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PRESSURE SAFETY VALVE DISCHARGE FORCES ON
PIPE RACK STRUCTURES FOR INDUSTRIAL FACILITIES
By
REED NEWCOMER
B.S., Colorado State University, 2012
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Civil Engineering
2015
ii
This thesis for the Master of Science
degree by
Reed Newcomer
has been approved for the
Civil Engineering Program
by
Chengyu Li, Chair
Peter Marxhausen
Frederick Rutz
April 21, 2015
iii
Newcomer, Reed (M.S., Civil Engineering)
Pressure Safety Valve Discharge Forces on Pipe Rack Structures for Industrial Facilities
Thesis directed by Professor Cheng Li
ABSTRACT
Pipe rack structures are used extensively within industrial facilities to support
pipes, cable trays, and equipment. Pressure safety valves are used in these industrial
facilities to prevent overpressures within the process system. Typically pressure safety
valves are mounted on these pipe rack structures and can produce sizable reaction forces
in the even that they are discharged. The force produced is similar to an impact that
imposes dynamic forces on the structure. Currently there are no code provisions that deal
with these forces on structural systems. The industry instead relies on information from
piping standards to determine equivalent static force on the supporting structure, which is
only intended to design pipes and fittings. The current practices need to be evaluated to
determine validity of these assumptions.
A literature review was conducted to determine the mechanics of a pressure safety
valve. From there, an evaluation was conducted on the reaction forces on supporting
structures due to the pressure safety valve discharge. The review also entails some
dynamic analysis pertinent to this topic.
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Multiple time history analyses on a generalized pipe rack were assessed to
determine the structural response due to various load cases involving safety valves.
Comparisons of the time history results were then related to current industry practices.
Conclusions were drawn to help engineers design pipe racks without a full blown
dynamic analysis.
The form and content of this abstract are approved. I recommend its publication.
Approved: Chengyu Li
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ACKNOWLEDGEMENT
I would like to thank first and foremost Dr. Chengyu Li for the support and
guidance in completion of this thesis. I would also like to thank Peter Marxhausen and
Dr. Frederick Rutz for participating on my graduate advisory committee. Lastly, I would
like to thank various work associates for their help with discussion of the topic.
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TABLE OF CONTENTS
List of Figures .................................................................................................................. viii
List of Tables .......................................................................................................................x
Chapter
1. Introduction ..............................................................................................................1
1.1 Pressure Safety Valves in Industrial Facilities.........................................................1
1.2 Pipe Racks in Industrial Facilities............................................................................4
2. Problem Statement ...................................................................................................7
2.1 Introduction ..............................................................................................................7
2.2 Significance of Research..........................................................................................8
2.3 Research Objective ..................................................................................................9
3. Literature Review...................................................................................................10
3.1 Introduction ............................................................................................................10
3.2 Pressure Safety Valves Mechanics ........................................................................10
3.3 Loads Caused by Pressure Safety Valves ..............................................................14
3.3.1 Open Discharge Systems ...............................................................................15
3.3.2 Closed Discharge Systems .............................................................................19
3.4 Single Degree of Freedom Structures ....................................................................24
3.4.1 Dynamic Load Factor ....................................................................................26
3.4.2 Impulse and Ramping Forcing Functions ......................................................29
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3.5 Multi Degree of Freedom Systems ........................................................................33
4. Research Plan .........................................................................................................34
5. Pipe Rack Analysis ................................................................................................36
5.1 Generalized Pipe Rack ...........................................................................................36
5.2 Structural Damping ................................................................................................39
5.3 Load Cases .............................................................................................................41
5.3.1 Case I .............................................................................................................44
5.3.2 Case II ............................................................................................................45
5.3.3 Case III ...........................................................................................................46
5.3.4 Case IV...........................................................................................................47
5.4 Base Support Conditions........................................................................................48
6. Comparison of Results ...........................................................................................49
7. Conclusions ............................................................................................................59
References ..........................................................................................................................62
Appendix A – STAAD Input Pressure Safety Valve Time History Analysis ...................64
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LIST OF FIGURES
Figure
1-1 Pressure Safety Valve Discharge (Pressure Relief Valve Scenarios) ................2
1-2 Typical Four-Level Pipe Rack (Bendapudi) ......................................................4
1-3 Typical Cross Section of Pipe Rack...................................................................5
3-1 Typical Direct Spring Safety Valve Components (Pentair) .............................11
3-2 Valve at Set Pressure (Pentair) ........................................................................13
3-3 Discharge of a Pressure Safety Valve (Leser) .................................................14
3-4 Pressure Safety Valve with Open Discharge (API 520) ..................................16
3-5 Time History of Safety Valve Relieving Force ...............................................18
3-6 Pressure Safety Valve with Closed Discharge (API 520) ................................19
3-7 Net Idealized Safety Valve Force Acting on a Pipe ........................................20
3-8 Typical Piping Expansion Loop.......................................................................22
3-9 Single Degree Freedom System .......................................................................25
3-10 Dynamic Load Factors .....................................................................................26
3-11 Response Spectrum for Ramping-Constant Force (Chopra) ...........................30
3-12 Triangular Pulse Function ................................................................................31
3-13 Response Spectrum for Triangular Pulse Force (Chopra) ...............................33
5-1 Isometric View of Typical Pipe Rack Used for Analysis ................................38
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5-2 Damping Effect on Structural Response ..........................................................40
5-3 Pipe Rack with Pressure Relief Discharge Applied .........................................43
5-4 Case I Forcing Function ...................................................................................44
5-5 Case II Forcing Function .................................................................................45
5-6 Case III Forcing Function ................................................................................46
5-7 Case IV Forcing Function ................................................................................47
6-1 Mode 1 Deformed Shape .................................................................................50
6-2 Mode 2 Deformed Shape .................................................................................51
6-3 Mode 3 Deformed Shape .................................................................................51
6-4 Response Due to Case I Load ..........................................................................52
6-5 Response Due to Case II Load .........................................................................54
6-6 Response Due to Case III Load .......................................................................55
6-7 Response Due to Case IV Load .......................................................................56
6-8 Response Due to Case V Load .........................................................................57
7-1 Response Spectrum for Pressure Safety Valves on Pipe Racks ......................60
x
LIST OF TABLES
Table
6-1 Modal Mass Participation ................................................................................49
6-2 Summary of Load Cases .................................................................................58
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1. Introduction
1.1 Pressure Safety Valves in Industrial Facilities
Pipelines have been proven to be the safest and most economical means of
transporting liquid and gaseous products from one point to another. Within a
petrochemical facility these pipes can be seen running in every direction to and from
various process units carrying valuable products. The pipes typically are under a
tremendous amount of temperature and pressure. Thus, the piping becomes a pressure
vessel. Safety valves are put into place to protect the system from an over-pressure
potentially causing catastrophic failure, damage to property, damage to the product, or
death and injury of personal.
The primary purpose of a pressure or vacuum relief valve is to protect life and
property by venting process fluid from an over-pressurized vessel or adding fluid, to
prevent formation of a vacuum strong enough to cause a storage tank to collapse. A
safety valve is a pressure relief valve characterized by rapid opening or closing and
normally used to relieve pressure of compressible fluids. Safety valves are mainly
installed in chemical plants, electric power boilers, and gas storage tanks. Safety valves
are designed to open and relieve excess pressure from vessels or equipment and to reclose
and prevent the further release of fluid after normal conditions have been restored. The
safety valve is a safety device and in many cases the last line of defense. It should have to
operate for one purpose only: overpressure protection.
2
There are a number of reasons why the pressure in a vessel or system can exceed
a predetermined limit. American Petroleum Institute (API) identifies six detailed
guidelines about causes of overpressure. The most common are blocked discharge,
exposure to external fire, thermal expansion, chemical reaction, heat exchanger tube
rupture, and cooling system failure. Each of the events may occur individually or
concurrently. Each source of overpressure also will generate a different mass or volume
flow to be discharged, for example minor mass flow for thermal expansion and big mass
flow in the event of a chemical reaction.
Figure 1-1 Pressure Safety Valve Discharge (Pressure Relief Valve Scenarios)
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When the pressure is released through a pressure safety valve a sudden inertia of
mass is released causing large forces. Since this force acts as an impulse it shall be
considered a dynamic load on the supporting structure. Figure 1-1 shows a pressure
safety valve during a discharge. The magnitude of the force can be noted by the vertical
height that the fluid is projected.
Pressure safety valve (PSV) and pressure relief valve (PRV) are commonly used
terms to identify pressure relief devices. These terms are often used interchangeably
however, it shall be noted that these terms differ. Pressure relief valves are used to
describe relief device on a liquid filled piping systems. For such a valve the opening is
proportional to increase in the system pressure. Hence the opening of valve is not abrupt
if the pressure is increased gradually. Pressure safety valves on the other hand are used to
describe a relief device on a compressible fluid or gas filled system. For such a valve the
opening is almost instantaneous. The opening of the pressure safety valve is often
described as a pop to define the sudden valve opening, and the sounds the valve makes
during its discharge. When the set pressure of the valve is reached, the valve opens
almost fully. For the purpose of this investigation Pressure Safety Valves will be
examined.
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1.2 Pipe Racks in Industrial Facilities
Pipe racks are structures used in various types of process facilities to support
pipes, cable trays and miscellaneous equipment. The pipe racks serve as a highway to
transport the piping and product from one process to another, or to storage.
Pipe racks are typically long narrow structures that consist of a series of
transverse moment framed bents connected by longitudinal struts. The pipes and cable
tray runs in the longitudinal direction through the transverse bents. Figure 1.2 shows a
typical multi-level process facility pipe rack.
Figure 1-2 Typical Four-Level Pipe Rack (Bendapudi)
Due to the structure stability, pipe routing, and maintenance access corridors; pipe
racks generally entail moment-resisting frames in the transverse direction. These frames
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resist any gravity loads as well as lateral loads from either process loads or wind and
seismic loads. Longitudinal lateral loads are typically collected into the struts and resisted
by a single braced bay at each column line. Figure 1.3 shows a typical transverse frame of
a pipe rack.
Figure 1-3 Typical Cross Section of Pipe Rack
Pressure Safety Valves are typically located on top of the pipe rack platforms
such that in a case of their discharge to the atmosphere, they will not damage process
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equipment or be in the vicinity of personal. In closed discharge systems (see section
3.3.2) the release of the fluids undergo further processing and need to be transported via a
header to knock out drums, and flares. Since the pipe headers run along the pipe rack,
pressure safety valves are typically placed on the rack.
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2. Problem Statement
2.1 Introduction
Industrial facilities typically have pipes and utilities running throughout the plant
which require large and lengthy pipe racks. Many of these pipes are under a significant
amount of heat and pressure which require a means to prevent the over pressure of the
fluid potentially causing damage to the product, or plant. The pressure safety valve is the
last line of defense to release pressure in the system by discharging the excess fluid
pressure and decreasing the internal pressure back to the predetermined limits.
In the event of a pressure safety valve discharge a large amount of inertia forces
and static forces are released through the valve and cause a large reaction force that needs
to be resisted by the pipe rack. These forces can vary in size from a couple hundred
pounds to upwards of 10 kips. The larger safety valves require special bracing
considerations within the pipe rack from structural engineers.
Pipe racks are considered non-building structures; code referenced documents
will usually not cover the design and analysis of these types of structures. Process
Industry Practices Structural Design Criteria (PIP STC01015) has developed a uniform
standard for design but it should be noted that this is not considered a code document.
Furthermore, structural references due to pressure safety valve relief forces on structural
systems do not exist.
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American Society of Mechanical Engineers (ASME) B31.1 has developed
equations to estimate the maximum static force on a piping system due to a discharge.
The code also provides some insight on dynamic effects which basically entails a load
factor of 2.0.
The lack of code referenced documents can lead to confusion in the design of pipe
racks. Ultimately, piping systems and structural systems will react differently to the loads
caused by a pressure safety valve discharge, and they both need to be explored.
2.2 Significance of Research
The maximum reaction force due to a pressure safety valve discharge is a
dynamic problem because the applied mass has enough acceleration in comparison to the
resisting structure's natural frequency such that the load effect is generally amplified. If a
load is applied sufficiently slowly, the inertia forces can be ignored and the analysis can
be simplified to a static analysis. Since the pressure safety valve force is sudden, it
warrants a dynamic analysis. A dynamic response of a structure can be significantly
different than a static analysis. The particular structure may exhibit more than twice the
static deflection for the same load, or both positive and negative flexural moments, or the
load can send the structure into a resonance. Thus it is important to understand the
response behavior due to this load effect.
Current industry practice treats this load as a static load with a 2.0 multiplier to
account for any dynamic effects. While this will be shown to be conservative within this
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thesis, it does not give any insight to the structural response. Since there is limited
research on the structures response due to a pressure valve discharge, this thesis will look
at these forces applied to a pipe rack structure. In most, if not all cases, this investigation
will show that the multiplier for use in a static analysis can be reduced.
Reducing one typical pressure safety valve force will have negligible effect on the
design of the structure, however in most cases there are multiple pressure relief valves
sometimes sharing the same beam that have the potential to discharge at the same time.
Some pipe racks could have upwards of 30 pressure safety valves on one structure. In this
case there can be a significant amount of savings as the result of reduced loads.
2.3 Research Objective
The main purpose of this thesis will be to analyze the dynamic effects on a typical
pipe rack structure due to pressure safety valve discharge load cases. The dynamic effects
will be compared to a static pressure safety valve discharge load case in order to
determine a range of dynamic load factors associated with the release of fluids. The paper
will have some recommendations for engineers to determine a dynamic load factor used
in static analysis and design. This paper is intended to provide insight on pressure safety
valve forces, and provide economical solutions to engineering analysis, member design,
and connection design.
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3. Literature Review
3.1 Introduction
This section will focus on review of the available literature on the subject of
pressure safety valve loading. The core attention of this literature review will be the
mechanics of pressure safety valves, and the forces imposed by various discharge
systems, followed by characteristics of the loading types. These loads will be further
implemented in the analysis of a generalized pipe rack. Layout guidelines for pipe racks
will also be reviewed, as their dynamic properties have a major influence on the loads
they must resist.
3.2 Pressure Safety Valves Mechanics
A pressure safety valve is a safety device designed to protect a pressurized vessel,
system or piping during an overpressure event; which refers to any condition which
would cause pressure in a vessel, system, pipe or storage tank to increase beyond the
specified design pressure or maximum allowable working pressure. The pressure safety
valve also needs to remain sealed during normal operations to keep the system closed.
The most widely used pressure safety valve is a direct spring safety valve as shown in
Figure 3-1. The direct spring pressure relief valve shown in Figure 3-1 is designed to
work exclusively on compressible media such as gases or steam.
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Figure 3-1 Typical Direct Spring Safety Valve Components (Pentair)
The direct spring safety valve consists of a nozzle, sealed by a disk that is
attached to a spring that is preloaded. The adjustable preload in the spring is fixed to the
set pressure. The set pressure is the design pressure at which the inlet pressure acting on
the disk area overcomes the spring force causing the disk to lift. The disk is connected to
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a guide that only allows for vertical movement. Most direct spring valves have a hand
pulled lever that is used for testing.
In a direct spring loaded safety valve, the closing force is applied by a helical
spring compressed by rotating an adjusting screw. The spring force is transmitted via the
spindle compressed on the disc. The disc seals against the nozzle as long as the spring
force is greater than the force created by the pressure acting against the disk inlet of the
valve.
In an upset situation a safety valve will open at a predetermined set pressure. As
the pressure below the valve increases above the set pressure the disc begins to lift, fluid
enters the huddling chamber exposing the internal pressure to an enlarged area, as shown
in Figure 3-2. This causes an incremental change in force, called the popping pressure,
which overcompensates for the increase in the downward spring force and allows the
valve to open at a rapid rate. This effect allows the valve to achieve maximum lift and
capacity within overpressures that will let this valve be set at the maximum allowable
working pressure and prohibit the accumulation pressure from exceeding Code mandated
levels. Overpressure is the pressure increase above the set pressure necessary for the
safety valve to achieve full lift and capacity. Codes and standards provide limits of the
maximum overpressure and is typically expressed as a percentage of the set pressure. A
typical value is 10 percent, ranging between 4 percent and 20 percent depending on the
code and application of the valve.
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Figure 3-2 Valve at Set Pressure (Pentair)
Because of the larger disc area exposed to the system pressure after the valve
achieves lift and the mass momentum due to the velocity of the fluid flow, the valve will
not close until system pressure has been reduced to some level well below the set
pressure. Blowdown is defined as the difference between the set pressure and the closing
point, or reseat pressure, and is frequently expressed as a percentage of set pressure. For
these safety valves, the typical performance curve is shown in Figure 3-3.
The operating pressure of the protected equipment should remain below the
reseating pressure of the valve. Manufacturers and codes and standards recommend the
reseating pressure and the operating pressure to be differentiated by 3 to 5 percent to
allow proper reseating of the valve and achieve good seat tightness again.
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Figure 3-3 Discharge of a Pressure Safety Valve (Leser)
3.3 Loads Caused by Pressure Safety Valves
One of the most common dynamic fluid forces often encountered in piping is the
relieving force from a safety relief valve. Safety valve relieving systems are generally
divided into two categories: open discharge and closed discharge. In an open discharge
system, the fluid is simply discharged into the atmosphere. The closed discharge system
collects the discharge fluid in a drum or header for proper recycling or disposal. In both
systems the discharge of a pressure safety valve will impose a reaction force due to the
effects of both momentum and static pressure. The magnitude of the reaction force will
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differ substantially depending on whether the installation is an open or a closed
discharge.
Many Codes and Standards are published throughout the world dealing with the
reaction forces of pressure relief valves. In part they are all pretty much the same. All
forces are approximations of the maximum steady state force during a release. The most
widely used pressure relief valve voluntary standard in the United States is published by
the American Petroleum Institute (API) 520. This standard provides recommended
practices for pressure relief valve construction, sizing, installation and maintenance.
The American Society of Mechanical Engineers (ASME) B31.1 Code also
provides specific rules governing the application of overpressure protection,
determination of and allowable tolerance on set pressure, allowable overpressure,
required blowdown, application of multiple valves, sizing for fire, requirements for
materials of construction, and rules for installation.
3.3.1 Open Discharge Systems
For non-toxic, non-hazardous fluids, the over-pressured fluid may be discharged
to the atmosphere either directly or through a separate vent pipe or a silencer. Figure 3-4
shows the most basic installation of the open discharge safety valve. The over-pressure
fluid is simply discharged to the atmosphere. In this case, the most apparent dynamic
force is the reaction of the discharge fluid momentum. “[Since,] the operating pressure is
generally much higher than twice the atmospheric pressure, the flow is sonic [speed of
16
sound = 1,125 ft/s] at the valve orifice location, and is most likely either sonic or
supersonic at the valve elbow exit location,” (Peng).
Figure 3-4 Pressure Safety Valve with Open Discharge (API 520)
Therefore, a pressure force also exists at the end of the safety valve elbow. The total force
at the end of the discharge elbow becomes:
17
where P1 is the exit pressure and Pa is the atmospheric pressure. Subscript 1 corresponds
to the end of the safety valve elbow location.
In the preceding equation, valve vendors generally supply the flowrate, so the
main undertaking is to find the elbow exit pressure and the exit velocity. Since the flow
inside the valve chamber is a very complicated phenomenon, the calculation of this exit
pressure and velocity is complex and uncertain. Therefore, it is useful to estimate these
quantities concretively, or to avoid calculating them at all. Because the valve elbow is
short, the friction can be assumed to be negligible. In this case, the flow inside the valve
chamber and elbow is isentropic, which preserves the impulse function based on the
momentum equation. That is,
where subscript * denotes sonic condition at the throat or the valve orifice, and AT is the
valve orifice flow area. The mass rate is the same at both the orifice and elbow exit. V* is
the sonic velocity at the throat.
The equation depends on the upstream stagnation pressure and the valve orifice
flow area. The orifice flow area and the flow rate are items generally supplied by the
valve vendor. When using the vendor supplied flow rate to calculate the reaction force, it
is important to note that the vendor flow rate is generally taken as 90% of the maximum
rate. Therefore, the vendor’s specified flow rate shall be increased by 1.11 times before
being used in the calculation of the reaction force.
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American Society of Mechanical Engineers (ASME) B31.1 – Power Piping uses
semi-empirical formulas and steam tables to calculate P1 and V1 from the flow rate and
stagnation enthalpy of the inlet steam. Essentially they produce the same result.
The general shape of the reliving force time history is shown in Figure 3-5. Due to
the inertia of the fluid column before the valve, the flow starts out proportionally more
slowly than the actual valve opening, but overshoots somewhat when the valve is fully
opened. The force calculated from above, or the ASME B31.1 method is the sustained
steady-state force. In many designs, the force is idealized as a ramp-sustained force. The
actual force or the idealized ramp force can be directly applied in a time-history analysis.
Figure 3-5 Time History of Safety Valve Relieving Force
Generally the valve is open for very short periods on the order of micro seconds
to a couple seconds. Unfourtanally this period cannot be determined simply. Detailed
cases by case analysis would need to be completed to determine the time between a valve
opening and reseating. If the over pressure is sustained, however rare, the valve can be
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open for minutes until normal operations are resumed. Continuing overpressures are
generally caused by fire senarios.
3.3.2 Closed Discharge Systems
When dealing with hazardous or toxic fluids such as radioactive steam and
Figure 3-6 Pressure Safety Valve with Closed Discharge (API 520)
most hydrocarbons, the over-pressured fluid is relieved to a closed system for recycling,
treatment, or proper disposal as seen in Figure 3-6. In a closed system, the maximum
flow is generally the same as the open discharge system, unless it is choked by the
friction of excessive piping length. Therefore, the maximum reaction force produced by
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the fluid leaving an elbow, and the impulse force produced by the fluid entering an elbow
can be considered the same as the F1 force calculated for the open discharge system. If
the friction force is neglected, then the force can be considered the same throughout the
system.
Figure 3-7 Net Idealized Safety Valve Force Acting on a Pipe Leg
For a piping leg between points n and n+1 as shown in Figure 3-7, there is an F1
shape force acting on end n and another F1 shape force acting at end n+1. These forces
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have the same maximum magnitudes, but are in opposite directions. Under the steady-
state condition, there is no net force because the two forces balance each other
out. This balanced situation is maintained even if the friction is included. The situation is
different during a transient condition, such as the initial phase of the safety valve
releasing.
When the pressure safety valve starts to pop open, the flow starts from nothing to
a maximum as shown in Figure 3-5. The flow compresses the fluid inside the discharge
pipe and transmits downstream either by wave motion or actual flow velocity. In any
case, for a safety valve discharge, “the wave speed and the flow speed are considered the
same,” (Peng). The force will have the same time history shape throughout the piping,
but the arriving time is different at each point. This is called a traveling wave. Because of
this arriving time difference, each pipe leg experiences a net force, whose magnitude
mainly depend on the length of the pipe leg.
Because the force wave travels at sonic velocity, the force arriving time at both
ends differ by
where a is the sonic velocity with respect to the fluid inside the pipe. If the pipe leg is
sufficently long, the time difference becomes greater than the valve opening time. In this
case, the force at n-end reaches the maximum before n+1 end has any force to
counterbalence it. When ∆t is smaller than the effective valve opening time to, the
maximum net force can be calculated as F1 times the ratio of ∆t to to. The net force starts
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to reduce when the initial flow reaches the second elbow and eventually reduces to zero
when the flow through the leg is steady.
Although there are no code mandated maximum length of pipe runs, general
practices limit the amount of thermal expansion to 2-3 inches. To prevent excessive pipe
stresses, and excessive elongation of longitudional pipe runs during thermal expansion,
pipe designers place expansion loops reducing the axial stiffness of the piping as shown
in Figure 3-8. The expansion loops consist of four elbows creating a break between the
longitudional pipe runs. The pipe elongation is based on the temperature differental
between the installment and the operating temperature. As a minimum for process
facilities the thermal differental is taken as 115 degrees Fahrenheit. Based on the criteria,
a steel pipe will run at a maximum 400 feet (~ 120 meters) between expansion loops.
Typical pipe runs range from 50 to 175 feet (~ 15 to 50 meters). Since a maximum pipe
run can be determined quickly, the maximum steady state force from a pressure safety
valve can be expressed as the effective valve opening time.
Figure 3-8 Typical Piping Expansion Loop
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A similar net force in the reverse direction also results from the closing of the
valve. The opening force and the closing force are generally well seperated by a few
seconds, so they are considered as two independent events in the structural response.
3.3.2.1 Pressure Safety Valve Opening Times
The opening time of the pressure safety valve was shown to be the main variable
in determining the maximum force exerted on closed discharge pressure reliving systems.
The opening time is the time it takes the valve to become fully open. Studies have shown
that there is no single opening time for a valve; instead their response is primarily
dependent on the level of overpressure, and in part their size. Since the expected rates of
pressure rise are slow compared to the expected response of the valves, we typically do
not expect overpressures over 10% above the set pressure of the valve.
The opening time can generally be obtained from most valve vendors.
Manufactures data and research papers suggest the opening times vary between 0.050 to
0.100 seconds for safety relief valves in a normal industrial context. The valve opening
time is often assumed to be 0.040 seconds but test results have shown shorter opening
times. Based on some tests conducted in 1982, Auble found that the typical opening time
for Crosby valves was 0.010 seconds, and for Dresser valves was 0.015 seconds. Very
fast response (0.004 seconds) was shown possible by Pipeline Simulation and Integrity
with unrealistically high overpressures.
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Conservatively we can say that for the 10% overpressure condition, the shortest
valve opening time is 0.010 seconds. By multiplying this value by the sonic flow
velocity, the critical length of a pipe run can be determined such that a shorter length
would have a smaller reaction to the valve discharge. This value is 11 feet, which is not
very practical in industrial settings. However with an opening time of 0.100 seconds the
critical length becomes 113 feet. For closely spaces expansion loops, the reaction can be
cut in half.
Ultimately, in most closed discharge piping systems it may not be desirable, or
practicable to reduce the valve reaction forces. Conservatively a closed discharge system
can be analyzed with the same maximum forces as an open system. The difference
between the two discharge systems will become evident during dynamic analyses which
showcase the effects of the excitation period.
3.4 Single Degree of Freedom Structures
The relationship between piping and structures is very complicated. Typically
structural engineers take the load out of the piping systems and put them into the
structural systems. Although piping can have a significant amount of strength and
stiffness we generally ignore their existence with exception to their weight.
Structures are mainly idealized linear systems; most of the structural behaviors
are linear combinations of the behaviors of simple systems. The simplest system is the
Single Degree of Freedom system. The fundamental characteristic of the single degree of
25
freedom structures are the basics of structural dynamics. The mass or the nodes in a
single degree of freedom system have six degrees of freedom: three in translation and
three in rotation, along and about the three orthogonal coordinate axes. Generally not all
six degrees of freedom are participating significantly in any one particular load case. The
single degree of freedom system has only one independent displacement (translational or
rotational) to define the motion of the structural system. Figure 3-9 shows a single degree
of freedom idealized beam with its mass lumped in the middle and the response described
by the mid span displacement.
Figure 3-9 Single Degree Freedom System
From the general free body diagram shown in Figure 3-9 the general equation of
motion can be derived from the equilibrium forces. The characteristic equation of motion
to describe any unique condition is thus:
where m is the mass of the system, c is the viscous damping cofficient, and k is the
stiffness of the system. All of these terms are a function of the displacement and
derrivatives of the position of the structure.
26
3.4.1 Dynamic Load Factor
As seen previously a pressure safety valve time history is that of an impact load
that typically has a very short period of occurrence. When a load is not applied gradually
to a structural system, it will impact the system just like other sudden changes we are
familiar with. If the application of a load is very fast, it is regarded as a shock. Figure 3-
10 shows a simple single degree freedom structural system that will be used to explain
the effect of an impact. The simple single degree of freedom structure is a spring, having
negligible mass in compared to the mass of the weight, and having one end anchored to
the floor.
Figure 3-10 Dynamic Load Factors
With a static loading, as shown in Figure 3-10(a), the spring deforms slowly and
gradually while the load is applied slowly. When the weight load reaches its preset
magnitude, W, the deformation reaches, ∆st, which is determined by
where k is the spring constant of the spring or structure and ∆st is the static deformation
of the spring. Because the load is applied very gradually, the velocity of the weight is
27
negligible. The deformation stops as soon as the load stops increasing. Therefore, ∆st is
the final deformation corresponding to the weight load that was applied.
One impact type that is often used as a benchmark comparison is the suddenly
applied load. As shown in Figure 3-10(b), the whole weight load, W, is held at the top of
the spring before being released suddenly. In this case, the spring force balances the
weight load when it deforms to ∆st. The weight at this balanced position, however, is not
still, but is instead at its maximum velocity. This velocity pushes the weight and the
spring further downward, generating an additional push up spring force. This unbalanced
push up spring force caused the weight to decelerate. The weight keeps moving
downward until its velocity decreases to zero. The spring force at this final deformed
shape is larger than the weight, thus pulling the weight back up. The motion of the weight
oscillates back and forth in a cyclic form. The oscillating amplitude will be gradually
reduced to zero by the damping of the system. The weight will eventually settle at the
static balanced position, ∆st. The largest deformation can be calculated by energy balance.
Because the kinetic energy of the weight at its maximum displacement location is zero,
due to zero velocity, the participating energy at this point includes only potential energy
and stored internal structural energy. They are balanced as follows:
or
28
where 1/2*k*∆1 is the average spring force. By comparing the static load with a load
suddenly applied, it is clear that the suddenly applied load has 2.0 times the deformation
of the static load with the same magnitude. This 2.0 factor is called the dynamic load
factor (DLF). That is,
where ∆ is the maximum deformation produced by the load and ∆st is the deformation
due to the static load of the same magnitude. Thus, for a linear elastic system the
suddenly applied load can be considered as the static load with twice the magnitude.
Which means that a suddenly applied load can be treated in a static analysis as a static
load having twice the magnitude of the actual load.
When a load reaches the structural system with an initial velocity like the weight
load with a free drop, it is called an impact load. The dynamic load factor of the weight
load with a free drop can be calculated using the energy balance; however for this
application it will not be shown.
Because ∆st is independent of the free drop gap, h, the dynamic load factor
increases as the gap increases. When the free drop gap is zero, the dynamic load factor is
2.0 as derived previously. With a free drop gap greater than zero, the dynamic load factor
is always greater than 2.0.
29
The above examples are applicable only to a sustained constant load with the
magnitude of the load maintaining the same throughout the deformation. In such cases
the dynamic load factor is always greater than 1.0. However, depending on the duration
of the load and the mass of the structure, the dynamic load factor can also be smaller than
1.0, and sometimes even approaches zero.
3.4.2 Impulse and Ramping Forcing Functions
For open and closed Pressure Safety Valve discharge systems the forcing function
consisted of a ramping function with a plateau as seen in Figure 3-5. Eventually this force
will dissipate with a similar idealized ramp decreasing to zero. For the closed discharge
system the time duration of the forcing functions is less, resembling an impulse.
Since the opening times of the valves are very short the forcing function can be
idealized as an impulse similar to the example used previously presented in Figure 3-
10(b). This suddenly applied force was shown to have a dynamic load factor of 2.0. This
creates the upper bound for pressure safety valve relieving forces because there is never
an initial velocity or initial acceleration associated with the discharge.
If we go back to a ramping forcing function followed by a constant force that is
infinitely long we can expect the dynamic load factor between 1.0 and 2.0. Through
rigorous mathematics the dynamic load factor can be calculated. Chopra expresses the
dynamic load factor for this excitation as:
30
where tr is the rise time, for our case the effective valve opening time for the Pressure
Safety Valve, and Tn is the natural period of the structure. The natural period of the
structure is defined as the time for the structure to complete one complete oscillation
during free vibration. In mathematical terms it is defined as
where m is the mass of the structure and k is the structures stiffness. Figure 3-11 shows
how the dynamic load factor varies in terms of the valve opening time and the natural
period of the structure.
Figure 3-11 Response Spectrum for Ramping-Constant Force (Chopra)
As seen from Figure 3-11 a decay function would envelope the dynamic load
factor spectrum. Since the valve opening time is for the most part known, as the structural
31
natural period gets smaller the amplification force decreases. Thus, by increasing the
stiffness of the structure, the time in which the structure responds becomes faster, and the
amplification factor for the forcing function is reduced.
For closed discharge systems the forcing function resembles an impulse force. An
impulse force is a large force that acts in a very short period. The time it takes the sonic
flow in a closed discharge system to get from one elbow to another is the time of the
impulse function. Based on the pipe length between elbows and the effective valve
opening time the impulse function can resemble a triangular pulse force or a ramp
sustained ramp force. The triangular pulse force is shown in Figure 3-12 and the ramp
sustained ramp force is shown in Figure 3-7.
Figure 3-12 Triangular Pulse Function
To understand the behavior of the structure during the discharge and for
simplicity we will focus on a triangular pulse function. In the previous example where the
mass was released on the spring without free fall, the dynamic load factor was
determined to be 2.0. Similarly, with this triangular pulse function, there is not initial
32
velocity or acceleration. Therefore the dynamic response’s upper bound is the same, 2.0.
Unlike the ramp sustained force, the response due to the triangular pulse function can be
less than the static response of the structure. The dynamic load factor can also be
determined through rigorous mathematics for this function. Chopra expresses the
dynamic load factor as:
where td is the pulse duration, and t represents the phase. Figure 3-13 shows the response
of the structure due to the triangular pulse function. The amplifications decrease as the
pulse duration increases past twice the natural frequency of the structure. The overall
maximum response occurs when the pulse duration equals the natural period of the
structure.
33
Figure 3-13 Response Spectrum for Triangular Pulse Force (Chopra)
As seen in the two response spectrums that were developed the main drivers are
the effective valve opening time of the pressure safety valve and the natural period of the
structure. With the effective valve opening time know, the natural period of the structure
can be modified to change the response due to the discharge force.
3.5 Multi Degree of Freedom Systems
Very few structures can be realistically simplified down to a single degree of
freedom system. Many systems are too complex to be represented by a single degree of
freedom model. Generally, pipe racks have hundreds of members with varying stiffness’s
and orientations. The number of independent displacements required to define the
displaced structure relative to their equilibrium position is the number of degrees of
freedom. These degrees of freedom all have their own dynamic properties such as natural
period. To determine the structural response of the structure the modes of the structure
need to be combined based on their participation due to the external load.
34
4. Research Plan
A literature review was first conducted to gather and review the available
information pertaining to pressure safety valve reaction forces. There was a lot of
information from vendors and code documents that reference a maximum steady state
force due to a pressure valve discharge; however there is not any information regarding
the length of time that the force exists on a structure.
Current practice applies a dynamic load factor of 2.0 to the steady state force as a
result of an impact load since the time is generally unknown. The time that the force is
applied and the natural period of the structure have a significant effect on the dynamic
load factor. Because 2.0 is the largest dynamic load factor possible for this type of load, it
is potentially too conservative.
A general plan for research that was conducted in presented here and is described
as follows:
1. Describe a typical pipe rack to be used for comparison of various load cases.
2. Develop general load cases to describe a spectrum of possible scenarios due to
pressure safety valve discharges.
3. Develop a general pipe rack in STAAD.Pro V8i model that can be used for time
history, and benchmark static analyses.
4. Determine and validate the structures displacements form the analysis model.
35
5. Compare the results to the static condition, and the current dynamic load factor
used in practice.
36
5. Pipe Rack Analysis
5.1 Generalized Pipe Rack
A typical pipe rack will be developed in order to determine general behavior of
the structure due to the discharge loads. Pipe racks natural period varies from structure to
structure and where the load is applied on the structure. The pipe rack was based on
idealized conditions, based on typical layouts. This analysis covers the general
performance of typical structures. Since this analysis is not tremendously sensitive, there
is no need for multiple analysis models.
A typical pipe rack will have one bay in the transverse direction varying 15 to 25
feet between column lines. These widths allow for vehicular traffic within the pipe rack
corridor. About 15 feet of vertical clearance is required for vehicular equipment, and 20
feet above internal roadways provided for access of maintenance and firefighting
equipment. The pipe racks are set on concrete piers or driven steel piles which are
extruded from grade 12 to 18 inches. For the purposes of this thesis the pipe rack will be
20 feet wide between column lines, and the first piping level will be 20 feet high.
As stated before, these transverse structures are connected together using moment
frames to allow for longitudinal access of pipes, cable trays, mechanical, and vehicular
equipment. It would be uneconomical to fix every level of the pipe rack; therefore
common practice is a moment connection at the first level and the top level of the pipe
rack. Levels of the pipe rack are assumed to be fully loaded with pipe, and when the
pipes need to exit the rack to the side to connect with equipment, a flat turn cannot be
37
achieved as this would clash with the other pipes at the same level. Thus the exiting pipe
is typically routed to turn either up or down and then out of the rack. Figure 1-3 shows a
transverse frame with an exiting pipe. A spacing of 5 feet is generally used between pipe
rack levels to permit room for pipes to enter and exit the pipe rack. If the pipe rack
carries larger pipes, additional room may be required between levels. This level spacing
should be determined with the help of the piping and process engineers. Spacing
between pipe rack levels of 5 feet will be used in this thesis.
The transverse moment frames are connected together by longitudinal struts
consisting of several bays. The longitudinal struts are placed between the piping
elevations to support pipes entering and exiting the pipe rack.
Overall lengths of pipe racks depend on the plant site layout. Due to thermal
considerations, they are generally kept less than 200 feet. In long pipe runs, multiple pipe
racks are placed next to each other separated with an expansion joint. A central
longitudinal bay within the pipe rack is typically braced to provide longitudinal stability
and allow the rack to expand and contract due to thermal loads. If multiple bays were
braced in the same column line, then large thermal loads would be imposed on the
structure.
Each bay of the pipe rack is equally spaced at 15 to 20 feet. This spacing is
chosen by pipe engineers based on the maximum allowable span of the pipes being
supported. The span is governed by the strength and deflection of the piping. Generally a
38
2 inch pipe is the smallest line supporter on a pipe rack. A 2 inch schedule 80 pipe full of
water can span 20 feet. Therefore in this thesis the bay spacing will be 20 feet.
On pipe racks with pressure safety valve discharges a platform is provided at the
top level for personal to access these valves. Generally there are multiple safety valves
on the platform.
Figure 5-1 Isometric View of Typical Pipe Rack Used for Analysis
Figure 5-1 shows the idealized pipe rack used for this analysis. Members were
selected based on normal pipe rack deflection limitations. For occupied pipe racks, like
39
this one with the access platform, Process Industry Practices (STC01015) limit
deflections from wind to H/200. This requirement will make sure the structure is within
the relative stiffness of existing pipe racks, which is also a good indication of the natural
period.
The full pipe rack was chosen to be modeled to account for the mass and stiffness
of the entire structure. Critical mode shapes could have been overlooked in a frame
analysis.
5.2 Structural Damping
Damping is a critical and very complex phenomenon in dynamic analysis. It is
critical to have the right initial parameters to obtain logical dynamic analysis results.
Damping is a term used to describe the means by which the response motion of a
structural system is reduced at each cycle as a result of energy absorption. The energy is
absorbed by friction within steel connections, and opening and closing of micro-cracks
which are summarized mathematically as a viscous damping coefficient.
Damping is determined through analysis by hysteresis loops, which shows the
energy absorption within each cycle. In general the damping coefficients of metals
depend on the stress amplitudes and temperature. At stresses well above yield, steel
structures can expect a damping coefficient of 7 to 20 percent as would be the case in an
extreme seismic event. In the event where stresses are well below the yield stress the
damping coefficient can be 0.5 percent.
40
For this analysis it is expected that the stresses induced by the pressure safety
valve will be well below yield for the main lateral resisting system, therefore a damping
ratio of 2 percent was selected.
In a side study within this thesis, a damping ration of 0.5 percent, 2 percent and 5
percent were analyzed for this pipe rack under open discharge loading. Theses damping
coefficients were selected because they correlate to stresses that are less than yielding.
Figure 5-2 shows the time-displacement summary for the open discharge loading with
varying damping coefficients.
0
0.5
1
1.5
2
2.5
3
3.5
0 0.25 0.5 0.75
Dis
pla
cem
ent
(in
.)
Time (s)
0.005
0.02
0.05
Static
Figure 5-2 Damping Effect on Structural Response
In all cases the peak response occurred in the third cycle. A half percent
damping shows 7 percent higher response than 2 percent damping. The 5 percent
damping on the other hand shows an 11 percent lower response than 2 percent damping.
41
Damping effects on the first cycle are shown negligible which is consistent with the
definition of damping.
For these types of excitations damping has a considerable effect on the maximum
response of the structure. Because the members are not being designed for plastic
ductility for this particular load case we are only concerned with the peak response. The
damping coefficient of 2 percent will give a good approximation of the structural
behavior but will not be the exact solution for this problem. The stress for this type of
load will be considered less than half the yield stress.
5.3 Load Cases
A couple load cases were considered in this analysis. Multiple load cases were run
for each open discharge and closed discharge systems. Only the load case was considered
because the load combinations are a variety of linear combinations which can be added
into the results later if necessary.
In practice, engineering judgment should be used in determining all applicable
loads. The following discussion is meant to define loads only for analysis and
comparison of general pipe racks response due to pressure safety valve loads and
therefore certain simplifications are made to facilitate analysis but still provide results
that are typical of pipe racks.
The effective mass of the structure is important for determination of the mode
shapes and mass participation in the dynamic analysis. The effective mass is any load that
42
is on the structure that can contribute to the inertia forces in the structure. The masses are
applied in all three orthogonal directions. For a pipe rack the loading changes drastically,
however Process Industry Practices (STC01015) provides design values for the design
dead, and operating loads. In most cases a fully loaded pipe rack will have 40 pounds per
square foot distributed at each level. This is equivalent to 8-inch diameter, schedule 40
pipes, full of water, at 15 inch spacing. Since this is an upper bound we will consider 60
percent of this to be the effective mass. Live loads and snow loads will not be considered
in an effective mass since their probability of being on the rack when there is a discharge
is small. The pipe rack used for analysis and comparison of load cases will have
consistent effective mass between cases to limit the number of variables.
Four load cases were selected for the pressure safety valve discharge, two cases
for open discharges, and two cases for closed discharge systems. Each load case will have
the same maximum force of 2.5 kips. The load that is being applied has no value to the
results except that it is the same for all test cases. However, in Case IV the maximum
force will be less than 2.5 kips because the pipe runs are short in the closed discharge
system. The discharge forces will be applied to the top beam at mid-span. Figure 5-3
shows the beam that the load is applied.
43
Figure 5-3 Pipe Rack with Pressure Relief Discharge Applied
The valve opening time in all cases will is assumed to be the industry standard
0.040 seconds. This short of a period has little effect on open discharge systems. Closed
systems on the other hand could benefit in reduced forces based on the valve opening
time. For consistency of the results, and due to the wide range of tested opening times,
the assumed industry standard of 0.040 seconds will be defaulted to.
44
5.3.1 Case I
The first load case considered is an open discharge system, in which the discharge
force is constant for long periods of time. This situation arises when a processes rate of
pressure release for the valve is similar to the rate that pressure is being added to the
system. This could be due to a fire, or a cooking process that continuously adds heat.
Figure 5-4 shows the idealized input forcing function that is used for this analysis.
For this loading, the response is expected to be at a maximum within the first second. To
capture the response between the maximum and the static steady state solution, 10
seconds of the structural behavior will be recorded. Figure 5-4 only shows the first two
seconds but the function can be upwards of 30 minutes long.
0
1
2
3
0 0.5 1 1.5 2
Fo
rce
(kip
)
Time (s)
Figure 5-4 Case I Forcing Function
45
5.3.2 Case II
The second load case considered is an open discharge system, in which the
discharge force has a very short period. This situation arises when a processes rate of
pressure release for the valve is much greater than the rate that pressure is being added to
the system. This could be due to external thermal loads such as heat from the sun. A
process engineer could indicate an estimated discharge time for a particular system.
Figure 5-5 shows the idealized input forcing function that is used for this analysis.
For this loading, the length of the time for the discharge is based on the system volume,
and amount of overpressure. In short bursts, the overpressure remains low. For this case a
short burst was considered to last 0.5 seconds. In reality a burst can be shorter than the
valve opening time.
0
1
2
3
0 0.5 1 1.5 2
Fo
rce
(kip
)
Time (s)
Figure 5-5 Case II Forcing Function
46
5.3.3 Case III
The third load case considered is a closed discharge system. The closed discharge
force was shown to be a function of the pipe length. To estimate a higher bound response
the maximum pipe length will be estimated as 400 feet which is consistent with common
practice. It should be noted that there is no code mandated maximum. Based on sonic
flow the total discharge period will be approximately 0.40 seconds
Figure 5-6 shows the idealized input forcing function that is used for this analysis.
It appears that the forcing function for cases II and III are the same except that the period
is a tenth of a second less for case II. These cases were considered to be lumped into one
analysis but it is beneficial to see how sensitive the response is to small changes to the
length of time that the force is applied.
0
1
2
3
0 0.5 1 1.5 2
Fo
rce
(kip
)
Time (s)
Figure 5-6 Case III Forcing Function
47
5.3.4 Case IV
The fourth load case considered is a closed discharge system in which the piping
length is very short. This load case applies where the discharge piping dumps the fluid
into large piping headers or drums. Typically when this is done the discharge piping only
travels a few feet because it is located very close to these large volumes. Since the flow
rate is conserved, the larger volumes reduce the velocity to a point where inertia forces
within the header or drums are neglected. Likewise, the static pressure is reduced because
pressure is inversely related to volume.
Figure 5-7 shows the idealized input forcing function that is used for this analysis.
For this case the longest pipe run is typically the width of the pipe rack which is 20 feet.
Within just 20 feet there is not enough pipe run distance to achieve the full unbalanced
force of 2.5 kips. In this case the force was reduced by nearly 44 percent.
0
1
2
0 0.5 1 1.5 2
Fo
rce
(kip
)
Time (s)
Figure 5-7 Case IV Forcing Function
48
5.4 Base Support Conditions
Column support conditions are affected by various factors, and they significantly
affect the behavior of the structure. End fixity for columns in actual conditions can be
very hard to accomplish but can see significant savings in member sizes within the
superstructure. Additional considerations to the foundation and anchorage design could
offset any savings. In reality a support condition is never truly fixed nor pinned. Varying
foundation types and anchorage layouts can significantly affect the rotational stiffness of
the structure.
Fixed base moment frames will also typically see a reduction in deformations due
to the additional moment capacity generated by the base fixity. Pinned base moment
frames on the other hand will typically require heavier members and experience
potentially larger deformations compared to similar fixed base moment frames. The fixed
base will also cause the structure to become stiffer, reducing the natural period of
vibration, and ultimately making the structure more responsive to suddenly applied loads
like a pressure safety valve discharge. Pinned base support condition models were
considered the design standard for these type of structures and will be used in the analysis
of the load cases, however a Case I will also be analyzed with fixed base supports to see
the impact it has on the structural response.
49
6. Comparison of Results
To verify that the dynamic analysis has adequate response the modal mass
participation was evaluated. Table 6-1 shows a summary of the modal periods of
vibrations, and their respective mass participation. The pressure safety valve load cases
applied were solely in the “X” direction. We can expect the general response of the
structure in the same direction. The table below shows that the behavior of the structure
due to these loads and it can be summarized be three modes; Mode 1, Mode 3, and Mode
6. The three modes contribute to 98.8 percent of the response. For the load cases
examined later, 99.2 percent of the response was captured within these 10 modes.
Table 6-1 Modal Mass Participation
Mode Frequency
(Hz)
Period
(s)
Participation (%) Type
X Y Z
1 0.659 1.517 35.501 0.000 0.000 Elastic
2 0.672 1.487 0.046 0.000 0.000 Elastic
3 0.672 1.487 33.869 0.000 0.000 Elastic
4 0.692 1.445 0.017 0.000 0.000 Elastic
5 0.728 1.373 0.049 0.000 0.000 Elastic
6 0.728 1.373 29.455 0.000 0.000 Elastic
7 1.551 0.645 0.000 0.000 34.284 Elastic
8 1.763 0.567 0.132 0.001 0.000 Elastic
9 4.172 0.24 0.205 0.000 0.000 Elastic
10 4.195 0.238 0.000 0.000 0.000 Elastic
99.274 0.001 34.284
Figures 6-1 through Figures 6-3 show the mode shape of Mode 1, Mode 3, and
Mode 6 respectively. Mode 1 oscillates with a natural period of 1.517 seconds. In which
50
the centeral bay translates along the “X” axis. Mode 3 and Mode 6 oscillates slightly
slower in the exterior bays, but are considered to have about the same natural period as
Mode 1.
Figure 6-1 Mode 1 Deformed Shape
Since this is a translational problem and the load is applied symetricly on the pipe
rack, the mode shapes seem reasonable to descibe the response of the structue. Given the
height of the structure, the fundamental natural viberation period of the pipe rack appears
to fit a typical design.
51
Figure 6-2 Mode 3 Deformed Shape
Figure 6-3 Mode 6 Deformed Shape
In the analysis of the load cases 10 seconds of the response was captures at the
point on the structure where the load was applied. The response varies between the load
52
cases due to the applied load. The maximum response for all cases occurred within the
first second after the load was applied. Thus, the dynamic load factor will be at a
maximum for a discharge durration longer than one second.
Case I would be the considered for a open discharge flow durration longer than
one second. The time displacement history for Case I is shown in Figure 6-4. The doted
line shows the dynamic displacement while the solid line shows an expected deformation
with a static analysis. The figure shows that the structure has a lag initially when the load
is applied but then over shoots the static deflection due to inertia forces. As the time goes
on the damping within the structure lessen the amplitude of the structure and converges
around the static deflection. There are two distinct cycles that are occuring in the
structure displacement, the first is the moment
0
0.5
1
1.5
2
2.5
3
3.5
0 1 2 3 4 5 6 7 8 9 10
Dis
pla
cem
ent
(in
)
Time (s)
Figure 6-4 Response Due to Case I Load
53
resisting frame, while the second peak is the beam which the load is applied. These
deformations do not have the same period. At the peak of the response the structure had a
maximum displacement 1.84 times greater than the static displacement.
The 2.0 dynamic load factor is established using similar load cases as presented in
Case I but with more flexible structures. The actual response of the idealized pipe rack
shows that the established dynamic load factor is about 10 percent conservative. This
extra conservation may be small, however it seems to be redundant when combined with
Load and Resistance Factored Design (LRFD). On the other hand 2.0 is a nice number to
use in engineering calculations.
Case II is similar to Case I in the sense that they are both open discharges.
Case II occurs when the rate of pressure build up is small, like thermal heating of the
fluid from the sun. Since anything over one second will have a similar response to Case I,
the period of the discharge is selected as 0.5 seconds. Figure 6-5 shows the time
displacement history of Case II within the first 10 seconds. The initial response is similar
to Case I except that the peak deformation occurs within the first cycle. After the first
cycle has been completed, the load effect is gone. The mass momentum forces the
structure to oscillate, but this time they develop both positive and negative displacements.
The negative maximum displacement for this case is similar to the maximum positive
displacement, which may need to be considered in design. Ultimately the displacement
for this case after a minute will be zero. At the peak of the response the structure had a
maximum displacement 1.44 times greater than the static displacement.
54
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5 6 7 8 9 10
Dis
pla
cem
ent
(in
)
Time (s)
Figure 6-5 Response Due to Case II Load
When the period of the discharge is decreased, the dynamic load factor will also
decrease. This relationship is seen between Case I and Case II. For periods less than a
quarter second the load factor can become less than one. In most cases it is not practical
to design for periods that are significantly smaller than a half second.
Case III deals with the upper bound envelope for closed discharge systems. It was
shown that the maximum force was based on the pipe run length. For this case the
maximum pipe run length provided a time similar to that of the discharge period for Case
II. This case was considered to be neglected due to its similar nature to the previous load
case; however it was decided to analyze this load case to show the sensitivity of the load
factor to small changes in the time that the force exists. Figure 6-6 shows the time
55
displacement history for Case III. Ultimately the maximum positive displacement was
almost the exact same as Case II. The maximum negative displacement increased
significantly, but is still less than the positive displacement. Generally the response due to
a time change of a tenth of a second is the same. At the peak of the response the structure
had a maximum displacement 1.44 times greater than the static displacement.
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5 6 7 8 9 10
Dis
pla
cem
ent
(in
)
Time (s)
Figure 6-6 Response Due to Case III Load
Case IV is a lower bound for pressure safety valve discharge forces. It deals with
short pipe runs which release the discharged fluids into large volumes like headers and
tanks. The pipe run was selected as the width of the pipe rack. In that small of a distance
the force does not have enough time to develop before the other end of the pipe starts to
balance out the force. Because the pipe length is so small the maximum for on the pipe
rack is 44 percent of the total design force. Figure 6-7 shows the time displacement
56
history of Case IV within the first 10 seconds. Because the period is so short (0.036
seconds) the structure does not have any time to react to the applied load. At the peak of
the response the structure had a maximum displacement 0.13 times the static
displacement due to the applied load. Since this is the same pressure safety valve as all
the other load cases the displacement should be compared with the maximum load due to
an open discharge. In that case the maximum displacement is 0.059 times the static
displacement for the open system.
Figure 6-7 Response Due to Case IV Load
Case V is the same as Case I with exception that the support conditions are fixed
instead of pinned. This analysis will show the effect on the natural period of the structure.
In essence the mode shapes and the modal mass participation factors are similar to the
pinned support cases presented in Table 6-1. The difference is that the fundamental
57
natural period of vibration decreased to 0.870 seconds; more than a 60 percent reduction.
Figure 6-8 presents the time displacement history for Case V. It is clear that the static
deflection is much less than the pinned condition, as such is the dynamic response. The
stiffer structure reaches its maximum displacement on the first cycle vs. the second cycle
in Case I. Otherwise the response is similar. At the peak of the response the fixed base
structure had a maximum displacement 1.76 times greater than the static displacement.
0
0.5
1
1.5
2
2.5
3
3.5
0 1 2 3 4 5 6 7 8 9 10
Dis
pla
cem
ent
(in
)
Time (s)
Figure 6-8 Response Due to Case V Load
Table 6-2 shows a summary of the results for each load case. The dynamic
load factor for all cases analyzed is less than the standard 2.0. Even with large changes of
the structural period the dynamic load factor calculated only changed by about 5 percent.
To get the dynamic load factor close to 2.0 the structure would need to become more
58
flexible, which is not practical because the pipe rack would then not satisfy drift
limitations.
Table 6-2 Summary of Load Cases
Case Period
(s) Static
Deformation (in)
Dynamic
Deformation (in)
Dynamic Load
Factor
I 1.517 1.853 3.414 1.842
II 1.517 1.853 2.674 1.443
III 1.517 1.853 2.674 1.443
IV 1.517 1.853 0.110 0.059
V 0.87 1.522 2.673 1.756
For open discharge systems the dynamic load factor is similar to the ones
specified for pipes in ASME B31.1. For closed discharge systems, depending on pipe
length, the dynamic load factor can be reduced to 1.5, or in some cases, neglected
altogether.
59
7. Conclusions
For the representative pipe rack model with extremely enveloped cases for both
open and closed pressure safety valve discharge systems the dynamic load factor that
would be used in a static analysis was found to be less than 2.0. Because we determined
that 2.0 is the maximum load factor for forces with zero initial acceleration and velocity,
it can be conservatively be the upper bound for all pressure valve discharges.
The dynamic load factor was determined by the ratio of the dynamic displacement
to the static displacements that is a function of the steady state load. The steady state
force is generally provided by process engineers and typically has a dynamic load factor
of 2.0 already built in. The force can also be calculated conservatively with the valve size
known.
A response spectrum was created with the five load cases analyzed with a time
history in this thesis. It should be noted that a response spectrum is not necessarily a
design spectrum because more cases would need to be evaluated to determine the
response of the structure at any case. The response spectrum with the cases analyzed give
a good approximation of the structural response due to the pressure safety valve
discharges. Figure 7-1 shows the response spectrum for the load cases examined in this
thesis. The dynamic load factor was plotted against the ratio of the excitation period to
the structural period. The excitation period was simply the time that the force was
applied. For Case I and Case V the excitation period is infinitely long. To get reasonable
60
results the excitation period was taken as the time at the maximum response because for
any force longer than that the dynamic load factor would be unchanged.
A cubic regression was applied to the five data points to show the general
behavior of the structure to any excitation period. The regression shown in Figure 7-1 has
an additional 12 percent conservation built in and is limited to 2.0. The additional 12
percent would be a recommendation for design to encompass any errors within this
analysis due to damping, or approximate effective masses.
0
0.5
1
1.5
2
0.01 0.1 1 10
Dy
na
mic
Lo
ad
Fa
cto
r
td/T
Figure 7-1 Response Spectrum for Pressure Safety Valves on Pipe Racks
This subject has room for future research in creating a design spectrum. Load
cases would need to be evaluated between the cases discussed in this thesis with multiple
structures of varying frequencies. Actual pressure safety valve discharge time histories
would be recommended for the design spectrum. Actual time histories were not used in
61
this thesis due to a lack of available information. Laboratory testing shall also be
conducted to verify the methods used to create the design spectrum.
Sometimes in design the information needed to conduct a thorough analysis is not
available. In these cases engineering judgment shall be used. For open discharge systems
when the excitation period is not supplied it is conservative to assume that it is infinitely
long and use a dynamic load factor of 2.0. For closed discharge systems the dynamic load
factor can generally be very small, and in most cases negligible.
62
REFERENCES
Bendapudi, K. V. (2010, February). Structural Design of Steel Pipe Support Structures.
Retrieved from Wermac: http://www.wermac.org/pdf/steel1.pdf
Buchanio, M. (2010, September 26). Q&A: Pressure Relief Valve Scenarios. Retrieved
from Flow Control Network: http://www.flowcontrolnetwork.com/articles/q-a-
pressure-relief-valve-scenarios
Chopra, A. K. (2012). Dynamics of Structures Theory and Applications to Earthquake
Engineering. Upper Saddle River: Prentice Hall.
Engineering The technical handbook. (n.d.). Retrieved from Leser:
http://www.leser.com/en/tools/engineering.html?et_cid=14&et_lid=31&et_sub=
MT_EN_Engineering_Startseite
Munson, B. R., Young, D. F., Okiishi, T. H., & Huebsch, W. W. (2009). Fundamentals of
Fluid Mechanics. Hoboken: Wiley.
Peng, L.-C., & Peng, T.-L. (2009). Pipe Stress Engineering. Fairfield: ASME Press.
Pentair Pressure Relief Valve Engineering Handbook Anderson Greenwood, Crosby and
Varec Products. (2012). Pentair Valves and Controls.
(2007). PIP STC01015 Structural Design Criteria. In Process Industry Practices. Austin:
Process Industry Practices.
(2002). Testing and analysis of relief device opening times. In Pipeline Simulation and
Integrity Ltd. Colegate: Crown.
63
(2012). Power Piping ASME Code for Pressure Piping (ASME B31.1). New York: The
American Society of Mechnical Engineers.
(2000). Sizing, Selection, and Installation of Pressure-Relieving Devices in Refineries
(API 520). Washington D.C.: American Petroleum Institute.
Solken, W. (2008). Introduction to Pipe Racks. Retrieved from Wermac:
http://www.wermac.org/steel/piperacks.html
(2010). Specification for Structural Steel Buildings (ANSI/AISC 360-10). In American
Institute of Steel Construction (AISC). Chicago: American Institute of Steel
Construction, Inc.
STAAD.Pro V8i "Technical Refrence Manual". (2007). Yorba Linda: Bentley Systems,
Inc.
64
Appendix A – STAAD Input Pressure Safety Valve Time History
Analysis
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 3/6/15
JOB NAME Thesis
JOB CLIENT UCD
JOB REV 0
ENGINEER NAME JRN
END JOB INFORMATION
INPUT WIDTH 79
*********************************************************************
* GEOMETERY
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 20 0 0; 3 0 20 0; 4 20 20 0; 5 0 22.5 0; 6 20 22.5 0; 7 0 25 0;
8 20 25 0; 9 0 27.5 0; 10 20 27.5 0; 11 0 30 0; 12 20 30 0; 13 0 32.5 0;
14 20 32.5 0; 15 0 35 0; 16 20 35 0; 17 10 35 0; 18 10 39 0; 19 6 35 0;
20 14.25 35 0; 21 15.75 35 0; 22 0 0 20; 23 0 20 20; 24 0 22.5 20; 25 0 25 20;
26 0 27.5 20; 27 0 30 20; 28 0 32.5 20; 29 0 35 20; 30 20 0 20; 31 20 20 20;
32 20 22.5 20; 33 20 25 20; 34 20 27.5 20; 35 20 30 20; 36 20 32.5 20;
37 20 35 20; 38 6 35 20; 39 10 35 20; 40 14.25 35 20; 41 15.75 35 20;
42 10 39 20; 43 0 0 40; 44 0 20 40; 45 0 22.5 40; 46 0 25 40; 47 0 27.5 40;
48 0 30 40; 49 0 32.5 40; 50 0 35 40; 51 20 0 40; 52 20 20 40; 53 20 22.5 40;
54 20 25 40; 55 20 27.5 40; 56 20 30 40; 57 20 32.5 40; 58 20 35 40;
59 6 35 40; 60 10 35 40; 61 14.25 35 40; 62 15.75 35 40; 63 10 39 40;
64 0 0 60; 65 0 20 60; 66 0 22.5 60; 67 0 25 60; 68 0 27.5 60; 69 0 30 60;
70 0 32.5 60; 71 0 35 60; 72 20 0 60; 73 20 20 60; 74 20 22.5 60; 75 20 25 60;
76 20 27.5 60; 77 20 30 60; 78 20 32.5 60; 79 20 35 60; 80 6 35 60;
81 10 35 60; 82 14.25 35 60; 83 15.75 35 60; 84 10 39 60; 85 0 0 80;
86 0 20 80; 87 0 22.5 80; 88 0 25 80; 89 0 27.5 80; 90 0 30 80; 91 0 32.5 80;
92 0 35 80; 93 20 0 80; 94 20 20 80; 95 20 22.5 80; 96 20 25 80; 97 20 27.5 80;
98 20 30 80; 99 20 32.5 80; 100 20 35 80; 101 6 35 80; 102 10 35 80;
103 14.25 35 80; 104 15.75 35 80; 105 10 39 80; 106 0 0 100; 107 0 20 100;
108 0 22.5 100; 109 0 25 100; 110 0 27.5 100; 111 0 30 100; 112 0 32.5 100;
113 0 35 100; 114 20 0 100; 115 20 20 100; 116 20 22.5 100; 117 20 25 100;
118 20 27.5 100; 119 20 30 100; 120 20 32.5 100; 121 20 35 100; 122 6 35 100;
123 10 35 100; 124 14.25 35 100; 125 15.75 35 100; 126 10 39 100;
127 0 11.25 50; 128 20 11.25 50; 129 20 27.5 50; 130 0 27.5 50; 131 0 32.5 50;
132 20 32.5 50; 133 10 39 50;
65
MEMBER INCIDENCES
1 1 3; 2 3 5; 3 5 7; 4 7 9; 5 9 11; 6 11 13; 7 13 15; 8 2 4; 9 4 6; 10 6 8;
11 8 10; 12 10 12; 13 12 14; 14 14 16; 15 3 4; 16 7 8; 17 11 12; 18 15 19;
19 19 17; 20 17 20; 21 20 21; 22 21 16; 23 17 18; 24 18 19; 25 22 23; 26 23 24;
27 24 25; 28 25 26; 29 26 27; 30 27 28; 31 28 29; 32 30 31; 33 31 32; 34 32 33;
35 33 34; 36 34 35; 37 35 36; 38 36 37; 39 23 31; 40 25 33; 41 27 35; 42 29 38;
43 38 39; 44 39 40; 45 40 41; 46 41 37; 47 39 42; 48 42 38; 49 43 44; 50 44 45;
51 45 46; 52 46 47; 53 47 48; 54 48 49; 55 49 50; 56 51 52; 57 52 53; 58 53 54;
59 54 55; 60 55 56; 61 56 57; 62 57 58; 63 44 52; 64 46 54; 65 48 56; 66 50 59;
67 59 60; 68 60 61; 69 61 62; 70 62 58; 71 60 63; 72 63 59; 73 64 65; 74 65 66;
75 66 67; 76 67 68; 77 68 69; 78 69 70; 79 70 71; 80 72 73; 81 73 74; 82 74 75;
83 75 76; 84 76 77; 85 77 78; 86 78 79; 87 65 73; 88 67 75; 89 69 77; 90 71 80;
91 80 81; 92 81 82; 93 82 83; 94 83 79; 95 81 84; 96 84 80; 97 85 86; 98 86 87;
99 87 88; 100 88 89; 101 89 90; 102 90 91; 103 91 92; 104 93 94; 105 94 95;
106 95 96; 107 96 97; 108 97 98; 109 98 99; 110 99 100; 111 86 94; 112 88 96;
113 90 98; 114 92 101; 115 101 102; 116 102 103; 117 103 104; 118 104 100;
119 102 105; 120 105 101; 121 106 107; 122 107 108; 123 108 109; 124 109 110;
125 110 111; 126 111 112; 127 112 113; 128 114 115; 129 115 116; 130 116 117;
131 117 118; 132 118 119; 133 119 120; 134 120 121; 135 107 115; 136 109 117;
137 111 119; 138 113 122; 139 122 123; 140 123 124; 141 124 125; 142 125 121;
143 123 126; 144 126 122; 145 5 24; 146 24 45; 147 45 66; 148 66 87;
149 87 108; 150 6 32; 151 32 53; 152 53 74; 153 74 95; 154 95 116; 155 10 34;
156 34 55; 157 55 129; 158 76 97; 159 97 118; 160 9 26; 161 26 47; 162 47 130;
163 68 89; 164 89 110; 165 13 28; 166 28 49; 167 49 131; 168 70 91; 169 91 112;
170 14 36; 171 36 57; 172 57 132; 173 78 99; 174 99 120; 175 18 42; 176 42 63;
177 63 133; 178 84 105; 179 105 126; 180 17 39; 181 39 60; 182 60 81;
183 81 102; 184 102 123; 185 20 40; 186 40 61; 187 61 82; 188 82 103;
189 103 124; 190 21 41; 191 41 62; 192 62 83; 193 83 104; 194 104 125;
195 16 37; 196 37 58; 197 58 79; 198 79 100; 199 100 121; 200 43 127;
201 51 128; 202 127 66; 203 128 74; 204 64 127; 205 127 45; 206 72 128;
207 128 53; 208 129 76; 209 130 68; 210 131 70; 211 132 78; 212 74 129;
213 129 53; 214 76 132; 215 132 55; 216 45 130; 217 66 130; 218 47 131;
219 131 68; 220 133 84;
*********************************************************************
* PROPERTIES AND SPECIFICATIONS
MEMBER RELEASE
16 17 23 24 40 41 47 48 64 65 71 72 88 89 95 96 112 113 119 120 136 137 143 -
144 TO 174 180 TO 201 204 TO 207 212 TO 219 START MY MZ
16 17 24 40 41 48 64 65 72 88 89 96 112 113 120 136 137 144 TO 156 -
158 TO 161 163 TO 166 168 TO 171 173 174 180 TO 199 202 TO 219 END MY MZ
175 TO 179 START MY
66
175 176 178 179 220 END MY
DEFINE MATERIAL START
ISOTROPIC STEEL
E 4.28151e+006
POISSON 0.3
DENSITY 0.489024
ALPHA 1.2e-005
TYPE STEEL
STRENGTH FY 5288.19 FU 8517.08 RY 1.5 RT 1.2
DAMP 0.02
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
15 TO 22 39 TO 46 63 TO 70 87 TO 94 111 TO 118 135 TO 142 TABLE ST W12X40
1 TO 14 25 TO 38 49 TO 62 73 TO 86 97 TO 110 121 TO 134 TABLE ST W12X58
23 47 71 95 119 143 175 TO 179 220 TABLE ST W8X24
24 48 72 96 120 144 TABLE LD L30303 SP 0.03125
200 TO 207 212 TO 219 TABLE LD L60606 SP 0.03125
145 TO 174 208 TO 211 TABLE ST W10X33
180 TO 199 TABLE ST W12X26
CONSTANTS
MATERIAL STEEL ALL
*********************************************************************
* BASE SUPPORT CONDITIONS
SUPPORTS
1 2 22 30 43 51 64 72 85 93 106 114 PINNED
CUT OFF MODE SHAPE 10
*********************************************************************
* CASE I
*DEFINE TIME HISTORY DT 0.001
*TYPE 1 FORCE
*0 0 0.04 2.5 10 2.5
*********************************************************************
* CASE II
*DEFINE TIME HISTORY DT 0.001
*TYPE 1 FORCE
*0 0 0.04 2.5 0.46 2.5 0.5 0 10 0
*********************************************************************
* CASE III
*DEFINE TIME HISTORY DT 0.001
*TYPE 1 FORCE
*0 0 0.04 2.5 0.356 2.5 0.396 0 10 0
67
*********************************************************************
* CASE IV
DEFINE TIME HISTORY DT 0.001
TYPE 1 FORCE
0 0 0.01778 1.111 0.03556 0 10 0
*********************************************************************
ARRIVAL TIME
0
DAMPING 0.02
*********************************************************************
* PRIMARY LOAD CASES
*********************************************************************
LOAD 1 STATIC
JOINT LOAD
133 FX 2.5
*********************************************************************
LOAD 2 TIME HISTORY
*EFFECTIVE WEIGHT
* 0.6*40PSF*20FT=0.48K/FT
MEMBER LOAD
15 TO 22 39 TO 46 63 TO 70 87 TO 94 111 TO 118 135 TO 142 175 TO 179 -
220 UNI GX 0.48
15 TO 22 39 TO 46 63 TO 70 87 TO 94 111 TO 118 135 TO 142 175 TO 179 -
220 UNI GY 0.48
15 TO 22 39 TO 46 63 TO 70 87 TO 94 111 TO 118 135 TO 142 175 TO 179 -
220 UNI GZ 0.48
* DYNAMIC LOAD EFFECT
SELFWEIGHT X 1
SELFWEIGHT Y 1
SELFWEIGHT Z 1
TIME LOAD
133 FX 1 1
*********************************************************************
PERFORM ANALYSIS
FINISH