Underground Blast Info

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Response of Underground Pipes due to BlastLoads by Simulation An Overview

A. J. Olarewaju

N.S.V. Kameswara Rao

M. A. Mannan

Civil Engineering Program, School of Engineering and Information Technology, SKTM,

Universiti Malaysia Sabah, UMS, 88999, Kota Kinabalu, Malaysia Fax: 088-320348

Corresponding authors e-mail address;- (1) [email protected]

ABSTRACTThis paper has analytically and numerically examined the static and dynamic responses of underground pipesdue to blast loads. The various components of blast considered are blast load, ground media, pipes and soil-pipeinteraction. Using Unified Facilities Criteria (2008), blast energy and ground movement parameters for varioustypes of explosion for short distance were estimated. Other numerical tools for predicting blast energy andsolving dynamic equation were equally suggested. Available technical manuals for designing structures to resistthe effects of accidental explosion were given. Methods of analysis of simulated buried pipes subjected to blastloads were considered. Analytical method may not provide accurate result owing to its limitations; consequently,numerical methods overcome the limitations of analytical methods. Numerical methods considered for solvingdynamic equilibrium equation are the central difference and finite element methods. The solutions to thedynamic equations using these two numerical methods can be achieved using ABAQUS numerical code. Apartfrom Abaqus code, other numerical tools that could be used to study the response of underground structures(pipes) by modeling/simulations were also suggested.

KEYWORDS: Blast; Underground; Pipes; Analytical; Numerical; Overpressure; Blast; Response, Simulation

INTRODUCTION

Underground structures are divided into two major categories, firstly, fully buried structures, andsecondly, partially buried structures. These two can be any structures of divers shapes, shelters,basement structures, underground mall facilities, underground parking spaces, silos, storage facilities,retention basins, shafts, tunnels, pipes, underground railway, metro stations to mention a few.Underground structures are constructed of different materials, of which the static and dynamic properties can be determined. These materials are: metals, structural steel, high strength low alloysteel, reinforcing steel, high carbon content steel, concrete, timber, etc. Underground pipes are used forwater supply, convey sewage, storm, oil and gas supply, irrigation, etc. Pipelines are also used to carry


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acid, industrial and domestic wastes, liquid gas, etc. Filling stations and depots have undergroundstorage cylindrical tanks to store petroleum products. It is important to consider the severity ofdestruction due to explosion; blast can create sufficient tremors to damage substructures over a largearea. It has been reported that at 138kpa of blast wave, reinforced concrete structures will be leveled.Peak overpressure is the extent the pressure in the blast wave exceeds atmospheric pressure of 10 5 Pa(Marusek, 2009). Consequent upon these phenomena are loss of lives and property. In the

manufacturing industry, it leads to disruption in production, land degradation, air pollution, etc. As aresult of these, there is need to study the relationship and consequences of blasts in undergroundstructures specifically pipes. This is with a view to designing protective underground structuresspecifically pipes to resist the effects of blast and to suggest possible mitigation measures. Theconstituents of blast are basically the explosive, ground media, intervening layer, structuralcomponents (pipes), and blast characteristics (Robert, 2002). In studying soil-pipe interaction mostespecially in this study through modeling, experimental results are required in other to simulate the prevailing situations between all the constituent materials (Ganesan, 2000). These data are bestobtained from field tests, laboratory tests, theoretical studies, work done in related fields and extensionof work done in related fields Newmark and Haltiwanger, 1962)

A lot of works have been done on dynamic soil-structure interaction majorly for linear,

homogeneous, and semi-infinite half space. The response of elastic half space was first carried out byLamb (1904), Newcomb (1951) and Converse (1953) derived empirical relation for the determinationof resonance frequency in vibrated soil. It was established that softer soils have lower naturalfrequency. The natural frequency is higher at lower bearing pressures on soil. Hard clays have lessnatural frequency than sand stones. Ronanki [49] obtain the responses of buried circular pipes underthree-dimensional static and seismic loading. Method used is the finite element based softwarepackage, SAP-80. Parametric studies were equally carried out. Boh et al (2007) used nonlinear finiteelement analysis to study the responses of structures in the oil and gas industry. They came up withrecommendations for design to resist blast and explosion to help in overcoming the limitations ofcommonly used analytical methods. George et al, (2007) proposed analytical method for calculation of blast-induced strains to buried pipelines. The result provided an improved accuracy at no majorexpense of simplicity as well as accounting for the effect of local soil condition. James Marusek,

(2008) also used finite element analysis to studied underground shelters due to blast loadings fromconventional weapon detonation. Elasticity was chosen to model the behavior of the soil material.Blast load was represented as short duration and parametric studies carried out. Husabei (2009)recently obtained the responses of subway structures under blast loading using commercially finiteelement code, Abaqus. The subway was placed in different soil layers and numerical simulationscarried out. Mitigation measure used to improve ground stiffness and strength was also analyzed.

CONSTITUENTS OF THE BLAST

In underground structures, the constituents of the blast comprises of rock/soil media, structures (inthis study pipes), intervening medium, blast and blast characteristics.

ROCK/SOIL MEDIA

The rock media depends on the geotechnical properties of the ground medium. It ranges from intactrocks like schist to average quality to poor quality rocks. Rocks are formed as a result of variousnatural processes such as the cooling of molten magma, the precipitation of inorganic materials, thedeposition of shells of various organisms, etc. Rocks are classified into three; igneous rocks, e.g.granite, volcanic-basalt, etc, sedimentary rocks, e.g. sandstone, limestone, shale, conglomerate, or


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metamorphic rocks, e.g. schist, slate. Rocks undergo geologic action namely denudation, deposition,and earth movement which leads to the formation of soil. Soil is a material which disintegrates intoindividual grains by mechanical means like agitation in water, application of flow pressure, etc. Soilsare identified by various methods, these are: Bureau of Soils (1890-1895), Atterberg 1905, MIT 1931,

USDA 1938, AASHO 1970, Unified Soil Classification System 1952, ASTM 1967 (Peck, 1974).There are three major types of soils. These are; frictional soils ( > 0, c = 0), cohesive soils ( = 0, c >

0), and frictional-cohesive soils (c > 0, > 0). c and are the shear strength parameters of the soil (c

is sometimes called cohesion and is called angle of shearing resistance orinternal friction angle asif they were unrelated properties of a soil; they are, in fact, two parameters of shear strength).Cohesive soils contain clay minerals; the major groups are montmorillonite, kaolinite and illite. Theminor groups are allophone, chlorite, vermiculite, attapulgite, palygorkite, and sepiolite (Grim, 1953).It must be noted that most natural soils are anisotropic; soils that have different geotechnical propertiesin different direction. Isotropic soils are those having geotechnical properties that do not vary withdirection. Homogeneous soils have the same kind of constituent elements, same uniform compositionor structure. Two layers of soils can be considered as a single homogeneous anisotropic layer. This

depends on the equivalent isotropic coefficients for the two layers. In isotropic media, the relationshipsbetween elastic constants and velocities as given by Sheriff and Geldart (1995) are presented in Table1. In other to determine geotechnical properties of soils and rocks as outline in Table 1, the followingamong other steps are identified: surface exploration, geological survey, subsurface exploratory, fieldclassification, laboratory investigation, miscellaneous laboratory tests, rock cores, etc.

STRUCTURES - PIPES

There are cylindrical shapes in engineering field like pipes, shafts, etc which is used for serviceslike water supply, transportation, dewatering and drainage, sewerage, oil and gas supply, storagefacilities, piling for jetties berths and foundations, caissons, surface and underground main lines forirrigation, penstocks for hydro-electric projects, etc. Pipes are made of steel, cast-iron, ductile steel,

reinforced concrete, polyvinylchloride, clay, fibre glass, etc manufactured to different standard sizesand thicknesses according to codes of practice. These pipes are also called cylindrical shells namely,thin cylindrical shells, and thick cylindrical shells.

Table 1: Relations between elastic constant and velocities in an isotropic media (after Sheriff and Geldart, 1995)


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If the thickness of the wall is less than between 0.10 to 0.067 of its diameter, it is known as thinshell. Walls of such vessels are thin compared to their diameter. If otherwise, it is known as thick shellas shown in Figure 1 (Khurmi, 2002).

Figure 1: Cross-section of pipe in different soil layers

In thin cylindrical shell, stresses are assumed to be uniformly distributed. In thick shell, stressesare no longer uniformly distributed, in that case it becomes complex. When thin cylindrical shell issubjected to pressure, there are two modes of failure. Is either, it split into two troughs (failure due tohoop stress), or it split into two cylinders (failure due to longitudinal stress). In more complicated problems like blast, the analytical approach may not give accurate results. In that case numericalmethods must be employed if acceptable accuracy is to be obtained. In analytical solution, one


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approach is the stress function concept. This was first proposed by Sir George B. Airy, latergeneralized for three-dimensional case by Clerk Maxwell.

2 = 0 or 4 = 0 (1)

Eq. 1 is referred to as the biharmonic equation. The is known as Airy stress function. Indesigning thin cylindrical shells, certain parameters must be designed for. This is to determine therequired thickness of cylindrical shell for a given diameter, length and a given maximum pressureintensity. It must be remembered that lateral strain is always accompanied by linear strain. It meansthat lateral strain will always accompany cylindrical shell subjected to an internal pressure. Similarlythere will be change in volume due to increased internal pressure. Therefore,

= = = l + 2 c (2)

Change in volume is given as,

= V ( l + 2 c) (3)

c = Circumferential strain, l = Longitudinal strain, V = Volume of the pipe, l = Length of thepipe and d = diameter of the pipe. In determining the thickness, t of cylindrical shell, efficiency of the joint should be given priority. According to Cheung and Yeo (1979) and Demeter (1996) for pipesubjected to internal pressure (Fig. 1), analytical solution for radial stress is given as,

r = (1 - ) (4)

Hoop or Tangential Stress is given as,

t = (1 + ) (5)

While the radial displacement is given as,

Ur = (6)

a is the internal radius, b is the external radius, r is radius from center to the middle of the

thickness of the pipe and P is the applied pressure. is the Poissons ratio and E is the Youngsmodulus of the pipe material. If internal pressure and external pressure is applied in a thick-walled

cylinder with closed ends, according to Fenner (1986), the longitudinal stress set up in the cylinderwall is,


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= (7)

Hoop stress, , at the internal surface is given as

- (8)

K = = (9)

Based on Lame theory, for thick-walled cylinders subjected to internal pressure only, neglecting

radial temperature variations, the equation of radial stress, r, is given as

r= -P (10)

Internal hoop stress is given by

hi = -P (11)

External hoop stress is given by

he = (12)

R1 is the internal radius and R2 is the external radius. These equations yield the stressdistributions. The maximum values of both the radial stress and hoop stress are at the inside radius(Ross, 1996).

SOIL-PIPE INTERACTION

In the analysis of soil-pipe interaction through modeling, it involves determination of soil forcessuch as pressure, displacement, strain stresses, mises, etc around the pipe. Analytically, axial frictionforce is determined using this expression,

F = (Wp 2 DH + Wp) ( ) (13)

F = Axial friction force (N/mm), = Coefficient of friction between pipe and soil. = Density ofbackfill soil (kg/m3), D = Outside diameter of pipe (m), H = Depth of soil cover to top of pipe (m),

Wp = Weight of pipe and content (kg/m). The soil density and friction coefficient are obtained fromsoil tests. Where no data are available, the following friction coefficient can be used, Silt = 0.3, Sand =0.4, and Gravel = 0.5. Three different lateral soil forces are normally encountered in pipeline analysis;upward lateral force, downward lateral force, and sideways lateral force. There are two stages in thelateral force; elastic stage, where resistance is proportional to pipe displacement and plastic stagewhere resistance remains constant regardless of displacements. The lateral soil force, U, is estimatedas


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U = (H + D) 2 tan2 (45 + ) (14)

U is the ultimate soil resistance, kg/m and other notations as previously defined. The active lengthof the pipe line can be determined as,

L = (15)

L is the active length (m), F is the anchor force or expansion force (kN), Q is the end resistanceforce (kN), f is the soil friction force (kN/m). End deflection, y is calculated as,

y = (F Q) 2 (16)

A is the cross-sectional area of the pipe. The first is to estimate the end displacement,

y = (17)

M = (18)

Elastic constant

K = (19)

= (20)

By substitution,

Q = C - (21)

Where

C = F + (22)

y is end displacement (mm), Q is end force (kN), K is soil elastic constant, E is modulus of elasticityof pipe (kN/m2), I is moment of inertia of pipe (mm4) and M is end bending moment (kNm) (Liang-Chaun, 1978; Lester, 2008). The response of underground pipe due to blast is non-linear and can be

suitably and easily solved by direct non-linear simulation (modeling).

BLAST CHARACTERISTICS

The typically adopted constitutive relations of soils are elastic, elasto-plastic, or visco-plastic.Under blast load the initial response of the constituents is the most important because it of short duration(transient). It involves some plastic deformation of soil that takes place within the vicinity of the explosion. As a


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result of this one could take the ground media to be an elasto-plastic material, beyond which the soil can beconsidered as elastic material at distance from the explosion. Visco-elasticsoils exhibit elastic behavior uponloading which is followed by slow and continuous strain increase at decreasing rate (Boh, 2007; Greg, 2008).Blast can take place above the ground surface, on the ground surface, underground or inside the structures(pipes). During explosion, surface waves and body waves are generated. Consequently there are isotropiccomponent and deviatory component of the stress pulse. Transient stress pulse causes compression and dilation

of soil and rock. This is accompanied by particle motion which is known as compression or P-waves while thedeviatory component causes shearing stress which is known as shear or S-wave. On the surface of the ground,the particles adopt circular motion known as Rayleigh or R-wave. Compression, P-waves and shear, S-waves happens in underground explosions and they move within short range due to intervening medium which issoil or rock. On the other hand, Rayleigh, R-waves dominates above-ground explosions; as a result they havelong range (Kameswara, 2000). Energy impulse from explosion decreases as it travels for two reasons, firstly,due to geometric effect i.e. by three dimensional dispersion of blast energy. Secondly, due to energy dissipationi.e. result of work done in plastically deforming the soil matrix.

BLAST ENERGYThe majority of explosives are formed from Carbon, Hydrogen, Nitrogen and Oxygen. The general

chemical formula is Cx HyNw Oz. The three categories of blast are: free-air blast, air burst, and surface burst.

Energy imparted to the ground by the explosion is the main source of ground shock (Ngo et al, 2007). For asufficiently deep underground explosion, there is no blast wave. The detonation of a high explosive generates hotgases under pressure. Consequently, a layer of compressed air known as blast wave is formed. Blast waveinstantaneously increases to a value of pressure above the ambient pressure. This is referred to as the side-onoverpressure which decays as the shock wave expands outward from the explosion source. After a short time, the

pressure behind the front may drop below the ambient pressure. TNT (trinitrotoluene) equivalent values are usedto relate the performance of different explosives. This is the mass of TNT that would give the same blast

performance as the mass of the explosive compound in question [34]. Conversion factors obtained fromRemennikov (2003) for various explosives are; TNT (trinitrotoluene) - 1.000, RDX (Cyclonite) - 1.185, PETN -1.282, Compound B (60% RDX 40% TNT) - 1.148, Pentolite 50/50 - 1.129, Dynamite - 1.300, and Semtex -1.250.

Types of Blast

When a detonation occurs adjacent to and above a protective structure such that no amplification of theinitial shock wave occurs between the explosive source and the protective structure, then the blast loads actingon the structure are free-air blast pressures. The air burst environment is produced by detonations which occurabove the ground surface and at 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.The charge detonates above the ground and the blast wave propagates with a spherical wave front. Theclasses of these charges falls within the aerially-delivered munitions with fuses designed to operateabove the ground. A surface burst explosion will occur when the detonation is located close to or onthe ground so that the initial shock is amplified at the point of detonation due to the ground reflections.The charge detonates as it comes in contact with the ground and the blast wave propagates with ahemispherical wave front. The initial wave of the explosion is reflected and reinforced by the ground

surface to produce a reflected wave. Examples of these types are the terrorist vehicle bombs ormilitary munitions fused to detonate on impact with the ground. Unlike the air burst, the reflectedwave merges with the incident wave at the point of detonation to form a single wave similar to airburst but essentially hemispherical in shape. A comparison of these parameters with those of free-airexplosions indicate that, at a given distance from a detonation, giving the same weight of explosive, allof the parameters of the surface burst environment are larger than those for the free-air environment(UFC, 2008). For a conservative design, surface burst is considered being the worst of the three typesof blast. The charge weight of the explosive material under consideration is increased by the required


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factor of safety, UFC (2008) allows for an increase of 20%. In multiple explosions, two or moreexplosions of similar material occur several milliseconds (ms) apart, the blast wave of the initialexplosion will propagate ahead of the waves resulting from the subsequent explosions. The phasing ofthe propagation of these latter waves is governed by the initiation time and orientation of the

individual explosives. If the time delay between explosions is not too large, the blast waves producedby the subsequent explosions will eventually overtake and merge with that of the initial detonation.After all the waves have merged, the pressures associated with the common or merged wave will havea pressure-time relationship which is similar to that produced by a single explosion. Shock waveparameters as obtained from UFC (2008) for spherical TNT explosion in free air and hemisphericalTNT explosion on the surface can be determined using Figures 2 and 3 after scaled distance has beendetermined.

Figure 2: Positive and Negative Shock Wave Parameters for a Spherical TNT Blast in Free Air at Sea Level


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Figure 3: Positive and Negative Shock Wave Parameters for a Spherical TNT Blast in Free Air at Sea Level

There are three methods available for predicting blast loads on structures. These are; Empiricalmethods using US Army Technical Manuals (TM 5-1300, TM 5-855-1, etc) now Unified FacilitiesCriteria (UFC, 2008) shown in Figures 2 and 3, CONWEP shown in Figure 4 (Peter and Andrew,2009), etc.


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Figure 4: Result from CONWEP of pressure against distance for 1000kg TNT hemispherical charge: Defence Academy ofthe Unite Kingdom

Semi-empirical methods based on simplified models of physical phenomena attempt to model theunderlying important physical processes in a simplified way as shown in Figure 5 (Peter and Andrew,2009). Examples of software codes used are BLASTXW for multiple roomed buildings, SPIDS fortunnel and ducting complexes, SPLIT-X, BLAPAN, etc.

Figure 5: Result of semi-empirical prediction with experimental data: Defence Academy of the United Kingdom

Numerical methods are based on mathematical equations describing the basic laws of physics

governing a problem. Eulerian numerical techniques have been developed using finite volume andfinite difference solvers like SHAMRC, ANSYS, AUTODYN (2D and 3D), Air3d shown in Figures 6(a and b) (Peter and Andrew, 2008), AutoReaGas (3D), MADER, etc.


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a bFigure 6: (a) Result of experimental data from Sheffield University and AUTODYN; (b) Experimental data from Cranfield

University and Air3D simulation

In these above case, air is treated as an ideal gas and the detonation is modeled using anappropriate suitable equation of state for the explosive material. There is universal normalizeddescription of blast effects known as blast wave scaling laws. It is general practice to express thecharge weight, W as an equivalent mass of TNT. Results are given as a function of the dimensionaldistance parameter (scaled distance),

Z (23)

R is the actual effective distance from the explosion. W is the weight of the explosion generally expressed inkilograms. TNT equivalent for unconfined detonations (UFC, 2008) is given as

WE = WEXP (24)

Where WE is the effective charge weight, WEXP is the weight of explosive in question, is the heat of

detonation of explosive in question and is the heat of detonation of TNT. The TNT equivalent of

confined explosions is given as

WEg = WEXP (26)

Where WEg is the effective charge weight for gas pressure, is the heat of combustion of TNT,

is the heat o combustion of explosive in question, is the TNT conversion factor, is the heat ofdetonation of TNT, is the heat of detonation of explosion in question and WEXP is the weight of

explosion in question.

Ground Movement Parameters for Surface Blast

To predict blast pressure, according to Brode (1955), peak overpressure, Pso, is given as,

Pso = (bars) (27)

according to Mills (1987), Pso = = (kPa) (28)

while according to Newmark and Hansen (1961), Pso = 67841/3

(bars) (29)


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Maximum reflected pressure, Pr, when the blast encounter an obstacle is given as

Pr= 2Pso (30)

where Po is the ambient pressure. For design purposes, reflected overpressure, Pr, can be idealized by anequivalent triangular pulse ofmaximum peak pressure, Pr, and time duration, td(related directly to the time takenfor overpressure to be dissipated), which yields the reflected positive phase impulse, ir

ir= c Pr td (c varies between 0.2 and 0.5) (31)

Air-induced ground shock results when the air-blast waves compresses the ground surface and send a stresspulse into the ground under-layers. The maximum velocity (m/s) at the ground surface expressed in terms of thepeak incident overpressure (UFC, 2008) is given as

Vs = (32)

and Cp are the mass density and compression seismic wave velocity (Table 2) in the soil respectively, Vs

is the maximum vertical velocity of the ground surface.

Table 2: Compression Wave Seismic Velocities for Soils and Rocks (after UFC, 2008)

Material Seismic Velocity (m/s)

Loose and dry soilsClay and wet soilsCoarse and compacted soilsSand stone and cemented soilsShale and marlLimestone-chalkMetamorphic rocks

Volcanic rocksSound plutonic rocksJointed graniteWeathered rocks

182.88 - 1005.84762 - 1920.24914.4 - 2590.8914.4 - 4267.21828.8 - 5334

2133.6 - 6400.83048 - 6400.8

3048 - 68583962.4 - 7620243.84 - 4672609.6 - 3048

Integrating velocity Vs with time gives maximum vertical displacement (m) of the ground surface given as,

Dv = (33)

where i is the unit positive impulse. Assuming linear velocity increase during a rise time being equal to onemillisecond, and increasing acceleration by 20 percent to account for nonlinearity, vertical accelerations, A v is

expressed as

Av = (34)

Where g is the gravitational constant = 9.81 m/s2. Expressing vertical motions as a function of seismicvelocity of soil and shock wave,


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DH = Dv tan {sin-1 (35)

VH = Vv tan {sin-1 (36)

AH = Av tan {sin-1 (37)

where U is the shock front velocity and other parameters as previously defined.

In the case of loads from direct ground shock, the peak vertical displacement in m/s at the groundsurface (UFC, 2008) is given as,

Dv(rock) = 0.025 (m/s), (38)

Dv(soil) =0.17 (m/s) (39)

Where DH = 0.5 DV (40)

For dry or saturated soil, DV = DH (41)

Maximum vertical velocity for all ground media is given by VV = (42)

While VH = VV (43)

Maximum vertical acceleration, Av, in m/s2for all ground media is given by,

Av = (44)

Duration td is related directly to the time taken for overpressure to be dissipated. Horizontal acceleration isgiven by AH = 0.5 AV (45)

But for wet soil or rock media, AH = AV (46)

The arrival time tAG of shock load is a function of seismic velocity in the soil and it is expressed as

tAG = (47)

All the parameters as previously defined. For surface blast, these parameters as calculated are shown in Figure 7.


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Figure 7: Ground shock parameters for surface blast

Ground Movement Parameters for Underground Blast

Ground shock parameters are equally known as the soil movement parameters which translate intoloading which the soil delivers to the buried structures. These parameters are peak particledisplacementwhich is caused by a buried explosive at a location a distance from the structure and

peak particle velocity which depends on both the seismic velocity and peak particle velocity(Kameswara, 2000; Husabei, 2009). For a totally or partially buried charge located at distance R fromthe structure, peak particle displacement,x caused is estimated by,

x = 60 (1n) (48)Peak particle velocity, u, is given by

u = 48.8 Fc(-n) (49)

x is measured in meters and W is the charge mass in kg. Fc is a dimensionless coupling factor for theexplosive charge which depends on explosive charge burial depth (usually taken as = 1). c is the soil seismic

velocity in m/s and, n, is the dimensionless attenuation coefficient (usually taken to be = 2.75). R is the radialdistance (m) measured from the center of the charge weight, W. The value of the loading wave velocity, Cp(m/s) is given by seismic velocity, c and peak particle velocity, u and is given as,

Cp (fully saturated clays) = 0.6 c + u (50)


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Cp (sands) = c + u (51)

The specific impulse is then evaluated using this,

io = Cpx (52)

is the density in kg/m3 and io is measured in Ns/m2 (TM5-855-1, 1986, UFC, 2008, Zhenweng,1997). All these parameters as calculated are shown in Figure 8.

Figure 8: Ground shock parameters for underground blast

METHODS OF ANALYSIS

Many methods are available to determine the responses of underground structures most especiallypipes due to blast load. These are the analytical methods and the numerical methods. The analyticmethod is deterministic such as empirical phenomenological and computational fluid mechanicsmodels which are used for blast load prediction. The problems are; it designs for elastic response orlimited plastic response, and it does not allow for large deflection and unstable responses. There areseveral numerical methods for assessing the response of structures due to dynamic loadings. These areiteration, series methods, weighted residuals (least square methods), finite increment techniques (step by step or time integration procedure) usually referred to as finite difference, Newmark, Wilson,Newton, Houbolt, Eular, Runge-Kuta and Theta methods. Finite difference is popularly used to solveordinary and partial differential equations, in particular, dynamic problems. Using this method,solution domain is replaced by a number of discrete points called mesh points or nodes. Solution to theproblem is obtained at these points by converting the differential equation into an algebraic equationapproximately satisfying the differential equation and the boundary conditions. The algebraic


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equations can be obtained in terms of forward, backward or central difference formulae but centraldifference formulae are preferred due to their higher accuracy (Kameswara, 1998).

Central Difference Method

In the central difference equations for a function U (t), in which the grid points, (i = 1, 2n) along

the independent coordinate, t are equally spaced with step length t = h, using Taylors series, valuesof functions Ui+1 and Ui 1 can be expressed in terms of Ui as

Ui+1 = Ui + h i + i + (53)

Ui 1 = Ui - h i + i + (54)

i = |t = ti= (Ui+1 Ui - 1) + 0(h2) (55)

i = |t = ti = (Ui+1 2Ui + Ui- 1) + 0(h2) (56)

Most of the numerical methods in dynamic analysis are based on finite difference approach. Theequation of motion is given as

[m] [ ] + [c] [ ] + [k] [U] = [P] (57)for U (t = 0) = Uo (58)

(t = 0) = o = vo (59)

Where m, c, and k are element mass, damping and stiffness matrices and t is the time. U and P are

displacement and load vectors while dot indicate their time derivatives. The time duration (period) forthe numerical solution can be divided into n intervals of time t (h). It should be noted that with nodamping

t

for stable and satisfactory solution or with damping

t (

is the maximum natural frequency, is the critical damping factor. Stability limitis the largest time increment that can be taken without the method generating large rapid growing

errors. The accuracy of the solution depends on the time step t = h. However, there are someconditionally stable methods where any time step can be chosen on consideration of accuracy only andneed not consider stability aspect. Accordingly, the unconditionally stable methods allow a muchlarger step for any given accuracy. Replacing Eq.57 by Eq. 55 and 56, we have


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m {{U (i+1) 2 i + U(i-1)} / } + c{{U(i+1) U(i-1)} / }} kUi = P (60)

where Ui = U(ti) and Ui+1 can be written as

U(i+1) = [ Ui + U(i-1) + Pi] (61)

This is the recurrence formula which gives the value of Ui+1 in terms of Ui, Ui 1 and Pi.Repeated use of the recurrence equation gives the response of U of the system in the entire domain ofinterest. This is also called an explicit integration method since Ui+1 is obtained by using the dynamicequilibrium of the system at ti as given in Eq. 60. The solution can not start by itself, because to obtainUi (i = 0) from Eq 61, there is need to get the values Uo and U-1. Uo is given by the initial condition

in Eq. 58, U-1 has to be generated using the other initial conditions o given by Eq. 60 and thegoverning equation of motion (Eq. 57) is given by

o = (m)-1 (Po c o kUo) (62)

From the difference equations (Eqs. 55 and 56), we obtained

U -1 = Uo - h o + o (63)

where o is known from the given initial conditions as expressed by Eq. 62, i is increment numberof an exp[licit dynamic step and dots indicate their time derivatives. This could be solved usingAbaqus dynamic explicit which uses explicit central difference operator that satisfies the dynamicequilibrium equations at the beginning of the increment, t, the acceleration calculated at time, t are

used to advance the velocity solution to time, t + and displacement solution to time, t + t. DynamicIn direct-integration dynamics of time integration in the Abaqus Explicit, the equations of motion(Eqs. 57, 60, 61, 62 and 63) of the system is integrated through out time. This makes it unnecessary for

the formation and inversion of the global mass and stiffness matrices [M], [K]. It also simplifies thetreatment of contact and requires no iteration. This means that each increment is relatively inexpensivecompared to the increments in an implicit integration scheme. It performs a large number of smallincrements efficiently. Explicit are used for the analysis of large models with relative short dynamicresponse times and extremely discontinuous events or processes (Abaqus Manual, 2009; Olarewaju etal, 2010). Other numerical tools like ANSYS, AUTODYN 2D and 3D, FLAC 2000, etc could suitablybe used.

Technical Design Manuals for Blast-Resistant Design

Structures to Resist the Effects of Accidental Explosions, TM 5-1300 (U.S. Departments of the Army,Navy,, and Air Force, 1990),

A Manual for Prediction of Blast and Fragment Loadings on Structures, DOE/TIC-11268 (U.S.Department of Energy, 1992),Protective Construction Design Manual, ESL-TR-87-57 (Air Force Engineering and Services Center,1989),Fundamentals of Protective Design for Conventional Weapons, TM 5-855-1 (U.S. Department of theArmy, 1986),The Design and Analysis of Hardened Structures to Conventional Weapons Effects (DAHS CWE,1998),Structural Design for Physical Security State of the Practice Report (ASCE, 1995),


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Vol. 15 [2010], Bund. G 849

Principles and Practices for Design of Hardened Structures, Number AFSWC-TDR-62-138, (Air ForceDesign Manual, Technical Documentary Report, 1962),Unified Facilities Criteria (2008), Structures to Resist the Effects of Accidental Explosions, UFC 3-340-02, Department of Defense, US Army Corps of Engineers, Naval Facilities Engineering Command,

Air Force Civil Engineer Support Agency, United States of America (This manual supersede TM5-1300(1990)), etc.

MODELING

Methods of structural analysis and design are broadly divided into three, firstly, theoreticalmethods which carrying out analysis and the use of design codes, secondly, by testing full sizestructure using experimental method and thirdly by the use of models (simulations) where the first twofailed like structures of complicated shapes. Structural problems demanding model studies(simulations) are to:

predict the behavior of complicated structures with irregular boundaries,design structures with complex supports and loading conditions,get direct aids in design i.e. it offer short cut to design,check the design of very important and expansive structures such as large span bridges, prestige

buildings, atomic reactors, cooling towers, shell structures,check the validity of analytical procedures,investigate failure of structures caused by wrong assumptions, crude approximations of structural

behavior, etc, andmake qualitative demonstrations of structural behavior vis--vis simple structural action, deformedshapes, points of contraflexure, modes of buckling and collapse, reciprocal theorem, principle ofsuperposition, etc (Ganesan, T. P, 2000).

Finite Element Modeling

In finite element model, real continuous structure is idealized into assemblage of discreteelements. Force-displacement relations and stress distributions are determined or assumed. Thecomplete solution is obtained for the entire structure by combining the individual elements into anidealized structure. Conditions of equilibrium and compatibility are satisfied at the junctions of theseelements. One-dimensional, two-dimensional and three-dimensional finite elements such as triangular,rectangular, hexahedron, tet, wedge, etc can be used. Advantages of this method are; much greaterflexibility both in fitting boundary shapes, in arranging internal distributions of nodal points to suit particular problems, and lastly it provides a great deal of information concerning the variations ofunknowns at points within the region of interest. The disadvantage is the expertise required andsubstantially increased storage requirement for equation coefficients. In modeling, damping may bespecified as part of a material definition that is assigned to a model. Abaqus has elements such asdashpots, springs and connectors that serve as dampers, all with viscous and structural damping

factors. It equally allows the specification of global damping factors for both viscous (Rayleighdamping) and structural damping (imaginary stiffness matrix). One can use choose to model theviscous damping matrix by using material damping properties and/or damping elements (such asdashpot or mass element) (Abaqus Manual, 2009). Contrary to our usual engineering intuition,introducing damping to the solution reduces the stable time increment. Raleigh damping is meant toreflect physical damping in the actual material. A small amount of numerical damping could beintroduced in the form of bulk viscosity to control high frequency oscillations. The boundaryconditions of the finite element model for displacements could be fixed at the base and roller on all the


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four sides. This is to simulate infinity of the soil medium despite the short duration of the blastproblem, and to allow the energy to dissipate away without reflecting back into the soil and buriedpipes (Olarewaju et al, 2010). Apart from Abaqus numerical code, FLAC, ANSYS, AUTODYN 2D,AUTODYN 3D, etc could be used to study the response of underground structures (pipes) bymodeling while SAP program could be used to study linear response.

CONCLUSION

This paper has highlighted the basic steps in the study of response of underground pipes due toblast loads. Blast characteristics were also critically examined. Analytical and numerical methods ofanalysis were considered for the response of underground pipes due to static and dynamic load. It must be noted that soil exists as a semi-infinite half space. Numerical methods to be employed mustincorporate the notion of infinity in the formation. Integral equation method and boundary elementmethod can handle infinite domain naturally. Finite difference and finite element methods are domaindescritization methods. They can not be applied to semi-infinite domain directly. A way out ofhandling such infinite domains is by considering a finite domain for descritization with approximateboundary conditions. Exact solutions to general partial differential equations are difficult to obtain dueto irregular and geometrically complicated domains. There is difficulty in applying finite differencemethod and variational methods, this difficulties lies in considering approximate functions of thedependent variable. These functions need to satisfy the geometric boundary conditions on irregulardomains which are suitably considered in numerical tool like ABAQUS software package. Othersoftware packages like ANSYS, AUTODYN 2D and 3D, PLAXI, FLAC 2000, etc could suitably beused for linear and non-linear response.

ACKNOWLEDGEMENT

The project is funded by Ministry of Science, Technology and Innovation, MOSTI, Malaysiaunder e-Science Grant no. 03-01-10-SF0042.

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