Structural Response to Explosions - HySafe - Safety of Hydrogen as

Structural Response - Shepherd 1

Structural Response to Explosions

Joseph E ShepherdCalifornia Institute of Technology

Pasadena, CA USA 91125

Presented at

1st European Summer School on Hydrogen SafetyUniversity of Ulster, August 2007


Summary

This lecture will cover the fundamentals of structural response to internal and external loading of structures by explosions of fuel-air mixtures. There will be two parts to the lecture. The first part will review the generation and characterization of pressure waves by deflagrations, detonations, transition from deflagration to detonation inside of vessel, and blast waves from unconfined vapor cloud explosions and detonations of fuel-air clouds. The second part will cover structural response of simple structures with an emphasis on single degree of freedom models and integral characterization of the pressure loading.


Outline• Overview• Determining Structural Loads• Determining Structural Response • Examples


Overview

• Why carry out structural response analysis? • How do explosions damage structures?• Motivations for considering structural failure in a

safety assessment• Elements of a structural response analysis


Why Study Structural Response?

• Before an event as part of a safety assessment activity

– Will structural failure happen?

• After an event as part of an incident investigation

– Why did structural failure happen?


How do explosions damage structures?

• Bend, break, or displace load-bearing panels, posts, and beams, possibly causing structural collapse

• Distort and possibly rupture pressure vessels. pipes, valves, and instrumentation, releasing hazardous (toxic or explosive) materials into the environment

• Shock and vibration can break nonstructural components (e.g., glass windows) far from incident.

• Create fragments which can travel long distances, causing facility damage and bodily injury.

• Start fires due to thermal radiation from fireballs and heat transfer from combustion products.


Pasadena TX 1989 – C2H4 Flixborough 1974 - cyclohexane

Port Hudson 1974 – C3H8

(20 Kg H2 )


Nuclear Blast Wave Damage – 5 psi (34 kPa)


Nuclear Blast Wave Effects

1.7 psi (11.7 kPa)5 psi (34 kPa)


Motivations for studying structural response

• Immediate life safety consequence – damage to critical structures will lead to injury or death. Examples: Pressure vessels and piping systems containing toxic materials.

• Creating a potential hazard – release of combustible or flammable material could result in fire or explosion that has life safety and secondary hazard generation consequences.

• Economic loss – destruction of high value processing equipment, loss of product, plant downtime, environmental cleanup, compensation of victims, litigation costs.


Pasadena TX 1989


Elements of Structural Response Analysis

• Define explosion hazard or sequence of events in an actual accident.

– HAZOP or FEMA • Develop a model for the type of explosion that takes

place.– Validate explosion model against existing data or new tests

• Estimate the structural loading• Develop a model for the structure and loading capacity• Estimate response of structure to loading

– Validate structural model against existing data or new tests• Establish pass-fail criteria based on material properties

and maximum deformations or stresses– Use existing databases or carry out material testing


Related Subjects• Earthquake engineering

– Strong ground motion excites building motion• Terminal ballistics

– Projectile impact creates stress waves and vibration• Crashworthiness

– Vehicle crash mitigation • Weapons effects

– Conventional (High explosive and FAE)– Nuclear and nuclear simulation testing

TIP – Many recent studies on structural response to blasts have been sponsored to counter terrorism – the results are often restricted to government agencies or official use only.



Determining structural loads

• Load generally means “applied force” in this context. The primary load is usually thought of as due to pressure differences created by the explosion process. Pressure differences across components of a structure create forces on the structure and internal stresses.

• Three simple cases– External explosion– Blast wave interaction– Internal explosion


External Explosion• Explosion due to accidental

vapor cloud release and ignition source starting a combustion wave

• Flame accelerates due to instabilities and turbulence due to flow over facility structures

• Volume displacement of combustion (“source of volume”) compresses gas and creates motion locally and at a distance– Blast wave propagates away

from source Unconfined Vapor Cloud Explosion (UVCE)


Blast Wave Interaction

• Blast wave consists of– Leading shock front– Flow behind front

• Pressure loading – Incident and reflected

pressure behind shock– Stagnation pressure from

flow

• Factors in loading– Blast decay time– Diffraction time– Distance from blast origin


Internal Explosion

• Can be deflagration or detonation• Deflagration

– Pressure independent of position, slow• Detonation

– Spatial dependence of pressure– Local peak associated with detonation wave

formation and propagation


Type of Combustion• The computation of structural loading requires determining the time history of

the pressure applied to the structure. There are two generic situations– Internal explosion– External explosion

• The mode of combustion is important in both situations– Deflagration – slow speed combustion (1-1000 m/s)– Detonation – high speed combustion (1500-3000 m/s)– Deflagration-to-detonation transition (DDT) – accelerating combustion wave with

localized pressure spikes• The mode of combustion depends on many factors

– Composition of mixture: amount of fuel, oxidizer and diluent– Initial temperature and pressure– Type of ignition source– Presence of flame accelerating elements such internal obstructions in tubes, pipe

racks, grates, etc.– Distance of propagation (size of pipe, vessel, or fuel-air cloud)


Pressure Generation Mechanism

• Volume expansion due to combustion– Displaces surrounding gas– Confinement due to

• Inertia of gas• Surrounding structure limits motion

• Pressure rise due to – Confinement– Compression of surrounding gas– Generation of blast waves


Combustion and Pressure Waves

• Overall Combustion Reaction – major speciesH2 + ½(O2 + 3.76 N2) H2O + 1.88N2

• Combustion results in temperature rise due to conversion of chemical to thermal energy

• Temperature rise creates– Volume expansion (low speed flames)– Pressure rise in constant volume combustion– Pressure rise and flow in detonation and high speed

flames


Creation of Pressure Waves by Explosions• Expansion of combustion

products due to conversion of chemical to thermal energy in combustion and creation of gaseous products in high explosives

• Expansion ratio for gaseous explosions depends on thermodynamics

• Expansion rate depends on chemical kinetics and fluid mechanics– Flame speeds – Detonation velocity


Creation of flow by Explosions I.• Flames create flow due to expansion of products

pushing against confining surfaces• Consider ignition at the closed-end of a tube

– Expansion ratio

– Flame velocity

– Flow velocity

Burned (u =0) Vf Unburned u > 0

flame

ST

effTfTf SAASV σσ == /

b

uρ

ρσ =

effT

effTf SSVU )1( −=−= σ

Blast wave

u = 0


Creation of flow by Explosions II• Detonations and shock waves create flow due to

acceleration by pressure gradients in waves• Consider ignition of detonation at the closed-end of a

tube

Burned (u =0) Burned u >0 Unburned u = 0

Detonation wave

Expansion wave

u

x


Loading Histories

• Pressure-time histories can be derived from several

sources

– Experimental measurements

– Analytical models with thermodynamic computation of

parameters

– Detailed numerical simulations using computation fluid

dynamics

– Empirical correlations of data

– Approximate numerical models of blast wave

propagation (Blast-X)

• Characterizing pressure-time histories

– Single peak or multiple peaks

– Rise time

– Peak pressure

– Duration

Slow flame in vessel

High speed flame in vessel

Nonideal vapor cloud explosion

Ideal vapor detonation


Pressure Loading Characterization

• Structural response time T vs. loading and unloading time scales τI

• Peak pressure Δ P vs. Capacity of structure• Loading regimes

– Slow (quasi-static), typical of flame inside vessels T << τL or τu

– Sudden, shock or detonation waves τL << T• Short duration – Impulsive τU << T• Long duration - Step load T << τU

τload τunload


Preview• Structural response in simplistic terms

– What are structural response times?• Large spectrum for a complex structure• Single value for simple structure

– How do these compare to loading and unloading times of pressure wave?

• Loading time• Unloading time

– Estimate peak deflection and stresses based on these time scale comparisons and peak load

– Compare capacity of structure with expected peak load. Failure can occur to do either

• Excessive stress – plastic deformation or fracture makes structure too weak for service

• Excessive deformation – structure not useable due to leaks in fittings or misfit of components (rotating shafts, etc).


Ideal Blast Wave SourcesSimplest form of pressure loading – due concentrated, rapid release of energyHigh explosive or “prompt” gaseous detonation. Main shock wave followed bypressure wave and gas motion, possibly secondary waves.


Blast Wave from Hydrogen-Air Detonation


Blast and Shock Waves

• Leading shock front pressure jump determined by wave speed – shock Mach number.

• Gas is set into motion by shock then returns to rest

• Wave decays with distance

• Loading determined by– Peak pressure rise– Impulse– Positive and negative

phase durations

Δ P

τ-τ+

Specific impulse!


Scaling Ideal Blast Waves I.

• Dimensional analysis (Hopkinson 1915, Sachs 1944, Taylor-Sedov)– Total energy release E = Mq

• M = mass of explosive atmosphere (kg)• q = specific heat of combustion (J/kg)

– Initial state of atmosphere Po or ρo and co

• Limiting cases– Strength of shock wave

• Strong Δ P >> Po

• Weak Δ P << Po

– Distance from source• Near R ~ Rsource

• Far R >> Rsource


Scaling Ideal Blast Waves II.• Scale parameters

– Blast length scale Rs = (E/Po)1/3

– Time scale Ts = Rs/co

– Pressure scale• Close to explosion Pexp (usually bounded by PCJ) • Far from explosion Po

• Nondimensional variables– pressure Δ P/Po

– distance R/Rs

– time t/Ts

– Impulse (specific) I/(Po Ts)

Relationships:

Δ P/Po = F(R/Rs)I/(Δ P Ts) = G(R/Rs)


Cube Root Scaling in Standard atmosphere

• Simplest expression of scaling (Hopkinson)– At a given scaled range R/M1/3, you will have the same

scaled impulse I/M1/3 and overpressure Δ P– When you increase the charge size by K, overpressure

will remain constant at a distance KR, and the duration and arrival time will increase by K.


TNT Equivalent• Ideal blast wave from gaseous explosion equivalent to

that from High Explosive (TNT) when energy of gaseous explosive is correctly chosen

• Universal blast wave curves in far field when expressed in Sachs’ scaled variables

• For ideal gas explosions (detonations) E is some fixed fraction of the heat of combustion (Q = qM)

• For nonideal gas explosions (unconfined vapor clouds), E is quite a bit smaller. Key issues:– How to correctly select energy equivalence?– How to correctly treat near field?


Scaling of Blast Pressure – Ideal Detonation

Comparison of fuel-air bag tests to high explosives

Work done at DRES (Suffield, CANADA) in 1980s


Scaling of Impulse – Ideal Detonation

Air burst

Surface burst

For the same overpressure or scaled impulse at a given distance, M(surface) = 1/2 M(air)


Energy scaling of H2-air blast

Energy Equivalence

100 MJ/kg of H2

or

2.71 MJ/kg of fuel-air mix for stoichiometric.


Hydrogen-air Detonation in a Duct

• Blast waves in ducts decay much more slowly than unconfined blasts

Δ P ~ x-1

• Multiple shock waves created by reverberation of transverse waves within duct

• Pressure profile approaches triangular waveshape at large distances.


Interaction of Blast Waves with Structures

Blast-wave interactions with multiple structures LHJ Absil, AC van den Berg, J. Weerheijm p. 685 - 290,Shock Waves, Vol. 1, Ed. Sturtevant, Hornung, Shepherd, World Scientific, 1996.


Idealized Interactions

Enhancement depends:

Incident wave strength

Angle of incidence


Nonideal Explosions

• Blast pressure depends on magnitude of maximum flame speed

• Flame speed is a function of– Mixture composition– Turbulence level– Extent of confinement

• There is no fixed energy equivalent– E varies from 0.1 to 10% of Q

• Impulse and peak pressure depend on flame speed and size of cloud – Sachs’ scaling has to be expanded to include these


Pressure Waves from Fast FlamesSachs’ scaling with addition parameter – effective flame Mach number Mf. Numericalsimulations based on ‘porous piston’ model and 1-D gas dynamics.

Tang and Baker 1999


What is Effective Flame Speed?

Dorofeev 2006

Consider volume displacementof a wrinkled (turbulent) flame growing ina mean spherical fashion.

Expansionratio


Internal Explosion - Deflagration

• Limiting pressure determined by thermodynamic considerations– Adiabatic combustion process– Chemical equilibrium in products– Constant volume

• Pressure-time history determinedby flame speed

fuel-air mixture

Products

Combustion wave

SLVf = Sf + u


Burning Velocity

Depends on substance, composition, pressure, temperature


Expansion ratio and Flame temperature

• Related to flame temperature through gas law

• E will depend on composition • For fuel-air mixtures, Emax ~7

reactantsreactantsreactants TNTN

VV

E productsproductsproducts ==

NRTPV=


Expansion ratio


Adiabatic Flame Temperature

• Temperature of products if there are no heat lossesHreactants(Treactants) = Hproducts(Tproducts)

• Simple approximation for lean mixture:Tproduct ~ Treactants + fHc/Cp

Hc = heat of combustion of fuel (42 MJ/kg fuel)Cp = heat capacity of products (including N2, …)

• For stoichiometric HC fuel-air mixtures: Tproducts ~ 2000oC• Decreases for off-stoichiometric, and diluted mixtures, 1100-

1400 oC at flammability limit.• Values are similar for all HC fuels when expressed in terms of

equivalence ratio.


Pressure in Closed Vessel ExplosionPeak pressure limited by heat transfer during burn and anyVenting that takes place due to openings or structural failure


Adiabatic Explosion Pressure

• Pressure of products if there are no heat losses and complete reaction occurs

• Energy balance at constant volumeEreactants(Treactants) = Eproducts(Tproducts)

Vreactants = Vproducts

Pp = Pr (NpTp/NrTr)• Products in thermodynamic equilibrium• For stoichiometric HC fuel-air mixtures: Pp ~ 8-10 Pr

• Decreases for off-stoichiometric, and diluted mixtures, • Values are similar for all HC fuels when expressed in terms of

equivalence ratio.• Upper bound for peak pressure as long as no significant flame

acceleration occurs


Measured Peak Pressure vs Calculated



Forces, Stresses and Strains

• Loading becomes destructive when forces are sufficient to displace structures that are not anchored or else the forces (or thermal expansion) create stresses that exceed yield strength of the material.

• Important cases– Rigid body motion – fragments and overturning– Deformation due to internal stresses

• Bending, beams and plates• Membrane stresses, pressure vessels


Rigid Body Forces due to Explosion

• Pressure varies with position and time over surface – has to be measured or computed

• Local increment of force on surface due to pressure only in high Reynolds’ number flow

Geometry and distribution of pressure will result in moments as well as forces! Be sure to add in contributions from body forces (gravity) to get total force.


Consequence of Forces I.

• Rigid body motions– Translation

– Rotation

X’ = X – Xcm distance from center of mass


Internal Forces Due to an Explosion

• Force on a surface element dS

• Stress tensor σ


Consequence of forces – small strains (<0.2 %)

• Elastic deformation • Elastic strain

• Elastic shear

Youngs’ modulus E, shear modulus E, and Poisson ratio ν are material properties


Consequences of forces – large strains

• Onset of yielding for σ ~ σY

• Necking occurs in plastic regime σ > σY

• Plastic instability and rupturefor σ > σu

• Energy absorption by plastic deformation Plot is in terms of engineering stress and strain, apparent

maximum in stress is due to area reduction caused by necking


Stress-Strain Relationships


Yield and Ultimate Strength

• Yield point σYP determined by uniaxial tension test• Yielding is actually due to stress differences or shear.

Extension of tension test to multi-axial loading:– Maximum shear stress model τmax < σYP/2– Von Mises or octahedral shear stress criterion

• Onset of localized permanent deformation occurs well before complete plastic collapse of structure occurs.


Some Typical Material Properties

ρ E G ν σy σu εruptureMaterial (kg/m3) (GPa) (GPa) (MPa) (MPa)Aluminum 6061-T6 2.71 x 103 70 25.9 0.351 241 290 0.05Aluminum 2024-T4 2.77 x 103 73 27.6 0.342 290 441 0.3Steel (mild) 7.85 x 103 200 79 0.266 248 410-550 0.18-0.25Steel stainless 7.6 x 103 190 73 0.31 286-500 760-1280 0.45-0.65Steel (HSLA) 7.6 x 103 200 0.29 1500-1900 1500-2000 0.3-0.6Concrete 7.6 x 103 30-50 20-30 - 0Fiberglass 1.5-1.9 x 103 35-45 - 100-300 -Polycarbonate 1.2-1.3 x 103 2.6 55 60 -PVC 1.3-1.6 x 103 0.2-0.6 45-48 - -Wood 0.4-0.8 x 103 1-10 - 33-55 -Polyethylene (HD) 0.94-0.97 x 103 0.7 20-30 37 -


Internal Forces due to Explosions

• Stress waves– Longitudinal or transverse– Short time scale

• Flexural waves– Shock or detonation propagation inside tubes– Vibrations in shells

• tension or compression– Deforms shells

• shearing loads– Bends beams and plates


Statics vs. Dynamics

• Static loading T >> τl, τu– Loading and unloading times long compared to

characteristic structural response time– Inertia unimportant– Response determined completely by stiffness,

magnitude of load. • Dynamic loading T · τ

– Loading or unloading time short compared to characteristic structural response time

– Inertia important– Response depends on time history of loading


Static Stresses in Spherical Shell

• Balance membrane stresses with internal pressure loading

• Force balance on equator

• Membrane stress

Validate only for thin-wall vessels h < 0.2 R

R

R


Static Stresses in Cylindrical Shells

• Biaxial state of stress• Longitudinal stress due to

projected force on end caps.

• Radial (hoop) stress due to projected force on equator


Bending of Beams

• Force on beam due to integrated effects of pressure loading

• Pure bending has no net longitudinal stress

• Deflection for uniform loading


Stress Wave propagation in Solids

• Dynamic loading by impact or high explosive detonation in contact with structure

• Two main types– Longitudinal (compression, P-waves) – Transverse (shear, S-waves

• Stress-velocity relationship (for bar P-waves) Cl exact for bar


Is direct stress wave propagation important?

• Time scale very fast compared to main structural response T ~ L/C

– Average out in microseconds (10-6 s)

• Stress level low compared to yield stress

σ ~ Δ P ~ 10 MPa << σY = 200- 500 MPa

31553205Aluminum32056100Steel

Cs(m/s)Cl (m/s)

Direct stress propagation within the structural elements is usually not relevant for structural response to gaseous explosions.


Structural motions

• Element vibrations– Membranes or shells

– Plates or beams

– Modes of flexural motion• Standing waves, frequencies ωi

• Propagating dispersive waves ω(k)

• Coupled motions of entire structure


Two Special Situations

• Loading on small objects– Represent forces as drag coefficients dependent on shape and

orientation and function of flow speed.F = ½ ρ V2 CD(Mach No, Reynolds No) x Frontal Area

• Thermal stresses. – Thermal stresses are stresses that are created by differential

thermal expansion caused by time-dependent heat transfer from hot explosion gases. This is distinct from the loss of strength of materials due to bulk heating, which is a very important factor in fires which occur over very much longer durations than explosions.

ε = σ/E + α Δ T


Determining structural response

• Issues– Static or dynamic

• depends on time scale of response compared to that of load– impulsive (short loading duration)– sudden (short rise time)– quasi-static (long rise time)

– Elastic or elastic-plastic• depends on magnitude of stresses and deformation

– yield stress limit appropriate for vessels designed to contain explosions

– maximum displacement or deformation limit appropriate for determining or preventing leaks or rupture under accident conditions


Simple estimates• Strength of materials approach assuming equivalent static load

– Useful only for very slow combustion (static loads) and negligible thermal load

• Theory of elasticity and analytical solutions– static solutions for many common vessels and components (Roarke’s Handbook)– dynamic solutions available for simple shapes – mode shapes and vibrational periods are tabulated.– Energy methods with assumed mode shapes (Baker et al method)– Analytical models for traveling loads available for shock and detonation waves– Transient thermo-elastic solutions available for simple shapes

• Theory of plasticity – rigid-plastic solutions available for simple shapes and impulsive loads.– Energy methods can provide quick bounds on deformation

• Empirical correlations– Test data available for certain shapes (clamped plates) and impulsive loads– Pressure-impulse damage criteria have been measured for many items and people subjected to blast

loading

• Spring-mass system models– single degree of freedom – multi-degree of freedom– elastic vs plastic spring elements


Simple Structural Models

• Ignore elastic wave propagation within structure• Lump mass and stiffness into discrete elements

– Mass matrix M– Stiffness matrix K– Displacements Xi– Applied forces Fi

• Equivalent to modeling structure as coupled “spring-mass” system

• Results in a spectrum of vibrational frequencies ωI corresponding to different vibrational modes– Fundamental (lowest) mode usually most relevant


Single Degree of Freedom Models (SDOF)

• Example - radial oscillation of a shell.

• Allow only for radial displacement x of tube surface

• Assumes radial and axial symmetry of load

• Elastic oscillations only• Results in harmonic oscillator

equation (no damping)

P(t)

xh

frequency

R

τp

t

period


SODF - Square Pulse

Pulse length τ: 100μsPulse length τ: 10μs


SDOF – Static Regime

• Very slow application of load – (quasi-static) no oscillations

T << τu or τL

• Static deflection

FMax

T time

force

displacement

τl


SDOF -Impulsive Regime• Sudden load application, short

duration of loading τ << T• Linear scaling between

maximum strain/ displacement and impulse in elastic regime:

• Impulse generates initial velocity

• Energy conservation determines maximum deflection


SDOF – Sudden regime

• Quick application of load and long duration τu >> T

• Peak deflection is twice static value for same maximum load

FMax

T time

force

displacement

τ


SDOF - Dynamic load factor (DLF)


Considerations about material properties

• Simple models: – perfectly plastic, – elastic perfectly plastic

• More realistic models– Strain hardening σY (ε)

– Strain rate effects, σY(dε/dt)

– Temperature effects σY(T)

σ

ε

σ

ε

ε.

.


SDOF - Plasticity

• Replace kX with nonlinear relationship based on flow stress curve σ(ε)

• Energy absorbed by plastic work is much higher than elastic work

• Peak deformation for impulsive load scales with impulse squared.


SDOF – Pressure- Impulse (P-I)

• Alternative representation of response• For fixed Xmax and pulse shape, unique relation

between peak pressure (P) and impulse (I)

Shock wave with exponential tail


Numerical simulation

• Finite element models• static• vibration: mode shape and frequencies• dynamic

– transient response to specified loading– elastic – plastic/fracture

• Numerical integration of simple models with complex loading histories

– spring-mass systems– Elasticity with assumed mode shape


Example

• Blast loading of a cantilever beam

– Giordona et al elastic response

– Van Netton and Dewey plastic response

– Baker et al energy method


Initial stages of shock diffraction over a cantilever beam

Giordano et al, Shock Waves 14 (1-2), 103-110, 2005.


Later stages of diffraction over a cantilever beam



Applied Load and Oscillations of Beam



Plastic Deformation of Blast loaded Cantilever

Van Netten and Dewey, Shock Waves (1997) 7: 175–190


Blast Loading



Shock tube experiments



Structural Response of Piping to Internal Gaseous Detonation


Detonations in Piping

• Accidental explosions• Potential hazard in

– Chemical processing plants– Nuclear facilities

• Waste processing• Fuel and waste storage• Power plants

• Test facilities– Detonation tubes used in laboratory facilities– Field test installations (vapor recovery systems)


Hamaoka-1 NPP

Brunsbuettel KBB

Recent Accidental Detonations

Both due to generation of H2+1/2O2 by radiolysis and accumulation in stagnant pipe legs without high-point vents or off-gas systems.


Outline• Basic detonation facts• Elastic response of tubes to detonation• Fracture of tubes with detonation loading• Bounding loads

– Deflagration to detonation transition– Reflection of detonation

• Plastic deformation• Interaction with bends and tees• Role of ASME code


What is a Detonation Wave?

A supersonic combustion wave characterized by a unique coupling between a shock front and a zone of chemical energy release referred to as the “reaction zone.”


Detonation Concepts

• Steadily propagating wave (CJ)• Shock-induced chemical reaction (ZND)• Propagating pressure wave• Induces a flow and pressure variation behind

detonation• Instability of front


Chapman-Jouguet (CJ) Model

Thermodynamics and elementary gas dynamicsAdequate to predict ideal wave speed

Combustion wave moves at minimum speed consistent with conservation of mass, momentum and energy across the wave front. Equivalent to productsaway from wave front with a relative velocity equal to the speed of sound “sonicor CJ condition”



ZND Model

UCJReactantsProductsProducts

•Steady reactive flow behind nonreactive shock•Shock-induced chemical reaction•1D “smooth” flow – no instabilities

Radicals

shoc

k


Chemical Length and Time Scales

0.8 1 1.2 1.4

10-2

10-1

100

Normalized velocity, U/UCJ

Indu

ctio

n Zo

ne le

ngth

, cm

0 0.5 10

1000

2000

3000

0

0.01

0.02

0.03

0.04

0.05

OH

T

Distance, cm

Tem

pera

ture

, K

OH

mol

e fra

ctio

n2H2-O2-60%N2

Δ


Measured Pressures in Tube


Taylor-Zeldovich Expansion Wavecl

osed

end

Lx

particle path

t

0

open

end

2

1 - at rest

3

detonation

expansion fan

Stationary region


Propagating Pressure Wave


Wave Front Has Structure

End plate soot foil


Summary on Detonation Facts

• Detonations have– Characteristic minimum speed (CJ model)– Characteristic peak pressure (CJ model)– Characteristic length scale (ZND model)

• Measure cell width

• Imposes traveling load on tube– Sudden jump in pressure– Decrease in pressure followed by uniform region


Detonations Excite Elastic Waves


Modeling Structural Response To Detonations

• SDOF model for hoop oscillations• Simplified traveling wave model

– Beam on an elastic foundation• Analytical shell models

– (Tang) with rotary inertia• Numerical simulation

– Shell models (Cirak)– FEM models (LS-Dyna)

Need to add mathematical equations


Flexural Waves in Tubes

• Coupled response due to hoop oscillations and bending

• Traveling load can excite resonance when flexural wave group velocity matches wave speed

• Can be treated with analytical and FEM models

Measured strain (hoop)

t (ms)0 2 4 6 8

10-4

Amplification factor

U (m/s)


Measuring Elastic Vibration


rigid collets

stiff I-Beam

straingages

Precision test rig


120o

S1

S2

S3S4 S5 Detonationwave

D=41mm

20mm 20mm

Strain gages:radial spacing: S1, S2, S3axial spacing: S3, S4, S5

vibrometer

S4

S3S5

vibrometer

Gage locations


Comparison of shell model with experiment

15o location


Fracture

Fracture

External Blast


Strain Gage

Locations

Strain Response of Fracturing TubesStrain Response of Fracturing Tubes



Post-test Al 6061-T6 Specimens (Pcj = 6.2 MPa)

Surface Notch Length = 1.27 cm

Outer diameter: 41.28 mm, Wall thickness: 0.89 mm, Length: 0.914 mSurface notch dimensions: Width: 0.25 mm, Notch depth: 0.56 mm, Lengths: 1.27 cm, 2.54 cm, 5.08

cm, 7.62 cm

Detonation wave direction




Fracture Behavior is a Strong Function of Initial Flaw Length


Fracture Threshold Model

Flat Plate Model

analyzed by Newman and Raju (1981)

Actual tube

surface

Fracture Condition:

(ΦΔpR/h)√(πd)/KIc > √(Q)/F

where Q, F = functions of flaw length (2a), flaw depth (d), and wall

thickness (h)

Approximate


Note: 1) Parameters on the axes are

non-dimensional2) Threshold is a 3-D surface

ΔP = Pcj - PatmR = Tube mean radiush = Tube wall thicknessd = Surface notch depth2a = Surface notch lengthKIc = Fracture toughnessΦ = Dynamic

Amplification factor

• Tube material: Al6061-T6• Wall thickness: 0.089 to 0.12

cm• d/h: 0.5 to 0.8• Pcj: 2 to 6 MPa• Axial Flaw Length: 1.3 to 7.6

cm• O.D.: 4.13 cm

RuptureNo RuptureThreshold Theory

Fracture Threshold of Flawed Tubes under Detonation Loading


Using Prestress to Control Crack Propagation Path

Detonation direction


Incipient Crack Kinking

Detonation Direction

Torque Direction(right-hand rule)

Initial Notch

HoopStress

ShearStress

HoopStress

ShearStress

Kinked Incipient Forward and Backward Cracks

Image from Shot 153

Initial Notch


Mixed-Mode Fracture

• Experimental data are compared with numerical data by Melin (1994) using a local kII = 0 criteria

Circles: Forward CracksDeltas: Backward Cracks

Stress Intensity Factors


Effect of Reflected Shear Wave: Crack Path Direction Reversal

• Cracks initially kinked at angles consistent with principal stresses

• The cracks then reversed directions due to reflected shear waves

• Shear wave travel time: 150 μs

Shot 143


Effect of Reflected Shear Wave: Crack Path Direction Reversal

Shear Strain Reversal

Detonation Wave Direction

Rosette 1 (solid) Rosette 2 (dotted)


Effect of Reflected Shear Wave: Additional Kinked Crack

Shot 142


Application to Pulse Detonation

• Pulse detonation engine use repeated detonations to generate thrust

• In development as primary thrust generator (ramjet-type device) and high pressure combustion chamber for jet engines


Testing at WPAFB

Thanks to John Hoke, Royce Bradley and Fred Schauer


Crack Opening – deep flaw

After 4700 cycles After 7500 cycles


DDT

• Deflagration to detonation transition is a common industrial hazard with gaseous explosions

• Compression of gas by flame increases pressure when detonation finally occurs “pressure piling”.

• Represents upper bound in severity of pressure loading.


The path of DDT


burned unburned

1. A smooth flame with laminar flow ahead

2. First wrinkling of flame and instability of upstream flow

3. Breakdown into turbulent flow and a corrugated flame

4. Production of pressure waves ahead of turbulent flame

5. Local explosion of vortical structure within the flame

6. Transition to detonation


Slow Flame (Deflagration)


Fast Flame


DDT after Flame Acceleration Period


Rapid onset of DDT


Structural Response to DDT

Thick walled vessels for elastic responseThin-walled vessels for plastic response and failure

Use bars or tabs as “obstacles” to cause flame acceleration


Reflection of near-CJ Detonation

30% H2 in H2-N2O mixture at 1 atm initial pressure


DDT near end flange

15% H2 in H2-N2O at 1 atm initial pressure


Summary of results for H2-O2 Mixtures

Strains and pressures are a strong function of composition, peak occurs whenDDT is close to the end of the tube.



Computations of Detonation Reflection

• 3-in Schedule 40 316L pipe 1-m long, 38 mm diam, 4.5 mm wall 240 MPa yield stress

• Reflected CJ detonation. CJ Velocity 2600 m/s, PCJ/Po = 26

• Three initial pressures 3, 6, 9 atm

• LS-DYNA simulation with traveling load model of waves

Structural Response - Shepherd 1413 atm




Spatial distribution of Effective Plastic Strain

3 atm

6 atm

9 atm


Plastic Deformation

• It is useful to use plastic deformation to accommodate rare events.

• Need to have more data and modeling to determine peak allowable impulses and pressures to avoid rupure.


Bends and Tees

• Limited data available• Important for plants and facilities• Some enhancement of hoop load due to wave

reflections• Transverse loads can be quite significant

– Creates bending in tubes– Supporting structures (hangers) can fail– Flange bolts can fail in shear due to transverse loads


Detonations and ASME Code Rules

• Not covered under current BPVC VIII or Piping Code B31• Proposed code case for impulsively loaded vessels is under

development by ASME Task Force on Impulsively Loaded Vessels, SWG/HPV, ASME VIII.

• Current impulsive loading code case intended to cover vessels used to contain high explosive detonation.– many common elements associated with dynamic loading– Further work needed to treat gas detonation specific issues


Issue for Gaseous Detonation

• Loading is more difficult to define for gases than for HE detonation– More testing is needed to have generic results

• Mixed loading regime, not purely impulsive.• Plastic deformation will require considering entire

loading history.• Traveling load aspects of gaseous detonation


Extending the Code

• Ad hoc design practices can be standardized

• Analysis of accidents and DDT harder to standardize

• Designers and analysts might be able to use extended code as a basis for building vessels and

piping to contain gaseous detonation

– Elastically for high frequency or intentional events

– Plastically for rare events or one-time use

• Much work has already been done for impulsively loaded vessels code case development

– Dynamic response of materials

– Stain hardening, strain rate effects

– Fracture safe design

– Plastic instability limits (incomplete)

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