A Novel Offshore Platform Blast Wall Design with Energy ... · A Novel Offshore Platform Blast Wall...

A Novel Offshore Platform Blast Wall Design

with Energy Absorption Mechanism

by

JinJing LIAO BEng(Hons)

This thesis is presented for the degree of

Master of Philosophy of

The University of Western Australia

School of Civil, Environmental and Mining Engineering

August 2017

THESIS DECLARATION

I, JinJing Liao, certify that:

This thesis has been substantially accomplished during enrolment in the degree.

This thesis does not contain material which has been accepted for the award of any other

degree or diploma in my name, in any university or other tertiary institution.

No part of this work will, in the future, be used in a submission in my name, for any other

degree or diploma in any university or other tertiary institution without the prior approval

of The University of Western Australia and where applicable, any partner institution

responsible for the joint-award of this degree.

This thesis does not contain any material previously published or written by another

person, except where due reference has been made in the text.

The work(s) are not in any way a violation or infringement of any copyright, trademark,

patent, or other rights whatsoever of any person.

JinJing Liao

28th March 2017


Abstract The University of Western Australia

i

Abstract

The research carried out in this thesis aims to demonstrate the feasibility and effectiveness

of a novel blast wall design concept with energy absorption mechanism and provide

guidance for achieving optimal design with the proposed design scheme.

Blast walls are important structures installed in offshore platform topsides to protect

personnel and critical equipment. The traditional designs dissipate energy mainly with large

deformation of wall panels. In congested topside module, possible clash with adjacent

equipment or structural components may occur during an explosion, which can lead to an

event escalation. In addition, the welded connections are vulnerable to rupture under

strong blasts. To this end, the present study proposes an energy absorption blast wall

design by using flexible supports filled with polymethacrylimide foam and rotational

springs, allowing the wall to slide/rotate a certain distance/angle to reduce the high stresses

at supports and meanwhile dissipate blast energy through foam deformations so that both

rupture and deflection can be limited.

An analytical model based on vibration theory and virtual work theory is developed to

demonstrate the concept. The boundary conditions at each support are simplified as a

translational spring and a rotational spring. The translational spring simulates the foam

compressive characteristics with elastic-plastic deformation and unloading, while the

rotational spring provides resistance to prevent large support rotations. The dynamic

responses including panel elastic-plastic flexural behaviour and the time history of

deflection at the mid-span are also captured, with material strain hardening effect

accounted for. Finite element method is applied to validate the accuracy of the analytical

model. Both analytical and numerical results show that significant reductions around 40%

in blast wall deflections are achieved compared to their counterparts with traditional

designs, which demonstrates the effectiveness of the proposed design. In addition, the

panel deflections predicted by numerical method are in reasonable agreement with those

calculated with the analytical model, which validates the analytical model as a quick

assessment tool for blast wall deflection estimation in preliminary design stage.


Abstract The University of Western Australia

ii

Numerical studies also examine the plastic strains at support welded connections, which is

a key design focus in detail design stage. Comparisons suggest that traditional design is

vulnerable to weld tear-out due to high membrane forces generated during strong blasts,

the plastic strains at support can be as high as 25% under a ductility level blast of 3 bar

overpressure. However, with the proposed design the plastic strains can be effectively kept

under the failure criterion 5% due to the releasing of high stresses through controlled

displacements and rotation at supports. This proposed design is particularly useful for

strong blasts.

After the demonstration of the blast mitigation capability of the proposed design scheme,

parametric studies are also carried out to evaluate the influences of five key parameters

(translational spring elastic stiffness Kt1, translational spring hardening stiffness Kt2,

rotational spring stiffness Kr, blast pressure profile and duration) on the performances of

the proposed design. Discussions are carried out to provide guidance for achieving optimal

design, and suggestions are presented for future investigations.


Table of Contents The University of Western Australia

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Table of Contents

ABSTRACT ........................................................................................................................................... I

TABLE OF CONTENTS .................................................................................................................. III

ACKNOWLEDGEMENTS ................................................................................................................. V

LIST OF FIGURES ........................................................................................................................... VI

LIST OF TABLES .......................................................................................................................... VIII

....................................................................................................................................... 1-1

INTRODUCTION ................................................................................................................................................................. 1-1

BACKGROUND ........................................................................................................................................................ 1-1

RESEARCH GOALS .................................................................................................................................................. 1-5

THESIS ORGANISATION ........................................................................................................................................ 1-5

...................................................................................................................................... 2-1

LITERATURE REVIEW AND PROPOSED DESIGN CONFIGURATION ......................................................................... 2-1

OVERVIEW............................................................................................................................................................... 2-1

BLAST LOADING..................................................................................................................................................... 2-1

BLAST WAVE .......................................................................................................................................... 2-1

BLAST PRESSURE.................................................................................................................................... 2-2

BLAST WALL DESIGN REQUIREMENTS AND ANALYSIS METHODS .............................................................. 2-4

DESIGN REQUIREMENTS ..................................................................................................................... 2-4

ANALYSIS METHODS............................................................................................................................. 2-6

(a) Single degree of freedom (SDOF) method ...................................................................................... 2-7

(b) Finite element method (FEM) .......................................................................................................... 2-10

(c) Regulations for using FEM analysis ................................................................................................. 2-15

REVIEW OF RESEARCH ON OFFSHORE BLAST WALL .................................................................................... 2-16

STIFFENED PANELS ............................................................................................................................ 2-17

PROFILED (CORRUGATED) PANELS ................................................................................................. 2-18

REVIEW OF ENERGY ABSORPTION DESIGN ................................................................................................... 2-21

PRINCIPLES OF ENERGY ABSORPTION DESIGN ............................................................................ 2-21

ENERGY ABSORPTION DESIGNS ...................................................................................................... 2-21

(a) Sandwich panel .................................................................................................................................... 2-22

(b) Other structural forms ........................................................................................................................ 2-26

PROPOSED DESIGN SCHEME ............................................................................................................................. 2-28


Table of Contents The University of Western Australia

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...................................................................................................................................... 3-1

ANALYTICAL MODELLING .............................................................................................................................................. 3-1

OVERVIEW OF ANALYTICAL MODEL ................................................................................................................. 3-1

BEAM ELASTIC STAGE ........................................................................................................................................... 3-3

BEAM PLASTIC STAGE ........................................................................................................................................... 3-7

CASE STUDY AND RESULT DISCUSSION ........................................................................................................... 3-11

EXAMPLE DESCRIPTION ..................................................................................................................... 3-11

RESULTS AND DISCUSSIONS............................................................................................................... 3-14

(a) Comparison between linear and nonlinear support springs ....................................................... 3-14

(b) Comparison between proposed design and traditional design ................................................... 3-17

SUMMARY ............................................................................................................................................................... 3-21

...................................................................................................................................... 4-1

NUMERICAL MODELLING ................................................................................................................................................ 4-1

INTRODUCTION ...................................................................................................................................................... 4-1

FINITE ELEMENT ANALYSIS SETUPS .................................................................................................................. 4-1

MODEL DESCRIPTION AND MESH SIZE ............................................................................................ 4-1

MATERIAL PROPERTY ........................................................................................................................... 4-2

BOUNDARY CONDITIONS AND MODELLING OF ENERGY ABSORPTION SUPPORTS ................ 4-4

LOADING AND ANALYSIS ..................................................................................................................... 4-4

RESULTS AND DISCUSSION ................................................................................................................................... 4-6

DEFLECTION AND ENERGY ABSORPTION ....................................................................................... 4-6

PLASTIC STRAIN AT CONNECTIONS ................................................................................................. 4-21

SUMMARY ............................................................................................................................................................... 4-28

...................................................................................................................................... 5-1

PARAMETRIC STUDY ......................................................................................................................................................... 5-1

INTRODUCTION ...................................................................................................................................................... 5-1

PARAMETRIC STUDIES AND RESULT DISCUSSIONS .......................................................................................... 5-1

TRANSLATIONAL SPRING ELASTIC STIFFNESS KT1 .......................................................................... 5-1

TRANSLATIONAL SPRING HARDENING STIFFNESS KT2 .................................................................. 5-4

ROTATIONAL SPRING STIFFNESS KR .................................................................................................. 5-6

BLAST PRESSURE PROFILES ................................................................................................................. 5-8

BLAST DURATION TD ........................................................................................................................... 5-11

SUMMARY ............................................................................................................................................................... 5-15

...................................................................................................................................... 6-1

CONCLUDING REMARKS .................................................................................................................................................. 6-1

MAIN FINDINGS ...................................................................................................................................................... 6-1

RECOMMENDATIONS FOR FUTURE WORK ......................................................................................................... 6-2

REFERENCES


Acknowledgements The University of Western Australia

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Acknowledgements

I would like to express my deep and sincere gratitude to my supervisor, Professor Guowei

Ma, who supported me persistently during the period of this research. Professor Ma always

helps me solve problems and provides useful advice, which keeps my research moving

forward continuously. The origin of the ideas in this thesis took shape in his incisive

thinking. What I learned from him will benefit me greatly in the rest of my life.

This research was supported by an Australian Government Research Training Program

(RTP) Fees Offset that covers the tuition fees, to which I would like to express my

gratitude and appreciation.

Many thanks to Dr. Hongyuan Zhou from Beijing University of Technology, who helped

review my paper and gave great suggestions.

I would also like to thank Atkins, which is the company I am working in. Atkins provides

me all kinds of technical resources during the period of this research.

I wish to express my sincere thanks to my family, especially my grandfather who passed

away in 2016, for their constant love and care. Without their support, I could not have

done it.

At last, I would like to acknowledge the global oil price slump in 2014, which has ultimately

made me make up my mind to pursue a high research degree. As said by one of the famous

scholars Hu Shih (胡適) in modern China, never giving up learning knowledge, knowledge

will never fail people and the efforts of learning knowledge will never go in vain.


List of Figures The University of Western Australia

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List of Figures

Figure 1-1. General view of offshore platform types ............................................................... 1-2

Figure 1-2. Offshore explosion accidents .................................................................................. 1-2

Figure 1-3. Corrugated blast wall structural configuration [6] ................................................ 1-4

Figure 2-1. Propagation of blast waves [7] ................................................................................. 2-2

Figure 2-2. Typical blast wave types [7] ...................................................................................... 2-3

Figure 2-3. Equivalent SDOF system with nonlinear resistance [15] .................................... 2-7

Figure 2-4. Peak response chart for an SDOF system under triangular loading [15] .......... 2-9

Figure 2-5. Stiffened panel blast wall of North Rankin Alpha Platform ............................. 2-17

Figure 2-6. Profiled panel blast wall in Belanak FPSO .......................................................... 2-19

Figure 2-7. Sandwich panel core topologies [41] .................................................................... 2-22

Figure 2-8. Comparison of foam core failure in simulation and experiment [45] .............. 2-23

Figure 2-9. Sandwich panels with metallic tubes as cores...................................................... 2-24

Figure 2-10. Friction damper with a spring and its force-displacement curve [50] ........... 2-25

Figure 2-11. Failure modes of empty and alumina filled sandwich panels [51] .................. 2-26

Figure 2-12. Passive impact barrier behind a blast wall [53].................................................. 2-27

Figure 2-13. Comparison of traditional design and the proposed design ........................... 2-31

Figure 3-1. Overview of the analytical model ............................................................................ 3-2

Figure 3-2. Two idealised blast pressure profiles [6] ................................................................ 3-5

Figure 3-3. Beam left support free body diagram and foam reaction curve ......................... 3-6

Figure 3-4. Application of virtual work theory .......................................................................... 3-7

Figure 3-5. Mid-span plastic bending moment curve ............................................................... 3-9

Figure 3-6. Beam plastic boundary surface and mid-span plastic hinge length .................. 3-10

Figure 3-7. Corrugated blast wall panel section ...................................................................... 3-10

Figure 3-8. A single strip of corrugated blast wall panel ........................................................ 3-12

Figure 3-9. Comparison between linear and nonlinear translational support springs ....... 3-15

Figure 3-10. Comparison between linear and nonlinear rotational support springs.......... 3-17

Figure 3-11. Comparison of blast wall panel mid-span deflections – 1 bar case ............... 3-18

Figure 3-12. Comparison of blast wall panel mid-span deflections – 1.5 bar case ............ 3-18


List of Figures The University of Western Australia

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Figure 3-13. Comparison of blast wall panel mid-span deflections – 2 bar case ................3-20

Figure 3-14. Comparison of blast wall panel mid-span deflections – 2.5 bar case .............3-20

Figure 3-15. Comparison of blast wall panel mid-span deflections – 3 bar case ................3-20

Figure 4-1. Mesh size sensitivity test ........................................................................................... 4-2

Figure 4-2. Stress-strain curves for AIS316L stainless steel ..................................................... 4-3

Figure 4-3. Boundary conditions for traditional and proposed design .................................. 4-5

Figure 4-4. Deflection and stress contour plots – 1 bar traditional design case ................... 4-7

Figure 4-5. Deflection and stress contour plots – 1 bar proposed design case .................... 4-8

Figure 4-6. Energy absorption of various components – 1 bar case ...................................... 4-9

Figure 4-7. Deflection and stress contour plots – 1.5 bar traditional design case ..............4-10

Figure 4-8. Deflection and stress contour plots – 1.5 bar proposed design case ...............4-11

Figure 4-9. Energy absorption of various components – 1.5 bar case .................................4-12

Figure 4-10. Deflection and stress contour plots – 2 bar traditional design case ...............4-13

Figure 4-11. Deflection and stress contour plots – 2 bar proposed design case ................4-14

Figure 4-12. Deflection and stress contour plots – 2.5 bar traditional design case ............4-15

Figure 4-13. Deflection and stress contour plots – 2.5 bar proposed design case .............4-16

Figure 4-14. Deflection and stress contour plots – 3 bar traditional design case ...............4-17

Figure 4-15. Deflection and stress contour plots – 3 bar proposed design case ................4-18

Figure 4-16. Energy absorption of various components – 2, 2.5 and 3 bar cases ..............4-20

Figure 4-17. PEEQ contour plots at support connection – 2 bar case................................4-23

Figure 4-18. Section force time histories at support connection – 2 bar case ....................4-24

Figure 4-19. PEEQ contour plots at support connection – 2.5 bar case ............................4-26

Figure 4-20. PEEQ contour plots at support connection – 3 bar case................................4-27

Figure 5-1. Parametric study result plots for Kt1 ....................................................................... 5-3

Figure 5-2. Dynamic crushing behaviour of sandwich tubes [70] ........................................... 5-4

Figure 5-3. Parametric study result plots for Kt2 ....................................................................... 5-5

Figure 5-4. Parametric study result plots for Kr ........................................................................ 5-7

Figure 5-5. Parametric study result plots for blast pressure profiles ....................................5-10

Figure 5-6. An example of blast overpressure exceedance curve [2] ....................................5-11

Figure 5-7. Fundamental mode shapes and natural periods of the blast wall panel ...........5-12

Figure 5-8. Parametric study result plots for blast durations .................................................5-14


List of Tables The University of Western Australia

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List of Tables

Table 2-1. Criticality category and performance for SCEs [3] ................................................ 2-5

Table 3-1. Blast wall panel section and material properties ................................................... 3-12

Table 3-2. Polymethacrylimide foam mechanical properties ............................................... 3-13

Table 3-3. Support translational and rotational spring properties ....................................... 3-13

Table 3-4. Result summary for translational spring comparison study .............................. 3-15

Table 3-5. Result summary for rotational spring comparison study ................................... 3-16

Table 3-6. Summary of the peak blast wall panel deflections and maximum foam

displacements ............................................................................................................................... 3-21

Table 4-1. Material parameters for AIS316L [67] ..................................................................... 4-3

Table 4-2. Summary of the peak blast wall panel deflections ............................................... 4-20

Table 4-3. Summary of energy absorption of various components ..................................... 4-21

Table 4-4. Summary of the maximum PEEQ at support connections ............................... 4-25

Table 5-1. Result summary for parametric study of Kt1 ........................................................... 5-3

Table 5-2. Result summary for parametric study of Kt2 ........................................................... 5-6

Table 5-3. Result summary for parametric study of Kr ............................................................ 5-8

Table 5-4. Result summary for parametric study of blast pressure profiles .......................... 5-9

Table 5-5. Result summary for parametric study of blast durations .................................... 5-14


Chapter 1 The University of Western Australia

1-1

Introduction

Background

Offshore oil and gas exploration can be dated back to more than one hundred years ago.

Around 1891, the first submerged oil well was drilled from platforms built on piles in the

fresh waters of the Grand Lake in Ohio, U.S. Around 1896, the first submerged oil well in

seawater was drilled in the portion of the Summerland oil field extending under the Santa

Barbara Channel in California, U.S. The wells were drilled from piers extending from land

out into the channel. Despite of those early expeditions, the worldwide boom for the

search of oil and gas into the oceans did not happen until the end of World War II. One

remarkable case was in 1946, Magnolia Petroleum (now ExxonMobil) drilled at a site

29 km off the coast, erecting a platform in 5.5 m of water off St. Mary Parish, Louisiana,

U.S.

Since the success of the first oil, technology evolution in offshore exploration develops

exponentially as research and scientific programs are well funded by oil companies. After

60 years development, a full range of offshore drilling and processing platforms have been

invented to suit the expedition in different water depths. Figure 1-1 shows a general view

of various types of offshore platforms. In the last 20 years, due to the rising demand, oil

and gas explorations have been extended to the deep-water region (deeper than 2000m)

that far away from shore, which requires the platforms to carry on multiple functions, such

as processing, storage and offloading. These requirements have resulted in more equipment

on platforms and hence more congested layouts. Currently, Royal Dutch Shell is building

the world’s first Floating Liquefied Natural Gas (FLNG) platform Prelude, which fits every

element of a conventional LNG facility into a ship area roughly one quarter of the original

size, whilst maintaining appropriate levels of safety.



1-2

1, 2) conventional fixed platforms; 3) compliant tower; 4, 5) vertically moored tension leg

and mini-tension leg platform; 6) spar; 7,8) semi-submersibles; 9) floating production,

storage, and offloading facility; 10) sub-sea completion and tie-back to host facility

Figure 1-1. General view of offshore platform types

In each type of the drilling and processing platform, handling of hydrocarbons and other

flammable materials can lead to an explosion. Although such incidents may be relatively

rare, they do occur and the consequences are extremely severe, especially for those with

congested layouts, as more obstacles tend to produce more turbulence and hence enhance

the flame accelerations [2]. A large number of offshore explosion accidents have been

witnessed globally, among them the Piper Alpha explosion in North Sea and BP

Deepwater Horizon explosion in the Gulf of Mexico (shown in Figure 1-2) are on top of

the list of the most catastrophic in the industry.

(a) Piper Alpha explosion (b) Deepwater Horizon explosion

Figure 1-2. Offshore explosion accidents

The Piper Alpha explosion happened on 6 July 1988 due to a leakage of gas condensate.

The explosion, and the resulting oil and gas fires, destroyed the topside modules, and killed

167 people including two crewmen of a rescue vessel, only 61 people survived. The total

insured loss was about US$3.4 billion. At the time of the disaster, the platform took up



1-3

approximately ten percent of North Sea oil and gas production, and the accident was the

worst offshore oil disaster in terms of life lost and industry impact. The Deepwater

Horizon drilling rig explosion occurred on April 20, 2010, the explosion and subsequent

fire caused the Deepwater Horizon to burn and sink. Although the casualties of this

explosion was not as disastrous as Piper Alpha (11 workers died and 17 others injured), the

accident was remarkable as it also caused a massive offshore oil spill in the Gulf of Mexico,

which is considered the largest accidental marine oil spill in the world, and the largest

environmental disaster in U.S. history.

Although prevention is the best protective measure against explosion, blast resistant design

and structural strengthening also play necessary and important roles in mitigation of

explosion risks, as structural survivability is the last barrier to prevent human life loss if an

explosion does occur. Therefore, it is of great interest to carry out research on the

performances of blast resistant structures under extreme loading to ensure their

survivability.

Blast wall in the offshore platform is a typical protective structure against blasts and fires. It

is also a safety critical structural element in an offshore topside module to protect

personnel and critical equipment, and prevent further escalation of events after explosions.

Traditionally, concrete is used for land based blast protective structures, due to its huge

weight and high stiffness. In offshore circumstances, alternative structural configurations

are required as weight is an important factor in design. Therefore, the requirements for

offshore blast walls are more rigorous, they must be light, robust and able to maintain their

integrity during blast events. Currently, stiffened panels and corrugated panels are widely

used in offshore constructions, among them stainless steel corrugated (or profiled) panels

are the most popular choices due to the advantages of easy fabrication, considerable ductile

ability and good corrosion and fire resistance properties. A typical structural configuration

for corrugated blast panel is shown in Figure 1-3, the blast wall connections to topside

modules consist of welded end plates (normally angle sections) that are extended from the

supporting plate girders.



1-4

Figure 1-3. Corrugated blast wall structural configuration [6]

In 2003 (revised in 2007 afterwards), Oil and Gas UK together with Health & Safety

Executive (HSE) [3] issued a new guidance for offshore explosion risk mitigation and blast

resistance design. Similar to earthquake assessments, the guidance requires blast walls to be

designed for two level of events, strength level blast (SLB) and ductility level blast (DLB).

The new guidance adopts the performance-based design approach that requires the blast

walls to meet certain criterions of deflection, strength and weld rupture for two different

level blasts. Especially for the DLB event, blast wall shall be designed to maintain its

integrity against blasts and subsequent fires, and meanwhile not to provoke an event

escalation by clashing with critical equipment, pipelines and structural members located

nearby. Therefore, as concluded by Boh et al. [4], design of blast walls under extreme

loadings involves three major aspects: large plastic deformations, possible clash with

adjacent equipment or structural components and weld tearing out.

Due to the uncertainty of accurately predicting blast loadings, back in 1990s most of the

blast walls were designed for a nominal overpressure of 0.5 bar, in which the traditional

design configuration shown in Figure 1-3 is efficient and economical because blast walls

often remain in elastic responses and the angle sections at supports have certain flexibility

to absorb energy. However, according to Selby and Burgan [5], recent large scale explosion

tests on a typical topside module revealed that the blast overpressures can be as high as 4

bar. Under such condition, the traditional design configuration is no longer valid because

blast wall is highly likely to undergo large plastic deformation and the end connections are

vulnerable to weld rupture. Therefore, blast wall design for high overpressures becomes a

challenging task for structural engineers.



1-5

Research Goals

Since traditional blast wall configuration can hardly satisfy the design requirements for high

overpressure blasts, this study aims to develop a novel design concept and demonstrate its

feasibility and effectiveness, meanwhile, provide guidance on design optimisation.

The research goals are detailed and listed as follows:

1. To develop a novel blast resistant structural scheme that can reduce panel

deflection and minimise weld rupture at the same time using energy absorption

concept.

2. To develop an analytical model using beam vibration theory and energy method to

prove the effectiveness of the proposed design in reducing panel deflections.

3. To verify the calculations of the analytical model against numerical simulations so

that the analytical solution can act as a quick design calculation tool.

4. To demonstrate the capability of the proposed design in minimising weld rupture

by comparing support plastic strains with traditional design.

5. To evaluate the performances of the proposed design and provide guidance on

design optimisation by carrying out parametric studies on key parameters.

Thesis Organisation

This thesis comprises six chapters. The five chapters following this introductory chapter

are arranged as follows:

Chapters 2 presents an extensive literature review on the blast dynamics, current blast wall

design practices in offshore industry and previous research on structural responses of blast

resistant panels. It covers blast loading, structural analysis methods against blast, research

of various types of blast wall and energy absorption design. At the end of this chapter,

details of the proposed design scheme are presented and explained.

Chapters 3 presents the derivation of the theoretical model to calculate the blast wall

deflection of the proposed design. The simplifications and assumptions in modelling,

together with theories and formulations at different stages are explained in this chapter. At

the end, several calculation examples are provided to demonstrate the privilege of the

proposed design in reducing panel deflection.



1-6

In chapter 4, finite element models are created for both traditional and proposed design

schemes. Nonlinear dynamic analyses in time domain are carried out for two purposes: to

validate the calculations of analytical model and to compare plastic strains at supports for

both design schemes.

After demonstrating the capability of the proposed design analytically and numerically,

chapter 5 focuses on investigating the influences of some key parameters on the structural

performances and behaviours of the proposed design. Therefore, parametric studies of

certain key factors are carried out to optimise the proposed design.

Finally, Chapter 6 summarises the main outcomes of this research, and makes suggestions

for future studies.



2-1

Literature Review and Proposed Design Configuration

Overview

This chapter presents an extensive literature review on the blast dynamics, current blast

wall design practices in offshore industry and previous research on structural responses of

blast resistant panels. It covers blast loading, structural analysis methods against blast

loading, research on various types of blast walls and studies of energy absorption designs.

At the end of this chapter, details of the proposed design are presented.

Blast Loading

Blast Wave

An explosion is defined as process of a rapid increase in volume and release of energy in an

extreme manner, usually associated with the generation of high temperatures and strong

shock waves [7]. The vast majority of explosions in offshore or marine environments are

chemical explosions with high rate of energy release and hence producing high blast

overpressures. In a microscopic view, a chemical explosion is caused by external heat or

shock that rapidly decomposes and rearranges the atomic formations of the original

substance, such as TNT (Trinitrotoluene) and hydrocarbons, during which large amount of

energy is released [7].

In free air, the blast effects of an explosion are a series of shock waves that propagate

outward from the explosion source into the surrounding air. Blast waves are normally

followed by a blast wind of negative pressure, which sucks items back, towards the

explosion centre [7]. Figure 2-1 depicts the propagation of blast waves.



2-2

Figure 2-1. Propagation of blast waves [7]

If a shock wave impinges on a rigid surface oriented at an angle to the direction of the

wave, a reflected wave is generated and travels back. Therefore, the reflected wave might

meet the incident wave and generate complex interactions. Mach stem formation is one of

the typical interactions. It occurs when a blast wave reflects off the ground and the

reflection catches up with the original shock front, creating a high pressure zone that

extends from the ground up to a certain point called the triple point at the edge of the blast

wave. Anything in this area experiences peak pressures that can be several times higher

than the peak pressure of the incident shock wave [7].

Blast Pressure

According to TM5-1300 [8], when a blast wave impinges and then engulfs a target

structure, the magnitude and distributions of the blast loads are affected by the following

factors: (1) explosive properties of the charge (weight, type, shape); (2) the standoff

distance between the detonation and the structure; (3) the interactions between the incident

wave and the reflected wave.

Figure 2-2 illustrates two typical blast types, detonation and deflagration. A detonation is

supersonic explosion that drives the reaction front move at the same speed of shock front.

Detonation profiles are characteristics of high explosives (such as TNT) or impulses from

gas or vapour cloud explosion in the far field [7]. A deflagration usually occurs in

petrochemical plants or offshore processing platforms as a result of leakage of flammable

gases. The delay of ignition results in a slow burning process, in which the reaction front

moves at a subsonic speed behind the shock front. It should be noted that under certain

circumstance, the deflagration process can be transformed to detonation, during which the

subsequent supersonic propagation can generate more intensive shock waves [7].



2-3

(a) Detonation (b) Deflagration

Figure 2-2. Typical blast wave types [7]

In engineering practice, it is generally convenient to express different explosive charges and

their weight as an equivalent TNT weight W. For simplicity the blast pressures are often

characterised with scaled distance Z shown in Eq.(2-1), where R is the standoff distance

between explosive centre and the target structure.

3 W

RZ = (2-1)

Blast induced overpressures and shock wave propagations in unconfined atmosphere have

been intensively studied. Many empirical formulae are available to predict the peak

overpressure Ps0 in free air, for example, Brode [9] and Henrych [10] both derived their

predictions in terms of scaled distance, shown in Eq.(2-2) and Eq.(2-3) respectively.

>=

≤≤−++=

barPforbarZ

P

barPbarforbarZZZ

P

soso

soso

0.10),(7.6

0.100.1),(019.085.5455.1975.0

3

32 (2-2)

≤<++=

≤<++=

≤≤+++=

100.1),(288.305.4662.0

0.13.0),(1324.23262.01938.6

3.005.0),(00625.03572.05397.50717.14

32

32

432

ZforbarZZZ

P

ZforbarZZZ

P

ZforbarZZZZ

P

so

so

so

(2-3)

As stated above, when a shock wave hits a rigid surface oriented at an angle to the direction

of the wave, a reflected pressure is instantly developed on the surface. The pressure acting

on the surface is amplified to a value exceeding Pso. The peak reflected pressure Pr is a

function of Pso and the angle between surface and the shock front. Empirical formulae are



2-4

also available to calculate Pr, the example shown in Eq.(2-4) is proposed by Henrych [10] as

an ideal case with normal shock wave incidence.

so

so

so

r PPP

PPP

0

0

7

148

+

⋅+⋅= (2-4)

In engineering practice, in unconfined conditions Pso and Pr are usually obtained by using

design charts provided in the U.S. design codes such as TM5-1300 [8]. In a confined

environment (explosions occur inside the structure), it is very difficult to establish empirical

formulae or design charts to predict peak pressures as the contributing factors to the

calculation are multiple and varied. Additional to wave reflection enhancement, the degree

of confinement, temperatures and ventilation also have influences on blast pressure on

structures. Under such condition, Pso and Pr are usually obtained by performing detail

computational fluid dynamics (CFD) simulations.

Blast Wall Design Requirements and Analysis Methods

Design Requirements

As the result of Piper Alphas accident, in 1992 Interim Guidance Notes (IGNs) were

issued by Steel Construction Institute (SCI), UK to provide guidance for offshore platform

topside design against fires and explosions. Since then, extensive research and

investigations have been carried out for getting better understanding of the risk associated

with fires and blasts in order to obtain more accurate prediction of structural response

under blast events. In 2003 (revised in 2007 afterwards), Oil and Gas UK together with

Health & Safety Executive (HSE) collated the latest research outcomes and issued a new

guidance (Fire and Explosion Guidance) [3] for explosion risk mitigation and blast

resistance design for platforms in North Sea. Almost at the same time (2006), American

Petroleum Institute (API) also developed a similar design standard (API-RP-2FB

Recommended Practice for the Design of Offshore Facilities against Fire and Blast

Loading) [11] for platforms in the Gulf of Mexico.

Both UK and US standards cover all related topics in fire and explosion design, such as

hazard mechanism and management, determination of explosion loads and regulations for

structural responses to explosions. This study mainly adopts the guidance from the sections

of the structural response design of the UK standard [3].



2-5

For blast resistant structural design, the UK standard adopts an element or system based

performance standard. To start with, this approach requires an identification and

categorisation of the safety critical elements (SCEs), such as structural elements, structural

systems, processing equipment or pipelines that required detailed assessments, into three

levels according to their criticalities. After classification, the measurable performance

standards are defined for each category with respect to their functionality, availability and

survivability. The categories and performances of SCEs are reproduced in Table 2-1:

Table 2-1. Criticality category and performance for SCEs [3]

Category Performance Standard

Criticality 1 Items whose failure would

lead to direct impairment of the Temporary

Refuge (TR) or evacuation escape and

rescue (EER) systems including the

associated supporting structure.

These items must not fail during the

strength and ductility blasts. Ductile

response of the support structure is

allowed during the ductility blasts.

Criticality 2 Items whose failure could lead to

major hydrocarbon release and escalation

affecting more than one module or

compartment. (Indirect impact on the TR is

possible through subsequent fire).

These items must have no functional

significance in an explosion event and

these items and their supports must

respond elastically under the strength

level blast.

Criticality 3 Items whose failure in an

explosion may result in module wide escalation,

with potential for inventories outside the

module contributing to a fire due to blowdown

and or pipework damage.

These items have no functional

significance in an explosion event and

must not become or generate projectiles.

Performance standards considered for structural elements normally involve the following

aspects: strength, deformation limit, connection rupture and global collapse. In analogy to

earthquake assessments, two level of blast assessments are recommended by the design

standard, they are strength level blast (SLB) and ductility level blast (DLB). A SLB

represents a relatively more frequent design event (10-3 exceedance per annual), in which

blast resistant design pays more attention to structural functionality. It is required the



2-6

responses of all SCEs remain in elastic range. In additional, the Criticality 1 and 2 SCEs

shall remain operational without significant repair after the events.

A DLB represents an extreme design event with high consequence (10-4 to 10-5 exceedance

per annual) that structural survivability shall be analysed carefully. Generally, it is required

the structure to respond in a ductile manner that the primary structure shall not collapse,

the integrity of TR shall not compromise and emergency evacuation routines shall remain

accessible after the event. To be specific, in order to dissipate energy and achieve an

economic design, plastic deformation of the structure is acceptable provided that

progressive collapse does not occur. Barriers must remain in-place and are able to resist the

subsequent fires, which requires the barrier connections to respond without rupture. In

certain areas, structural element deflections need to be limited to avoid collision with vital

equipment or pipework, which may provoke an event escalation.

For blast wall design, according to the regulations above, a blast wall falls into SCE of

Criticality 2 in which design for global collapse is not required. In terms of strength, blast

wall shall not exceed its yield limit and generate unrecoverable deflection during a SLB

event. While for DLB event, deformation limit and connection rupture are more important

aspects than strength. In terms of deformation limit, the design codes do not give an exact

value, however, in engineering practice, it is common to limit the panel deflection to the

minor of 300mm and the clearance to critical equipment, pipelines and structural members

located nearby. For weld rupture, structural integrity is often assessed by using strain limits

which if exceeded will cause rupture. Current prevailing design codes, such as ISO 19902

[12], set the allowable plastic strain limits at 5% at welded connections and 15% in parent

material away from welding.

Analysis Methods

The differences between structural responses under static and dynamic loads are the inertial

component in the equation of motion and the additional kinematic energy due to motions

[13]. The pressure loading generated by explosion is time-dependant, hence the analysis of

the structural behaviours under blasts involves solving the equation of motion in time

domain. There are two basic analysis methods currently being employed in both

engineering practice and research: single degree of freedom (SDOF) method and finite

element method (FEM).



2-7

(a) Single degree of freedom (SDOF) method

The SDOF method is also known as the Biggs’ method [14], which has been incorporated

in many design codes such as Fire and Blast Information Group Technical Note 5 (FABIG

TN5) [1] and DNV-RP-C204 (Design against Accidental Loads) [15]. It is a quick and

useful assessment tool that is normally used in preliminary design stage, as it avoids

complex dynamic analysis by just converting the dynamic time dependent pressure into an

equivalent static load. The equivalent static load is obtained by multiplying the specific peak

blast pressure by a dynamic load factor (DLF), which is defined as the ratio of the

maximum dynamic reaction to the static reaction computed from the peak blast pressure.

The nature of SDOF method is to approximately idealise the real system into an equivalent

spring mass model based on an assumed deflection shape of the structure. Figure 2-3

shows the simplification of blast wall structure to an equivalent SDOF spring system.

(a) Equivalent SDOF system

(b) Elastic-perfectly plastic structural resistance function

Figure 2-3. Equivalent SDOF system with nonlinear resistance [15]



2-8

With the assumed deflection shape ξ(x), the equivalent mass Me, force Fe and stiffness Ke

of the equivalent system can be calculated in Eq.(2-5) through Eq.(2-7). Subsequently, the

equivalent system can be established by constructing the equation of motions shown in

Eq.(2-8):

∫ ⋅=l

e dxxM0

2)(ξρ (2-5)

∫ ⋅=l

e dxxxFF0

)()( ξ (2-6)

t

e

LeF

FKKKK =⋅= (2-7)

)(tFyKyM eee =⋅+⋅ && (2-8)

e

e

K

MT π2= (2-9)

where ρ is the mass density per unit length of the actual structure; K is the stiffness of the

actual structure; Ft is the total force acting on the actual structure; T is the natural period of

the equivalent system.

The method can also account for blast wall plastic responses. By assuming an elastic-

perfectly-plastic response, which is shown in Figure 2-3, the stiffness term in Eq.(2-8) can

be replaced by the maximum resistance force Rm and the formula is rewritten as Eq.(2-10).

)( tFRyM eme =+⋅ && (2-10)

In engineering practice, it is convenient to read off the peak responses of the blast wall

panels from design charts that are available in design codes. A typical design chart for a

simply-supported beam under triangular pressure load is reproduced in Figure 2-4 from

DNV-RP-C204 [15].



2-9

Figure 2-4. Peak response chart for an SDOF system under triangular loading [15]

Although SDOF method is a quick and simple approach to calculate structural response of

blast walls, it has limitations. Some of the major drawbacks are summarised as follows [16]:

1. Since the approximation is based on an assumed deflection shape, it is not easy to

covert complex structures (e.g. loading profile, mass distribution) into equivalent

spring mass model.

2. Peak deflections read from design charts do not consider any local ductility and

local instabilities that may govern the design of the blast wall, which means it may

not be applicable for non-compact sections.

3. The design charts and transformation factors are established for pinned and fixed

cases, which fails to account for the stiffness of end restraints. The end restrain

stiffness can affect the calculation accuracy of structural natural period and peak

responses.

4. Strain rate effects cannot be accurately addressed, they are simply accounted by

increasing yield strength (by 20%-30%).



2-10

(b) Finite element method (FEM)

The FEM is introduced in the 1960s, it is a numerical approximation for finding the

boundary values for partially differential equations [17]. After decades of development,

FEM has become a very reliable tool for structural analysis in civil, mechanical and

aerospace engineering. Some advantages of FEM are listed as follows [17]:

1. Accurate representation of complex geometry.

2. Inclusion of nonlinear material properties.

3. Easy representation of the total solution.

4. Capture of local effects.

In a general term, a typical FEM subdivides a large problem into smaller, simpler parts,

called finite elements. These finite elements represented by simpler equations are then

assembled into a larger system of equations that models the entire problem. In solid

mechanics, the nature of FEM is to find the strains or stresses in the finite element model

by minimising an energy function. The minimum of the energy function is found by setting

the differentiation of the energy function with respect to the unknown displacements to

zero [18]. To be specific, an FEM procedure generally consists of four steps [18]:

Step (1): Dividing or discretising the original structure in to small elements and finding

the element properties.

For an elastic body or structure subjected to body and surface forces, a single finite element

extracted from the body has the nodal displacement vector {q}. Displacements {u} at

some point inside the finite element can be determined using nodal displacements {q} and

shape functions [N] that contains interpolation functions.

{ } [ ]{ }qNu = (2-11)

With that, strains {ε} can be determined from the nodal displacements {q} as shown in

Eq.(2-12), in which [B] is displacement differentiation matrix. And stresses {σ} are related

to strains {ε} by Hooke’s Law in Eq.(2-13), in which [E] is the material operator matrix.

{ } [ ]{ }qB=ε (2-12)

{ } [ ]{ }εσ E= (2-13)



2-11

Using relations for stresses and strains, the element total potential energy π can be

expressed through nodal displacements {q}:

[ ]{ } [ ]{ }( ) { } [ ]{ }( ) { }dSpqNdVpqNVdqBS

T

S

VT

VV ∫∫∫ −−=2

1π (2-14)

where {pV} is the vector of body forces; {pS} is the vector of surface forces. The first term

on the left hand side of this equation represents the internal strain energy and the second

and third terms are, respectively, the potential energy contributions of the body force loads

and distributed surface loads.

As mentioned earlier, the minimum of the energy function is determined by differentiation

of the element total potential energy π in respect to nodal displacements {q}, ∂π/∂q=0,

which produces the equilibrium equations for a single finite element in Eq.(2-15), so that

element properties such as element stiffness [k] and element force vectors {f} can be

obtained:

[ ]{ } { }fqk = (2-15)

[ ] [ ] [ ][ ]dVBEBkT

V∫= (2-16)

{ } [ ] { } [ ] { }dSpNdVpNf ST

S

VT

V ∫∫ += (2-17)

Step (2): Assembling the element equations to form an approximate system of the

original structure and imposing boundary conditions.

The aim of assembly is to form the global equation shown in Eq.(2-18), where [K], {Q}

and {F} are global stiffness matrix, displacement vector and load vector, using element

equations in Eq.(2-15).

[ ]{ } { }FQK = (2-18)

Three simple assemblies of element force vectors {Fd}, displacement vectors {Qd} and

stiffness matrices [Kd] are formed in advance through summation of element properties

derived from the previous step.

{ } { } { }{ }⋅⋅⋅= 21 qqQ d (2-19)

{ } { } { }{ }⋅⋅⋅= 21 ffFd (2-20)

[ ][ ]

[ ]

⋅⋅⋅

=

00

00

00

2

1

k

k

K d (2-21)



2-12

Then the task is changed to establish a relationship between [K] and [Kd], {F} and {Fd}.

This can be achieved by presenting the total potential energy for the structure Π as a sum

of element potential energies πi , Π =∑ πi , and similarly applying the derivation of

minimum of the global total potential energy to global displacement, ∂Π/∂Q=0. This will

produce the global equation system in form of [Kd] and {Fd}:

[ ] [ ][ ]{ } [ ] { }d

T

d

TFAQAKA = (2-22)

where [A] is the matrix providing transformation from global to local enumeration.

Thereby, the assembly the global stiffness matrix [K] and the global load vector {F} are

given below:

[ ] [ ] [ ][ ]AKAK d

T= (2-23)

[ ] [ ] { }d

TFAF = (2-24)

For a dynamic analysis, the matrices representing a continuous distribution of mass of the

structure [M] and viscous damping [C] are added to the global equation system:

[ ]{ } [ ]{ } [ ]{ } { }FQKQCQM =++ &&& (2-25)

Displacement boundary conditions are not accounted for in the function of the total

potential energy. They can be applied to the global equation system after the assembly, for

example, Qm=d, by defining the global displacement at point m to a known value of d.

Step (3): Solving the global equation system

With the global equation system and boundary conditions, unknown displacements can be

solved by inverting the global stiffness matrix:

{ } { }[ ] 1−= KFQ (2-26)

In a linear static analysis, solution methods for linear equation systems can be divided into

two large groups: direct methods and iterative methods [19]. Direct solution methods, such

as Gaussian elimination method, yield the exact solution in a finite number of elementary

arithmetic operations, but the accuracy is affected by the rounding-off errors. This method

is usually used for problems of moderate size. For large problems iterative methods require

less computing time and hence is preferable. Iterative methods, such as Newton-Raphson

method, start with an initial approximation, and by applying a suitable algorithm, can lead



2-13

to better approximations when the process converges. The accuracy and the rate of

convergence of iterative methods depend on the algorithm chosen.

In terms of dynamic analysis, solutions can be determined by either implicit or explicit

integration schemes. The essential difference between these two methods is whether

directly solving the dynamic differential equations at each time step. Implicit method

attempts to find solutions by solving the equations which involving both the current state

(displacement Qn, velocity Q�

n and acceleration Q�

n at step n) of the system and the next

one (Qn+1 , Q� n+1 and Q�

n+1 at step n+1). A typical example is the Newmark’s Method

which is based on the following equations:

11

])1[( ++ ⋅∆⋅+⋅∆⋅−+= nnnn QtQtQQ &&&&&& γγ (2-27)

1

22

1])5.0[( ++ ⋅∆⋅+⋅∆⋅−+⋅∆+= nnnnn QtQtQtQQ &&&&& ββ (2-28)

where β and γ are two parameters defining the variation of acceleration over time step. For

β=1/4 and γ=1/2, acceleration over the time step is assumed to be constant. For β=1/6

and γ=1/2, acceleration over the time step is assumed to vary linearly.

Substituting the terms into the equation of motion, iterative process is needed for solving

the equations. In addition, implicit analysis requires to invert the stiffness matrix once or

even several times over a time step, which makes implicit analysis an expensive, time-

consuming operation, especially for large models.

In contrast, explicit techniques avoids solving the governing dynamic differential equations

at each time step, it only requires the results of the current and previous states of motion

(step n and n-1) to evaluate the next state (step n+1). Centre difference method with

constant time step is a typical explicit integration technique. Take a SDOF problem for

example, discretise time-dependent displacement Q(t) and force F(t) with a sufficiently

small constant time step ∆t, the velocity and acceleration can be expressed by displacement

Q(t) using centre difference approximation.

t

QQQ nn

n∆⋅

−= −+

2

11& (2-29)

2

11 2

t

QQQQ nnn

n∆

+⋅−= −+&& (2-30)

Substitution into the equation of motion and rearrange the terms, the displacement at n+1

step Qn+1 can be determined by displacement results from previous steps:



2-14

]2

[

]2

[]2

[

2

212

1

t

C

t

M

Qt

MKQ

t

C

t

MF

Qnnn

n

∆⋅+

∆

⋅∆

⋅−−⋅

∆⋅+

∆−

=−

+ (2-31)

Initial conditions (Q0, Q�

0 and Q�

0) and a special starting point Q�� are required to derive

the following steps.

M

QKQCFQ 000

0

⋅−⋅−=

&&& (2-32)

0

2

0012

1QtQtQQ &&& ⋅∆⋅+⋅∆−=−

(2-33)

As compared above, it is clear that the key advantage of explicit techniques over implicit is

the high computational efficiency, as explicit approach does not require iteration and

inversion of global stiffness matrix. It also has other privileges in dealing with problems

with large deformation, contact and nonlinearities. However, unlike the implicit solution

scheme, which is unconditionally stable for large time steps, the explicit scheme is stable

only if the time step size is sufficiently small. An estimate of the time step that required to

ensure stability shall be less than the Courant time step (time it takes a sound wave to travel

across an element):

C

Lt s=∆ (2-34)

where Ls is the characteristic element size of the smallest element and C is the speed of

sound waves in the material.

Step (4): Calculating element stresses and strains.

Solution of the global equation system provides displacements at nodes of the finite

element model. Using disassembly, which is the reverse of the assemble process in Step 2,

nodal displacement for each element can be obtained. Strains inside an element are

determined with the displacement differentiation matrix, referring to Eq.(2-12). Stresses are

calculated with the Hooke’s law as Eq.(2-13).

It should be noted that for an individual elements, displacements are derived at the

integration points. In order to build a continuous field of strains or stresses, it is necessary

to extrapolate result values from integration points to all nodes of the element. For

adjacent elements, the extrapolated nodal stresses and strains are usually non-continuous



2-15

across the element boundaries. Therefore, it is also necessary to smoothen the stress and

strain fields by averaging the contributions from the neighbouring finite elements, which

shall be performed for all nodes of the finite element model to produce a continuous field

of structural responses.

(c) Regulations for using FEM analysis

Although FEM yields more accurate structural responses, cautions must be exercised when

using it. As explained above, the FEM calculations are heavily depended on the pre-

processing steps, such as geometry modelling, boundary conditions and mesh properties.

In order to obtain reliable results for structural design and avoid misusing, design codes,

such as DNV-RP-C208 [20], have given regulations for using FEM.

First of all, the software package selected for design must be tested and meanwhile be

capable of handling nonlinear behaviours including nonlinear material, nonlinear geometry

and complex contact problems. ABAQUS is a commercial FEM software, which is famous

for its capability to solve complex and highly nonlinear problems. It contains three parts,

pre-processor ABAQUS/CAE, implicit solver ABAQUS/STANDARD and explicit solver

ABAQUS/EXPLICIT. All numerical analyses in this thesis are performed with ABAQUS.

For geometry modelling, take blast wall for example, the local finite element model shall

include sufficient structural details such as dimensions, offsets and stiffenings to simulate

the correct structural stiffness, and also include sufficient adjacent supporting structures to

ensure that the end-restraints are captured realistically. However, rather than constructing a

model with a high level of detailing, engineering judgements shall be made to omit certain

small details, such as rat-holes and fillet welds, to save computational efforts as long as

their influences on the behaviours structural components of interest are minimal. For a

buckling analysis, initial imperfections shall be seeded in accordance to the mode shapes of

the structure. For material curve, the selected material model shall be able to represent both

linear and non-linear behaviours (elastic and plastic) for increasing and decreasing loads

(unloading). For transient analysis such as blast wall assessment, strain rate effects shall also

be included.

As blast wall design is normally a local analysis, which is separate from the global model,

decision of the boundary conditions for the local model requires careful considerations.

The design codes recommend two options: the selected model boundary condition needs

to represent the real condition in a way that will lead to results that are either accurate or



2-16

conservative. In order to be accurate, in engineering practice, the adjacent structural

components to the blast wall shall be modelled and the truncation locations shall be far

away from the area of interest. Prescribed displacements extracted from global model shall

be applied to these truncation locations as boundary conditions. In some cases where

global results are absent, the truncation locations can be either pinned or fixed because they

have little effects on the structure of interest due to the distance. On the other hand, to

save computational costs, pinned or fixed supports can be directly applied to the blast wall

edges, and blast wall are designed for the worse responses generated from these two

boundary conditions.

As derived above, finite element properties make great influences on the calculation results.

According to the experiences of Louca and Boh [16], first order reduced integration shell

elements (S4R) in ABAQUS are appropriate for general blast assessment purpose. S4R

elements can accommodate both thick and thin shell theories, and exhibit good ability to

account for problems with large displacements, nonlinear behaviours and finite membrane

strains. Reduced integration (i.e. S4R has single integration point) can yield good results

with less computational efforts as long as the elements are not distorted. Their experiences

[16] also suggest that at least approximately 4000 to 8000 elements are required to capture

good global responses of a corrugated blast wall panel. In addition, design code DNV-RP-

C208 [20] requires more and finer mesh elements to be placed in the yield zones and

connection regions to have good strain estimates, as well as buckling zones to capture the

failure shapes. Although transitions are inevitable between coarse and fine meshes, abrupt

changes to mesh densities shall be avoided as numerical errors may be generated. As per

Louca and Boh [16], the mesh shall be graded in a way that for individual element, the

aspect ratio shall be kept less than 2 and Jacobian ration no less than 0.25 to eliminate

distorted elements. For adjacent elements, there shall not be large dimensional

discrepancies between them, generally equivalent dimensions are not greater than 2.

Review of Research on Offshore Blast Wall

Currently, two types of blast walls are widely used in offshore industry, they are stiffened

panels and profile (corrugated) panels. Both of them are light-weight blast walls with high

strength for offshore environments. The research on their structural behaviours under blast

loading are reviewed and summarised in this section.



2-17

Stiffened Panels

A typical stiffened panel (shown in Figure 2-5) is a thin-walled plate with welded ribs or

stiffeners (typically flat bar or T-section) to increase its bending capacity. Its application in

civil engineering is wide and hence intensive research have been carried out to study its

performance.

Figure 2-5. Stiffened panel blast wall of North Rankin Alpha Platform

A large amount of investigation attention has been drawn to the interactions between plate

and stiffeners. Schleyer et al. [21-22] developed simplified analytical model to predict

dynamic responses of both stiffened and unstiffened panels against hydrocarbon

explosions, the outcomes of the study have been incorporated in design guidance [23]

issued by Health & Safety Executive (HSE). Similarly, Louca et al. [24] also managed to

create analytical models based on Lagrange equation, additionally, FEM was adopted to

study the effects of in-plane boundary conditions, initial imperfection and local buckling of

the stiffeners. Another piece of work by Louca et al. [25] focused on the effects of various

blast pressure profiles on a T-stiffened panel, the significant influences of boundary

conditions were again highlighted. A comparison was made between the responses

obtained from FEM and SDOF method, good correlations were observed in the region

where plastic deformations were relatively small. Other than theoretical and numerical

measures, Pan and Louca [26] also carried out experiments to study the resistant capacities

of T-stiffened panels under blast loadings. It was reported that the contribution of

stiffeners was mainly influenced by the second moment of the cross section. Boh [6]

reproduced the explosion tests on two flat bar stiffened panels executed by Advantica

(formerly British Gas) using finite element models. With proper modelling setups, major

features and global deflections were captured well. It was also concluded that the governing

failure criteria of stiffened panel was buckling of the stiffeners. Stiffeners were important in



2-18

the early phase of the responses because they effectively increased the load bearing capacity,

however, after buckling the plate behaved as an unstiffened panel.

Since the load bearing capacities of stiffened panels are heavily depended on the buckling

of the stiffeners, other than flat bar and T-section, various of studies have been carried out

using different sections of stiffeners to improve the buckling capacity. Arendsen et al. [27]

derived an optimisation code which made use of linking and scaling of variables to design

stiffened panel with buckling and post buckling constraints. Four stiffener configurations

(T, J, L and Hat-stiffeners) were selected as examples to demonstrate its efficiency.

Barkanov et al. [28] numerically studied three rib bays stiffened panels with T, I and HAT-

stiffeners to investigate their buckling behaviours in dependence on skin and stiffener lay-

ups, stiffener height, stiffener top and root width. Optimisation studies were performed to

achieve the minimum weights while maintaining the buckling capacities.

Other than stiffener types, dynamic responses of stiffened panels due to stiffeners layouts

have also been investigated. Geol et al. [29] performed numerical comparison study

between ten stiffener layouts and their equivalently thickened unstiffened plates. It was

concluded that by strategically placing stiffeners the central point displacements of the

plates could be significantly reduced. Similar study was repeated by Tavakoli and

Kiakojouri [30] with another seven configurations, it was found out that with the same

number of stiffeners, intersected stiffening layouts produced less midpoint displacements

than parallel arrangements, and it was more advantageous to place more stiffeners close to

the panel centre at an even spacing. Chen and Hao [31] developed a new design using

multi-arch double-layered panels and carried out numerical parametric studies on the

influences of various stiffener configurations, boundary conditions, stiffener dimensions,

strain rate sensitivities and blast intensities of the new design performances. It was also

demonstrated that the strategic arrangement of stiffeners with appropriate boundary

conditions could maximise the reduction of dynamic response of the panels to blast

loadings.

Profiled (Corrugated) Panels

A profiled panel is a plate folded in certain profiles (typically trapezoidal profile) or

corrugated shapes to increase its second moment of inertia and thus bending capacity. The

advantages such as, easy fabrication, low costs, considerable ductile ability and good

corrosion and fire resistance properties, have made profile/corrugated panels very popular



2-19

choices for blast wall in the offshore industry. A typical profiled blast wall is shown in

Figure 2-6.

Figure 2-6. Profiled panel blast wall in Belanak FPSO

Substantial studies on the dynamic structural performances of profiled blast wall panels

have been conducted as well. Louca et al. [32] performed numerical comparison study

between three different profiled sections (shallow, medium and deep) made up of stainless

steel SS2205 and SS316 to identify their distinguished blast responses against those

predicted by FABIG TN5 [1]. Similar to the findings in stiffened panels, good correlations

were observed up to the elastic limit, but due to the extensive inelastic deformation, it was

concluded that FABIG TN5 [1] was not reliable after stresses fell into plastic region. This

research also suggested that the guidance in FABIG TN5 [1] might be too conservative,

because considerable reserve capacity before instability was obtained for a slender section,

while plastic response was not permit for such section in accordance with FABIG TN5 [1].

The considerable advantages in the use of SS316 material for blast wall design, such as,

better energy absorption capability, prevention of the sudden onset of instability, were also

emphasised in this study. Similarly, Boh et al. [33] compared the structural responses

among the three section (shallow, medium and deep) themselves rather than to FABIG

TN5 [1] predictions. Results revealed that the performance of the intermediate section

outweighed the other two sections in ductile deformation as no instability was observed up

to the overpressures investigated in the research. Additionally, recommendations and

guidelines of using finite element method for design and analysis of corrugated blast panels

were laid down in this study. In order to understand the relationship between panel

dimension and energy absorption, Faruqi et al. [34] performed numerical parametric studies

on the various elements (e.g., compression flange, depth, angle of corrugation, etc.) of a

corrugated blast wall. It was found out that a ratio of the compression flange to the wall



2-20

thickness between 16 and 18 was optimal in producing an efficient static ductility ratio. If

deflection were not limited, the ratio of the span length to panel depth was optimised to be

kept around 17 for good static ductility ratios.

For studies presented above, the barrier responses were isolated from the influences of

connections for illustrative purposes. In engineering practice, the detailed modelling of the

connections is necessary and corresponding studies have been carried out as well. Boh et al.

[35] conducted numerical assessments to study the weld tear out failure along with the

strain rate effects. In this investigation, continuous weld failure was governed by the

equivalent plastic strain, while interaction equation between shear and tensile forces was

taken as the failure criteria for spot welds. Results indicated that the enhancement due to

strain rates on the deflection of the panel were effective before weld failure, after which the

enhancements effects might be affected by different weld failure criteria and the extent of

weld failure was highly dependent on the rate effects.

The flexibility of the end plates also plays an important role in the dynamic behaviour of

blast wall. A series of field tests were carried out by Schleyer et al. [36] using 1/4 scale

stainless steel profile panels to show the panel responses on the reflected blast loading and

investigate the influence of the connection detail on the overall performance of the panel.

Three sizes of panel with different flexible angle end plate lengths (60, 120, and 180 mm)

were tested by two level of pressures (low and high) individually. It was found out that the

support restraint decreased as the angle plate length increased, and larger displacements

were produced by the higher flexibility supports as they were bended more easily. For very

high blast loadings, the panels adopted a membrane-type mode in which high in-plane

forces could be generated in the supports and thus increased the risk of weld rupture

failure. However, the increased flexibility of the panels with longer angle connections

delayed the onset of membrane behaviour. Other than experimental method, Langdon and

Schleyer [37-39] systematically studied the deformation patterns and influences of

boundary conditions of profiled stainless steel blast wall by employing not only

experimental but also analytical and numerical methods. Results were cross-compared and

good correlations were obtained, which built a milestone in the blast wall research. The

findings of these three approaches revealed that membrane actions would increase the

design capacity but would also jeopardise the weld integrity.



2-21

Review of Energy Absorption Design

Principles of Energy Absorption Design

Based on structural global behaviours, engineering structures can be generally categorised

into two groups: strong and soft structural forms [6]. Strong structures rely on high

strength and stiffness to resist imposed loads, such as concrete blast walls. These structures

are usually heavy and bulky, their responses to blast loads are normally in the elastic range

with little damage and deformation, however, large amount of blast energy will be

transferred to supports and primary structural components [6]. Soft structures adopt an

opposite approach known as energy absorption, in which most of the blast energy is

dissipated in form of large geometry changes, material strain hardening effects, strain-rate

effects and various interactions between different deformation modes such as bending and

stretching [40].

The book “Energy absorption of structures and materials” written by Lu and Yu [40] has

systematically summarised the recent research outcomes such as energy dissipation by axial

crushing, lateral crushing and ductile tearing of thin-walled tubes, mechanical properties of

cellular materials (foam and honeycomb) and energy absorbing capacity of composite

materials. In conclusion, Lu and Yu [40] give some general principles for energy absorption

design that are listed and explained below:

1. Irreversible energy conversion, which means the structures should be able to

convert most of the input kinetic energy into irreversible plastic strain energy.

2. Restricted and constant reactive force, which requires the energy absorbers’

reaction forces remain constant during plastic deformation to minimise the impact

on major structural components.

3. Long stroke, which requires the plastic deformation stage should be sufficiently

long to maximise the energy absorption.

4. Stable and repeatable deformation mode.

5. Light weight and high specific energy-absorption capacity.

6. Low cost and easy installation.

Energy Absorption Designs

The traditional blast walls such as stiffened and profiled panels mentioned above are

monolithic structures that the input energy is only absorbed by the deformation of blast

wall panel itself. However, given the circumstance that an offshore platform is a very



2-22

congested construction that are full of pipelines and equipment, large plastic deformations

tend to increase the probability of clash with adjacent equipment or structural components,

which might escalate the blast event. While adopting deeper or larger sections for design

will make the connection at supports more vulnerable for rupture which is a more

catastrophic failure. Therefore, intensive research efforts have been spent on developing

designs with additional energy absorbing materials or devices to achieve code compliance.

Various studies on energy absorbing systems have been reported below.

(a) Sandwich panel

Sandwich panels are efficient energy absorption designs, a typical sandwich panel is made

up by two face sheets and various kinds of energy absorption cores, which have been

summarised by Zhu et al. [41] and shown in Figure 2-7. Investigations on blast resistant

performances of sandwich panels are numerous.

Figure 2-7. Sandwich panel core topologies [41]

Aluminium foam can react very quickly to impulse or blast loadings, which makes it a

perfect material for sandwich panel core. Jing et al. [42] performed numerical investigations

on the dynamic response and energy absorption capability of cylindrical sandwich shells

filled with closed-cell aluminium foam cores. After validating the numerically predicted

deflections and failure modes against experiments, the findings indicated that the thickness

of back face-sheet had a greater contribution than the front face-sheet in blast mitigation. It

was also concluded that reducing foam core relative density and panel curvature radius

could enhance the energy absorption capability of sandwich shells. Ma and Ye also studied



2-23

the energy absorption capacity of aluminium foam for blast alleviation. One of their work

[44] employed the shock wave propagation theory to derive the foam deformations under

blast loadings. Two non-dimensional parameters of the foam cladding were introduced to

express the maximum deflection of the protected structure. With that, the foam cladding

could be appropriately designed to achieve structural retrofit against blast loads and

meanwhile the maximum allowable blast load for the structure under the protection of a

particular foam cladding could be predicted. Another study [43] also adopted similar

analytical model with the assumption of a rigid-perfectly plastic-locking foam model. The

responses of different configurations of the layered foam cladding were calculated with the

model. Numerical simulations were carried out to justify the results and good agreements

were found between the analytical and the numerical results, which validated the analytical

solution as an efficient tool for quick assessment. Apart from numerical and theoretical

approaches, experimental measures have also been used to study the dynamic behaviours

of sandwich panels with metallic foam core. Zhu et al. [45] carried out an experimental

investigation into the responses of square sandwich panels with aluminium foam cores

under blast loading, the front sheet showed inwardly curved dishing deformation and the

core exhibited a progressive crushing damage rather than debonding at interfaces. The

following FE analysis repeated this failure pattern (see Figure 2-8) and thus numerical

parametric studies on the impulse level, face sheet thickness, foam core density and

thickness were carried out subsequently. It was concluded that the energy absorption

capacities were improved with thicker face sheet and increasing core density, in addition,

the back sheet deflection would reduce with thicker foam cores. Nemat-Nasser et al. [46]

performed two series of experiments to investigate both the mechanical characteristics of

aluminium foam under impact loads and the dynamic responses of sandwich panels filled

with aluminium foam core, it was reported that the foam core was compressed very fast

from the front face and showed good energy absorption ability.

Figure 2-8. Comparison of foam core failure in simulation and experiment [45]

Metallic cores are also excellent energy absorbers, their applications in sandwich panels for

blast mitigations also attract researchers’ attentions. Theobald and Nurick [47] carried out



2-24

experimental and numerical analyses using square tubes as sandwich cores. Two materials

(aluminium and mild steel) and three layouts (four, five and nine tubes placed axially

between face sheets) were firstly tested by using a ballistic pendulum, and followed by a

numerical parametric study (see Figure 2-9). It was observed that most of the input energy

was dissipated through the progressive buckling of the tubes and little deformation was

noticed for the top face sheets. In addition, aluminium tubes experienced more crushing

distance than mild steel tubes under the same amount of input energy. The FE parametric

study evaluated the effect of load uniformity and concluded that the energy absorptions

were sensitive to load uniformity for different layouts. Similar study also been conducted

by Xiang et al. [48] with the difference of placing tubes laterally as sandwich cores (see

Figure 2-9). In their study, three types of cores were tested, namely, closely arranged

identical circular tubes, spaced circular tubes and spaced square tubes. Different masses of

TNT with various standoff distances were applied to the sandwich beams. FE simulations

were also performed afterwards for comparison, and the results suggested that the global

bending of the sandwich beam started after the lateral crushing of the tube cores, and thus

over half of the input energy was consumed by tube cores and the rest was undertaken by

beam global bending, which suggested a good energy absorbing potential. Sandwich panels

with a combination of foam and metallic cores were investigated by Yazici et al. [49].

Experiments using shock tubes were performed for sandwich panels consist of five layers

of corrugated sheets and eight foam filling configurations. After cross-validation results

between experiments and FE simulations, another 24 configurations were studied

numerically. It was concluded that soft/hard arrangement (front to back) was the most

effective for blast resistivity as it produced the smallest back face sheet deflection. Results

also implied that increasing the number of foam layers would reduce the deflection of front

face sheet but had little effect on the back face sheet as it was governed by the panel overall

responses.

(a) Theobald and Nurick [47] study (b) Xiang et al. [48] study

Figure 2-9. Sandwich panels with metallic tubes as cores



2-25

Compared to metallic foam cores that rely on core plastic deformations, such as axial and

lateral crushing, to dissipate energy, friction damper is a relatively new concept that it has

the privilege of high energy absorbing capability over a relatively small core displacement,

which helps to maintain the panel’s operational function. Chen and Hao [50] developed a

design using a modified friction damper device with a spring (RFHDS) (see Figure 2-10) as

the core for sandwich panels to mitigate blast loading effects. Numerical simulations were

applied to investigate the feasibility of the new design. The mechanical property (force-

displacement curve) of the RFHDS was determined initially with a local FE model and

then applied to sandwich panel as a nonlinear spring. Findings were compared to the plain

panel and demonstrated that the majority of the blast energy was absorbed by the friction

dampers so that significant reduction in the panel peak and permanent deformations

together with the reaction forces were achieved. Since the friction damper device was

equipped with a spring, panel could also be pushed back to its original position somehow.

Figure 2-10. Friction damper with a spring and its force-displacement curve [50]

Although the majority of the explosion hazards are caused by blast overpressures, the

damages due to projectiles shall not be overlook either, as they can easily penetrate thin-

walled blast walls and cause casualties. Blast wall design against projectile penetrations is

also an attractive topic to researchers. Wadley et al. [51] conducted experiments to studied

the mechanisms of projectile penetration of two aluminium triangular corrugated core

sandwich panels, one was empty filled the other was alumina filled. The failure mode of the

empty filled panel was observed to be a complete penetration (termed shear-off or

plugging), which was ineffective at dissipating the projectiles kinetic energy. On the

contrast, the panel’s penetration resistance was significantly increased with inserted alumina.

The presence of alumina prevented shear-off mechanism and the failure mode was

changed to be severe fragmentation of the projectile and fracture of the ceramic.



2-26

Furthermore, pairs of parallel cracks were noticed in the rear back sheet at core web-sheet

nodes. A comparison of these two failure modes are shown in Figure 2-11. The subsequent

study by Wadley et al. [52] continued this design philosophy but extended the experiments

to a series of corrugated cores, such as triangular, trapezoidal or rectangular cross-sections,

with their voids filled by alumina. The projectiles were hard steel spheres in the velocity

range from 570 to 1800m/s. The penetration mechanisms were observed through X-ray

tomography and high-speed cameras. The findings suggested that the projectile exit

velocity was significantly reduced when replacing triangular prisms with trapezoidal prisms.

Additionally, it was also found that decreasing the number of web members and hence

increasing the width of the inserted alumina would result in higher probability of projectile

defeat due to higher contact area and thus higher contact pressure. However, the back

sheet plastic deflections were noticed to increase due to less constraints from the core webs.

Figure 2-11. Failure modes of empty and alumina filled sandwich panels [51]

(b) Other structural forms

Apart from sandwich panels, other energy absorbing designs, such as installing passive

impact barrier, adoption of reinforcements and utilising flexible supports, have also been

developed and studied by researchers.

Boh et al. [53] proposed a method to increase the energy absorption capacity by

introducing passive impact barriers, shown in Figure 2-12. The research illustrated a

technique where the response of blast walls was modified by the inclusion of a passive

barrier system at a certain offset distance from the wall. It was observed that with the



2-27

diagonal impact barriers, the capacities of the blast walls were greatly enhanced, which had

in term increased the overall robustness of the blast wall and reduced the possibility of an

escalation of event. The impact barriers can also delay the tearing of the horizontal welds.

However, one major drawback of this design scheme is that more space may be required

for installation of the barriers.

Figure 2-12. Passive impact barrier behind a blast wall [53]

Nwankwo et al. [54] developed both analytical and numerical models to demonstrate the

effectiveness of a strengthening technique, which involved placing carbon fibre reinforced

plastic (CFRP) patches in the central region of the panel. The findings revealed that energy

absorption capacity of the strengthened wall was higher than that of its un-strengthened

counterpart, and both mid-span displacements and strains at the connections were reduced.

Moreover, the proposed analytical method enabled the users to quickly assess the effect of

adding a composite patch.

Soft support is another possible way to absorb blast energy. The double-layered plate with

spring connectors combined with soft support has been investigated both experimentally

and numerically by Hao [55]. The support forms included spring and rubber supports of

different stiffness. The numerical models calibrated by laboratory data were used to

simulate responses of these supports against blast and impact loads. It was found that the

structure with soft-supports performed well in mitigating impact loading effects. Du et al.

[56] also performed numerical simulations of a beam with two spring supports to study the

effectiveness of soft support on blast energy absorption for structure protection, and

defined the optimal supporting stiffness with respect to the blast loading duration. It was

also concluded that soft support could effectively mitigate blast loading effect on structures.



2-28

Proposed Design Scheme

Traditional design of blast walls to resist large impact and blast load tends to increase the

structural strength and stiffness by strengthening or thickening, which will inevitably make

the structure heavy and bulky. As a result, it increases not only the construction and

material cost, but also the maintenance cost and operational efficiency. This design concept

is extremely not compatible in offshore or marine circumstances as weight is important

factor in design. Therefore, the focus of this study shifts from blast wall to its supports,

focusing on developing novel support configuration with energy absorption concept to

improve the blast resistant capacity of blast wall.

As mentioned before, in traditional design, blast wall connections consist of welded end

plates that are extended from the supporting plate girders, shown in Figure 2-13. This

design may be adequate for weak to medium blasts with overpressure less than 1 bar as the

end plates have certain degree of energy absorption capacity. However, recent large scale

explosion tests confirmed that the blast wall overpressure can be as high as 4 bar for a

typical offshore topside module [5]. Under such conditions, the traditional design

configuration is no longer sufficient because blast wall will undergo large plastic

deformation that develops membrane actions.

Under high blast loading, the connections are subjected to translational shear forces and in-

plane membrane (axial) forces. Connections shall be designed to provide high rotational

ductility for blast wall to deform and absorb energy, which makes the connections

extremely vulnerable for rupture. Local strengthening with gusset plates may be required

for strong blasts (e.g. 2 bar to 4 bar), shown in Figure 2-13. Although this strengthened

connection configuration can significantly reduce the blast wall deflections, it will also

attract larger membrane forces at welded connections, where hot spots of high stress may

occur and propagate progressively, leading to a shift of failure mode from ductile bending

failure to brittle shear or tensile rupture at supports. These behaviours coincide with the

monolithic structural failure mode II (tensile tearing) and mode III (shear failure) at

supports defined by Menkes and Opat [57]. Therefore, it is relatively difficult and tedious

to achieve a balanced blast wall design between wall deflections and weld tearing out

resistance.

The present study proposes a novel design concept of offshore blast wall by using energy

absorption devices at the supports, aiming to limit blast wall deflection and minimise weld



2-29

tearing out risk simultaneously. In the proposed design, instead of end plates, energy

absorption devices will be installed between the supporting plate girders and the blast wall

panels to form a flexible support, shown in Figure 2-13. The devices can be welded to both

girders and panels at site. It is comprised of three parts: sliding core, roller core and outer

crust that all made of steel, the details are illustrated in Figure 2-13. There are voids or gaps

between these three components hence it has certain degree of freedom to slide and rotate

at the same time, which is specially designed to avoid mode II and mode III failures [57].

Similar devices are widely used at bridge supports to mitigate earthquake or thermal loads.

Certain modifications are made to suit the energy absorption purpose, it is designed to fill

the sliding voids with polymethacrylimide foam (about 50-100mm) and restrain the roller

core with linear springs. Thereby, the sliding core of the support together with the blast

wall panel can move back in a certain distance (but not beyond the edge of the plate girder)

to avoid peak pressure and meanwhile blast energy can be dissipated through foam

deformations. Another advantage of using polymethacrylimide foam is that its crushing

plateau over a certain distance can act as a cushion to prevent sudden and rapid increase of

reaction shear forces, which will reduce the likelihood of weld shear failure. The linear

springs around roller core allow certain rotations to release high stresses, which helps to

minimise the risk of weld tensile tearing out. In addition, the linear spring rollers provide

substantial resistance to prevent large support rotation especially during strong blasts,

which can further reduce the panel deflection and thus avoid possible event escalation due

to clash.

In the following chapters, both analytical and numerical models using corrugated blast wall

as example are developed to demonstrate the feasibility and effectiveness of this energy

absorption blast wall design concept.



2-30

(a) Blast wall top view

(b) Blast wall side view



2-31

(c) Traditional design support configuration

(d) Energy absorption device of the proposed design

Figure 2-13. Comparison of traditional design and the proposed design



3-1

Analytical Modelling

Overview of Analytical Model

The theoretical model of a reinforced concrete beam under blast loading with linear springs

and damping at supports has been derived by Song et al. [58] and Chen et al. [59]. Song et

al. [58] focused on the effects of different blast pressure profiles and discovered that the

beam displacement under rectangular pulse was always larger than triangular pulse under

the same support constraints and peak pressure loads. Chen et al. [59] concentrated on

studying the beam failure modes with different support stiffness and found out that with

increasing support stiffness, beam failure modes transferred from flexural failures to shear

failures. Furthermore, both studies have concluded that with relatively small support

stiffness the beam peak responses (i.e. deflection, moment and shear force) were delayed

even to the free vibration phase. However, there are two major drawbacks in using linear

supports. On the one hand, their results suggested that peak responses were delayed rather

than reduced, as energy was stored rather than absorbed. On the other hand, to delay peak

responses, the stiffness of the linear soft support needed to be small, which would result in

large support displacements that are not applicable in engineering practice.

In the proposed design, polymethacrylimide foam is selected as the sliding support filling

material because of its excellent energy absorption capability. Its typical compressive

behaviour is illustrated in Figure 3-1. As reviewed above, the mechanics and energy

absorbing ability of foams have been well studied and proven efficient for blast mitigation.

In addition, the good corrosion and fire resistant abilities of polymethacrylimide foam also

make it perfect for blast mitigation in offshore environment. However, the majority of

applications of cellular foam are as cores of sandwich panels [45] or sandwich beams [71],



3-2

there is limited study on the energy absorption effect by placing cellular foams at blast

panel supports.

The present study further advances the theoretical model of linear spring supports to

account for the elastic, crushing and unloading behaviours of polymethacrylimide foam.

Additional efforts are also made to incorporate the material nonlinearity, such as strain

hardening of steel after plastic hinge formation, so that both the maximum and permanent

(ultimate) deflections can be predicted by this improved model. It should be noted that the

friction effect is considered minimal and hence neglected in this study.

A general view of the analytical model for blast wall with flexible supports is shown in

Figure 3-1. The corrugated blast wall panel is simplified as a beam model, each of the

support is modelled with two sets of springs (translational and rotational). The beam length

is denoted as L and unit length mass as m. Kt is the stiffness of the translational spring, Kr

is the stiffness of the rotational spring and ms is the mass of each support.

Figure 3-1. Overview of the analytical model



3-3

Beam Elastic Stage

In elastic stage, the beam deflection can be derived with the simultaneous beam vibration

governing equation and the equation of motion. The complete derivation details are

described below.

The blast loading is defined in Eq.(3-1), where p0 is the peak blast loading magnitude, F(t)

is the blast loading time characteristic function, f(x) is blast loading distribution function. t

is time starting from the arrival of the blast load on the beam and x is the coordinate

originating from the left support shown in Figure 3-1.

)()(),( 0 xftFptxp ⋅⋅= (3-1)

For simplicity of derivation, it is assumed that both supports move spontaneously with the

same displacements. The beam total displacement Y(x,t) is the summation of support

displacement u(t) and beam deflection y(x,t) as in Eq.(3-2), where Φi(x) is the ith mode shape

and Ti(t) is its corresponding scalar in generalised space.

∑∞

=

⋅+=+=1

)()()(),()(),(i

ii tTxtutxytutxY φ (3-2)

The support displacement u(t) is the rigid body motion mode of the beam and can be

considered as the zeroth mode of the beam deflection. Under blast loading, the governing

equation of beam vibration in its translational direction can be expressed as:

),(2

2

4

4

txpt

Ym

x

YEI =

∂

∂+

∂

∂ (3-3)

where EI is the bending stiffness of the beam.

Substituting Eq. (1) and Eq. (2) into Eq. (3) and simplifying with mode shape orthogonality

yields:

)(,0)(),()()()(

)()(

01 1

2

2

2

2

4

4

jidxxtxpxdt

tTdm

dt

tudmtT

dx

xdEI j

l

i i

i

i

i

i ≠=⋅

−++∫ ∑ ∑

∞

=

∞

=

φφφ

(3-4)

The governing equation of beam vibration then takes the following form:

∫∫∫ ∫ =++⋅l

i

l

i

l l

iiii dxxxftFpdxxtTmdxxtumdxxxftT0

00

2

0 0)()()()()()()()()()( φφφφ &&&& (3-5)



3-4

A typical vapour cloud explosion in oil and gas industry a spherical blast wave front with

small curvature. Thus, it is considered to be uniformly distributed over the beam span (i.e.

f(x) =1). Galerkin method is adopted to simplify Eq.(3-5):

)()()()( 0

222tFptumtTtT iiiii ⋅⋅=⋅⋅+⋅+ ωωω &&&&

(3-6)

∫

∫ ⋅=

l

i

l

i

i

dxxm

dxxxf

0

2

02

)(

)()(

φ

φω (3-7)

where ωi is the ith natural frequency of mode shape Φi(x).

According to Biggs [14], rather than summation of an adequate number of characteristic

vibration modes, the deflection of a beam subjected to uniformly distributed blast load can

be represented by its deflection shape under unit static uniformly distributed load with

sufficient accuracy:

lxl

x

l

x

l

xx ≤≤

+−= 0,

2

5

16)(

3

3

4

4

φ (3-8)

The blast overpressures are often idealised as triangular with different rise time, shown in

Figure 3-2. Type 1 is caused by high explosives detonation or far-field loading from a

hydrocarbon explosion while Type 2 is due to ignition of flammable vapour clouds, whose

rising time is normally taken as half of the total blast duration in engineering practice.

Negative phase is normally neglected in engineering design. As stated above, under certain

condition, Type 2 blast (deflagration) could be transformed to Type 1 blast (detonation).

This is because after a gas explosion that generates significant overpressures, a pressure

pulse will propagate into the surrounding atmosphere. The convection in the flow will tend

to make pressure disturbances at the back of the pulse catch up with those at the leading

edge of the pulse. This has the effect of decreasing the duration of the positive part of the

pulse and steepens the leading edge. A blast wave with near zero rise time will then develop.

This wave is normally referred as far-field effect. The blast loading profiles of Type 1 and

Type 2 can be denoted as F1(t) and F2(t) respectively and given below:

)(1 tF = ,0

/1 dtt− )(

)0(

d

d

tt

tt

>

≤≤ (3-9)



3-5

)(2 tF =��

,0

,/1

,/

d

d

tt

tt

− )(

)5.0(

)5.00(

d

dd

d

tt

ttt

tt

>

≤<

≤≤

(3-10)

Figure 3-2. Two idealised blast pressure profiles [6]

With free body diagram shown in Figure 3-3, the equation of motion of the left support is:

)(),0( tumRtV s&&⋅=− (3-11)

where V(0,t) is the shear force at the left end of the beam and it takes the form of:

)(5

192),(),0(

3

0

3

3

tTEIlx

txyEItV

x

⋅=∂

∂−=

=

(3-12)

R is the reaction force from the translational spring. Rather than using linear spring [58-59],

a tri-linear model is adopted to approximate the compressive behaviour of

polymethacrylimide foam. The support reactions at foam elastic, crushing and unloading

phases are denoted in Figure 3-3 as R1, R2 and R3 respectively, with analytical expressions:

uKR t ⋅= 11 (3-13)

elttt uKKuKR ⋅−+⋅= )( 2122

(3-14)

)()( max2113 elttt uuKKuKR −⋅−−⋅= (3-15)



3-6

where uel is the foam elastic limit, uult is the ultimate crushing limit when foam is fully

compressed (i.e. densification), umax is the maximum displacement when foam stops and

starts unloading (i.e. velocity u� �t�=0). Since the ideal energy absorption zone is located in

the crushing plateau, the behaviour after densification is not considered in this study.

Reaction (i.e. R1, R2, R3) shall be substituted in Eq.(3-11) along with the foam compression

phases.

(a) Free body diagram at the left support; (b) Foam reaction curve at different phases.

Figure 3-3. Beam left support free body diagram and foam reaction curve

Combining Eq.(3-6) and Eq.(3-11) gives the forced vibration equations as in Eq.(3-16):

��

0)(5

192)(

)1()()()(

3

0

222

=⋅−+⋅

−⋅=⋅+⋅+

tTEIl

Rtum

t

tptumtTtT

s

d

&&

&&&& ωωω

(3-16)

Free vibration is a special case of the forced vibration when p0 is zero. This is a linear

nonhomogeneous second order differential system of equations, which can be solved by

the available commercial mathematical software, such as MATHEMATICA. The at-rest

initial conditions for this system of equations can be described as:



3-7

0)0(,0)0(,0)0(,0)0( ==== TTuu &&

The beam mid-span deflection in elastic stage can be calculated:

)()2

( tTl

y el ⋅= φ (3-17)

Beam Plastic Stage

When at time t = τ, beam mid-span bending moment M(0.5l, τ) reaches the section plastic

moment Mp of the profiled blast wall panel, beam enters plastic stage, thus:

pM

x

lyEIlM =

∂

∂−=

2

2),5.0(

),5.0(τ

τ (3-18)

The formulation for beam plastic stage is based on virtual work theory under two

assumptions:

1) Plastic hinge length is assumed to be constant throughout the blast;

2) The support rotation θ has rigid-plastic characteristics after plastic hinge formation.

Assume at time t = τ (τ < td), beam forms plastic hinge under blast loading. According to

virtual work theory, as shown in Figure 3-4, summation of work of the internal and

external forces done by virtual displacements δθ and δu is zero:

0)(),(5.0

0

5.0

0=⋅⋅−⋅−⋅⋅+⋅−⋅⋅ ∫∫ δθθδθδθθδθ

l

rmid

l

KMxdxuxmxdxtxp &&&& (3-19)

0)()(),(5.0

0

5.0

0=⋅+−⋅⋅+⋅−⋅⋅ ∫∫

l

s

l

uRumdxuuxmdxutxp δδθδ &&&&&& (3-20)

Figure 3-4. Application of virtual work theory

The system of equations of beam plastic stage Eq.(3-19) and Eq.(3-20) for forced vibration

phase (τ < t < td) can then be simplified as:



3-8

��

)1(2

1)(

8

1)()

2

1(

)1(8

1)(

8

1)(

24

1

0

2

2

0

23

d

s

mid

d

r

t

tlptmlRtumml

Mt

tlptumlKtml

−=⋅++⋅+

−−=⋅+⋅+⋅

θ

θθ

&&&&

&&&&

(3-21)

For free vibration phase (t ≥ td), p0 is zero. Due to continuity, the initial conditions of

plastic stage equal to the terminal conditions of elastic stage at time t = τ.

)(,0),(),( 0000 τθθθττ &&&& ==== uuuu

The initial rotational velocity at supports can be determined by the momentum

conservation law between the end of elastic stage and the onset of plastic stage:

∫∫ ⋅=⋅ll

xdxmdxxym5.0

00

02),( θτ &&

)(30

3

0

0τθ T

EI

lp&& = (3-22)

Mmid is the plastic bending moment at the mid-span of the beam. Since stainless steel has

high ductility, the material strain hardening effect shall also be considered in formulation.

Therefore, rather than elastic-perfectly-plastic moment profile, the additional moment

carried by the corrugated blast wall panel due to strain hardening is modelled by using a

moment hardening parameter Kθ. Mmid1 and Mmid2 represent the mid-span bending moment

hardening and unloading behaviours.

Kθ was first introduced by Langdon and Schleyer [38] as an attempt to capture the buckling

process of the beam. The procedure that involving solving complex equations was difficult

for theoretical calculation, furthermore, there was no moment unloading curve after peak

rotation defined, leading to the predicted beam post-peak responses not agreeable with

numerical results [54].

The present study simplifies the procedure and assigns a moment unloading path shown in

Figure 3-5. Thereby, both maximum and permanent deflections can be predicted by this

analytical model. The formula for Kθ is derived from bending moment formula shown in

Eq.(3-23):

θκσ θ ⋅+=⋅⋅+= ∫ KMdAzzEM phymid )(1

(3-23)



3-9

where σy is the material yield stress (0.2% proof stress for stainless steel); Eh is the material

hardening modulus; z is the distance to equal area axis; κ is the curvature at mid-span of

the beam.

Figure 3-5. Mid-span plastic bending moment curve

As the beam translational deflection increases, the in-plane membrane force tends to

increase with the shedding of plastic bending moment at mid-span. Reflecting on the

plastic boundary surface (full yield surface) shown in Figure 3-6, the state of internal forces

at the beam mid-span tends to shift from location A to location B along the boundary

surface curve. Hence, the plastic hinge at mid-span will develop further due to strain

hardening.

According to Nonaka [60], the plastic hinge was assumed to have a length equalled to the

thickness of the plate at the onset of plastic hinge formation (i.e. M=Mp, N=0) and further

developed to two times of the plate thickness when a beam reached fully membrane state

(i.e. M=0, N=Np), afterwards, the beam behaved as a string. Jones [61] used the average

values of the two extremes as a simplified approach to derive the threshold impulse for a

beam rupture due to excessive tensile strain and compared favourably with experimental

results.



3-10

Figure 3-6. Beam plastic boundary surface and mid-span plastic hinge length

This study follows this simplified approach and thus assumes plastic hinge length to be 1.5

times of the plate thickness throughout the blast duration. However, this assumption is

made for rectangular sections. Considering the section has already formed plastic hinge at

this stage, equivalent thickness teq is computed by equating the plastic section modulus Zp

of the corrugated section to a same width W rectangular section with the same magnitude.

An example is given below and shown in Figure 3-7 (Zp is given in Table 3-1).

,4

2

eq

p

tWZ

⋅= .7.46,640 mmtmmW eq ==

Figure 3-7. Corrugated blast wall panel section



3-11

According to Jones [61], κ can be expressed as Eq.(3-24), in which Lh is the plastic hinge

length at mid-span. Kθ can be obtained by substituting Eq.(3-24) into Eq.(3-23).

eqh tL 5.1

θθκ == (3-24)

h

h

L

IEK

⋅=θ (3-25)

The slope of the moment unloading curve is assumed to be close to the elastic stiffness of

the beam. In this study, 20Kθ is adopted for the following calculations and hence the

moment unloading process is deduced to be:

)(20maxmax2

θθθ θθ −⋅⋅−⋅+= KKMM pmid (3-26)

where θmax is the maximum rotation at the support (i.e. angular velocity θ� �t�=0).

Eq.(3-21) can be solved in a similar manner as Eq.(3-16), accounting for varying

expressions of R and Mmid in different phases. With the continuity condition in-between the

different phases, the maximum mid-span deflection of the beam is calculated to be:

2)()

2( maxmax

lT

ly ⋅+⋅= θτφ (3-27)

Case Study and Result Discussion

Example Description

An example using a single strip of the corrugated blast wall panel (highlighted in Figure 3-8)

is presented to demonstrate the effectiveness of the proposed energy absorption design

concept.

The panel section dimensions are given in Figure 3-7, according to FABIG TN5 [1] and

Eurocode 3 [62], the selected panel section is classified as Class 1 plastic section, in which

plastic hinge can form without premature local bucking. It also has sufficient rotational

capacity to allow redistribution of bending moments in the structure.



3-12

Figure 3-8. A single strip of corrugated blast wall panel

AISI316L stainless steel is adopted due to its good energy dissipation ability. The strain rate

effect is taken into account by using the Cowper-Symonds relationship:

q

y

dy

D

/1

1

+=

ε

σ

σ & (3-28)

where σdy is the dynamic yield stress at plastic strain rate ε�; σy is the static yield stress. The

material constants D and q are coefficients for different materials. According to FABIG

TN6 [63], D=240s-1 and q=4.74 are taken for AISI316L. The plastic strain rate ε� is taken as

0.422s-1 [64]. Table 3-1 summarises the detailed section and material properties of the blast

wall panel.

Table 3-1. Blast wall panel section and material properties

Section Property Values &Units

Material Property Values &

Units

Beam length L 3m Young’s modulus E 200GPa

Beam unit weight m 39.2kg/m Density ρ 7850kg/m3

Support weight ms 10kg Static yield stress σy 250MPa

Cross section area A 4994mm2 Dynamic yield stress σdy 316MPa

Moment of Inertia I 2.9×107mm4 Hardening modulus Eh 2.35GPa

Plastic section modules Zp 3.5×105mm3

For simplicity and demonstration purpose, blast pressure profile Type 1 in Figure 3-2 is

adopted in the present calculations. The blast duration td is taken as 50ms, which is a typical



3-13

medium duration for hydrocarbon explosion. Five blast overpressures, i.e. 1 bar, 1.5 bar, 2

bar, 2.5 bar and 3 bar (1bar=100kPa) are used to test the performances of the proposed

design. Generally, 1 bar is a typical overpressure for a SLB event, 3 bar can be taken as

DLB event overpressure.

Correspondingly, five pieces of foam are used to resist these overpressures respectively. It

is designed to place foams with 62.5mm in height, 100mm in depth along the direction of

blast incidence, the width is 640mm same as the width of the panel strip.

Polymethacrylimide foam is a closed cell foam that the elastic unloading is typically taken at

a compressive strain of 5% as per Arezoo et al. [65], the other foam mechanical properties

are computed according to Gibson and Ashby [66] and listed in Table 3-2. The stiffness at

foam crushing plateau is assumed as 1/100 of the stiffness in foam elastic stage. The linear

springs around the rotation core forms a push-pull mechanism, which generates rotational

stiffness. The rotational stiffness for these five overpressures are constant to be

Kr=400kNm/rad. A detailed description of the properties of translational and rotational

springs is shown in Table 3-3.

Table 3-2. Polymethacrylimide foam mechanical properties

Foam name Relative density

Plastic collapse stress (MPa)

Elastic unloading

strain

Compressive Young’s modulus

(MPa)

Densification strain

Foam for 1 bar 10% 2.50 5% 50.0 86% Foam for 1.5 bar 13% 3.13 5% 62.6 82% Foam for 2 bar 16% 4.06 5% 81.2 78% Foam for 2.5 bar 19% 5.00 5% 100.0 73% Foam for 3 bar 22% 6.25 5% 125.0 69%

Table 3-3. Support translational and rotational spring properties

Spring name Translational spring Kt Rotational spring

Kt1 (kN/m) Kt2(kN/m) uel (mm) uult (mm) Kr (kNm/rad)

Spring for 1 bar 20000 200 5 86 400

Spring for 1.5 bar 25000 250 5 82 400

Spring for 2 bar 32500 325 5 78 400

Spring for 2.5 bar 40000 400 5 73 400

Spring for 3 bar 50000 500 5 69 400



3-14

Results and Discussions

In preliminary design stage, the primary concern is to determine the blast wall deflection so

that the blast wall layout and section can be decided. In this study, the dynamic structural

responses of the proposed design are compared with their counterparts of traditional

design to demonstrate its advantages. In addition, the results computed from linear springs

are compared to nonlinear springs to illustrate the differences.

(a) Comparison between linear and nonlinear support springs

In this section, the mid-span deflection time-histories and support displacements computed

from linear support springs and nonlinear support springs (i.e. foam) are plotted for

comparison. Translational and rotational springs are assessed individually with the other

one set to zero. In order to capture the beam responses in both forced and free vibration

phases, analyses are performed for a total duration of 2td=0.1s (i.e. one td of blast followed

by another td of free vibration) for cases. All calculations are performed with the assistance

of the commercial mathematical software MATHEMATICA 10.

For translational support spring, the nonlinear spring for 2 bar case listed in Table 3-3 is

adopted with rotational stiffness Kr=0. The same elastic stiffness Kt1 is taken as its

counterpart, no plastic deformation and stiffness are required for linear support spring. The

support details are summarised in Table 3-4. The overpressure for this test is 2 bar. Figure

3-9 shows the effects of nonlinear springs over linear spring. The maximum panel

deflection under linear translational support spring is 120.2mm and it is almost the same as

the deflection calculated from pinned boundary condition (see Table 3-6), which implies

that linear translational spring has limited energy absorbing capability. This can be verified

by looking at the support displacement time history plots in Figure 3-9. Linear support

springs reach the maximum displacements at about 7.3mm, in which the energy is absorbed

and stored by the springs, and then springs rebound to zero displacement and vibrate

about it, in which the stored energy is released. Therefore, the linear springs are energy

storage devices rather than energy absorbers, they have limited effects in reducing blast wall

panel defection over a controllable support displacements. In contrast, the maximum panel

deflection under nonlinear translational support springs is effectively reduced to 92.5mm

with a peak plastic support displacement at 51.9mm, which indicates that due to the energy

consumption of support plastic deformations, less share of energy is absorbed by blast wall

panel and hence less deflection. Detail result summary is given in Table 3-4. This

comparison study of translational springs proves that the energy absorption capability of



3-15

nonlinear support spring outweighs linear support spring in reducing peak panel deflection,

because nonlinear spring is energy absorber rather than energy storage device.

(a) Panel deflection comparison

(b) Support deformation comparison

Figure 3-9. Comparison between linear and nonlinear translational support springs

Table 3-4. Result summary for translational spring comparison study

Compare items Translational spring Kt Max panel

deflection

Support displacement

Kt1 (kN/m) Kt2(kN/m) uel (mm) Max. Residual

Linear spring 32500 \ \ 120.2mm 7.3mm 0

Nonlinear

spring (foam) 32500 325 5 92.5mm 51.9mm 46mm



3-16

Similarly, in order to assess the energy absorption effect of the rotational support spring,

translational spring stiffness Kt is set to zero. The properties of linear and nonlinear

rotational support springs are listed in Table 3-5. The testing overpressure is still 2 bar.

Figure 3-10 illustrates the panel mid-span deflection and support rotation time-history

plots. Different from translational spring, the linear rotational springs generate less panel

deflection than nonlinear springs, the peak values are 89.2mm for linear rotational springs

and 97.7mm for nonlinear springs. This is because linear springs can provide more

restraints to support rotations. Since panel permanent deflection has occurred, linear

rotational springs are not able to release energy to restore to the original positions.

Therefore, the linear rotational springs have become energy absorbers instead. Comparing

to nonlinear rotational springs, linear springs are more stiffer and hence can provide more

resistance to restrain or reduce the support rotations, as shown in Eq.(3-27), panel

deflection is directly proportional to support rotation in plastic stage. Detail result summary

is given in Table 3-5. This comparison study of rotational springs emphasises on the

restraining effects of linear support springs.

Table 3-5. Result summary for rotational spring comparison study

Compare items Translational spring Kr Max panel

deflection

Support rotation

Kr1 Kr2 θel Max. Residual

Linear spring 400kNm/rad \ \ 89.2mm 4.02deg 2.7deg

Nonlinear spring 400kNm/rad 4kNm/rad 1.5deg 97.7mm 4.57deg 3.4deg

(a) Panel deflection comparison



3-17

(b) Support rotation comparison

Figure 3-10. Comparison between linear and nonlinear rotational support springs

(b) Comparison between proposed design and traditional design

In this section, the mid-span deflection time histories for both traditional and proposed

design are plotted for comparison. As per FABIG TN5 [1] and Louca et al. [32] blast wall

panels are normally assumed to be simply supported in preliminary design, which is a

special case of the analytical model with all Kt and Kr equal to zero. Similarly, all

calculations are performed for a total duration of 2td=0.1s by using MATHEMATICA 10.

Figure 3-11 shows the blast wall panel mid-span deflection time histories predicted by the

analytical model for 1 bar traditional design case. As shown, the panel peak deflection is

computed to be 23.8mm. As no damping is applied, the panel keeps vibrating after the

blast pulse and the permanent deflections is observed to be 6mm approximately, which

implies that larger sections are needed to satisfy the code requirements following the

traditional design procedure. As in design requirements describe above, blast wall shall not

yield (i.e. deform plastically) in a SLB event. In contrast, it is not necessary to switch to

larger sections by adopting the proposed design. The similar deflection time history with

the proposed design case is also shown in Figure 3-11. It is observed that the centre point

of the panel has a peak deflection of 15.2mm, which is 36% less than traditional design and

there is nearly no permanent deflection as it vibrates around zero displacement after the

blast loading, which indicates that the blast wall panel remains in its elastic range. The

maximum foam displacement is computed to be 30.9mm, within the energy absorption

zone. Therefore, rather than switching to larger sections, the energy absorption supports



3-18

mitigate the blast effects and satisfy code-compliant limits of the blast wall deflection and

strength.

Figure 3-11. Comparison of blast wall panel mid-span deflections – 1 bar case

Similarly, the deflection time history for 1.5 bar traditional and proposed design cases are

illustrated in Figure 3-12. The energy absorption supports reduces the panel peak

deflection from 58.1mm by traditional design to 30.8mm, which is 47% reduction. The

permanent deflections are approximately 35mm for traditional design and 10mm for the

proposed design. The maximum foam displacement is computed to be 57.1mm, within the

energy absorption zone.

Figure 3-12. Comparison of blast wall panel mid-span deflections – 1.5 bar case



3-19

From an explosion risk and hazard point of view, 1 bar and 1.5 bar are relatively medium

blast overpressures with high occurrence probability, structures shall remain operational

with minimum repairment, which has been proven satisfactory with the proposed design

scheme. For strong and rare blasts (2 bar to 4 bar), event escalation must not be provoked.

Under such strong blast overpressures, the effects of energy absorption supports become

more obvious.

The structural responses of three typical strong blast overpressures (2 bar, 2.5 bar and 3 bar)

are examined, whose deflection time histories are shown in Figure 3-13 for 2 bar case,

Figure 3-14 for 2.5 bar case and Figure 3-15 for 3 bar case. As shown, for traditional design,

large panel deflections are required to dissipate the blast energy: the peak deflections are

119.4mm for 2 bar case, 196.0mm for 2.5 bar case and 278.7mm for 3 bar case. The

permanent deflections for 2 bar case bounces back to around 90mm while 2.5 bar and 3

bar cases remain almost the same as peak deflections.

As mentioned above, the code allowable deflection for blast wall is 300mm or the clearance

to critical equipment, pipelines and structural members if they are located in the vicinity.

Although the peak deflection for 3 bar case is 278.7mm, which is approximately 20mm less

than the code requirement 300mm, the collision risk is still high as offshore platforms or

floating process units like FLNG are very congested constructions. By adopting the

proposed design, the peak deflections are significantly decreased to 61.9mm for 2 bar case,

119.7mm for 2.5 bar case and 171.5mm for 3 bar case, which are generally 40% less

compared to traditional designs. Especially for the 3 bar case, the proposed design cuts off

the peak panel deflection by around 100mm, which demonstrates its excellent performance

against strong blasts. The maximum foam displacements for 2 bar, 2.5 bar and 3 bar cases

are 64.7mm, 53.7mm and 69.4mm respectively.



3-20


Figure 3-14. Comparison of blast wall panel mid-span deflections – 2.5 bar case




3-21

Table 3-6 summarises the peak deflections and maximum foam displacements for the five

blast examples comparing the traditional and proposed designs. To sum up, it is

demonstrated analytically that the proposed design is efficient in mitigating the blast effects

by reducing the blast wall panel deflections (around 40% reduction).

Table 3-6. Summary of the peak blast wall panel deflections and maximum foam displacements

Blast Cases

Traditional design Proposed design Panel

deflection

reduction

Panel deflection

(mm)

Panel deflection

(mm)

Foam displacement

(mm)

1 bar 23.9 15.2 30.9 -36%

1.5 bar 58.1 30.8 57.1 -47%

2 bar 119.4 72.3 64.7 -39%

2.5 bar 196.0 129.9 53.7 -34%

3 bar 278.7 177.2 69.4 -36%

Summary

This chapter derives an analytical model to demonstrate the effectiveness of the proposed

design. The analytical model covers both beam elastic and plastic stages. In beam elastic

stage, the model is established by beam vibration theory, while virtual work theory is

adopted for the formulation in beam plastic stage. The boundary conditions at each

support are simplified as a translational spring and a rotational spring. The translational

spring simulates the foam compressive characteristics with elastic-plastic deformation and

unloading, while the rotational spring provides resistance to prevent large rotations. The

dynamic responses including beam elastic-plastic flexural behaviour and the time history of

panel deflection at the mid-span are also captured, with material nonlinearity and strain

hardening accounted for.

Case studies using a single strip of blast wall panel is presented for demonstration. Five

explosion cases with various blast overpressures are computed to examine the performance

of the proposed design configuration. Significant reductions in deflections (around 40%)

are observed compared with their counterparts of traditional designs.

In addition, the energy absorbing effects of linear and nonlinear support springs (both

translational and rotational) are also assessed separately. Results suggest that linear

translational springs have lower energy absorption capability than nonlinear translational



3-22

springs because they tend to rebound and release the energy stored previously. However,

linear rotational springs are stiffer than nonlinear rotational counterparts so that they can

efficiently restrain the support rotations in plastic stage and hence limit the panel

deflection.



4-1

Numerical Modelling

Introduction

In this chapter, numerical simulations are performed to demonstrate the effectiveness of

the proposed design concept. Finite element models by ABAQUS are created for both

traditional and proposed design schemes. Dynamic analyses in time domain are carried out

for two purposes: to validate the calculations of the analytical model in the previous

chapter and to compare plastic strains at supports for both design schemes. The FEA setup

details such as mesh size, material properties and boundary conditions are presented,

furthermore, FEA results such as blast wall panel deflection, energy absorption

breakdowns and plastic strains at support connections are compared and discussed.

Finite Element Analysis Setups

Model Description and Mesh Size

The numerical models are constructed by using the commercial FEA software ABAQUS

6.14-2. Similar to analytical study, one strip of corrugated blast wall panel with a length of

3m (see Figure 3-8) is modelled with S4R linear quadrilateral reduced integration shell

elements. The cross section dimensions are shown in Figure 3-7.

Since the computational accuracy depends on the element size, an initial convergence test

using element size of 30mm, 20mm and 10mm are performed. Figure 4-1 shows the mid-

span deflection time histories for different mesh size models. Generally, the results are in

close agreement, which indicates that reducing mesh sizes has negligible impact on

computational accuracy in this study. In order to save simulation time, mesh size 20mm is

selected for the subsequent analyses. The blast wall FE model with 20mm mesh size



4-2

consists of a total of 7121 elements. The element numbers here are within the

recommended range (approximately 4000 to 8000 elements) as per Louca and Boh [16] for

capturing good global responses of a corrugated blast wall panel.

Figure 4-1. Mesh size sensitivity test

Material Property

The full range of stainless steel stress-strain curves have been derived by Rasmussen [67], in

which the traditional stress-strain curves based on the Ramberg-Osgood expressions were

deemed inaccurate for stresses beyond the 0.2% proof stress and hence the formulations

were modified based on experimental testing results. The midfield Ramberg-Osgood

expressions are shown in Eq.(4-1), which is given in nominal stress-strain relationships.

ε =��

,)(

,002.0

2.0

2.0

2.0

2.0

2.00

m

u

u

n

E

E

−

−+

−

⋅−

σσ

σσε

σσ

σ

σσ

)(

)(

2.0

2.0

σσ

σσ

>

≤

(4-1)

In this study, the above approach is adopted to generate the material input for numerical

simulations. The material properties of AISI316L is provided in Table 4-3, which are

obtained from Rasmussen [67]. It should be noted that the 0.2% proof stress σ0.2 in Table

4-3 has taken into account of the strain rate effect, which is equivalent to the dynamic yield

stress σdy in Table 3-1. As required by ABAQUS, the engineering stress-strain relationships



4-3

are converted to true stress-strain date following the procedure given by the software

manual [68]. Figure 4-2 shows a comparison between the nominal and true stress-strain

curves. For analytical analysis, the post yield stress-strain relationship is simplified as a

linear strain hardening behaviour, the hardening modulus used for analytical solutions (see

Table 3-1) is obtained by curve fitting shown in Figure 4-2.

Table 4-1. Material parameters for AIS316L [67]

Parameters Values Units

Initial Young’s modulus E 200 GPa

Tangent modulus at 0.2% strain E0.2 23.541 GPa

0.2% proof stress σ0.2 316 MPa

Ultimate plastic stress σu 616 MPa

Ultimate plastic strain εu 0.487 \

Strain hardening constant n 5.88 \

Strain hardening constant m 2.8 \

Figure 4-2. Stress-strain curves for AIS316L stainless steel



4-4

Boundary Conditions and Modelling of Energy Absorption Supports

In traditional design, blast wall panels are fillet-welded to plate girder through end plates,

therefore the assumption of simply support conditions for the end connections is suffice

for a preliminary design [16]. Rather than directly applying pinned supports to all the end

edge nodes, kinematic coupling with all degrees of freedoms is applied to tie all end nodes

to the panel section centroid node at each end, at which translations in three orthogonal

directions (i.e. global UX, UY and UZ) are restrained. In this way, it eliminates the

additional moments caused by eccentricity and renders the numerical model in the same

condition as analytical model so that the results can be compared. The corrugated edges of

the panel are restrained in global UX, RY and RZ directions to form symmetrical

boundaries, which allows the model to behave as a continuous panel.

For proposed design, similar to Chen and Hao [50] who used spring elements with discrete

force-displacement curves to simulate the behaviour of sandwich cores, the energy

absorption supports are simplified as nonlinear springs using connector elements in

ABAQUS. The springs are defined by the key word card *CONNECTOR SECTION,

AXIAL, ROTATION, which enables independent and instantaneous translational and

rotational movements. With the derived force-displacement relationships, the spring

behaviours are modelled by using key word card *CONNECTOR BEHAVIOR, which can

simulate elastic, plastic, unloading and stopping mechanisms. The connector elements are

used to link the centroid nodes of the panel, at which global UZ is restrained, to boundary

nodes, at which all degrees of freedom are fixed.

Figure 4-3 illustrates the FE models and their boundary conditions for both traditional

design and proposed design.

Loading and Analysis

The blast loadings are applied to all shell elements of the blast panel as distributed

pressures normal to their faces by using the key word card *DLOAD. Similar to analytical

solutions, Type 1 blast profile with a duration of 50ms are analysed.

No damping and imperfections are incorporated in the model. The analyses are performed

with ABAQUS\Explicit.



4-5

(a) Traditional design

(b) Proposed design

Figure 4-3. Boundary conditions for traditional and proposed design .



4-6

Results and Discussion

Deflection and Energy Absorption

As mentioned previously, the FEA results are used to validate the calculations of analytical

model in the last chapter. At the same time, the energy absorptions by various components

can also be extracted and compared through FEA analyses. Therefore, similar to analytical

calculations, FE analyses are performed for five blast cases listed in Table 3-3 and results

are discussed below.

Figure 4-4 shows the blast wall panel mid-span deflection time histories predicted by both

analytical and numerical models for 1 bar traditional design case. The corresponding von-

Mises stress contour plot at the time of peak deflection is also depicted, the maximum

envelop stresses are taken out of top and bottom surfaces of the panel, and deformation

amplification factor is set as 1.0. As shown, the panel peak deflection predicted by

ABAQUS is 24.0mm (23.8mm by analytical model). As no damping is applied, the panel

keeps vibrating after the blast and the permanent deflections is observed to be 6mm

approximately. The maximum stress is 331.9MPa, implying that the panel is subjected to

plastic deformation, which confirms the conclusions made in the previous chapter. Similar

deflection time histories and stress contour plots for 1 bar case with the proposed design

are shown in Figure 4-5. It is observed that the centre point of the panel has a peak

deflection of 16.8mm numerically (15.2mm analytically) and there is nearly no permanent

deflection. The maximum stress is reduced to 319MPa, which is just marginally over the

yield stress. Again, the numerical results verify the efficiency of the energy absorption

supports without switching to larger sections.

In addition, for the 1 bar case, the deflection time histories of both traditional and

proposed design predicted by the analytical model show good correlations with those

computed by numerical simulations in not only the peak values (maximum difference less

than 2mm) but also the post-peak responses.



4-7

(a) Comparison of mid-span deflection time history plots

(b) Stress contour plot at peak deflection

Figure 4-4. Deflection and stress contour plots – 1 bar traditional design case



4-8



Figure 4-5. Deflection and stress contour plots – 1 bar proposed design case

During the blast, the explosion energy is transferred to the blast wall as it displaces and

deforms. Neglecting the energy loss, the total energy Et input to the structure system (i.e.

work of external forces in this study) is basically a summation of kinetic energy Ek and

internal energy Ei (i.e. strain energy in this study). The kinetic energy will quickly reduce to

its minimal around zero with time, while the internal energy of the system will rise to

absorb the blast energy. Figure 4-6 compares the internal energy absorbed by different

structural components for both 1 bar traditional and proposed design cases, with the

percentages of energy absorption labelled above the bar charts. For the traditional design,



4-9

as the only component, the blast wall undertakes almost all the input energy (99%).

However, for the proposed design, the total energy is mainly absorbed by

polymethacrylimide foams (76%), hence the panel strain energy is significantly reduced,

with a reduction of 58.5% compared to the traditional design.

Figure 4-6. Energy absorption of various components – 1 bar case

Similarly, the deflection time history and stress contour plots for 1.5 bar traditional and

proposed design cases are illustrated in Figure 4-7 and Figure 4-8 respectively. As shown,

the proposed design has mitigated the blast effect by limiting the panel peak deflection

from 54.6mm of traditional design by ABAQUS (58.1mm by analytical model) to 28.7mm

by ABAQUS (30.8mm by analytical mode), which is a 48% reduction. The maximum stress

has been reduced from 365MPa in traditional design to 339.5MPa in the proposed design.

The analytical solutions match favourably with numerical results with a maximum

difference only 3mm. From the energy absorption point of view, the contribution of

polymethacrylimide foams is substantial. From Figure 4-9, the majority portion (78%) of

the input energy is absorbed by polymethacrylimide foams, leading to a 68% reduction of

panel strain energy comparing to the traditional design.



4-10



Figure 4-7. Deflection and stress contour plots – 1.5 bar traditional design case



4-11



Figure 4-8. Deflection and stress contour plots – 1.5 bar proposed design case



4-12

Figure 4-9. Energy absorption of various components – 1.5 bar case

For the three strong blast overpressures (2 bar, 2.5 bar and 3 bar), their deflection time

histories and stress contour plots are shown in Figure 4-10 and Figure 4-11 for 2 bar case,

Figure 4-12 and Figure 4-13 for 2.5 bar case, and Figure 4-14 and Figure 4-15 for 3 bar

case. As shown, for traditional design, large deflections are required to dissipate the blast

energy, the numerically computed peak deflections are 110.5mm for 2 bar case, 192.8mm

for 2.5 bar case and 290.7mm for 3 bar case, permanent deflections remain almost at the

same level as peak deflections. By adopting the proposed deign, the peak deflections have

significantly decreased to 61.9mm for 2 bar case, 119.7mm for 2.5 bar case and 171.5mm

for 3 bar case, which are generally 40% less compared to their corresponding traditional

designs. Especially for the 3 bar case, the proposed design cuts off the panel deflection by

around 120mm, which demonstrates its excellent performance against high blasts.



4-13






4-14






4-15



Figure 4-12. Deflection and stress contour plots – 2.5 bar traditional design case



4-16



Figure 4-13. Deflection and stress contour plots – 2.5 bar proposed design case



4-17






4-18






4-19

From the corroboration between the numerical and analytical results, it is found that the

correlations for 2.5 bar and 3 bar cases are relatively strong, and the analytical solutions can

also capture the beam post-peak responses accurately. The differences of the peak

deflections between the numerical and analytical results are 2% for 2.5 bar traditional

design case, 8% for 2.5 bar proposed design case, and less than 4% for both 3 bar

traditional and proposed design cases. However, it is noticed that the correlations are

slightly weaker in the peak deflections for 2 bar cases. It could be caused by the

simplification in material strain hardening modelling. In the analytical model, material

behaviour is simplified to be a bi-linear model with linear strain hardening assumed for the

post-yield behaviour. Therefore, as shown in Figure 4-2 with stress approaching 400MPa

(which is the peak stress of 2 bar case) the analytical material model gives larger strain than

that from ABAQUS material under the same stress, leading to around 10mm deflection

difference between analytical and numerical results for 2 bar case. Nevertheless, for a quick

hand calculation in preliminary design stage, the difference is still acceptable.

Figure 4-16 is the energy absorption breakdown bar charts for 2 bar, 2.5 bar and 3 bar

cases. Together with the energy absorption breakdown bar charts for 1 bar and 1.5 bar

shown in Figure 4-6 and Figure 4-9, it is observed that with the increase of blast

overpressures, the energy absorptions by polymethacrylimide foams decrease, and

oppositely the energy absorptions by the panel increase. This caused by the transfer of blast

wall primary deform modes. For the selected blast wall section in this study, the loadings

under 2 bar are relatively mild compared to the panel bending stiffness, the primary blast

wall deform mode is similar to a rigid body movement, which can be confirmed by their

respective deform shapes in Figure 4-5, Figure 4-8 and Figure 4-11, and hence the majority

of the energy is absorbed by foam deformations (more than 60%). However, with the

increasing blast overpressure, blast wall primary deform mode has shifted to bending,

which is clearly illustrated in Figure 4-15 for the 3 bar case. This is because the plastic hinge

formation occurs too early for the foam to absorb sufficient energy during the blast (e.g. at

2.8ms for 3 bar case), which causes the panels to attract more energy and hence the energy

absorptions by the panels tend to increase with overpressures (up to 60% for 3 bar case).

Noticeably, the energy absorptions by the linear springs also rise along with blast

overpressures (from 1% in 1 bar case up to 7% for 3 bar case) due to larger support

rotations. Nevertheless, the foams together with the linear springs can still absorb around

40% of the input energy for 2.5 bar and 3 bar cases.



4-20

Figure 4-16. Energy absorption of various components – 2, 2.5 and 3 bar cases

Table 4-2 and Table 4-3 compare the peak deflections and maximum energy absorptions of

different components for the five cases between traditional and proposed designs. To sum

up, it is demonstrated both analytically and numerically that the proposed design is efficient

in mitigating the blast effects by absorbing energy and hence limiting the blast wall panel

deflections (40% reduction in average). As the predictions with analytical model are in

reasonable agreement with those with numerical simulation, it can be used as a quick

assessment tool for blast wall deflection estimation in preliminary design.

Table 4-2. Summary of the peak blast wall panel deflections

Blast Cases

Traditional design Proposed design Reduction (by

numerical) Numerical

(mm)

Analytical

(mm)

Numerical

(mm)

Analytical

(mm)

1 bar 24.0 23.9 16.8 15.2 -30%

1.5 bar 54.6 58.1 28.7 30.8 -48%

2 bar 110.5 119.4 61.9 72.3 -44%

2.5 bar 192.3 196.0 119.8 129.9 -38%

3 bar 290.7 278.7 171.5 177.2 -41%



4-21

Table 4-3. Summary of energy absorption of various components

Blast

cases Total Et

(kJ)

Panel Foam Spring

Ei,panel (kJ) Ration Ei,foam (kJ) Ration Ei,panel (kJ) Ration

1 bar 5.77 1.26 22% 4.41 76% 0.07 1%

1.5 bar 17.01 3.30 19% 13.30 78% 0.25 1%

2 bar 32.68 11.97 37% 19.64 60% 1.03 3%

2.5 bar 53.31 31.02 58% 18.80 35% 3.46 6%

3 bar 90.35 54.35 60% 29.34 32% 6.64 7%

Plastic Strain at Connection

For a DLB event, another important design requirement is the maintenance of the overall

integrity of the blast wall, i.e. the blast wall must remain in-place to resist the consequent

fire or blasts, which is typically addressed in detailed design stage with the aids of FEA

tools.

As stainless steel has high rupture strain, the rupture failures normally happen at the

support welds. In engineering design, to capture a more realistic representation of the

support stiffness, the blast wall supporting endplates and details of surrounding structures

shall be incorporated in numerical modelling. However, in the present study, for simplicity

and demonstration purpose, only the endplates are modelled, while the surrounding

structures are assumed to be rigid for both traditional and proposed designs. The endplates

are normally angle sections shown in Figure 2-13, with dimensions 100mm in height,

240mm in length and 15mm in thickness for the current study.

The welds between the blast wall panel and endplates are not explicitly modelled, instead,

strain based failure criterion is often used as a guide to estimate the integrity of welded steel

structures with a maximum plastic strain rupture value taken as 5%, which is given in

current prevailing design codes, such as ISO 19902 [12]. Plastic strains can be very sensitive

to the density of the mesh. Louca and Friis [69] performed a strain sensitivity study on

mesh density, in which four mesh densities were tested under the same blast condition. The

peak deflections obtained were more or less the same, but peak plastic strains varied

dramatically. It was concluded that the plastic strains in elements connected to the

supporting angle were more sensitive to the density of the mesh. Therefore, in this study,

the mesh around the weld connections are refined for both traditional and proposed

designs.



4-22

According to the latest FEA design code DNV-RP-C208 [20], on the endplate the

elements within a length of 10t (t is the thickness of the panel) adjacent to the panel and

endplate connections, together with the elements located up to 200mm on the panel, have

been refined. The mesh sizes are reduced to 7-8mm and quad-mesh are used as many as

possible in the critical areas. The aspect ratios of these elements are set close to 1 and the

element angles are made close to 90° so that the elements are almost in a square shape and

not distorted. Transition mesh is used between fine and coarse mesh to avoid excessive

discrepancy. Only the strong blast cases (2 bar, 2.5 bar and 3 bar) are carried out. To obtain

reasonable stress and strain prediction, the analyses are performed in ABAQUS/Standard

implicit numerical scheme.

Figure 4-17 compares the equivalent plastic strains (PEEQ) at the panel and endplate

connections between the traditional and proposed designs for 2 bar case. Although the

locations of the largest plastic strains (i.e. at two corners of the compression flange) are the

same for both designs, the strain magnitudes are remarkably different. The proposed

design has significantly reduced the plastic strain to 0.61% compared to 10.4% of

traditional design. For traditional design, it is very likely that weld tearing will initiate at the

two corners of the compression flange and quickly develop through all welds as the loading

continues, which is a catastrophic failure.

Figure 4-18 compares the time history plots of the panel axial force in global Z direction Fz

and shear force in global Y-axis Fy at the connection for 2 bar case. These two forces are

responsible for the plastic strains as well as mode II (tensile tearing) and mode III (shear

failure) failures at supports. As shown, the axial force Fz at connections in the proposed

design is 58% less than that in the traditional design, and shear force Fy is 20% less. The

springs form a semi-fix connection with rotation flexibility to control or delay membrane

force build-up, on the other hand the foam crushing plateau acts as a cushion to prevent

sudden and rapid increase of reaction shear forces. Therefore, through controllable

rotations and displacements, the linear springs together with the foams help to reduce

section forces at supports, hence prevent the plastic strain build-up and enhance the

integrity of the blast wall from weld tear out.



4-23

(a) PEEQ contour plots for traditional design

(b) PEEQ contour plots for proposed design

Figure 4-17. PEEQ contour plots at support connection – 2 bar case



4-24

(a) Support axial force Fz comparison

(b) Support shear force Fy comparison

Figure 4-18. Section force time histories at support connection – 2 bar case



4-25

Similarly, dramatic decreases in plastic strain are also observed from 2.5 bar and 3 bar

cases, shown in Figure 4-19 and Figure 4-20. By adopting the proposed design, maximum

plastic strains at connections are predicted to decrease from 15.83% of traditional design to

1.46% for 2.5 bar case, and from 24.77% to 4.74% for 3 bar case. The maximum plastic

strain remains under the 5% limit for 3 bar case during a DLB event.

Table 4-4 summarises and compares the maximum plastic strains between traditional and

proposed designs. To sum up, it is demonstrated numerically that the proposed design

substantially reduces the plastic strains at connections, due to the release of high stresses

through controllable support displacements and rotations. By adjusting the stiffness of

linear springs at supports, the proposed design shows advantage over traditional design in

controlling the membrane forces and shear forces at supports to minimise weld tearing out

risk.

Table 4-4. Summary of the maximum PEEQ at support connections

Blast cases

Plastic strain

Traditional Proposed

2 bar 10.4% 0.61% 2.5 bar 15.83% 1.46% 3 bar 24.77% 4.74%



4-26



Figure 4-19. PEEQ contour plots at support connection – 2.5 bar case



4-27



Figure 4-20. PEEQ contour plots at support connection – 3 bar case



4-28

Summary

This chapter employs numerical method to demonstrate the effectiveness of the proposed

design. FE models with S4R elements are created by ABAQUS and dynamic transient

analyses are performed under the same loading and support conditions as the analytical

models.

From five blast cases, the proposed design is efficient in absorbing blast energy and hence

limiting peak blast wall deflections by approximate 40% compared to those with traditional

design. At the same time, the numerical predictions are in reasonable agreement with those

computed from analytical model in the previous chapter, which validates the analytical

model as a quick assessment tool for blast wall deflection estimation in preliminary design

stage.

Comparison study has also been conducted to investigate the weld tearing out at supports

for both traditional and proposed designs, especially under strong blast loadings. Results

reveal that with controllable support displacements and rotations by the proposed design,

the build-up of blast wall panel membrane forces and shear forces can be delayed and kept

acceptable, which effectively reduces plastic strains at supports to avoid weld tearing out.



5-1

Parametric Study

Introduction

Since the feasibility and effectiveness of the proposed design have been successfully

demonstrated analytically and numerically in the previous chapters, this chapter focuses on

investigating the contributions of some key factors to the structural performances and

behaviours of the proposed design through parametric studies. Five key parameters are

selected for investigation in this chapter, they are translational spring initial stiffness in

foam elastic range Kt1, translational spring stiffness in foam crushing stage Kt2, rotational

spring stiffness Kr, blast duration td and blast overpressure profiles. Discussions based on

results aim to provide guidance on how to achieve an optimal design.

Parametric Studies and Result Discussions

Translational Spring Elastic Stiffness Kt1

According to Gibson and Ashby [66], the linear elastic stiffness of foam depends on the

relative density (denoted as ρ*/ρs). In general, the foam elastic stiffness increases along

with the increase of the relative density but it reaches the densification strain earlier as the

porosity in foam becomes less.

From a microscopic view, the mechanism of foam linear elasticity differs from open-cell

foams to closed-cell foams [66]. At low relative densities, open-cell foam deforms primarily

by cell wall bending, with increasing relative densities, tension or compression of the cell

wall begins to dominate and hence the linear elastic stiffness of the foam tends to be stiffer.

For closed-cell foam such as polymethacrylimide foam adopted in this study, the behaviour

is more complex as it is affected by the fraction of solids contained in cell edges (denoted



5-2

as Φ). If too many materials are concentrated in cell edges that leaving only a thin

membrane cross the cell faces, it will rupture early and behave similar to open-cell foam

afterwards. In contrast, if substantial fraction of materials are contained in cell faces, the

foam linear elastic stiffness will become significantly high with increasing relative densities.

Gibson and Ashby [66] derived two Φ values Φ=0.8 and Φ=0.6 (indicating 20% and 40%

of solids contained in the cell faces respectively) through curve fitting among a substantial

amount of experimental data, and the empirical formulae are given below [66]:

)(

2**

cellopenE

E

ss

−

=

ρ

ρ (5-1)

)()/1(

)21()1(

*

**2

*

2

*

cellclosedE

p

E

E

ss

at

sss

−−

−+

−+

=

ρρ

υ

ρ

ρφ

ρ

ρφ (5-2)

s

Dρ

ρε

*

4.11−= (5-3)

where E* is Young’s modulus of the foam; Es is Young’s modulus of the foam solids; pat is

atmospheric pressure (0.1MPa); υ* is Poisson’s ratio of the foam, taken as 1/3.

This study uses 2 bar overpressure to investigate the influences of Kt1 by studying three

stiffness values (small, medium and large) which are directly taken from the springs for 1

bar, 2 bar and 3 bar cases in Table 3-3, rotational spring stiffness are kept the same as the

previous chapters. The rest of the structural properties also remain the same as the

previous chapters. The panel deflection and foam displacement plots are shown in Figure

5-1, corresponding results are summarised in Table 5-1.

As shown, for large Kt1 stiffness case (Case 3 in Figure 5-1), similar to the translational

linear spring case discussed in Section 3.4.2 (a), the foams reach a maximum displacement

of 4.96mm (less than the foam elastic limit uel=5mm) and bounce back, which indicates

that the large Kt1 stiffness is too stiff in this case. As a result, the foams act like a linear

spring without energy absorption capability and hence the maximum panel deflection is

90.1mm. For low Kt1 stiffness case (Case 1 in Figure 5-1), the panel deflection soars steeply

from 16.8mm to 117.8mm over a period of 10ms (highlighted in Figure 5-1). This

excessive deflection is caused by the additional impact due to the sudden stopping

mechanism when the foams reach the full densification strain, which suggests that a low

Kt1 stiffness may limit the energy absorption ability. By comparison, it is clear that the

selection of Kt1 has significant influence on the results. In order to achieve an optimal



5-3

design, the selection of Kt1 values shall allow the foam to stop in the plastic crushing

plateau, where is ideal for energy absorption zone.

(a) Three Kt1 springs and the maximum foam displacements

(b) Deflection plots for three Kt1 springs

Figure 5-1. Parametric study result plots for Kt1

Table 5-1. Result summary for parametric study of Kt1

Case Kt1

(kN/m)

Densification

displacement uult (mm)

Panel max.

deflection (mm)

Foam max.

displacement (mm)

Case 1 20000 86 117.8 86

Case 2 32500 78 61.9 60

Case 3 50000 69 90.1 4.97



5-4

Translational Spring Hardening Stiffness Kt2

Generally, the stiffness in foam crushing stage (termed hardening stiffness Kt2 in this study)

is relatively low, or even as a flat line. However, since the energy absorbed by foam equals

to the area below its force-displacement curve, researchers have endeavoured to increase

Kt2 values in order to enhance energy absorption. This can be achieved by using other

foam materials with suitable relatively densities or developing hybrid energy absorption

devices. For instance, Fan et al. [70] performed both experimental and numerical

investigations into the dynamic lateral crushing behaviour of short sandwich circular tubes

filled by aluminium foam. Results suggested that due to interaction effects, the force-

displacement curve tended to increase after plastic limit, which is shown in Figure 5-2.

Figure 5-2. Dynamic crushing behaviour of sandwich tubes [70]

The parametric study in this section investigates the effect of hardening stiffness Kt2 using

the 2 bar spring in Table 3-3 as an example, the testing overpressure is also 2 bar. Three

cases with Kt2 equals to 1/500, 1/100 and 1/50 of Kt1 are studied, rotational spring

stiffness and support ultimate displacement are unchanged. The panel deflections are

illustrated in Figure 5-3 and corresponding results are summarised in Table 5-2.

As shown, generally the deflections for the three cases are around 61-62mm, the

differences of using three Kt2 values are minor (within 2mm). In all three cases, supports

stop in the ideal energy absorption zones, and a larger Kt2 (Case 3 in Figure 5-3) tends to

make the foams stop earlier at less displacement (51.8mm). While in the low Kt2 case (Case

1 in Figure 5-3) the foams have almost been compressed to the full densification strain at a

displacement of 76.1mm. Therefore, it is concluded that under the same blast loading, the

effects of hardening stiffness Kt2 are relatively minor. Although foams with lower Kt2 may



5-5

deform more and hence produce less panel deflection, it is more conservative to adopt a

medium Kt2 stiffness in design as it has more safety back zone.

(a) Three Kt2 springs and the maximum foam displacements

(b) Deflection plots for three Kt2 springs

Figure 5-3. Parametric study result plots for Kt2



5-6

Table 5-2. Result summary for parametric study of Kt2

Case Kt1

(kN/m)

Kt2

(kN/m)

Panel max.

deflection (mm)

Foam max.

displacement (mm)

Case 1 32500 1/500· Kt1 60.9 76.1

Case 2 32500 1/100· Kt1 61.9 60.0

Case 3 32500 1/50· Kt1 63.0 51.8

Rotational Spring Stiffness Kr

As stated earlier, the function of rotational spring is to a semi-fix connection with certain

degree of bending stiffness and rotation flexibility to simultaneously control the panel

membrane force and end moments. The comparison study performed in Section 3.4.2 (a)

concludes that the advantage of using linear rotational springs at supports outweighs

nonlinear springs because linear springs become energy absorbers in beam plastic stage and

provide more restraints to support rotations. However, its influence to the plastic strains at

support connections cannot be overlooked.

The parametric study in this section evaluates the influences of the rotational spring

stiffness Kr for both aspects of panel deflection and plastic strain at connections. The 3 bar

case in Table 3-3 is taken as an example for investigation. Three cases with Kr equals to 200,

400 and 800kNm/rad are studied, similarly, the other structural properties such as

translational spring stiffness and support ultimate displacement are kept the same for the

three cases. The mid-span defection and support plastic strain contour plots are depicted in

Figure 5-4 and corresponding results are summarised in Table 5-3.

As shown, the effect of rotational spring stiffness on panel behaviour is significant. The

maximum panel deflections computed from Kr=200 (Case 1), 400 (Case 2) and

800kNm/rad (Case 3) are 207.2mm, 171.5mm and 133.8mm respectively. Therefore, it is a

clear trend that with the increasing Kr stiffness the panel deflections are reduced

substantially. However, a different trend is noticed for plastic strains at support, with the

increasing Kr stiffness the maximum plastic strains at supports also increase. They are 3.86%

for Case 1, 4.74% for Case 2 and 5.53% for Case 3. As mentioned above, plastic strains

cannot exceed the 5% rupture limit for design, therefore, Case 3 is an invalid design despite

it yields the lowest panel deflection. In conclusion, a larger Kr stiffness is beneficial for

reducing panel deflection but it also attracts large end forces at supports, an optimal design

shall consider and balance both aspects according to actual design conditions.



5-7

(a) Deflection plots for three Kr springs

(b) Support plastic strain contour plots for three Kr springs

Figure 5-4. Parametric study result plots for Kr



5-8

Table 5-3. Result summary for parametric study of Kr

Case Translation spring

properties

Kr

(kNm/rad)

Panel max.

deflection (mm)

Plastic strain at support

connection

Case 1 Spring for 3 bar case 200 207.2 3.86%



Blast Pressure Profiles

As descried before, there are normally two types of blast pressure profiles, which is shown

in Figure 3-2. Type 2 is a typical blast profile for hydrocarbon explosion if the blast wave

directly hit the structures in a close range. If a pulse propagates into atmosphere with

obstacles, the interactions are complex and may alter both the blast profile and pressure

magnitude.

According to Fire and Explosion Guidance [3], in an open field without obstacles the peak

overpressure of blast wave will decrease with distance while the blast wave duration will

typically increase and hence the impulse will decrease more slowly than the overpressure.

However, in an offshore platform, the blast wave will be affected by other confining

objects, such as decks, blast walls and accommodation blocks that will result in reflection

and diffraction of the blast wave. In some cases this may affect the decay of the blast wave,

and in some cases it can increase local overpressures up to twice of the incident pressure

where a blast wave is reflected from a surface or object, this process is referred to as

“pressure doubling”. In some other cases depending on the leak size and fire effects, the

convection in the flow will tend to make pressure disturbances at the back of the pulse

catch up with the disturbances at the leading edge of the pulse, which will as a result

decrease the duration of the positive part of the pulse and steepen the leading edge. This

process is referred to as “far field effect”.

The parametric study in this section aims to study the effects of different blast pressure

profiles on the structural responses of blast wall. As shown in Figure 5-5, three types of

blast profiles are selected for investigation. Type 1 and 2 have been explained in Figure 3-2,

Type 3 is identically the same as Type 2 except for a negative pressure phase. The pressure

magnitude of the negative phase is taken as 0.2 times of p0 in the positive phase and the

negative duration is taken as 2td (td=50ms) with a total duration of 3td. The structural



5-9

responses for both traditional and proposed design schemes are investigated using an

overpressure of 2 bar.

For the proposed design cases, support displacements for all cases rest in the foam plastic

crushing stage, therefore, as discussed above they are all effective designs. Results of both

traditional and proposed design schemes are summarised in Table 5-4 and corresponding

mid-span deflection time histories are shown in Figure 5-5. As shown, for traditional

design the peak deflection is 110.5mm at the time of 15ms for Type 1 blast compared to

70.4mm at 34ms for Type 2 blast. Type 3 blast yields the same peak deflection in the

positive pulse phase, but less permanent (or ultimate) deflection at roughly 47mm

compared to 50mm for Type 2 because of the reverse pressure in the negative phase.

Similar result patterns are also observed for proposed design cases. When comparing peak

deflections between traditional and proposed designs under the same pressure profile,

significant reductions are also observed: 44% reductions for Type 1 which has been

concluded in the last chapter, and 43% reduction for Type 2 (Type 3 similar) from 70.4mm

of traditional design to 39.8mm of the proposed design.

Based on the results of this parametric study, a few conclusions can be drawn on the

influences of blast pressure profiles on the blast wall structural responses:

• Under the same blast pressure magnitude and duration (impulse), Type 1 profile is

more onerous because it tends to produce larger panel deflections than Type 2. Its

effect shall not be overlooked in engineering design.

• The negative phase of blast pressure can reduce panel permanent deflection but

has no effect on panel peak deflection.

• The proposed energy absorption supports are effective in mitigating blast effects

for both pressure profiles.

Table 5-4. Result summary for parametric study of blast pressure profiles

Blast

Profile

Traditional design Proposed design

Max.

deflection

ymax (mm)

Time at ymax

(ms)

Permanent

deflection

yperm (mm)

Max.

deflection

ymax (mm)

Time at ymax

(ms)

Permanent

deflection

yperm (mm)

Type 1 110.5 15 88 61.9 10 40

Type 2 70.4 34 50 39.8 31 20

Type 3 70.4 34 47 39.8 31 17



5-10

(a) Three blast pressure profiles

(b) Deflection plots under three blast profiles

Figure 5-5. Parametric study result plots for blast pressure profiles



5-11

Blast Duration td

In engineering design, blast overpressure and duration are determined by detailed explosion

risk analysis, the results are usually provided in the form of exceedance curves with Y-axis

representing event occurrence annual frequency and X-axis as explosion overpressure, an

example is given in Figure 5-6. Each design overpressure is normally associated with three

blast durations: short, mean and long. Due to the dynamic nature of an explosion, all three

durations shall be taken into account for blast wall design.

Figure 5-6. An example of blast overpressure exceedance curve [2]

According to Fire and Explosion Guidance [3], the response of a structure to a dynamic

load is commonly characterised by the ratio of the load duration td to the natural period Tn

of the structure. Depending on the value of this ratio, three response regimes are defined.

They are impulsive (td/Tn<0.3), dynamic (0.3<td/Tn<3.0) and quasi-static (td/Tn>3.0).

Therefore, prior to the parametric study, a natural frequency analysis is carried out in

ABAQUS using the keyword *FREQUENCY, EIGENSOLVER = SUBSPACE. The first

six modes are indicated in Figure 5-7, the first mode is an out-of-plane bending with a

natural period of 14.8ms. Given that most of the explosion durations are ranging from

20ms to 200ms [39], the responses of this blast wall panel are expected to be dynamic or

quasi-static, in which the blast duration can significantly affect the structural behaviour of

the panel especially when the responses are plastic.



5-12

Figure 5-7. Fundamental mode shapes and natural periods of the blast wall panel

The parametric study in this section assesses the responses of the blast wall panel under

three blast durations: 30ms (short), 50ms (mean) and 100ms (long) using 2 bar

overpressure and the 2 bar springs given in Table 3-3. Both traditional and proposed design

schemes are studied with both Type 1 and 2 blast profiles, the results are summarised in

Table 5-5 and corresponding mid-span deflection time-histories are shown in Figure 5-8.

For traditional design, the peak deflections under Type 1 blast profile are predicted to be

89.4mm for 30ms duration, 110.5mm for 50ms duration and 134.4mm for 100ms duration.

It is a clear trend that with the increase of blast duration, the panel tends to deform more



5-13

due to more input blast energy. In all three blast duration cases, the time steps when blast

wall panels reach their peak deflections (denoted as tpeak) are relatively concentrated in the

range of 13ms to 16ms. Similar trend of increases in panel deflections along with the

increase of duration is also observed from traditional design Type 2 results, peak

deflections are 58.3mm, 70.4mm and 85.8mm for short, mean and long durations

respectively. However, a different pattern of tpeak is noticed from results of Type 1. Rather

than concentrating in a small interval of time, tpeak for Type 2 are scattered and occur after

0.5td. The shapes of deflection curves for mean and long durations are also varied, a

relatively flat stage is observed prior to the rapid rising. However, this flat stage is not very

clear for the short duration case, which may be caused by the relatively short rising time in

Type 2 profile.

Compared to traditional design results, the peak deflections obtained from the proposed

designs decrease substantially. Peak deflections are 58.2mm (Type 1) and 39.7mm (Type 2)

for short duration case, 61.9mm (Type 1) and 39.8mm (Type 2) for mean duration case,

and 88.6mm (Type 1) and 37.0mm (Type 2) for long duration case. They are at least 32%

less when comparing to their counterparts from traditional designs, which has again

demonstrated the excellent energy absorption capability of the proposed design. Especially

under Type 2 profile, more deflection reductions (up to 57%) are achieved with the

increase of blast duration that implies more input energy. This finding together with the

results in previous chapters demonstrate the efficiency of using the proposed design in high

energy blast scenarios from both blast duration and overpressure perspectives. Similar

trend can be seen in short and mean duration results of Type 1 profile, but long duration

result shows otherwise. By studying the deflection shape of Type 1 100ms case, two peaks

and a climbing pattern in between are noticed between the time intervals of 10ms and

23ms. This is because the safety back zone has been completely consumed and the

supports have finally been compressed to the full densification limit of 78mm at time step

of 18ms. As a result, panel deflection rises again to absorb the remaining input blast energy.

In conclusion, based on td/Tn ratios, blasts with long durations are more onerous in design

as they are possess of more energy and the panel tends to response in a quasi-static way. In

additional, the results in this section again emphasise the importance of sufficient safety

backup zone. This can be achieved by using material with long stroke or larger Kt2 stiffness.



5-14

Table 5-5. Result summary for parametric study of blast durations

Blast Profile

td Traditional Design Proposed Design

Deflection reduction Max. deflection

(mm) tmax (ms)

Max. deflection (mm)

tmax (ms)

umax

(mm)

Type 1

30ms 89.4 13 58.2 10 36.7 -35%

50ms 110.5 15 61.9 10 60 -44%

100ms 134.4 16.5 88.6 23 78 -34%

Type 2

30ms 58.3 22 39.7 19 18.3 -32%

50ms 70.4 34 39.8 31 20.8 -43%

100ms 85.8 60 37.0 61 52.5 -57%

(a) Traditional design deflection plots for three blast durations

(b) Proposed design deflection plots for three blast durations

Figure 5-8. Parametric study result plots for blast durations



5-15

Summary

In order to shed some lights on achieving an optimal design, this chapter investigates the

performances of the proposed design scheme by conducting parametric studies on five key

parameters (Kt1, Kt2, Kr, blast pressure profiles and durations). The following conclusions

are made:

• Kt1 is important in absorbing energy and reducing panel deflection. In an optimal

design, the selection of Kt1 shall render the supports to stop in foam plastic crushing

zone where is ideal for energy absorption. The value of Kt1 can be controlled by

adjusting the foam relative densities.

• Kt2 is less influential than Kt1 in reducing panel deflections. Under the same blast

overpressure and duration, a relatively steeper Kt2 tends to make the support stop

earlier and farther away from the foam densification limit and hence leaves more safety

backup zone. The value of Kt2 can be controlled by using hybrid energy absorption

devices.

• Increasing Kr can efficiently reduce blast wall panel deflections, but it also causes

membrane force build-up at support connections and thus higher plastic strains.

Therefore, a balanced consideration is required in engineering design.

• Offshore blast walls are usually designed for hydrocarbon blast profile, however, under

the same blast overpressure and duration, far field blast profile is more onerous that

shall not be overlooked. The negative phase of blast pressure has no effect on the panel

peak response but may affect the panel ultimate deflection.

• Since the natural periods of blast wall panels are relatively short, long duration blasts

are more onerous in blast wall design due to td/Tn ratios. A sufficient safety backup

zone may be required to prevent the supports from being compressed to densification.

This can be achieved by using material with long stroke or larger Kt2 stiffness.



6-1

Concluding Remarks

Main findings

When subjected to strong blasts, blast wall with traditional design may undergo excessive

deflection at mid-span and suffer to weld ruptures at connections. To overcome these

drawbacks, the research carried out in this thesis adopts analytical and numerical methods

to demonstrate the feasibility and effectiveness of a novel blast wall design concept with

energy absorption mechanism applied at blast wall supports. The major contributions and

findings achieved in this research are summarised as follows.

1. After systematically reviewing the current offshore platform blast wall design

requirements in the industry along with the latest research on blast mitigation using

energy absorption concept, this research proposes a novel design concept for offshore

blast walls specifically against the common failure modes in traditional design, aiming to

limiting blast wall deflection and minimising weld tearing out risk simultaneously. An

energy absorption blast wall design is proposed by using flexible supports filled with

polymethacrylimide foam and rotational springs, which allow the wall to slide/rotate a

certain distance/angle to reduce the high stresses at supports and meanwhile dissipate

blast energy through foam deformations so that both support rupture and panel

deflection can be limited.

2. An analytical model is developed based on beam vibration theory and virtual work

theory to investigate the feasibility and effectiveness of the proposed design. The

boundary conditions at each support are simplified as a translational spring and a

rotational spring. The translational spring simulates the foam compressive characteristics

such as elastic-plastic deformation and unloading, while the rotational spring provides

resistance to prevent large rotations. The dynamic responses including the elastic-plastic

flexural behaviour and deflection time history at the mid-span of blast wall panel are



6-2

included in the model, with material strain hardening accounted for. Case studies with

various explosion overpressures demonstrate that the maximum panel deflections can

be significantly reduced by around 40% with the proposed design compared to

traditional design.

3. The case studies are then numerically simulated with the commercial FEA software

ABAQUS to verify the analytical model. Reasonable agreements between analytical and

numerical results are obtained, which validates the analytical model as a quick

calculation tool for blast wall deflection estimation in preliminary design stage.

4. Numerical results also show that traditional design is vulnerable to weld tearing out due

to high membrane forces generated under strong blast, for example, the plastic strains

at supports can be as high as 25% under 3 bar overpressure. However, with the

proposed design the plastic strains can be effectively kept under the failure criterion 5%

due to the release of high stresses through controllable displacements and rotations at

supports.

5. In order to achieve an optimal design, parametric studies are conducted for five key

parameters (translational spring elastic stiffness Kt1, translational spring hardening

stiffness Kt2, rotational spring stiffness Kr, blast pressure profiles and durations). It is

concluded that Kt1 and Kr play important roles in reducing blast wall panel deflections.

The selection of Kt1 shall render the supports to stop in foam plastic crushing zone and

the selection of Kr shall achieve a balance between panel deflections and plastic strains

at support connections. Kt2 is less influential than Kt1 and Kr in reducing panel

deflections, but it affects the safety backup that prevents the foams from full

densification. Studies on blast pressure profiles also suggest that far field blast profile

and longer blast durations are more onerous in engineering design.

Recommendations for future work

The feasibility and effectiveness of the proposed design scheme using profile blast wall

panel as example have been demonstrated analytically and numerically in this thesis.

Further investigations for the future study are outlined below:

1. In order to further validate the results, experimental tests can be carried out. As the

support is relatively difficult to fabricate in small workshops, a simplified scheme

without rotational springs is allowed.

2. As explained in Chapter 3, in the plastic stage of the analytical model, the plastic

hinge length is assumed to be a constant value 1.5teq throughout the blast process.



6-3

This has caused some discrepancies in predicting panel deflection for relatively

strong blast cases. More insights are required in this aspect.

3. This study uses polymethacrylimide foam as the energy absorption material for

translational displacement. The Kt1 of polymethacrylimide foam can be adjusted by

its relative density, however, Kt2 is relatively flat and there is a lack of controllable

measure for adjustment, if a design requires certain degree of spring hardening after

plastic crushing point. Some research has already looked into this aspect with

hybrid materials and structural forms to increase Kt2 to absorb more energy.

Further efforts should make the new designs adjustable and results tabulated.

4. In this thesis blast overpressures are assumed to be uniformly distributed loads

over the entire panel. Localised blast effect and projectile penetrations are not

considered. Thin-walled panels such as profiled blast walls are very unlikely to resist

projectile penetrations even with the energy absorption supports. Therefore,

sandwich panels together with the energy absorption supports may be a promising

scheme to resist projectile penetration. Further research can investigate the

effectiveness of this combined design.


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