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.

iv

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

vii

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

viii

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

ix

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

1

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)

3

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.

4

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

5

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

6

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.

7

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.

8

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

9

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.

10

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.

11

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

12

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.

13

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.

14

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

15

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.

18

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

19

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

20

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

21

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

22

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.

24

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