = Overall height of Chimney = 25.0 m

R = Radius of Chimney at base = 0.355 m

L/R = Ratio of Height to Radius for Chimeny = 70.42

Since L/R > 50 & Stack is Stiffened then Beam Theory can Be used

SUMMARY OF OVALING DYNAMIC CHECK WIND SPEED TO DETERMINE IF OVALING IS POSSIBLEIf V/Vr > 1 then Vortex shedding is Possible per CICIND 2010 Para 7.2.5.2

Elev d t E Rho fo Vr V Unity

V/Vr

m m mm MPa Kg/m^3 Hz m/s m/s

----- ----- ---- ---------- ------ ------ ----- ----- -----

25.00 0.710 6.30 209,947.00 7,849 32.318 57.36 29.58 0.52

24.80 0.710 6.30 209,947.00 7,849 32.318 57.36 29.53 0.51

21.80 0.710 6.30 209,947.00 7,849 32.318 57.36 28.70 0.50

18.80 0.710 6.30 209,947.00 7,849 32.318 57.36 27.78 0.48

15.80 0.710 6.30 209,947.00 7,849 32.318 57.36 26.74 0.47

12.80 0.710 6.30 209,947.00 7,849 32.318 57.36 25.53 0.45

12.50 0.710 6.30 209,947.00 7,849 32.318 57.36 25.39 0.44

9.50 0.710 6.30 209,947.00 7,849 32.318 57.36 23.91 0.42

8.00 0.710 6.30 209,947.00 7,849 32.318 57.36 23.02 0.40

7.00 0.710 6.30 209,947.00 7,849 32.318 57.36 23.02 0.40

6.00 0.710 6.30 209,947.00 7,849 32.318 57.36 23.02 0.40

5.00 0.710 6.30 209,947.00 7,849 32.318 57.36 23.02 0.40

4.00 0.710 6.30 209,947.00 7,849 32.318 57.36 23.02 0.40

3.00 0.710 6.30 209,947.00 7,849 32.318 57.36 23.02 0.40


Page 111 of 180

2.00 0.710 6.30 209,947.00 7,849 32.318 57.36 23.02 0.40

1.00 0.710 6.30 209,947.00 7,849 32.318 57.36 23.02 0.40

0.31 0.710 6.30 209,947.00 7,849 32.318 57.36 23.02 0.40

d = Outer diameter of shell t = Corroded thickness of shell

E = Modulus of Elasticity Rho = Density of Material

fo = Ovalling Freq Eqn 7.46 Vr = Critical Wind Speed: fo*d /(2*St)

V = Wind Speed at Elevation Unity = If < 1 then no chance of Ovalling Vibration

** PASS, Vr > V for all Elevations, no potential for ovalling vibration. **

PERMISSIBLE STRESS SUMMARY FOR STACK PER CICIND 2010 MODEL CODE FOR STEEL CHIMNEYSValues for each section of stack needed for Stability And Carrying Capacity checks

Bot El r t r/t Fyld AlphaN AlphaB Scr fk

m mm mm MPa MPa MPa

------ ------ ---- ------ ------ ------ ------ -------- ------

24.80 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

21.80 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

18.80 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

15.80 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

12.80 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

12.50 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

9.50 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

8.00 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

7.00 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

6.00 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

5.00 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

4.00 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

3.00 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

2.00 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

1.00 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

0.31 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

0.00 348.70 6.30 55.349 355.00 0.666 0.729 2,294.85 355.00

1. r = Radius of Cylindrical Shell t = Corroded Thkness of Stack Shell

2. Fyld = Yield Stress based on 't' (Table 6.1) Scr = Crit Elas Buckling: .605*E*t/r

3. AlphaN= Equations 8.9a and 8.9b AlphaB= Equation 8.9b

4. fk = Limit Stress: Fyld1 (Material Factor applied in each load case)


Page 112 of 180

Section C - BS 6399 Wind + BS 4076


Page 113 of 180

Wind Loads Per BS 6399 Part 2 - 1997


≔Vb 30 ― ≔Sa 1 ≔Sd 1 ≔Ss 1 ≔Sp 1 ≔Terrain ≔Fac_BS4076 1

≔h 25 ≔b 1.25 ≔t1 4 ≔ρa 1.25 ――3

≔ξs 0.002 ≔n 2.039

≔dz Vfill (( ,14 b)) ≔tz Vfill ⎛⎝ ,14 t1⎞⎠

≔z

2523211917151311975310

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=dz

1.251.251.251.251.251.251.251.251.251.251.251.251.251.25

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=tz

44444444444444

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔ϕ

1.000 1.0000.890 0.6170.780 0.2440.671 −0.0990.564 −0.3860.461 −0.5920.363 −0.7040.272 −0.7150.190 −0.6340.120 −0.4840.064 −0.3010.024 −0.1280.003 −0.016

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔mz =mcyl⎛⎝ ,,z dz tz⎞⎠

122.65122.65122.65122.65122.65122.65122.65122.65122.65122.65122.65122.65122.65

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

――

≔Vs =⋅⋅⋅⋅⋅Fac_BS4076 Vb Sa Sd Ss Sp 30 ―

≔H

025

1015203050

100

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Sb

1.481.481.651.781.851.901.962.042.12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔_Sb((h)) linterp (( ,,H Sb h)) ≔_Ve

((z)) ⋅Vs _Sb((z))

≔Sb =_Sb((h)) 1.93 ≔_qs ((z)) ⋅0.613 ――

3_Ve

((z))2

≔gt 3.44 (Gust Peak factor looked up from Table 24)

≔Ca 0.909 (with a = 25 m and class A, look up from Fig 4)

≔Tt 1 (Table 22, Lookup based upon h and 0 Km from sea)

≔St 0.1235 (Table 23, Lookup based upon h and 0 Km from sea)

≔Sc 1.355

≔Sh 0 (Ref 3.2.3.4.2, no topography so use 0)

≔Sg =+1 ⋅gt St 1.42 (Eqn C.1 for country terrain)

≔Kt 1.33 (Terrain correction factor at sea)

≔So =⋅Sc⎛⎝ +1 Sh

⎞⎠ 1.36

≔a =‾‾‾‾‾‾‾‾‾‾‾

+(( ⋅0.5 b))2

h 2 25.01

≔KhKb =⋅⋅⋅Kt

⎛⎜⎜⎝―――

⋅20 So

⋅n2

a

⎞⎟⎟⎠

―2

3⎛⎜⎜⎝―――Vs

⋅24 ξs

⎞⎟⎟⎠

⎛⎜⎝ ⋅

――−1

3――−1

3

⎞⎟⎠ 339.19

≔CT =⋅⎛⎜⎜⎝

−1 ――1

Sg2

⎞⎟⎟⎠

⎛⎜⎜⎝

−‾‾‾‾‾‾‾‾

+1 ――KhKb

601

⎞⎟⎟⎠

0.8

‖ |


Page 114 of 180

≔DynFac =‖‖‖‖‖‖‖‖

|||||||

|||||

|

if

else

≤≤0 CT 0.25‖‖‖

⎛⎝ +1 CT⎞⎠

‖‖ ((1))

1

≔Hd 0 2.5 10 99999[[ ]]

≔θ

0102030405060708090

100120140160180

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Table7

1 1 1 10.9 0.9 0.9 0.90.7 0.7 0.7 0.7

0.35 0.35 0.35 0.350 0 0 0

−0.5 −0.5 −0.7 −0.7−1.05 −1.05 −1.2 −1.2−1.25 −1.25 −1.4 −1.4−1.3 −1.3 −1.45 −1.4−1.2 −1.2 −1.4 −1.4

−0.85 −0.85 −1.1 −1.1−0.4 −0.4 −0.6 −0.6

−0.25 −0.25 −0.35 −0.35−0.25 −0.25 −0.35 −0.35−0.25 −0.25 −0.35 −0.35

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔_Cpe (( ,Ang Hd_ratio)) DoubleInterp ⎛⎝ ,,,,θ THd Table7 Ang Hd_ratio⎞⎠

≔_Cpe_tot(( ,H D))

‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖

|||||||||||||||||||||

|

←rows rows ((θ))for ∊ |

|||||||||||||||

j , ‥0 1 −rows 2‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖

←θavg ⋅0.5 ⎛⎝

+θj

θ+j 1

⎞⎠

←θdif ⎛⎝

−θ+j 1

θj⎞⎠

←Cj

_Cpe

⎛⎜⎝

,θj

―H

D

⎞⎟⎠

←Arcj

――――⋅⋅D θdif

360←Cseg

j⋅⋅cos ⎛⎝ ⋅θavg 1 ⎞⎠ C

jArc

j

←Ctot +Ctot Csegj

←Ctot ―――⋅2 Ctot

D⎛⎝Ctot

⎞⎠

≔Cpe =_Cpe_tot(( ,h b)) 0.756


Page 115 of 180

≔Sb =_Sb((z))

1.931.921.911.891.871.851.821.791.751.71.651.541.481.48

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ve =_Ve((z))

57.957.5457.1856.756.155.554.6653.8252.6251.0649.546.144.444.4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― ≔qs =_qs ((z))

20552030200419711929188818311776169715981502130312081208

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔_Funi ((Z))‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((Z)) 2‖‖‖

←Cj

⋅⋅⋅⋅⋅qsjdz

j_Cpe_tot

⎛⎜⎝

,h dzj⎞⎟⎠

Ca DynFac ⎛⎝

−zj

z+j 1

⎞⎠

((C))

≔Cpe1 ((Z))‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((Z)) 2‖‖‖

←Cj

_Cpe_tot⎛⎜⎝

,zjdz

j⎞⎟⎠

((C))

≔Funi =_Funi ((z))

3530348634433385331432433146305029152745258022381038

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Vsum ⎛⎝Funi⎞⎠ 38112 =Cpe1 ((z))

0.7560.7560.7560.7560.7560.7560.7560.7390.7180.6960.6740.6540.654

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Vortex Shedding Analysis per BS 4076


Page 116 of 180


≔K 3.5 ≔In_Town “True” ≔Dt =b 1.25

≔S2 1.019 ≔V 45.34 ―

≔w =⋅―――――――⋅

⎛⎝ −b

2 ⎛⎝ −b ⋅2 t1⎞⎠2 ⎞

⎠

4490 ――

3122.9 ――

≔ρ 1.25 ――m

3≔∆ 0.0094

Equivalent BS CP 3 Ch V wind:

≔S1 1 ≔S3 1 ≔Vs =⋅⋅⋅V S1 S2 S3 46.2 ―

≔Vcrit =⋅⋅5 Dt n 12.74 ―

≔C =+0.6 ⋅K

⎛⎜⎜⎜⎜⎜⎜⎝

+――――

⋅10⎛⎜⎝――Dt

1

⎞⎟⎠

2

⎛⎜⎜⎜⎝

――w

1 ――

⎞⎟⎟⎟⎠

―――⋅1.5 ∆

Dt

⎞⎟⎟⎟⎟⎟⎟⎠

1.08

≔Result =‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||

|

||||||||

|

if

else if

else

<C 1‖‖ ←res “Oscillations Unlikely”

≥C 1.3‖‖ ←res “Use Strakes or Damper”

‖‖ ←res “Multiply wind loads by C”

((res))

“Multiply wind loads by C”

Allowable Stress per BS 4076


Page 117 of 180


≔D =⎛⎝ −b t1⎞⎠ 1.246 ≔t =t1 0.004 ≔he 50 ≔Fy 354.94

=dz

1.251.251.251.251.251.251.251.251.251.251.251.251.251.25

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Dz =−dz tz

1.2461.2461.2461.2461.2461.2461.2461.2461.2461.2461.2461.2461.2461.246

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ratio =‖‖‖‖‖‖‖‖‖

|||||||

|

for ∊ |||||

|

j , ‥0 1 −rows ⎛⎝Dz⎞⎠ 1

‖‖‖‖‖‖

←rj

――

Dzj

tzj

((r))

311.5311.5311.5311.5311.5311.5311.5311.5311.5311.5311.5311.5311.5311.5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔A =‖‖‖‖‖‖‖‖‖‖‖

||||||||||

for ∊ ||||||||

j , ‥0 1 −rows ⎛⎝Dz⎞⎠ 1

‖‖‖‖‖‖‖

←Aj

――――――――1

⎛⎜⎜⎜⎝

+0.84⎛⎜⎜⎝

⋅0.019 ――he

Dzj

⎞⎟⎟⎠

2 ⎞⎟⎟⎟⎠

((A))

0.7040.7040.7040.7040.7040.7040.7040.7040.7040.7040.7040.7040.7040.704

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔B =‖‖‖‖‖‖‖‖

||||||

|

for ∊ ||||

|

j , ‥0 1 −rows ⎛⎝Dz⎞⎠ 1

‖‖‖‖‖

←Bj

⋅⋅270⎛⎜⎜⎝

―――1

Ratioj

⎞⎟⎟⎠

⎛⎜⎜⎝

−1 ⋅67⎛⎜⎜⎝

―――1

Ratioj

⎞⎟⎟⎠

⎞⎟⎟⎠

((B))

0.680.680.680.680.680.680.680.680.680.680.680.680.680.68

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Allow =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ⎛⎝Dz⎞⎠ 1

‖‖‖

←Alj

⋅⋅⋅0.5 Fy AjB

j

((Al))

84.9584.9584.9584.9584.9584.9584.9584.9584.9584.9584.9584.9584.9584.95

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 118 of 180



File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\BS6399\ v5314_BS6399_25m.Stk

INPUT PARAMETERS:




Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------25.0 1.25 | 25.0 4.0 | 25.0 0.0

Materials:


---- -------------- ---- ------ -------- -------- ---------- ------- 25.0 S355 (FE 510A) 20.0 354.94 209,947 1.15E-05 122.73 7,849 20.0 354.94 209,947 1.15E-05 122.73 7,849

Design Codes


Stress Criteria: British Standard BS 4076: 1989 'Specification for Steel Chimneys'

Wind Load Criteria: British BS 6399 'Loading for Buildings'

Vb = Basic Wind Speed = 30.00 m/s Ds = Attitude above Sea level (Ref Para 2.2.2.2) = 0.0 m Se = Effective slope of Topographic feature (Para 2.2.2.2.3) = 0.000 s = Topographic location factor (Ref Para 2.2.2.3) = 0.000 Ss = Seasonal Factor (Ref 2.2.2.4) = 0.000 Sp = Probability Factor (Ref 2.2.2.5) = 1.000 Kt = Terrain correction factor (Country=1.33,Town=0.75) = 1.330 DistToSea = Closest distance to the sea referenced in Eqn C.3 = 0.0000 Km Terrain = Terrain referenced in Table 4 (A = Country, B = Town) = A

Vortex Shedding Criteria: British Standard BS 4076: 1989 'Specification for Steel Chimneys'

K = Factor based upon type Of construction = 3.500 In_Town = Is Stack Located In Town? = True

Fatigue Criteria: Fatigue None


Section Properties



Page 119 of 180


----- ----- -------- ----- -------- ------ ----------- ----------- --------- 25.00 23.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 23.00 21.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 21.00 19.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 19.00 17.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 17.00 15.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 15.00 13.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 13.00 11.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 11.00 9.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 9.00 7.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 7.00 5.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 5.00 3.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 3.00 1.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 1.00 0.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5

Weight Detail




----- ---- ----- ----- ---- ------- -------- ---------- ---------- 25.00 0.00 12.50 1.250 4.00 7,849 3,072.4 122.90 3,072.4

----- ---- ----- ----- ---- ------- -------- ---------- ---------- Total 12.50 3,072.4 3,072.4

Weight Summary



----------------------------------- -------- ------- Total 12.500 3,072.4

Wind Areas

Wind Area Summary


----- --------- ----- ------ -------- ------ ------- ------ 25.00 0.756 0.945 0.000 0.000 0.000 0.945 1.890 23.00 0.756 0.945 0.000 0.000 0.000 0.945 1.890 21.00 0.756 0.945 0.000 0.000 0.000 0.945 1.890 19.00 0.756 0.945 0.000 0.000 0.000 0.945 1.890 17.00 0.756 0.945 0.000 0.000 0.000 0.945 1.890 15.00 0.756 0.945 0.000 0.000 0.000 0.945 1.890 13.00 0.756 0.945 0.000 0.000 0.000 0.945 1.890 11.00 0.739 0.924 0.000 0.000 0.000 0.924 1.849 9.00 0.718 0.897 0.000 0.000 0.000 0.897 1.794 7.00 0.696 0.870 0.000 0.000 0.000 0.870 1.740 5.00 0.674 0.843 0.000 0.000 0.000 0.843 1.685 3.00 0.654 0.817 0.000 0.000 0.000 0.817 1.634 1.00 0.654 0.817 0.000 0.000 0.000 0.817 0.817

----- --------- ----- ------ -------- ------ ------- ------ Total 22.746

Frequency Summary


Page 120 of 180

Description Direction Mode Mode Mode # 1 # 2 # 3 Deg Hz Hz Hz

----------------- --------- ----- ------ ------ Cold & Uncorroded 0.00 2.039 12.725 35.391

Wind Conversion from Code_Wind_BS6399 to Code_Wind_BSCP3A Code is being utilized in this analysis which is based upon Code_Wind_BSCP3

Vortex Shedding Wind Conversion Vortex shedding analysis has a different wind basis than the wind criteria selected so convert the wind into a basis which is consistent with the Vortex Shedding Criteria Convert Wind Speed Basis per ASCE 7-10 Figure 26.5-1

Description Code Velocity Gust Recurrence m/s Sec Years

-------------- ------------------ -------- ---- ---------- Original Wind BS 6399 Pt 2 30.00 3600 50.0 Converted Wind BS CP 3 Ch V Pt II 45.34 3 50.0

Terrain = BS 6399 'Site in Town' assumed equivalent to BS CP3 Terrain 1 = 1 S1 = Assumed to be equal to BS 6399 value for Sa = 1.000 S2 = Ground Roughness Factor per BS CP 3 Table 3 = 1.019 S3 = Assumed to be equal to BS 6399 value for Sp = 1.000 Vs = Design Wind Velocity: V * S1 * S2 * S3 = 46.21 m/s

Vortex Shedding Analysis per BS 4076 : 1989Ref Appendx B 'Wind-excited Oscillations'

K = Factor based upon type Of construction = 3.500 In_Town = Is Stack Located In Town? = True W = Average uniform mass Of top 1/3 Of stack = 122.90 kg/m Rho = Air density = 1.250 Kg/m^3 Dt = Average OD Of top 1/3 Of stack = 1.25 m f = Natural Frequency Of Mode # 1 = 2.039 Hz Vcrit = Critical Strouhal Velocity: 5 * Dt * f {Para B.3.1} = 12.75 m/s Vcrit = Critical Strouhal Velocity: 5 * Dt * f {Para B.3.1} = 28.5 mph Vs = Design Wind Velocity: V * S1 * S2 * S3 = 46.21 m/s

Vcrit <= Vs, the tendency to oscillate may be estimated by C

Defl = Deflection resulting from 1 KN/m^2 applied to the stack = 0.094 m C = 0.6 + K * (10 * Dt^2 / W + 1.5 * Defl / Dt) {Para B.3.3 = 1.438

Since C > 1.3 And the stack Is in an Urban area Add A Damper Or Helical Strakes

--> Since the stack Is Not fitted with a Damper Or Strakes the Vortex Shedding Is Failing

BS 6399: Part 2-97 Wind Loads - Standard MethodPressures calculated based upon lowest natural frequency

Vb = Basic Wind Speed = 30.00 m/s Ds = Attitude above Sea level (Ref Para 2.2.2.2) = 0.0 m Se = Effective slope of Topographic feature (Para 2.2.2.2.3) = 0.000 s = Topographic location factor (Ref Para 2.2.2.3) = 0.000 Ss = Seasonal Factor (Ref 2.2.2.4) = 0.000 Sp = Probability Factor (Ref 2.2.2.5) = 1.000 Kt = Terrain correction factor (Country=1.33,Town=0.75) = 1.330 DistToSea = Closest distance to the sea referenced in Eqn C.3 = 0.0000 Km Terrain = Terrain referenced in Table 4 (A = Country, B = Town) = A

He = Effective Height of structure = 25.0 Hz Bs = Structural Damping = 0.0020 NF = Natural Frequency = 2.039 Hz Sa1 = 1 + 0.001 * Ds {Eqn 9} = 1.000 Sa2 = 1 + 0.001 * Dt + 1.2 * Se * s {Eqn 10} = 1.000 Sa = Greater of Sa1 or Sa2 = 1.000 Ss = Seasonal Factor (See 2.2.2.4) = 1.000


Page 121 of 180

Sp = Probability Factor (See 2.2.2.5) = 1.000

Vs = Vb * Sa * Sd * Ss * Sp {Eqn 8} = 30.00 m/s

Sc = Fetch factor per Table 22 = 1.355

St = Turbulence factor per Table 22 = 0.124

Tc = Fetch adjustment factor per Table 23 = 0.695

Tt = Turbulence factor per Table 23 = 1.920

Sh = Topographic Increment (per Table 25) = 0.000

gt = Gust peak factor (a = Stack Ht) = 3.440

Sg = Gust Factor (1+gt*St) [Annex C.1.2] = 1.425

So = Sc * (1 + Sh) = 1.355

a = Diagonal dimension for entier stack height = 25.03 m

Ca = Size effect factor from Table 4 = 0.909

KhxKb = (Kt*(20*So/(NF^2)*a)^0.667)*(Vs / (24*Bs)) = 338.91

Ct = Dynamic Augmentation Factor {Eqn C.1} = 0.801

The Ct value has exceeded 0.25 or h > 300m, so the structure is

more than mildly dynamic, and simply applying the factor (1+Ct)

is overly conservative and less accurate. BS6399 Para C1.3

and Fig 1 recommends following references [1] through [4] for

dynamic structures. No factors will be applied to the loads in

MecaStack, if a factor is to be applied please include that factor

by manually adjusting the load factor in the load combinations.

DynFac = Struc is dynamic, more rigorous calc's needed, use 1.0 = 1.000

Elev Elev Sb Ve qs Cpe Area Force Force Force

Uniform Total Total

m ft m/s kPa m^2/m N/m lb N

----- ----- ----- ----- ----- ----- ----- ------- ----- ------

25.00 82.02 1.930 57.90 2.055 0.756 0.945 1,764 793 3,528

23.00 75.46 1.918 57.54 2.030 0.756 0.945 1,742 783 3,485

21.00 68.90 1.906 57.18 2.004 0.756 0.945 1,721 774 3,441

19.00 62.34 1.890 56.70 1.971 0.756 0.945 1,692 761 3,384

17.00 55.77 1.870 56.10 1.929 0.756 0.945 1,656 745 3,312

15.00 49.21 1.850 55.50 1.888 0.756 0.945 1,621 729 3,242

13.00 42.65 1.822 54.66 1.831 0.756 0.945 1,572 707 3,145

11.00 36.09 1.794 53.82 1.776 0.739 0.924 1,491 671 2,983

9.00 29.53 1.754 52.62 1.697 0.718 0.897 1,384 622 2,767

7.00 22.97 1.702 51.06 1.598 0.696 0.870 1,263 568 2,526

5.00 16.40 1.650 49.50 1.502 0.674 0.843 1,150 517 2,300

3.00 9.84 1.537 46.10 1.303 0.654 0.817 967 435 1,934

1.00 3.28 1.480 44.40 1.208 0.654 0.817 897 202 897

----- ----- ----- ----- ----- ----- ----- ------- ----- ------

Total 8,305 36,944

Sb = Terrain and Building Factor per Table 4

Vs = Vb * Sa * Sd * Ss * Sp {Eqn 8}

Ve = Vs * Sb

qs = (0.613*Ve^2)

Cpe = Summation Of Cpe values In table 7 (D > 1 m) Or Fig 25 (D <= 1m)

Force Uniform = qs * Ca * DynFac * Area {Shape Factor incl In Wind Area Calculation}

Force Total = Press * Ht



Elev Elev Load Load

m m N/m N

----- ----- ------- ------

25.00 23.00 1,764 3,528

23.00 21.00 1,742 3,485

21.00 19.00 1,721 3,441

19.00 17.00 1,692 3,384

17.00 15.00 1,656 3,312

15.00 13.00 1,621 3,242

13.00 11.00 1,572 3,145

11.00 9.00 1,491 2,983

9.00 7.00 1,384 2,767


Page 122 of 180

7.00 5.00 1,263 2,526

5.00 3.00 1,150 2,300

3.00 1.00 967 1,934

1.00 0.00 897 897

----- ----- ------- ------

Total 36,944

Allowable Stress Summary per BS 4076 : 1989Ref BS 4076 : 1989 'Specification for Steel Chimneys' Para 4.3.3

Elev Elev he D t he/D D/t Fy A B Allow Allow

Comp Comp

m ft m m m MPa ksi MPa

----- ----- ----- ----- ----- ----- ------ ------ ----- ----- ----- -----

23.00 75.46 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

21.00 68.90 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

19.00 62.34 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

17.00 55.77 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

15.00 49.21 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

13.00 42.65 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

11.00 36.09 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

9.00 29.53 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

7.00 22.97 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

5.00 16.40 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

3.00 9.84 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

1.00 3.28 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

0.00 0.00 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

D = Mean Diameter (Corroded) t = Corroded Thickness of Stack Shell

he = Effective Ht for Buckling Fyld = Yield stress of Stack

A = 1 / [0.84 + (0.019*he/D)^2] {he/D > 21} -OR- 1 {he/D <= 21}

B = 270 *(t/D) *(1-67*(t/D)) {D/t > 130} -OR- 1 {D/t <= 130}

Allow = Allowable Compressive stress (Para 4.3.3): 0.5 * Fyld * A * B


Page 123 of 180

Section D - BS CP 3 Ch V Wind + BS 4076


Page 124 of 180

Wind BS CP 3 Ch V Part II - 1972


≔V 45 ― ≔S1 1 ≔S3 1 ≔Use_BS4076 1

≔h 25 ≔b 1.25 ≔t1 4 ≔ρa 1.25 ――3

≔ξs 0.002 ≔n 2.039


≔z

2523

21191715

131197

5310

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=dz

1.251.25

1.251.251.251.25

1.251.251.251.25

1.251.251.251.25

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=tz

44

4444

4444

4444

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔ϕ

1.000 1.0000.890 0.6170.780 0.244

0.671 −0.0990.564 −0.3860.461 −0.5920.363 −0.704

0.272 −0.7150.190 −0.6340.120 −0.4840.064 −0.301

0.024 −0.1280.003 −0.016

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


122.65122.65122.65

122.65122.65122.65122.65

122.65122.65122.65122.65

122.65122.65

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

――

≔Class =_Class ((h)) “C” ≔S2 =Calc_S2(( ,h Class)) 1.02 =S1 1 =S3 1

≔_Vs((Z)) ⋅⋅⋅V S1 Calc_S2

(( ,Z Class)) S3

=_Vs⎛⎝z

0⎞⎠

45.87 ―

≔Vs ((Z))‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((Z)) 2‖‖‖

←Vsj

_Vs⎛⎝Z

j⎞⎠

⎛⎝Vs⎞⎠

≔_q ((Z)) ⋅0.613 ――3

_Vs((Z))

2

≔q ((Z))‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((Z)) 2‖‖‖

←qj

_q ⎛⎝Z

j⎞⎠

((q))

≔Cf(( ,d z))

‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖

|||||||||||||||||||||||||

|||||||||||||||||||||||

|

if

else

＝Use_BS4076 1‖‖

((0.6))

‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖

←DVs ⋅d _Vs((z))

||||||||||||||||

|

if

else if

else if

else

<DVs 6 ――2

‖‖

((1.2))

≥DVs 33 ――2

‖‖

((0.8))

<≤6 ――2

DVs 12 ――2

‖‖

((0.6))

‖‖

((0.7))

≔_Ftot ((Z))‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((Z)) 2‖‖‖

←Fj

⋅⋅⋅_q ⎛⎝Z

j⎞⎠

⎛⎝

−Zj

Z+j 1

⎞⎠

Cf⎛⎜⎝

,dzjZ

j⎞⎟⎠

dzj

((F))

45.87⎡ ⎤ 1290⎡ ⎤ 1934⎡ ⎤


Page 125 of 180

=Vs ((z))

45.8745.5745.2844.98

44.6944.3943.7343.08

41.739.6137.51

35.3335.39

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― =q ((z))

1290127312571240

1224120811721138

1066962863

765768

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ftot =_Ftot ((z))

1934191018851860

1836181217591706

159914421294

1147576

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Vsum ⎛⎝Ftot⎞⎠ 20761



Page 126 of 180


≔K 3.5 ≔In_Town “True” ≔ρ 1.25 ――m

3≔∆ 0.094 ≔Dt =b 1.25

≔w =⋅―――――――⋅

⎛⎝ −b

2 ⎛⎝ −b ⋅2 t1⎞⎠2 ⎞

⎠

4490 ――

3122.9 ――

≔Vcrit =⋅⋅5 Dt n 12.74 ―

≔C =+0.6 ⋅K

⎛⎜⎜⎜⎜⎜⎜⎝

+――――

⋅10⎛⎜⎝――Dt

1

⎞⎟⎠

2

⎛⎜⎜⎜⎝

――w

1 ――

⎞⎟⎟⎟⎠

―――⋅1.5 ∆

Dt

⎞⎟⎟⎟⎟⎟⎟⎠

1.44

≔Result =‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||

|

||||||||

|

if

else if

else

<C 1‖‖ ←res “Oscillations Unlikely”

≥C 1.3‖‖ ←res “Use Strakes or Damper”

‖‖ ←res “Multiply wind loads by C”

((res))

“Use Strakes or Damper”



Page 127 of 180


≔D =⎛⎝ −b t1⎞⎠ 1.246 ≔t =t1 0.004 ≔he 50 ≔Fy 354.94

=dz

1.251.25

1.251.251.251.25

1.251.251.251.25

1.251.251.251.25

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Dz =−dz tz

1.2461.246

1.2461.2461.2461.246

1.2461.2461.2461.246

1.2461.2461.2461.246

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ratio =‖‖‖‖‖‖‖‖‖

|||||||

|

for ∊ |||||

|

j , ‥0 1 −rows ⎛⎝Dz⎞⎠ 1

‖‖‖‖‖‖

←rj

――

Dzj

tzj

((r))

311.5311.5

311.5311.5311.5311.5

311.5311.5311.5311.5

311.5311.5311.5311.5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔A =‖‖‖‖‖‖‖‖‖‖‖

||||||||||

for ∊ ||||||||

j , ‥0 1 −rows ⎛⎝Dz⎞⎠ 1

‖‖‖‖‖‖‖

←Aj

――――――――1

⎛⎜⎜⎜⎝

+0.84⎛⎜⎜⎝

⋅0.019 ――he

Dzj

⎞⎟⎟⎠

2 ⎞⎟⎟⎟⎠

((A))

0.7040.704

0.7040.7040.7040.704

0.7040.7040.7040.704

0.7040.7040.7040.704

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔B =‖‖‖‖‖‖‖‖

||||||

|

for ∊ ||||

|

j , ‥0 1 −rows ⎛⎝Dz⎞⎠ 1

‖‖‖‖‖

←Bj

⋅⋅270⎛⎜⎜⎝

―――1

Ratioj

⎞⎟⎟⎠

⎛⎜⎜⎝

−1 ⋅67⎛⎜⎜⎝

―――1

Ratioj

⎞⎟⎟⎠

⎞⎟⎟⎠

((B))

0.680.68

0.680.680.680.68

0.680.680.680.68

0.680.680.680.68

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Allow =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ⎛⎝Dz⎞⎠ 1

‖‖‖

←Alj

⋅⋅⋅0.5 Fy AjB

j

((Al))

84.9584.95

84.9584.9584.9584.95

84.9584.9584.9584.95

84.9584.9584.9584.95

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 128 of 180



File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\BS CP3 ChV\ v5314_BSCP3.Stk

INPUT PARAMETERS:




Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------25.0 1.25 | 25.0 4.0 | 25.0 0.0

Materials:


---- -------------- ---- ------ -------- -------- ---------- ------- 25.0 S355 (FE 510A) 20.0 354.94 209,947 1.15E-05 122.73 7,849 20.0 354.94 209,947 1.15E-05 122.73 7,849

Design Codes


Stress Criteria: British Standard BS 4076: 1989 'Specification for Steel Chimneys'

Wind Load Criteria: British BS CP 3 Ch V Part 2 : 1972 'Wind Loads'

V = Basic Wind Speed = 45.00 m/s S1 = Topography Factor {Para 5.4} = 1.000 S3 = Factor based upon Statistical Concepts {Para 5.6} = 1.000 Terrain = Ground Roughness as defined in Para 5.5.1 = 1 Use_BS4076= Apply the criteria from BS 4076 Para 4.1 = True

Vortex Shedding Criteria: British Standard BS 4076: 1989 'Specification for Steel Chimneys'

K = Factor based upon type Of construction = 3.500 In_Town = Is Stack Located In Town? = True



Section Properties



----- ----- -------- ----- -------- ------ ----------- ----------- --------- 25.00 23.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 23.00 21.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 21.00 19.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5 19.00 17.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5


Page 129 of 180

17.00 15.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5

15.00 13.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5

13.00 11.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5

11.00 9.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5

9.00 7.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5

7.00 5.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5

5.00 3.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5

3.00 1.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5

1.00 0.00 1,250.00 4.00 440.53 156.58 6,210.0 4,861.8 303,860.5

Weight Detail






----- ---- ----- ----- ---- ------- -------- ---------- ----------

25.00 0.00 12.50 1.250 4.00 7,849 3,072.4 122.90 3,072.4

----- ---- ----- ----- ---- ------- -------- ---------- ----------

Total 12.50 3,072.4 3,072.4

Weight Summary


m kg

----------------------------------- -------- -------


----------------------------------- -------- -------

Total 12.500 3,072.4

Wind Areas

Wind Area Summary

Elev Shape Fac Riser Ladder Platform Piping Tot Uni Total

Riser Area Area Area Area Area Area

m m^2/m m^2/m m^2/m m^2/m m^2/m sq m

----- --------- ----- ------ -------- ------ ------- ------

25.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

23.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

21.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

19.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

17.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

15.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

13.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

11.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

9.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

7.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

5.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

3.00 0.600 0.750 0.000 0.000 0.000 0.750 1.500

1.00 0.600 0.750 0.000 0.000 0.000 0.750 0.750

----- --------- ----- ------ -------- ------ ------- ------

Total 18.750

Frequency Summary


# 1 # 2 # 3

Deg Hz Hz Hz

----------------- --------- ----- ------ ------

Cold & Uncorroded 0.00 2.039 12.725 35.391

Vortex Shedding Analysis per BS 4076 : 1989

Ref Appendx B 'Wind-excited Oscillations'

K = Factor based upon type Of construction = 3.500


Page 130 of 180

In_Town = Is Stack Located In Town? = True W = Average uniform mass Of top 1/3 Of stack = 122.90 kg/m Rho = Air density = 1.250 Kg/m^3 Dt = Average OD Of top 1/3 Of stack = 1.25 m f = Natural Frequency Of Mode # 1 = 2.039 Hz Vcrit = Critical Strouhal Velocity: 5 * Dt * f {Para B.3.1} = 12.75 m/s Vcrit = Critical Strouhal Velocity: 5 * Dt * f {Para B.3.1} = 28.5 mph Vs = Design Wind Velocity: V * S1 * S2 * S3 = 46.35 m/s

Vcrit <= Vs, the tendency to oscillate may be estimated by C

Defl = Deflection resulting from 1 KN/m^2 applied to the stack = 0.094 m C = 0.6 + K * (10 * Dt^2 / W + 1.5 * Defl / Dt) {Para B.3.3 = 1.438

Since C > 1.3 And the stack Is in an Urban area Add A Damper Or Helical Strakes

--> Since the stack Is Not fitted with a Damper Or Strakes the Vortex Shedding Is Failing

Wind Load Calculations

per BS CP 3: Chapter V Part II 1972

V = Basic Wind Speed = 45.00 m/s S1 = Topography Factor {Para 5.4} = 1.000 S3 = Factor based upon Statistical Concepts {Para 5.6} = 1.000 Terrain = Ground Roughness as defined in Para 5.5.1 = 1 Use_BS4076= Apply the criteria from BS 4076 Para 4.1 = True

BS 4076 Para 4.1(b) --> Since Stack Ht > 10 m (32.8 ft) And < 80 m (262.5 ft) Increase S2 for Class C by the term (S2b-S2c)*(80-h)/70 S2b = S2 for Class B, S2c = S2 for Class C, And H Is Stack Ht in Meters Fac_BS4076= Wind Speed Factor determined from BS 4076 Appendix B3.1 = 1.000

Elev Elev S2 S2 S2 Vs q Area Force Force Force Class C Add Uniform Total Total m ft m/s kPa m^2/m N/m lb N

----- ----- ------- ----- ----- ----- ----- ----- ------- ----- ------ 25.00 82.02 0.980 0.039 1.019 45.87 1.290 0.750 967 435 1,934 23.00 75.46 0.972 0.039 1.011 45.51 1.270 0.750 952 428 1,904 21.00 68.90 0.964 0.039 1.003 45.15 1.249 0.750 937 421 1,874 19.00 62.34 0.956 0.039 0.995 44.79 1.230 0.750 922 415 1,844 17.00 55.77 0.948 0.039 0.987 44.43 1.210 0.750 907 408 1,815 15.00 49.21 0.940 0.039 0.979 44.07 1.190 0.750 893 401 1,786 13.00 42.65 0.924 0.039 0.963 43.35 1.152 0.750 864 388 1,728 11.00 36.09 0.908 0.039 0.947 42.63 1.114 0.750 835 376 1,671 9.00 29.53 0.876 0.039 0.915 41.19 1.040 0.750 780 351 1,560 7.00 22.97 0.828 0.039 0.867 39.03 0.934 0.750 700 315 1,401 5.00 16.40 0.780 0.039 0.819 36.87 0.833 0.750 625 281 1,250 3.00 9.84 0.730 0.039 0.769 34.62 0.735 0.750 551 248 1,102 1.00 3.28 0.730 0.039 0.769 34.62 0.735 0.750 551 124 551

----- ----- ------- ----- ----- ----- ----- ----- ------- ----- ------ Total 4,591 20,420 S2_ClassC = S2 for Class C at Elevation {per BS4076 Para 4.1(b)} S2_Add = Additional amount added to S2 {per BS4076 Para 4.1(b)} S2 = S2_ClassC + S2_Add {per BS 4076 Para 4.1 (b)} Vs = V * S1 * S2 * S3 * Fac_BS4076 {Para 4.3 with Adjustment per BS 4076 App B.3.1} q = 0.613 * Vs^2 {Para 4.3}


Top Bot Uniform Total Elev Elev Load Load m m N/m N

----- ----- ------- ------ 25.00 23.00 967 1,934 23.00 21.00 952 1,904 21.00 19.00 937 1,874 19.00 17.00 922 1,844 17.00 15.00 907 1,815 15.00 13.00 893 1,786


Page 131 of 180

13.00 11.00 864 1,728

11.00 9.00 835 1,671

9.00 7.00 780 1,560

7.00 5.00 700 1,401

5.00 3.00 625 1,250

3.00 1.00 551 1,102

1.00 0.00 551 551

----- ----- ------- ------

Total 20,420

Primary Loads

Abbreviations are used in Load Combinations

Number Description Abbreviation

------ ----------- ------------

1 Dead D

2 Live L

3 Operating O

4 Thermal Hot TH

5 Wind W

Load Combinations

Column Abbreviations can be found in the Primary Load Table

Num Description S D FS Dir D L O TH W

Deg

--- ----------- - - ----- --- --- --- --- -- ---

6 D+W X X 1.000 0 1.0 1.0

7 D+L+W X X 1.000 0 1.0 1.0 1.0

8 D+O+P X 1.000 0 1.0 1.0

FS: Factor of Safety used in Conjunction with the Stress Code which has been selected.

Dir: Direction of Lateral Loads (WInd/Seismic/Vortex), 0 Deg = X Axis, 90 Deg = Z Axis

P: Internal Pressure (Doesn't produce support loads only used for stack tensile stress)

S: Include in Stress Analysis D: Calculate Deflextions

Stress Criteria: BS 4076 (Limit States)

Wind Criteria: BS CP 3 CH V (ASD)

Seismic Criteria: None ()

Wind Combinations:

The BS CP 3 Ch V code is referenced in BS 4076, and so load factors of 1 are

used since BS 4076 provides no specific load combinations.

Allowable Stress Summary per BS 4076 : 1989

Ref BS 4076 : 1989 'Specification for Steel Chimneys' Para 4.3.3


Comp Comp


----- ----- ----- ----- ----- ----- ------ ------ ----- ----- ----- -----

23.00 75.46 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

21.00 68.90 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

19.00 62.34 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

17.00 55.77 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

15.00 49.21 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

13.00 42.65 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

11.00 36.09 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

9.00 29.53 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

7.00 22.97 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

5.00 16.40 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

3.00 9.84 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

1.00 3.28 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95

0.00 0.00 50.00 1.246 0.004 40.13 311.50 354.94 0.704 0.680 12.32 84.95



A = 1 / [0.84 + (0.019*he/D)^2] {he/D > 21} -OR- 1 {he/D <= 21}

B = 270 *(t/D) *(1-67*(t/D)) {D/t > 130} -OR- 1 {D/t <= 130}

Allow = Allowable Compressive stress (Para 4.3.3): 0.5 * Fyld * A * B


Page 132 of 180

Section E - Indian Standards


Page 133 of 180

Indian Standard


≔Vb 30 ― ≔k1 1 ≔k3 1 ≔Terrain 1 Class=B

≔f 2.387 ≔h 25 ≔ρa 1.25 ――3

≔ξs 0.002 ≔C 0.7 ≔Ht 25

≔d1 1.5 ≔t1 4

≔q ((V)) ⋅⋅0.6 V2 ―――⋅ 2

2≔Vz ((k)) ⋅⋅⋅Vb k1 k k3 ≔dz Vfill ⎛⎝ ,14 d1⎞⎠ ≔tz Vfill ⎛⎝ ,14 t1⎞⎠

≔z

2523211917151311975310

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=dz

1.51.51.51.51.51.51.51.51.51.51.51.51.51.5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=tz

44444444444444

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔k2

1.1151.1091.1031.0941.0821.0701.0541.0381.0301.0301.0301.0301.0301.030

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Vz =Vz ⎛⎝k2⎞⎠

33.4533.2733.0932.8232.4632.131.6231.1430.930.930.930.930.930.9

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― ≔qz =q ⎛⎝Vz⎞⎠

671664657646632618600582573573573573573573

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔hz =‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((z)) 2‖‖‖

←hj

−z,j 0

z+j 1

((h))

2222222222221

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Pst =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((z)) 2‖‖‖

←Pj

⋅⋅⋅C qzjhz

jdz

j

((P))

141013951380135713281298126012221203120312031203

602

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Pst_tot =Vsum ⎛⎝Pst⎞⎠ 16063

Location A

≔mk_El

010204060

100200350

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔mk_val

0.60.6

0.550.480.460.420.380.35

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔mk =linterp (( ,,mk_El mk_val Ht)) 0.53

294.51⎡ ⎤ 1.000⎡ ⎤


Page 134 of 180

≔mz =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖

←Sj

⋅Wt_Cylinder ⎛⎜⎝

,dzjtzj⎞⎟⎠hz

j

((S))

294.51294.51294.51294.51294.51294.51294.51294.51294.51294.51294.51294.51147.26

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔γ

1.0000.8900.7800.6710.5640.4610.3630.2720.1900.1200.0640.0240.003

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔M =Vsum ((m)) 400

≔Num =∑=j 0

−rows ⎛⎝hz⎞⎠ 1

⋅⋅γjPst

jmk 3864 ≔Den =∑

=j 0


⋅γj

2 mzj

1073

≔n =‖‖‖‖‖‖‖

|||||

|

for ∊ |||

|

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖‖

←nj

⋅γj

――Num

Den

((n))

3.63.212.812.422.031.661.310.980.680.430.230.090.01

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

―2

≔Ex

0.025.05

.0750.1

0.1250.15

0.1750.2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ey

1.32.53.13.5

3.754.14.34.54.7

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔T =―1

f0.42 ≔ξ1 =―――

⋅T Vb

12000.01 ≔ξi =linterp ⎛⎝ ,,Ex Ey ξ1⎞⎠ 1.8

≔ν 0.7 (Table 7)

≔Pdyn =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖

←Pj

⋅⋅⋅mzjξi n

jν

((P))

133911911044

898755617486364254161

8632

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Pwind =+Pst Pdyn

274825862424225520831915174615861457136412891235

604

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Pdyn_tot =Vsum ⎛⎝Pdyn⎞⎠ 7229 ≔Ptot =Vsum ⎛⎝Pwind⎞⎠ 23292

Wind Excited Oscillations (IS 6533 Annex A)


Page 135 of 180

Wind Excited Oscillations (IS 6533 Annex A)

Para 8.4, Unlined Stack

≔Vcr =⋅⋅5 d1 f 17.9 ― ≔Vtop =Vz033.45 ― ≔Vlwr =⋅0.33 Vtop 11 ―

≔qcr =⋅――Vcr

2

16

⎛⎜⎜⎝―――

⋅ 2

2

⎞⎟⎟⎠

20.03 ≔Vupr =⋅0.8 Vtop 27 ―

Since Vlwr <= Vcr <= Vupr --> Then Resonance Must be checked per IS 6533 Annex A

Calculate Resonance Forces per Para A-4:

≔Cy 0.25

≔Fe1 =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖

←Fj

⋅⋅⋅Cy qcr dzjhz

j

((F))

151515151515151515151515

7.5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔F1 =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖

←Fj

⋅γjFe1

j

((F))

15131210

875432100

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Calculate Along Wind Loads acting at Vcr Wind Speed:

≔Pst_vcr =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((z)) 2‖‖‖

←Pj

⋅⋅⋅C qcr hzjdz

j

((P))

42424242424242424242424221

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Numvcr =∑=j 0


⋅⋅γjPst_vcr

jmk 121 ≔Denvcr =∑

=j 0


⋅γj

2 mzj

1073

≔nvcr =‖‖‖‖‖‖‖‖

|||||||

for ∊ |||||

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖‖

←nj

⋅γj

―――Numvcr

Denvcr

((n))

0.110.10.090.080.060.050.04⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

――2

≔Ex

0.025.05

.0750.1

0.1250.15

0.1750.2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ey

1.32.53.13.5

3.754.14.34.54.7

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅T Vcr1


Page 136 of 180

≔ν 0.7≔T =―

1

f0.42 ≔ξ1 =―――

⋅T Vcr

12000.006 ≔ξi =linterp ⎛⎝ ,,Ex Ey ξ1⎞⎠ 1.6 (Table 7)

≔Pdyn_vcr =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖

←Pj

⋅⋅⋅mzjξi nvcr

jν

((P))

3733292521171410

74210

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Pvcr =+Pst_vcr Pdyn_vcr

79757167635956524947444321

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Pdyn_vcr_tot =Vsum ⎛⎝Pdyn_vcr⎞⎠ 201 ≔Ptot_vcr =Vsum ⎛⎝Pvcr⎞⎠ 727

Ovalling Stiffening Rings per Annex A Para A-9

As long as t > D / 300 then stiffening rings not required:

≔tmin =――d1

3005

≔Rings_Reqd =‖‖‖‖‖‖‖‖

|||||||

|||||

|

if

else

>t1 tmin‖‖ ((“Rings Not Reqd: t1>tmin”))

‖‖ ((“Rings Req'd: t1 < tmin”))

“Rings Req'd: t1 < tmin”

Stiffener at top and lower down if h > 20*d

≔hmax =⋅20 d1 30

≔Num_of_Rings_Reqd =‖‖‖‖‖‖‖‖

|||||||

|||||

|

if

else

>h hmax‖‖ ((“Multiple Rings Reqd”))

‖‖ ((“Ring @ Top Reqd”))

“Ring @ Top Reqd”

Max Stiffening Ring Spacing (If Multiple Rings Req'd)

≔Lrings =⋅1500 t1 6

Minimum Moment of Inertia Required of rings:

≔treqd =――d1

2007.5 ≔Wreqd =⋅16 treqd 120 ≔Ireqd =―――――

⋅Wreqd treqd3

120.42 4

Try flat bar Stiffener:

≔Dstiff 50 ≔tstiff 6 ≔Istiff =――――⋅tstiff Dstiff

3

126.25 4

=Unity ⎛⎝ ,Ireqd Istiff⎞⎠0.07

“Pass”⎡⎢⎣

⎤⎥⎦


Page 137 of 180

Seismic Loads per IS 1893 - 1975

Zone = V ≔I 2 ≔β 1.5 ≔Fo 0.4 ≔δ =⋅⋅2 ξs 0.0126 ≔T =―1

f0.42

≔h' =Ht 25 ≔rg =Sect_Prop_Cylinder_r ⎛⎝ ,d1 t1⎞⎠ 0.529

≔K =―h

rg47.27 ≔Cv 1.5 ≔Sa_g 0.26 ≔Ah =⋅⋅⋅β I Fo Sa_g 0.312

≔x' =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖

←xj

−h 0.5 ⎛⎝

+zj

z+j 1

⎞⎠

((x))

13579

1113151719212324.5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔xh =―x'

h'

0.040.120.20.280.360.440.520.60.680.760.840.920.98

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔W =⋅mz

2888288828882888288828882888288828882888288828881444

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Vseis =‖‖‖‖‖‖‖

|||||

|

for ∊ |||

|

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖‖

←Vj

⋅⋅⋅Cv Ah Wj

⎛⎜⎝

−⋅―5

3xh

j⋅―

2

3xh

j

2⎞⎟⎠

((V))

89257415560694817928

10271115119212571310

671

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Vtot =Vsum ⎛⎝Vseis⎞⎠ 10331

Allowable Compressive Stress per IS 6533 Annex C


Page 138 of 180

Allowable Compressive Stress per IS 6533 Annex C

≔Fy 248.21 ≔he =⋅2 h 50

≔hed =―he

dz

33.3333.3333.3333.3333.3333.3333.3333.3333.3333.3333.3333.3333.3333.33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔dm =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖

←xj

−dzj

tzj

((x))

1.51.51.51.51.51.51.51.51.51.51.51.51.5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Dt =‖‖‖‖‖‖‖‖‖

|||||||

|

for ∊ |||||

|

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖‖‖‖

←xj

――

dmj

tzj

((x))

374374374374374374374374374374374374374

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔A =‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖

|||||||||||||

|

for ∊ |||||||||||

|

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖‖‖‖‖‖‖‖‖‖

|||||||||

|

if

else

<hedj

21

‖‖‖

←Aj

1

‖‖‖‖‖

←Aj

―――――――1

+0.84 ⎛⎝

⋅0.019 hedj⎞⎠

2

((A))

0.810.810.810.810.810.810.810.810.810.810.810.810.81

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔B =‖‖‖‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||||||

for ∊ ||||||||||||

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||

if

else

<Dtj

130

‖‖‖

←Bj

1

‖‖‖‖‖

←Bj

⋅⋅270⎛⎜⎜⎝

――1

Dtj

⎞⎟⎟⎠

⎛⎜⎜⎝

−1 ⋅67⎛⎜⎜⎝

――1

Dtj

⎞⎟⎟⎠

⎞⎟⎟⎠

((B))

0.5930.5930.5930.5930.5930.5930.5930.5930.5930.5930.5930.5930.593

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Allow =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ⎛⎝hz⎞⎠ 1‖‖‖

←Allowj

⋅⋅⋅0.5 Fy AjBj

((Allow))

59.2659.2659.2659.2659.2659.2659.2659.2659.2659.2659.2659.2659.26

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Calculate Actual Wind Stresses per Para 9.1:


Page 139 of 180

Calculate Actual Wind Stresses per Para 9.1:

Stability Check: 1.6 * Stress_Wind - 0.9 * Stress_Dead < 1.8 * Stress_Allow

≔Mw =Mstack ⎛⎝ ,z Pwind⎞⎠

2.7510.8323.9241.763.889.91

119.68152.78188.92227.89269.51313.65336.64

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅ ≔Vd =⋅Vstack ⎛⎝ ,z mz⎞⎠

2.895.788.66

11.5514.4417.3320.2223.1125.9928.8831.7734.6636.1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Sstk =Sect_Prop_Cylinder_S ⎛⎝ ,dz tz⎞⎠

7012.27012.27012.27012.27012.27012.27012.27012.27012.27012.27012.27012.27012.27012.2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

3 ≔Astk =Sect_Prop_Cylinder_A ⎛⎝ ,dz tz⎞⎠

187.99187.99187.99187.99187.99187.99187.99187.99187.99187.99187.99187.99187.99187.99

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

2

≔σw =σbend ⎛⎝ ,Mw Sstk⎞⎠

0.391.543.415.959.1

12.8217.0721.7926.9432.538.4344.7348.01

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σd =σaxial ⎛⎝ ,Vd Astk⎞⎠

0.150.310.460.610.770.921.081.231.381.541.691.841.92

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σact =+⋅1.6 σw ⋅0.9 σd

0.772.755.87

10.0715.2521.3528.2835.9744.3553.3863.0173.2278.54

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ratio =‖‖‖‖‖‖‖‖‖‖‖

||||||||||

for ∊ ||||||||

j , ‥0 1 −rows ⎛⎝Mw⎞⎠ 1‖‖‖‖‖‖‖‖

←U Unity ⎛⎜⎝

,σactjσallow

j⎞⎟⎠

←R,j 0

U0

←R,j 1

U1

((R))

0.01 “Pass”0.03 “Pass”0.06 “Pass”0.09 “Pass”0.14 “Pass”0.2 “Pass”0.27 “Pass”0.34 “Pass”0.42 “Pass”0.5 “Pass”0.59 “Pass”0.69 “Pass”0.74 “Pass”

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σallow =⋅1.8 Allow

106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 140 of 180

In MecaStack the loads are multiplied by the factors rather than the stresses, but as can be seen in this validation the resulting stress ratios are identical. Here are the loads/stresses as reported by MecaStack w/factors applied.

=⋅1.6 Mw

4.417.3338.2866.71

102.09143.86191.49244.45302.28364.62431.21501.83538.62

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅ =⋅0.9 Vd

2.65.27.8

10.41315.618.220.7923.3925.9928.5931.1932.49

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=⋅1.6 σw

0.632.475.469.51

14.5620.5227.3134.8643.115261.4971.5776.81

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=⋅0.9 σd

0.140.280.410.550.690.830.971.111.241.381.521.661.73

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σw =σbend ⎛⎝ ,⋅1.6 Mw Sstk⎞⎠

0.632.475.469.51

14.5620.5227.3134.8643.115261.4971.5776.81

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σd =σaxial ⎛⎝ ,⋅0.9 Vd Astk⎞⎠

0.140.280.410.550.690.830.971.111.241.381.521.661.73

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σact =+σw σd

0.772.755.87

10.0715.2521.3528.2835.9744.3553.3863.0173.2278.54

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Ratios ⎛⎝ ,σact σallow⎞⎠

0.01 “Pass”0.03 “Pass”0.06 “Pass”0.09 “Pass”0.14 “Pass”0.2 “Pass”0.27 “Pass”0.34 “Pass”0.42 “Pass”0.5 “Pass”0.59 “Pass”0.69 “Pass”0.74 “Pass”

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Calculate Actual Seismic Stresses per Para 9.1:


Page 141 of 180

Calculate Actual Seismic Stresses per Para 9.1:

Stability Check: 1.6 * Stress_Seismic - 0.9 * Stress_Dead < 1.8 * Stress_Allow

≔Ms =⋅1.6 Mstack ⎛⎝ ,z Vseis⎞⎠

0.140.842.615.94

11.2719.0329.5743.2560.3581.14

105.85134.67150.66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅ ≔Vd =⋅⋅0.9 Vstack ⎛⎝ ,z mz⎞⎠

2.65.27.8

10.41315.618.220.7923.3925.9928.5931.1932.49

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σs =σbend ⎛⎝ ,Ms Sstk⎞⎠

0.020.120.370.851.612.714.226.178.61

11.5715.119.221.49

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0.140.280.410.550.690.830.971.111.241.381.521.661.73

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σact_seis =+σs σd

0.160.40.791.42.33.545.197.279.85

12.9516.6220.8623.21

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Ratios ⎛⎝ ,σact_seis σallow⎞⎠

0 “Pass”0 “Pass”0.01 “Pass”0.01 “Pass”0.02 “Pass”0.03 “Pass”0.05 “Pass”0.07 “Pass”0.09 “Pass”0.12 “Pass”0.16 “Pass”0.2 “Pass”0.22 “Pass”

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Calculate Actual Vortex Stresses per Para 9.1:


Page 142 of 180

Calculate Actual Vortex Stresses per Para 9.1:

Stability Check: 1.6 * Stress_Vortex - 0.9 * Stress_Dead < 1.8 * Stress_Allow

≔Malong =⋅1.6 Mstack ⎛⎝ ,z Pvcr⎞⎠

0.130.51.111.942.974.215.627.218.96

10.8712.9215.1116.25

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅ ≔Vd =⋅⋅0.9 Vstack ⎛⎝ ,z mz⎞⎠

2.65.27.8

10.41315.618.220.7923.3925.9928.5931.1932.49

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Macross =⋅1.6 Mstack (( ,z F1))

0.020.090.20.350.520.720.941.171.421.671.932.192.32

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅ ≔Mv =‾‾‾‾‾‾‾‾‾‾‾‾‾‾+Malong2 Macross

2

0.130.511.131.973.024.275.77.319.07

10.9913.0615.2616.42

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅

≔σv =σbend ⎛⎝ ,Mv Sstk⎞⎠

0.020.070.160.280.430.610.811.041.291.571.862.182.34

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0.140.280.410.550.690.830.971.111.241.381.521.661.73

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σact_vor =+σv σd

0.160.350.580.831.121.441.782.152.542.953.383.844.07

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66106.66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Ratios ⎛⎝ ,σact_vor σallow⎞⎠

0 “Pass”0 “Pass”0.01 “Pass”0.01 “Pass”0.01 “Pass”0.01 “Pass”0.02 “Pass”0.02 “Pass”0.02 “Pass”0.03 “Pass”0.03 “Pass”0.04 “Pass”0.04 “Pass”

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 143 of 180


Calculations Prepared by: Calculations Prepared For: Meca Enterprises Client: 0 816 W. Elgin St Project #: 0 Broken Arrow, OK, 74012 Location: 0 Date: Apr 23, 2017 Description: Designer: 0 0

File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\Indian\ Indian_25m_Rev0.Stk

INPUT PARAMETERS:




Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------25.0 1.5 | 25.0 4.0 | 25.0 0.0

Materials:


---- -------- ---- ------ -------- -------- ---------- ------- 25.0 A-36 37.8 248.21 199,603 1.22E-05 114.45 7,849 37.8 248.21 199,603 1.22E-05 114.45 7,849

Design Codes

Comprehensive Design Standard: Indan Standards: IS 6533 and IS 875 Pt 3

Stress Criteria: Indian Code IS6533: 1989 'Code of Practice for Design and Construction of Steel Chimneys'

Wind Load Criteria: Indian IS 875 (Part 3): 1987 'Code of Practice for Design Loads for Buildings and Structures'

V = Wind Speed = 30.00 m/s K1 = Topography Factor (Ref IS 875 Para 5.3.1) = 1.00 K3 = Probability Factor (Ref IS 875 Para 5.3.2) = 1.00 Terrain = Terrain Roughness (Ref IS 875 Para 5.3.2.1) = 1 Modes = Number of Modes to be Considered in Dynamic Load Calculations = 1

Vortex Shedding Criteria: Indian IS 875 (Part 3): 1987 'Code of Practice for Design Loads for Buildings and Structures'

Group = Stack Arrangement = Single Spacing = Center to center stack spacing (Only used for 'Group') = 0.0 m S = Manually entered Strouhal Number (Only used for 'Custom') = 0.0000


Seismic Criteria: IS:1893-1975 Criteria for Earthquake Resistant Design of Struc Zone = Seismic Zone = V I = Important Factor = 2.000 Beta = Beta: Soil-Foundation Coeffient = 1.50




Section Properties



----- ----- -------- ----- -------- ------ ----------- ----------- --------- 25.00 23.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 23.00 21.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 21.00 19.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 19.00 17.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 17.00 15.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 15.00 13.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 13.00 11.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 11.00 9.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5


Page 144 of 180

9.00 7.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 7.00 5.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 5.00 3.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 3.00 1.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5 1.00 0.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

Weight Detail




----- ---- ----- ----- ---- ------- -------- ---------- ---------- 25.00 0.00 12.50 1.500 4.00 7,849 3,688.9 147.56 3,688.9

----- ---- ----- ----- ---- ------- -------- ---------- ---------- Total 12.50 3,688.9 3,688.9

Weight Summary



----------------------------------- -------- ------- Total 12.500 3,688.9

Wind Areas

Wind Area Summary


----- --------- ----- ------ -------- ------ ------- ------ 25.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 23.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 21.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 19.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 17.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 15.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 13.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 11.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 9.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 7.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 5.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 3.00 0.700 1.050 0.000 0.000 0.000 1.050 2.100 1.00 0.700 1.050 0.000 0.000 0.000 1.050 1.050

----- --------- ----- ------ -------- ------ ------- ------ Total 26.250

Thermal Expansion of Stack

+ values are in the +Y (up) direction and - values are in the -Y (Down)

Elev Installation Hot Hot Cold Cold Temperature Temperature Expansion Temperature Expansion m Deg C Deg C mm Deg C mm

----- ------------ ----------- --------- ----------- --------- 25.00 70.00 37.78 5.10 37.78 5.10 23.00 70.00 37.78 4.69 37.78 4.69 21.00 70.00 37.78 4.28 37.78 4.28 19.00 70.00 37.78 3.88 37.78 3.88 17.00 70.00 37.78 3.47 37.78 3.47 15.00 70.00 37.78 3.06 37.78 3.06 13.00 70.00 37.78 2.65 37.78 2.65 11.00 70.00 37.78 2.24 37.78 2.24 9.00 70.00 37.78 1.84 37.78 1.84 7.00 70.00 37.78 1.43 37.78 1.43 5.00 70.00 37.78 1.02 37.78 1.02 3.00 70.00 37.78 0.61 37.78 0.61 1.00 70.00 37.78 0.20 37.78 0.20 1. Elevation for Y restraint = 0.0 m

Frequency Summary

Description Direction Mode Mode Mode # 1 # 2 # 3 Deg Hz Hz Hz

----------------- --------- ----- ------ ------ Hot & UnCorroded 0.00 2.387 14.865 41.220 Cold & Uncorroded 0.00 2.387 14.865 41.220




Page 145 of 180

(2.387 Hz) (14.865 Hz) (41.22 Hz) m Normalized Normalized Normalized

--------- ---------- ----------- ---------- 25.00 -1.000 1.000 -1.000 23.00 -0.890 0.617 -0.371 21.00 -0.780 0.244 0.187 19.00 -0.671 -0.099 0.559 17.00 -0.564 -0.386 0.656 15.00 -0.461 -0.592 0.468 13.00 -0.363 -0.704 0.080 11.00 -0.272 -0.715 -0.355 9.00 -0.190 -0.634 -0.673 7.00 -0.120 -0.484 -0.762 5.00 -0.064 -0.301 -0.607 3.00 -0.024 -0.128 -0.306 1.00 -0.003 -0.016 -0.044 0.00 0.000 0.000 0.000


Elevation Mode # 1 Mode # 2 Mode # 3 (2.387 Hz) (14.865 Hz) (41.22 Hz) m Normalized Normalized Normalized

--------- ---------- ----------- ---------- 25.00 -1.000 1.000 -1.000 23.00 -0.890 0.617 -0.371 21.00 -0.780 0.244 0.187 19.00 -0.671 -0.099 0.559 17.00 -0.564 -0.386 0.656 15.00 -0.461 -0.592 0.468 13.00 -0.363 -0.704 0.080 11.00 -0.272 -0.715 -0.355 9.00 -0.190 -0.634 -0.673 7.00 -0.120 -0.484 -0.762 5.00 -0.064 -0.301 -0.607 3.00 -0.024 -0.128 -0.306 1.00 -0.003 -0.016 -0.044 0.00 0.000 0.000 0.000

Wind Loadings per Indian Standards (Design Wind Speed Vb)Ref IS 875 (Part 3) 1987 and IS 6533 Part 2: 1989

V = Wind Speed = 30.00 m/s K1 = Topography Factor (Ref IS 875 Para 5.3.1) = 1.00 K3 = Probability Factor (Ref IS 875 Para 5.3.2) = 1.00 Terrain = Terrain Roughness (Ref IS 875 Para 5.3.2.1) = 1 Modes = Number of Modes to be Considered in Dynamic Load Calculations = 1 Class = Stack Height is between 20m and 50 m = B Vz = Design Wind Speed in Imperial Units = 67.1 mph

Static Wind Load (Design Wind Speed Vb)

z z h K2 Vz q Area Pst Pst m ft m m/s kPa sq m lb N

----- ----- ----- ----- ----- ------- ------ ----- ------ 25.00 82.02 2.00 1.115 33.45 671.342 2.100 317 1,410 23.00 75.46 2.00 1.109 33.27 664.136 2.100 314 1,395 21.00 68.90 2.00 1.103 33.09 656.969 2.100 310 1,380 19.00 62.34 2.00 1.094 32.82 646.291 2.100 305 1,357 17.00 55.77 2.00 1.082 32.46 632.191 2.100 298 1,328 15.00 49.21 2.00 1.070 32.10 618.246 2.100 292 1,298 13.00 42.65 2.00 1.054 31.62 599.895 2.100 283 1,260 11.00 36.09 2.00 1.038 31.14 581.820 2.100 275 1,222 9.00 29.53 2.00 1.030 30.90 572.886 2.100 270 1,203 7.00 22.97 2.00 1.030 30.90 572.886 2.100 270 1,203 5.00 16.40 2.00 1.030 30.90 572.886 2.100 270 1,203 3.00 9.84 2.00 1.030 30.90 572.886 2.100 270 1,203 1.00 3.28 1.00 1.030 30.90 572.886 1.050 135 602

----- ----- ----- ----- ----- ------- ------ ----- ------ Total 25.00 26.250 3,611 16,063 K2 = Factor for Elevation (IS 875 Table 2) | Vz = Vb * K1 * K2 * K3 (IS 875 Para 5.3) q = 0.6 * Vz^2 (IS 875 Para 5.4) | Area = Eff Wind Area incl Shape Factors Pst = Static Force: Area*q (IS 6533: 8.2.3) | h = Height of Element

Dynamic Wind Loading for Mode # 1 (Design Wind Speed Vb)

n = Natural Frequency = 2.387 Hz T = Period of vibration = 0.419 s E1 = (1/n) * Vb / 1200 = 0.010 Coef_E1 = Coefficient of Dynamic Influence (Table 5) = 1.80 Coef_v = Space Correlation Coefficient (Table 7) = 0.70

z z Pst Phi Num Den M mk N Pdyn Pdyn m ft N kg m/s^2 lb N

----- ----- ------ ------ -------- ------- ------- ----- ----- ----- ----- 25.00 82.02 1,410 -1.000 -750.73 295.11 295.1 0.533 3.75 314 1,398 23.00 75.46 1,395 -0.890 -669.57 233.69 295.1 0.539 3.34 280 1,244 21.00 68.90 1,380 -0.780 -588.12 179.56 295.1 0.547 2.93 245 1,090 19.00 62.34 1,357 -0.671 -505.55 132.93 295.1 0.555 2.52 211 938 17.00 55.77 1,328 -0.564 -423.32 93.99 295.1 0.565 2.12 177 789 15.00 49.21 1,298 -0.461 -344.18 62.73 295.1 0.575 1.73 145 644


Page 146 of 180

13.00 42.65 1,260 -0.363 -267.46 38.87 295.1 0.585 1.36 114 507 11.00 36.09 1,222 -0.272 -197.71 21.83 295.1 0.595 1.02 85 380 9.00 29.53 1,203 -0.190 -137.33 10.68 295.1 0.600 0.71 60 266 7.00 22.97 1,203 -0.120 -86.69 4.26 295.1 0.600 0.45 38 168 5.00 16.40 1,203 -0.064 -46.08 1.20 295.1 0.600 0.24 20 89 3.00 9.84 1,203 -0.024 -17.26 0.17 295.1 0.600 0.09 8 33 1.00 3.28 602 -0.003 -1.00 0.00 147.6 0.600 0.01 0 2

----- ----- ------ ------ -------- ------- ------- ----- ----- ----- ----- Total 16,063 -4034.99 1075.02 3,688.9 1,697 7,550 M = Mass Of Element | mk = Coef Of Pulsation (Is 6533 Table 6) Num = Phi * Pst * mk (IS 6533: 8.3.4) | Phi = Normalized Mode Shape Den = Phi^2 * M (IS 6533: 8.3.4) | Pst = Static Wind Load N = Deduced Accel: Phi * Num_Total / Den_Total (IS 6533: 8.3.4) Pdyn = Dynamic Force: M * E1 * N * v (IS 6533: 8.3.2)

Total Static + Dynamic Wind Load (Design Wind Speed Vb)

z z Pst Pdyn Ptot Ptot All Modes Mode # 1 m ft N N lb N

----- ----- --------- -------- ----- ------ 25.00 82.02 1,410 1,398 631 2,808 23.00 75.46 1,395 1,244 593 2,639 21.00 68.90 1,380 1,090 555 2,470 19.00 62.34 1,357 938 516 2,295 17.00 55.77 1,328 789 476 2,116 15.00 49.21 1,298 644 437 1,943 13.00 42.65 1,260 507 397 1,767 11.00 36.09 1,222 380 360 1,602 9.00 29.53 1,203 266 330 1,469 7.00 22.97 1,203 168 308 1,371 5.00 16.40 1,203 89 291 1,292 3.00 9.84 1,203 33 278 1,236 1.00 3.28 602 2 136 603

----- ----- --------- -------- ----- ------ Total 16,063 7,550 5,308 23,612 Pst = Static Wind Load (IS 6533: 8.2.3) | Pdyn = Dynamic Wind Load (IS 6533:8.3.2) Ptot = Total Force: Pst + Summation for all Modes[(Pdyn)^2]^0.5 (IS 6533: Para 8.3.7)

Total Wind Loads (Static + Dynamic) Applied to the Model


----- ----- ------- ------ 25.00 23.00 1,404 2,808 23.00 21.00 1,319 2,639 21.00 19.00 1,235 2,470 19.00 17.00 1,148 2,295 17.00 15.00 1,058 2,116 15.00 13.00 971 1,943 13.00 11.00 884 1,767 11.00 9.00 801 1,602 9.00 7.00 735 1,469 7.00 5.00 685 1,371 5.00 3.00 646 1,292 3.00 1.00 618 1,236 1.00 0.00 603 603

----- ----- ------- ------ Total 23,612

Static Wind Loads Applied to the ModelPer IS 6533 Para 7.4, Deflection Limit only needs to be evaluated for Static Loads


----- ----- ------- ------ 25.00 23.00 705 1,410 23.00 21.00 697 1,395 21.00 19.00 690 1,380 19.00 17.00 679 1,357 17.00 15.00 664 1,328 15.00 13.00 649 1,298 13.00 11.00 630 1,260 11.00 9.00 611 1,222 9.00 7.00 602 1,203 7.00 5.00 602 1,203 5.00 3.00 602 1,203 3.00 1.00 602 1,203 1.00 0.00 602 602

----- ----- ------- ------ Total 16,063

Seismic Loads Per IS 1893 - 1975

All seismic loads are based upon the lowest period calculated of First Mode.

Zone = Seismic Zone = V I = Important Factor = 2.000 Beta = Beta: Soil-Foundation Coeffient = 1.50


Page 147 of 180

Fo = Seismic Zone Factor per IS 1883 Table 2 = 0.40

Del = Log Decrement Damping = 0.0126

n = Natural Frequency = 2.387 Hz

T = Period of Vibration = 0.419 s

h' = Height of Stack = 25.0 m

rg = Radius of Gyration of Base Section = 0.53 mm

K = h' / rg = 47.27

Cv = Coefficient from Table 5 = 1.5

Sa/g = Averge Acceleration Spetra from Fig 2 = 0.261

Ah = Seismic Coefficient: Beta * I * Fo * Sa/g (Para 3.4.2.3.b) = 0.3137

Elev x' x'/h' W W V V

m m lb N lb N

------ ------ ----- ----- ------ ----- ------

24.000 1.000 0.040 651 2,894 20 88

22.000 3.000 0.120 651 2,894 58 256

20.000 5.000 0.200 651 2,894 93 413

18.000 7.000 0.280 651 2,894 125 558

16.000 9.000 0.360 651 2,894 155 692

14.000 11.000 0.440 651 2,894 183 814

12.000 13.000 0.520 651 2,894 208 924

10.000 15.000 0.600 651 2,894 230 1,024

8.000 17.000 0.680 651 2,894 250 1,111

6.000 19.000 0.760 651 2,894 267 1,187

4.000 21.000 0.840 651 2,894 281 1,252

2.000 23.000 0.920 651 2,894 293 1,305

0.500 24.500 0.980 325 1,447 150 669

------ ------ ----- ----- ------ ----- ------

Total 8,133 36,176 2,314 10,293

Elev = Elevation to mid-point of element

x' = Distance from the top of stack: h' - Elev (Ref 5.3.4)

W = Weight of Element

V = Seismic Shear force: Cv*Ah*W*[ (5/3)*(x'/h') - (2/3)*(x'/h')^2] (Para 5.3.4)



Elev Elev Load Load

m m N/m N

----- ----- ------- ------

25.00 23.00 44 88

23.00 21.00 128 256

21.00 19.00 206 413

19.00 17.00 279 558

17.00 15.00 346 692

15.00 13.00 407 814

13.00 11.00 462 924

11.00 9.00 512 1,024

9.00 7.00 556 1,111

7.00 5.00 594 1,187

5.00 3.00 626 1,252

3.00 1.00 653 1,305

1.00 0.00 669 669

----- ----- ------- ------

Total 10,293


Mode Configuration Frequency Crit Wind Reduced Defl Defl Defl Comment


Hz m/s kg m m

---- ----------------- --------- --------- ------- ----- ----- ----- -------------------

1 Hot & UnCorroded 2.387 17.90 921 0.002 0.006 0.26 Pass

1 Cold & Uncorroded 2.387 17.90 921 0.002 0.006 0.26 Pass

2 Hot & UnCorroded 14.865 111.49 896 0.000 0.006 0.00 Pass: Vcr>26.76 m/s

2 Cold & Uncorroded 14.865 111.49 896 0.000 0.006 0.00 Pass: Vcr>26.76 m/s

3 Hot & UnCorroded 41.220 309.15 851 0.000 0.006 0.00 Pass: Vcr>26.76 m/s

3 Cold & Uncorroded 41.220 309.15 851 0.000 0.006 0.00 Pass: Vcr>26.76 m/s


Vortex Shedding AnalysisRef IS 6533 (Part 1) 1989

f = Natural Frequency for Mode # 1 = 2.387 Hz

Bs = Structural Damping = 0.0020

Dt = Top 1/3 Average Diameter = 1.5 m

Sc = Scruton Number (Provided for information only) = 0.00

Vcr = Critical Wind Speed: 5 * Dt *f (IS 6533: Annex A, Eqn 1) = 17.90 m/s

Vcr = Critical Wind Speed in Imperial Units = 40.0 mph

qcr = Speed Thrust for Vcr: Vcr^2 / 16 (IS 6533: Annex A, Eqn 2) = 20.025 kPa

Check for Resonance: (IS 6533: Pt 2, 8.4)

Lined = Is the Stack Lined = False

Class = Stack Height is between 20m and 50 m = B

k2 = Height Factor per IS 875 Table 2 = 1.115

Vz = Wind Speed at Elevation 25.0 m = 33.45 m/s

Vlwr = Lower Limit for Design Wind for Lined Chimney: 0.33*Vz = 11.04 m/s

Vupr = Upper Limit for Design Wind for Lined Chimney: 0.8*Vz = 26.76 m/s


Page 148 of 180

Since Vlwr <= Vcr <= Vupr then Resonance must be checked per IS 6533 Annex A

Resonance Forces per IS 6533 (Part 2) 1989 Annex A Para A-4

Forces are acting perpendicular to the direction of the wind

z z h d Cy Fe1 Phi F1 F1 m ft m m N lb N

----- ----- ----- ---- ---- --- ------ --- --- 25.00 82.02 2.00 1.50 0.25 15 -1.000 -3 -15 23.00 75.46 2.00 1.50 0.25 15 -0.890 -3 -13 21.00 68.90 2.00 1.50 0.25 15 -0.780 -3 -12 19.00 62.34 2.00 1.50 0.25 15 -0.671 -2 -10 17.00 55.77 2.00 1.50 0.25 15 -0.564 -2 -8 15.00 49.21 2.00 1.50 0.25 15 -0.461 -2 -7 13.00 42.65 2.00 1.50 0.25 15 -0.363 -1 -5 11.00 36.09 2.00 1.50 0.25 15 -0.272 -1 -4 9.00 29.53 2.00 1.50 0.25 15 -0.190 -1 -3 7.00 22.97 2.00 1.50 0.25 15 -0.120 0 -2 5.00 16.40 2.00 1.50 0.25 15 -0.064 0 -1 3.00 9.84 2.00 1.50 0.25 15 -0.024 0 0 1.00 3.28 1.00 1.50 0.25 8 -0.003 0 0

----- ----- ----- ---- ---- --- ------ --- --- Total 25.00 188 -18 -81 z = Elevation to top of Element | h = Height of Element d = Outer Diameter | Cy = Coeff of Transverse Force Fe1 = Cy * qcr * d * h (Eqn 2) | Phi = Mode Shape F1 = Aerodynamic Force: Fe1 * Phi (Eqn 2)

Along Wind Loads @ Vcr Acting During Vortex SheddingRef IS 875 (Part 3) 1987 and IS 6533 Part 2: 1989

V = Wind Speed = 17.90 m/s K1 = Topography Factor (Ref IS 875 Para 5.3.1) = 1.00 K3 = Probability Factor (Ref IS 875 Para 5.3.2) = 1.00 Terrain = Terrain Roughness (Ref IS 875 Para 5.3.2.1) = 1 Modes = Number of Modes to be Considered in Dynamic Load Calculations = 1 Class = Stack Height is between 20m and 50 m = B Vcr = Design Wind Speed in Imperial Units = 40.0 mph

Static Wind Load (Critical Wind Speed Vcr)

z z h K2 Vcr qcr Area Pst Pst m ft m m/s kPa sq m lb N

----- ----- ----- ----- ----- ------ ------ --- --- 25.00 82.02 2.00 1.115 17.90 20.025 2.100 9 42 23.00 75.46 2.00 1.109 17.90 20.025 2.100 9 42 21.00 68.90 2.00 1.103 17.90 20.025 2.100 9 42 19.00 62.34 2.00 1.094 17.90 20.025 2.100 9 42 17.00 55.77 2.00 1.082 17.90 20.025 2.100 9 42 15.00 49.21 2.00 1.070 17.90 20.025 2.100 9 42 13.00 42.65 2.00 1.054 17.90 20.025 2.100 9 42 11.00 36.09 2.00 1.038 17.90 20.025 2.100 9 42 9.00 29.53 2.00 1.030 17.90 20.025 2.100 9 42 7.00 22.97 2.00 1.030 17.90 20.025 2.100 9 42 5.00 16.40 2.00 1.030 17.90 20.025 2.100 9 42 3.00 9.84 2.00 1.030 17.90 20.025 2.100 9 42 1.00 3.28 1.00 1.030 17.90 20.025 1.050 5 21

----- ----- ----- ----- ----- ------ ------ --- --- Total 25.00 26.250 118 526 K2 = Factor for Elevation (IS 875 Table 2) | Vz = Vb * K1 * K2 * K3 (IS 875 Para 5.3) q = 0.6 * Vz^2 (IS 875 Para 5.4) | Area = Eff Wind Area incl Shape Factors Pst = Static Force: Area*q (IS 6533: 8.2.3) | h = Height of Element

Dynamic Wind Loading for Mode # 1 (Critical Wind Speed Vcr)

n = Natural Frequency = 2.387 Hz T = Period of vibration = 0.419 s E1 = (1/n) * Vcr / 1200 = 0.006 Coef_E1 = Coefficient of Dynamic Influence (Table 5) = 1.60 Coef_v = Space Correlation Coefficient (Table 7) = 0.70

z z Pst Phi Num Den M mk N Pdyn Pdyn m ft N kg m/s^2 lb N

----- ----- --- ------ ------- ------- ------- ----- ----- ---- ---- 25.00 82.02 42 -1.000 -22.39 295.11 295.1 0.533 0.12 9 39 23.00 75.46 42 -0.890 -20.19 233.69 295.1 0.539 0.10 8 35 21.00 68.90 42 -0.780 -17.93 179.56 295.1 0.547 0.09 7 30 19.00 62.34 42 -0.671 -15.66 132.93 295.1 0.555 0.08 6 26 17.00 55.77 42 -0.564 -13.41 93.99 295.1 0.565 0.07 5 22 15.00 49.21 42 -0.461 -11.15 62.73 295.1 0.575 0.05 4 18 13.00 42.65 42 -0.363 -8.93 38.87 295.1 0.585 0.04 3 14 11.00 36.09 42 -0.272 -6.80 21.83 295.1 0.595 0.03 2 11 9.00 29.53 42 -0.190 -4.80 10.68 295.1 0.600 0.02 2 7 7.00 22.97 42 -0.120 -3.03 4.26 295.1 0.600 0.01 1 5 5.00 16.40 42 -0.064 -1.61 1.20 295.1 0.600 0.01 1 2 3.00 9.84 42 -0.024 -0.60 0.17 295.1 0.600 0.00 0 1 1.00 3.28 21 -0.003 -0.03 0.00 147.6 0.600 0.00 0 0

----- ----- --- ------ ------- ------- ------- ----- ----- ---- ---- Total 526 -126.54 1075.02 3,688.9 47 210 M = Mass Of Element | mk = Coef Of Pulsation (Is 6533 Table 6) Num = Phi * Pst * mk (IS 6533: 8.3.4) | Phi = Normalized Mode Shape


Page 149 of 180

Den = Phi^2 * M (IS 6533: 8.3.4) | Pst = Static Wind Load N = Deduced Accel: Phi * Num_Total / Den_Total (IS 6533: 8.3.4) Pdyn = Dynamic Force: M * E1 * N * v (IS 6533: 8.3.2)

Total Static + Dynamic Wind Load (Critical Wind Speed Vcr)

z z Pst Pdyn Ptot Ptot All Modes Mode # 1 m ft N N lb N

----- ----- --------- -------- ---- ---- 25.00 82.02 42 39 18 81 23.00 75.46 42 35 17 77 21.00 68.90 42 30 16 72 19.00 62.34 42 26 15 68 17.00 55.77 42 22 14 64 15.00 49.21 42 18 13 60 13.00 42.65 42 14 13 56 11.00 36.09 42 11 12 53 9.00 29.53 42 7 11 49 7.00 22.97 42 5 11 47 5.00 16.40 42 2 10 45 3.00 9.84 42 1 10 43 1.00 3.28 21 0 5 21

----- ----- --------- -------- ---- ---- Total 526 210 165 736 Pst = Static Wind Load (IS 6533: 8.2.3) | Pdyn = Dynamic Wind Load (IS 6533:8.3.2) Ptot = Total Force: Pst + Summation for all Modes[(Pdyn)^2]^0.5 (IS 6533: Para 8.3.7)

Calculated Deflection for Vortex Shedding

Dx = Deflection due to Along Wind = 0.06 mm Dz = Deflection due to Across Wind (Perpendicular to Wind Direction) = 0.01 mm Dtot = Max Resultant Deflection: (Dx^2 + Dz^2)^0.5 = 0.06 mm

Resonance Loads Applied to Stack (Loads Perpendicular to Wind Direction)


----- ----- ------- ----- 25.00 23.00 -8 -15 23.00 21.00 -7 -13 21.00 19.00 -6 -12 19.00 17.00 -5 -10 17.00 15.00 -4 -8 15.00 13.00 -3 -7 13.00 11.00 -3 -5 11.00 9.00 -2 -4 9.00 7.00 -1 -3 7.00 5.00 -1 -2 5.00 3.00 0 -1 3.00 1.00 0 0 1.00 0.00 0 0

----- ----- ------- ----- Total -81

Along Wind Loads Applied to the Model (Critieral Wind Speed Vcr)


----- ----- ------- ----- 25.00 23.00 40 81 23.00 21.00 38 77 21.00 19.00 36 72 19.00 17.00 34 68 17.00 15.00 32 64 15.00 13.00 30 60 13.00 11.00 28 56 11.00 9.00 26 53 9.00 7.00 25 49 7.00 5.00 23 47 5.00 3.00 22 45 3.00 1.00 21 43 1.00 0.00 21 21

----- ----- ------- ----- Total 736

Primary Loads


Number Description Abbreviation------ ------------- ------------

1 Dead D 2 Live L 3 Operating O 4 Thermal Hot TH 5 Wind W 6 Wind Static WS 7 Seismic E 8 Vortex Across V


Page 150 of 180

9 Vortex Along V

Load Combinations


Num Description S D FS Dir D L O TH W WS E V V

Deg

--- ----------------- - - ----- --- --- --- --- -- --- -- --- --- -

10 0.9*D+1.6*W X 1.800 0 0.9 1.6

11 0.9*D+1.6*L+1.6*W X 1.800 0 0.9 1.6 1.6

12 D+W X 1.800 0 1.0 1.0

13 0.9*D+1.6*E X 1.800 0 0.9 1.6

14 0.9*D+1.6*L+1.6*E X 1.800 0 0.9 1.6 1.6

15 D+E X 1.800 0 1.0 1.0

16 D+O+P X 1.000 0 1.0 1.0

17 0.9*D+1.6*V X 1.100 0 1.0 1.0 1.6




S: Include in Stress Analysis D: Calculate Deflextions

Stress Criteria: IS 6533 (ASD)

Wind Criteria: IS 875: Part 3 (ASD)

Seismic Criteria: IS 1893 (ASD)

Wind Combinations:

Use the standard load combinations in IS 6533 Part 2, Para 6.5 and 9.1.

Seismic Combinations:

Use the standard combinations in IS 6533 Part 2, Para 6.5 and 9.1.

IS 6533 Part 2 Para 9.1 only addresses wind, but it is applied to seismic also

Check Stiffening Rings per IS 6533 (Part 2) : Para A-9

If thickness > 300 * d then no rings are required

Elev d t tmin Ratio Result

tmin/t

m m mm mm

----- ----- ---- ---- ------ -----------------------------------

25.00 1.500 4.00 5.00 1.25 Fail, Ring Req'd (t less than tmin)













All checks did NOT pass, therefore the stack requires stiffening at one or more locations.

ALLOWABLE STRESS SUMMARY FOR STACK PER INDIAN CODE

Ref IS 6533 Part 2: 1989 'Code of Practice for Design and Construction of Steel Chimney'


Comp Comp


----- ----- ----- ----- ----- ----- ------ ------ ----- ----- ----- -----

23.00 75.46 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

21.00 68.90 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

19.00 62.34 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

17.00 55.77 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

15.00 49.21 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

13.00 42.65 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

11.00 36.09 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

9.00 29.53 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

7.00 22.97 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

5.00 16.40 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

3.00 9.84 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

1.00 3.28 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15

0.00 0.00 50.00 1.496 0.004 33.42 374.00 248.21 0.804 0.593 8.58 59.15



A = 1 / [0.84 + (0.019*he/D)^2] {he/D > 21} -OR- 1 {he/D <= 21}

B = 270 *(t/D) *(1-67*(t/D)) {D/t > 130} -OR- 1 {D/t <= 130}

Allow = Allowable Compressive stress (Annex C): 0.5 * Fyld * A * B

STACK COMPRESSIVE STRESSES PER IS 6533, Load # 10Load Combination #10 0.9*D+1.6*W (Hot & UnCorroded), FS = 1.8

Bot El Shear Moment Vert fa fb ftot ftot Allow Allow Case1

Comp Comp

ft KN KN-m KN MPa MPa MPa ksi ksi MPa

------ ----- ------ ------ ----- ----- ------ ------ ----- ------ -----

75.46 2.25 8.98 -1.30 -0.07 1.28 1.21 0.18 15.44 106.48 0.01

68.90 6.60 3.95 -3.91 -0.21 0.56 0.36 0.05 15.44 106.48 0.00

62.34 10.69 9.80 -6.51 -0.35 1.40 1.05 0.15 15.44 106.48 0.01


Page 151 of 180

55.77 14.50 31.73 -9.12 -0.48 4.53 4.04 0.59 15.44 106.48 0.04

49.21 18.03 61.31 -11.72 -0.62 8.74 8.12 1.18 15.44 106.48 0.08

42.65 21.28 97.93 -14.33 -0.76 13.97 13.20 1.92 15.44 106.48 0.12

36.09 24.25 141.05 -16.93 -0.90 20.12 19.21 2.79 15.44 106.48 0.18

29.53 26.94 190.07 -19.53 -1.04 27.11 26.07 3.78 15.44 106.48 0.24

22.97 29.40 244.38 -22.14 -1.18 34.85 33.67 4.88 15.44 106.48 0.32

16.40 31.67 303.50 -24.74 -1.32 43.28 41.96 6.09 15.44 106.48 0.39

9.84 33.80 367.09 -27.35 -1.45 52.35 50.90 7.38 15.44 106.48 0.48

3.28 35.82 434.87 -29.95 -1.59 62.02 60.42 8.76 15.44 106.48 0.57

0.00 37.30 509.51 -31.91 -1.70 72.66 70.96 10.29 15.44 106.48 0.67

fa = Axial Stress | fb = Bending Stress

ftot = Combined Axial & Bend: fa + fb | Case1 = Unity Ratio: ftot / Allow

Allow = Allowable Compressive Stress Calculated from Annex A multiplied by FS

STACK COMPRESSIVE STRESSES PER IS 6533, Load # 11

Load Combination #11 0.9*D+1.6*L+1.6*W (Hot & UnCorroded), FS = 1.8


Comp Comp


------ ----- ------ ------ ----- ----- ------ ------ ----- ------ -----

75.46 2.25 8.98 -1.30 -0.07 1.28 1.21 0.18 15.44 106.48 0.01

68.90 6.60 3.95 -3.91 -0.21 0.56 0.36 0.05 15.44 106.48 0.00

62.34 10.69 9.80 -6.51 -0.35 1.40 1.05 0.15 15.44 106.48 0.01

55.77 14.50 31.73 -9.12 -0.48 4.53 4.04 0.59 15.44 106.48 0.04

49.21 18.03 61.31 -11.72 -0.62 8.74 8.12 1.18 15.44 106.48 0.08

42.65 21.28 97.93 -14.33 -0.76 13.97 13.20 1.92 15.44 106.48 0.12

36.09 24.25 141.05 -16.93 -0.90 20.12 19.21 2.79 15.44 106.48 0.18

29.53 26.94 190.07 -19.53 -1.04 27.11 26.07 3.78 15.44 106.48 0.24

22.97 29.40 244.38 -22.14 -1.18 34.85 33.67 4.88 15.44 106.48 0.32

16.40 31.67 303.50 -24.74 -1.32 43.28 41.96 6.09 15.44 106.48 0.39

9.84 33.80 367.09 -27.35 -1.45 52.35 50.90 7.38 15.44 106.48 0.48

3.28 35.82 434.87 -29.95 -1.59 62.02 60.42 8.76 15.44 106.48 0.57

0.00 37.30 509.51 -31.91 -1.70 72.66 70.96 10.29 15.44 106.48 0.67




STACK COMPRESSIVE STRESSES PER IS 6533, Load # 16

Load Combination #16 D+O+P (Hot & UnCorroded), FS = 1


Comp Comp


------ ----- ------ ------ ----- ---- ------ ------ ----- ----- -----

75.46 0.00 0.00 -1.45 -0.08 0.00 -0.08 -0.01 8.58 59.15 0.00

68.90 0.00 0.00 -4.34 -0.23 0.00 -0.23 -0.03 8.58 59.15 0.00

62.34 0.00 0.00 -7.24 -0.38 0.00 -0.38 -0.06 8.58 59.15 0.01

55.77 0.00 0.00 -10.13 -0.54 0.00 -0.54 -0.08 8.58 59.15 0.01

49.21 0.00 0.00 -13.02 -0.69 0.00 -0.69 -0.10 8.58 59.15 0.01

42.65 0.00 0.00 -15.92 -0.85 0.00 -0.85 -0.12 8.58 59.15 0.01

36.09 0.00 0.00 -18.81 -1.00 0.00 -1.00 -0.15 8.58 59.15 0.02

29.53 0.00 0.00 -21.71 -1.15 0.00 -1.15 -0.17 8.58 59.15 0.02

22.97 0.00 0.00 -24.60 -1.31 0.00 -1.31 -0.19 8.58 59.15 0.02

16.40 0.00 0.00 -27.49 -1.46 0.00 -1.46 -0.21 8.58 59.15 0.02

9.84 0.00 0.00 -30.39 -1.62 0.00 -1.62 -0.23 8.58 59.15 0.03

3.28 0.00 0.00 -33.28 -1.77 0.00 -1.77 -0.26 8.58 59.15 0.03

0.00 0.00 0.00 -35.45 -1.89 0.00 -1.89 -0.27 8.58 59.15 0.03





Page 152 of 180

Section F - Canadian (NBCC 2010) Wind & Seismic


Page 153 of 180

q = Refrence Wind pressure 0.5 Kpa

No = Natural Frequency of Structure 2.387 Hz

Exp = Exposure (A-Open or Std, B-Rough) A

Iw = Importance Factor 1.25

Cp = External Pressure Coeff 1.00

H = Overall Height of Structure 82.021 ft 25.00 m

W = Width of Structure 4.92 ft 1.50 m

Flex = Flexible Struc (No<=1Hz, H/w>4) TRUE

Override= Override Flexible Criteria FALSE

V = Reference wind speed (q/660e-6)^0.5 62.0 mph 27.74 m/s

Ceh = Exposure Factor at Top of Stack 1.29

Vh = Hourly Mean Wind Speed @ Top 70.5 mph 31.53 m/s

K = Surface Roughness Coeff of Terrain 0.080

B = Background Turbulence Factor (Fig I-18) 1.436

S = Size reduction factor (Fig I-19) 0.081

xo = Paramater from Fig I-20 to calc F 92.36

F = Gust Energy Ratio (Fig I-20) 0.049

Beta = Critical Damping Ratio (Struc+Aero) 0.0068

v = No*(s*F/(S*F+Beta*B))^0.5 1.283

COV = ((K/Ce)*(B+s*F/Beta))^0.5 0.354

gp = Peak Factor of Loading Effect (Fig I-21) 4.248

Cg = N/A Since 0.25 <= No < 1 Hz 2.502

Ce

(ft) (m) (psf) (kPa)

82.021 25.00 1.292 42.22 2.021

65.6168 20.00 1.214 39.67 1.899

49.2126 15.00 1.120 36.60 1.752

32.8084 10.00 1.000 32.67 1.564

Note: p = Iw*q*Ce*Cg*Cp

Elevation Pressure

NBC 2010 Wind


Page 154 of 180

General Parameters:

Sa(0.2): Spectral Response Acceleration at 0.2 sec period = 0.12

Sa(0.5): Spectral Response Acceleration at 0.5 sec period = 0.056

Sa(1.0): Spectral Response Acceleration at 1 Second Period = 0.02

Sa(2.0): Spectral Response Acceleration at 2 Second Period = 0.006

Ht Overall Height of Structure 25.0 m

82.0 ft

NF Natural Frequency of Structure 2.39 Hz

Ta Fundamental Period of Structure 0.419 sec

Importance Category

Site Class =

Ie: Importance Factor = 1.30

Fa - Site Coefficient from Table 11.4-1 = 1.30

Fv - Site Coefficient from Table 11.4-2 = 1.40

Para 4.1.8.7, if any of these are true then simplified procedure acceptable

Check # 1 Ie*Fa*Sa(0.2) < 0.35 0.20 TRUE

Check # 2 Is Ht < 60 m and Period < 2sec TRUE

Simplified Procedure of Para 4.1.8.11 can be Followed:

Para 4.1.8.4 (6)

T <= 0.2s Fa*Sa(0.2) 0.156

T = 0.5s Smaller of Fv*Sa(0.5) or Fa*Sa(0.2) 0.0784

T = 1.0s Fv*Sa(1.0) 0.0322

T = 2.0s Fv*Sa(2.0) 0.0084

T >=4.0s Fv*Sa(2.0) / 2 0.0042

S(Ta) Refer to above chart based upon Ta 0.099

Rd Table 4.1.8.9 for "Other Steel SFRS" 1.00

Ro Table 4.1.8.9 for "Other Steel SFRS" 1.00

Mv Table 4.1.8.11 "other systems" 1.00

Para 4.1.8.11(2) Minimum Lateral EQ Force

V1 S(Ta) * Mv * Ie * W / (Rd * Ro) 0.129 * W 1,050

V shall not be less than

Vmin S(2.0) * Mv * Ie * W / (Rd * Ro) 0.0078 * W 63

V Greater of V1 or Vmin 0.129 * W 1,050

Para 4.1.8.11(6) Force Ft acting at top of Structure

Ft1 0.07 * ta * V 0.004 * W 31

Ftmax Need not exceed 0.25*V 0.032 * W 263

Ft Force to be applied at top of structure 0.000 * W -

NBC 2010 Seismic for Steel Stack


Page 155 of 180

W = 8,129 lbs 36176 N

V = 1,050 lbs 4673 N

Ft = - lbs


Page 156 of 180


Calculations Prepared by: Calculations Prepared For: Meca Enterprises Client: 0 816 W. Elgin St Project #: 0 Broken Arrow, OK, 74012 Location: 0 Date: Apr 23, 2017 Description: Designer: 0 0

File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\NBCC2010\ v5314_NBCC.Stk

INPUT PARAMETERS:




Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------25.0 1.5 | 25.0 4.0 | 25.0 0.0

Materials:


---- -------- ---- ------ -------- -------- ---------- ------- 25.0 A-36 37.8 248.21 199,603 1.22E-05 114.45 7,849 37.8 248.21 199,603 1.22E-05 114.45 7,849

Design Codes

Comprehensive Design Standard: Canadian Standards


Wind Load Criteria: National Building Code of Canada 2010

q = Wind Pressure = 0.500 kPa Iw = Importance Factor = 1.250 Exposure = Ground Roughness = A Override = Override Criteria that Struc Dynamic if H/W>4 **OR** NF<1 Hz = False


Arrange = Stack Arrangement = Single Spacing = Center to center stack spacing (Only required for Group) = 3.048 m S = Manually entered Strouhal Number (Only used for Custom) = 0.2000 Const_Dia = Force the analysis to use Constant Diameter Equations. = False Life = Fatigue life of the stack = 50.0 Years Cycles = Override Number of Cycles (0 to calculate automatically) = 0


Elev Cat Cf Fth 2a w tp m MPa mm mm mm

---- --- -- --- -- -- --


Page 157 of 180

Seismic Criteria: National Building Code of Canada 2010

Sa(0.2) = Spectral Response Acceleration at 0.2 Sec Period = 0.120




IE = Importance Factor for Earthquake Loads and Effects = 1.300

SiteClass = Site Classification for Seismic Site Reponse {Table 4.1.8.4.A} = D

Rd = SFRS Ductility-Related Force Mod Factor {Table 4.1.8.9} = 1.00

Ro = SFRS Overstrength-Related Force Mod Factor {Table 4.1.8.9} = 1.00

Section Properties


Top Bot Outer Thick Rad of Area Plastic Elastic Mom of

Elev Elev Diameter Gyration Sec Modulus Sec Modulus Inertia

m m mm mm mm sq cm cm^3 cm^3 cm^4

----- ----- -------- ----- -------- ------ ----------- ----------- ---------

25.00 23.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

23.00 21.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

21.00 19.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

19.00 17.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

17.00 15.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

15.00 13.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

13.00 11.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

11.00 9.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

9.00 7.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

7.00 5.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

5.00 3.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

3.00 1.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

1.00 0.00 1,500.00 4.00 528.92 187.99 8,952.0 7,012.2 525,912.5

Weight Detail






----- ---- ----- ----- ---- ------- -------- ---------- ----------

25.00 0.00 12.50 1.500 4.00 7,849 3,688.9 147.56 3,688.9

----- ---- ----- ----- ---- ------- -------- ---------- ----------

Total 12.50 3,688.9 3,688.9

Weight Summary


m kg

----------------------------------- -------- -------


----------------------------------- -------- -------

Total 12.500 3,688.9

Wind Areas

Wind Area Summary

Elev Shape Fac Riser Ladder Platform Piping Tot Uni Total

Riser Area Area Area Area Area Area

m m^2/m m^2/m m^2/m m^2/m m^2/m sq m

----- --------- ----- ------ -------- ------ ------- ------

25.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

23.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

21.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

19.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

17.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500


Page 158 of 180

15.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

13.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

11.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

9.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

7.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

5.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

3.00 0.500 0.750 0.000 0.000 0.000 0.750 1.500

1.00 0.500 0.750 0.000 0.000 0.000 0.750 0.750

----- --------- ----- ------ -------- ------ ------- ------

Total 18.750

Frequency Summary


# 1 # 2 # 3

Deg Hz Hz Hz

----------------- --------- ----- ------ ------

Hot & UnCorroded 0.00 2.387 14.865 41.220

Cold & Uncorroded 0.00 2.387 14.865 41.220

Wind Loads per National Building Code of Canada 2010


q = Wind Pressure = 0.500 kPa

Iw = Importance Factor = 1.250

Exposure = Ground Roughness = A

Override = Override Criteria that Struc Dynamic if H/W>4 **OR** NF<1 Hz = False

H = Overall Height of the Structure = 25.00 m

W = Width of Structure at Base = 1.5 m

H/W = Aspect Ratio: H / W = 16.67

No = Natural Frequency of Structure for Mode # 1 = 2.387 Hz

Dynamic = Structure is considered dynamic if No <= 1 Hz and H / w > 4 = True

CeH = Exposure at the top of the stack = 1.292

Vref = Equivalent Wind Speed: (q / 0.00065)^0.5 = 27.74 m/s

Vh = Mean Hourly Wind Speed at Top of Stack = 31.53 m/s

V3 = Equivalent speed for 3 sec gust, 50 yr Recurrence = 41.91 m/s

Vref = Equivalent Wind Speed: (q / 0.00065)^0.5 = 62.0 mph

Vh = Mean Hourly Wind Speed at Top of Stack = 70.5 mph

V3 = Equivalent speed for 3 sec gust, 50 yr Recurrence = 93.8 mph

Bs = Structural Damping for Mode # 1 = 0.0020

Ba = Aerodynamic Damping calcualted per ASCE 7-10 = 0.0048

Beta = Total Damping: Bs + Ba = 0.0068

b = Background Turbulence Factor (Fig I-18) = 1.436

K = Surface Roughness Coefficient of Terrain = 0.080

S = Size reduction factor (Fig I-19) = 0.081

xo = Parameter from Fig I-20 to calculate F = 92.34

F = Gust Energy Ratio (Fig I-20) = 0.049

v = No * (s * F / (S * F + Beta * B))^0.5 = 1.280

COV = ((K / Ce) * (B + s * F / Beta))^0.5 = 0.353

gp = Peak Factor of Loading Effect (Fig I-21) = 4.248

Cg = Dynamic Structure: 1 + gp * COV = 2.501

Elev Elev Ht Area Total Ce Press Press Force Force

Area

m ft m m^2/m sq m psf kPa lb N

----- ----- ----- ----- ------ ----- ----- ----- ----- ------

25.00 82.02 2.00 0.750 1.500 1.292 42.19 2.020 681 3,030

23.00 75.46 2.00 0.750 1.500 1.263 41.22 1.973 665 2,960

21.00 68.90 2.00 0.750 1.500 1.231 40.18 1.924 649 2,886

19.00 62.34 2.00 0.750 1.500 1.197 39.07 1.871 631 2,806

17.00 55.77 2.00 0.750 1.500 1.160 37.87 1.813 611 2,720

15.00 49.21 2.00 0.750 1.500 1.120 36.57 1.751 590 2,626

13.00 42.65 2.00 0.750 1.500 1.076 35.13 1.682 567 2,523


Page 159 of 180

11.00 36.09 2.00 0.750 1.500 1.027 33.53 1.605 541 2,408

9.00 29.53 2.00 0.750 1.500 1.000 32.64 1.563 527 2,344

7.00 22.97 2.00 0.750 1.500 1.000 32.64 1.563 527 2,344

5.00 16.40 2.00 0.750 1.500 1.000 32.64 1.563 527 2,344

3.00 9.84 2.00 0.750 1.500 1.000 32.64 1.563 527 2,344

----- ----- ----- ----- ------ ----- ----- ----- ----- ------

Total 24.00 18.000 7,045 31,336

Area = Represents the effective width of each element incl shape factor

Total Area = Total wind area for each element: Area * Ht

Ce = Exposure factor based upon elevation

Press = Wind Pressure: q*Ce*Cg*Iw (Shape factors incl in Wind Area) {Para 4.1.7.1}

Force = Press * Total Area



Elev Elev Load Load

m m N/m N

----- ----- ------- ------

25.00 23.00 1,515 3,030

23.00 21.00 1,480 2,960

21.00 19.00 1,443 2,886

19.00 17.00 1,403 2,806

17.00 15.00 1,360 2,720

15.00 13.00 1,313 2,626

13.00 11.00 1,262 2,523

11.00 9.00 1,204 2,408

9.00 7.00 1,172 2,344

7.00 5.00 1,172 2,344

5.00 3.00 1,172 2,344

3.00 1.00 1,172 2,344

----- ----- ------- ------

Total 31,336

IBC 2012/ASCE 7-10 Seismic Analysis

All seismic loads are based upon the lowest period calculated for Mode 1.





IE = Importance Factor for Earthquake Loads and Effects = 1.300

SiteClass = Site Classification for Seismic Site Reponse {Table 4.1.8.4.A} = D

Rd = SFRS Ductility-Related Force Mod Factor {Table 4.1.8.9} = 1.00

Ro = SFRS Overstrength-Related Force Mod Factor {Table 4.1.8.9} = 1.00

Fa = Site Coefficient based upon Sa(0.2) per Table 4.1.8.4.B = 1.30

Fv = Site Coefficient based upon Sa(1.0) per Table 4.1.8.4.C = 1.40

Ta = Lowest period of First mode of vibration = 0.419 s

Verify that Equivalent Static Force Procedure can be Used {4.1.8.11}

Param1 = Paramater per 4.1.8.7(a): IE * Fa * Sa(0.2) = 0.203

Check1 = Is Param1 < 0.35 {Para 4.1.8.7(a)} = True

Check2 = Is h < 60 m and Ta < 2 sec? = True

Since either Check1 or Check2 are true, the structure can be analyzed

using the equivalent Static Force procedure described in 4.1.8.11

S(T) = Design Spectral acceleration values from Para 4.1.8.4(7) = 0.099

Mv = Higher Mode Factor {'Other System per Table 4.1.8.11} = 1.00

W = Total Weight of Structure = 36176 N

V1 = Mimimum Lateral Force: S(Ta)*Mv*IE*W/(Rd*Ro) {8.1.8.11(2)} = 4672 N

Vmin = V not less than: Sa(2.0)*Mv*Ie*W/(Rd*Ro) {8.1.8.11(2c)} = 282 N

V = The greater of V1 or Vmin = 4672 N

Ft1 = Force acting at the topof the structure {Para 4.1.8.11(6)} = 137 N

Ftmax = Ft need not exceed 0.25*V {Para 4.1.8.11(6)} = 1168 N

Ft = Since Ta < 0.7 sec, then Ft can be taken as zero {4.1.8.11(6)} = 0.00


Page 160 of 180

hx Wx Wx*hx Fx

m N N-m N

------ ------ --------- -----

24.000 2,894 69,457.4 718

22.000 2,894 63,669.2 658

20.000 2,894 57,881.1 598

18.000 2,894 52,093.0 538

16.000 2,894 46,304.9 478

14.000 2,894 40,516.8 419

12.000 2,894 34,728.7 359

10.000 2,894 28,940.6 299

8.000 2,894 23,152.5 239

6.000 2,894 17,364.3 179

4.000 2,894 11,576.2 120

2.000 2,894 5,788.1 60

0.500 1,447 723.5 7

------ ------ --------- -----

Total 36,176 452,196.3 4,672

Fx = (V - Ft) * Wx * hx / (Summation(Wx*hx)) {4.1.8.11}



Elev Elev Load Load

m m N/m N

----- ----- ------- -----

25.00 23.00 359 718

23.00 21.00 329 658

21.00 19.00 299 598

19.00 17.00 269 538

17.00 15.00 239 478

15.00 13.00 209 419

13.00 11.00 179 359

11.00 9.00 150 299

9.00 7.00 120 239

7.00 5.00 90 179

5.00 3.00 60 120

3.00 1.00 30 60

1.00 0.00 7 7

----- ----- ------- -----

Total 4,672


Page 161 of 180

Section G - Eurocodes


Page 162 of 180

Eurocode Standard - Wind Loads Per EN 1991-1-4:2005+A1:2010


≔Vbo 30 ― ≔Cdir 1 ≔Cseason 1 ≔Co 1 ≔ki 1 ≔k 0.02 ≔Terrain 0

≔h 25 ≔b 1.5 ≔t1 4 ≔ρa 1.225 ――3

≔ξs 0.002 ≔n 2.387


≔z

2523211917151311975310

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=dz

1.51.51.51.51.51.51.51.51.51.51.51.51.51.5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=tz

44444444444444

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔ϕ

1.000 1.0000.890 0.6170.780 0.2440.671 −0.0990.564 −0.3860.461 −0.5920.363 −0.7040.272 −0.7150.190 −0.6340.120 −0.4840.064 −0.3010.024 −0.1280.003 −0.016

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


147.26147.26147.26147.26147.26147.26147.26147.26147.26147.26147.26147.26147.26

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

――

≔Vb =⋅⋅Cdir Cseason Vbo 30 ― ≔Zcr =⋅⎛⎜⎝―5

6

⎞⎟⎠h 20.83

_Terrain

0

1

2

3

4

5

_zo

(( ))

0.003

0.01

0.05

0.3

1

1

_zmin

(( ))

1

1

2

5

10

10

≔zo =||_zo ⟨

Terrain

⟩ || 0.003

≔zmin =||_zmin ⟨

Terrain

⟩ || 1

≔zo2 0.05

≔kr =⋅0.19⎛⎜⎜⎝――zo

zo2

⎞⎟⎟⎠

0.07

0.156

≔_Cr ((z))‖‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||||

||||||||||

if

else

≤z zmin‖‖‖‖

←C⎛⎜⎜⎝

⋅kr ln⎛⎜⎜⎝――zmin

zo

⎞⎟⎟⎠

⎞⎟⎟⎠

‖‖‖‖

←C⎛⎜⎝

⋅kr ln⎛⎜⎝―z

zo

⎞⎟⎠

⎞⎟⎠

((C))

≔Cr ((Z))‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((Z)) 2‖‖‖

←Cj

_Cr ⎛⎝Zj⎞⎠

((C))

≔_Vm ((z)) ⋅⋅_Cr ((z)) Co Vb

≔Vm ((z)) ⋅⋅Cr ((z)) Co Vb

≔σv =⋅⋅kr Vb ki 4.68 ―

≔zt 200 ≔Lt 300

⎛ zo ⎞1.409⎡ ⎤ 42.26⎡ ⎤ ‖ ||


Page 163 of 180

≔α =+0.67 ⋅0.05 ln⎛⎜⎝――zo

1

⎞⎟⎠

0.38 ≔_L ((Z))‖‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||||

||||||||||

if

else

≥Z zmin‖‖‖‖

←L ⋅Lt⎛⎜⎝―Z

zt

⎞⎟⎠

α

‖‖‖‖‖

←L ⋅Lt

⎛⎜⎜⎝――zmin

zt

⎞⎟⎟⎠

α

((L))

≔_fl (( ,z n)) ―――⋅n _L ((z))

_Vm ((z))

=Cr ((z))

1.4091.3961.3811.3661.3491.3291.3071.2811.2491.211.1581.0780.906

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Vm ((z))

42.2641.8741.4440.9840.4639.8739.238.4237.4836.334.7332.3427.19

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― ≔_Sl (( ,z n)) ―――――――⋅6.8 _fl (( ,z n))

⎛⎝ +1 ⋅10.2 _fl (( ,z n))⎞⎠

―5

3

≔L ((Z))‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((Z)) 2‖‖‖

←Lj

_L ⎛⎝Zj⎞⎠

((L))

≔_B2 ((z)) ―――――――1

+1 ⋅0.9⎛⎜⎝―――

+b h

_L ((z))

⎞⎟⎠

0.63≔_Iv ((z))

‖‖‖‖‖‖‖‖‖‖‖‖‖

||||||||||||

||||||||||

if

else

<z zmin‖‖‖‖

←l ――――σv

_Vm ⎛⎝zmin⎞⎠

‖‖‖‖

←l ―――σv

_Vm ((z))

((l))

≔va ⋅15 10 −6 ――2

≔qb =⋅⋅0.5 ρa Vb2 551.25

≔_qp ((z)) ⋅⋅⋅⎛⎝ +1 ⋅7 _Iv ((z))⎞⎠ 0.5 ρa _Vm ((z))2

≔_Ce ((z)) ―――_qp ((z))

qb≔Iv ((Z))

‖‖‖‖‖‖‖

||||

|

for ∊ ||

|

j , ‥0 1 −rows ((Z)) 2‖‖‖

←Ij

_Iv ⎛⎝Zj⎞⎠

((I))≔_V ((z))‾‾‾‾‾‾‾‾――――

⋅2 _qp ((z))

ρa≔_Re (( ,b z)) ―――

⋅b _V ((z))

va

≔_cfo1 (( ,b z)) ―――――0.11

⎛⎜⎝――――_Re (( ,b z))

10 6

⎞⎟⎠

1.4≔_cfo2 (( ,,k b z)) +1.2 ――――――――

⋅0.18 log⎛⎜⎝

⋅10⎛⎜⎝―k

b

⎞⎟⎠

⎞⎟⎠

+1 ⋅0.4 log⎛⎜⎝――――_Re (( ,b z))

106

⎞⎟⎠

≔λ =_λ (( ,h b)) 15.238

≔_cfo (( ,,k b z)) max ⎛⎝ ,_cfo1 (( ,b z)) _cfo2 (( ,,k b z))⎞⎠

≔ψλ =_ψλ (( ,h b)) 0.746

_δa (( ,,k b z))

≔_cf (( ,,k b z)) ⋅_cfo (( ,,k b z)) ψλ

Calculate parameters at Zs:

≔Zs =⋅0.6 h 15 ≔Lzs =_L ⎛⎝Zs⎞⎠ 112.24 ≔Czs =_Cr ⎛⎝Zs⎞⎠ 1.329 ≔Vzs =_Vm ⎛⎝Zs⎞⎠ 39.87 ―

=_fl ⎛⎝ ,Zs n⎞⎠ 6.72 =_Sl ⎛⎝ ,Zs n⎞⎠ 0.039 ≔Vms =_Vm ⎛⎝Zs⎞⎠ 39.87 ― ≔Vs =_V ⎛⎝Zs⎞⎠ 53.81 ―

=_λ ⎛⎝ ,Zs b⎞⎠ 10 ≔ψλ =_ψλ ⎛⎝ ,Zs b⎞⎠ 0.7

=_Re ⎛⎝ ,b Zs⎞⎠ ⋅5.38 10 6 =_cfo ⎛⎝ ,,k b Zs⎞⎠ 0.66 ≔cf =_cf ⎛⎝ ,,k b Zs⎞⎠ 0.492


Page 164 of 180

=_qp ⎛⎝Zs⎞⎠ 1773.81 =_Iv ⎛⎝Zs⎞⎠ 0.117 =_cfo1 ⎛⎝ ,b Zs⎞⎠ 0.01

≔mode 1

≔me =Calc_me ⎛⎝ ,,z mz ϕ⟨⟨ −mode 1⟩⟩⎞⎠ 147.3 ――

≔δa =―――――⋅⋅⋅cf ρa b Vms

⋅⋅2 n me0.0513 ≔δs =⋅⋅ξs 2 0.0126 ≔δ =+δa δs 0.0639

≔ηh =⋅―――⋅4.6 h

_L ⎛⎝Zs⎞⎠_fl ⎛⎝ ,Zs n⎞⎠ 6.885 ≔Rh =−―

1

ηh⋅―――

1

⋅2 ηh2

⎛⎝ −1 ⋅−2 ηh⎞⎠ 0.135

≔ηb =⋅―――⋅4.6 b

_L ⎛⎝Zs⎞⎠_fl ⎛⎝ ,Zs n⎞⎠ 0.413 ≔Rb =−―

1

ηb⋅―――

1

⋅2 ηb2

⎛⎝ −1 ⋅−2 ηb⎞⎠ 0.773

≔R2 =⋅⋅⋅――2

⋅2 δ_Sl ⎛⎝ ,Zs n⎞⎠ Rh Rb 0.313 ≔B2 =_B2 ⎛⎝Zs⎞⎠ 0.734

≔v =

‖‖‖‖‖‖‖‖‖

||||||||

←v ⋅n‾‾‾‾‾‾‾‾―――

R2

+B2 R2|||

if <v 0.08‖‖ ←v 0.08

((v))

1.305 ≔T 600 ≔kp =+‾‾‾‾‾‾‾‾‾⋅2 ln (( ⋅v T)) ―――――0.6

‾‾‾‾‾‾‾‾‾⋅2 ln (( ⋅v T))3.815

≔cscd =――――――――――+1 ⋅⋅⋅2 kp _Iv ⎛⎝Zs⎞⎠ ‾‾‾‾‾‾‾+B2 R2

+1 ⋅7 _Iv ⎛⎝Zs⎞⎠1.052 ≔qb =⋅⋅―

1

2ρa Vb

2 0.551

≔cs =―――――――+1 ⋅7 _Iv ⎛⎝Zs⎞⎠ ‾‾‾B2

+1 ⋅7 _Iv ⎛⎝Zs⎞⎠0.935 ≔cd =――――――――――

+1 ⋅⋅⋅2 kp _Iv ⎛⎝Zs⎞⎠ ‾‾‾‾‾‾‾+B2 R2

+1 ⋅⋅7 _Iv ⎛⎝Zs⎞⎠ ‾‾‾B2

1.125

=z

2523211917151311

975310

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Cr ((z))

1.4091.3961.3811.3661.3491.3291.3071.2811.2491.211.1581.0780.906

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Vm ((z))

42.2641.8741.4440.9840.4639.8739.238.4237.4836.334.7332.3427.19

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― =Iv ((z))

0.1110.1120.1130.1140.1160.1170.1190.1220.1250.1290.1350.1450.172

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Ce ((z))

3.5233.4723.4173.3573.2913.2183.1353.0392.9252.7862.6042.3391.812

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=qp ((z))

1.9421.9141.8841.8511.8141.7741.7281.6751.6131.5361.4361.2890.999

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=qp ((z))

40.5639.9839.3538.6537.8937.05

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

≔Fw (( ,,k b Z))‖‖‖‖‖‖‖

||||||

for ∊ ||||

j , ‥0 1 −rows ((Z)) 2‖‖‖

←Fj

⋅⋅⋅⋅_cf ⎛⎝

,,k bjZj⎞⎠cscd _qp ⎛

⎝Zj⎞⎠dz

j⎛⎝

−zj

z+j 1

⎞⎠

((F))

0.664⎡ ⎤ 0.495⎡ ⎤


Page 165 of 180

≔Fw1 =Fw ⎛⎝ ,,k dz z⎞⎠

303329872937288328232756268125952492236722041968

750

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=cfo ⎛⎝ ,,k dz z⎞⎠

0.6640.6630.6620.6620.6610.660.6590.6580.6570.6550.6520.6480.639

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=cf ⎛⎝ ,,k dz z⎞⎠

0.4950.4940.4940.4940.4930.4920.4920.4910.490.4880.4870.4840.476

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Fwind =Vsum ⎛⎝Fw1⎞⎠ 32475

Vortex Shedding Per EN 1991-1-4:2005+A1:2010 Annex E - Para E.1.5.2 'Approach 1'


Page 166 of 180


≔St 0.18 =Zcr 20.83 =δs 0.0126 =me 147.26 ―― =ρa 1.23 ――3

≔_Vcrit(( ,b n)) ――

⋅b n

St≔Vcrit =_Vcrit

(( ,b n)) 19.89 ― ≔Vm =_Vm ⎛⎝Zcr⎞⎠ 41.41 ―

=⋅1.25 Vm 51.76 ―

≔Check_Vortex =‖‖‖‖‖‖‖‖

|||||||

|||||

|

if

else

>Vcrit ⋅1.25 Vm

‖‖ ((“No Vortex Check Req'd:Vcrit >1.25*Vm”))

‖‖ ((“Vortex Check Required”))

“Vortex Check Required”

≔Sc =――――⋅⋅2 δs me

⋅ρa b21.34 ≔Re =―――

⋅Vcrit b

va⋅1.99 10 6

=va⎛⎝ ⋅1.5 10 −5⎞⎠ ――

2

≔Mode 1

Evaluate Nodes and AntiNodes:

≔ϕi =ϕ⟨⟨ −Mode 1⟩⟩

10.890.780.670.560.460.360.270.190.120.060.020

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔N =Nodes ⎛⎝ ,z ϕi⎞⎠ 0[[ ]] ≔n1 =rows ((N)) 1 ≔N =Vshift (( ,,N 0 h))25

0⎡⎢⎣

⎤⎥⎦

≔M =AntiNodes ⎛⎝ ,z ϕi⎞⎠ 25[[ ]] ≔m1 =rows ((M)) 1

Iteratively solve for yf:

Gu

ess

Valu

es

Co

nst

rain

tsSo

lver

≔yf 0.1

≔Kw _Kw⎛⎝ ,,,,,N M ϕi z yf b⎞⎠

≔K _K ⎛⎝ ,,,N M ϕi z⎞⎠

＝―yf

b⋅⋅⋅⋅――

1

St2――

1

ScK Kw _clat (( ,,n b M))

≔ =⎛⎝ ⎞⎠ 0.519

=―b

0.35 ≔Lj =_Lj ⎛⎝ , b⎞⎠ 13.42 =_Lj_range ⎛⎝

,Lj M0⎞⎠

2511.58

⎡⎢⎣

⎤⎥⎦

≔Vm =_Vm ⎛⎝Zcr⎞⎠ 41.41 ―

17176⎡ ⎤ 34352⎡ ⎤


Page 167 of 180

=_Fw ⎛⎝ ,,,mz n ϕi ⎞⎠

17176152871339711525

9687791862354672326320611099

41252

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― ≔Fwtot =_Fwtot ⎛⎝ ,,,,mz n ϕi z⎞⎠

34352305742679523050193751583612470

9344652741222199

82452

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Vsum ⎛⎝Fwtot⎞⎠ 185520

≔ε0 0.3 ≔Tyrs 20 ≔vo =⋅0.2 Vm 8.28 ― ≔T =⋅⋅⋅3.2 10 7 Tyrs 1 sec ⎛⎝ ⋅6.4 10 8 ⎞⎠

=Vm 41.41 ― =Vcrit 19.89 ― ≔N =⋅⋅⋅⋅⋅2 T n ε0

⎛⎜⎜⎝――Vcrit

vo

⎞⎟⎟⎠

2⎛⎜⎜⎝−


vo

⎞⎟⎟⎠

2⎞⎟⎟⎠ ⋅1.651 10 7



Page 168 of 180


=Sc 1.34 =Re ⋅1.99 106 =St 0.18 =me 147.26 ――

≔Ka =_Ka_max((Re)) 1 ≔Cc =_Cc

((Re)) 0.01 ≔aL 0.4

≔kp =⋅‾‾2⎛⎜⎜⎝

+1 ⋅1.2 atan⎛⎜⎜⎝

⋅0.75⎛⎜⎝―――

Sc

⋅⋅4 Ka

⎞⎟⎠

4 ⎞⎟⎟⎠

⎞⎟⎟⎠

1.41

≔c1 =⋅――aL

2

2

⎛⎜⎝

−1 ―――Sc

⋅⋅4 Ka

⎞⎟⎠

0.07 ≔c2 =⋅⋅⋅―――⋅ρa b2

me――aL

2

Ka

――Cc

2

St4―b

h⋅1.7117 10−5

≔σy =⋅⎛⎜⎝

‾‾‾‾‾‾‾‾‾‾‾+c1

‾‾‾‾‾‾+c12 c2

⎞⎟⎠ b 0.57

≔ymax =⋅σy kp 0.8 For comparison, Method 1 gave ==> = 0.52 =――ymax

1.55

≔ε0 0.15 ≔Tyrs 20 ≔vo =⋅0.2 Vm 8.28 ― ≔T =⋅⋅⋅3.2 10 7 Tyrs 1 sec ⎛⎝ ⋅6.4 10 8 ⎞⎠

≔N =⋅⋅⋅⋅⋅2 T n ε0


vo

⎞⎟⎟⎠

2⎛⎜⎜⎝−


vo

⎞⎟⎟⎠

2⎞⎟⎟⎠ ⋅8.254 10 6

=_Fw ⎛⎝ ,,,mz n ϕi ymax⎞⎠

265762365320730178331498912252

96477229505031891701

63880

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

― ≔Fwtot =_Fwtot ⎛⎝ ,,,,mz n ϕi ymax z⎞⎠

531534730641459356662997824503192941445810099

637834021276

80

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Vsum ⎛⎝Fwtot⎞⎠ 287052

=Sstack ⎛⎝ ,z Fwtot⎞⎠

53.15100.46141.92177.58207.56232.07251.36265.82275.92282.29285.7286.97287.05

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Mstack ⎛⎝ ,z Fwtot⎞⎠

53.15206.76449.14768.64

1153.791593.422076.842594.023135.753693.964261.954834.625121.63

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅


Page 169 of 180

Stress Analysis per EN 1993-1-6

≔Fab_Class “C” ≔Case 4 ≔E 199603 ≔fyk 248.21 ≔γM1 1.1

≔Fw_fac =⋅1.5 Fw1

4.554.484.414.324.234.134.023.893.743.553.312.951.13

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Sstack ⎛⎝ ,z Fw_fac⎞⎠

4.559.03

13.4317.7621.9926.1330.1534.0437.7841.3344.6447.5948.71

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Mstk =Mstack ⎛⎝ ,z Fw_fac⎞⎠

4.5518.1340.5971.78

111.53159.65215.93280.11351.93431.04517609.23657.38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅

≔Fw_tot =Vsum ⎛⎝Fw_fac⎞⎠ 48.71

≔Vstk =⋅1.1 Vstack ⎛⎝ ,z mz⎞⎠

3.186.359.53

12.7115.8819.0622.2425.4228.5931.7734.9538.1239.71

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Nb =_Nb⎛⎝Vstk

⎞⎠

3.186.359.53

12.7115.8819.0622.2425.4228.5931.7734.9538.1239.71

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Ncrit =_Ncrit⎛⎝ ,Vstk Mstk

⎞⎠

7264.647264.647264.647264.647264.647264.647264.647264.647264.647264.647264.647264.647264.64

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Nratio =――Nb

Ncrit

00.0010.0010.0020.0020.0030.0030.0030.0040.0040.0050.0050.005

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔ηb =_ηb ⎛⎝Nb⎞⎠

0.040.060.080.090.10.110.120.120.130.140.140.150.15

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Mb' =_Mb' ⎛⎝ ,,ηb Nratio Mstk⎞⎠

4.5518.1340.5971.78

111.53159.65215.93280.11351.93431.04517609.23657.38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(( ⋅ ))

=ω

456.44456.44456.44456.44456.44456.44456.44456.44456.44456.44

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Cxb

111111111⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Cx =_vCx⎛⎝ ,Mstk Vstk

⎞⎠

0.60.60.60.60.60.60.60.60.6⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

_

3.42⎡ ⎤ 0.245⎡ ⎤


Page 170 of 180

_

≔∆wk =_v∆wk

3.423.423.423.423.423.423.423.423.423.423.423.423.42

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔αx =_vαx ⎛⎝ ,tz ∆wk⎞⎠

0.2450.2450.2450.2450.2450.2450.2450.2450.2450.2450.2450.2450.245

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔λxo =_vλxo ⎛⎝ ,Mb' Vstk⎞⎠

0.20.20.20.20.20.20.20.20.20.20.20.2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔λpx =_vλpx ⎛⎝αx⎞⎠

0.7830.7830.7830.7830.7830.7830.7830.7830.7830.7830.7830.7830.783

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σxRcr =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝Cx⎞⎠ 1

‖‖‖

←Vj

_σxRcr ⎛⎜⎝

,,Cxjdz

jtzj⎞⎟⎠

((V))

386.43386.43386.43386.43386.43386.43386.43386.43386.43386.43386.43386.43386.43

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔λx =_λx ⎛⎝σxRcr⎞⎠

0.80.80.80.80.80.80.80.80.80.80.80.80.8

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔χx =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝αx⎞⎠ 2‖‖‖

←Vj

_χx ⎛⎜⎝

,,,,λxjλxo

jαx

jβ η⎞

⎟⎠

((V))

0.380.380.380.380.380.380.380.380.380.380.380.38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σxRk =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝χx⎞⎠ 1‖‖‖

←Vj

_σxRk ⎛⎜⎝χx

j⎞⎟⎠

((V))

94.8394.8394.8394.8394.8394.8394.8394.8394.8394.8394.8394.83

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σxRd =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝χx⎞⎠ 1‖‖‖

←Vj

_σxRd ⎛⎜⎝χx

j⎞⎟⎠

((V))

86.2186.2186.2186.2186.2186.2186.2186.2186.2186.2186.2186.21

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=Cθ

0.60.60.60.60.6⋮

⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

=αθ

0.50.50.50.50.5⋮

⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

≔Cθs =_vCθs

0.60.60.60.60.6⋮

⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

≔λθo =_vλθo

0.40.40.40.4⋮

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

1.118⎡ ⎤ 1.57⎡ ⎤


Page 171 of 180

≔λpθ =_vλpθ ⎛⎝αθ⎞⎠

1.1181.1181.1181.1181.1181.1181.1181.1181.1181.1181.1181.1181.118

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σθRcr =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ((z)) 2‖‖‖

←Vj

_σθRcr ⎛⎜⎝

,,h dzjtzj⎞⎟⎠

((V))

1.571.571.571.571.571.571.571.571.571.571.571.571.57

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔λθ =_λθ ⎛⎝σθRcr⎞⎠

12.5712.5712.5712.5712.5712.5712.5712.5712.5712.5712.5712.5712.57

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔χθ =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝αx⎞⎠ 2‖‖‖

←Vj

_χθ ⎛⎜⎝

,,,,λθjλθo

jαθ

jβ η⎞

⎟⎠

((V))

0.00320.00320.00320.00320.00320.00320.00320.00320.00320.00320.00320.0032

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σθRk =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝χx⎞⎠ 1‖‖‖

←Vj

_σθRk ⎛⎜⎝χθ

j⎞⎟⎠

((V))

0.790.790.790.790.790.790.790.790.790.790.790.79

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σθRd =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝χθ⎞⎠ 1‖‖‖

←Vj

_σθRd ⎛⎜⎝χθ

j⎞⎟⎠

((V))

0.710.710.710.710.710.710.710.710.710.710.710.71

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔fxEN =‖‖‖‖‖‖‖‖‖

|||||||

|

for ∊ |||||

|

j ‥0 −rows ⎛⎝Vstk⎞⎠ 1

‖‖‖‖‖‖

←A _Acyl⎛⎜⎝

,dzjtzj⎞⎟⎠

←Vj

_fxEN ⎛⎜⎝

,VstkjA⎞

⎟⎠

((V))

0.170.340.510.680.841.011.181.351.521.691.862.032.11

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔fxEM =‖‖‖‖‖‖‖‖‖

|||||||

|

for ∊ |||||

|


‖‖‖‖‖‖

←S _Scyl ⎛⎜⎝

,dzjtzj⎞⎟⎠

←Vj

_fxEM ⎛⎜⎝

,Mb'jS⎞

⎟⎠

((V))

0.652.595.79

10.2415.9122.7730.7939.9550.1961.4773.7386.8893.75

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0.82⎡ ⎤ 0.24⎡ ⎤


Page 172 of 180

≔fxEd =‖‖‖‖‖‖‖

||||||

for ∊ ||||


‖‖‖

←Vj

_fxEd ⎛⎜⎝

,,,Mb'jVstk

jdz

jtzj⎞⎟⎠

((V))

0.822.926.3

10.9116.7523.7831.9841.351.7163.1675.5988.9195.86

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔fθEd =‖‖‖‖‖‖‖

||||||

for ∊ ||||


‖‖‖

←Vj

_fθEd ⎛⎜⎝

,,,,_qp ⎛⎝zj⎞⎠

0 h dzjtzj⎞⎟⎠

((V))

0.240.230.230.230.220.220.210.20.20.190.170.160.12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σxRd =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝χx⎞⎠ 1‖‖‖

←Vj

_σxRd ⎛⎜⎝χx

j⎞⎟⎠

((V))

86.2186.2186.2186.2186.2186.2186.2186.2186.2186.2186.2186.21

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔σθRd =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝χx⎞⎠ 1‖‖‖

←Vj

_σθRd ⎛⎜⎝χθ

j⎞⎟⎠

((V))

0.710.710.710.710.710.710.710.710.710.710.710.71

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔kx =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝χx⎞⎠ 1‖‖‖

←Vj

_kx ⎛⎜⎝χx

j⎞⎟⎠

((V))

1.541.541.541.541.541.541.541.541.541.541.541.54

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔kθ =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝χx⎞⎠ 1‖‖‖

←Vj

_kθ ⎛⎜⎝χθ

j⎞⎟⎠

((V))

1.251.251.251.251.251.251.251.251.251.251.251.25

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔ki =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝χx⎞⎠ 1‖‖‖

←Vj

_ki ⎛⎜⎝

,χxjχθ

j⎞⎟⎠

((V))

⋅1.464 10−6

⋅1.464 10−6

⋅1.464 10−6

⋅1.464 10−6

⋅1.464 10−6

⋅1.464 10−6

⋅1.464 10−6

⋅1.464 10−6

⋮

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≔Unity =‖‖‖‖‖‖‖

||||||

for ∊ ||||

j ‥0 −rows ⎛⎝χx⎞⎠ 1‖‖‖

←Vj

Unity_Eqn8_19 ⎛⎜⎝

,,,,,,,,,σxRdjσθRd

j1 fxEd

jfθEd

j0 kx

jkθ

j1 ki

j⎞⎟⎠

((V))

0.250.250.260.280.310.360.430.530.650.810.991.2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


Page 173 of 180


Calculations Prepared by: Calculations Prepared For: Client: Joe Mama Date: Aug 01, 2017 Project #: BR 549 Designer: CR Location: Europe Description: Euro Stack standard 25 m

File Location: C:\Users\Chris Rosencutter\Documents\Software\MecaStack\Validation\Eurocode\ v5334_Euro_25m.Stk

INPUT PARAMETERS:

* Linear Static Analysis * Stress analysis based upon Allowable Stress Design



Stack Geometry:


--------- -------------- - --------- --------- - --------- ---------------25.0 1.5 | 25.0 4.0 | 25.0 0.0

Materials:


---- -------- ---- ------ -------- -------- ---------- ------- 25.0 A-36 37.8 248.21 199603 1.22E-05 114.45 7849 37.8 248.21 199603 1.22E-05 114.45 7849

Design Codes

Comprehensive Design Standard: EuroCode Standards

Stress Criteria: Eurocde 3 - Design of Steel structures: Part 3-2: Towers, masts and chimneys

Wind Load Criteria: Eurocode: EN 1991-1-4:2005 'Actions on Structures'

No valid wind criteria was specified

Vortex Shedding Criteria: Eurocode: EN 1991-1-4:2005 'Actions on Structures'

Spacing = Center to center stack spacing (Only required for Group) = 0.0 m Group = Stack Arrangement = Single S = Manually entered Strouhal Number (Only used for Custom) = 0.2 Simplified= Use the simplified method found in Para E.1.5.3 'Approach 2' = True

Fatigue Criteria: Eurocde 3 - Design of Steel structures: Part 3-2: Towers, masts and chimneys

Seismic Criteria: Eurocode 8 - Design of Structures for earthquak resistance: Part 6: Towers, masts and chimneys




Damping Criteria:

Structural Damping Criteria: Criteria: ASME STS-1-2016 Stack Type: UnLined Support: RIGID

Mode Bs Ds Struc Damping Log Decrement Damping

---- ------------- --------------------- 1 0.002 0.0126 2 0.002 0.0126 3 0.002 0.0126 4 0.002 0.0126 5 0.002 0.0126

Log Decrement Damping = 2 * PI * Bs

Aerodynamic Damping Criteria: (Only used in Along wind analysis, Not considered for vortex shedding)

Criteria: ASME STS-1-2016

Weight Summary


Page 1

file:///C:/Program%20Files%... 8/1/2017 11:48:10 AM


Page 174 of 180

m kg

----------------------------------- -------- ------

Cylinders (Corroded Wt = 3688.9 kg) 12.5 3688.9

----------------------------------- -------- ------

Total 12.5 3688.9





----- ---- ----- --- --- ------- -------- ---------- ----------

25.00 0.00 12.50 1.5 4.0 7849 3688.9 147.56 3688.9

----- ---- ----- --- --- ------- -------- ---------- ----------

Total 12.50 3688.9 3688.9

Wind Area Summary

Elev Shape Fac Riser Ladder Platform Piping Attach Tot Uni Total

Riser Area Area Area Area Area Area Area

m m^2/m m^2/m m^2/m m^2/m m^2/m m^2/m sq m

----- --------- ----- ------ -------- ------ ------ ------- ------

25.00 0.495 0.742 0.000 0.000 0.000 0.000 0.742 1.484

23.00 0.494 0.742 0.000 0.000 0.000 0.000 0.742 1.483

21.00 0.494 0.741 0.000 0.000 0.000 0.000 0.741 1.482

19.00 0.494 0.740 0.000 0.000 0.000 0.000 0.740 1.481

17.00 0.493 0.740 0.000 0.000 0.000 0.000 0.740 1.479

15.00 0.492 0.739 0.000 0.000 0.000 0.000 0.739 1.477

13.00 0.492 0.738 0.000 0.000 0.000 0.000 0.738 1.475

11.00 0.491 0.736 0.000 0.000 0.000 0.000 0.736 1.473

9.00 0.49 0.735 0.000 0.000 0.000 0.000 0.735 1.469

7.00 0.488 0.733 0.000 0.000 0.000 0.000 0.733 1.465

5.00 0.487 0.730 0.000 0.000 0.000 0.000 0.730 1.460

3.00 0.484 0.725 0.000 0.000 0.000 0.000 0.725 1.451

1.00 0.476 0.714 0.000 0.000 0.000 0.000 0.714 0.714

----- --------- ----- ------ -------- ------ ------ ------- ------

Total 0.000 18.394

Section Properties


Top Bot Outer Thick Rad of Area Plastic Elastic Mom of Section

Elev Elev Diameter Gyration Sec Mod Sec Mod Inertia Props

m m mm mm mm sq cm cm^3 cm^3 cm^4

----- ----- -------- ----- -------- ------ ------- ------- -------- -------

25.00 23.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

23.00 21.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

21.00 19.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

19.00 17.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

17.00 15.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

15.00 13.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

13.00 11.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

11.00 9.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

9.00 7.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

7.00 5.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

5.00 3.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

3.00 1.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

1.00 0.00 1500.0 4.0 528.92 187.99 8952.0 7012.2 525912.5 Ideal

Section Props - If a number is shown, then the section contains something unusual

(Breech, nozzle, stiffeners, etc..) and so the propoerties were

calculated using numerical methods.

- 'Ideal' indicates the properties calculated using ideal equations

Thermal Expansion of Stack

+ values are in the +Y (up) direction and - values are in the -Y (Down)

Elev Installation Hot Hot Cold Cold

Temperature Temperature Expansion Temperature Expansion

m Deg C Deg C mm Deg C mm

----- ------------ ----------- --------- ----------- ---------

25.00 21.11 3.21 5.1 3.21 5.1

23.00 21.11 3.21 4.69 3.21 4.69

21.00 21.11 3.21 4.28 3.21 4.28

19.00 21.11 3.21 3.88 3.21 3.88

17.00 21.11 3.21 3.47 3.21 3.47

15.00 21.11 3.21 3.06 3.21 3.06

13.00 21.11 3.21 2.65 3.21 2.65

11.00 21.11 3.21 2.24 3.21 2.24

9.00 21.11 3.21 1.84 3.21 1.84

7.00 21.11 3.21 1.43 3.21 1.43

5.00 21.11 3.21 1.02 3.21 1.02

3.00 21.11 3.21 0.61 3.21 0.61

1.00 21.11 3.21 0.2 3.21 0.2

1. Elevation for Y restraint = 0.0 m

Frequency Summary

Description Direction Mode Mode Mode Mode Mode

# 1 # 2 # 3 # 4 # 5

Deg Hz Hz Hz Hz Hz

Page 2



Page 175 of 180

----------------- --------- ----- ------ ------ ------ -------

Hot & UnCorroded 0.0 2.387 14.865 41.220 79.674 129.491

Cold & Uncorroded 0.0 2.387 14.865 41.220 79.674 129.491



(2.387 Hz) (14.865 Hz) (41.220 Hz) (79.674 Hz) (129.491 Hz)

m Normalized Normalized Normalized Normalized Normalized

--------- ---------- ----------- ----------- ----------- ------------

25.00 -1.000 1.000 -1.000 -1.000 -1.000

23.00 -0.890 0.617 -0.371 -0.124 0.109

21.00 -0.780 0.244 0.187 0.514 0.664

19.00 -0.671 -0.099 0.559 0.643 0.314

17.00 -0.564 -0.386 0.656 0.248 -0.444

15.00 -0.461 -0.592 0.468 -0.350 -0.712

13.00 -0.363 -0.704 0.080 -0.705 -0.173

11.00 -0.272 -0.715 -0.355 -0.552 0.561

9.00 -0.190 -0.634 -0.673 0.004 0.650

7.00 -0.120 -0.484 -0.762 0.573 -0.013

5.00 -0.064 -0.301 -0.607 0.767 -0.690

3.00 -0.024 -0.128 -0.306 0.498 -0.664

1.00 -0.003 -0.016 -0.044 0.083 -0.131

0.00 0.000 0.000 0.000 0.000 0.000



(2.387 Hz) (14.865 Hz) (41.220 Hz) (79.674 Hz) (129.491 Hz)

m Normalized Normalized Normalized Normalized Normalized

--------- ---------- ----------- ----------- ----------- ------------

25.00 -1.000 1.000 -1.000 -1.000 -1.000

23.00 -0.890 0.617 -0.371 -0.124 0.109

21.00 -0.780 0.244 0.187 0.514 0.664

19.00 -0.671 -0.099 0.559 0.643 0.314

17.00 -0.564 -0.386 0.656 0.248 -0.444

15.00 -0.461 -0.592 0.468 -0.350 -0.712

13.00 -0.363 -0.704 0.080 -0.705 -0.173

11.00 -0.272 -0.715 -0.355 -0.552 0.561

9.00 -0.190 -0.634 -0.673 0.004 0.650

7.00 -0.120 -0.484 -0.762 0.573 -0.013

5.00 -0.064 -0.301 -0.607 0.767 -0.690

3.00 -0.024 -0.128 -0.306 0.498 -0.664

1.00 -0.003 -0.016 -0.044 0.083 -0.131

0.00 0.000 0.000 0.000 0.000 0.000


Mode Configuration Frequency Crit Wind Reduced Defl Defl Defl Comment


Hz m/s kg m m

---- ----------------- --------- --------- ------- ------ ----- ----- -------

1 Hot & UnCorroded 2.387 19.89 921 0.8022 0.075 10.70 8266143

1 Cold & Uncorroded 2.387 19.89 921 0.8022 0.075 10.70 8266143

2 Hot & UnCorroded 14.865 0.00 0 0.0 0.0 0.00

2 Cold & Uncorroded 14.865 0.00 0 0.0 0.0 0.00

3 Hot & UnCorroded 41.220 0.00 0 0.0 0.0 0.00

3 Cold & Uncorroded 41.220 0.00 0 0.0 0.0 0.00

4 Hot & UnCorroded 79.674 0.00 0 0.0 0.0 0.00

4 Cold & Uncorroded 79.674 0.00 0 0.0 0.0 0.00

5 Hot & UnCorroded 129.491 0.00 0 0.0 0.0 0.00

5 Cold & Uncorroded 129.491 0.00 0 0.0 0.0 0.00


Vortex Shedding Analysis per Eurocode: EN 1991-1-4:2005+A1:2010

Ref Para E.1.5.3 Approach 2 for calculation of cross wind amplitudes, Valid only for Mode 1

Zcr = (5/6) * Stack Ht + Dist from Grade To Base Of Stack = 20.83 m

Co = Orography factor based upon user entered data @ Elev = 20.83 m = 1.0

Vm = Mean Wind Velocity: Cr * Co * Vb {Eqn 4.3} = 41.41 m/s

Bs = Structural Log Decrement Damping = 0.0126

St = Strouhal number for a single stack = 0.18

Vc = Critical Wind Speed: b * n / S = 19.89 m/s

b = Average Outer Diameter of top 1/3 of stack = 1.5 m

n = Natural Frequency of Mode # 1 = 2.387 Hz

Rho = Air Density = 1.225 Kg/m^3

me = Equivalent mass per unit length = 147.56 kg/m

Sc = Scruton Number: 2 * me * Bs / (Rho * D1^2) = 1.35

d = Diameter used for Reynolds Number Calculation = 1.5 m

V = V used for Reynolds Number Calculation = 55.42 m/s

Re = Reynolds Number: d * V / 0.000015 {Eqn 7.15} = 5.54E+06

Cc = Constant taken from Table E.6 = 0.01

Kamax = Constant From Table E.6 = 1.0

Ka = Convservatively assume equal to Kamax, per Eqn E.16 Note 4 = 1.0

aL = Constant taken from Table E.6 = 0.4

c1 = Constant: (aL^2 / 2) * (1 - Sc / (4 * PI * Ka)) {Eqn E.16} = 7.143E-02

c2 = Constant: (Rho*b^2/me)*(aL^2/Ka)*(Cc^2/St^4)*(b/h) {Eqn E.16} = 1.708E-05

Sy = Std Deviation of Displ: (c1+(c1^2+c2)^0.5)^0.5 * b {Eqn E.15} = 0.5672

ymax = Max Displacement: Sy * kp = 0.8022 m

Eo = Bandwidth factor: 0.15 {Para E.1.5.3 Note (7)} = 0.15

Vcrit = Critical Wind Speed corresponding to Clat calcualtion = 19.89 m/s

Vm = Mean Wind Speed corresponding to Clat calcualtion = 41.41 m/s

Vo = Weibull probability distribution * (2)^0.5: 0.2*Vm {Eqn E.10} = 8.28 m/s

Page 3



Page 176 of 180

Tyrs = Life of Structure in Years = 20.0 Years

T = Life of Structure in Seconds: 3.2*10^7 * T = 6.40E+08

N = Num Cycles: 2*T*n*Eo*(Vcrit/Vo)^2*Exp(-(Vcrit/Vo)^2) {Eqn E.10} = 8.266E+06

Elev Elev m Phi Ht Fw Fw Fw

Uniform Total Total

m ft kg/m m N/m lb N

----- ----- ------ ------ --- ------- ------ -------

25.00 82.02 147.56 -1.0 2.0 -26617 -11967 -53233

23.00 75.46 147.56 -0.89 2.0 -23685 -10649 -47371

21.00 68.90 147.56 -0.78 2.0 -20762 -9335 -41523

19.00 62.34 147.56 -0.671 2.0 -17864 -8032 -35728

17.00 55.77 147.56 -0.564 2.0 -15021 -6754 -30042

15.00 49.21 147.56 -0.461 2.0 -12271 -5517 -24543

13.00 42.65 147.56 -0.363 2.0 -9660 -4343 -19320

11.00 36.09 147.56 -0.272 2.0 -7238 -3255 -14477

9.00 29.53 147.56 -0.19 2.0 -5064 -2277 -10128

7.00 22.97 147.56 -0.12 2.0 -3197 -1437 -6393

5.00 16.40 147.56 -0.064 2.0 -1699 -764 -3398

3.00 9.84 147.56 -0.024 2.0 -636 -286 -1273

1.00 3.28 147.56 -0.003 1.0 -73 -17 -73

----- ----- ------ ------ --- ------- ------ -------

0.00 0.00 0.00 0.0 0.0 0 -64633 -287502

Fw = m * (2*PI*n)^2 * Phi * ymax {Eqn E.6} m = Uniform mass



Elev Elev Load Load

m m N/m N

----- ----- ------- -------

25.00 23.00 -26617 -53233

23.00 21.00 -23685 -47371

21.00 19.00 -20762 -41523

19.00 17.00 -17864 -35728

17.00 15.00 -15021 -30042

15.00 13.00 -12271 -24543

13.00 11.00 -9660 -19320

11.00 9.00 -7238 -14477

9.00 7.00 -5064 -10128

7.00 5.00 -3197 -6393

5.00 3.00 -1699 -3398

3.00 1.00 -636 -1273

1.00 0.00 -73 -73

----- ----- ------- -------

Total -287502

Primary Loads


Number Description Abbreviation

------ ------------- ------------

1 Dead D

2 Live L

3 Operating O

4 Thermal Hot TH

5 Wind W

6 Seismic E

7 Vortex Across V

Load Combinations


Num Description S D FS Dir D L O TH W E V

Deg

--- ---------------------------- - - --- --- ---- ---- ---- -- --- --- ---

8 A: 0.9*D X 1.1 0 0.9

9 A: 1.1*D+1.5*W+0.9*L+0.9*O X 1.1 0 1.1 0.9 0.9 1.5

10 B: 1.35*D+1.5*W+0.9*L+0.9*O X 1.1 0 1.35 0.9 0.9 1.5

11 B: 1.0*D X 1.1 0 1.0

12 C: 1.0*D+1.3*W+0.78*L+0.78*O X 1.1 0 1.0 0.78 0.78 1.3

13 C: 1.0*D X 1.1 0 1.0

14 Serviceability: 1.0*D+1.0*W X 1.1 0 1.0 1.0

15 1.0*D+1.0*S+1.0*L+1.0*O X 1.1 0 1.0 1.0 1.0 1.0

16 1.0*D+1.0*S X 1.1 0 1.0 1.0

17 Serviceability: 1.0*D+1.0*S X 1.1 0 1.0 1.0

18 D+O+P X 1.0 0 1.0 1.0

19 D+V X 1.0 0 1.0 1.0 1.0




S: Include in Stress Analysis D: Calculate Deflections

Stress Criteria: EN 1993-3-2 (ASD)

Wind Criteria: EN 1991-1-4 ()

Seismic Criteria: EN 1998-6 ()

Wind Combinations:

Service wind loads specified, use combos per EN 1990 Appendix A

Seismic Combinations:

Seismic code is based upon ultimate loads, and so are EN 1990 Table A1.3 combinations, so we use the factors from that table

Boundary Conditions Specified for System

'F' is translational restraint and 'M' is rotational restrant

Page 4



Page 177 of 180

Joint # Description X Y Z Fx Fy Fz Mx My Mz m m m

------- ----------- --- --- --- ----- ----- ----- ----- ----- ----- 14 Stack Base 0.0 0.0 0.0 Fixed Fixed Fixed Fixed Fixed Fixed

Verify that Beam Theory can be Used per EN 1993-3-2 [5.1]

For unstiffened chimney need LR >= LRmin to use Beam Theory

L = Overall height of Chimney = 25.0 m rm = Radius of Chimney at base = 0.75 m LR = Ratio of Height to Radius for Chimeny: L / rm = 33.33 LRmin = Minimum Ratio fof L/rm to ignore shell bending: 0.14*rm/t + 10 = 36.25

Since LR < LRmin & Stack is Un-Stiffened then Beam Theory Can't Be used Either peform a Finite Element Analysis of the Chimney or Add Stiffening Rings

Finite Element Analysis will lead to reduction in compression stress at the chimney base or immediately above changes in diameter, but will increase compression stresses elsewhere. Similarly, this will lead to increases in tensile stresses at the base and immediately above changes in chimney diameter, which will be important in deriving bolt tensions. The increase in tensile stress in these regions is approximated in the table below (Ref CICIND 2010):

Buckling Summary Per EN 1993-1-6:2007, Load # 9

Load Combination #9 A: 1.1*D+1.5*W+0.9*L+0.9*O (Hot & Corroded), FS = 1.1

Bot Moment Vert f_xEN f_xEM f_xEd f_qEd s_xRd s_qRd kx kq Unity Elev Ratio m KN-m KN MPa MPa MPa MPa MPa MPa

----- ------ ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- 23.00 4.55 1.59 0.08 0.65 0.73 0.24 86.21 0.73 1.537 1.252 0.25 21.00 9.17 4.78 0.25 1.31 1.56 0.23 86.21 0.73 1.537 1.252 0.24 19.00 31.78 7.96 0.42 4.53 4.96 0.23 86.21 0.73 1.537 1.252 0.25 17.00 63.14 11.14 0.59 9.00 9.60 0.23 86.21 0.73 1.537 1.252 0.26 15.00 103.08 14.33 0.76 14.70 15.46 0.22 86.21 0.73 1.537 1.252 0.30 13.00 151.40 17.51 0.93 21.59 22.52 0.22 86.21 0.73 1.537 1.252 0.35 11.00 207.91 20.69 1.10 29.65 30.75 0.21 86.21 0.73 1.537 1.252 0.42 9.00 272.36 23.88 1.27 38.84 40.11 0.20 86.21 0.73 1.537 1.252 0.51 7.00 344.50 27.06 1.44 49.13 50.57 0.20 86.21 0.73 1.537 1.252 0.63 5.00 423.99 30.24 1.61 60.46 62.07 0.19 86.21 0.73 1.537 1.252 0.79 3.00 510.45 33.43 1.78 72.79 74.57 0.17 86.21 0.73 1.537 1.252 0.97 1.00 603.40 36.61 1.95 86.05 88.00 0.16 86.21 0.73 1.537 1.252 1.18 0.00 657.46 39.79 2.12 93.76 95.88 0.12 86.21 0.73 1.537 1.252 1.28

f_xEN = Axial Stress: V/A [D.12] f_xEM = Bending Stress: M / S [D.12] f_xEd = Total Stress: f_xEN+f_xEM [D.12] f_qEd = Circum Stress: (qeq+qs)*(r/t) [D.30] s_xRd = Buckling Resistance: s_xRk/SF [8.11] s_qRd = Buckling Resistance: s_qRk/SF [8.11] kx = Buckling Param: 1+Xx^2 [8.20] kq = Buckling Param: 1+Xq^2 [8.21] Unity = (f_xEd/s_xRd)^kx - ki(f_xEd/s_xRd)*(f_qEd/s_q_Rd)+(r_qEd/s_qRd)^kq [8.19]

Check to see if Secondary Moments need to be Considered, Load # 9

Per EN 1993-3-2: Para 5.2.3

Nb = Design Value of Total Vertical Load at Base [5.5] = 8.95 KN Ncrit = Elastic Critical Value for Failure at Base: s_xRcr*A [5.5] = 7264.68 KN Nratio = Nb / Ncrit [5.5] = 0.001

Since Nratio <= 0.1 then Secondary Moments need not be considered.

Meridional Buckling Per EN 1993-1-6:2007, Load # 9


Bot w Cxb Cx Dwk ax Lxo Lpx Lx Xx s_xRcr s_xRk s_xRd Elev m mm MPa MPa MPa

----- ----- --- --- ---- ----- --- ----- ----- ----- ------ ----- ----- 23.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 21.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 19.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 17.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 15.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 13.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 11.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 9.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 7.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 5.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 3.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 1.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21 0.00 456.4 1.0 0.6 3.42 0.245 0.2 0.783 0.801 0.382 386.43 94.83 86.21

ax = 0.62 / (1+1.91*(Dwk/t)^1.44) [D.14] Q = Quality Parameter: 16 [Table D.2] w = Length Param: l/(r*t)^0.5 [D.1] Dwk = Imperfect Amp: (1/Q)*(r/t)^0.5*t [D.15] Cx = Meridional Param: [D.2 to D.13] Cxb = Meridional Param [Table D.1] Lpx = (ax / (1-B))^0.5 [8.16] Lx = (fyk / s_xRcr)^0.5 [8.17] Lxo = 0.2 + 0.1*(s_xEM / s_xEd) [D.17] s_xRcr = 0.605*E*Cx*(t/r) [D.2] s_xRk = Xx * fyk [8.12] s_xRd = s_xRk / SF [8.11]

Circumferential Buckling Per EN 1993-1-6:2007, Load # 9


Bot Cq Cqs aq Lq Lpq Kw qw_max qeq Xq f_qEd s_qRcr s_qRk s_qRd Elev m kPa kPa MPa MPa MPa MPa

----- --- --- --- ----- ----- ---- ------ ----- ------ ----- ------ ----- ----- 23.00 0.6 0.6 0.5 12.44 1.118 0.65 1.942 1.262 0.0032 0.24 1.60 0.80 0.73 21.00 0.6 0.6 0.5 12.44 1.118 0.65 1.914 1.244 0.0032 0.23 1.60 0.80 0.73

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19.00 0.6 0.6 0.5 12.44 1.118 0.65 1.884 1.225 0.0032 0.23 1.60 0.80 0.73 17.00 0.6 0.6 0.5 12.44 1.118 0.65 1.851 1.203 0.0032 0.23 1.60 0.80 0.73 15.00 0.6 0.6 0.5 12.44 1.118 0.65 1.814 1.179 0.0032 0.22 1.60 0.80 0.73 13.00 0.6 0.6 0.5 12.44 1.118 0.65 1.774 1.153 0.0032 0.22 1.60 0.80 0.73 11.00 0.6 0.6 0.5 12.44 1.118 0.65 1.728 1.123 0.0032 0.21 1.60 0.80 0.73 9.00 0.6 0.6 0.5 12.44 1.118 0.65 1.675 1.089 0.0032 0.20 1.60 0.80 0.73 7.00 0.6 0.6 0.5 12.44 1.118 0.65 1.613 1.048 0.0032 0.20 1.60 0.80 0.73 5.00 0.6 0.6 0.5 12.44 1.118 0.65 1.536 0.998 0.0032 0.19 1.60 0.80 0.73 3.00 0.6 0.6 0.5 12.44 1.118 0.65 1.436 0.933 0.0032 0.17 1.60 0.80 0.73 1.00 0.6 0.6 0.5 12.44 1.118 0.65 1.289 0.838 0.0032 0.16 1.60 0.80 0.73 0.00 0.6 0.6 0.5 12.44 1.118 0.65 0.999 0.649 0.0032 0.12 1.60 0.80 0.73

Cq = Ext Buckling Fac Short Cyl [Table D.3] Cqs = Ext Bcklng Fac Med Cyl [Table D.4] aq = Param based on Fab Quality [Table D.5] Lq = (fyk/s_qRcr)^0.5 [8.17] Lpq = (aq / (1-B))^0.5 [8.16] Lqo = 0.4 [D.26] s_qRk = Xq * fyk [8.12] f_qEd = Circ Stress: (qeq+qs)*(r/t) [D.30] s_qRcr= Elastic Crit Circ Bcklng Stress [D.1.3.1] s_qRd = s_qRk / SF [8.11] qeq = Kw*qw_max [D.28] qs = Internal vacum: 0 [D.30] Kw = 0.46*(1+0.1*(Cq*r/(w*t))^0.5 [D.29] qw_max= Max Widn Pressure [D.28]

Static Summation of Forces

Total sum of all loads acting on system, Shear = (Fx^2+Fz^2)^0.5

Ld Load Case Dir Fx Fy Fz Shear (P)=Primary, (C)=Combination Deg KN KN KN KN

-- ------------------------------ --- ----- ----- ------- ------ 1 (P)Dead 0 0.00 36.18 0.00 0.00 2 (P)Live 0 0.00 0.00 0.00 0.00 3 (P)Operating 0 0.00 0.00 0.00 0.00 4 (P)Thermal Hot 0 0.00 0.00 0.00 0.00 5 (P)Wind 0 32.48 0.00 0.00 32.48 6 (P)Seismic 0 75.60 0.00 0.00 75.60 7 (P)Vortex Across 0 0.00 0.00 -287.50 287.50 8 (C)A: 0.9*D 0 0.00 32.56 0.00 0.00 9 (C)A: 1.1*D+1.5*W+0.9*L+0.9*O 0 48.72 39.79 0.00 48.72 10 (C)B: 1.35*D+1.5*W+0.9*L+0.9*O 0 48.72 48.84 0.00 48.72 11 (C)B: 1.0*D 0 0.00 36.18 0.00 0.00 12 (C)C: 1.0*D+1.3*W+0.78*L+0.78* 0 42.22 36.18 0.00 42.22 13 (C)C: 1.0*D 0 0.00 36.18 0.00 0.00 14 (C)Serviceability: 1.0*D+1.0*W 0 32.48 36.18 0.00 32.48 15 (C)1.0*D+1.0*S+1.0*L+1.0*O 0 75.60 36.18 0.00 75.60 16 (C)1.0*D+1.0*S 0 75.60 36.18 0.00 75.60 17 (C)Serviceability: 1.0*D+1.0*S 0 75.60 36.18 0.00 75.60 18 (C)D+O+P 0 0.00 36.18 0.00 0.00 19 (C)D+V 0 0.00 72.35 -287.50 287.50

Support Loading Notes

These notes apply to the support loading data which follows

1) Joint = Joint Number, Load = Load Number, Type = P (Primary) or C (Combination) 2) Dir= Direction of Lateral Loads (Wind/Seismic/Vortex), 0 Deg=+X axis, 90 Deg=+Z Axis 3) Fx/Fy/Fz = Forces in X/Y/Z directions, Mx/My/Mz = Moments about X/Y/Z Axis 4) Y is the vertical direction, and + loads are acting up, and - is acting down.



Ld Load Case Dir Fx Fy Fz Mx My Mz (P)=Primary,(C)=Combo Deg KN KN KN KN-m KN-m KN-m

-- ----------------------- --- ----- ----- ------- -------- ---- -------- 1 (P)Dead 0 0.00 36.18 0.00 0.00 0.00 0.00 2 (P)Live 0 0.00 0.00 0.00 0.00 0.00 0.00 3 (P)Operating 0 0.00 0.00 0.00 0.00 0.00 0.00 4 (P)Thermal Hot 0 0.00 0.00 0.00 0.00 0.00 0.00 5 (P)Wind 0 32.48 0.00 0.00 0.00 0.00 -438.31 6 (P)Seismic 0 75.60 0.00 0.00 0.00 0.00 -1295.14 7 (P)Vortex Across 0 0.00 0.00 -287.50 -5129.75 0.00 0.00 8 (C)A: 0.9*D 0 0.00 32.56 0.00 0.00 0.00 0.00 9 (C)A: 1.1*D+1.5*W+0.9*L 0 48.72 39.79 0.00 0.00 0.00 -657.46 10 (C)B: 1.35*D+1.5*W+0.9* 0 48.72 48.84 0.00 0.00 0.00 -657.46 11 (C)B: 1.0*D 0 0.00 36.18 0.00 0.00 0.00 0.00 12 (C)C: 1.0*D+1.3*W+0.78* 0 42.22 36.18 0.00 0.00 0.00 -569.80 13 (C)C: 1.0*D 0 0.00 36.18 0.00 0.00 0.00 0.00 14 (C)Serviceability: 1.0* 0 32.48 36.18 0.00 0.00 0.00 -438.31 15 (C)1.0*D+1.0*S+1.0*L+1. 0 75.60 36.18 0.00 0.00 0.00 -1295.14 16 (C)1.0*D+1.0*S 0 75.60 36.18 0.00 0.00 0.00 -1295.14 17 (C)Serviceability: 1.0* 0 75.60 36.18 0.00 0.00 0.00 -1295.14 18 (C)D+O+P 0 0.00 36.18 0.00 0.00 0.00 0.00 19 (C)D+V 0 0.00 72.35 -287.50 -5129.75 0.00 0.00

Load Load Case Vertical Shear Moment (P)=Primary,(C)=Combo KN KN KN-m

---- ----------------------- -------- ------ ------- 1 (P)Dead 36.18 0.00 0.00 2 (P)Live 0.00 0.00 0.00 3 (P)Operating 0.00 0.00 0.00 4 (P)Thermal Hot 0.00 0.00 0.00 5 (P)Wind 0.00 32.48 438.31 6 (P)Seismic 0.00 75.60 1295.14 7 (P)Vortex Across 0.00 287.50 5129.75 8 (C)A: 0.9*D 32.56 0.00 0.00

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9 (C)A: 1.1*D+1.5*W+0.9*L 39.79 48.72 657.46

10 (C)B: 1.35*D+1.5*W+0.9* 48.84 48.72 657.46

11 (C)B: 1.0*D 36.18 0.00 0.00

12 (C)C: 1.0*D+1.3*W+0.78* 36.18 42.22 569.80

13 (C)C: 1.0*D 36.18 0.00 0.00

14 (C)Serviceability: 1.0* 36.18 32.48 438.31

15 (C)1.0*D+1.0*S+1.0*L+1. 36.18 75.60 1295.14

16 (C)1.0*D+1.0*S 36.18 75.60 1295.14

17 (C)Serviceability: 1.0* 36.18 75.60 1295.14

18 (C)D+O+P 36.18 0.00 0.00

19 (C)D+V 72.35 287.50 5129.75





Maximum Support Loads for All Load Combinations

Maximum loading for each restraint, but loads may not necessarily occur at the same time

Restraint Description Vertical Uplift Shear Moment Torsion

KN KN KN KN-m KN-m

--------------------- -------- ------ ------ ------- -------

Stack Base 0.00 72.35 287.50 5129.75 0.00

Vertical = Max downward acting Fy (Largest negative Fy value)

Uplift = Max upward acting Fy (Largest postive Fy value)


Moment = Resultant Overturning Moment: (Mx^2 + Mz^2)^0.5

Torsion = Maximum magnitude of My

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