Wind induced vibrations on tall vessels are a concern and AutoPIPE Vessel currentley has two methods to address this design consideration.

# Method 1 - Kanti Mahajan

- Obtain first the natural frequency ‘f’ of your vessel

- Check for vibration possibility

If W/LDr^{2 }<=20, issue message in report “Vibration analysis necessary to be performed as W/LDr^{2 }<=20”

vibration analysis must be performed and proceed to step 3

where W= corroded wt. in lb, L=length in ft., Dr=average internal diameter of top half of the structure in ft.

If W/LDr^{2 }>20, issue message in report “Vibration analysis not necessary to be performed as W/LDr^{2 }>20”

- Perform stability investigation,

Calculate damping factor D_{F}=Wδ/LDr^{2}

Default value of δ = 0.03 (Table 3)

If D_{F} <= 0.75, issue message in report “system is unstable and vortex shedding vibration analysis is required as D_{F} <= 0.75”, proceed to step 4

If D_{F} > 0.75, issue message in report “Vortex shedding vibration analysis not necessary to be performed as D_{F} > 0.75”

- Calculate critical wind velocity being where
*S*= 0.226 (Strouhal number)

*Justification: In the book this formula appears as ** **in mph, but Dr is introduced in feet, f in Hz, so the number of Strouhal is affected by a conversion factor from feet to miles and hours to seconds. To reverse this change and work in consistent units we have to divide by 1.466 (1mph = 5280/3600 = 1.466 ft/s). And to place S in the denominator, we inverse the coefficient: *

- Calculate maximum wind velocity at top of the structure (H = 10m or 30ft)

V_{w} =V_{b}(L/H)^{0.143 }

Where V_{b} = wind velocity at H

- Apply gust factor, 1.3 to obtain maximum gust velocity = 1.3V
_{w}

- If V
_{c }<=maximum gust velocity, perform amplitude calculations go to step 8.

If V_{c }> maximum gust velocity, issue message in report “Vibration amplitude no need to be calculated as V_{c }> maximum gust velocity”.

- Calculate amplitude of maximum dynamic deflection, Z=L
^{5}V_{C}^{2}/W δ D_{r }(10)^{-6}(0.00243) in.

- Check if Z < Maximum Deflection input by user, issue message in report “structure is safe:amplitude of maximum dynamic deflection < maximum allowable deflection”

If Z> Maximum Deflection input by user, issue message in report “structure is unsafe:amplitude of maximum dynamic deflection exceeds maximum allowable deflection, please modify design”

Reference: Kanti Mahajan, PE. (1979). *Design of Process Equipment*. Tulsa, OK: Pressure Vessel Handbook Publishing.

# Method 2: Bednar

- Check vibration criteria. See points 1 and 2 in Kanti Mahajan.

- Calculate maximum wind velocity at top of the structure. See points 5 and 6 in Kanti Mahajan.

- Calculate first and second critical wind velocity, V
_{1 }and V_{2}

where *S* = 0.2 (Strouhal number)

V_{2} = 6.25.

If the design wind velocity V>V_{1}, perform the detailed vibration analysis from step 3 below with V_{1 }

If the design wind velocity V>V_{2}, perform the detailed vibration analysis from step 3 below with V_{2}

If V_{1 }or V_{2} > 60 mph/27 m/s, ignore the calculation for that V value as there is cutoff for maximum critical wind speed for design consideration (page 120).

- Consider default value of lift coefficient, C
_{L}=0.5, normally lift coefficient is 0.4-0.6 (page 111) for practical engineering. Allow users to input C_{L}as optional to default.

- Create table 4.3 (page 116) in 2 dropdown selections for evaluation of logarithmic decrement, δ and Magnification Factor, M.F. Allow 2 dropdowns selections

- Selection of soil types (Soft – I, Stiff – II, Rock, Very Stiff – III)
- Equipment type (Tall process columns, unlined stacks, lined stacks)

Determine δ and M.F. from the above selected combinations

- Determine equivalent force F=0.00086 (C
_{L }x M.F.)(d x H x V_{1}^{2}) (*replace V*)_{1}by V_{2}as needed

Where, d = outside projected dia., ft., H= total column height, ft.

- This force F acts on the top of the column so calculate the moments at each level in same way as static wind load analysis.

Note in the example only the moment at the junction of column and skirt is considered but APV has different approach of calculating at each individual level

- Perform fatigue analysis at the bottom tangent line weld joint

- Evaluate the cyclical stress range S
_{R}=2S, where S is the bending stress due to F acting at the top. We’ll be taking the maximum of |σ_{M}| , |σ_{z}| , |σ_{eq}| .The calculation of S_{R }in the example is confusing and we need to follow the standard practice of calculating the bending stress as implemented in the program S= σ = M/section modulus, section modulus = π (r_{2}^{4}-r_{1}^{4})/4r_{2} - calculate number of cycles to failure, N = (K/βS
_{R})^{n }(page 118) - where, n=5, K=780,000 (these values are for carbon steel, no other value specified for any other grade or type of steel)

stress intensification factor, β as follows

β = 1.2, shell plate with smooth finish,

β = 1.8, butt weld

β = 3.0 fillet weld

- Allow users to select value for β from dropdown, default β = 2.0
- Allow user to specify safety factor, S.F. Default S.F.=20.
- Safe vibration time or service life considering weld fatigue due to wind vortex induced vibration = (N/S.F.)x T/3600 x(1/24)x(1/365) years
- T=time period (seconds per cycle) = 1/f

Reference: Henry H. Bednar, PE. (1991). *Pressure Vessel Design Handbook Second Edition*. Malabar, FL: Krieger Publishing Company.