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Structural Response to Blast Loads
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THE UNIVERSITY OF ADELAIDE DEPARTMENT OF CIVIL, ENVIRONMENTAL
AND MINING ENGINEERING
C&ENVENG 4099 Structural Response to Blast Loads (3 pts)
C&ENVENG 7059 Structural Response to Blast Loads (3 pts)
A/P. Chengqing Wu Room N236
Civil & Env. Eng. Phone: 8313-4834
[email protected]
Hanout Contents:
Part 1: Syllabus
Course objectives, methods of instruction, assessment, time
table
Part 2: Lecture Course
Introduction Characterization of Blast Loading Structural
Dynamics Blast Resistant Capacity Analysis Structural Design
Against Blast Loading Retrofitting Structures against Blast
Loading
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Structural Response to Blast Loads
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THE UNIVERSITY OF ADELAIDE DEPARTMENT OF CIVIL &
ENVIRONMENTAL ENGINEERING
C&ENVENG 4099 Structural Response to Blast Loads (3 pts)
C&ENVENG 7059 Structural Response to Blast Loads (3 pts)
LECTURER: A/P. Chengqing Wu Room N236 Civil & Env. Eng. .
Phone: 8313-4834 Email: [email protected] 10/02/2015
COURSE OBJECTIVES: Recently terrorist attacks are becoming more
and more realistic threats to society. These terrorist attacks may
cause severe damage and even collapse of structures. As a
consequence, there is always not only enormous economic loss, but
also injuries and fatalities. To prevent catastrophic collapse of
structures under blast loading, there is a need to be able to
develop blast resistant systems that can be applied to the
buildings to protect the its occupants. The objective of this
course is intended to understand the fundimenatal characteristics
of blast loads, the basic principles of dynamic analysis of
structures and blast resistant design. Topics covered include:
blast loads; dynamic analysis method and structural design againt
blast loads. It is also to develop the ability to communicate
through report writing and to encourage original thought which are
essential attributes to practising engineers.
ASSESSMENT: Tutorial Quizzes: Design Project I: Design Project
II:
10% 50% 10% 30%
LEARNING RESOURCES: Textbook: None. Lecture note: Avaliable from
myuni. References: Design of Blast Resistant Buildings in
Petrochemical Facilities (ASCE 1997); UFC Structures to Resist the
Effects of Accidental Explosion 2008; Structural Dynamics (Mario
Paz 1985).
Assignments: will be handed out in class. MYUNI: Staff contact
details; Lecture notes; Assignments; Design project in pdf format;
Reference papers
Time table: Please note that these are approximate lecture times
that can be increased or decreased depending on the progress.
Day Time Venue Description Tuesdays 3-4pm Ligertwood 231 Law
Lecture Theatre 1 Lectures Thursdays 3-4pm Barr Smith South 3029
Flentje Lecture
Theatre Lecture/Tutorial/quiz
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Structural Response to Blast Loads
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Chapter 2
Characterization of Blast Loading
2.1 Blast induced ground vibrtions Surface and underground mine
blasting (see Fig. 2.1), quarry blasting, construction blasting and
underground ammunition storage blasting will generate ground
vibrations
Fig. 2.1 Underground explosion
2.1.1 Basic Theory of Wave Propagation
i. Characteristics of sinusoidal wave
Dropping a stone in the water, it will create a water wave that
results in the motion of a bobbing cork. The water wave that
excites the cork is described by a sinusoidal wave either as time
history at a given position (Fig. 2.2a), or in terms of the
location at a given instant of time (Fig. 2.2b). The
characteristics of the sinusoidal wave include its wavelength, , or
the distance between wave crests, the speed, c , at which it
travels outward from the stones impact; and the frequency, f , or
the number of times the cork bobs up and down in 1 minute. The wave
propagation velocity c should not be confused with the particle
velocity u& because c is the speed with which the water wave
passes by the cork, whereas u& is the speed at which the cork
moves up and down while the wave passes by.
-2.5
0
2.5
0 15
T
U u
time t
Fig. 2.2a. Sinusoidal displacement at a fixed point
(x=constant)
Fig. 2.2b. Sinusoidal displacement at one instant
(t=constant)
-2.5
0
2.5
0 15
U distance x u
Ground Inhabited building distance DSD
Ds
Dc
Rock Mass
Soil Mass
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Structural Response to Blast Loads
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The time and distance descriptions can be illustrated in a
generalized mathematical equation of the displacement u
)sin(max tKxuu += (2-1) where maxu is the maximum displacement,
K is a constant called the wave number, is also a constant called
the circular natural frequency, and t is time. If time and
frequency are constants, the variation of displacement u with the
distance can be described as
)sin(max constKxuu += (2-2) If the wavelength is defined as the
distance at which the wave repeats, K must be equal to
pi /2 to result in the sine function to repeat every time x
increases by an amount equal to .
On the other hand, if the location and wavelength are constants,
the variation with time at a fixed point becomes
)sin(max tconstuu += (2-3) Since the period, T, is the time
between repetitions, must be equal to T/2pi to cause the sine
function to repeat when time advances by one period.
Since the wave repeats after a time called the period T, the
frequency f or number of times the wave repeats itself each second
is then 1/T and the circular natural frequency (which has a unit of
radians) is
fT
pipi 212 =
= (2-4)
The frequency f (which has a unit of herz or second-1), is not
the same as the circular natural frequency and should not be
confused when calculating peak accelerations and displacements of
sinusoidal waves.
For the sinusoidal waves, the wavelength and the propagation
velocity c are related through the period T as
fccT1
== (2-5)
Since velocity is defined as the change in displacement per unit
time, the first order derivative of Equation 2-1 with respect to
time will give the particle velocity u& as
)cos(max tKxudtdu
u +==& (2-6)
accordingly the acceleration u&& as
)sin(2max tKxudtud
u +==&
&& (2-7)
Blast induced waves such as compressive wave can also be
described as by their wave length, propagation velocity and
frequency in the same fashion as the water wave. There is one
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Structural Response to Blast Loads
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difference between a surface water wave and one that propagates
along the ground; however, it does not affect any of the forgoing
relations. The particle motion for a water wave is progressive,
while the solid is retrogressive. In other words, at top of a
surface water wave the cork will be moving in the direction of the
propagation, whereas at the top of a surface ground wave, soil
particles will be moving in a direction opposite to the propagation
direction.
ii. Plane waves and plane-wave equations The simplest geometry
for plane wave propagation is that the propagation occurs only in
the direction down a long bar. As a wave travels outward from the
source in Fig. 2.3, the front of the wave becomes less curves with
the increasing distance. At the distance where the particle motion
is parallel along the structure of concern (such as house), the
wave front is said to behave as a plane.
Fig. 2.3 Plane-wave geometry (Plan view)
Plane wave approximation is reached if a difference of 5%
between the particle motion vectors is tolerable. For example, the
plane wave approximation would be appropriate for a 10 m wide house
if it is located at 15 m away from a blast centre. As shown in the
Fig. 2.3, if the vector between the source and the midpoint p of
the structure is of length R, the edge vector e which is 5 m away
is 105% R when the angle between p and e is 180. Thus 5/R = tan 180
and R is approximately 15 m. Compressive (longitudinal) waves
generate particle motions in a direction that are parallel to the
direction of wave propagation as shown in Fig. 2.4.
Fig. 2.4 Particle motion variation with wave type for
compressive wave
When compressive wave travels in the direction of its
propagation, the distance the wave travels between times t1 and t2
is the product of the time interval and propagation velocity as
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Structural Response to Blast Loads
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shown in the Fig. 2.5. Then, the generalized mathematical
description of displacements caused by a plane-wave travelling in
the positive direction
)sin(max ctKxuu = (2-8) With K=1, the particle velocity is
expressed as
)cos(max ctxcudtdu
u ==& (2-9)
Fig. 2.5 Transmission of wavefronts for sinusoidal pulse
iii. Strains induced by stress wave Strain is usually defined as
the change in length L divided by the original length L, or in
engineering term, L/L. This ratio is the same as the change in
displacement per unit distance. Therefore, the first derivative of
Equation with respect to position x yields the strain
)cos(max ctxudxdu
== (2-10)
With K=1, the strain can be expressed in terms of particle
velocity u&
cu /&= (2-11)
Positive particle velocities towards the right then produce
negative strains. Therefore negative strains are compressive. Thus
for plane waves, ground strains can be calculated directly for the
particle velocities if the compressive-wave propagation velocity is
known.
Example
-1.5
0
1.5
0 8
distance
c(t2-t1)
u&
The strains plotted in the Figure can defined mathematically as
follows (C = 300 m/s)
)sin(max ctxuu =
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Structural Response to Blast Loads
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To determine particle velocity, acceleration and displacement
history
iv. Wave propagation velocity and dynamic stress The
relationships between plane wave propagation velocity and particle
velocity, and the resulting normal stress as well as Youngs modulus
E can be derived by considering a stress wave propagation down a
straight bar as shown in Fig. 2-6. Based on Newtons second law, the
force F, mass m and acceleration u&& have following
relation
umF &&= (2-12)
Substitute appropriate values from Figure 2-6, it yields
t
uxAA
= &)( (2-13)
where is the mass density of the material in the bar, A the
cross-sectional area, and x the distance travelled by the wave in
time t . Then by deleting the area in the Equation 2-9, the
relationship between stress and particle velocity can be written
as
ut
x&
= (2-14)
maxucc & = (2-15) where maxu& is the maximum particle
velocity and tx / is the propagation velocity of the longitudinal
wavefront cc . The maximum stress then occurs at the maximum
particle velocity
maxu& . For elastic material, its Youngs modulus can be
expressed as
E=
(2-16)
Thus
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Structural Response to Blast Loads
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Etx
uEc
uEc
=
==
/&&
(2-17)
Substitution of Equation 2-17 to Equation 2-14
EucEc
uc
c
&& = (2-18)
E
cc = (2-19)
Fig. 2.6 A stress wave propagation down a straight bar
The wave velocity is found from the Youngs modulus E and the
mass density .
v. Shear plane-wave velocity Shear waves generate particle
motions in one direction that are perpendicular to the direction of
propagation. They are distinguished from longitudinal waves with
particle motions parallel to the direction of travel. Stresses and
strains from shear waves can be calculated from particle velocities
in a manner similar to that used form longitudinal waves due to
body waves producing particle velocities in one direction.
Derivations of these equations are similar to those used for
compressive waves. The shear wave propagation velocity sc is
G
cs = (2-20)
where G is the shear modulus and is related to Youngs modulus
by
)1(2 +=EG (2-21)
in which is Poissons ratio.
+
x
uu && +
u&
Area A
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Structural Response to Blast Loads
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vi. Transmission and reflection Consider two identical
longitudinal compressive waves moving toward each other along a bar
as shown in Fig. 2.7. When the two waves collide, the velocity
vectors of the particles will momentarily cancel and become zero
along line aa and stresses will be doubled. The waves retain their
initial shape and continue to propagate after passing. When a
compression and a tension wave collide, the particle velocity
vectors point in the same direction and add, but the stresses
cancel as shown in Fig. 2.7. In collision case of compressive
waves, the velocity at line aa is zero. Therefore the bar can be
thought of as being fixed at that position, meaning that a single
compressive wave colliding with a fixed boundary results in the
reflection of another compressive wave, travelling in the opposite
direction. For the collision case of tension and compression waves,
line aa has zero stress and it can be thought of being free or
unbounded.
u& +
Fixed
a
a
Free
a
a
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Structural Response to Blast Loads
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Fig. 2.7. Two identical longitudinal compressive waves moving
toward each other along a bar
Wave transmission can be generalized for transmission from one
medium ( 11c ) to another ( 22c ) as shown in Fig. 2.8. The
incident wave I , travelling done a bar with properties 1 and 1c
intersects another material, with properties 2 and 2c , produced a
reflected wave R and a transmitted wave T . At the interface the
sum of the particle velocities on both sides must be equal.
Thus
TRI uuu &&& =+ )( (2-22) Reflected velocity is
negative because it is travelling in the opposite direction. Since
particle velocity can be directly related to stress by Equation,
thus
221111 ccc
TRI
=
+ (2-23)
Furthermore, the stress on each side of the interface must be
equal and thus TRI =+ )( (2-24)
Then the transmitted and reflected stresses can be calculated
from the incident stress as
1122
22 )(2cc
cIT
+= (2-25)
1122
1122 )(cc
ccIR
+
= (2-26)
When 22c is many times that of 11c , R is equal to I and the
interface is similar to a fixed boundary. Similarly, if 22c is many
times smaller that of 11c , the reflected wave should be similar to
that of a free boundary.
22c
Fig. 2.8. Wave transmission from one medium to another
2.1.2. Characteristics of blast-induced ground motions
Blast-generated ground vibrations can be divided into body wave
types, compressive, P, shear S, and surface wave, R as shown in the
Fig. 2.9(a). To describe the motions completely, three
perpendicular components of motion must be measured as shown in
Fig. 2.9(b). The longitudinal component, L, is usually oriented
along a horizontal radius to the source. It follows then, that the
other two perpendicular components will be vertical, V, and
transverse, T, to the radial direction. The three main types can be
divided into two varieties: body waves, which propagate through the
body of the rock and soil, and surface waves, which are transmitted
along a surface. Body waves can be further subdivided into
compressive waves denoted as P and shear waves denoted as S. At
small distances, explosion generates predominately body waves.
These body
11cc
22c
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Structural Response to Blast Loads
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waves propagate outward in a spherical manner until they reach a
boundary such as another layer (rock or soil) or the ground
surface. At this intersection, shear and surface waves are produced
and the reflected surface (Rayleigh) waves become important at
larger transmission distances. At the small distances, all three
wave types will arrive together and greatly complicate wave
identification whereas at large distances, the more slowly moving
shear and surface waves begin to separate from the compressive.
Fig. 2.9 Compressive, P, shear S, and surface wave
The three wave types produce radically different patterns in
soil and rock particles as they pass. As a result, structures built
on or in soil (or rock) will be deformed differently by each type
of wave. In each case, the wave is propagating or moving to the
right as shown in the Fig. 2.10. The longitudinal (compressive)
wave produces particle motions in the same direction as it is
propagating. On the other hand, the shear wave produces motions
perpendicular to its direction of propagation: either horizontal,
as shown, or vertical. The Rayleigh wave (the most complicated)
produces motions both in the vertical direction and parallel to its
direction of propagation.
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Structural Response to Blast Loads
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Fig. 2.10 Particle motion variation with wave type: (a)
compressive; (b) shear; (c) Rayleigh
A close-in explosion produces the single-spiked pulse, A, by
direct transmission to the transducer at position A as shown in
Fig. 2.11. But most blasting problems involve the transducer
position B and result in relatively sinusoidal waves B as shown in
Fig. 2.11. The idealized waves shown are typical for blasting where
the close-in blasting produces transient pulses that last 1 to 2 ms
and 10 to 100 ms at relatively large distances. Combinations of
these single pulses produce the commonly observed blast-induced
sinusoidal wave strains.
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Structural Response to Blast Loads
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Fig. 2.11 Close-in and relative far explosion
Calculation with the sinusoidal approximation as shown in Fig.
2.12.
Fig. 2.12 The sinusoidal approximation
)sin(max tKxuu +=
)cos(max tKxudtdu
u +==&
)sin(2max tKxudtud
u +==&
&&
For sinusoidal wave:
In most circumstance, only the absolute value of the maximum
motion is of interest:
maxmax uu =
fuuu pi 2maxmaxmax ==&max
22max
2maxmax 24 uffuuu &&& pipi ===
(2-27)
(2-28)
(2-29)
(2-30)
(2-31)
(2-32)
smmu /15max =&
Recorded blast vibration time histories
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Structural Response to Blast Loads
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Theoretically, integration of the transverse particle velocity
record would give the displacement. For example, the area between
time 1 and 3 in Fig. 2.12 is 0.54 mm and is found by summing the
product of the average velocity between two timing lines and
interval between timing lines. The entire velocity time history
must be integrated from zero to find the true displacement. In this
case, the local integrated value of 0.54mm is the displacement the
occurs between the negative peak displacement and the positive peak
displacement. Therefore the zero-to-peak displacement is one-half
this value, 0.27mm. The maximum velocity is 15 mm/s. The period T
is the twice the half period which is the time difference between
times 1 and 3 is 0.06 s. Therefore,
Theoretically, acceleration is the maximum just after time 1,
when the particle velocity slope is a maximum, which the slope of
the line between points 1 and 2
The maximum acceleration with sinusoidal approximation is
estimated as follows. The appropriate period T would be four times
the one-fourth period, which calls the rise time of the pulse,
which in this case is the time between times 1 and 2. Therefore
The principal frequency is defined as that associated with the
greatest amplitude pulse as shown in Fig. 2.13.
mmssmmTu
fu
u 286.02
)06.02(/1522maxmax
max =
===
pipipi
&&
2/125001.0
/5.12smm
s
smm
t
u==
&
2maxmaxmax /1178)02.04(
)2(/1522 smmsmmT
ufuu =
===
pipipi
&&&&
(2-33)
(2-34)
(2-35)
Fig. 2.13 Principal frequency definition
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Structural Response to Blast Loads
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The principal frequency is also defined as that associated with
the averaged amplitude pulse as shown in Fig. 2.14.
Fig. 2.14 Principal frequency definition (averaged)
Comparison of dominant frequencies from construction blasting
with those by other segments of the blasting industry as shown in
Fig. 2.15.
2.1.3 Prediction of blast induced vibrations The most important
information of blast induced vibrations includes principal
frequency (PF), and peak values such as peak particle acceleration
(PPA)/ (peak particle velocity) PPV.
-0.8
-0.4
0
0.4
0.8
0 0.02 0.04 0.06 0.08 0.1
t(s)
Velo
city (m
/s)
Frequency (Hz)
0
10
20
0 500 1000 1500 2000
1F 2F
PF 2maxF
maxF
Fo
urier
spectra
(m/s)
Fig. 2.15 Comaprision of dominant frequencies
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Structural Response to Blast Loads
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Frequency-based criteria for control of vibrations require
methods to predict both peak particle velocity (PPV) in Fig. 2.16
and principal frequency (PF) in Fig. 2.17. Prediction of PPV can be
approached from scaling relations which are often associated with
blasting in rock or soil. Scaling describes decay with distance
that is normalized (hence scaled) by the source energy, and is most
useful when the same source at the same distance may release
variable energies, as in blasting and dynamic compaction
activities. When comparing blast wave the concept of scale distance
is commonly used. Scale distance is a comparative measure which can
be used to scale down the amount of explosive needed to create the
same blast wave. This is useful in experimental testing as large
blasts can be simulated by locating a smaller charge weight at a
closer distance to the specimen. The relationship given for
calculating scale distance:
where R is the blast range in metres and W is the equivalent
weight of TNT in kilograms. This relationship has been used in this
study to scale down potential explosive loads to a scale that can
be analysed in a blast chamber.
Blast Range 0.5 m
Blast Range 1 m
Blast Range 10 m
Scale Distance Charge Weight (kg of TNT)
0.50 1.000 8.000 8000.0 0.75 0.296 2.370 2370.4 1.00 0.125 1.000
1000.0 1.25 0.064 0.512 512.0 1.50 0.037 0.296 296.3 2.00 0.016
0.125 125.0
Fig. 2.16. PPV attenuation with scaled charge weights and
radical distance
Scale Distance (SD) 3 WR
= m/kg1/3
Peak
pa
rtic
le v
elo
city
m
/s
Peak
pa
rtic
le v
elo
city
m
/s
0.01
0.1
1
0.1 1 10 100
0.01
0.1
1
0.1 1 10 100
m
QRAPPV
= 3/1
PPV = Peak Particle Velocity (m/s); Q = Equivalent TNT charge
weight (kg); R = Radial distance (m) measured from the charge
center to the point of interest on the ground surface; A is initial
value at scaled range, R/Q1/3 = 1.0 ;and m is the attenuation
coefficient.
Prediction of peak particle velocity (PPV)
Scaled range m/kg1/3 Scaled range m/kg1/3
(2-36)
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Structural Response to Blast Loads
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Fig. 2.17. PF attenuation with scaled charge weights and radical
distance
2.2 Free air blast When a detonation occurs adjacent to and
above a protective structure such that no amplification of the
initial shock wave occurs between the explosive source and the
protective structure, then the blast loads acting on the structure
are free-air blast pressures (see Fig.2.18).
Fig. 2.18. Free air burst environment
n
QRBPF
= 3/1
Prediction of principal frequency (FF)
PF = Principal Frequency (Hz); Q = Equivalent TNT charge weight
(kg); R = Radial distance (m) measured from the charge center to
the point of interest on the ground surface; B is initial value at
scaled range, R/Q1/3 = 1.0 ; n is the attenuation coefficient.
10
100
1000
0.1 1 10Prin
cipa
l fre
quen
cy H
z
Scaled range m/kg1/3
(2-37)
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Structural Response to Blast Loads
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The shock front, termed the blast wave, is characterized by an
almost instantaneous rise from ambient pressure to a peak incident
pressure Pso (also called overpressure, see Figure). At any point
away from the blast, the pressure disturbance has the shape shown
in Fig. 2.19. The shock front arrives at a given location at time
tA and, after the rise to the peak value, Pso the incident pressure
decays to the ambient value in time to which is the positive phase
duration. This is followed by a negative phase with a duration to-
that is usually much longer than the positive phase and
characterized by a negative pressure (below ambient pressure)
Fig. 2.19. Free field pressure time variation
The incident impulse density associated with the blast wave is
the integrated area under the pressure-time curve and is is denoted
as for the positive phase and iS- is- for the negative phase.
i. Dynamic Pressure (Drag)
The pressure shock front travels radially from the burst point
with a diminishing shock velocity U (C) which is always in excess
of the sonic velocity of the medium. Gas molecules behind the front
move at a lower flow velocities, term particle velocities u. These
latter particle velocities are associated with the dynamic pressure
whose pressure formed by the winds produced by the passage of the
shock front (blast wind) . Those parameters which vary as the peak
incident pressure varies are presented in Fig. 2.20.
Fig. 2.20. Peak incident pressure versus peak dynamic pressure
and particle velocity
Important parameters: Peak overpressure Pso Duration Impulse
Arrive time
Other parameters: Peak dynamic pressure (Blast wind) Shock front
velocity U or C Blast wave length Lw
Wave length Lw=Ut0
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Structural Response to Blast Loads
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ii. Reflected pressure history As the incident wave moves
radically away from the center of the explosion, it will impact
with the structure, and, upon impact, the initial wave (pressure
and impulse) is reinforced and reflected (see Fig. 2.21). The
reflected pressure pulse of the Fig. 2.21 is typical for infinite
plane reflectors.
Fig. 2.21. Reflected free field pressure time variation
When the shock wave impinges on a surface oriented so that a
line which describes the path of travel of the wave is normal to
the surface, then the point of initial contact is said to sustain
the maximum (normal reflected) pressure and impulse. Fig. 2.22
shows possitive and negative phase shock wave parameters for a
spherical TNT explosion in free air at see level. Fig. 2.23
presents reflected pressure as a function of angle of
incidence.
Fig. 2.22. Possitive and negative phase shock wave parameters
for a spherical TNT explosion in free air at see level
Scaled Distance Z = R/W^(1/3)
Figure 2-7. Positive phase shock wave parameters for aspherical
TNT explosion in free air at sea level
0.1 1.0 10 100.001
.01
0.1
1.0
10
100
1000
10000
1.E+5Pr, psiPso, psiIr, psi-ms/lb^(1/3)Is, psi-ms/lb^(1/3)ta,
ms/lb^(1/3)to, ms/lb^(1/3)U, ft/msLw, ft/lb^(1/3)
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Structural Response to Blast Loads
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Fig. 2.23. Variation of reflected pressure as a function of
angle of incidence
Example For Q = 1.1 tonne and H = 90m, determine peak
overpressure Pso, peak reflected pressure Pr, duration t0, arrival
time ta, shock wave velocity U.
Angle of Incidence, Degrees
Pra
(psi)
Figure 2-9. Variation of reflected pressure as a function of
angle of incidence
0 10 20 30 40 50 60 70 80 901.0
10
100
1000
10000
1.E+5
1.E+6Scaled height of charge
0.30.60.81.93.0
5.37.28.911.914.3
Scaled height of the charge RA/Q1/3
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Structural Response to Blast Loads
21
3. Surface explosion A charge located on the ground surface is
considered to be a surface burst as shown in Fig. 2.24. The initial
wave of the explosion is reflected and reinforced by the ground
surface to produce a reflected wave. Unlike the air burst, the
reflected wave merges with the incident wave at the point of
detonation to form a single wave, similar to the mach wave of the
air burst but essentially hemispherical in shape. Fig. 2.25.
Possitive and negative phase shock wave parameters for a
hemispherical TNT explosion on the surface at see level
Fig. 2.24. Surfance burst blast environment
Fig. 2.25. Possitive and negative phase shock wave parameters
for a hemispherical TNT explosion on the surface at see level
The air burst environment (see Fig. 2.26) is produced by
detonations which occur above the ground surface and a distance
away from the protective structure so that the initial shock wave,
propagating away from the explosion, impinges on the ground surface
prior to arrival at the structure. As the shock wave continues to
propagate outward along the ground surface ,
Scaled Distance Z = R/W^(1/3)
Figure 2-15. Positive phase shock wave parameters for
ahemispherical TNT explosion on the surface at sea level
0.1 1.0 10 100.001
.01
0.1
1.0
10
100
1000
10000
1.E+5
1.E+6Pr, psiPso, psiIr, psi-ms/lb^(1/3)Is, psi-ms/lb^(1/3)ta,
ms/lb^(1/3)to, ms/lb^(1/3)U, ft/msLw, ft/lb^(1/3)
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Structural Response to Blast Loads
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a front known as the Mach front is formed by the interaction of
the initial wave and the reflected wave as shown in Fig. 2.27.
Fig. 2.26. Air burst blast environment
The height of the Mach front increases as the wave propagates
away from the center of the detonation. This increase in height is
referred to as the path of the triple point and is formed by the
intersection of the initial, reflected and Mach waves as shown in
Fig. 2.28 and Fig. 2.29. A protected structure is considered to be
subjected to a plane wave when the height of the triple point
exceeds the height of the structure. If the height of the triple
point does not extend above the height of the structure, the
magnitude of the applied loads will vary above the height of the
structure.
Fig. 2.27. Pressure time variation for air burst
Above the triple point, the pressure-time variation consists of
an interaction of the incident and reflected incident wave
pressures resulting in a pressure-time variation different from
that of the Mach incident wave pressures.
-
Structural Response to Blast Loads
23
Fig. 2.29. Variation of reflected pressure as a function of
angle of incidence
Scaled Horizontal Distance from Charge, ft/lb^(1/3)
Sca
led
Heig
ht o
f Trip
le Po
int,
ft/lb
^(1/
3)
Figure 2-13. Scaled height of triple point
2 4 6 8 10 12 14 16 180
2
4
6
8
10
12
11.5
2
2.53
3.5
4
5
67
Numbers adjacent to curvesindicate scaled chargeheight,
Hc/W^(1/3)
Scaled height of the charge: H/Q^1/3
Fig. 2.28 Scaled height of triple point
Scaled height of the charge: H/Q^1/3
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Structural Response to Blast Loads
24
4. External blast loading on structures (ASCE)
The procedures presented here for the determination of the
external blast loads on structures are restricted to rectangular
structures positioned above the ground surface where the structure
will be subjected to a plane wave shock front as shown in Fig. 2.31
and Fig. 2.32. The forces acting on a structure associated with a
plane shock wave are dependent upon both the peak pressure and the
impulse of the incident and dynamic pressures acting in the free
field. For design purposes, it is necessary to establish the
variation or decay of both the incident and dynamic pressures with
time since the effects on the structure subjected to a blast
loading depend upon the intensity-time history of the loading as
well as on the peak intensity.
The blast loading on a structure caused by a high-explosive
detonation as shown in Fig. 2.30 is dependent on several
factors:
The magnitude of the explosion. The location of the explosion.
The geometrical configuration of the structure. The structure
orientation with respect to the explosion and the ground surface
(above, flush with or below the ground.
W
H
L
Blast wave
Fig. 2.30 Blast loading on a structure
Fig. 2.31 Schematic of blast wave interaction with a rectangular
building
Fig. 2.32 Blast loading general arrangement for a rectangular
building
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Structural Response to Blast Loads
25
Ps = Ps0 +CDq
Front wall loads
US
tc3
=
At the moment the incident shock front strikes the front wall,
the pressure immediately rises from zero to the normal reflected
pressure Pr which is a function of the incident pressure as shown
in Fig. 2.33. The clearing time tC (reflected pulse time) required
to relieve the reflected pressure is represented as:
where S is clearing distance and is equal to H or W/2 whichever
is the smallest H is height of the structure; W is the width of the
structure; U is sound velocity in the reflected region.
The pressure acting on the front wall after time tc is the
algebraic sum of the incident pressure PS and the drag pressure
CDq:
P = PS +CDq
The duration of the reflected overpressure effect tc should not
exceed that of the free positive overpressure td.
The bilinear pressure-time curve can be simplified to an
equivalent triangle and the duration of the equivalent triangle is
determined
te = 2i/Pr = (td-tc)Ps/Pr+tc
A value of CD = 1 for the front wall is considered adequate for
plane pressure wave.
Fig. 2.33 Front wall blast loading (2-38)
(2-39)
(2-40)
Roof and side walls The general form of side wall blast loading
is shown in Fig. 2.34. As the shock front travels along the length
of a structural element, the peak overpressure will not be applied
uniformly. It varies with both time and distance. For example, if
the length of the side wall equals the length of the blast wave,
when the peak overpressure reaches the far end of the wall, the
overpressure at the near end has returned to ambient. A reduction
factor, Ce is used to account for this effect in design (see Fig.
2.35).
The equation for the side walls is as follows
Pa = CePs0 +CDq
The side wall load has a rise time equal to the time it takes
for the blast wave to travel across the element being considered.
The overall duration is equal to the rise time plus the duration of
the free field overpressure.
The roof blast loading is similar to the side wall
Fig. 2.34 Roof and side wall blast loading
Fig. 2.35 Effective overpressure values
(2-41)
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Structural Response to Blast Loads
26
Peak dynamic pressure (MPa)
0--0.172 (MPa)
0.172--0.344 (MPa)
0.344--0.896 (MPa)
Drag coefficient CD -0.4 -0.3 -0.2 Table for Drag coefficient vs
peak dynamic pressure
Example The dimensions of the building are 20m*10m*3m. Determine
blast loading on the components of the building subjected to an
explosion 1000 lb at the scaled distance of 3 m/kg1/3. The blast
wave will be applied normal to the long side of the building. It is
assumed that the peak incident pressure and duration do not change
significantly over the length of the building.
Rear wall loading
The shape of the rear wall loading is similar to that for side
and roof loads, however, the rising time is and duration are
influenced by a not well known pattern of spillover from the roof
and side walls and from ground reflection effects. The rear wall
blast load lags that for the front wall by L/U, the time for the
blast wave to travel the length L of the building (see Fig. 2.36).
The effective peak overpressure is similar to that for the side
walls (Pb is normally used to designate the real wall peak
overpressure instead of Ps).
Pb = CePs0 +CDq
Fig. 2.36 Rear wall blast loading
(2-41)
20m
3m
10m
blast
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Structural Response to Blast Loads
27
5. External blast loading on structures (TM5-1300)
The interaction of the incident blast wave with a structure is a
complicated process. To reduce the complex problem of blast to
reasonable terms, it will be assumed that (1) the structure is
generally rectangular in shape; (2) the incident pressure of
interest is 1.4 MPa or less; (3) the structure being loaded is in
the region of the Mach stem; and (4) the Mach stem extends above
the height of the building (see Fig. 2.38).
Fig. 2.38 Air burst blast environment
The form of incident blast wave is shown in the Fig. 2.37. For
design purposes, the actual decay of the incidental pressure may be
approximated by the rise of an equivalent triangular pressure
pulse. The actual positive duration is replaced by a fictitious
duration which is expressed as a function of the total positive
impulse and peak pressure:
pitof /2=
For determining the pressure-time data for the negative phase, a
similar procedure as used in the evaluation of the idealized
positive phase may be utilized. The equivalent negative
pressure-time curve will have a time of rise equal to 0.25 to
whereas the fictitious duration is given by the triangular
equivalent pulse equation:
= pitof /2
Fig. 2.37 Idealized pressure-time variation
(2-42)
(2-43)
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Structural Response to Blast Loads
28
Front wall loads
At the moment the incident shock front strikes the front wall
(Fig. 2.39), the pressure immediately rises from zero to the normal
reflected pressure Pr which is a function of the incident pressure.
The clearing time tC (reflected pulse time) required to relieve the
reflected pressure is represented as:
r
c CRS
t )1(4
+=
where S is clearing distance and is equal to H or W/2 whichever
is the smallest H is height of the structure; R is ratio of S/G
where G is equal to H or W/2 whichever is larger Cr is sound
velocity in the reflected region.
Fig. 2.39 Front wall blast loading
(2-44)
Front wall loads
The pressure acting on the front wall after time tC is the
algebraic sum of the incident pressure PS and the drag pressure
CDq:
P = PS +CDq
A value of CD = 1 for the front wall is considered adequate for
the pressure ranges in TM5.
At higher pressure ranges, the above procedure may yield a
fictitious pressure-time curve due to the extremely short pressure
pulse duration involved. Thus, the pressure-time curve constructed
must be checked to determine its accuracy. The comparison is made
by constructing a second curve (dotted triangle as indicated in the
Fig) using the total reflected pressure impulse ir for a normal
reflected shock wave. The fictitious duration trf for the normal
reflected wave is calculated from:
trf = 2ir/Pr
whichever curve gives the smallest value of the impulse (area
under curve)
(2-45)
(2-46)
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Structural Response to Blast Loads
29
If the shock front approaches the structure at an oblique angle,
the peak pressure will be a function of the incident pressure and
the incident angle between the front wall as shown in Fig.
2.40.
Front wall loads
Angle of incidence, Degrees
Cr =
Pr
/ P
so
Figure 2-193. Reflected pressure coefficientversus angle of
incidence
0 10 20 30 40 50 60 70 80 900
1.5
3
4.5
6
7.5
9
10.5
12
13.5
Peak Incident Overpressure,
psi5000300020001000500400300200150100
70503020105210.50.2
Fig. 2.40 Reflected pressure coefficient versus angle of
incidence
Roof and side walls As the shock front traverses a structure a
pressure is imparted to the roof slab, and side walls equal to the
incident pressure at a given time at any specified point reduced by
a negative drag pressure. The portion of the surface loaded at a
particular time is dependent upon the magnitude of the shock front
incident pressure, the location of the shock front and the
wavelength of the positive and negative pulses.
As the shock wave traverses the roof, the peak value of the
incident pressure decays and the wave length increases. The
equivalent uniform pressure will increase linearly from time tf
when the blast wave reaches the beginning of the element (point f)
to time td when the peak equivalent uniform pressure is reached
when the shock front arrives at point D. The equivalent uniform
pressure will then decrease to zero where the blast load at point b
on the element decreases to zero. td is rising time. Fig. 2.41
shows average pressure-time variation for roof and side wall.
d D
UD
td =
Fig. 2.41 Average pressure-time variation for roof and side
wall
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Structural Response to Blast Loads
30
The peak value of the pressure acting on the roof PR is the sum
of contribution of the equivalent uniform pressure and drag
pressure
Fig. 2.42 The equivalent load factor CE and location of the peak
equivalent uniform
pressure vs wavelength-span ratio
Rear walls
The blast loads on the rear wall is calculated using equivalent
uniform method used for computing the blast loads on the roof and
side wall as shown in Fig. 2.43. The peak pressure of the
equivalent uniform pressure-time curve is calculated using the peak
pressure Psob. The equivalent uniform load factors CE are based on
the wave length of the peak pressure and the height of the rear
wall HS as are the time rises and duration of both the positive and
negative phase.
Like the roof and side walls, the blast loads acting on the rear
wall are a function of the drag pressures in addition to the
incident pressure. The dynamic pressure of the drag corresponds to
that associated with the equivalent pressure CEPsob, while the
recommended drag coefficients are the same as used for the roof and
side walls.
PR = CEPsob +CDqob
Fig. 2.43 Average pressure-time variation for rear wall
PR = CEPsof +CDqof
where Psof is the incidental pressure occurring at point f and q
is the dynamic pressure corresponding to CE Pof. CE is equivalent
load factor. The drag coefficient CD for the roof and side walls is
a function of the peak dynamic pressure. Recommended values are as
shown in Fig. 2.42
.
(2-47)
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Structural Response to Blast Loads
31
6. Confined and partially confined explosions (TM5-1300)
6.1 Effects of Confinement When an explosion occurs within a
structure, the peak pressures associated with the initial shock
front (free-air pressures) will be extremely high and, in turn,
will be amplified by their reflections within the structure. In
addition, and depending upon the degree of confinement, the effects
of the high temperatures and accumulation of gaseous products
produced by the chemical process involved in the explosion will
exert additional pressures and increase the load duration within
the structure. The combined effects of these pressures may
eventually destroy the structure unless the structure is designed
to sustain the effects of the internal pressures. Provisions for
venting of these pressures will reduce their magnitude as well as
their duration.
The use of cubicle-type structures (Figure 2-44a) or other
similar barriers with one or more surfaces either sufficiently
frangible or open to the atmosphere will provide some degree of
venting depending on the opening size. This type of structure will
permit the blast wave from an internal explosion to spill over onto
the surrounding ground surface, thereby, significantly reducing the
magnitude and duration of the internal pressures. The exterior
pressures are quite often referred to as "leakage" pressures while
the pressures reflected and reinforced within the structure are
termed interior "shock pressures." The pressures associated with
the accumulation of the gaseous products and temperature rise are
identified as "gas" pressures. For the design of most fully vented
cubicle type structures, the effects of the gas pressure may be
neglected.
49 Fig. 2.44 Patially confined blast environment
Detonation in an enclosed structure with relatively small
openings (Figure 2-44b) is associated with both shock and gas
pressures whose magnitudes are a maximum. The duration of the gas
pressure and, therefore, the impulse of the gas pressure is a
function of the size of the opening. It should be noted that the
onset of the gas pressure does not necessarily coincide with the
onset of the shock pressure. Further, it takes a finite length of
time after the onset for the gas pressure to reach its maximum
value. However, these times are very small and, for design purposes
of most confined structures, they may be treated as
instantaneous.
In the following paragraphs of this section, a simple cantilever
barrier as well as cubicletype and containment type structures will
be discussed. The cubicles are assumed to have one or
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Structural Response to Blast Loads
32
more surfaces which are open or frangible while the containment
structures are either totally enclosed or have small size openings.
The effects of the inertia of frangible elements of these
structures will be discussed in subsequent sections. 50 6.2 Shock
Pressures. 6.2.1 Blast Loadings. When an explosion occurs within a
cubicle or containment-type structure, the peak pressures as well
as the impulse associated with the shock front will be extremely
high and will be amplified by the confining structure. Because of
the close-in effects of the explosion and the reinforcement of
blast pressures due to the reflections within the structure, the
distribution of the shock loads on any one surface will be
nonuniform with the structural surface closest to the explosion
subjected to the maximum load.
An approximate method for the calculation of the internal shock
pressures has been developed using theoretical procedures based on
semi-empirical blast data and on the results of response tests on
slabs. The calculated average shock pressures have been compared
with those obtained from the results of tests of a scale-model
steel cubicle and have shown good agreement for a wide range of
cubicle configurations. This method consists of the determination
of the peak pressures and impulses acting at various points of each
interior surface and then integrating to obtain the total shock
load. In order to simplify the calculation of the response of a
protective structure wall to these applied loads, the peak
pressures and total impulses are assumed to be uniformly
distributed on the surface. The peak average pressure and the total
average impulse are given for any wall surface. The actual
distribution of the blast loads is highly irregular, because of the
multiple reflections and time phasing and results in localized high
shear stresses in the element. The use of the average blast loads,
when designing, is predicated on the ability of the element to
transfer these localized loads to regions of lower stress.
Reinforced concrete with properly designed shear reinforcement and
steel plates exhibit this characteristic.
The parameters which are necessary to determine the average
shock loads are the structure's configuration and size, charge
weight, and charge location. Figure 2-45 shows many possible simple
barriers, cubicle configurations, and containment type structures
as well as the definition of the various parameters pertaining to
each. Surfaces depicted are not frangible for determining the shock
loadings.
Because of the wide range of required parameters, the procedure
for the determination of the shock loads was programmed for
solutions on a digital computer. The results of these calculations
are presented in UFC (Figures 2-52 to 2-100) for the average peak
reflected pressures pr (Figures 2-101 through 2-149) in UFC for the
average scaled unit impulse ir/W1/3. These shock loads are
presented as a function of the parameters defining the
configurations presented in Figure 2-45. Each illustration is for a
particular combination of values of h/H, l/L, and N reflecting
surfaces adjacent to the surface for which the shock loads are
being calculated. The wall (if any) parallel and opposite to the
surface in question has a negligible contribution to the shock
loads for the range of parameters used and was therefore not
considered.
The general procedure for use of the above illustrations is as
follows: 51
-
Structural Response to Blast Loads
33
1. From Figure 2-45, select the particular surface of the
structure which conforms to the protective structure given and note
N of adjacent reflecting surfaces as indicated in parenthesis.
2. Determine the values of the parameters indicated for the
selected surface of the structure in Item 1 above and calculate the
following quantities: h/H, l/L, L/H, L/RA, and ZA = RA/W1/3.
3. Refer to Table 2-3 for the proper peak reflected pressure and
impulse charts conforming to the number of adjacent reflected
surfaces and the values of l/L and h/H of Item 2 above, and enter
the charts to determine the values of pr and ir/W1/3.
In most cases, the above procedure will require interpolation
for one or more of the parameters which define a given situation,
in order to obtain the correct average reflected pressure and
average reflected impulse. Examples of this interpolation procedure
are given in Appendix 2A.
Because of the limitations in the range of the test data and the
limited number of values of the parameters given in the above shock
load charts, extrapolation of the data given in UFC (in Figures
2-52 through 2-149) may be required for some of the parameters
involved. However, the limiting values as given in the charts for
other parameters will not require extrapolation. The values of the
average shock loads corresponding to the values of the parameters,
which exceed their limiting values (as defined by the charts), will
be approximately equal to those corresponding to the limiting
values. The following are recommended procedures which will be
applicable in most cases for either extrapolation or establishing
the limits of impulse loads corresponding to values of the various
parameter which exceed the limits of the charts:
-
Structural Response to Blast Loads
34
2-45
-
Structural Response to Blast Loads
35
1. To extrapolate beyond the limiting values of ZA, plot a curve
of values of pr versus ZA for constant values of L/RA, L/H, h/H and
l/L. Extrapolate curve to include the value of pr corresponding to
the value of ZA required. Repeat similarly for value of ir
/W1/3.
2. To extrapolate beyond the limiting values of L/RA,
extrapolate the given curve of pr versus L/RA for constant values
of ZA, L/H, h/H and l/L to include the value of pr corresponding to
the value of L/RA required. Repeat this extrapolation for value of
ir /W1/3.
3. Values of pr and ir /W1/3 corresponding to values of L/H
greater than 5 shall be taken as equal to those corresponding to
L/H = 6 for actual values of ZA, h/H, and l/L but with a fictitious
value of L/RA in which RA is the actual value and L is a fictitious
value equal to 5H.
4. Values of pr and ir /W1/3 corresponding to values of l/L less
than 0.10 and greater than 0.75 shall be taken as equal to those
corresponding to l/L = 0.10 and 0.75, respectively.
5. Values of pr and ir /W1/3 corresponding to values of h/H less
than 0.10 and greater than 0.75 shall be taken as equal to those
corresponding to h/H = 0.10 and 0.75, respectively.
A protective element subjected to high intensity shock pressures
may be designed for the impulse rather than the pressure pulse only
if the duration of the applied pressure acting on the element is
short in comparison to its response time. However, if the time to
reach maximum displacement is equal to or less than three times the
load duration, then the pressure pulse should be used for these
cases. The actual pressure-time relationship resulting from a
pressure distribution on the element is highly irregular because of
the multiple reflections and time phasing. For these cases, the
pressuretime relationship may be approximated by a fictitious peak
triangular pressure pulse. The average peak reflected pressure of
the pulse is obtained in UFC (from Figures 2-52 through 2-100) and
the average impulse in UFC (from Figure 2-101 through 2-149) and a
fictitious duration is established as a function of the reflected
pressure pr and impulse ir acting on the element.
to = 2ir / pr The above solution for the average shock load does
not account for increased blast effects produced by contact
charges. Therefore, if the values of the average shock loads given
in UFC (in Figures 2-52 through 2-149) are to be applicable, a
separation distance between the element and explosive must be
maintained. This separation is measured between the surface of the
element and the surface of either the actual charge or the
spherical equivalent, whichever results in the larger normal
distance between the element's surface and the center of the
explosive (the radius of a spherical TNT charge is r = 0.136 W1/3).
For the purposes of design, the following separation distances are
recommended for various charge weights:53
(2-48)
-
Structural Response to Blast Loads
36
The above separation distances do not apply to floor slabs or
other similar structural elements placed on grade. However, a
separation distance of at least one foot shouldbe maintained to
minimize the size of craters associated with contact
explosions.
It should be noted that these separation distances do not
necessarily conform to those specified by other government
regulations; their use in a particular design must be approved by
the cognizant military construction agency.
Average shock loads over entire wall or roof slabs were
discussed above. An approximate method may be used to calculate
shock loads over surfaces other than an entire wall. These surfaces
might include a blast door, panel, column, or other such items
found inside any shaped structure.
The method assumes a fictitious strip centered in front of the
charge having a width equal to the normal distance RA and a height
equal to that of the structure. This is the maximum representative
area that may be considered. Average shock loads can be determined
on entire area or any surface falling within the boundaries of the
representative area.
The procedure for determining the shock loads consists of
partitioning the surface under consideration into subareas. These
subareas do not need to be the same size. The angle of incidence to
the center of each subarea is calculated. The reflected pressure
and scaled impulse are determined for each subarea using Figure
2-25, respectively. A weighted average with respect to area is
taken for both pressure and scaled impulse.
Both the pressure and the impulse are multiplied by a factor of
1.75 to account for secondary shocks.
6.3 Gas Pressures. 63.1 Blast Loadings. When an explosion occurs
within a confined area, gaseous products will accumulate and
temperature within the structure will rise, thereby forming blast
pressures whose magnitude is generally less than that of the shock
pressure but whose duration is significantly longer. The magnitude
of the gas pressures as well as their durations is a function of
the size of the vent openings in the structure. For very small
openings or no openings at all, the duration of the gas pressures
will be very long in comparison to the fundamental periods of the
structure's elements and, therefore, may be considered as a long
duration load similar to that associated with a nuclear event.
These conditions usually occur in total or near containment type
structures. In the former, the internal blast pressures must be
contained because of the presence of toxic or other harmful
materials in the structure. In near containment structures, the
leakage of pressure flow out of the structure usually must be
limited because either personnel or frangible structure are located
immediately adjacent to the donor structure. In other cases,
however, openings in structures may be quite large, thereby
minimizing the products' accumulation and limiting the temperature
rise, hence producing gas pressures with limited duration or no
duration at all. The structures without gas pressure buildup are
referred to as fully vented structures.
-
Structural Response to Blast Loads
37
A typical pressure-time record at a point on the interior
surface of a partially vented chamber is shown in UFC (in Figure
2-151 in UFC). The high peaks are the multiple reflections
associated with shock pressures. The gas pressure, denoted as pg,
is used as the basis for design and is a function of the charge
weight and the contained net volume of thechamber. UFC (Figure
2-152 in UFC) shows an experimentally fitted curve based upon test
results of partially vented chambers with small venting areas where
the vent properties ranged between:
0 Af /V2/3 0.022 The values of A and Vf are the chamber's total
vent area and free volume which is equal to the total volume minus
the volume of all interior equipment, structural elements, etc. The
maximum gas pressure, Pg, is plotted against the charge weight to
free volume ratio.
UFC (Figures 2-153 through 2-164 in UFC) provides the
relationship of the gas pressure scaled impulse ig /W1/3 as a
function of the charge weight to free volume ratio W/Vf, scaled
value of the vent opening A/Vf 2/3, the scaled unit weight of the
cover WF/W1/3 over the opening, and the scaled average reflected
impulse ir /W1/3 of the shock pressures acting on the frangible
wall or a non-frangible wall with a vent opening. The curves in UFC
(in Figures 2-153 through 2-164 in UFC) for WF/W1/3 = 0 were
obtained from data with A/Vf 2/3 1.0. Extrapolated values, for
which there is less confidence, are dashed. Curves for WF/W1/3 >
0 are not dashed at A/Vf 2/3 > 1.0 because they are not strongly
dependent on the extrapolated portion of the curve for WF/W1/3 = 0.
Even lightweight frangible panels displace slowly enough that the
majority of the gas impulse is developed before significant venting
(A/Vf 2/3 > 1) can occur.) For a full containment type structure
the impulse of the gas pressure will be infinite in comparison to
the response time of the elements (long duration load). For near
containment type structures where venting is permitted through vent
openings without covers, then the impulse loads of the gas
pressures are determined using the scaled weight of the cover equal
to zero. The impulse loads of the gas pressures corresponding to
scaled weight of the cover greater than zero relates to frangible
covers and will be discussed later. The effects on the gas pressure
impulse caused by the shock impulse loads will vary. The gas
impulse loads will have greater variance at lower shock impulse
loads than at higher loads. Interpolation will be required for the
variation of gas impulse as a function of the shock impulse loads.
This interpolation can be performed in a manner similar to the
interpolation for the shock pressures.
The actual duration and the pressure-time variation of the gas
pressures is not required for the analysis of most structural
elements. Similar to the shock pressures, the actual pressure-time
relationship can be approximated by a fictitious peak triangular
pulse. The peak gas pressure is obtained in UFC (from Figure 2-152
in UFC) and the impulse in UFC (from Figures 2-153 through 2-164 in
UFC) and the fictitious duration is calculated from the
following:
tg = 2ig / Pg 58 UFC (Figure 2-165a in UFC) illustrates an
idealized pressure-time curve considering both the shock and gas
pressures. As the duration of the gas pressures approaches that of
the shock pressures, the effects of the gas pressures on the
response of the elements diminishes until the duration of both the
shock and gas pressures are equal and the structure is said to be
fully vented.
If a chamber is relatively small and/or square in plan area then
the magnitude of the gas pressure acting on an individual element
will not vary significantly. For design purposes the gas pressures
may be considered to be uniform on all members. When the chamber is
quite
(2-49)
(2-50)
-
Structural Response to Blast Loads
38
long in one direction and the explosion occurs at one end of the
structure, the magnitude of the gas pressures will initially vary
along the length of the structure. At the end where the explosion
occurs, the peak gas pressure is Pg1 (Figure 2-165bin UFC) which
after a finite time decays to Pg2, and finally decays to zero. The
gas pressure Pg2 is based on the total volume of the structure and
is obtained in UFC (from Figure 2-152 in UFC) while the time for
this pressure to decay to zero is calculated from Equation 2-4
where the impulse is obtained in UFC (from Figures 2-153 through
2-164 in UFC) again for the total volume of the structure. The peak
gas pressure Pg1 is obtained in UFC (from Figure 2-152 in UFC)
based on a pseudo volume in UFC (Figure 2-165bin UFC) whose length
is equal to its width and the height is the actual height of the
structure. The time tp for the gas pressure to decay from Pg1 to
Pg2 is taken as the actual length of the structure minus the width
divided by the velocity of sound (1.12 fpms). At the end where the
explosion occurs, the peak gas pressures (Pg1, Figure 2-165b in
UFC) will be a maximum and, after a finite time, they will decay to
a value (Pg2, Figure 2-165b in UFC) which is consistent with full
volume of the structure; after which they will decay to zero. The
magnitude of the peak gas pressures (Pg1) may be evaluated by
utilizing in UFC (Figure 2-152 in UFC) and a pseudo volume whose
length is equal to its width and the height is the actual height of
the chamber. The length of time tp between the two peak gas
pressures may be taken as the length minus the width of the
structure divided by the velocity of sound.
UFC
-
Structural Response to Blast Loads
39
Chapter 3
Structural Dynamics (Single Degree of Freedom Model)
3.1 Analysis of Free Vibrations
3.1.1 Single Degree-of-Freedom system
The structures modelled as systems with a single displacement
coordinate as shown Fig. 3.1 called Single Degree Of Freedom
System
Fig. 3.1 Components of the Basic Dynamic System
Equation of motion of the basic dynamic system
fI(t)+ fD(t)+fS(t) = p(t) (3-1) Then
)(tpkyycym =++ &&& (3-2)
3.1.2 Analysis of Undamped Free Vibrations
For undampted free vibrations, fD(t)= 0 p(t) = 0, Eq. 3.2 can be
simplified as 0=+ kyym && (3-3)
To solve the second-order differential equation, a trial
solution given by tAy cos= or tBy sin= and then submitting them to
Eq 3.3, it has
0cos)( 2 =+ tAkm (3-4)
Thus the solution is tBtAy sincos += . The velocity is
Initial conditions and 0yA = , /0vB =
fD(t)
fS(t) fI(t)
y(t) y(t) fD(t) Damping force = cy(t) fS(t) Spring force = ky(t)
fI(t) Inertia force = my(t) ..
.
m
k=
2 mk /=
tBtAy cossin +=&
0,0 yyt == 0vy =&
-
Structural Response to Blast Loads
40
The final solution is
(3-5)
Cyclic frequency pi 2/=f and Period (reciprocal) 1/f
3.1.3 Analysis of Damped Free Vibrations
0=++ kyycym &&& (3-6)
A trial solution given by exponential function: stCey = .
Then
(3-7) The characteristic equation for the system:
02 =++ kcsms (3-8) The roots of this quadratic equation
(3-9)
General solution is give by the superposition of the two
possible solutions:
(3-10) Three cases corresponding to radical being zero, positive
or negative.
i. Critical Damped System
The quantity under the radical is zero:
(3-11)
The roots of this quadratic equation
General solution:
ii. Overdamped System when c > ccr, the quantity under the
radical is positive. Thus the two roots of the characteristic
equation are real and distinct, consequently, the solution is given
directly:
(3-12)
It should be noted that for the overdamped system and the
critical damped system, the resulting motion is not oscillatory;
the magnitude of the oscillation decays exponentially with time to
zero.
iii. Damped System when c < ccr, the quantity under the
radical is negative. The roots of this quadratic equation
tv
tyy
sincos 00 +=
02 =++ ststst kCecCseemCs
m
km
c
m
cs
=2
2,1 22
tsts eCeCy 21 21 +=
02
2
=
m
km
ccr kmccr 2=
m
cs cr
22,1=
tsts eCeCy 21 21 += tmccretCCy )2/(21 )( +=
tsts eCeCy 21 21 +=
2
2,1 22
=
m
c
m
kim
cs
-
Structural Response to Blast Loads
41
(3-13)
For this case, it is convenient to make use of Eulers equations
which relate exponential and trigonometric functions:
General solution: (3-14)
where D the damped frequency of the system, that can be
expressed as
in which
With initial conditions
(3-15)
Alternatively, where
In practice, the natural frequency for a damped system may be
taken to be equal to the undamped natural frequency. Fig. 3.2 shows
free vibration response with under-critically damped system.
iv. Logarithmic Decrement
In free vibration At point points
Logarithmic decrement :
Fig. 3.2 Free vibration response with under damped system
y(t)
y0
tCe
xixeix sincos += xixe ix sincos =
)sincos()2/( tBtAey DDtmc +=
2
2
=
m
c
m
kD
21 =D
mk /=crc
c=
0,0 yyt == 0vy =&
)sincos( 000 tyv
tyey DD
Dt
++=
)cos( = tCey Dt
( )2
2002
0D
yvyC
++=
0
00 ||tany
yv
D
+=
2122
pi
pi
==
DDT
)cos( = tCey Dt1
1tcey = 22
tcey =
Dt
t
Tce
ce
yy
===
2
1
lnln2
1
-
Structural Response to Blast Loads
42
For small damp ratio:
Fig. 3.3 Curve showing peak displacement and displacements at
points of tangency
3.2 Response to Harmonic Loading
3.2.1 Undamped Harmonic Excitation Motion equation for undamped
harmonic excitation
(3-16)
The complementary solution of the free vibration: 3-17)
Particular solution:
(3-18)
Fig. 3.4 a: Undampeted harmionic excited SDOF system. b: Free
body diagram
where
= the ratio of the applied forced frequency to the natural
frequency of vibration of the system
y(t)
y1
y2
tce
Tangent points ]1)[cos( = tD
2122
pi
pi
==
DDT 21
2
pi
=
pi 2
tFkyym sin0=+&&
tBtAtyc sincos)( +=tYtyp sin)( =
tFtkYtYm sinsinsin 02
=+
20
20
1/ ==kF
mkFY
-
Structural Response to Blast Loads
43
General solution:
(3-19)
(3-20)
Since in a practical case, damping will cause the last term to
vanish eventually. The forcing frequency term (transient response)
is
(3-21)
3.2.2 Damped Harmonic Excitation
Motion equation for damped harmonic excitation (3-22)
The complementary solution of the free vibration:
(3-23)
(3-24)
For the system starting from rest (t=0 y0=0, v0=0): A= 0
201/
=
kFB
F0/k = yst is the displacement which would be produced by the
load F0 applied statically
is the dynamic magnification factor (DMF) representing the
amplification factor of the harmonically applied loading. It can be
seen, when forcing frequency is equal to the natural frequency, the
amplitude of the motion becomes infinitely large. In this case, the
system is said at resonance.
211== sty
YD
Using polar coordinate form iecmkicmk 222 )()( +=+
)()()( tytyty pc +=
tkF
tBtAty sin1/
sincos)( 20
++=
)sin(sin1
/)( 20 ttkF
ty =
tYtkFty sinsin1/)( 20 =
=
tFkyycym sin0=++ &&&
)sincos()( tBtAety DDtc +=
Particular solution: tCtCty p sincos)( 21 +=
Follow Eulers relation: tite ti sincos +=
Rewritten motion equation tieFkyycym 0=++ &&&
Particular solution: tip Cey
=
titititi eFkCeeicCemC 02
=++
icmkFC
+= 2
0
icmkeFy
ti
p += 2
0
-
Structural Response to Blast Loads
44
The response to the force is then the imaginary component
(3-25)
or
General solution:
(3-26) The dynamic magnification factor
(3-27)
3.2.3 Resonant Response It is clear that the steady-state
response amplitude of a undamped system tends towards infinity as
the frequency ratio approaches unity. When the frequency of the
applied loading equals the undamped natural vibration frequency, is
called resonance. It is also seen that at resonance the dynamic
magnification factor is
i
ti
pecmk
eFy222
0
)()( +=
222
)(0
)()(
cmkeFy
ti
p+
=
2tan mk
c
=
222 )2()1()sin(
+
=
tyy stp
212
tan
=where kFyst /0=crc
c=
=
222 )2()1( +=styY
2220
)()()sin(
cmk
tFyp+
=
)sin( = tYy p 2220
)()( cmkFY
+=
222 )2()1()sin()sincos()(
+
++= ty
tBtAety stDDt
222 )2()1(1
+== styYD
21
1 ==D
222 )2()1(1
+== pyYD
-
Structural Response to Blast Loads
45
Fig. 3.5 Dynamic magnification factor for damped system
3.3 Response to General Dynamic Loading
3.3.1 Impulsive Loading and Duhamels Integral
Newtons Law of Motion:
Lets consider this impulse F()d acting on the structure
represented by the undamped oscillator. At the time the oscillator
will experience a change of velocity given by dv. This change of
velocity is then introduced in the equation
as the initial velocity v0 together with y0 = 0 at time
producing a displacement at a later t in the following
(3-28)
Fig. 3.6 General load function as impulse loading
)(sin)()(
= t
m
dFtdy
dtFm
tyt )(sin)(1)(
0= (Duhamels Integral)
Considering initial conditions: 0,0 yyt == 0vy =& , the
total displacement of undamped SDOF an arbitrary load:
212
tan
=
)(
Fddv
m =m
dFdv )(=
tv
tyy
sincos 00 +=
dtFm
tv
tytyt )(sin)(1sincos)(0
00 ++=
-
Structural Response to Blast Loads
46
3.3.2 Constant Force For a constant force applied suddenly to
the undamped oscillator at time t = 0, both initial displacement
and initial velocity equal to zero,
(3-29)
and integration yields:
Fig. 3.7 Response of undampted SDOF system to a suddenly applied
constant force
3.3.3 Rectangular Loading
For a constant force applied suddenly but only during a limited
time duration td, at td time, displacement and velocity are
)cos1(0 dd tkFy =
dd tkF
v sin0=
For the response after time td, it follows free vibrations.
tv
tyy
sincos 00 +=
Replacing t by t-td, and y0 and v0 respectively, it has
)(sinsin)(cos)cos1()( 00 dddd tttkF
tttkF
ty += )]cos)([cos)( 0 tttkF
ty d =
If the dynamic load factor (DLF) is defined as the displacement
at any time t divided by static displacement
dtttDLF = cos1 dd tttttDLF = cos)(cos
dttTtDLF =
'2cos1 pi dd ttT
t
Tt
TtDLF =
'2cos)(2cos pipior
Fig. 3.8
(3-30)
(3-31)
(3-32)
dtFm
tyt )(sin1)(0 0
=
tt
m
Fty 02
0 )(cos)(
= )cos1()cos1()( 0 tytkF
ty st ==
(3-33)
-
Structural Response to Blast Loads
47
3.3.4 Triangle Loading
The displacement and velocity at time td as
For the response after time td, it follows free vibrations.
Replacing t by t-td, and y0 and v0 respectively, it has
or
3.3.5 Duhamels Integral-Undamped System
For a triangular force which has an initial value F0 and
decreases linearly to zero at a time td:
=
dtFF 1)( 0
and the initial conditions by 00 =y 00 =v
dtFm
tyt )(sin)(1)(
0=
substitute these values in Eq.
and integration gives ( )
+= t
t
ktF
tkFy
d
sincos1 00
dddst
ttt
t
TtTtTt
yyDLF +==
/2)/2sin()/2cos(1
pi
pipi
or in terms of the dynamic load factor and dimensionless
parameters
Fig. 3.9
(3-34)
(3-35)
(3-36)
(3-37)
= d
d
dd tt
t
kFy
cos
sin0
+=
dd
ddd tt
tt
kF
v1cos
sin0
tv
tyy
sincos 00 +=
( ) tkF
ttttk
Fy dd
cos)(sinsin 00 =
ttttt
DLF dd
cos))(sin(sin1 =
Tt
Tt
Tt
Tt
TtDLF d
d
pipipipi
2cos))(2sin2(sin/2
1=
dtFm
tyt )(sin)(1)(0
= Total displacement:
dFm
tdFm
ttytt
sin)(1coscos)(1sin)(00 =
or ttBttAty cos)(sin)()( =
(3-38)
(3-39)
(3-40)
(3-41)
(3-42)
(3-43)
-
Structural Response to Blast Loads
48
Example
Determine the dynamic response of a tower subjected to a blast
loading. The idealization of the structure and the blast loading
are shown in Fig. Neglecting damping.
where
The calculation of Duhamels integral requires the evaluation of
the integrals A(t) and B(t) numerically. Several numerical
integration techniques have been used for this evaluation. In these
techniques the integrals are replaced by a suitable summation of
the function v()=F()cos under the integral and evaluated for
convenience at n equal time increments, as shown in this Fig. The
numerical integral A(t) can now be used as follows:
=t
dFm
tA0
cos)(1)( =
tdF
mtB
0sin)(1)(
F()
F0
v() = F()cos
F1 F3 F4
F5 F6
F2
v0 v1
v2
0
v3 v4
v6
Fig. 3.10 Duhamels interal
Using any of these equations, A(t) can be obtained directly for
any specific value of n indicated. However, usually the entire
time-history of response is required so that one must evaluate A(t)
for successive values of n until the desired time-history of
response is obtained. For this purpose, it is more efficient to use
these equations in their recursive forms:
Simple summation: i = 1, 2, 3, 11 )()(
+= iii vm
tAtA
Trapezoidal rule: i= 1, 2, 3, )(2)()( 11 iiii vvmtAtA +
+=
nnnnn ttBttAty cos)(sin)()( =Displacement response:
(3-47)
(3-48)
(3-49)
y(t)
F(t) F(t)
n = 1, 2, 3, )...()( 110 +++
= nn vvvm
tA
Simple summation:
Trapezoidal rule: n = 1, 2, 3, )2...2(2)( 110 nnn vvvvmtA
++++
=
Simpson rule: n = 2, 4, 6, )4...24(3)( 1210 nnn vvvvvmtA
+++++
=
(3-44)
(3-45)
(3-46)
-
Structural Response to Blast Loads
49
3.4 Nonlinear Structural Response
3.4.1 Nonlinear SDOF Model
3.4.2 Linear Acceleration Step-by-Step Method
FI(ti)+ FD(ti)+FS(ti) = F(ti) The equilibrium of these forces is
expressed as
and at short time t later as FI(ti+t)+ FD(ti+t)+FS(ti+t) =
F(ti+t) Subtracting the above equations results in the differential
equation of motion in terms of
FI+ FD+ FS = F where the incremental forces in this equation are
defined as FI = FI(ti+t) - FI(ti) FD = FD(ti+t) - FD(ti) FS =
FS(ti+t) - FS(ti) F = F(ti+t) - F(ti)
Fig. 3.11 Dampted SDOF system
(3-50)
(3-51)
(3-52)
(3-53)
If assuming the damping force is a function of the velocity and
the spring force a function of displacement as shown graphically in
Fig. while the inertia force remain proportional to the
acceleration, the incremental forces can be expressed as
iI ymF &&= iiD ycF &= iiS ykF =
Nonlinear stiffness Nonlinear damping
yi = y(ti+t) - y(ti) yi = y(ti+t) - y(ti) . . . yi = y(ti+t) -
y(ti) .. .. .. where the incremental displacement yi, incremental
velocity yi, and incremental acceleration yi are given by
. ..
iiiiii Fykycym =++ &&&The incremental equation can
be written as
FI+ FD+ FS = F
Fig. 3.12 Nonlinear stiffness and damping
(3-54)
(3-55)
In the linear acceleration method, it is assumed that the
acceleration may be expressed by a linear function of time during
the time interval t as
)()( iii tttyyty
+=&&
&&&&
t Integrating above equation, the velocity is 2)(
21)()( iiii ttt
yttyyty
++=&&
&&&&
the displacement then is given as
iiiiii Fykycym =++ &&&
Fig. 3.13 Linear acceleration method
(3-56)
(3-57)
-
Structural Response to Blast Loads
50
32 )(61)(
21)()( iiiiii ttt
yttyttyyty
+++=&&
&&&
The above equations at time t = ti +t gives
tytyy iii += &&&&& 21
22
61
21
tytytyy iiii ++= &&&&&
Now to use the incremental displacement y as the basic variable
in the analysis
Fig. 3.14 Acceleration, velocity and displacement
(3-58)
(3-59)
(3-60)
iiii yyty
ty &&&&& 3662
= tytyy iii += &&&&& 21Substitute to
22
61
21
tytytyy iiii ++= &&&&&From iiii yyty
ty &&&&& 3662
=
iiii ytyy
ty &&&
233
=
iiiiii Fykycym =++ &&&The incremental equation:
iiiiiiiiii Fykytyy
tcyy
ty
tm =+
+
&&&&&
2333662
Transferring all the terms containing the unknown incremental
displacement y to the left
iii Fyk =
in which ki is the effective spring constant _
t
c
t
mkk iii +
+=
362
and Fi is the effective incremental force _
++
+
+= iiiiiii ytycyy
tmFF &&&&&&
2336
.
(3-61)
(3-62)
(3-63)
(3-64)
(3-65)
(3-66)
(3-67)
iii Fyk =The equation is equivalent to the static incremental
equilibrium equation
The incremental displacement is simply determined by i
ii k
Fy =
iii yyy +=+1The displacement yi+1 at time ti+1 is obtained
by
iiii ytyy
ty &&&
233
=The incremental velocity is given by
The velocity at time ti+1 is obtained by iii yyy &&&
+=+1
The acceleration yi+1 at the end of the time step is obtained
directly from the differential equation of motion
..
{ })()()(1 1111 ++++ = iSiDii tFtFtFm
y&&
.
(3-67)
(3-68)
(3-69)
(3-70)
(3-71)
-
Structural Response to Blast Loads
51
3.4.3 Elasto-Plastic Behavior
3.4.4 Algorithm for Step-by-Step Solution for Elastoplastic SDOF
System
Elasto-plastic behaviour is a simplified model by assuming a
definite yield point beyond which additional displacement takes
place at a constant value for the restoring force without any
further increase in the load.
Referred to the Fig. 3.15, assuming the initial conditions are
zero for the unloaded structure. Initially, as the load is applied,
the system behaves elastically along curve E0. The displacement yt
at which plastic behaviour in tension may be initiated and
displacement yc at which plastic behaviour in compression may be
initiated are calculated
kRy tt /= kRy cc /=The system will remain on curve E0 as long as
the displacement y satisfies tc yyy 0. When y < 0, the system
reverses to elastic behaviour on a curve such as E1 with new
yielding point given by
maxyyt = kRRyy ctc /)(max =
. .
Fig. 3.15 Elasto-plastic behaviour (3-72)
(3-73)
Conversely, if y decreases to yc, the system begins to behave
plastically in compression along curve C as shown; it remains on
curve C as long as the velocity y < 0. When y > 0, the system
reverses to elastic behaviour on a curve such as E2 with new
yielding point given by
minyyc = kRRyy ctt /)(min +=
.
The restoring force on an elastic phase of the cycle is
calculated as
kyyRR tt )( =
on a plastic phase in tension tRR = on a plastic phase in
compression cRR =
(3-74)
(3-75)
iii Fyk = tc
t
mkk iii +
+=
362
++
+
+= iiiiiii ytycyy
tmFF &&&&&&
2336
After the displacement, velocity and acceleration have been
determined at time ti+1, the outlined procedure is repeated to
calculate these quantities at the following time step ti+2 and the
process is continued to any desired final value of time.
.
-
Structural Response to Blast Loads
52
(3-76)
(3-77)
(3-78)
(3-79)
-
Structural Response to Blast Loads
53
(3-80)
(3-81)
(3-82)
(3-83)
(3-84) (3-85)
(3-86)
(3-87)
(3-88)
(3-89)
-
Structural Response to Blast Loads
54
Example To illustrate the hand calculation in applying the
step-by-step integration method described above, consider the
single degree-of-freedom system in Fig. 7.5 with elastoplastic
behavour subjected to loading history as shown. For this example,
we assume that the damping coefficient remains constant ( = 0.087).
Hence the only nonlinearities in the system arise from the changes
in stiffness as yielding occurs.
-
Structural Response to Blast Loads
55
3.5 Generalized Single Degree of Freedom System 3.5.1 General
comments on SDOF system
3.5.2 Dynamic transformation factors
In formulating the SDOF equations of motion and response
analysis procedures, it is assumed that the structure under
consideration has a single lumped mass that can move only in a
single fixed direction. The equation of motion for a linear elastic
SDOF system under blast loads is presented as )(tpKyyM
=+&&
However, the analysis of most real systems requires the use of
more complicated idealization. In the case of structures having
distributed elasticity, the SDOF shape restriction is merely an
assumption because the distributed elasticity actually permits an
infinite variety of displacement patterns to occur. However, when
the system motion is limited to a single form of deformation, it
only has a SDOF in a mathematical sense. Therefore, when the
generalized mass, damping, and stiffness properties associated with
this degree of freedom have been evaluated, the structure may be
analyzed in the same way as a true SDOF system.
p (t)
y
x
L
M
R = KX M F
Fig. 3.16 SDOF system for structural member
In order to define an equivalent one-degree system, it is
necessary to evaluate the parameters of that system; namely, the
equivalent mass ME, the equivalent spring constant KE and the
equivalent load FE. The equivalent system is selected usually so
that the deflection of the concentrated mass is the same as that
for a significant point of the structure. The
single-degree-of-freedom approximation of the dynamic behavior of
the structural element may be achieved by assuming a deflected
shape for the element which is usually taken as the shape resulting
from the static application of the dynamic loads. The assumption of
a deflected shape establishes an equation relating the relative
deflection of all points of the element.
p (t)
y
x
L
M
In most cases, a structure can be replaced by an idealized
system which behaves timewise in nearly the same manner as the
actual structure. The distributed masses of the given structure are
lumped together in to a concentrated mass. A structural system
having distributed mass can be modeled as a SDOF system.
y
x
L
ME
FE KE
Fig. 3.17 SDOF syste for a distributed mass of a structural
memeber
-
Structural Response to Blast Loads
56
3.5.3 Load Factor
The load factor is derived by setting the external work done by
the equivalent load FE on the equivalent system equal to the
external work done by the actual load F on the actual element
deflecting to the assumed deflected shape. For a structure with
distributed loads;
==L
ED dxxxpFW 0max )()( where max = maximum deflection of