Vibration Of Tall Towers And Stacks [1727
Tall cylindrical stacks and towers may be susceptible to windinduced oscillations as a result of vortex shedding. This phenomenon, often referred to as dynamic instability, has resulted in severe oscillations, excessive deflections, structural damage, and even failure. Once it has been determined that a vessel is dynamically unstable, either the vessel must be redesigned to withstand the effects of windinduced oscillations or external spoilers must be added to ensure that vortex shedding does not occur.
The deflections resulting from vortex shedding are perpendicular to the direction of wind flow and occur at relatively low wind velocities. When the natural period of vibration of a stack or column coincides with the frequency of vortex shedding, the amplitude of vibration is greatly magnified. The frequency of vortex shedding is related to wind velocity and vessel diameter. The wind velocity at which the frequency of vortex shedding matches the natural period of vibration is called the critical wind velocity.
Windinduced oscillations occur at steady, moderate wind velocities of 2025 miles per hour. These oscillations commence as the frequency of vortex shedding approaches the natural period of the stack or column and are perpendicular to the prevailing wind. Larger wind velocities contain highvelocity random gusts that reduce the tendency for vortex shedding in a regular periodic manner.
A convenient method of relating to the phenomenon of wind excitation is to equate it to fluid flow around a cylinder. In fact this is the exact case of early discoveries related to submarine periscopes vibrating wildly at certain speeds. At low flow rates, the flow around the cylinder is laminar. As the stream velocity increases, two symmetrical eddies are formed on either side of the cylinder. At higher velocities vortices begin to break off from die main stream, resulting in an imbalance in forces exerted from the split stream. The discharging vortex imparts a fluctuating force that can cause movement in the vessel perpendicular to the direction of the stream.
Historically, vessels have tended to have many fewer incidents of windinduced vibration than stacks. There is a variety of reasons for this:
1. Relatively thicker walls.
2. Higher first frequency.
3. External attachments, such as ladders, platforms, and piping, that disrupt the wind flow around the vessel.
4. Significantly higher damping due to:
a. Internal attachments, trays, baffles, etc.
b. External attachments, ladders, platforms, and piping.
c. Liquid holdup and sloshing.
e. Foundation.
f. Shell material.
g. External insulation.
Damping Mechanisms
Internal linings are also significant for damping vibration; however, most tall, slender columns are not lined, whereas many stacks are. The lining referred to here would be the refractory type of linings, not paint, cladding, or some protective metal coating. It is the damping effect of the concrete that is significant.
Damping is the rate at which material absorbs energy under a cyclical load. The energy is dissipated as heat from internal damping within the system. These energy losses are due to the combined resistances from all of the design features mentioned, i.e., the vessel, contents, foundation, internals, and externals. The combined resistances are known as the damping factor.
The total damping factor is a sum of all the individual damping factors. The damping factor is also known by other terms and expressions in the various literature and equations and expressed as a coefficient. Other common terms for the damping factor are damping coefficient, structural damping coefficient, percent critical damping, and material damping ratio. In this procedure this term is always referred to either as factor DF or as /3.
There are eight potential types of damping that affect a structure's response to vibration. They are divided into three major groups:
Resistance:
Damping from internal attachments, such as trays. Damping from external attachments, such as ladders, platforms, and installed piping. Sloshing of internal liquid.
Base support: Soil.
Foundation.
Energy absorbed by the shell (hysteretic): Material of shell. Insulation. Internal lining.
Karamchandani, Gupta, and Pattabiraman give a detailed account of each of these damping mechanisms (see Ref. 17)
for process towers (trayed columns). They estimate the "percent critical damping" at 3% for empty vessels and 5% for operating conditions. The value actually used by most codes is only a fraction of this value.
Design Criteria
Once a vessel has been designed statically, it is necessary to determine if the vessel is susceptible to windinduced vibration. Historically, the rule of thumb was to do a dynamic wind check only if the vessel L/D ratio exceeded 15 and the POV was greater than 0.4 seconds. This criterion has proven to be unconservative for a number of applications. In addition, if the critical wind velocity, Vc, is greater than 50mph, then no further investigation is required. Wind speeds in excess of 50 mph always contain gusts that will disrupt uniform vortex shedding.
This criterion was amplified by Zorrilla [18], who gave additional sets of criteria. Criterion 1 determines if an analysis should be performed. Criterion 2 determines if the vessel is to be considered stable or unstable. Criterion 3 involves parameters for the first two criteria.
Criterion 1
• If W/LD^ < 20, a vibration analysis must be performed.
• If 20 < VV/LD^ < 25, a vibration analysis should be performed.
• If W/LDf > 25, a vibration analysis need not be performed.
Criterion 2
• If 0.75 < W<5/LD^ < 0.95, the vessel is probably unstable.
Criterion 3
This criterion must be met for Criteria 1 and 2 to be valid.
• Vt. > 50 mph; vessel is stable and further analysis need not be performed.
Criterion 4
An alternative criterion is given in ASME STS12000, "Steel Stacks." This standard is written specifically for stacks. The criterion listed in this standard calculates a "critical vortex shedding velocity," V.^,. This value is then compared to the critical wind speed, Vc, and a decision made.
• If Vc < Vzcrit, vortex shedding loads shall be calculated.
• If Vz(.nt < V,, < 1.2V7(;rit, vortex shedding loads shall be calculated; however, the loads may be reduced by a factor of (Vzcrit/Vc)2
• If Vc > 1.2 Vzcrit, vortex shedding may be ignored.
Equations are given for calculating all of the associated loads and forces for the analysis. This procedure utilizes the combination of two components of one fi for aerodynamic damping, and one for steel damping, The two values are combined to determine the overall f3.
This standard does not require a fatigue evaluation to be done if the stack is subject to windinduced oscillations.
Criterion 5
An alternative criterion is also given in the Canadian Building Code, NBC. The procedure for evaluating effects of vortex shedding can be approximated by a static force acting over the top third of the vessel or stack. An equation is given for this value, Fl, and shown is this procedure.
Dynamic Analysis
If the vessel is determined by this criterion to be unstable, then there are two options:
a. The vessel must be redesigned to withstand the effects of windinduced vibration such that dynamic deflection is less than 6in./100ft of height.
b. Design modifications must be implemented such that windinduced oscillations do not occur.
Design Modifications
The following design modifications may be made to the vessel to eliminate vortex shedding:
a. Add thickness to bottom shell courses and skirt to increase damping and raise the POV.
b. Reduce the top diameter where possible.
c. For stacks, add helical strakes to the top third of the stack only as a last resort. Spoilers or strakes should protrude beyond the stack diameter bv a distance of d/12 but not less than 2 in.
d. Crossbrace vessels together.
e. Add guy cables or wires to grade.
f. Add internal linings.
g. Reduce vessel below dynamic criteria.
Precautions
The following precautions should be taken.
a. Include ladders, platforms, and piping in your calculations to more accurately determine the natural frequency.
b. Grout the vessel base as soon as possible after erection while it is most susceptible to wind vibration.
c. Add externa] attachments as soon as possible after erection to break up vortices.
d. Ensure that tower anchor bolts are tightened as soon as possible after erection.
Definitions
Critical wind velocity: The velocity at which the frequency of vortex shedding matches one of the normal modes of vibration.
Logarithmic decrement: A measure of the ability of the overall structure (vessel, foundation, insulation, contents, soil, lining, and internal and external attachments) to dissipate energy during vibration. The logarithmic ratio of two successive amplitudes of a damped, freely vibrating structure or the percentage decay per cycle.
Static deflection: Deflection due to wind or earthquake in the direction of load.
Dynamic deflection: Deflection due to vortex shedding perpendicular to the direction of the wind.
Notes
1. See Procedure 33 to determine a vessel's fundamental period of vibration (POV).
2. See Procedure 44 to determine static deflection.
3. Vessel should be checked in the empty and operating conditions with the vessel fully corroded.
4. Concentrated eccentric loads can be converted to an additional equivalent uniform wind load.
5. I7D ratios for multidiameter columns can be determined as shown in Note 8.
6. A fatigue evaluation should be performed for any vessel susceptible to vortex shedding. A vessel with a POV of 1 second and subjected to 3 hours per day for 30 years would experience 120 million cycles.
7. This procedure is for cylindrical stacks or vessels only, mounted at grade. It is not appropriate for tapered stacks or vessels. There is a detailed procedure in ASME STS1 for tapered stacks. Multidiameter columns and stacks can be evaluated by the methods shown. This procedure also does not account for multiple vessels or stacks in a row.
8. L/D ratios can be approximated as follows:
ux where quantities L*DX are calculated from the top down.
Table 415
Summary of Critical Damping
Table 415
Summary of Critical Damping
Item 
Description 
Case 1 : Empty 
Case 2: Operating  
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