Lecture Note Blast2015

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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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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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


4

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


5

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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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 =


7

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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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


9

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


10

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


11

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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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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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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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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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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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)


17

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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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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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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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


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)


22

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.


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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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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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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)


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


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)


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)


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


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)


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


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


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:


34

2-45


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)


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.


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)


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


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 =&


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


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


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


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


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


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 ++=


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)


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)


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)


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)


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)


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.

.


52

(3-76)

(3-77)

(3-78)

(3-79)


53

(3-80)

(3-81)

(3-82)

(3-83)

(3-84) (3-85)

(3-86)

(3-87)

(3-88)

(3-89)


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.


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


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

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