Health and Safety Executive

Fluid structure interaction effects on and dynamic response of pressure vessels and tanks subjected to dynamic loading

Prepared by The Steel Construction Institute for the Health and Safety Executive 2007

RR527 Research Report

Health and Safety Executive

Fluid structure interaction effects on and dynamic response of pressure vessels and tanks subjected to dynamic loading Part 1: State-of-the-art review

The Steel Construction Institute Silwood Park Ascot Berks SL5 7QN

As part of a suite of work looking at fluid interaction effects on (and the dynamic response of ) pressure vessels and tanks subjected to dynamic loading, this report details the findings of a state-of-the-art review of the available literature, to consider analysis methodologies, dynamic loads and simplified procedures for the determination of the response of tanks and pressure vessels subjected to strong vibration. Strong vibration is defined as the shaking of a structure resulting from earthquake, blast or ship impact. The response of a tank/vessel under strong vibration can be split into three hydrodynamic components and simplified procedures exist for determining the response of fixed-base, vertical, cylindrical tanks/vessels. For other tank/vessel types, linear/non-linear finite element dynamic analyses need to be used, as no simple solution for the various hydrodynamic components are available.

This report and the work it describes were funded by the Health and Safety Executive (HSE). Its contents, including any opinions and/or conclusions expressed, are those of the authors alone and do not necessarily reflect HSE policy.

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CONTENTS

EXECUTIVE SUMMARY VI

1 INTRODUCTION 1

1.1 BACKGROUND 1

1.2 OBJECTIVES OF STUDY 1

1.3 SCOPE OF STUDY 2

2 STATE-OF-THE-ART REVIEW OF ANALYSIS METHODOLOGIES FOR

PRESSURE VESSELS AND TANKS ON TOPSIDES 3

2.1 INTRODUCTION 3

2.2 METHODOLOGIES FOR ANALYSIS OF PRESSURE VESSELS AND TANKS

ON TOPSIDES 4

2.3 SIMPLIFIED UNCOUPLED ANALYSIS 6

2.4 COUPLED ANALYSIS AND UNCOUPLED ANALYSIS USING DECK

RESPONSE SPECTRA 6

2.4.1 UNCOUPLED ANALYSIS USING DECK RESPONSE SPECTRA 9

2.4.2 COUPLED ANALYSIS 11

2.5 SUMMARY OF FINDINGS ON ANALYSIS METHODOLOGIES 11

3 REVIEW OF LOADING ON PRESSURE VESSELS AND TANKS ON TOPSIDES

FROM LATERAL EXCITATION 12

3.1 INTRODUCTION 12

3.2 EARTHQUAKE LOADING 12

3.2.1 DESCRIPTION OF EARTHQUAKE LOADING 12

3.2.2 EARTHQUAKE RESPONSE SPECTRA FOR ANALYSIS OF OFFSHORE

PLATFORMS 13

3.2.3 SECONDARY EARTHQUAKE RESPONSE SPECTRA FOR ANALYSIS OF

EQUIPMENT ON TOPSIDES 13

3.3 BLAST LOADING 14

3.3.1 DESCRIPTION OF BLAST LOADING 14

3.3.2 DECK RESPONSE SPECTRA FOR BLAST LOADING 14

3.4 LOADING ON PRESSURE VESSELS/TANKS FROM SHIP IMPACT 15

3.4.1 DESCRIPTION OF SHIP IMPACT LOADING 16

3.4.2 DECK RESPONSE SPECTRA FOR SHIP IMPACT LOADING 17

3.5 COMPARISON OF EARTHQUAKE, BLAST AND SHIP IMPACT

SECONDARY RESPONSE SPECTRA 18

3.6 SUMMARY OF FINDINGS ON LOADING ON TOPSIDES FROM

EARTHQUAKE, BLAST AND SHIP IMPACT 19

4 FLUID-STRUCTURE INTERACTION AND DYNAMIC RESPONSE OF PRESSURE

VESSELS/TANKS UNDER LATERAL EXCITATION 20

4.1 INTRODUCTION 20

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4.2 DESCRIPTION OF RESPONSE OF PRESSURE VESSELS AND TANKS 20

4.3 VERTICAL CYLINDRICAL FIXED BASE TANKS/VESSELS 22

4.3.1 CONVECTIVE COMPONENT 22

4.3.2 RIGID IMPULSIVE COMPONENT 24

4.3.3 FLEXIBLE IMPULSIVE COMPONENT 26

4.4 RECTANGULAR FIXED BASE TANKS 29

4.4.1 CONVECTIVE COMPONENT 29

4.4.2 RIGID IMPULSIVE COMPONENT 31

4.4.3 FLEXIBLE IMPULSIVE COMPONENT 32

4.5 FIXED BASE HORIZONTAL CIRCULAR CYLINDRICAL TANKS/VESSELS 32

4.6 TANKS WITH INTERNAL BAFFLES 34

4.7 ROCKING RESPONSE OF VESSELS/TANKS 35

4.8 COMBINATION OF PRESSURES AND RESULTANT STRESSES 36

4.9 SUMMARY OF DYNAMIC RESPONSE OF VESSELS/TANKS UNDER

LATERAL EXCITATION 37

5 CONCLUSIONS 39

6 REFERENCES 40

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EXECUTIVE SUMMARY

A state-of-the-art review was carried out on the analysis methodologies, dynamic loads and simplified procedures for the determination of the response of tanks and pressure vessels under strong vibration.

Strong vibration is defined as the strong shaking of a structure as may occur under dynamic loadings such as earthquake, blast and ship impact. This review covered the relevant offshore and nuclear codes of practice in addition to recent technical papers.

The strong vibration phenomenon is explicitly addressed in ISO/CD 19901-3 [4] where it is recommended that depending on the exposure level of the platform, all new installations or reassessment of existing installations be designed for the effects of strong vibration. The recommended approach for determining the response of deck appurtenances and equipment to strong vibration is based on whether the component can be classified as rigid or flexible. For rigid equipment, simplified uncoupled procedures can be used. Flexible or compliant equipment require the use of coupled analyses or uncoupled analyses using deck (or secondary) response spectra.

The dynamic loading from strong vibration can result from earthquake, blast and ship impact. The frequency content of the loading from these excitations is different and it was found that the peak values from blast and ship impact loading generally occur at higher frequencies than that from earthquake loading. In addition, it was noted that compared to onshore spectra, sea-floor spectra exhibit lower peak vertical and comparable peak horizontal accelerations although at different frequencies.

The response of a tank/vessel under strong vibration can be split into 3 hydrodynamic components termed convective or sloshing component, rigid-impulsive component and flexible-impulsive component. The frequencies associated with sloshing are usually quite low and generally only the first sloshing frequency is considered for design purposes. The rigid-impulsive component is due to rigid body motion of the fluid and is subjected to the maximum ground acceleration. The flexible-impulsive component accounts for the flexibility of the tank/vessel and is subjected to the acceleration pertaining to the relevant modes of vibration of the tank/fluid system.

It was found that simplified procedures exist for determining the response of fixed base vertical cylindrical tanks/vessels. For other tank types (rectangular, horizontal cylindrical), resort must be mode to linear/non-linear finite element dynamic analyses as no simple solution for all the hydrodynamic components are available.

Part 2 of the report will describe the numerical studies and results for the dynamic response of a typical pressure vessel on topsides under earthquake, ship impact and blast loading.

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1 INTRODUCTION

1.1 BACKGROUND

The Piper Alpha disaster in 1988, which resulted in fatalities, led to a complete re-assessment of safety on offshore platform structures. Lord Cullen in his subsequent report [1] on the disaster identified strong vibration as a major issue, which may have led to the failure of emergency systems. Strong vibration is defined as the strong shaking of the structure which can occur following blast loading, earthquakes, ship impacts or dropped objects that may result in breakdown or failure of appurtenances and equipment such as emergency shutdown systems, fire protection systems, pressure vessels and tanks. Such failures can lead to escalation effects and delay mitigation measures resulting in partial or total collapse of the structure.

The phenomenon of strong vibration has been the subject of several studies since the issue was raised in Lord Cullens report and has led to the extension of the Health and Safety at Work Act 1974 to include strong vibration hazards offshore. The Safety Case Regulations (SCR) 1992 [2] requires that all potential hazards that may lead to a major accident are identified and defines a major accident as any event involving major damage to the structure of the installation or plant affixed thereto. The risks associated with such major accidents must be reduced to as low as reasonably practicable (ALARP) through quantified risk assessment (QRA).

In the offshore industry, the main focus is usually directed at the design of the primary structure. A survey of suppliers of equipment to North Sea operators, carried out as part of an EATEC study [3], revealed that there were no specific vibration requirements set for the various equipments. However, the various items of equipment play a major role in ascertaining the overall safety of the structure and, as such, need to be designed for the loading scenarios expected during the lifetime of the structure.

The items of equipment considered in this study are pressure vessels and tanks on topsides. The various types of platform configurations that exist and the complexity of the fluid-structure interaction problem combined with the different fixity conditions of the equipments preclude the provision of standard simplified design guidelines. Instead, this study aims at providing an understanding of the loads that are associated with the strong vibration phenomenon and their subsequent treatment in the analysis of equipment on topsides. In addition, an assessment of the dynamic response of pressure vessels and tanks on topsides is carried out with particular regard to the quantification of the various hydrodynamic components governing the response under lateral excitation. Comparison against simplified codified procedures is also provided.

1.2 OBJECTIVES OF STUDY

This study is concerned with the loading and response of pressure vessels and storage tanks on topsides for both fixed jacket structures, jackups (mobile platforms) and floating platforms subjected to strong vibration. The main objectives of the study can be summarised as follows:

Assess state-of-the-art practice with respect to design of pressure vessels and tanks on topsides against lateral excitation

Provide guidance on modelling of tank-liquid and vessel-liquid systems Quantify the relative importance of the hydrodynamic components on the dynamic

behaviour of pressure vessels and tanks

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Assess magnitude of linear sloshing response due to lateral excitation.

1.3 SCOPE OF STUDY

To achieve the above objectives, the study is split into several stages as follows:

(a) Conduct a review of current practice for design of pressure vessels and storage tanks on topsides. The review includes all relevant codes of practice and relevant technical papers from the offshore industry and also, where appropriate, reference is made to the nuclear codes of practice. The review will address both the loading (from earthquakes, blast and ship collisions) and the description and quantification of the dynamic response of pressure vessels and tanks.

(b) Carry out linear dynamic finite element analysis on typical pressure vessels and tanks on topsides to determine both structural and fluid response.

(c) Carry out linear dynamic finite element analyses to assess relative importance of various hydrodynamic components on response of pressure vessels and tanks. The contribution of the various components to the total response is expected to differ depending on the configuration and geometry of the vessel/tank, the frequency content of the loading and the fixity conditions.

Chapter 2 provides an overview of the state-of-the-art practices in the analysis methodologies for pressure vessels and tanks subject to lateral excitation. The various analysis methodologies for determining the response of the equipment are discussed in relation to their range of applicability. The loading on the equipment arising from earthquake, ship impact and blast is described in Chapter 3. The procedures for computing the response of various types of tanks and vessels are outlined in Chapter 4. Finally the findings from this study are described in Chapter 5.

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2 STATE-OF-THE-ART REVIEW OF ANALYSIS

METHODOLOGIES FOR PRESSURE VESSELS AND TANKS ON

TOPSIDES

2.1 INTRODUCTION

This chapter provides a review of the analysis methodologies for design of pressure vessels and storage tanks on topsides. Both relevant papers and codes are assessed and current design practice is reported. Due to the vast amount of work carried out in the nuclear industry regarding the design of equipment (alternatively referred to as secondary systems), reference is also made to relevant nuclear codes of practice. The design process encompasses the following:

Modelling or analysis methodology for pressure vessels/tanks Loading definition and application on pressure vessels/tanks Dynamic response of the pressure vessels/tanks

The codes of practice reviewed as part of this study are as follows:

ISO/CD 19901-3 Petroleum and natural gas industries Specific requirements for offshore structures Part 3: Topsides structure [4]

API RP 2A-WSD Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms Working Stress Design [5]

NORSOK Standard N-003 Actions and Action Effects [6] NORSOK Standard N-004 Design of Steel Structures [7] ISO/CD 19902 Petroleum and Natural Gas Industries Fixed Steel Offshore Structures

[8] Guidelines for the Seismic Design of Oil and Gas Pipeline Systems ASCE [9] Seismic Design of Storage Tanks, Recommendations of a New Zealand Study Group [10] Eurocode 8: Design Provisions for Earthquake Resistance of Structures Silos, Tanks and

Pipelines [11] Department of Energy (HMSO) Offshore Installations: Guidance on design, construction

and certification [12] ASCE 4-98 Seismic Analysis of Safety-Related Nuclear Structures and Commentary [13] Committee on Nuclear Structures and Materials (ASCE) Structural Analysis and Design

of Nuclear Plant Facilities [14]

The subsequent sections in this chapter provide a review of the various procedures used in the analysis of pressure vessels and tanks subjected to lateral excitation. Description and quantification of the loading resulting from earthquake, blast and ship impacts are assessed in Chapter 3. Chapter 4 addresses the dynamic response of pressure vessels and tanks under lateral excitation and provides an overview of the various simplified procedures used to quantify the response for tanks/vessels of various configurations.

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2.2 METHODOLOGIES FOR ANALYSIS OF PRESSURE VESSELS AND TANKS ON

TOPSIDES

The terminology used in this report follows that adopted in the offshore and nuclear industries. Primary and secondary structures (or systems) are used to designate the whole platform structure and the equipment (i.e. pressure vessels, storage tanks etc.) respectively. Deck response spectra (variation of acceleration with frequency at deck level of platform) are used interchangeably with secondary response spectra.

The strong vibration phenomenon is explicitly addressed in ISO/CD 19901-3 [4], which recommends that, depending on the exposure level of the platform, all new installations or reassessment of existing installations be designed for the effects of strong vibration. General guidelines are provided for the assessment of deck appurtenances and equipment and fall into two categories namely:

Analytical methods finite element analysis to determine magnitude and frequency of loading. The risk to the items of equipment is quantified by identifying their significant modes, amplitudes and frequencies of vibration and, if required, by detailed analysis. The code recognises the fact that the loading arising from strong vibration, particularly relating to ship impacts and blast loading, is difficult to estimate accurately. Only limited studies (EATEC study [3]) have been carried out to quantify the loading on topsides arising from strong vibration. This issue is addressed further in Chapter 3.

Walkdown studies whereby safety-critical equipment are identified and a qualitative assessment of their adequacy to sustain strong vibration is carried out. The latter is not addressed in this report but details of walkdown assessments are provided in the Health and Safety Executive (HSE) report OTH 93 415 [15].

Although seismic events are part of the strong vibration phenomenon, recommendations relating to seismic resistance of deck equipment are dealt with separately in ISO/CD 19901-3 [4] and are similar to the guidelines provided in API RP 2A-WSD [5]. However, the analysis types defined for the seismic case can be equally applied to blast loading and ship impacts unless otherwise specified.

The approach to determining the response of deck appurtenances and equipment to seismic loads is based on whether the component can be classified as rigid or flexible. Components are classified as rigid if they satisfy the following requirements

Their horizontal natural frequencies of vibration fall in the high frequency tail region of the deck response spectra (earthquake loading only). For blast and ship impact, there is no clearly defined tail region as the spectral acceleration at high frequencies can be significantly higher than the maximum deck acceleration. In such cases, engineering judgement should be used to decide whether the equipment can be assumed to move in unison with deck motion.

Their support is sufficiently stiff so as not to allow for any dynamic amplification

Such components can be treated through simplified uncoupled analysis i.e. with the equipment assumed to have negligible effect on the stiffness properties of the primary structure (mass usually accounted for by point mass representation). Components that do not meet the above requirements have to be designed either through coupled analyses or uncoupled (decoupled) analyses using secondary response spectra. Figure 1 provides a summary of the analysis methodologies, which are detailed in the following sub-sections.

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Coupled Uncoupled

Uncoupled

Equipment

Classification of Equipment

Rigid Flexible

Analysis Analysis Using

Deck Response

Spectra or time

history Simplified

Analysis

Obtain

Forces/Moments

at Supports

Detailed FE

Model of

Primary

Structure and

Apply Loading Time

History or Response

Spectrum

Obtain

Forces/Moments,

Stresses at Nodal

Points in Equipment

and at Supports

Generate Deck

Response Spectra/time

history at Support Points

to Equipment

Apply loading to

Detailed FE Model

of Equipment

Obtain

Forces/Moments,

Stresses at Nodal

Points in Equipment

and at Supports

Figure 1 Analysis Methodologies For Equipment on Offshore Platforms

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2.3 SIMPLIFIED UNCOUPLED ANALYSIS

For simplified uncoupled analysis, the equipment is generally modelled as a point mass at the appropriate centre of gravity. The supports are then designed for the resulting forces and moments from the dynamic loads. ISO/CD 19901-3 [4] recommends that the forces/moments be calculated from the following steps:

(d) Determine the significant modes of vibration of the primary structure and extract the acceleration at the equipment support for each mode

(e) Compute the equipment acceleration by multiplying the acceleration at the support from (a) by a dynamic magnification ratio and

(f) Multiply the equipment mass by equipment acceleration found in step (b) to obtain forces/moments

The code specifies that the above method cannot be applied to flexible equipment requiring multi-degree-of-freedom (MDOF) treatment. Usually, items of equipment with fundamental natural frequencies of vibrations exceeding 20 Hz (in the case of earthquake loads) can be designed using the simplified uncoupled approach. This is, however, not the case for pressure vessels and tanks where the fundamental frequencies of the hydrodynamic components generally lie in the lower frequency ranges.

2.4 COUPLED ANALYSIS AND UNCOUPLED ANALYSIS USING DECK RESPONSE

SPECTRA

Components that do not meet the above requirements have to be designed either through coupled or uncoupled analyses using deck response spectra. In a coupled analysis, the component is modelled together with the structure and the stresses in the item of equipment and forces at supports can be directly obtained. For an uncoupled analysis, deck response spectra have to be generated or time histories of displacements/accelerations obtained at the deck level. These are then applied as loads to a model of the component from which forces and stresses can be extracted.

The nuclear code ASCE 4-98 [13] provides clear guidelines as to the criteria for selection of coupled/uncoupled analyses for items of equipment. ASCE 4-98 [13] recommends that coupled analysis is not required if the equipment (or secondary system) satisfies the following requirements:

Total mass of component is 1% or less of supporting primary structure. If components are identical and located together, their masses shall be lumped together.

Stiffness of component supported at two or more points does not restrict movement of primary system and

Static constraints do not cause significant redistribution of load in primary structure.

For tanks and vessels with single deck attachment (i.e. connected at one deck level only and with no significant separation between the support points so that the acceleration at the various points can be assumed to be the same), the selection of coupled analysis or uncoupled analysis is based on the frequency ratio and the modal mass ratio [13]. The frequency ratio is the ratio of the uncoupled modal frequency of the equipment to the uncoupled modal frequency of the primary structure. The modal mass ratio is defined numerically as [13]

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M 2

s , M pi = ( ) (1) 1 cii

M pi

where ci is the mode vector value from the primary systems modal displacement at the connection point to the equipment obtained from the ith mass (with respect to the primary structure mass matrix) normalised modal vector { pi} and Ms is the total mass of the secondary system. From the numerical values of the frequency ratio and modal mass ratio, the selection of the type of analysis can be carried out based on Figure 2 [13].

Figure 2 Criteria for Selection of Analysis Type for Equipment Attached to Primary

System [13]

The definition of the models is as follows:

Model A: Uncoupled Analysis In this model, the mass and stiffness of the equipment have negligible effect on the primary structures dynamic characteristics. The response (response spectrum or time history) of the primary structure at the attachment point to the equipment is obtained and subsequently applied to a model of the equipment to evaluate the forces/moments and stresses in the latter.

Model B: Uncoupled Analysis This model is similar to Model A except for the fact that the inertial loads due to the equipment are significant enough to warrant inclusion in the modelling of the primary structure. This is generally achieved via representation as a point mass at the appropriate centre of gravity similar to the simplified uncoupled case. The response (response spectrum or time history) of the primary structure at the attachment point to the equipment is

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obtained and subsequently applied to a model of the equipment to evaluate the forces/moments and stresses in the latter.

Model C: Coupled Analysis In this model, both the mass and stiffness properties of the equipment have a significant influence on the dynamic characteristics of the primary structure. The equipment model is included in the analysis of the primary structure so that the forces/moments and stresses in the equipment can be obtained directly.

For frequency ratios close to 1.0 and modal mass ratios exceeding approximately 0.05, a coupled analysis is generally required. For frequency ratios greater than approximately 1.5 and higher modal mass ratios (usually exceeding 0.15), an uncoupled analysis (Model B) with the mass of the equipment accounted for is adequate. Model A usually applies for low frequency ratios (less than 0.5) and high modal mass ratios (exceeding approximately 0.15).

In the latter two cases, a coupled analysis is sometimes carried out particularly in cases where a more accurate and less conservative result is sought.

For tanks and vessels with multi-deck attachment (i.e. connected at two or more deck levels), the differential accelerations have to be accounted for in the analysis. ASCE 4-98 [13] provides guidelines for determining whether the interaction of the secondary system with the primary structure is significant enough to warrant a coupled analysis or whether a decoupled analysis is permissible.

The procedure requires the determination of the modal mass ratio (as for equipment with single attachment point), which provides a measure of the degree of interaction of the masses of the equipment and the supporting structure in various modes. Numerically, for a primary structure mode i and an equipment mode j, it is defined as

2 r ={ [ ]} (2) [ ]ciij cj

where rij is the modal mass ratio, ci is a subvector of the uncoupled primary structures ith normalised modal vector comprising only of the connecting degrees of freedom (i.e. the degrees of freedom corresponding to the connection points between the equipment and the primary structure) and cj is a matrix of secondary system participation factors consisting of one term for each of the connecting degree of freedom. The values of rij are computed for all combinations of the modes and Figure 2 can be used to determine the analysis type required based on the value of rij used for the x-axis. Models A and B in the figure pertain to uncoupled analyses and Model C requires coupled analysis as described for equipment with single point attachment.

In addition, multi-point attachments can result in static constraints resulting in redistribution of loads and causing an increase in the primary systems modal frequencies. Generally, ASCE 4-98 recommends that if the ratio of the increased primary systems natural frequency to the uncoupled frequency exceeds 1.1, a coupled analysis should be performed.

The procedure for carrying out uncoupled analysis for equipment with multi-point attachment differ from that for single point attachment as the effect of the differential accelerations mentioned earlier have to be accounted for. A method for the uncoupled analysis of equipment with multi-point attachments is described in reference [14]. The procedure requires the determination of secondary response spectra at the various support points between the primary structure and the equipment. The various spectra are subsequently enveloped to produce a set of upper-bound spectra that can be used in the analysis of the equipment. In addition, an analysis should be performed to determine the effect of differential boundary displacements on the

8

equipment. This can be achieved through a static analysis whereby the maximum relative support displacements are obtained from the prior response spectrum analysis and imposed on the supported equipment in the most unfavourable condition.

Alternatively, loading can obtained at the various support points and are subsequently applied to the equipment. The force/moments and stresses in the equipment can then be obtained directly from the analysis.

2.4.1 UNCOUPLED ANALYSIS USING DECK RESPONSE SPECTRA

Deck response spectra are required for uncoupled analyses in the frequency domain. The problems associated with the generation of deck response spectra on offshore structures have been summarised by Kost and Sharpe [16]:

Uncertainties in the frequency content of the input loading Assumptions used in the modelling of the interaction between the component, structure,

water and foundation stiffness Assumptions used in the modelling of the dynamic characteristics and energy absorption

properties of the structure and Uncertainty in the representation of the non-linear stiffness and damping effects

Similar uncertainties and approximations exist in the analysis of equipment or secondary systems in nuclear structures. Detailed guidelines are provided for the generation of secondary response spectra (SRS) and time histories for uncoupled/decoupled analyses in ASCE 4-98 [13]. The secondary response spectra are generated from the linear time history accelerations at the deck or support equipment locations. The platform acts as a filter with peaks generally occuring at the frequencies corresponding to peaks on the ground motion and at the natural frequencies of the supporting structure as shown in Figure 3.

Figure 3 Structure Excitation and Response at Various Platform Levels

9

&& && x 2 and && In the figure, x g represents the ground acceleration and x1 , && x 3 are the resulting accelerations at the first, second and upper levels respectively. The response spectra exhibit amplified response at a frequency corresponding to the natural frequency of the structure (period Ts). In general, other major peaks will be present and are associated with higher vibrational modes.

ASCE 4-98 [13] recommends that the secondary response spectra be broadened prior to use in the analysis of secondary systems. Broadening, i.e. widening of the peaks associated with structural frequencies is carried out to account for the effects of the various uncertainties as shown in Figure 4. Craig et. al. [17], in their discussion of the API RP 2A 20th Edition Update, discuss a similar approach with regard to the generation of deck response spectra for design of equipment on offshore structures. However, no guidance is provided as to the amount of peak broadening required. ASCE 4-98 [13] suggests that the peaks be broadened by 15% in the frequency domain. This concept is illustrated in Figure 4 [14] where fj is 15%.

In conjunction with the broadening, ASCE 4-98 [13] also allows for a 15% reduction in the peak amplitude provided that the secondary system damping does not exceed 10%. The reduction is associated with the probability of exceedance of the various uncertainties described above. Similar procedures, as adopted in the nuclear industry, can be used for generation of deck response spectra for equipment on offshore platforms. It is noted, however, that for offshore sites, the level of uncertainties, particularly in the spectra/time histories used for analysis of the structure, may be greater than for onshore sites.

Figure 4 Broadened Spectrum

A procedure for carrying out uncoupled analyses of equipment on offshore platforms has been proposed by Bea and Bowen [18] based on an approach developed by Biggs and Roesset [19]. The method uses a combination of empirical (based on analysis results of conventional platforms) and theoretical approaches to derive Acceleration Magnification Ratio (AMR) curves which provide the ratio of the maximum acceleration of the equipment mass to the acceleration

10

of the structure at the equipment support as a function of the equipment period to structure period. The AMR is dependent on the damping ratios of the equipment and the structure and is also based on the assumption that when the equipment period exceeds 1.25 the structure period, the acceleration experienced by the equipment is governed by the ground motion rather than the support acceleration. It is noted, however, that it does not account for any of the uncertainties mentioned previously with respect to peak broadening for secondary response spectra.

Generally, decoupled analysis using secondary response spectra usually results in higher forces/moments and stresses in the equipment and supports than an equivalent coupled analysis. This is because for a coupled case, exact resonance cannot occur as the resonant frequencies shift away from each other [14]. For the uncoupled case, it is possible for a natural frequency of vibration of the equipment to correspond exactly to a natural frequency of the structure thereby leading to a resonant behaviour.

2.4.2 COUPLED ANALYSIS

For coupled analysis, ASCE 4-98 [13] recommends that a refined model of the component and a simplified model (instead of a detailed model) of the supporting structure may be used. The simplified model of the structure should, however, capture the significant natural frequencies and modes of vibration at the equipment support locations. This is ascertained by comparing the secondary response spectra from the simplified model of the supporting structure to that of the detailed model of the supporting structure at the support locations.

2.5 SUMMARY OF FINDINGS ON ANALYSIS METHODOLOGIES

In summary, review of the relevant codes and papers related to design of equipment on topsides led to the following findings:

Strong vibration is addressed in the ISO standard. Analytical methods and walkdown studies are described for the assessment of equipment on topsides.

Offshore codes ISO 19901-3 [4] and API 2A WSD [5] provide guidelines for analysis of equipment under seismic loading. Similar procedures can, however, be used for blast loading and ship impacts.

For rigid equipment, simplified uncoupled procedures can be used. Flexible or compliant equipment require the use of coupled analyses or uncoupled analyses using deck (or secondary) response spectra.

In the nuclear industry, input and modelling uncertainties are accounted for by broadening the secondary response spectra at the peaks corresponding to the structural frequencies.

For equipment with multi-point attachment, ASCE 4-98 [13] provides guidelines for the selection of the type of analysis. This requires the determination of the modal mass ratio and the frequency ratio.

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3 REVIEW OF LOADING ON PRESSURE VESSELS AND TANKS ON TOPSIDES FROM LATERAL EXCITATION

3.1 INTRODUCTION

Strong vibration from earthquakes, blast and ship impact results in lateral excitation of the decks that support the pressure vessels and storage tanks. Dynamic amplification occurs as a result of the inherent flexibility of the platform structure and depends on the dynamic characteristics of the platform and on the frequency content of the applied loading. As a result, the equipment may be subjected to higher acceleration values than free-field accelerations depending on the ratio of the natural frequencies of the equipment to that of the supporting structure. This review will address the current practice for determining the loading on topsides due to earthquakes, blast and ship collision.

3.2 EARTHQUAKE LOADING

The loading on pressure vessels and tanks due to earthquake arises from the vibratory response of the supporting deck structure. It is not the aim of this review to investigate the derivation of earthquake spectra used in the analysis of offshore platforms but rather to assess their application in the loading of pressure vessels and tanks on topsides. However, owing to the obvious relationship between the base spectra and the loading at deck level, a brief description of earthquake spectra used in the offshore sector is provided.

3.2.1 DESCRIPTION OF EARTHQUAKE LOADING

The shape and intensity of the response spectrum or time history loading at a particular site are governed by several parameters as described in reference [20]. Guidelines for generation of earthquake spectra are provided in the following codes of practice:

API RP 2A-WSD Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms Working Stress Design [5]

NORSOK Standard N-003 Actions and Action Effects [6] ISO/CD 19902 Petroleum and Natural Gas Industries Fixed Steel Offshore Structures

[8]

The codes recommend that the platform structure be assessed under two different level of earthquake intensity based on their probability of occurrence. ISO/CD 19902 [8] defines these two levels as the Strength Level Earthquake (SLE), associated with a ground motion which has a reasonable likelihood of not being exceeded at the site during the lifetime of the platform, under which the platform should sustain little or no damage and the Ductility Level Earthquake (DLE), associated with a rare intense earthquake return period of usually 1 in 10000 years, where considerable damage is allowed without any loss of life and/or major environmental damage. Specific clauses are included regarding the effect of SLE and DLE on deck appurtenances and equipment. The code requires that all equipment shall be designed and supported such that SLE actions can be resisted and that under DLE, safety critical systems and equipment shall be designed to be functional during and after the DLE event.

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3.2.2 EARTHQUAKE RESPONSE SPECTRA FOR ANALYSIS OF OFFSHORE

PLATFORMS

Typical spectra for use on offshore platforms are provided in the API [5] and NORSOK [6] codes. The API [5] code provides recommendations for the generation of offshore spectra, which involves seismotectonic and site characterisation, seismic exposure assessment, ground motion characterisation and design ground motion specification.

Generally offshore spectra are generated from earthquake forcing functions obtained or modified from onshore sites. However, Smith [21] reported that data gathered from the Seafloor Earthquake Measurement System (SEMS) Program showed that seafloor motions differed from ground motions from onshore sites. The study concluded that the peak vertical acceleration recorded on offshore sites was lower than that recorded from onshore sites but occur at equal frequencies while the horizontal accelerations exhibited comparable peaks with slight differences in frequencies. However, owing to the lack of sufficient data, a rigorous probabilistic analysis was not possible and the above conclusions can only be taken as indicative of a trend.

3.2.3 SECONDARY EARTHQUAKE RESPONSE SPECTRA FOR ANALYSIS OF

EQUIPMENT ON TOPSIDES

Deck response spectra can be generated from the acceleration response resulting from time-history analysis of the offshore platform structure. Only limited studies, e.g. EATEC [3], have been carried out to assess the acceleration response at deck levels of the structure. A typical deck response spectra, reproduced from the EATEC study, corresponding to a 1 in 10000 earthquake is shown in Figure 5.

The structure was representative of a fixed leg platform of the type commonly found in the North Sea and comprised of two deck levels in addition to a heliport. The acceleration spectrum exhibits amplification in regions where the structural frequencies are associated with high frequency content in the ground motion. Typically, for earthquake loading, this occurs in the lower frequency ranges (usually below 5 Hz).

13

Figure 5 Deck Response Spectra For a 1 in 10000 year Earthquake Loading [3]

3.3 BLAST LOADING

Blast loading on offshore structures results from the explosion of a mixture of hydrocarbon gas in air. The loading generated by the blast depends on several factors including the stochiometry of the hydrocarbon mixture, the ignition source location, the amount of congestion in the module and the amount of confinement. Blast generally results in two types of loading namely overpressure loading and drag loading. The latter is not addressed in this study as the focus is on lateral excitation of the platform, which is generally negligible in the case of drag loading.

3.3.1 DESCRIPTION OF BLAST LOADING

Explosion or blast loading as may arise from overpressure is caused by the increase in pressure that results from the expansion of combustion products. A typical idealised overpressure time history is shown in Figure 6 and is characterised by the rise time, peak overpressure and the area under the curve.

Figure 6 Idealised Overpressure Time History For Blast Loading [24]

Generation of the time history can be achieved through experimental modelling or from mathematical/numerical models. The complexity of the problem, however, implies that the loading generated can only be a best estimate of the actual loading. Renwick and Norman [22] reports on the differences found between experimental overpressure loading data from a full scale testing at the British Gas site at Spadeadam and theoretical overpressures derived by various explosion modelling agencies. They found that the theoretical models exhibited a wide scatter with significant underestimation of the peak overpressure (compared to the experimental data).

3.3.2 DECK RESPONSE SPECTRA FOR BLAST LOADING

Walker et. al. [23] studied blast-induced vibrations on topsides and suggested a response spectrum for the assessment of the ability of equipment to withstand severe vibration. The study considered the displacement and acceleration response at various locations on the deck structure from 3 overpressure time histories idealised to 1 bar with rise times of 0.1sec, 0.2sec and 0.3sec.

14

The results showed that the rise times did not have any significant impact on the peak response. In addition, the longer duration impulse resulted in maximum displacement at the remote locations whereas the shorter duration impulse led to larger maximum accelerations.

The authors noted that the acceleration response spectra generated at the remote locations could be used in the assessment of equipment to severe vibration. However, the above-mentioned authors did not address the issue of peak broadening and peak reduction as described in section 2.4.2. These issues still need to be addressed for offshore applications. A typical deck response spectrum from blast loading, reproduced from the EATEC study [3], is shown in Figure 7.

Figure 7 Deck Response Spectra For Blast Loading [3]

It is noted that the peak accelerations are higher and occur at higher frequencies than that corresponding to the earthquake secondary response spectrum. This implies that the various hydrodynamic components characterising the response of pressure vessels and tanks will have different relative contributions depending on the loading type.

3.4 LOADING ON PRESSURE VESSELS/TANKS FROM SHIP IMPACT

The problem of ship collisions with offshore platforms is generally addressed in terms of energy considerations. The dissipation of the impact energy in fixed platform structures may be achieved through the following processes [24]:

Ship deformation and/or rotation Local deformation or denting of member Elastic/plastic bending of member Fendering device Local framing distortions of platforms Global sway of platform

The elastic vibrations of platforms arising from ship impacts are generally conservatively neglected in the energy dissipation formulation of a ship striking a platform structure. This

15

assumption leads to a conservative design for all the external platform members likely to be subjected to the impact forces. Petersen and Pedersen [25] argue that for the case of ships hitting small offshore structures, the elastic vibrations of the platform may be significant, particularly where the generalised masses of the ship and platform are of the same order of magnitude and the lowest period of vibration of the platform is comparable to the duration of the collision. In addition, offshore structures having small dynamic stiffness will also experience a significant level of platform vibrations.

3.4.1 DESCRIPTION OF SHIP IMPACT LOADING

The codes of practice generally adopt the energy approach for the determination of the impact forces acting on offshore structures. The relevant codes that address this particular issue are:

API 2A WSD [5] ISO/CD 19902 [8] NORSOK N-003 [6]

The codes recommend that the kinetic energy of the vessel be calculated from

E = a 5 . 0 m v2 (3)

where m is the mass of the ship, v is the velocity of the ship at impact and a is the added mass coefficient which accounts for the hydrodynamic forces acting on the ship during the collision. A value of 1.1 is assumed for the added mass coefficient for bow/stern collision and 1.4 for broadside impacts. ISO/CD 19902 [5] specifies that the values pertain to large vessels (5000 tons displacement) and that for smaller vessels, the above values should be increased. Petersen and Pedersen [25] report that previous studies have shown that the value of the added mass coefficients also depends on the duration of the impact and the relation between the collision force and the deformation. The authors pointed out that for broadside collisions, the added mass of 40% of vessel mass is a reasonable approximation for very short duration impacts (approx. 0.5-1.0 sec). For longer durations, the value of the added mass coefficient for broadside collisions can approach 100% of the vessel mass.

NORSOK N-004 [7] specifies that equation (3) pertains to fixed offshore platforms and that for floating structures, the following equation should be taken as:

vi 21

-v 2E = 0.5 (m + v ) a s s s

s (4) s m + a s1 + s m + ai i

where ms and mi are the masses of the ship and installation respectively, as and ai are ship and installation added masses and vs and vi are the velocities of the ship and installation. The velocity of the installation is usually taken as zero. No values are, however, provided for the added masses to be used for the ship and the installation.

ISO/CD 19902 [8] also specifies that two energy levels should be used in the assessment of the platform structure namely a low energy level impact corresponding to a frequent occurrence and a high energy impact level corresponding to a rare event. Guidelines are provided for the selection of the vessel velocity at impact. For the low energy event, a value of 0.5m/s is

16

recommended while for the high energy impact level, a value of 2m/s is specified. However, no guidance is offered as to the amount of energy imparted to the offshore structure.

This issue is addressed in the Department of Energy document [12] where the total kinetic energy of the collision process is specified as 14MJ for broadside impacts and 11MJ for bow/stern impacts. The amount of energy imparted to the structure is taken to be equal to the total kinetic energy released or less depending on the relative stiffness of the vessel and structure. The guidance document recommends that for fixed steel structures, the amount of energy absorbed by the structure should not be less than 4MJ. No energy levels are specified for floating platform structures. In addition, no clear specifications are provided for determining the duration of the collision. Previous studies by EATEC [3] and Walker et. al. [24] have used values ranging from 0.097sec to 0.5sec. These values were based on the assumption of elastic displacement of the structure with a peak impact force of approximately of 30MN.

Detailed studies on the collision mechanics of vessels with offshore structures have been carried out by various researchers [25, 26, 27]. Collision models requiring numerical solutions have been proposed where the impacting ship is represented by non-linear springs and the variation of the added mass is accounted for. It is, however, beyond the scope of this study to assess the applicability of these models for loading on offshore platforms. Instead, the energy approach adopted by the codes of practice will be used in defining the loading time history from ship impacts. In this approach, the loading is specified in terms of a force time history based on the amount of energy imparted to the platform. The force time history is usually determined as a rectangular impulse with an assumed duration based on the magnitude of the energy.

3.4.2 DECK RESPONSE SPECTRA FOR SHIP IMPACT LOADING

Similar procedures to those described for the seismic and blast loading can be used for the case of vessel impact. Time histories of accelerations/displacements or secondary response spectra are generated at the supports to the equipment and depending on the magnitude and frequency content of the loading, the appropriate analysis type (i.e. quasi-static, coupled/decoupled) for design of the component and its supports can be selected.

A typical deck response spectrum from ship impact, reproduced from the EATEC study, is shown in Figure 8. Two impact scenarios were considered:

SHIP 1 Load applied steadily over 0.097sec then removed SHIP 2 Load applied steadily over 0.4 sec then removed

The deck acceleration response spectra corresponding to these two impact scenarios exhibit comparable peak responses at nearly equal frequencies. The structural peaks occur at different frequencies and have different magnitudes compared to that from blast and earthquake loading. However, the displacement of the structure was higher for the second case owing to the longer duration of loading (more comparable to the bending frequency of vibration of the structure) which resulted in greater energy absorption in the structure. Due consideration should therefore be given to the relative displacement between the equipment and the primary structure as this may affect the required amount of clearance between the equipment and any adjacent part of the structure (similar effect may arise from blast and earthquake loading).

17

Figure 8 Deck Response Spectra For Ship Impact [3]

3.5 COMPARISON OF EARTHQUAKE, BLAST AND SHIP IMPACT SECONDARY RESPONSE SPECTRA

The secondary response spectra reproduced from the EATEC study were derived from a series of finite element analyses carried out on a typical fixed jacket structure. The structure comprised of two main decks and a helicopter deck supported on a fixed leg jacket. Earthquake, blast and ship impact time histories were applied at the appropriate levels and the acceleration response at various points on the platform was obtained.

A comparison of the secondary response spectra, as reproduced from the EATEC study [3], from the various loading types is shown in Figure 9. It is observed that:

The secondary response spectra from blast loading and ship impact generally results in higher peak acceleration values than the corresponding spectra from earthquake loading

The significant peaks in the earthquake secondary response spectra generally occur at frequencies below 5Hz whereas for blast and ship impacts, structural peaks can occur at much higher frequencies (up to 15Hz).

18

Figure 9 Comparison Between Deck Response Spectra For Earthquake, Blast and Ship Impact Loading [3]

3.6 SUMMARY OF FINDINGS ON LOADING ON TOPSIDES FROM EARTHQUAKE,

BLAST AND SHIP IMPACT

Earthquake spectra used in analysis of offshore platforms are generally generated using onshore data [21]. Sea-floor spectra exhibit lower peak vertical accelerations and comparable peak horizontal accelerations although at different frequencies. Peaks in deck response spectra occur at low frequencies (usually below 5Hz)

Input spectra from blast loading are based on overpressure loading. Studies have shown that CFD and/or mathematical models can result in significant underestimation of the peak overpressure. Peak accelerations in deck response spectra from blast loading are usually higher than those from earthquake loading. Also, the peak values tend to occur at higher frequencies

Derivation of input loading on offshore platforms from ship impacts is based on energy considerations. It is recommended that the energy to be absorbed by the structure should not be less than 4MJ. Deck response spectra from ship impacts exhibit similar characteristics to those from blast loading. Peaks are higher than the earthquake values and occur at higher frequencies.

Only limited amount of data that relate to deck response spectra are available. Loading on topsides used in subsequent analysis (section 4.0) is mainly derived from the EATEC study [3].

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4 FLUID-STRUCTURE INTERACTION AND DYNAMIC RESPONSE OF PRESSURE VESSELS/TANKS UNDER

LATERAL EXCITATION

4.1 INTRODUCTION

This section provides an overview of the state-of-the-art practice for design of pressure vessels and tanks against lateral excitation as may arise from the phenomenon of strong vibration. The lateral excitation can result from earthquake loading, blast loading or ship impacts as described in the previous sections. The following sub-sections address the current practice for determining response of pressure vessels and tanks based on both simplified procedures from relevant codes of practice and technical papers.

Existing guidelines for the design of pressure vessels and tanks mainly address seismic loading and, in most cases, are applicable to onshore structures. However, the response of vessels and tanks on topsides under any lateral excitation is characterised by the same hydrodynamic components and the loading type will only influence the relative contribution of the components to the overall response so that similar procedures used in the determination of seismic response can be used for blast and ship impacts.

4.2 DESCRIPTION OF RESPONSE OF PRESSURE VESSELS AND TANKS

The dynamic behaviour and response of the pressure vessels and tanks under lateral excitation is generally non-linear in nature and has been the subject of extensive research. The problem was first addressed by Housner [28] for the case of a fixed base rigid upright cylindrical tank under seismic excitation. The motion of the liquid inside the tank results in hydrodynamic pressure loading on the tank walls and Housner assumed that the response of the rigid tank could be split into 2 hydrodynamic components namely:

Impulsive component due to rigid-body motion of the liquid. Under dynamic loading, part of the liquid moves synchronously with the tank as an added mass and is subject to the same acceleration levels as the tank. This is hereafter called the rigid-impulsive component.

Convective component due to sloshing of the liquid at the free surface. Under lateral excitation, oscillations of the fluid occur and this results in the generation of pressures on the walls, base and roof of the tank.

In addition to causing forces and moments in the tank wall, the hydrodynamic pressures on the walls in conjunction with the pressures on the base result in a net overturning moment on the tank. Based on the assumptions that

The liquid is incompressible and inviscid Motion of liquid is irrotational and satisfies Laplaces equation and Structural and liquid motions remain linearly elastic

in conjunction with the boundary conditions

Vertical velocity of liquid along tank base must equal corresponding ground velocity and Radial velocities of liquid and tank wall must be the same for rigid tank

20

Housner derived solutions for the rigid-impulsive and convective pressure components. The rigid-impulsive component of the solution satisfies the actual boundary conditions on the tank walls and base and the condition of zero hydrodynamic pressure at z = H where z is the vertical coordinate (datum is at base of tank) and H is the height of the liquid. The convective component of the solution corrects for the difference between the actual boundary condition at z = H (accounting for the effects of sloshing of the liquid) and the one used in the development of the rigid-impulsive solution.

Following from the solutions, Housner proposed a mechanical (spring-mass) model, as shown in Figure 10 below, for representing the response of the rigid tank-liquid system. The rigid-impulsive mass is assumed to be rigidly attached to the container walls while the convective

thmass is split into a series of sub-masses m1, m2, , mn associated with the 1st, 2nd, , nsloshing masses respectively. These latter masses are attached to the container wall via springs

thof stiffness k1, k2, , kn representing the 1st, 2nd, , n antisymmetric sloshing frequencies respectively.

kn/2 kn/2

m

m

m0

m1

mn

h0

h1

k1/2 k1/2

0 = impulsive mass

1 = 1st mode convective mass

h

mn = nth mode convective mass

n h0 = height at which impulsive mass attached to tank

h1 = height at which first mode convective mass attached to tank

hn = height at which nth mode convective mass attached to tank

Figure 10 Housners Mechanical Spring-Mass Model for Rigid Tanks

This simple model allows for the computation of the convective and rigid-impulsive pressures and associated base shears, overturning moments and stresses in tank wall. Housners procedure was included in the US Atomic Energy Commission in the TID-7024 regulations [29] and was developed into a practical design by Epstein [30]. However, since the 1964 Alaskan earthquake, where damage occurred to several tanks, several studies [31, 32, 33] were carried out to address the issue of tank flexibility and it was shown that the impulsive forces in a flexible tank are considerably higher than those computed from a rigid assumption.

This fluid-structure interaction effect whereby the flexibility of the tank results in the dynamic characteristics of the tank-fluid system to be significantly different from that of a rigid tank, has led to the inclusion of a third hydrodynamic component to quantify the dynamic response of flexible vessels and tanks namely the flexible-impulsive component. Methods for determining the contribution of the flexible-impulsive component to the total response (base shear, overturning moments, wall stresses) of vessel/tanks under seismic excitations have been proposed by various researchers [33, 34, 35, 36]. These methods pertain mainly to vertical cylindrical tanks supported on a fixed base and have been adopted by several codes of practice namely

Guidelines for the Seismic Design of Oil and Gas Pipeline Systems ASCE [1984] [9]

21

Seismic Design of Storage Tanks, Recommendations of a New Zealand Study Group [1986] [10]

Eurocode 8: Design Provisions for Earthquake Resistance of Structures Silos, Tanks and Pipelines [1998] [11]

Seismic Analysis of Safety-Related Nuclear Structures and Commentary ASCE 4-98 [13]

The latter guideline is mainly based on the rigid tank approach with general recommendations for flexible tanks. Reference is made to the works of Haroun and Housner [34] and Veletsos and Yang [33] but is not explicitly used in the expressions provided. Eurocode 8 [11] uses the ASCE (1984) [9] and Recommendations from New Zealand Study Group [10], as well as results from more recent papers e.g. Fisher et. al. [36, 37], as sources and provides the most comprehensive simplified design procedures for fixed based cylindrical pressure vessels. The following sub-sections provide a review of the procedures developed to date for the computation of the response of various tank types to dynamic lateral excitation. The tanks types considered are:

(g) fixed base vertical cylindrical (h) fixed base rectangular, (i) fixed base horizontal cylindrical (j) fixed base tanks (any type) with internal baffles

4.3 VERTICAL CYLINDRICAL FIXED BASE TANKS/VESSELS

Most of the studies carried out on the response of pressure vessels and tanks under lateral excitation pertain to the vertical cylindrical fixed base configuration. The pressure resultant on the tanks walls, base and roof can be divided into the 3 hydrodynamic components, convective, rigid-impulsive and flexible-impulsive as described above. The computation of these various components are discussed in turn in the following sub-sections for the case of vertical cylindrical fixed base tanks and pressure vessels.

4.3.1 Convective Component

When a tank or vessel is subjected to lateral excitations, the liquid in the upper part of the container moves independently of the tank wall. This liquid motion is known as sloshing and causes pressures to be induced on the tank walls and base. For a perfectly circular shell, the sloshing can be described by an infinite number of antisymmetric modes with the first four modes for a model of a typical tank [38] shown in Figure 11 below.

N = 1 0.135 Hz

N = 2 0.268 Hz

N = 3 0.339Hz

N = 4 0.397Hz

Figure 11 Antisymmetric modes and frequencies for tank with height to radius ratio =

0.5 [38]

22

Symmetrical modes of vibration about the vertical axis are also possible although in general, such modes are not excited by earthquake motions. The frequencies associated with the sloshing modes are usually low (fundamental frequencies usually in the range 0.1 - 0.2 Hz) and for a given mode vary with the tank dimensions and are generally independent of liquid height except for very shallow depths. The jth sloshing frequency in Hertz is given by

1 g

H

f = tanh

j (5) j j2 R

R

where g is the acceleration due to gravity, R is the tank/vessel radius, H is height of fluid in tank and j are the zeros of the first derivative of the Bessel function of the first kind and first order and values of j for the first four modes are given by

1 = 1.841 2 = 5.331 3 = 8.536 4 = 11.706 (6)

The hydrodynamic pressure induced by the sloshing modes is given by [11]

r

z

J1 cosh n

2R

R

R

n

pc = l ( ) J ( ) H cos S (7) 2 nn =1 n -1 1 n cosh

n

R

where l is the density of the fluid, r, and z are the radial, circumferential and vertical coordinates respectively, J1 is Bessel function of first order and Sn is the acceleration corresponding to the nth sloshing frequency. The convective pressure varies as a cosine function in the circumferential direction. Generally, only the fundamental sloshing mode is accounted for in the computation of convective pressures as the higher sloshing modes have low mass participations and consequently low associated pressures as shown in Figure 12 (reproduced from reference 11).

Figure 12 Variation of first two modes sloshing pressures along height and for various

height to radius ratios

23

Figure 13 Variation of first two sloshing frequencies with H/R ratio

The convective pressure is a maximum at the top of the tank and decreases to negligible values for tall, slender tanks (H/R 3) but maintain quite high values for squat tanks. The contribution of the second mode is observed to be negligible. Figure 13 shows that the sloshing frequencies ( is frequency in rad/sec) are nearly independent of the height to radius ratio for values of H/R greater than 1.0.

The maximum convective pressure, accounting for the fundamental sloshing mode only, occurs at z = H (datum at base of tank) and r = R and can be expressed as

p = 0.837 l S R 1 cos (8) c max

where S1 is the maximum spectral acceleration value determined from a response spectrum of the particular ground motion at the fundamental sloshing frequency and damping value. Usually, the damping in the sloshing modes is very low (around 0.5%) and is generally assumed to be zero (for light viscosity liquids and tanks with no internal baffles [37]) for design purposes as this leads to a conservative estimate of the pressures.

The convective component is, in general, assumed to be decoupled from the effects of tank wall flexibility [33, 34] so that the same expressions can be used for both rigid and flexible tanks for computing the pressure contributions from sloshing. In ASCE-1984 [9], based primarily on the work of Veletsos [31] and Veletsos and Yang [33], it is argued that

Convective effects arise from sloshing motion of liquid whereas impulsive effects are caused by the lateral motion of the tank and participating liquid. Consequently, those two effects exhibit only a weak coupling. The tank flexibility alters the lateral motion of the tank wall so that only the impulsive component is significantly affected.

There exists a wide frequency gap between motions associated with convective and impulsive effects. This further weakens the coupling between the two components and further reduces the sensitivity of one component to changes in the other.

These arguments were basically used to justify the assumption that the wall flexibility does not affect the convective component. This was confirmed by further studies by Haroun and Housner [34].

4.3.2 Rigid Impulsive Component

For a tank or vessel under lateral excitations, the liquid in the lower part of the container tend to move in unison with the tank/vessel and is subject to the maximum ground acceleration. The rigid impulsive pressure component is given in Eurocode 8 [11] as

x (t) (9) p = Ci H cos && i l m

where Ci is dependent on the dimensionless parameter z/H and the height to radius ratio and x (t) is the maximum ground acceleration. The variation of Ci (normalised with respect to the && m value at z = 0) along the height of the tank is shown in Figure 14. It is observed that the rigid impulsive pressure decreases from the base of the tank to the top. For tall and slender tanks, the coefficient assumes high values up to significant tank heights.

24

Explicit expressions for the coefficient Ci is provided in Eurocode 8 [11] and result from solution of the momentum equations subject to the assumptions and boundary conditions described in section 4.2. The solution is given in terms of a series expansion as follows:

(-1)n ( C ) = i ,

( cos I ) 1

n

(10) I ( / 2

n

)n = 0 1 n n

2n + 1 H r z ,where = = , = a nd = . I1and I denote the modified Bessel function of the n 1

2 R R H

first kind and its derivative. A simplified approximation more amenable to design calculations is

reported by Tedesco et. al. [35] and was first proposed by Housner [28] and used in TID-7024 [29]. The impulsive pressure component in this particular case is given by

Ci (z)/Ci (0)

Figure 14 Variation of Coefficient Ci (z) Along Tank Height For Various Tank Height to

Radius Ratios [9]

2

3

z

1

z

&& p = x H

1 -

-

1-

tanh 3 cos (11) i l m

H

2

H

R H

25

where the symbols have been previously defined. In the above expression, the impulsive pressure is given at the wall of the container so that the radial variation is not contained. The circumferential variation of the rigid impulsive pressure component is a cosine function similar to the convective pressure variation. In the radial direction, the variation is generally non linear

for tanks with low height to radius ratios but tend to a linear distribution for slender tanks as shown in Figure 15 [9].

Figure 15 Radial Variation of Rigid Impulsive Pressure Component For Tanks of Various Height to Radius Ratios [11]

4.3.3 Flexible Impulsive Component

The flexibility of the tank/vessel wall leads to an interaction between the fluid motion and the deformation of the wall. The impulsive pressure due to the coupled tank-fluid vibration usually lies in the high frequency range and results from the contribution of an infinite number of tank-liquid modes. Expressions for the flexible impulsive pressure component have been proposed by various researchers (Haroun & Housner [34]; Veletsos & Yang [33]; Tedesco et. al [35]; Fischer et. al [36]) and are reported in the various codes of practice (Eurocode 8 [11]; ASCE-1984 (9); NZSEE [10]).

The derivation of the expression is, in general, based on the assumption that only the first circumferential m = 1 mode contributes significantly to the pressure. Fischer et. al. [36] reports that this assumption is valid since

26

Modes with m 1 do not contribute to the overturning moment which leads to the most common failure mode for tanks and

Higher order m = 1 modes have low participating mode factors.

The tank-fluid system is considered as a single degree of freedom system vibrating in an assumed configuration. Fischer et. al. [36] proposed the following expression for computing the flexible-impulsive component:

( ) 1 I

j

j

1 ( )cos (12) ( )j R 2 f d= p fin g cosl j I1 2 2 j

0

where = x/H0, j = (j)/(2), = H0/R, I1 = (j = 1, 3, 5,.). The symbols used in the dI d / 1 j expression are defined as follows:

pfin flexible-impulsive pressure component normalised to an excitation of 1g

H0 tank height

x axial coordinate

) ( f f() amplitude of vibration mode with max =1

A similar expression is used in Eurocode 8 [11]. The determination of the flexible-impulsive component requires an iterative procedure. It starts with an assumed mode shape from which an updated effective mass of the tank can be calculated. The next iterative step involves computing the new mode shape based on the updated value of the tank mass. This procedure is repeated until convergence is achieved.

An expression for the flexible impulsive component more amenable to design was proposed by Tedesco et. al. [35] is given by

2

z

z

( && S x - )H 1 cos (13) p fi = - l a m

H

H

0

where H0 is the height of the tank, (z/H0) is an assumed deflection configuration and Sa is the spectral acceleration corresponding to the natural frequency of the flexible tank-liquid system vibrating in the assumed configuration. Based on a statistical analysis of the mode shapes associated with free vibration of tank-liquid systems, Tedesco et. al. [35] have proposed simplified approximations to the deflection configuration parameter for tanks filled and half-filled with fluid.

The approach adopted by Veletsos and Yang [33] was to use a single expression to compute the total impulsive (both rigid and impulsive) pressure and is given by

pit =C l A H ( )cos (14) ti 0 where pit is the total impulsive pressure, is the counterpart of Ci in the expression for rigid Ci impulsive pressure component and A0 (t) is the pseudo-acceleration evaluated for the natural frequency and damping ratio of the flexible tank-fluid system. In addition to the height to radius

27

ratio and z/H, the coefficient also depends on the assumed deflection configuration and the Ci ratio of the mass of the tank to the mass of the fluid.

The deflection configuration of the tank-liquid system depends on the geometry of the container, the material properties of the tank/vessel, the depth of the liquid and the boundary conditions at the top and bottom of the tank/vessel. Veletsos and Yang [33] summarised the deflection characteristics of an empty tank based on the H/R ratio as follows:

For large values of H/R, the tank deflects in a cantilever flexural beam mode with practically no distortion of the cross-section

For smaller values of H/R, the deflection shape is as a cantilever shear beam with no distortion of the cross-section

For yet smaller values of H/R, the deflection configuration of the tank can be represented as a series of ovalling modes and

For extremely small values of H/R, the tank/vessel wall behaves as a series of independent cantilever flexural strips.

The vibrational characteristics of the tank-liquid system are similar to those of the empty tank. Kana [38] reports that flexural modes are more important in tanks with H/R > 1 while more than one ovalling mode may be important in tanks of all sizes.

Expressions for the fundamental natural frequency of the flexible impulsive mode of vibration have been proposed by various researchers based on different approaches including finite element methods and the Rayleigh-Ritz procedure. Rammerstorfer et. al. [39] provide a comparison between the various expressions and conclude that, in general, they exhibit good agreement. The approximation provided in Eurocode 8 [11] is as follows:

1 s E 3 1 2f = g ( )= 0.01675 - 0.15 + 0.46 (15) s g R 2 ( ) H

where E is the Youngs Modulus of the material of the tank/vessel, s is the thickness at 1/3 height and is the density of the liquid.

In the various procedures developed for computing the flexible impulsive response, an important assumption is that the cross section of the tank is truly circular. The effect of out-of-roundness and other imperfections would be to excite modes involving more than a single sine wave in the circumferential direction, which may significantly affect the response of the system [39].

In summary, the main difference between the response of a rigid and flexible tank pertains to the nature of the acceleration component. For a rigid tank, the response is proportional to the maximum ground acceleration whereas for a flexible tank, the response is governed by the spectral acceleration corresponding to the fundamental frequency of the tank-liquid vibration and associated damping ratio. The flexible impulsive frequencies usually lie in the range 2 20 Hz. Within this region, the spectral acceleration, for a given damping ratio, is typically greater than the maximum ground acceleration. In general, the damping in the flexible impulsive mode is assumed to be approximately 2% (Rammerstorfer et. al. [39]).

28

4.4 RECTANGULAR FIXED BASE TANKS

Results for rectangular tanks have originally been proposed by Housner [28] based on the same structure and fluid assumptions as for a rigid vertical cylindrical fixed base tank. The response of the rigid tank is again considered to be made up of a convective part and a rigid impulsive part.

4.4.1 Convective Component

The convective pressure component results from the contribution of an infinite number of sloshing modes. However, as for the case of a cylindrical tank, the dominant contribution arises from the first sloshing mode. The fundamental sloshing frequency for rectangular tanks can be expressed as [11]:

H

tanh

1 2

2 L

f1 = (16) 2 g L

where g is the acceleration due to gravity and L is the half-width of the tank in the direction of the excitation. Plots of the first two sloshing frequencies (dimensionless period in this case) for

both cylindrical and rectangular tanks are shown in Figure 16.

29

Figure 16 Variation of First and Second Mode Sloshing Periods For Cylindrical and

Rectangular Tanks With Height to Radius Ratio (Cylindrical) or Height to

Half Width Ratio (Rectangular)

The slosh frequencies pertaining to the rectangular tanks are lower than for the cylindrical case. In both cases, the frequencies tend to become independent of the height to half-width (rectangular tanks) or height to radius (cylindrical tanks) ratios.

The convective pressure on the tank/vessel wall, perpendicular to the direction of motion, from the fundamental sloshing mode is given by [11]:

p = q S L 1 (17) c 1

where q1 is a dimensionless convective pressure corresponding to the first sloshing mode (Figure 17) and S1 is the spectral acceleration corresponding to a response spectrum for the ground motion under consideration at the fundamental sloshing frequency and damping ratio.

Figure 17 Dimensionless Convective Pressure For First Sloshing Mode on

Rectangular Tank Walls

The variation of the dimensionless convective pressure on the rectangular tank walls is similar to that of the convective pressure distribution for cylindrical tanks (Figure 12). The same

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variation is observed with H/R whereby the convective effects become more pronounced at the base of the tank as H/R decreases i.e. for squat tanks.

Again, as for the cylindrical tank, the damping ratio in the sloshing modes of vibration is usually negligible and is assumed to be zero for conservative results. Also, the wall flexibility does not affect the convective component so that the same expressions can be used for both rigid and flexible tanks.

4.4.2 Rigid Impulsive Component

The approach for determining the rigid impulsive component is similar to that for cylindrical tanks and the pressure component in this case is given by [11]:

0 z && pi = q ( ) x L (18) m where the function q0 (z) is shown in Figure 18. All other symbols have been previously defined. The variation is similar to that of the rigid impulsive pressure coefficient for cylindrical tanks.

Figure 18 Dimensionless Impulsive Pressures on Rectangular Tank Wall Perpendicular to Direction of Excitation

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4.4.3 Flexible Impulsive Component

The wall tank flexibility results in an interaction between the deformation of the wall and the motion of the liquid. In the case of rectangular tanks, however, no closed form expression for determining the magnitude of the flexible impulsive component is available. Solutions can, however, be obtained through linear finite element dynamic analysis.

An approximation reported in Eurocode 8 [11] is to use the same expression as for the rigid rectangular tank with the spectral acceleration Sfi replacing the maximum ground acceleration. The spectral acceleration is evaluated at the fundamental frequency and damping ratio of the tank-liquid mode based on the response spectrum of the excitation under consideration.

The fundamental frequency of the tank-liquid mode can be approximated as [11]:

1 gf = (19)

2 d f

where g is the acceleration due to gravity, df is the deflection of the wall on the vertical centreline and at the height of the impulsive mass, when the wall is loaded by a uniform load in the direction of ground motion and of magnitude mi g / 4 B H, mi is the impulsive mass and 2B is the tank width perpendicular to the direction of loading.

4.5 FIXED BASE HORIZONTAL CIRCULAR CYLINDRICAL TANKS/VESSELS

For the case of horizontal circular cylindrical tanks, the response is further complicated by

Cross sectional curvature and Possible significant length differences longitudinally compared to laterally.

The slosh configurations would differ along the tank axes so that both the longitudinal and lateral components of the excitation must be taken into account.

Few studies have been carried out to investigate the response of horizontal circular cylindrical tanks. Experimental studies on the slosh response are reported by Kana [38] and compared to predictions based on expressions for rectangular tanks. These studies showed that non-linearities have a significant effect on the response of horizontal circular cylindrical tanks.

For the impulsive component, Eurocode 8 [11] recommends that the pressures in both the longitudinal and lateral directions be computed from rectangular tank solutions by assuming an equal dimension at the liquid level and in the direction of excitation and at a depth required to give equal liquid volume. The approximation is considered valid for the range 0.5 H/R 1.6. For values greater than 1.6, the tank should be assumed to be full.

An approximate expression for determining the rigid-impulsive response is provided in Eurocode 8 [11] for partially filled tanks and for a seismic excitation perpendicular to the length of the tank and is given by

p ( )= q ( ) A R ( ) (20) ti 0 g where R is the radius of the tank, Ag (t) is the input acceleration at time t, and the pressure function q0 () is given by

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n1 -1H ( )

q0 ( ) = 2 2n sin (21) n =1 ( ) -12n The function is plotted in Figure 19. No closed form expressions have been proposed for determining the convective component of response. However, the sloshing frequency in the first mode can be obtained from Figure 20.

Figure 19 Rigid-Impulsive Pressure Variation on Horizontal Cylinder (H = R) Subjected

To Transverse Excitation [11]

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Figure 20 Dimensionless First Convective Mode Frequency for Rigid Tanks of Various

Shapes [11] 4.6 TANKS WITH INTERNAL BAFFLES

The design of tanks of various types with internal baffles is addressed in the aerospace monographs [40, 41] and is mainly concerned with slosh suppression. Baffles increase the effective fluid damping thereby reducing the duration of free oscillations and magnitude of forced oscillations. The variation of the effective damping ratio with baffle location for rigid ring baffles in a circular cylindrical tank is shown in Figure 21.

1 is the maximum sloshing wave height measured from the quiescent liquid surface at tank wall. The results are based on the assumption that the baffle remains in the fluid during sloshing. It is observed that

For a fixed baffle location, the effective damping ratio increases with increasing baffle size i.e. as a0/a decreases and

For a fixed baffle size, the effective damping ratio increases as the baffle location moves up along the tank height.

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Figure 21 Variation of Effective Damping Ratio With Baffle Location

The latter observation can be readily explained by the fact that the sloshing effects are more prominent near the top of the liquid surface so that the effectiveness of the baffles in suppressing the sloshing motion would be more evident nearer the top of the fluid surface. For a series of ring baffles, reference 41 suggests that the combined damping ratio can be obtained by adding the contribution of each ring. This approximation has, however, not been thoroughly verified. It is noted that the baffles in this particular case are rigid and that flexible baffles may provide more damping than rigid ones.

In addition to increasing the effective damping ratio, the effect of the baffles is also to reduce the masses associated with sloshing and increase the sloshing frequencies. Similar effect is achieved by compartmentalising the tank i.e. subdividing it into smaller regions and also by internal components within the tank. Ma et. al. [42] carried out a finite element analysis on a reactor tank with internal components and found that the presence of internals (e.g. pumps) can significantly alter the dynamic characteristics of the sloshing motion. Notably, the fundamental frequency is considerably increased with the sloshing wave height reduced. The hydrodynamic pressure was also found to increase.

This can be explained as follows. In the low frequency region (usually

model for determining the rocking response. Explicit values are provided for the various parameters characterising the system based on the additional assumption that the deformed shape of the tank (for flexible tanks) is taken as sin (z/2H) where z is the vertical coordinate and H is the height of the liquid in the tank.

Veletsos and Tang [45] considered the same problem and proposed approximate expressions similar to that for tanks under purely lateral excitation. By decomposing the rocking motion of the tank into a rotation of wall with no base rotation and rotation of base with no wall rotation, it was shown that the impulsive component of wall pressure for tall tanks is dominated by rocking action of the wall and for broad tanks, it is dominated by the rocking action of the base. A similar conclusion was postulated for the pressures acting on the base.

The above studies pertained to upright cylindrical tanks only and results for tanks of other configurations are not available.

4.8 COMBINATION OF PRESSURES AND RESULTANT STRESSES

Two different procedures are widely used in the literature for combination of the impulsive and convective components namely a direct numerical addition of the absolute maxima and the SRSS rule. The direct summation of the absolute maxima clearly results in a conservative result as it is based on the assumptions that

The maximum values of the convective acceleration and the flexible impulsive pseudo-acceleration occur simultaneously with the maximum ground acceleration and

Maximum positive and maximum negative values of the convective acceleration and the flexible impulsive pseudo-acceleration are equally likely

In general, the convective fluid pressure does not occur simultaneously with the 2 impulsive fluid parts. An alternative superposition rule is based on the square-root-of-sum-of-squares (SRSS) method. Fischer et. al. [36] proposed three sets of different SRSS superposition rules based on the natural frequency of the flexible impulsive mode of vibration. A more general SRSS rule proposed by Fischer et. al. [36] and is given by

2 2 p = ( ) (p + pfi ) (22) pmax c + ri

where pc is the convective pressure and pri, pfi are the rigid impulsive and flexible impulsive pressure components respectively.

The total pressure on the tank wall should include both the hydrodynamic pressure and the hydrostatic pressure. The tank forces and stresses can then be calculated based on an appropriate shell theory. In general, only the maximum values of the circumferential, shearing and axial stresses in the tank walls are required. Following the Veletsos and Yang [33] procedure, as reported in ASCE-1984 [9], the maximum values of the stresses can be determined from:

P R maxMaximum Hoop or Circumferential Stress: ( ) = (23) max H

Maximum Shear Stress: = 2 Q max = 1 Q max (24) max A H R s

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( ) = M max R 1 M maxMaximum Axial Stress: = (25) z max I R 2 H

where Qmax is the maximum base shear, Mmax is the maximum overturning moment and I is the moment of inertia of the tank wall about a horizontal centroidal axis. The maximum base shear and overturning moment can be evaluated from expressions proposed by Veletsos and Yang [33] amongst others. Simplified approximations have been proposed by Malhotra et. al. [46] whereby the base shear and overturning moment are calculated from:

fM =m i h i S ( )+m h S ( ) (26) e f imp c c e con where mi, mc are the impulsive and convective masses respectively, hi, hc are the heights of the centroids of the impulsive and convective masses respectively, Se (fimp) is the spectral acceleration from a response spectrum of appropriate damping ratio at the tank-liquid frequency and Se (fcon) is the spectral acceleration from a response spectrum at the sloshing frequency. The masses and heights can be determined from figures presented in the paper.

Balendra et. al. [47] provides a good discussion of the stress resultants in flexible cylindrical tanks and reports that for broad and shallow tanks, the circumferential stress is the largest among all the membrane stress resultants. As the height-to-radius (H/R) ratio increases, the membrane shear stress becomes dominant and for tall and slender tanks (large H/R ratios), the axial stress predominates. It is pointed out, however, that the design criterion may not be set by the value of the maximum stress. This is because the hoop stress is tensile while the axial stress may be compressive which can result in buckling at lower stresses.

4.9 SUMMARY OF DYNAMIC RESPONSE OF VESSELS/TANKS UNDER LATERAL

EXCITATION

The response of a tank/vessel under lateral excitation can be split into three hydr