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 1

 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 2

 Outline
Overview
Determining Structural Loads
Determining Structural Response
Examples

Structural Response - Shepherd 3

 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?

Structural Response - Shepherd 5

 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 6

 Pasadena TX 1989 C2H4 Flixborough 1974 - cyclohexane

(20 Kg H2 )

Port Hudson 1974 C3H8

Structural Response - Shepherd 7
 Nuclear Blast Wave Damage 5 psi (34 kPa)

Structural Response - Shepherd 8

 Nuclear Blast Wave Effects

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

Structural Response - Shepherd 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.

Structural Response - Shepherd 10

 Pasadena TX 1989

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 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

Structural Response - Shepherd 12

 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 13
Structural Response - Shepherd 14
 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

Structural Response - Shepherd 15

 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)

Structural Response - Shepherd 16

 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

Structural Response - Shepherd 17

 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 18

 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)

Structural Response - Shepherd 19

 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

Structural Response - Shepherd 20

 Combustion and Pressure Waves

Overall Combustion Reaction major species

H2 + (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

Structural Response - Shepherd 21

 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

Structural Response - Shepherd 22

 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 = u
b

Flame velocity V f = ST A f / A = STeff

Flow velocity U = V f STeff = ( 1) STeff

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

flame Blast wave

Structural Response - Shepherd 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

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

Detonation
wave
u

x
Structural Response - Shepherd 24
 Loading Histories
Pressure-time histories can be derived from several

sources
Slow flame in vessel
Experimental measurements

Analytical models with thermodynamic computation of

parameters

Detailed numerical simulations using computation fluid

High speed flame in vessel
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
Nonideal vapor cloud explosion
Peak pressure

Duration

Structural Response - Shepherd 25

Ideal vapor detonation
 Pressure Loading Characterization

load unload

Structural response time T vs. loading and unloading time scales I

Peak pressure P vs. Capacity of structure
Loading regimes
Slow (quasi-static), typical of flame inside vessels T << L or u
Sudden, shock or detonation waves L << T
Short duration Impulsive U << T
Long duration - Step load T << U
Structural Response - Shepherd 26
 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 27

 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
Leading shock front
P 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!
Structural Response - Shepherd 30
 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 o and co
Limiting cases
Strength of shock wave
Strong P >> Po
Weak P << Po
Distance from source
Near R ~ Rsource
Far R >> Rsource

Structural Response - Shepherd 31

 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 Relationships:

pressure P/Po P/Po = F(R/Rs)

distance R/Rs I/( P Ts) = G(R/Rs)

time t/Ts
Impulse (specific) I/(Po Ts)

Structural Response - Shepherd 32

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 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.

Structural Response - Shepherd 33

 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 34
Scaling of Blast Pressure Ideal Detonation

Comparison of fuel-air
bag tests to high explosives

Work done at DRES

(Suffield, CANADA) in 1980s

Structural Response - Shepherd 35

 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 36
 Energy scaling of H2-air blast

Energy Equivalence

100 MJ/kg of H2

or

2.71 MJ/kg of fuel-

air mix for
stoichiometric.

Structural Response - Shepherd 37

 Hydrogen-air Detonation in a Duct
Blast waves in ducts
decay much more slowly
than unconfined blasts
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 38

 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.

Structural Response - Shepherd 39

 Idealized Interactions

Enhancement depends:

Incident wave strength

Angle of incidence

Structural Response - Shepherd 40

 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 41
 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 42

 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 43

 Internal Explosion - Deflagration

Limiting pressure determined by thermodynamic

considerations
Combustion wave
Adiabatic combustion process
Chemical equilibrium in products
Constant volume
Pressure-time history determined
Products
by flame speed
S
Vf = Sf + u L
fuel-air mixture

Structural Response - Shepherd 44

 Burning Velocity
Depends on substance, composition, pressure, temperature

Structural Response - Shepherd 45

 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

V products N productsTproducts
E= =
Vreactants N reactantsTreactants

Structural Response - Shepherd 46

 Expansion ratio

Structural Response - Shepherd 47

 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, 1100-
1400 oC at flammability limit.
Values are similar for all HC fuels when expressed in terms of
equivalence ratio.

Structural Response - Shepherd 48

 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

Structural Response - Shepherd 49

 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

Structural Response - Shepherd 50

 Measured Peak Pressure vs Calculated

Structural Response - Shepherd 51

Structural Response - Shepherd 52
 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

Structural Response - Shepherd 53

 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
Structural Response - Shepherd 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 55

 Internal Forces Due to an Explosion

Force on a surface element dS

Stress tensor

Structural Response - Shepherd 56

 Consequence of forces small strains (<0.2 %)

Elastic deformation
Elastic strain

Elastic shear

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

Structural Response - Shepherd 57

 Consequences of forces large strains
Onset of yielding for
~ Y
Necking occurs in
plastic regime > Y
Plastic instability
and rupture
for > u
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

Structural Response - Shepherd 58

 Stress-Strain Relationships

Structural Response - Shepherd 59

 Yield and Ultimate Strength
Yield point YP 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 max < YP/2
Von Mises or octahedral shear stress criterion

Onset of localized permanent deformation occurs well

before complete plastic collapse of structure occurs.

Structural Response - Shepherd 60

 Some Typical Material Properties

E G y u rupture
Material (kg/m3) (GPa) (GPa) (MPa) (MPa)
3
Aluminum 6061-T6 2.71 x 10 70 25.9 0.351 241 290 0.05
3
Aluminum 2024-T4 2.77 x 10 73 27.6 0.342 290 441 0.3
3
Steel (mild) 7.85 x 10 200 79 0.266 248 410-550 0.18-0.25
3
Steel stainless 7.6 x 10 190 73 0.31 286-500 760-1280 0.45-0.65
3
Steel (HSLA) 7.6 x 10 200 0.29 1500-1900 1500-2000 0.3-0.6
3
Concrete 7.6 x 10 30-50 20-30 - 0
3
Fiberglass 1.5-1.9 x 10 35-45 - 100-300 -
3
Polycarbonate 1.2-1.3 x 10 2.6 55 60 -
3
PVC 1.3-1.6 x 10 0.2-0.6 45-48 - -
3
Wood 0.4-0.8 x 10 1-10 - 33-55 -
3
Polyethylene (HD) 0.94-0.97 x 10 0.7 20-30 37 -

Structural Response - Shepherd 61

 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

Structural Response - Shepherd 62

 Statics vs. Dynamics

Static loading T >> l, u

Loading and unloading times long compared to
characteristic structural response time
Inertia unimportant
Response determined completely by stiffness,
magnitude of load.
Dynamic loading T
Loading or unloading time short compared to
characteristic structural response time
Inertia important
Response depends on time history of loading
Structural Response - Shepherd 63
 Static Stresses in Spherical Shell
Balance membrane
stresses with internal
pressure loading
Force balance on equator R

Membrane stress

Validate only for thin-wall vessels h < 0.2 R

Structural Response - Shepherd 64
 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

Structural Response - Shepherd 65

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

Pure bending has no net

longitudinal stress
Deflection for uniform
loading

Structural Response - Shepherd 66

 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

Structural Response - Shepherd 67

Is direct stress wave propagation important?
Time scale very fast compared to main
structural response T ~ L/C
Cl (m/s) Cs(m/s)
Steel 6100 3205
Aluminum 3205 3155
Average out in microseconds (10-6 s)
Stress level low compared to yield stress

~ P ~ 10 MPa << Y = 200- 500 MPa

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

Structural Response - Shepherd 68

 Structural motions
Element vibrations
Membranes or shells

Plates or beams

Modes of flexural motion

Standing waves, frequencies i
Propagating dispersive waves (k)
Coupled motions of entire structure

Structural Response - Shepherd 69

 Two Special Situations
Loading on small objects
Represent forces as drag coefficients dependent on shape and
orientation and function of flow speed.
F = 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.
= /E + T

Structural Response - Shepherd 70

 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

Structural Response - Shepherd 71

 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 (Roarkes 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

Structural Response - Shepherd 72

 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 I

corresponding to different vibrational modes
Fundamental (lowest) mode usually most relevant

Structural Response - Shepherd 73

 Single Degree of Freedom Models (SDOF)
Example - radial oscillation of
h
a shell. x R
p
Allow only for radial
displacement x of tube P(t)
surface
t
Assumes radial and axial
symmetry of load
Elastic oscillations only
Results in harmonic oscillator
equation (no damping)

frequency period
Structural Response - Shepherd 74
 SODF - Square Pulse

Pulse length : 10s Pulse length : 100s

Structural Response - Shepherd 75

 SDOF Static Regime

Very slow application of load (quasi-static) no

oscillations
T << u or L
Static deflection
l

FMax
force

displacement

time
T
Structural Response - Shepherd 76
 SDOF -Impulsive Regime
Sudden load application, short
duration of loading << T
Linear scaling between
maximum strain/ displacement
and impulse in elastic regime:
Impulse generates initial
velocity

Energy conservation
determines maximum
deflection

Structural Response - Shepherd 77

 SDOF Sudden regime
Quick application of load and long duration
u >> T
Peak deflection is twice static value for same
maximum load

FMax
force

displacement

time
T
Structural Response - Shepherd 78
 SDOF - Dynamic load factor (DLF)

Structural Response - Shepherd 79

 Considerations about material properties

Simple models:

perfectly plastic,
elastic perfectly plastic

More realistic models .

Strain hardening Y ()

.
Strain rate effects, Y(d/dt)

Temperature effects (T)

Y

Structural Response - Shepherd 80

 SDOF - Plasticity
Replace kX with nonlinear
relationship based on flow
stress curve ()
Energy absorbed by
plastic work is much
higher than elastic work
Peak deformation for
impulsive load scales with
impulse squared.

Structural Response - Shepherd 81

 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

Structural Response - Shepherd 82

 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
Structural Response - Shepherd 83
 Example
Blast loading of a
cantilever beam
Giordona et al elastic
response
Van Netton and Dewey
plastic response
Baker et al energy
method

Structural Response - Shepherd 84

 Initial stages of shock diffraction over a cantilever beam

Structural Response - Shepherd Giordano et al, Shock Waves 14 (1-2), 103-110, 2005. 85
 Later stages of diffraction over a cantilever beam

Structural Response - Shepherd Giordano et al, Shock Waves 14 (1-2), 103-110, 2005. 86
 Applied Load and Oscillations of Beam

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

Structural Response - Shepherd 87
 Plastic Deformation of Blast loaded Cantilever

Structural Response - Shepherd Van Netten and Dewey, Shock Waves (1997) 7: 175190 88
 Blast Loading

Structural Response - Shepherd Van Netten and Dewey, Shock Waves (1997) 7: 175190 89
 Shock tube experiments

Van Netten and Dewey, Shock Waves (1997) 7: 175190

Structural Response - Shepherd 90

 Structural Response of Piping to
Internal Gaseous Detonation

Structural Response - Shepherd 91

 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)

Structural Response - Shepherd 92

 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 93
 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

Structural Response - Shepherd 94

 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.

Structural Response - Shepherd 95

 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

Structural Response - Shepherd 96

 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
away from wave front with to predict
a relative idealequal
velocity wave speed
to the speed of sound sonic
or CJ condition

Structural Response - Shepherd 97

Structural Response - Shepherd 98
 ZND Model
Steady reactive flow behind nonreactive shock
Shock-induced chemical reaction
1D smooth flow no instabilities

UCJ
Products Products
Radicals Reactants

shock

Structural Response - Shepherd 99

 Chemical Length and Time Scales

3000 0.05
T
0.04

OH mole fraction
Temperature, K

2000
0.03
OH
0.02
1000

0.01

0 0
0 0.5 1
Distance, cm
Induction Zone length, cm

2H2-O2-60%N2
10 0

10 -1

-2
10

0.8 1 1.2 1.4

Normalized velocity, U/UCJ

Structural Response - Shepherd 100

Measured Pressures in Tube

Structural Response - Shepherd 101

 Taylor-Zeldovich Expansion Wave
t

particle path

open end
closed end

3
Stationary region

expansion fan
1 - at rest
0 x
L

Structural Response - Shepherd detonation 102

 Propagating Pressure Wave

Structural Response - Shepherd 103

 Wave Front Has Structure

End plate soot foil

Structural Response - Shepherd 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

Structural Response - Shepherd 105

 Detonations Excite Elastic Waves

Structural Response - Shepherd 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

Structural Response - Shepherd 107

 Flexural Waves in Tubes
Measured strain (hoop)

10-4

Coupled response due to hoop 0 2 4 6 8

oscillations and bending t (ms)
Traveling load can excite
resonance when flexural wave
group velocity matches wave
Amplification factor
speed
Can be treated with analytical
and FEM models

U (m/s)
Structural Response - Shepherd 108
 Measuring Elastic Vibration

Structural Response - Shepherd 109

 Precision test rig

rigid collets

strain
gages
stiff I-Beam

Structural Response - Shepherd 110

 Gage locations
120o

D=41mm
vibrometer S2

S4 S3 S5 Detonation
wave
S1

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

Structural Response - Shepherd 111

 Comparison of shell model with experiment

15o location

Structural Response - Shepherd 112

 Fracture

External Blast

Fracture

Structural Response - Shepherd 113

 Strain
StrainResponse
Responseof
ofFracturing
FracturingTubes
Tubes

Strain
Gage
Locations

Structural Response - Shepherd 114

Structural Response - Shepherd 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 116

Detonation wave direction
 Fracture Threshold Model

Flat Plate
Model
analyzed by
Newman and
Raju (1981)

Approximate

Fracture Condition:
(pR/h)(d)/KIc > (Q)/F
where Q, F = functions of flaw length
(2a), flaw depth (d), and wall Actual
thickness (h) tube
surface
Structural Response - Shepherd 117
 Fracture Threshold of Flawed Tubes under Detonation Loading
P = Pcj - Patm
R = Tube mean radius
h = Tube wall thickness
d = Surface notch depth
Rupture 2a = Surface notch length
No Rupture KIc = Fracture toughness
= Dynamic
Threshold Theory 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

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

Structural Response - Shepherd 119

 Incipient Crack Kinking

Detonation Direction
Initial Notch

Hoop Shear Hoop Shear Torque Direction

Stress Stress Stress Stress (right-hand rule)

Initial Notch

Image from Shot 153

Kinked Incipient Forward
and Backward
Structural Response - Shepherd 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

Structural Response - Shepherd
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 s

Structural Response - Shepherd 122

 Effect of Reflected Shear Wave: Crack Path Direction Reversal

Shear Strain Reversal

Rosette 1 (solid) Rosette 2 (dotted)

Structural Response - Shepherd 123

Detonation Wave Direction
 Effect of Reflected Shear Wave: Additional Kinked Crack

Structural Response - Shepherd 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

Structural Response - Shepherd 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

Structural Response - Shepherd 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.

Structural Response - Shepherd 128

 The path of DDT

Structural Response - Shepherd 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)

Structural Response - Shepherd 131

 Fast Flame

Structural Response - Shepherd 132

 DDT after Flame Acceleration Period

Structural Response - Shepherd 133

 Rapid onset of DDT

Structural Response - Shepherd 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.

Structural Response - Shepherd 138

Structural Response - Shepherd 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

Structural Response - Shepherd 3 atm 141
Structural Response - Shepherd 6 atm 142
Structural Response - Shepherd 143
9 atm
Spatial distribution of Effective Plastic Strain

6 atm

3 atm

9 atm

Structural Response - Shepherd 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.

Structural Response - Shepherd 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

Structural Response - Shepherd 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

Structural Response - Shepherd 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

Structural Response - Shepherd 148

 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)

Structural Response - Shepherd 149