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A REPORT ON Studies on Blast Response of Steel-Concrete Composite Panels Using Smoothed Particle Hydrodynamics (SPH) method BY RAVI KIRAN PUVVALA 2007A2PS586P Under the guidance of MRS.N.ANANDAVALLI Scientist, CSIR-SERC AT CSIR-STRUCTURAL ENGINEERING RESEARCH CENTRE CSIR Campus, Chennai, India A Practice School II station

Ravikiran Midsem Report Final (1)

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Page 1: Ravikiran Midsem Report Final (1)

A

REPORT

ON

Studies on Blast Response of Steel-Concrete Composite Panels Using Smoothed Particle

Hydrodynamics (SPH) method

BY

RAVI KIRAN PUVVALA 2007A2PS586P

Under the guidance of

MRS.N.ANANDAVALLI

Scientist, CSIR-SERC

AT

CSIR-STRUCTURAL ENGINEERING RESEARCH CENTRE

CSIR Campus, Chennai, India

A

Practice School II station

OF

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI

(March, 2011)

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BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, Pilani (Rajasthan)Practice School Division

A

REPORT

ON

Studies on Blast Response of Steel-Concrete Composite Panels Using Smoothed Particle

Hydrodynamics (SPH) method

BY

RAVI KIRAN PUVVALA 2007A2PS586P

Prepared in partial fulfillment of the

Practice School II Course

AT

CSIR-STRUCTURAL ENGINEERING RESEARCH CENTRE.

Chennai, India

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI

(March, 2011)

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ACKNOWLEDGEMENTS:

It gives me pride to profusely and gratefully render sincere thanks to my project guide Mrs.N.Anandavalli, Scientist,CSIR- SERC, for her valuable guidance in doing this project.

I thank Dr. Nagesh Iyer, Director, SERC and our project incharge, Dr. S Arunachalam, Advisor to Management , CSIR-SERC for giving us this opportunity to work at this premier institute, thereby, enabling us to gain insight into the latest research areas and build research perspective.

Finally, I thank my PS II Instructor, Dr. K Venkataraman, for his encouragement and moral support during the course of the project.

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Station: CSIR-STRUCTURAL ENGINEERING RESEARCH CENTRE Centre: CHENNAI

Duration: 6 months Date of Start: 5th January 2011

Date of Submission: 25th March 2011

Title Of The Project: Studies on Blast Response of Steel-Concrete Composite Panels Using Smoothed Particle Hydrodynamics (SPH) method

NAME OF THE STUDENT ID .No. Discipline

Ravi Kiran Puvvala 2007A2PS586P B.E (hons) Civil Engineering

NAME(s) & DESIGNATION(s) OF THE EXPERT

Mrs.N. Anandavalli , Scientist, CSIR-SERC

NAME OF THE PS FACULTY: Dr. K Venkataraman

PROJECT AREAS: Numerical Simulation techniques , Structural Dynamics, Smoothed Particle Hydrodynamics method, Blast effects on building, AUTODYN software.

ABSTRACT:

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An explosion within or immediately nearby a building can cause catastrophic damage on the building's external and internal structural frames, collapsing of walls, blowing out of large expanses of windows, and shutting down of critical life-safety systems. Loss of life and injuries to occupants can result from many causes, including direct blast-effects, structural collapse, debris impact, fire, and smoke. The indirect effects can combine to inhibit or prevent timely evacuation, thereby contributing to additional casualties. In addition, major catastrophes resulting from gas-chemical explosions result in large dynamic loads, greater than the original design loads, of many structures. Due to the threat from such extreme loading conditions, efforts have been made during the past three decades to develop methods of structural analysis and design to resist blast loads. The analysis and design of structures subjected to blast loads require a detailed understanding of blast phenomena and the dynamic response of various structures. This report presents a comprehensive overview of the effects of explosion on structures from the basics concepts involved in it. An explanation of the nature of explosions and the mechanism of blast waves in free air is also given. This report also introduces methods to estimate blast loads and structural response. The method we emphasis mainly on is the SMOOTHED PARTICLE HYDRODYNAMICS method, a meshfree Lagrangian analysis with discussing its edge over other simulating methods.

SIGNATURE OF THE STUDENTS SIGNATURE OF PS FACULTY

DATE: DATE:

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CONTENTS

Structural dynamics

Numerical simulation

SPH method

Blast effects on buildings

Autodyn modeling and analysis

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1. Structural Dynamics

Introduction:

Structural dynamics is the topic of the 21st century. It took almost the entire 20th century to bring the civil engineering profession to the point where buildings and other structures can be analyzed with a reasonable degree of confidence to evaluate their performance to real civil engineering types of forcing functions. The load that act upon structures received special attention starting in the 1970s with the aid of modern data recording and acquisition systems. For example, San Fernando Earthquake. A significant number of recordings (acceleration versus time histories) were taken for the first time. Gradually, more and increasingly accurate records of ground and building motions during earthquakes have been produced. Therefore, Structural Dynamics has entered with this foundation of good and improved analysis methods and records.

Definition:

Structural dynamics is a subset of structural analysis which covers the behavior of structures subjected to dynamic loading. Dynamic loading includes people, wind, waves, traffic, earthquakes and blasts.

Dynamic Analysis:

Any structure can be subjected to dynamic loading. Dynamic analysis can be used to find dynamic displacements, time history, and modal analysis. A static load does not vary with time. A dynamic load is one which changes with time. If it changes slowly, the structure’s response may be determined with static analysis, but if it varies quickly (relative to the structure’s ability to respond), the response must be determined with a dynamic analysis. Dynamic analysis for simple structures can be carried out manually, but for complex structures finite element analysis can be used to calculate the mode shapes and frequencies.

Time History Analysis

A full time history will give the response of a structure over time during and after the application of a load. To find the full time history of a structure’s response the structure’s equation of motions has to be solved. There are two kinds of analysis: i) linear ii) non-linear.We focus more on the linear analysis in the initial stages of study.

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Single-Degree-of-Freedom Linear Oscillator (SDOF)

The interdependence of the properties of the structure and the time history of ground motion is best demonstrated using an SDOF system. Also, many building code equations are based on the response of an SDOF system.For many dynamic systems the relationship between restoring force and deflection is approximately linear for small deviations about some reference. If the system is complex (e.g., a building that requires numerous variables to describe its properties) it is possible to transform it (using the normal modes of the system) into a number of simple 1- dimensional linear oscillator problems (SDOF).An SDOF system can be represented in many different forms as shown in Fig 1.1 and 1.2

Fig 1.2: SDOF System

Consider the SDOF system as shown in Fig 1.3 which is subjected to a ground acceleration. Four types of forces are acting on the mass

1.stiffness force (F s=−kx )

2.damping force ¿¿)

3.external force ¿¿) here it is the ground acceleration

4.inertial force

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Fig 1.1: SDOF System

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For any SDOF system we have,

∑ F=F s+Fd+F e= -kx - cx+F e= mx (1.1)

Rearranging the terms in Eq. 1.1gives

m x ( t )+c x+kx (t )=Fe(t ) (1.2)

This is called the Dynamic equilibrium equation and is often called the equation of motion for the system. As it is a second order linear differential equation, it requires two initial conditions to define its response. These initial conditions are generally the initial displacement of the mas, x(0) =x0 and the initial velocity of the mass, x (0 )= xo

The solution involves two parts: the homogeneous solution,xh (t ) , and the particular solution,x p(t).

System response, x(t) is calculated from these basic newton’s laws and solving the above differential equation, as in 1.2.

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Fig 1.3 Structure subjected to a ground acceleration

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2. Numerical Simulation: Introduction

Role of numerical simulation is increasingly become a very important approach for solving complex practical problems in engineering and science. It translates important aspects of a physical problem into a discrete form of mathematical description, recreates and solves the problem on the computer and reveals phenomena virtually according to the requirements of the analysts. It acts as a bridge between experimental models and theoretical predictions

Solution procedure of general numerical simulations

ODEs and PDEs are established in mathematical modeling. Domain discretization techniques may be different for different numerical methods. It

basically refers to representing a continuum problem domain with finite number of components, which form the computational frame for the numerical approximation. it may be a set of mesh or grid, which consists of a lattice of points or grid nodes to approximate the geometry of the problem domain. The grid nodes are the locations where field variables are evaluated, and their relations are defined by some nodal connectivity. Accuracy of the numerical approximation is closely related to the mesh cell size and the mesh patterns.

Fundamental frames for describing physical governing equations are:

I. Eulerian description: A spacial description, represented by finite difference method (FDM)

II. Lagrangian description: A material description, represented by finite element method (FEM)

Lagrangian grid: Fixed to or attached on the material in the entire computational process and therefore it moves with the material

Eulerian grid: Fixed on the space in which the simulated object is located and moves across fixed mesh shells in the grid

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Physical phenomenamathematical modeldomain discretizationnumerical algorithmcoding & implementationnumerical simulation

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Combined Lagrangian and Eulerian grids: This resulted in devolpment of coupled Eulerian Lagrangian(CEL)

Limitations of the grid/mesh-based methods: constructing a regular grid/mesh for irregular or complex geometry has been a difficult task.It usually requires special technique like rezoning which is a tedious and time consuming process.

Meshfree methods:

They are different in the means of function approximation and implementation process. Smoothened particle hydrodynamics(SPH), as a meshfree and particle method was originally invented for modeling astrophysical phenomena and later widely extended for applications to problems of continuum solid and fluid mechanics.

Advantages of meshfree particle methods

I. In MPMs, the problem domain is discretized with particles without a fixed connectivity. Treatment of large deformation is relatively much easier.

II. Discretization of complex geometry for MPMs is relatively simpler as only an initial discretization is required.

III. Refinement of particles is expected much easier to perform than the mesh refinementIV. Easy to obtain features of entire physical system through tracing the motion of particles

and easy identifying free surfaces, deformable boundaries and time history of field variables

Solution strategy of MPMs

I. Governing eqns with boundary conditions(BC) and/or initial conditions(IC)II. Domain discretization technique for creating particles

III. Numerical discretization techiniqueIV. Numerical technique to solve resultant ODE.

In computational mechanics an intense research activity is constantly conducted aiming to propose and improve numerical methods and procedures, to simulate particular mechanics conditions, such as high dynamic loading conditions, large displacements or rapid deformations. Indeed, these conditions can represent severe criticalities for classical numerical methods, such as Finite Element Methods (FEM) or Boundary Element Methods (BEM). Mesh distortion and stress concentration, in particular, represent two of the main issues that researchers try to overcome, introducing new numerical procedures. In case of seismic engineering, particularly complex analyses, especially for severe seismic excitations, can be subjected to such undesirable lacks of accuracy and consequently can need to be conducted via particular numerical procedures. Some of the most interesting numerical methods for severe dynamic conditions are the so-called meshless methods. This expression

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indicates a number of numerical methods that, using different approaches, do not need a mesh discretization of the continuum, as intended in the classical FEM approach. Alternatively, a number of points are defined, where all the variables and equations are evaluated. In this way, points, classically referred to as “particles”, can be considered as degrees of freedom, whose displacements do not introduce any mesh distortion and consequently numerical instabilities are avoided. On the contrary, meshless methods often suffer of lack of accuracy close to the boundary, which leads to the necessity of particular procedures, trying to enforce convergence of numerical solutions in every point of the domain.

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3. Smoothed particle hydrodynamics (SPH)

Introduction

In SPH method, the state of system is represented by a set of particles which posses individual material properties and move according to the governing conservation equations. It is a mesh free lagrangian particle method which is adaptive. This adaptability of SPH is achieved at the very early stage of the field variable approximation that is performed at each time step based on a current local set of arbitrarily distributed particles

SPH (Smoothed Particle Hydrodynamics) method was first introduced by Lucy (1977) and Gingold and Monaghan (1977) to address astrophysics problems. Then the method was used in a number of applications, mainly in fluid-dynamics; then it was successful introduced in mechanics and structural dynamics.

SPH as it is commonly attributed in literature to a group of meshless methods, sharing the same basic approach. Indeed, in all SPH procedures, aiming to approximate a function A(x), the following expression is considered as in equation 2.1

where δ(x) is the Dirac function and D is the domain of A(x); hence, a first approximation is introduced, through the following expression 2.2

where W(x), called kernel function, is substituted to approximate the Dirac function. Common kernel functions are Gaussian-like functions, whose expressions can be exactly that of Gaussian functions or also polynomial functions.

Method

I. The problem domain is represented by a set of arbitrarily distributed particles,if it is not yet in that form. No connectivity for them is needed

II. Integral representation function is used for field function approximation.III. The kernel approximation is then further approximated using particles.This is turned as

particle approximation in SPHIV. Particle approximation is done at every time step to all terms related to field functions in

PDE to get ODE in discretised form

V. ODEs are solved using an explicit integration algorithm.

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A (x) =∫A (x) δ(x) dΩ

A (x) = ∫A( x) W( x) dΩ

(Equation 3.1)

(Equation 3.2)

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Formulation of SPH

The formulation of the SPH is often divided into two key steps.

I. Integral representation of a function

The first step is integral representation or the so-called Kernel approximation of field functions. In this step integration of the multiplication of an arbitrary function and a smoothing kernel function gives the kernel approximation in the form of integral representation of the function

II. Particle approximation

The second step is the particle approximation. The integral representation obtained is then approximated by summing up the values of the nearest neighbor particles, which yields the particle approximation of the function at a discrete point or particle.

The concept of integral representation of a function f(x) used in the SPH starts from the following identity.f ( x )=∫ f ( x ' ) δ ( x−x ' ) dx '

Where f is a function of 3 dimensional position vector x, and δ is the Dirac Delta function given by

δ ( x−x ' )=1 for x=x '¿0 for x≠ x '

(Equation 3.4)

If the Delta function is replaced by smoothing function W ( x−x ' ,h ), the integral representation is given by

f ( x )≜ f ( x ' )W ( x−x ' , h ) dx ' (Equation 3.5)

Where W is so-called smoothing Kernel function. In the smoothing function, h is the smoothing length defining the influence area of the smoothing function W.

In SPH convention, the Kernel approximation is marked by the angle bracket < > and therefore the above equation is rewritten as

¿ f ( x )≥∫ f ( x ' ) W ( x−x ' , h ) dx ' (Equation 3.6)

Kernel function & its properties

In order to correctly reproduce the Dirac and in order to guarantee the convergence of the approximation method, the following properties are commonly requested to kernel functions

I. Positive in its domain

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(Equation 3.3)

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II. Unitary area within its domainIII. Delta function propertyIV. Defined in a compact support (Compact condition)

Normalization condition / the unitary condition states that

∫W ( x−x' , h ) d x '=1 (equation 3.7)

Delta function property is observed when the ‘h’ tends to ZERO

limh→ 0

W ( x−x ' , h )=δ (x−x ') (equation 3.7)

Compact condition gives

W ( x−x ' ,h )=0 when|x−x '|>kh (equation 3.8)

where k is a constant related to smoothing function for point at x.

Hence, a second approximation is introduced, substituting previous equation with its discrete expression:

where a partition of D has been introduced through a finite number N of spaces ΔΩ, whose centroids are the so called “particles”

APPROXIMATION OF DERIVATIVES

In order to obtain an approximated expression of derivatives of function A(x), a number of approaches has been developed and proposed. Classical one, applying Green formula and neglecting border contributions, expresses the first derivative of A(x) as

And in discrete form as:

Further derivatives, according to classical approach, are then evaluated reiterating the procedure

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A(x) = Σ A( x) W( x) ΔΩ (equation 3.9)

∇A x ≅ ∫{A( x) – A( x)}∇W (x) dx (equation 3.10)

∇A x =Σ[A( x) – A( x)]∇W (x) dΩ (equation 3.11)

∇2A(x) =Σ[∇A(x) −∇A(x)]∇W(x) ΔΩ (equation 3.12)

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In order to overcome lack of convergence, in particular occurring close to boundaries, different procedures have been proposed in literature. Starting from Taylor series expansion, Chen and Beraun (2000), proposed a generalized formulation leading to the following expressions for first and second derivatives:

Such a procedure, deriving from Taylor series expansion up to the first order, guarantees everywhere in the domain, even close to the boundary, a h order of convergence to the exact solution, where h represents the size of the particle discretization. In this case, no specific constraints are introduced for the kernel function. On the contrary, different approaches, also focusing on Taylor series expansion, aiming to manage the convergence error, indicate particular expressions for kernel functions. It is the case of Reproducing Kernel Particle Method (RKPM) from Liu and coworkers (1995) and Kulasegaram and Bonet (2000), where, forcing to zero n-th order momenti of kernel function, it is aimed to achieve higher order errors. However, boundary deficiencies in these cases are not completely eliminated and particular local procedures become necessary. To overcome such problems and obtain a second order error in approximation of derivatives, authors are currently working on an original formulation, which starts from a basic idea of Liu et al. (2005) and Zhang and Batra (2004). In this procedure Taylor series expansion up to the second order is projected against a kernel function an its derivative, obtaining the following linear system, providing the first and the second derivatives of the function A(x):

(B11 B12

B21 B22)( ∇ A ( x i )∇2 A ( x i ))=( ∫D [ A ( x )−A( xi)]W i ( x ) dx

∫D

[ A ( x )−A (x i)]∇W i ( x ) dx)Where

B11=[∫D

[ x−x i ]W i ( x ) dx]

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∇ A ( xi )=∑j=1

N

[ A ( x j )−A ( x i ) ]∇W i ( x j ) ∆ Ω j

∑j=1

N

(x j−x i)∇W i ( x j ) ∆ Ω j

∇2 A ( x i )=∑j=1

N

[ A ( x j )−A ( x i ) ]∇W i ( x j ) ∆ Ω j−∇ A ( x i )∑j=1

N

(x j−x i)W i ( x j ) ∆ Ω j

12∑j=1

N

(x j−x i)2W i ( x j ) ∆ Ω j

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B12=12 [∫

D( x−x i )

2W i ( x ) dx ]B21=[∫

D[ x−x i ]∇W i ( x )dx ]

B22=12 [∫

D( x−x i )

2∇W i ( x ) dx]Hence, properly choosing kernel function, enforcing for each particle that specific momenti of the kernel function and of its derivative are null, the error is moved to h2order, even for particles close to the boundary. In order to appreciate the enhancement introduced with this formulation, the method was used to solve in one dimension the following problem

defined in the domain D [0;1], where u is an unknown function and f is equal to:

NUMERICAL SPECTRA

In order to appreciate the capability of the investigated methods in reproducing high dynamic problems, it is useful to evaluate how higher vibration modes of a dynamic system are reproduced through the numerical approximations. A uniaxial elastic element is considered, of length L, longitudinal elastic stiffnessEA, and longitudinal density ρL. Dirichlet boundary conditions are applied. The governing equation is

EAδ 2 xδ X2 +ρL

δ2 xδ t 2 =0

where X and x are the initial and the actual configuration of the element, respectively. Introducing the celerity c as

c=√ EA

ρL

c2 δ 2 xδ X2 +

δ 2 xδ t 2 =0

In a stationary problem the following equation holds:

δ2 xδ t2 =ω2 x

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u” = − f

f = sin(πx) x)

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where ω is the frequency of vibration of the element; hence, equation can be rewritten as:

c2 δ 2 xδ X2 +ω2 x=0

The solution of this differential equation is given by:x = sin(α x)

SPH for simulating explosions

The combination of adaptive, meshfree and lagrangian nature of the SPH plus an explicit algorithm make the SPH method very attractive in treating highly dynamic phenomena with large deformations and large in-homogeneneties that occur in the extremely transient HE explosion process. The SPH formulation is performed based on the Euler equations, as the explosion is an extremely fast phenomenon. The JWL equation of state for high explosives is incorporated into the SPH equations.

Parameters before and after detonation shock D−U=V √( p−P)/ (V−v) U−U=(V−v)√(P−p)/(V−v) e−E=1/2( p+P)(V−v)

Euler equations

d ldt

= −(1/e)∇ p.v

dvdt

= -(1/e¿∇ p

Pressure of the explosion gas is given by

P=A (1−w )e−R

n +B (1−w ) e−R'

n +wnpe

Simplifies usage

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P= (Ɣ−1 ) le

This is called the Gamma law for ideal gas

SPH formulations are then applied on the pressure equations and then solution is obtained using the algorithm.

High dynamic problems represent a critical aspect of structural dynamics, which is currently attracting the efforts of many researchers of numerical community. In this field, meshless methods seem to be able to overcome the problems related to mesh distortions and numerical instabilities, which affect classical FEM, if employed under particular loading conditions.I’ve presented the basic idea of SPH method and summarized its basic formulations; in particular, the attention is focused on derivatives expressions, which play a fundamental role in deriving numerical framework to simulate dynamic structural systems. In this case, given the lack of accuracy occurring close to the boundary, different improvements have been proposed in literature and some of them are here cited. Then, the derivation of an original formulation is described. The approach here proposed appears to exhibit good results both in the direct approximation of derivatives, tested via a numerical problem, and in the derivation of the numerical spectrum. In particular, this also reveals that existing SPH formulations lose completely accuracy in predicting higher vibration frequencies. On the contrary, the proposed formulation, presented in two variants, is able to better reproduce higher modes(especially in case of the second formulation), providing results comparable with those of FEM in case of linear shape functions, without the classical deficiencies characterizing FEM method in high dynamic analyses. Authors are currently working on differences between proposed formulations and other existing SPH procedures in order to investigate the numerical spectra, conducting further numerical tests, and extending the formulations to multidimensional cases.

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4. Blast effects on buildings Explosive loading

Explosion effects on building structures can be divided into primary and secondary effects.

The primary effects include:

1. Air blastThe blast wave causes a pressure increase of the air surrounding a building structure and also a blast wind.

2. Direct ground shockAn explosive which is buried completely or partly below the ground surface, will cause a ground shock. This is a horizontal (and vertical, depending on the location of the explosion with regard to the structural foundation) vibration of the ground, similar to an earthquake but with a different frequency.

3. HeatA part of the explosive energy is converted to heat. Building materials are weakened at increased temperature. Heat can cause fire if the temperature is high enough.

4. Primary fragmentsFragments from the explosive source are thrown into the air at high velocity (for example wall fragments of an exploded gas tank). Fragments can hit people or buildings near the explosion. They are not a direct threat to the bearing structure of the building, which is usually covered by a facade. However, they may destroy windows and glass facades and cause victims among inhabitants and passers-by.

An overview of the explosion effects on buildings, summarized, is given in figure 4.1 below. Secondary explosion effects, such as secondary fragments and blast induced ground shock are not considered.

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Blast loading

During an explosion an oxidation reaction occurs that is called combustion. When explosive materials decompose at a rate below the speed of sound (subsonic), the combustion process is called deflagration. Gas and dust explosions are of this type. Under specific conditions a deflagration to detonation transition can occur. Detonation is the other form of reaction which produces a high intensity shock wave. The reaction rate is 4-25 times faster than the speed of sound (supersonic). An explosion of TNT is an example of a detonation. The two types of explosions have significantly different pressure-time profiles and will therefore be treated separately in this report.

Pressure time profile

The meaning of a few important blast parameters can be seen in figure 4.2. In this figure, t a is the arrival time of the blast, t d is the positive (overpressure) phase duration of the blast,t d , n is the negative (under-pressure i.e. negative overpressure) phase duration of the blast, p0 is the

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Fig 4.1: Explosion effects on structures

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ambient pressure, pm is the peak static overpressure, pn , m is the maximum value of under-pressure, i is the impulse of the positive phase of the pressure-time curve and in is the impulse of the negative phase of the pressure-time curve. The pressure-time profile in the figure is that of a detonation. The deflagration pressure-time profile is different as can be seen in figure 4.3. The deflagration pressure-time profile will transform to a detonation profile if the peak-static overpressure exceeds the value of approximately 3kPa (pm > 3 kPa).

Scaled distance

An important parameter for determination of air-blast pressure and impulse is the scaled distance z, which is dependent of the distance r from the charge center in meters and the charge mass M TNT expressed in kilograms of TNT:

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Fig 4.3: Detonation and deflagration pressure time history

Fig 4.2: Blast wave pressure time profile

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Other blast parameters can conveniently be plotted against the scaled distance. In figure 4.4 the peak static overpressure p, impulse i, time of blast arrival t a and positive phase duration t d are shown depending on the scaled distance z.

Dynamic blast pressure

Apart from a static overpressure p, there is also a dynamic pressure q (i.e. blast wind) associated with a blast wave. This dynamic pressure is higher than the static overpressure for small scaled distance and lower than the static overpressure for large scaled distance. The positive phase duration of the dynamic pressure t dq and static overpressure t dp is also different, but in this report it will be assumed that both durations are equal. In figure 4.5 the static overpressure p and the dynamic pressure q are displayed versus the scaled distance z.

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Fig 4.4: Blast parameters for TNT equivalent explosions

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Blast-structure interaction

Blast parameters are found for TNT blasts in free space (free-air burst) or half space (surface burst). In case of a surface burst, parameters for free-air bursts should be multiplied by a reflection factor of 1.8. Theoretically this factor should equal 2, but some energy is dissipated in the deformation of the surface. A reflection factor of 1.8 gives good agreement with experimental results.

Blast waves on an infinite rigid plane

If a blast wave with a certain (time dependent) static overpressure p(t), dynamic pressure q(t) and impulse i(t) encounters an infinite, rigid plane, it is reflected. Because the incident wave and the reflected wave coincide, the pressure on the rigid plane is higher than the pressure of the incident wave and is denoted by pr( t), reflected overpressure. The reflected impulse associated with the reflected overpressure is denoted by ir(t )

The reflected overpressure and impulse are dependent on the angle of incidence ∝i of the blast wave, which is the angle between the blast wavefront and the target surface. The reflection coefficient C r is defined as the ratio of the reflected overpressure and the incident overpressure (overpressure if the wave were not obstructed, sum of static overpressure p(t) and dynamic pressure q(t)). If the angle of incidence is 90deg, the blast wave travels alongside the plane and the overpressure is equal to the static overpressure, which is also referred to in as side-on overpressure. The dynamic pressure q(t) works in this case only in the direction parallel to the plane and is therefore (almost) not obstructed. The friction between the moving air and the rigid plane is negligible. For all ∝ibetween 0 – 90 deg, the reflected pressure pr (t )is dependent on the static overpressure p(t) and the dynamic pressure q(t). The reflection coefficient C ris shown versus the angle of incidence ∝i in figure 4.6 for a

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Fig 4.5: Static overpressure (p) vs Dynamic overpressure (q)

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detonation. As the figure points out, the reflection coefficient is dependent on the static overpressure p(t). The reflection coefficient also depends on the type of explosion, detonation or deflagration.

Starting at an angle of incidence ∝i of approximately 40deg, depending on the static overpressure, the reflection coefficient C rincreases and has a local maximum which is sometimes higher than the reflection coefficient at ∝i = 0 deg. This is due to Mach reflection, which occurs when the reflected wave catches up and fuses with the incident wave at some point above the reflecting surface to produce a third wavefront called the Mach stem. These local peak values are the result of theoretical derivations that could not be verified by experiments. Therefore it is suggested that these peak values are flattened for simple calculations.

Blast waves on a building

If a blast wave encounters a building, the building is loaded by a pressure, which is a summation of two parts. The first part is due to the static overpressure and the second part isdue to the dynamic pressure or blast wind. These pressures are shown in figure 4.7. Both static overpressure and dynamic pressure are a function of time, and also of the unobstructed distance to the charge center. Time and distance are coupled by the velocity at which the blast wave is propagating, although this velocity is not constant. Due to the coupling of time and distance, the part of translational pressure ptrans(t)which is due to the static overpressure p(t) depends on the size of the building (or other object) compared to the positive phase

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Fig 4.6: Reflection coefficient vs angle of incidence for a detonation

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duration of the blast t d. The translation pressure is defined as the net pressure due to blast loading.

If the building is very small in one direction (width or height), the overpressure on the front and the back of the building is approximately equal and hence the translational pressure is approximately zero (figure 4.8a, overpressure).

If the building is larger (width or height) compared to the blast duration and the angle of incidence is 0 deg, the front of the building is loaded with the reflected static overpressure pr ; st(t).

The sides are loaded with the static overpressure. However, this pressure works in opposite direction on both sides and does not result in translational pressure. The pressure in the blast wave traveling along the sides equals the static overpressure (dynamic pressure is considered further on). This pressure is lower than the reflected static overpressure, which causes large local pressure differences. These pressure differences cause the blast wave to diffract around the building. When the diffraction from the front to the sides of the building is completed, the static pressure on the front is decreased from the reflected static overpressure to the static overpressure. Similarly, pressure differences cause the blast wave to diffract from the sides to the back of the building, once the blast wave front passes by the back of the building.

Translational pressure resulting from diffraction is significant only when the time to diffract around the building is approximately equal to or larger than the positive phase duration of the blast. For this type of building and blast, the building is called a diffraction target. If the positive phase duration of the blast is much smaller than the time to diffract around the building, the building is loaded sequentially. A sequentially loaded building feels a pressure from the blast wave either at the front, at the sides or at the back, but not at the same time.

Summarizing the effects of static overpressure on a building, three essentially differenttypes can be distinguished:

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Fig 4.7(a): Static overpressure Fig 4.7(b): Dynamic overpressure

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No translational pressure (long positive phase duration compared to time of diffraction).

Translational pressure due to difference between overpressure on front and back (positive phase duration approximately equal to or larger than the time of diffraction).

Translational pressure due to local overpressure only (small positive phase duration compared to time of diffraction).

The effects of static overpressure determine the translational pressure on a building together with effects of the dynamic pressure or blast wind. Whereas the static overpressure is caused by an increased density of the air, the dynamic pressure is the result of the movement of air away from the blast source. Similar to ordinary wind, the blast wind causes a pressure on the front of a building and a negative pressure (suction) at the back of a building. Both pressures are translational in the same direction. Because of their translational nature, these pressure are called drag pressures. The drag pressure on a building or object is equal to the dynamic pressure multiplied by a drag factor Cd. The drag coefficient for suction at the back of the object is smaller and negative (Cd= −0.3 for a boxed-shape object and ∝i = 0deg).

The dynamic pressure on the front is reflected similar to the static overpressure. In literature, the reflected overpressure pr( t)includes both the reflected static overpressure pr : st( t) and the reflected dynamic pressure qr(t). This is rather confusing and leads to mistakes where the drag force is superimposed on the force from the reflected overpressure. If a building is small (height or width), the pushing and sucking drag pressures on front and back are applied at approximately the same time. Because of the small dimensions of the front face, no reflection occurs (figure 4.8a, dynamic pressure).

For the other types of buildings, the behavior is similar as for static overpressure (figure 4.8b-4.8c, dynamic pressure), except for the direction of the pressure on the back of the building, which is opposite of the direction of static overpressure. The reason for this similarity is found in the similarity of the pressure-time history of static overpressure and dynamic pressure.

Summarizing the effects of both static overpressure and dynamic pressure, the target(building with respect to blast) can be divided into three categories:

Drag target: translational pressure from dynamic pressure, crushing from static overpressure (long positive phase duration compared to time of diffraction). See figure 4.8a.

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Diffraction target: translational pressure due to both dynamic pressure and static overpressure (positive phase duration approximately equal to or larger than time of diffraction).

Sequentially loaded target: translational pressure due to both dynamic pressure and static overpressure, but only local (positive phase duration (much) smaller than time of diffraction). See figure c)

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Fig 4.8(a): Drag Target

Fig 4.8(b): Diffraction Target

Fig 4.8(c): Sequentially loaded target

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Other objects near the blast source as well as inequalities in the surface all influence the blast pressure-time history. It can be concluded that a lot of information is needed about the explosive and its location, the geometry of the target and its surroundings to be able to quantify the pressure-time history. In complex built environments, a lot of reflections can occur, leading to a complex pressure-time history. It is impossible to calculate all reflections and possible re-reflections under different angles of incidence as well as the influence of the geometry of the target and surrounding structures on the reflected overpressure analytically. In this case it is necessary to do experiments or to use CFD, Computational Fluid Dynamics, to simulate the blast. In uncoupled CFD buildings are modeled as rigid objects. In coupled CFD the influence of movement and failure of building elements on the blast wave propagation is also considered. However, a lot of input is required by the user as well as expertise in finite element modeling.

Blast distribution

Using the knowledge from the previous sections and the methods from literature to obtain quantitative information about the translational pressure and impulse on a building, the distribution of pressure and impulse over the facade of the building has not been taken into account. For blasts at large range, the distribution is approximately uniform. At shorter ranges, this assumption is not valid, since the blast propagation from a surface burst is hemispherical. At close range, the range (and scaled distance) to different heights on the facade is significantly unequal.

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5. AUTODYN

This part describes how 2D and 3D numerical analysis can be applied to the simulation of high explosives. The capabilities of AUTODYN-2D & AUTODYN-3D hydrocodes are described, with emphasis on the modeling of blast and explosive events. The results of a number of example analyses are presented, to illustrate the application of various modeling techniques to high explosive events. These examples describe the analysis of channeling of air blast down a street, structural response due to an explosion in a munitions store, an explosively formed penetrator, a shaped charge and a fragmenting warhead.

Introduction

Overview of analysis techniques

The objective is to show examples where 2-D and 3D numerical analysis software tools have been used in the simulation of high explosives. This part of report will concentrate on 5 particular case studies associated with explosion and blast problems, including the analysis of loading and response

High explosive loading and response problems involve highly non-linear transient phenomena. A great range of physical processes must be taken into account to enable accurate characterization of such events. It is the responsibility of the engineer/scientist/designer/assessor to consider these complex interacting phenomena using a range of appropriate techniques. Thera are 4 basic techniques that can be applied, together with more general skills such as experience and judgement, and these are outlined below. Firstly hand calculations can be applied; however, only the simplest highly idealized problems are practically solvable. More complex analytical techniques which are usually computer based or involve the use of look-up graphs and charts, are very useful in enabling consideration of many different cases relatively quickly. By their very nature analytical techniques are only applicable to a narrow range of problems; this is because they are based on a limited set of experimental data or particular gross simplifying assumptions. Difficulties in modeling highly non-linear phenomena mean that physical experiments play a vital role in the charaterisation of such problems. However, these experiments can be very costly, and are often difficult to instrument and interpretation of results is rarely straightforward.

Numerical software

This offers an alternative approach to high explosive blast and explosion phenomena. Their advantage is that they attempt to model the full physics of the phenomena. In other

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words, they are designed to solve from first principles the governing conservation equations that describe the behavior of the system. By their nature numerical techniques are suitable for solving a wider range of problems than a particular technique. They enable great savings to be made in the costs of investigative physical experiments and allow the analyst to look at ‘perfectly instrumented numerical eperiments’. Thus parameters that are virtually impossible to measure in physical experiments can be examined in detail.

In reality, numerical techniques for these highly non-linear phenomena are not able to model the complete physics and often the sub-models, which exist in all state-of-the art tools, are empirically based or require data which must be obtained through experimental validation. For example, equation of state parameter for HE are often determined using data from so called cylinder tests where the motion of the wall of a copper cylinder filled with explosive is measured. There are 2 major general problems to be faced in the numerical analysis of the types of events described in this report. Firstly, for problems of solid dynamics (example: Structural response) the chief problem is material characterization in terms of the models that are used and the data required for them. For fluid dynamics (example: The expansion of the high explosive products and blast) the chief problem is the lack of numerical resolution available for solving such problems. Much of the current research and development work related to numerical codes is concerned with better overcoming these 2 major issues.

Despite the computational requirements of numerical analysis, the increased power and availability of computers has led to the wide-spread use of numerical software tools for solving highly non-linear dynamic events. The barriers between experimentalists, analysts and designers are gradually breaking down as such tools become more widely used. Indeed, problems are most efficiently and effectively solved when a combined approach involving physical experimentation, analytical and numerical techniques are taken.

A more general problem faced by all techniques, but which becomes particularly apparent when developing numerical techniques, is that many areas of non-linear response are poorly understood; two notable examples are the details of dynamic material fracture and turbulent fluid flow. This poor understanding does not mean that modeling techniques are rendered useless, indeed numerical modeling is a major vehicle in developing our understanding of these complex phenomena.

The report will start by reviewing the current status of the AUTODYN numerical software used in the analyses illustrated here. Following this each of the applications will be described together with sample results from the analyses.

AUTODYN-2D & 3D

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The special features and capabilities of AUTODYN-2D & 3D are described below. Importantly, they both include all the required functions for model generation, analysis and display of the results in a single graphical menu-driven package. The codes can be run, with the same functionality albeit at varying speeds, on personal computers and engineering workstations through to mainframes and supercomputers. The codes are written in ANSI standard FORTRAN and C for portability. These codes are under constant and active development through industrial and academic research and development. Such developments are to a great extent driven by the feedback obtained from users of the codes

AUTODYN-2D & 3D are fully integrated engineering analysis codes specifically designed for non-linear dynamic problems. They are particularly suited to the modeling of impact, penetration, blast and explosion events. AUTODYN-2D & 3D are explicit numerical analysis codes, sometimes referred to as “hydrocodes” where the equations of mass, momentum and energy conservation coupled with materials descriptions are solved. Finite difference, finite volume, finite element and meshless methods are used depending on the solution technique (or processor ) being used. Alternative numerical processors are available and can be selectively used to model different regions of a problem. The currently available processors include Lanrange, typically used for modeling solid continua and structures, and Euler for modeling gases, fluids and the large distortion of solids. The Euler capacity allows for multi-material flow and material strength to be included. A fast single material high resolution Euler FCT processor in both 2D and 3D has also been developed, to more efficiently address which can be used to provide automatic rezoning of distorted grids; ALE rezoning algorithms can range from Langrangian (i.e. grid moves with material) to Eulerian (i.e. grid fixed in space). A Shell processor is available for modeling thin structures and both codes include an erosion algorithm that enhances the ability of the lagrange and shell processor to simulate impact problems where large deformations occur. Coupling between the processor types is available so that the best processor type for each region of a problem can be used. Various techniques, such as remapping which uses an initial explosion calculation to set up the initial conditions for a subsequent calculation stage, can be used to improve computational efficiency and solution accuracy.

The lagrange processor, in which the grid distorts with the material, has the advantage of being computationally fast and gives good definition of material interfaces. The Euler processor, which uses a fixed grid through which material flows, is computationally more expensive but is often better suited to modeling larger deformations and fluid flow.An SPH (Smooth Particle Hydrodynamics) processor is also available in the codes. SPH method is explained in the earlier parts. At present, SPH capability is best suited to the modeling of impact/penetration problems, although the rapid evolution of the SPH

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technique is likely to lead to a much wider range of applications for which SPH is a good choice. A description of the SPH technique and examples of impact and penetration simulations are also given below.

A large range of material equations (EOS) and constitutive models are available and the user can incorporate further options through the provided user-subroutine facilities. High explosives are usually modeled using the Jones Wilkins and Lee. This is an empirical material model with parameter typically derived from cylinder test data. The explosive is initiated at points or planes and a detonation wave propagates away from the ignite ion locations into the material at the detonation velocity. This process converts the explosive to high pressure detonation products. Alternatively, the Lee Tarver ignition and growth model can be used for more detailed explosive initiation studies

Application Examples

Numerical analysis methods can be used to simulate a wide range of explosive applications:

Explosions in air, underground, underwater and in other materials Shaped charges for the military, demolition and oil well perforation Materials forming, welding, cutting and powder compaction Other types of warhead such as explosively formed projectiles or fragmentation Numerical analyses of few example applications are described in this section

Example 1:GOAL:Model the explosive demolition of a concrete slab. The materials to be used are AIR, CONC-35MPA, TNT. The principal stress for tensile failure of CONC-35MPA is 5.0e3, take fracture energy=100.Take Internal energy of AIR=2.068e5≡ 1 atmosphere. Dimensions of the Structure are 500x1000.

Modelling:

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Fig 5.1 Model of the Problem

Final picture after the RUN

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Fig 5.1: After the detonation

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Fig 5.2: The 3D view of the PRESSURE DISRTIBUTION over the entire model after the detonation.

Fig 5.3 When the number of cells are HALFED

Fig 5.4: When the number of cells are DOUBLED

Example 2:

SHAPED CHARGE JET FORMATIONShaped charge warheads are fundamental to many weapons systems, as well as to civilian applications such as rock fracturing for oil drilling and demolition. Over the past four decades, enormous amounts of effort have been invested in attempting to maximise the performance of these warheads and to understanding the effects of material properties and manufacturing tolerances. Extensive experimental programmes have helped to identify the crucial factors in charge design, allowing geometries and dimensions to be optimised. Sophisticated measurement techniques have similarly given an understanding of the processes involved in the jet formation. This development has been well supportedby the availability of 2D and 3D numerical models capable of accepting readily available design data and generating simulations of shaped charge operation which allow

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visualisation of the jetting and penetration process. The information produced can be validated experimentally. The jet formation process within a shaped charge involves extremely high pressures, deformations and strain rates in the liner material at the jetting point and in the early stages of jet formation. For this reason, the numerical modelling of the jetting process is commonly carried out using the Euler processor. An alternative approach available in AUTODYN-2D is a combined numerical / analytical method where the liner is modelled using a Shell subgrid coupled to an Euler grid containing the explosive charge. The acceleration and deformation of the liner are calculated numerically until the liner reaches the symmetry axis. An analytical calculation is then used to predict the resulting jet and slug behaviour.The following example illustrates the application of the AUTODYN-2D Euler processor to analysis of a 90mm diameter precision shaped charge. The charge configuration is shown below, consisting of an Octol explosive fill and an OFHC copper liner. A single Euler grid of about 100,000 nodes (equivalent to a grid size of approximately 0.5mm) is used. The warhead configuration and resultant jet at 50 microseconds are shown in Figures 5.5 & 5.6

Fig 5.5 Shaped Charge Warhead

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Example 3:

Street channeled Air BlastIn recent years high explosive bomb attacks have been increasingly directed agains civil structures by various terrorist organizations. This has encouraged structural designers to look for new and improved methods of protecting civil structures without resorting to military style massive reinforced concrete construction[]. Such improved designs must ne based on design blast loads with an appropriate degree of accuracy. For conceptual design studies it is possible to use simple analysis methods such as analytical or 1 or 2 dimentional numerical methods. For detailed design of a protection system, better accuracy design loads are required, and for the typical complex geometries found in congestes urban environments this means that a 3D analysis tool must be used. However, if such a tool is to be used, it must first be validated to show that it can reproduce the complex effects associated with blast waves propagating and interacting with multiple obstracles. Whatever method is used to generate the blast loading predictions it must be capable of predicting load time histories on entire building facades and on structural components such as individual panels in a building facade

In order to validate the 3D blast calculations conducted using the Euler-FCT processor, and the remapping facilities in AUTODYN-2D & 3D, results from a series of numerical calculations were compared with results from a small scale experiment. The Pressure-Time history is shown in Fig 5.7. The plan of the experimental rigis shown in Figure 5.9. The test is 1/50 th scale and therefore represents a 1000kg TNT charge detonating at the centre of a typical street geometry.

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Fig 5.6: Shaped Charge Jet at 50µs

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As shown, a pressure guage was placed on one of the building faces opposite the end of the street. The experiments were conducted by the Royal Military College of scienceFigure 5.8 shows the numerical mesh used to model the experiment. This model used 2 symmetry planes to reduce the size of the numerical mesh and hence the calculation time, and with a cell size of 10mm within the street it contained approximately 360000 cells.

Pressure Time histories on the front face of the building opposite the end of the street are shown in the below figure. A 2D approximation of the street geometry results in a significant over-prediction of the peak pressure, while the full 3D calculation agrees well with the experimental results. The Conwep time history shows the effects of using a simple analytical calculation that neglects the channeling of the blast wave down the street.These calculations show that for a complex geometry with the blast wave channeled down a street only 3D numerical analysis that incorporated all the 3D geometry of the problem gave good agreement with experimental results. Comparison of blast wave predictions made using the Conwep program with the experimental results showed that neglecting the effects of channeling the blast wave along the street caused severe under prediction of the blast wave peak pressure and impulse. A protection system based on these blast loads would be very unsafe.The records we get are also shown below.

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300mm

200mm

300mm 240mm

150mmConcrete block

StreetBuilding

Charge 8g TNT equivalent

Guage Location

Fig 5.9: Plan of Experimental geometry

Concrete Block

Building

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Fig 5.7: Street channeled blast pressure time histories on front face of the building

Fig 5.8: 3D Street Channelled Blast model

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REFERENCES:

[1] Gary C Hart, Kevin Wong (2000) “Structural Dynamics for structural engineers”, John Wiley & Sons

[2] John M Biggs “Strcutural Dynamics”, McGraw-Hill (1964)

[3] Baker, W.E., Cox, P.A., Westine, P.S., Kulesz, J.J. and Strehlow, R.A., Explosion hazards and evaluation, Elsevier Scientific Publishing Company, New York, 1983.

[4] Brode, H.L., “Numerical solution of spherical blast waves”, Journal of Applied Physics, American Institute of Physics, Ney York, 1955.

[5] The initial velocities of fragments from bombs, shells, granades BRL Report 405

[6] CONDAT GmbH and century Dynamics “Split-X user manual 1995”

[7] Dobratz, B.M and Crawford, P.C., "LLNL Explosives Handbook"

[8] Nitesh, N., Moon. (2009)" Prediction of Blast Loading and Its Impact on Buildings ", M.T.thesis, National Institute of Technology, Roukema.

[9] Khadid et al. (2007), “Blast loaded stiffened plates” Journal of Engineering and Applied Sciences, Vol. 2(2) pp. 456-461.

[10] Alexander M. Remennikov, (2003) “A review of methods for predicting bomb blast effects on buildings”, Journal of battlefield technology, Vol 6, no 3. pp 155-161.

[11] TM 5-1300(UFC 3-340-02) U.S. Army Corps of Engineers (1990), “Structures to Resist the Effects of Accidental Explosions”, U.S. Army Corps of Engineers, Washington, D.C.

[12] A.K. Pandey et al. (2006) “Non-linear response of reinforced concrete containment structure under blast loading” Nuclear Engineering and design 236. pp.993-1002.

[13] T. Ngo, P. Mendis, A. Gupta & J. Ramsay, (2007)," Blast Loading and Blast Effects on Structures", Int., J. Struc Eng., Australia, pp.76-91.

[14] Clough, Ray.W., and Penzien, J., Dynamics of Structures, Volume 2, McGraw Hill, New York,N.Y., 2003.

[15] Ansys Autodyn Workshop Manual

[16] www.ansys.com

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[17] www.nptel.iitm.ac.in (structural dynamics)

[18] G R Liu, M B Liu, “Smoothened Particle Hydrodynamics”

[19] W.Benz., E.Asphaug “Simulations of Brittle Solids using SPH”

[20] J.Bonet, S.Kulasegaram “Correction and stabilization of SPH method with applications in metal forming simulations”

[21] J K Chen, J E Beraun “A generalized SPH method for nonlinear dynamic problems”

[22] W K Liu,S jun, S Li “Reproducing kernel particle methods for structural dynamics”

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