Structure Blast Wave

Structural Response to Explosions Joseph E Shepherd California Institute of Technology Pasadena, CA USA 91125 Presented at 1st European Summer School on Hydrogen Safety University of Ulster, August 2007

Structural Response - Shepherd

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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. Structural Response - Shepherd

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Outline • • • •

Overview Determining Structural Loads Determining Structural Response Examples


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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


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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?


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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. Structural Response - Shepherd

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Pasadena TX 1989 – C2H4

Flixborough 1974 - cyclohexane

(20 Kg H2 )


Port Hudson 1974 – C3H8

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Nuclear Blast Wave Damage – 5 psi (34 kPa)


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Nuclear Blast Wave Effects

5 psi (34 kPa) Structural Response - Shepherd

1.7 psi (11.7 kPa) 9

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.


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Pasadena TX 1989


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Elements of Structural Response Analysis •

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

•

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 –

•

HAZOP or FEMA

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


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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. Structural Response - Shepherd

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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


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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)


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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


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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 Structural Response - Shepherd

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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)


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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


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Combustion and Pressure Waves • Overall Combustion Reaction – major species H2 + ½(O2 + 3.76 N2) Æ 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


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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


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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

σ = ρu ρ

– Expansion ratio

V f = σST A f / A = σSTeff

– Flame velocity – Flow velocity

U = V f − STeff = (σ − 1) STeff ST

Burned (u =0)

flame Structural Response - Shepherd

b

Vf

Unburned u > 0

u=0

Blast wave 23

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)

Expansion wave Burned u >0

Unburned u = 0

Detonation wave u

x Structural Response - Shepherd

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Loading Histories •

Pressure-time histories can be derived from several sources –

Experimental measurements

–

Analytical models with thermodynamic computation of

Slow flame in vessel

parameters –

Detailed numerical simulations using computation fluid dynamics

–

Empirical correlations of data

–

Approximate numerical models of blast wave

High speed flame in vessel

propagation (Blast-X)

•

Characterizing pressure-time histories –

Single peak or multiple peaks

–

Rise time

–

Peak pressure

–

Duration


Nonideal vapor cloud explosion

Ideal vapor detonation

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Pressure Loading Characterization

τload

• • •

τunload

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


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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). Structural Response - Shepherd

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Ideal Blast Wave Sources Simplest form of pressure loading – due concentrated, rapid release of energy High explosive or “prompt” gaseous detonation. Main shock wave followed by pressure wave and gas motion, possibly secondary waves.


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Blast Wave from Hydrogen-Air Detonation


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Blast and Shock Waves ΔP τ+

τ-

• 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


Specific impulse!

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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 Structural Response - Shepherd

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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)

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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.


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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? Structural Response - Shepherd

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Scaling of Blast Pressure – Ideal Detonation Comparison of fuel-air bag tests to high explosives Work done at DRES (Suffield, CANADA) in 1980s


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Scaling of Impulse – Ideal Detonation Surface burst

Air burst For the same overpressure or scaled impulse at a given distance, M(surface) = 1/2 M(air) Structural Response - Shepherd

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Energy scaling of H2-air blast

Energy Equivalence 100 MJ/kg of H2 or 2.71 MJ/kg of fuelair mix for stoichiometric.


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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. Structural Response - Shepherd

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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.


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Idealized Interactions

Enhancement depends: Incident wave strength Angle of incidence


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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 Structural Response - Shepherd

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

Tang and Baker 1999 Structural Response - Shepherd

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What is Effective Flame Speed? Consider volume displacement of a wrinkled (turbulent) flame growing in a mean spherical fashion. Expansion ratio

Dorofeev 2006 Structural Response - Shepherd

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Internal Explosion - Deflagration • Limiting pressure determined by thermodynamic considerations Combustion wave – Adiabatic combustion process – Chemical equilibrium in products – Constant volume

• Pressure-time history determined by flame speed Vf = Sf + u


Products

S L

fuel-air mixture

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Burning Velocity Depends on substance, composition, pressure, temperature


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Expansion ratio and Flame temperature • Related to flame temperature through gas law

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

E=

V products Vreactants


=

N productsTproducts N reactantsTreactants

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Expansion ratio


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Adiabatic Flame Temperature •

Temperature of products if there are no heat losses Hreactants(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, 11001400 oC at flammability limit.

•

Values are similar for all HC fuels when expressed in terms of equivalence ratio.


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Pressure in Closed Vessel Explosion Peak pressure limited by heat transfer during burn and any Venting that takes place due to openings or structural failure


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Adiabatic Explosion Pressure •

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

•

Energy balance at constant volume Ereactants(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


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Measured Peak Pressure vs Calculated


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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


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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.54

Consequence of Forces I. • Rigid body motions – Translation – Rotation

X’ = X – Xcm distance from center of mass Structural Response - Shepherd

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Internal Forces Due to an Explosion • Force on a surface element dS

• Stress tensor σ


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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 Structural Response - Shepherd

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Consequences of forces – large strains • Onset of yielding for σ ~ σY • Necking occurs in plastic regime σ > σY • Plastic instability and rupture for 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


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Stress-Strain Relationships


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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.


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Some Typical Material Properties

Material Aluminum 6061-T6 Aluminum 2024-T4 Steel (mild) Steel stainless Steel (HSLA) Concrete Fiberglass Polycarbonate PVC Wood Polyethylene (HD)

ρ (kg/m3) 3 2.71 x 10 3 2.77 x 10 3 7.85 x 10 3 7.6 x 10 3 7.6 x 10 3 7.6 x 10 3 1.5-1.9 x 10 3 1.2-1.3 x 10 3 1.3-1.6 x 10 3 0.4-0.8 x 10 3 0.94-0.97 x 10


E (GPa) 70 73 200 190 200 30-50 35-45 2.6 0.2-0.6 1-10 0.7

G (GPa) 25.9 27.6 79 73

ν 0.351 0.342 0.266 0.31 0.29

σy σu εrupture (MPa) (MPa) 241 290 0.05 290 441 0.3 248 410-550 0.18-0.25 286-500 760-1280 0.45-0.65 1500-1900 1500-2000 0.3-0.6 20-30 0 100-300 55 60 45-48 33-55 20-30 37 -

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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


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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 Structural Response - Shepherd

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Static Stresses in Spherical Shell • Balance membrane stresses with internal pressure loading • Force balance on equator

R R

• Membrane stress

Validate only for thin-wall vessels h < 0.2 R Structural Response - Shepherd

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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


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Bending of Beams • Force on beam due to integrated effects of pressure loading

• Pure bending has no net longitudinal stress • Deflection for uniform loading


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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

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Is direct stress wave propagation important? • Time scale very fast compared to main structural response T ~ L/C Steel Aluminum

Cl (m/s) 6100 3205

Cs(m/s) 3205 3155

– Average out in microseconds (10-6 s)

• Stress level low compared to yield stress σ ~ Δ P ~ 10 MPa << σY = 200- 500 MPa Direct stress propagation within the structural elements is usually not relevant for structural response to gaseous explosions. Structural Response - Shepherd

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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


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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


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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


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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


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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 “springmass” system

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

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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

x

τ

h

R P(t)

t

frequency Structural Response - Shepherd

period 74

SODF - Square Pulse Pulse length τ: 10μs


Pulse length τ: 100μs

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SDOF – Static Regime • Very slow application of load – (quasi-static) no oscillations T << τu or τL • Static deflection τl force

FMax

displacement

T Structural Response - Shepherd

time 76

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


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SDOF – Sudden regime • Quick application of load and long duration τu >> T • Peak deflection is twice static value for same maximum load τ FMax

force

displacement

T Structural Response - Shepherd

time 78

SDOF - Dynamic load factor (DLF)


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Considerations about material properties • Simple models:

σ

– perfectly plastic, – elastic perfectly plastic

. ε

• More realistic models – Strain hardening σY (ε)

.

– Strain rate effects, σY(dε/dt)

ε

σ ε

– Temperature effects σ (T) Y


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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.


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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


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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


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Example •

Blast loading of a cantilever beam – – –

Giordona et al elastic response Van Netton and Dewey plastic response Baker et al energy method


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Initial stages of shock diffraction over a cantilever beam


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

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Later stages of diffraction over a cantilever beam


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

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Applied Load and Oscillations of Beam

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

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Plastic Deformation of Blast loaded Cantilever


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

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Blast Loading



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Shock tube experiments



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Structural Response of Piping to Internal Gaseous Detonation


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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)


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Recent Accidental Detonations

Hamaoka-1 NPP

Brunsbuettel KBB

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

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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


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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.”


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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


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Chapman-Jouguet (CJ) Model Combustion wave moves at minimum speed consistent with conservation of Thermodynamics and elementary gas dynamics mass, momentum and energy across the wave front. Equivalent to products Adequate predict idealequal wave speed away from wave front with to a relative velocity to the speed of sound “sonic or CJ condition”


97


98

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


shock

Products Products Radicals

UCJ

Reactants

99

Chemical Length and Time Scales

Δ 3000

0.05

Temperature, K

0.04 2000 0.03

OH 0.02 1000 0.01

Induction Zone length, cm

0

0

0.5

Distance, cm

1

OH mole fraction

T

0

2H2-O2-60%N2

10 0

10 -1

10

-2

0.8

1

1.2

1.4

Normalized velocity, U/UCJ Structural Response - Shepherd

100

Measured Pressures in Tube


101

Taylor-Zeldovich Expansion Wave t

open end

closed end

particle path

3 Stationary region 2

0

expansion fan

1 - at rest

x L


detonation

102

Propagating Pressure Wave


103

Wave Front Has Structure

End plate soot foil


104

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


105

Detonations Excite Elastic Waves


106

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


107

Flexural Waves in Tubes Measured strain (hoop) 10-4

• •

•

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

0

2

4 6 t (ms)

8

Amplification factor

U (m/s) Structural Response - Shepherd

108

Measuring Elastic Vibration


109

Precision test rig

rigid collets

strain gages stiff I-Beam


110

Gage locations 120o

vibrometer S4

S3

S2 S5 S1

D=41mm Detonation wave

20mm 20mm vibrometer Strain gages: radial spacing: S1, S2, S3 axial spacing: S3, S4, S5


S4

S5 S3

111

Comparison of shell model with experiment

15o location


112

Fracture

External Blast Fracture


113

Strain StrainResponse Responseof ofFracturing FracturingTubes Tubes

Strain Gage Locations


114


115

Fracture Behavior is a Strong Function of Initial Flaw Length Outer diameter: 41.28 mm, Wall thickness: 0.89 mm, Length: 0.914 m Surface notch dimensions: Width: 0.25 mm, Notch depth: 0.56 mm, Lengths: 1.27 cm, 2.54 cm, 5.08 cm, 7.62 cm

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

Surface Notch Length = 1.27 cm

Surface Notch Length = 2.54 cm Surface Notch Length = 5.08 cm

Surface Notch Length = 7.62 cm Structural Response - Shepherd

Detonation wave direction

116

Fracture Threshold Model

Flat Plate Model analyzed by Newman and Raju (1981) Approximate Fracture Condition: (ΦΔpR/h)√(πd)/KIc > √(Q)/F where Q, F = functions of flaw length (2a), flaw depth (d), and wall thickness (h) Structural Response - Shepherd

Actual tube surface 117

Fracture Threshold of Flawed Tubes under Detonation Loading ΔP = Pcj - Patm R = Tube mean radius h = Tube wall thickness d = Surface notch depth 2a = Surface notch length KIc = Fracture toughness Φ = Dynamic Amplification factor

Rupture No Rupture Threshold Theory

• • • • • •

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

Note: 1) Parameters on the axes are non-dimensional 2) Threshold is a 3-D surface Structural Response - Shepherd

118

Using Prestress to Control Crack Propagation Path

Detonation direction


119

Incipient Crack Kinking Detonation Direction Initial Notch Hoop Stress

Shear Stress

Hoop Stress

Shear Stress

Torque Direction (right-hand rule)

Initial Notch

Image from Shot 153 Kinked Incipient Forward Structural Response - Shepherd and Backward Cracks

120

Mixed-Mode Fracture Stress Intensity Factors

•

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


Circles: Forward Cracks Deltas: Backward Cracks

121

Effect of Reflected Shear Wave: Crack Path Direction Reversal

Shot 143

•

•

•

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


122

Effect of Reflected Shear Wave: Crack Path Direction Reversal

Shear Strain Reversal

Rosette 1 (solid) Rosette 2 (dotted)


123

Detonation Wave Direction

Effect of Reflected Shear Wave: Additional Kinked Crack


Shot 142

124

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


125

Testing at WPAFB

Thanks to John Hoke, Royce Bradley and Fred Schauer Structural Response - Shepherd

126

Crack Opening – deep flaw

After 4700 cycles


After 7500 cycles

127

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.


128

The path of DDT


129

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 Structural Response - Shepherd

130

Slow Flame (Deflagration)


131

Fast Flame


132

DDT after Flame Acceleration Period


133

Rapid onset of DDT


134

Structural Response to DDT Thick walled vessels for elastic response Thin-walled vessels for plastic response and failure

Use bars or tabs as “obstacles” to cause flame acceleration Structural Response - Shepherd

135

Reflection of near-CJ Detonation

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

136

DDT near end flange

15% H2 in H2-N2O at 1 atm initial pressure Structural Response - Shepherd

137

Summary of results for H2-O2 Mixtures

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


138


139

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

140


3 atm

141


6 atm

142


9 atm

143

Spatial distribution of Effective Plastic Strain 6 atm

3 atm

9 atm


144

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.


145

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


146

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


147

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


148

Extending the Code •

Ad hoc design practices can be standardized

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Analysis of accidents and DDT harder to standardize

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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

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Fracture safe design

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Plastic instability limits (incomplete)


149

Structure Blast Wave

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