Sielicki Piotr W Masonry Failure Under Unusual Impulse Loading (1)

Poznan University of TechnologyFaculty of Civil and Environmental Engineering

PhD Dissertation

Masonry Failure under UnusualImpulse Loading

Thesis by Piotr W. Sielicki

Supervisor: Prof. Tomasz odygowski

Pozna 2013

Contents

Preface v

Conversion of the measure units vii

1 Introduction 11.1 Motivation and aim of the work . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Phenomenon of explosion: experimental and numerical approach 31.2.2 Material behavior under dynamic loading . . . . . . . . . . . . . 61.2.3 Theoretical models for masonry . . . . . . . . . . . . . . . . . . . 8

1.3 Main goal formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Material behavior under dynamic loading 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Material under different strain rates . . . . . . . . . . . . . . . . . . . . 152.3 Engineering design of masonry under explosion . . . . . . . . . . . . . . 23

3 Explosion and blast waves action 273.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Explosion and blast wave features . . . . . . . . . . . . . . . . . . . . . 283.3 Thermal effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Basic concepts of thermodynamics for explosion . . . . . . . . . . . . . . 32

3.4.1 Introduction and governing equations . . . . . . . . . . . . . . . 333.4.2 Empirical prediction of explosive loading . . . . . . . . . . . . . . 41

3.5 Numerical modeling of explosion . . . . . . . . . . . . . . . . . . . . . . 533.5.1 ALE and CEL approaches . . . . . . . . . . . . . . . . . . . . . . 583.5.2 Mesh size vs. the results . . . . . . . . . . . . . . . . . . . . . . . 683.5.3 Comparison of numerical results with reality . . . . . . . . . . . 76

3.6 Loading distribution on structure . . . . . . . . . . . . . . . . . . . . . . 773.7 Concept of safety for structural design . . . . . . . . . . . . . . . . . . . 80

iii

iv CONTENTS

4 Modeling of masonry behavior under blast 854.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2 Description of failure for CMU . . . . . . . . . . . . . . . . . . . . . . . 854.3 Numeric validation of FE model behavior . . . . . . . . . . . . . . . . . 90

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.3.2 Mesh refinement study . . . . . . . . . . . . . . . . . . . . . . . . 924.3.3 FE masonry wall: job formulation . . . . . . . . . . . . . . . . . 944.3.4 FE masonry wall: results . . . . . . . . . . . . . . . . . . . . . . 97

4.4 Analyses due to engineer approach . . . . . . . . . . . . . . . . . . . . . 100

5 Reinforcing masonry to resist explosions 1035.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2 Job formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Conclusions 1096.1 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

A Blast and explosive properties 113

B Safety criteria and protection levels 127

Bibliography 131

Preface

Extremely high pace of development force engineers to handle complicated and unusualproblems. Commonly accepted design methodologies very often must undergo severechanges and improvements exceeding widely accepted standards to deliver the reliableresults. The author of this dissertation handles complex problem of explosive loadingalso known as high velocity impulse loading which occurs during an impact. The classof such problems very often refers to the unusual impulse which, in particular, canbe subjected to the masonry structures. These commonly used long structures canbe characterized as composite material composed of two distinct sections: bricks andmortar. Moreover, this material is commonly used to construct supporting elementsuch as walls, vaults and pillars. The wide range of possible dynamic loading acting onmasonry causes the recognition of dynamic material properties, which are importantfor the accurate prediction of failure or fragmentation phenomena. That is why thedynamic material model of masonry, both in discrete and homogeneous approach, is ofa crucial importance to handle complex unusual loading problems. The developmentof pioneering material model not yet present in the literature provides exact physicalexplanation of fast loading dynamic processes. The overall emphasis of this dissertationis to study the problems of the blast wave and impulse loading on a masonry wallstructures ultimately leading to its complete failure.

The dissertation consists of six chapters, preceded by a list of symbols and conver-sion of the measure units, bibliography and two appendices. The detailed aim of thework and actual state of the art on the masonry behavior are described in Chapter 1.Chapter 2 treats the overview of the established failure criteria under dynamic loadingfor brittle materials. The blast wave propagation in the air and its action on anyobstacle structure can be found in Chapter 3. Moreover, this chapter contains theallowable security criteria for structural elements as well as the whole structures.

The investigation on the new failure model for masonry under unusual impulseloading is presented in Chapter 4. The considerations comes partially from the au-thor laboratory results showed in the Chapter 2. Following Chapter 5 deals with thefull scale numerical examples verifying the assumptions formulated in the previous

v

vi PREFACE

chapters. The results show the effects of the explosive forces on the brittle walls.Moreover, Chapter 5 contains the reinforcing and retrofitting methods introduced tomasonry structural elements. The studies show the preliminary tests to the real fieldsexperiment. The dissertation is concluded in Chapter 6 with the final remarks andperspectives. Moreover, there are two appendices in addition to the above presentedchapters which include useful data for designers.

Concluding this preface, author would like to express grateful acknowledgementsto all who have assisted along the long way towards successful finishing on the disser-tation. The author wishes to thank in particular Professor Tomasz odygowski for hisstimulating suggestions, wise guidance, numerous productive discussions and patiencein reading and correcting the manuscript. Without him this work would not reach sucha level of excellence. My deepest words of appreciation to my university colleagues fortheir friendship and will to sophisticated discussions. Very special thanks to my familyfor their patience and understanding.All presented tables, photos and figures come from the own author1 database.

Piotr W. SIELICKI

The support of the Ministry of Science and Higher Education under thegrant O (N506) R00 0097 12 The System of Passive Safety of the Criti-cal Infrastructure (polish title: Bezpieczestwo Infrastruktury Krytycznejpoprzez System Ochrony Pasywnej) is kindly acknowledged.

1 [email protected], www.cad.put.poznan.pl

Conversion of the measure units

Distance and Length1 m = 102 cm = 103 mm = 106 m

1 m = 3.281 foot = 39.37 inch

1 foot = 0.3048 m

1 inch = 0.0254 m

Mass1 kg = 103 gram = 2.2046 lb

1 lb = 0.4535924 kg

Pressure1 Pa = 0.00001 bar = 0.000009869 atm = 1 N/m2 = 0.000001 N/mm2 = 0.0001 lb/in2

1 bar = 100000 Pa

1 atm = 101325 Pa

1 PSI = 6894.757 Pa

1 MPa = 145.04 PSI

vii

viii CONVERSION OF THE MEASURE UNITS

Pressure Impulse (US to SI, and SI to US)1 Pa s = 6.895 PSI ms

1 PSI ms = 0.145 Pa s

Scaled Distance (US to SI, and SI to US)1 m/kg

13 = 0.367 foot/lb

13

1 foot/lb13 = 2.520 m/kg

13

Scaled Impulse (US to SI, and SI to US)1 Pa s/kg 13 = 8.974 PSI ms/lb 13

1 PSI ms/lb 13 = 0.111 Pa s/kg 13

Time1 s = 1000 ms = 1000000 s

Work (Energy)1 Joule = 0.001 kJ = 0.102 KGm = 8.8507 in lb

AbbreviationsALE Arbitrary Lagrangian EulerianCEL Coupled Eulerian LagrangianCFD Computational Fluid DynamicsCMU Concrete Masonry UnitDIF Dynamic Increasing FactorGPA Generalized Particle HydrodynamicsSPH Smooth Particle HydrodynamicsSHPB Split Hopkinson Pressure BarTNT TrinitrotolueneUFC Unified Facility Criteria

Chapter 1

Introduction

1.1 Motivation and aim of the work

Numerical simulations are extensively used for solving vast variety of dynamic problemsassociated with the explosions, such as the gaseous detonations followed by the pres-sure wave propagation. Utilizing these sophisticated methods it is possible to evaluatethe propagation direction of this pressure wave and its affect on different constructionstructures. For many decades researchers conducted and presented numerous real lifeexperiments along with the numerical explosive simulations for different types of ma-terials trying to capture the detailed mechanism of blast phenomenon and to obtaina credible numerical modeling method to predict the failure of the obstacles.

It is obvious that the behavior of structure subjected to the explosion depends onthe type and power of the charge. Varying these two elements can cause fundamen-tally different results. The current Chapter encloses the motivation of undertaking thetopic of this dissertation and presents the actual state of the art in the area of interest.Furthermore, it shows the path of conducted research and subsequently explains wherethe presented dissertation should be sited in sequence of thematic literature.

Current design procedures are gradually improved by various governmental agen-cies considering not only the simulation of the entire structures but also structures withmissing structural element (i.e. column, wall etc.). The application of this approachmay be useful in case of buildings particularly exposed to the possibility of explosionor blast loading. Among many others, objects such as embassies, banks, skyscrapers,hotels, airports and other objects of intense public use have the high level of risk ofexperiencing unexpected terrorist actions.

Nowadays, the fundamental threat is due to terrorist activities and may involvea combination of thermal, impact and explosive loads. The list of this unfortunatelygrowing terrorist activities in this millennium may start with the spectacular event

1

2 CHAPTER 1. INTRODUCTION

from September 11 in 2001. This particular tragedy serves to highlight the vulnera-bility of existing structures toward terrorist activities not only in the United States.The awareness among the politicians and governmental authorities rises making themaware of the consequences of an effective attacks on governmental facilities or otheralike targets. Many of these buildings are historical, with ornate meaning, constructedusing traditional techniques with masonry elevations, like an attacked in 2008 on theBombay hotel, therefore their improvement requires additional attention.

The primary goal of the dissertation is obtaining the reliable material model ofmasonry structure which reflects the behavior of a real brick structure subjected todifferent types of the high stain rate impact loading. In order to obtain the properdescription of the pressure wave propagation traveling in the air and acting on thestructure, it is necessary to consider the explosion and the wave motion as an impactloading problem. There are numerous recent papers which reflect the state of the artin the analysis and simulations of blast as well as the propagation of blast waves inthe air. These materials are closely presented in second part of Chapter 1. In mostof these articles the main attention circles only around plane examples which may betreated as the preliminary studies and can serve as a good introduction to the topic ofhigh velocity dynamic loadings.

The entire process of the blast simulation and its influence on any kind of obstacleis uncoupled into two phases that should be treated separately. The first one is thepropagation of the blast wave in three-dimensional air space with the crucial informa-tion on the pressure distribution on the all surfaces of the obstacle structure whichvaries in time and space. The second step considers the analysis of the stress wavepropagation in the masonry material finally leading to its failure due to fast dynamicloadings.

This dissertation will also focus on some available methods of the strengthening ex-isting masonry structures providing increased resistance to the effects of a blast attack.Furthermore, the retrofitting existing buildings to increase their explosive resistance isadditionally a great challenge for the engineers. For nearly sixty years, many companies(Baker Engineering and Risk Consultants Inc.1, Karagozian and Case2, Cintec Inter-national Ltd.3, any others) manufactured specialized elements providing additionalreinforcement, strengthening and repairing of all types of existing structures aroundthe world. The dissertation gives also some examples of new retrofitting methodswhich, in particular, increase masonry explosive resistance, see Chapter 5.

1 www.bakerrisk.com2 www.kcse.com3 www.cintec.com

1.2. LITERATURE REVIEW 3

1.2 Literature review

1.2.1 Phenomenon of explosion: experimental and numerical ap-proach

The history of research concerning explosions dates back to the 10th century. Thisevent deals with the military purpose of a explosive gunpowder in China. The firstexplosion used for industrial application was recorded about six hundred years later,that is in XVII century, where the explosives were used for rock blasting in Hungaryore mines.

The change of application from military to the industrial of the much strongercharges, such as the nuclear ones, was transferred much faster than in case of chemicalexplosives. Half a century after the first nuclear explosion in Hiroshima and Nagasakiin 1945, there exist about five hundred nuclear power plants in the world. Nowadays,the explosive energy is widely used for many industrial fields, such as rock blasting,sheet-metal forming, fast coupling of composites phases or electricity production. Foreach of these fields the safety of the structures under explosion is a major concerned.This section consists of the papers with crucial meaning for further explosive vulner-ability assessment of structures in the mechanical and blast analysis sense. Explosionterm is commonly used to describe a rapid release of energy established in variousforms. The terms like detonation and resulting wave propagation are strongly corre-lated to the explosion phenomenon.

The first tests of nuclear weapon, as a result of the Manhattan Engineering Dis-trict4, were led by the helm of the project R.J. Oppenheimer. As the power of explo-sion was increasing exponentially by far bypassing all expectations, the governmentsstrongly supported the expansion of this investigation. Arising interest followed by fastdevelopment of the explosion phenomenon continued mainly during the World War II,where the US government intercepted the greatest achievements and scientists workingon the military improvement of the explosions. The mathematical formulation of theexplosions and following wave propagation as well as the estimation of the mechanicaleffects on the surroundings and various structures were strongly investigated. Due tolack of any technical literature, the brightest minds in the field of applied mathemat-ics and physics joined their intellectual abilities to analyze and address the problem.The first papers [133, 134] were elaborated by Taylor and published for Civil DefenceResearch Committee in UK (1941). His work was concerned with the blast wave forma-tion and propagation after explosion. Another valuable studies [126, 142] were carriedout by Von Neumann (1941) and Sedov (1946).

At that time the only available reference point, crucial for further research, was thepioneering paper prepared by Rankine (1870) on the Rankine-Hugonit conditions [114].He first presented the front wave parameters for explosion in the air assuming it to bean ideal gas. Further, the equations for blast wave velocity, air density and maximum

4The full name for the US governments secret project to create a nuclear weapon (1942).


dynamic pressure were presented. Each of these equations are obtained for the case ofatmospheric pressure, sound velocity and peak static overpressure functions.

Taylor observed that the shock is the small discontinuity in the front surface ofthe wave, while the blast wave means the entire part of distributed gas behind theshock wave. Furthermore, Taylor noted that at a certain level of pressure the nuclearblast produce much more heat energy in comparison to the conventional bomb. Thisassumption allowed to formulate analytic shock theory [17] by Brinkley and Kirkwood(1947) and by Sachdev (1972) [120]. In 1950 Taylor obtained the numerical solutionof nonlinear shock theory, transformed from partial differential equations (PDE) ineulerian coordinates notation [134]. Although this approach gained significant sci-entific value, the following researchers like Zeldovich and Raizer [155] proposed thelagrangian formulation for the blast phenomenon which turned out to be as convenientas the eulerian (1967). Following this strong World War II research trend, in his in-dependent studies, Von Neumann obtained an analytic solution of the point explosionin lagrangian coordinates [142] (1941). This approach presented more explicit solutionthen those suggested by Taylor or Sedov. However, some strict parameter simplifica-tions were left without any physical interpretation at that time. In 1969 Laumbachand Probestein found an explicit analytic solution of the point explosion [80]. Tryingto obtain the more realistic behavior of the blast wave, without investigation of an-other model, which considered the attenuation of a spherical blast at the large distancefrom the centre of the explosion [146] (1950). In the same decade, McFadden provedthe considerable advantage for full spherical approach [99]. Furthermore, in 1958 hedeliberately pointed out the differences with respect to the plane model, which waspreviously investigated by Brode (1955) [18]. The continuation of McFadden work wasresearch performed by Friedman [46] which led to the phenomenon of secondary shockafter explosion of high compressed charges (1961). Another author who transferredthe blast analysis to more specialized field was Chisnell who considered channeling themoving pressure waves [29] and derived a relation between shock strength and areaof the channel (1957). Later on, Hayes considered a powerful explosions in the atmo-sphere [56] for a different altitude of detonation (1968).

Nevertheless, there is vast number of possible solutions and models predicting themost important blast wave parameter i.e. the peak of static overpressure. The vari-ation of peak value for a free air explosion is often presented as a function of scaleddistance, static pressure and velocity of sound. The simplified approaches for obtain-ing this peak were presented by Brode (1955) [18], TM-5 1900 technical manual (1969)[140], Henrych (1979) [57], Baker (1983) [6] or Kinney-Graham (1985) [73] and arecloser discussed in the Chapter 3.

Nowadays, scientific research presented in great number of relevant publicationsdeals also with more sophisticated examples of detonation, blast wave propagation orexplosion effect. The research shows the tendency of increasing the virtual analysisutilizing different numerical codes. Nevertheless, the real life experiments of a smaller


complexity are still performed to compare with the numerous approach. In 1994 Cham-pan et al. [27] analysed the effect of using the blast wall. Authors experimented on1:10 scale model of the cantilever beam obstacle located just behind protected struc-ture. They analyzed various scenarios for different size of charges, obstacle locationand geometry. The resulting derivations and obtained relations serve significantly inthe upgrade of safety of the structure elevation. Alia and Souli proved the accuracy ofthe combined Arbitrary Lagrangian eulerian approach [3] by showing their results ofreal field explosion of C4 charge and comparing them with the numerical solution. Atthe same time Schleyer et al. investigated the blast wall panels under shock pressureloading [8]. The authors obtained pressure values for different points on the panel ina shock tube device on 1:4 scale model. Another example performed on the scaledexample is presented by Baylot and Bevins [7]. Authors, in the numerical fashion,simulated behavior of RC frame filled with masonry with use of Autodyn code andcompared obtained results to the real experiment of 1:4 scale model. The advantageof such approach is provided by including ground interaction with a structure. Re-mennikov and Rose analyzed pressure loading on different building faces situated inhighly concentrated urban area [118]. They presented varying values of pressure asa function of building highs. Furthermore, they perform the simulations for differentvalues of a scaled distance where the numerical models includes significantly more then107 number of finite elements.

Another approach connected with the complex analysis of a highly urbanized down-town area was presented by Luccioni et al. [93]. Authors simulated numerically the3D explosive event which occurred in the past in Buenos Aires (1994). They analyzedbuilding resistance against different weight of TNT charges and varied distances fromthe explosion centre. The obtained results based on Autodyn code and compared withthe well known consequences. Furthermore, they performed detailed FE analysis ofpressure propagation based on the empirical formula of Friedlander equation or Kin-ney and Graham tables. The study concerning higher level of details of any structureare presented in Wei et al. [145]. Authors proved that the short time of shearing signif-icantly affects the behavior of laminated glazing under blast. In this case the numericalanalysis were performed with use of LS-DYNA and Abaqus codes. Furthermore, nu-merical simulation of blast wave interaction with structure columns is presented byShi et al. [152]. Once more with the use Autodyn code the numerical experiments ofpressure acting on a structure are performed. The task was connected with the vortexpropagation during action of a blast wave in vicinity to column faces. In this worthmentioning experiment performed for a lower scale model authors successfully verifiedthe numerical approach and obtained a pressure functions for a different column facesas a function of scaled distance.


1.2.2 Material behavior under dynamic loading

The material behavior differs considerably with the variation of the strain rates. Thisprinciple was observed and intensely studied by Hopkinson [60] in (1914). The generalattention of this dissertation is focused on masonry structures under unique dynamicloading. Hence, author finds it crucial to present short references considering thedynamic behavior of brittle materials.

Definition 1.1. Dynamic Loading for the need of this dissertation is going to beconsidered as a blast kind of loading. The ratio of the duration of the load phase tothe characteristic response time is lower than 0.25 for impact (e.g. for blast loads) andequals 106 for shock loads (e.g. for high energy explosion5). Nevertheless, the strainrate is higher than 102 s1 for each loading scheme. Furthermore, it could happenthat is higher than 105 s1 for a special case, when an equation of the state is reallynecessary to characterize material behavior [128].

Hopkinsons work inspired other researchers, such as Naunton and Waring (1938)and further Alexandrov and Lazurkin (1940) who carried out an experimental inves-tigation of the rate sensitivity of copper and lead. They submitted materials to veryhigh stresses which were maintained only for few microseconds. Kolsky was success-fully introduced the measurement method of the stress-strain behavior in strain ratesof the order of 2105 s1 [76]. Furthermore, he considered the theory of relaxation andmemory effects in the materials for different specimen sizes (1949). In 1992 Hanchaket al. [55] carried out triaxial pressure-shear experiments. In addition, he performedballistic perforation measurements for reinforced concrete slabs. The steel projectilehad initial velocity from 150 to 1000 m s1 for different tests. The following year,Holmquist et al. [59] developed a constitutive model where the failure criterion wasbased on cumulative strain measures. This approach is primarily based on the workof Johnson and Cook [69]. Grote et al. used split Hopkinson pressure bar (SHPB)for uniaxial and plate impact tests to measure rate dependent strength of the concreteand mortar. The considered strain rates were between 250 and 1700 s1 with compres-sive stress up to 1300 MPa for the mortar. Following the experiments the numericalanalysis performed with the extended Drucker-Prager theory showed good agreementof both approaches. In 2008 Clayton described in details a fragmentation phenomenonobserved in high-speed photographs produced during impact crushing [31]. He em-ployed a generalized particle algorithm (GPA) to perform numerical observations. Dueto possibilities of choosing a smoothing functions the GPA is more efficient then popu-lar smooth particle hydrodynamics (SPH). Furthermore, the GPA enables to associatethe particle velocities and trajectories directly to the free fragments. Nowadays, thelaboratory experiments allow to investigate the behavior of materials under the strainrate up to 5106 s1 [32]. Gilbert et al. [49] studied a response of a masonry bar-riers to low velocity impact. They obtained numerically and experimentally critical

5Explosions are measured in EMT that is equivalent megatons.


displacement values for different schemes of destruction mechanism. For wide rangeof analyzes there are considered vertical cracks and non-fracture sliding mechanism aswell.

Among other dynamic loading topics many researchers have taken into considera-tion also masonry elements. The vast majority utilizes the SHPB technique e.g. Shihet al. (2000) [127], Serva and Nemat-Nasser (2001) [122] or plate impacts e.g. Grady(1997) [50] or Bourne et al. (1998) [14]. It is generally considered that the failure modesin ceramic materials highly depend not only on material type but also on strain-rateregions. The failure schemes are significantly influenced by the occurring microscopicdefects. There are also other methods used by Ravivhandran and Subhash (1995)[115] and by Bhattacharya et al. (1998) [13]. These authors developed two materialmodels based on experimental observations. The micro-crack-nucleation-growth andenergy diffusion models mathematically describe the failure process in ceramics. In theRavichandran model, the source of damage represented by micro-cracks in a confinedcell is investigated. The failure of the material is defined when the damage densityreaches limit value. The energy model of Bhattacharya et al. describes the reductionof elastic energy from the initial state to the split state, where it must overcome surfaceenergy associated with new surface. Furthermore, the observations of dependency ofthe failure strength and loading pressure are explained.

Due to the fact that the description of the ceramics behavior under high speedimpacts is to complex for theoretical formulation [158], many authors have developedtype of stochastic formulations from the numerical point of view. These considerationsare base on the intrinsic defects in the initial state of brittle material. The model ofCamacho and Ortiz (1996) [24] describes this phenomenon where the failure process isdescribed by the growth of micro cracks, which are simulated by cohesive elements [2].The cohesive zone follows the concept of cohesive zone introduced by Dugdale (1960)[41]. This crack formulation is adopted in the finite element methods in a concept ofa cohesive finite element. The similar approaches have been presented by many otherauthors [43, 100, 154].

Above mentioned authors have presented two dimensional problem. Three dimen-sional numerical simulation with employed fracture was validated by Zhou and Molinari(2001) [157]. Another group of researchers have considered unique dynamic loading i.e.explosive loading and penetration directly for masonry. One of this work is performedby Wong and Karamanoglu (1999) [149], where the authors obtained numerical solu-tion for masonry subjected to explosion of gas mixture. The presented results were ina good agreement with the full scale experiment. Baylot and Bavins (2007) [9] obtainedthe result of a structural response of non full scale building under explosion of con-densed charge. Their model represents reinforced concrete frame filled with masonrywalls. Furthermore, they suggested a retrofit techniques. Davidson et al. (2005) [37]and Buchan and Chan (2007) [20] analysed a polymer fibres as an reinforcement of thepure masonry under explosion. They performed few real scale tests and confirmed the


experiments through numerical solutions in LS-Dyna code. Su et al. (2008) [131] per-formed some numerical calculations for the mitigation of blast effect on unreinforcedmasonry wall. In 2010 Cullis et al. [36] described an influence of explosion on differ-ent infrastructure. In particular during the last decade the development of buildingand structure vulnerability under blast loading occurred extensively. The masonrymaterial is one of these materials which, near concrete and steel, has been taken intoconsideration in design codes [140].

1.2.3 Theoretical models for masonry

A masonry is a complex material composed from evenly aligned bricks and mortar. Inmost general approach there exist two different ways to analyze the masonry. The firstone is a discrete method, where each of materials is taken into account individuallywith actual physical and geometrical properties. In this method additional attentionmust be payed to describe the contact between phases. This approach is often verifiedin laboratory tests for static and dynamic loading. However, due to very expensivecomputational requirements even for relatively small specimens [113] it has a seriouslimitations in computational research. The second approach considers the structure ascontinuous i.e. the heterogeneous brick structure is replaced by homogenized masonry.

Definition 1.2. Homogenization is the process of uniforming the global materialproperties by considering representative volume inhomogeneous elements (RVE) con-sisting of bricks and mortar. Utilizing homogenization techniques it becomes possibleto uniform the properties of the RVE. From the global point of view material can betreated as homogeneous and is described by homogenized material parameters. Thisprocess allows the global behavior of composite to be derived from separate propertiesof different components.

Several methods are used to derive a homogeneous masonry material properties.Pande et al. [110] in their work derived the equivalent elastic modulus (1989). Theyshowed expression for obtaining elastic properties of equivalent material and verifiedit with a brick panel subjected to uniaxial loading.

Pietruszczak and Niu [113] adopted two steps homogenization procedure, which isbased on the Mori-Tanaka method (material obtained by neglecting the presence ofthe horizontal beds of mortar), and the lamination theory. They fully described thehomogenized orthotropic elastic material (1991), provided the three dimensional for-mulation to average macroscopic properties of masonry and investigate the conditionsof failure. The above mentioned studies are extended and supported by extensive nu-merical study. In 1995 Anthoine [4, 5] proposed a global elastic coefficient for masonrybased on elementary cell and with the use of numerical methods. He has taken intoaccount a finite thickness of the masonry and subjected both brick and mortar to theisotropic damage. He proved that the approach is good for the elastic behavior ofmasonry but may be significantly affected by the mode of failure.


Lourenco [90] proposed another direct homogenization method (1996). His ap-proach allowed to obtain the average constitutive relation for periodically layeredcomposite by proceeding the proposed process for each of three general directionsof masonry. Luciano and Sacco [94] introduce the brittle damage model for old brickstructure (1997). The theory of homogenized material with periodic microstructureis used to derive the moduli of uncracked and cracked masonry. They also developeda numerical procedure to obtain elastic properties for two dimensional undamaged anddamaged brick structure as well as energy and local strength criteria for mortar.

In 2001 Ma et al. [95] obtained the elastic moduli and damage behavior throughfinite elements (FE) analyses. The average properties for a composite structure arederived numerically. The representative volume element (RVE) is modeled under dif-ferent stress conditions. Author applied a varied displacement boundary conditionson RVE faces. The model allows to obtain three different types of failure. The firstone is connected with mortar tension. The second one is based on the shear failure ofmortar or combined shear failure of brick and mortar. The last one arises from simplecompressive failure of bricks. The model found its biggest utilization in analyzing largescale masonry.

A non periodic masonry is homogenized by Cluni and Gusella [33] (2004). Theycame up with the medium stiffness tensor for masonry. The authors also replaced theconcept of periodic cell with the RVE. This volume is found by employing a finite sizetest-windows technique. The tensor of homogenized stiffness is derived by consideringthe hierarchy of estimates relative to essential and natural boundary conditions. A nu-merical application elucidates the effectiveness of the approach.

Another homogenization method is introduced in 2004 by Zuccini and Lourenco[160]. The cracking process is responsible for nonlinear behavior of masonry, due tothe low tensile strength. Authors proposed a composite fracture energy approach whichis in a good agreement with the discrete FE approach. Their study accomplished thecoupling of the elastic micro-mechanical model with a scalar damage model for jointsand units leading to the damage coefficients computations.

Brasile et al. (2007) introduced a multilevel, quasi-homogeneous approach for ma-sonry [44, 16]. Authors described a nonlinear behavior of masonry including damageevolution and friction toughness phenomenon. They performed two dimensional nu-merical tests for both simple walls and large scale structure. They assembled masonryof rigid bricks linked to one another by mortar interface. The authors represent thewall as a discrete Lagrange system in which fractures and all constitutive aspects lo-calize in the interface phase only.

Massart et al. developed another computational homogenization technique [98]for structural masonry (2007). This approach is more complex because it uses theanisotropy evolution and localization induced by mesostructural damage. The tech-nique includes a damage orientations which are strongly connected to periodic structureof the material. The results discussion is based on the computational example which


shows the cracking evolution and anisotropic damage. The FE experiment is performedon a two dimensional masonry wall under vertical static loading.

There also exists another group of homogenization methods which are able to de-scribe the structures behavior under dynamic loading. Wu and Hao [151] used a ho-mogenization approach for three dimensional model of masonry (2006). They includedorthotropic elastic properties and obtained a strength envelope. Authors character-ized damage parameter as scalar. The behavior of composite is described by tensile,shear and compressive failure, and additionally by high pressure failure. Finally, theysubjected two different kinds of walls (continuum and homogenized) to the blast. Thethree dimensional experiment is performed numerically with use of LS-Dyna code. Weiand Hao [144] investigated a behavior of masonry panels under strain rates up to200 s1 (2008). Authors used a modified Drucker-Prager strength criterion and per-formed a few numerical tests for orthotropic RVE. The obtained results were foundto be in a good agreement with the discrete approach. Additionally, new concept offailure bond values for stresses and strains for RVE of masonry was presented based onthe dynamically increasing factor (DIF) which is obtained in laboratory experiments.

1.3 Main goal formulation

Motivation to undertake this study is significant usage of masonry structures. Theprevalence of masonry in high occupancy structures [54] is proved in Figure 1.1. Author

Figure 1.1: Construction summary shows the prevalence of masonry in military buildings

formulates the fundamental concept of this dissertation, which is confirmed in the nextsections. It reads as follows:

1.3. MAIN GOAL FORMULATION 11

There are reinforcing or retrofitting methods that allow to greatly in-crease the strength of masonry exposed to blast waves. These approachesapply to newly constructed and existing facilities in both.

The main interest of this research circles around the word: failure. The abovestatement during high speed dynamic loading, where the quality of the results (i.e.failure initiation and propagation) depends directly on the number of degree of freedom.Due to above assumption which should be reached, author achieves the secondary goals.There are in particular:

engineering simplification for prediction of blast pressure features

obtaining safety threshold for non-reinforcement CMU structure under explosion

increasing of CMU safety trough introducing some reinforcement method

obtaining the critical FE size for valuable results

introducing material behavior in strain rate effect functions into subroutine pro-cedure

introduction to numerical design of explosion phenomenon

coupling of complex engineering problem in one FE numerical example

Chapter 2

Material behavior under dynamicloading

2.1 Introduction

Structural materials under specific rate of loading exhibit hardening or softening whichusually forerun the failure and fracture phenomena. This process may be very rapidfor both quasi-static as well as dynamic loadings. Considering such material behaviorone has to define mathematical formulation followed by the computational solutionof various applications with extraordinary care. Particularly important is the correctdefinition of the solved problem; any discrepancies from the definition of well posedproblem may result in serious consequences especially in the system of the governingequations. As an example one may refer to the static cases usually governed by ellipticequations. It is well known that with such approach softening results in not positivelydefinite constitutive matrices and changes the type of equations into hyperbolic ones.Such changes have further severe impact on the exactness of the numerical solutionhence it is crucial to address them at the earliest stage of theoretical definition. Lackof mathematical knowledge on correct problem definition through the initial boundaryvalue problem leads to therefore called pathological mesh dependency. It simply meansthat the obtained results are meaningless and by no means should not be presentednor discussed.

There are several ways to avoid the change of equation type during the compu-tations. In general such procedure is called regularization process. The motivationbehind the regularization process may have various backgrounds. One of them is phys-ical, e.g. viscoplasticity or higher order media for soil mechanics. Another one comesfrom artificial manipulations, also on the level of numerical discretization, that simplykeeps the type of governing equation unchanged during the entire incremental pro-

13

14 CHAPTER 2. MATERIAL BEHAVIOR UNDER DYNAMIC LOADING

Table 2.1: Speed of deformation for specific physical phenomena [84]

Type of phenomenon Velocity of deformation [s1]Creep from 1010 to 105

Creep beyond the yield criteria from 105 to 101

Hot drawing from 101 to 101

High speed drawing from 101 to 103

Machining from 103 to 105

Drawing with use of explosion > 105

Type of loading Velocity of deformation [s1]Earthquake from 103 to 101

Car crash from 102 to 100

Plane crash from 5102 to 2100Hard hit from 100 to 5101Projectile hitting from 102 to 106

Explosive loading from 106 to ...Type of deformation Velocity of deformation [s1]Geological movements 1010Creeping 106One axial tension test 104Drilling, rolling, drawing 100Test with Hopkinsons bar 103High velocity impacts 106

cess. For static problems the system remains elliptic whereas for dynamic ones itshyperbolic. Discussion around the type of loading, quasi-static or dynamic, should bebalanced based on own experience compared with particular rates of deformation fortypical processes, see Table 2.1. [84].

The scope of this work focuses on fast dynamic processes such as explosions orblasts which generate the structural rate of deformations of an order between 103 upto 106 s1. Furthermore, especially for the static cases, the computations require anykind of imperfections, such as missing element or slightly changed plastic strain local-ization, in order to enforce the fracture initiation. With such a setup the first signs offracture occur in those places hence using the specific failure criterion is of fundamentalimportance. In this part the author decided to concentrate the attention on the de-scription and analysis of particular types of loadings. The subsequent consideration onthe analysis accuracy of the brittle-like structures subjected to an impact or explosiveloading, employ the knowledge of crack initiation and its evolution. Furthermore, lackof experimental verification rises the complexity of considered problems. Difficultieswith performing and obtaining some experimental results are due to several reasons.The most common issues are economical and practical i.e. the physical impossibility tomeasure some desired values. In such circumstances the support of numerical analysisbecomes crucial. This Chapter contains separate criteria for failure of the material

2.2. MATERIAL UNDER DIFFERENT STRAIN RATES 15

Table 2.2: Load classification [140]

Load classification T Type of the loadQuasi-static >4 Conventional testingQuasi-static 1 Transient loading on structuresImpact


Generally, the author considers solid bricks with dimensions of 0.25 m by 0.125 mby 0.06 m, typical hollow ceramic brick 0.37 m by 0.25 m by 0.24 m, and silicateunit 0.33 m by 0.24 m by 0.2 m. Figure 2.1 shows different failure scenarios of brickscaused by variously applied compression force. The presented cases show where thecompressible boundary were loaded with various velocities. In initial setup the loadingvelocity equals to 0.001 m per second, whereas the final one equals 0.01 m per second.The average loading strain rates were 0.02 and 0.17 s1, respectively what is an ex-ample of quasi-static study. Nevertheless, increase of the value of the critical strengthin comparison to the static loading was observed. This behavior is even more evidentwhen the strain rate becomes higher and equals to 100 or 1000 s1. High strain ratesrequire particular experimental approach such as split Hopkinson pressure bar tech-nique (SPHB) [28, 51, 60, 76, 92, 119]. In general, this system consists of two differentbars. The first one is the incident and the second one the transmitter bar, where thematerial specimen sandwiched between them. This is widely used to characterise thedynamic behavior under rapid compression of brittle materials.

Furthermore, the author performed some residual dynamic test for solid, hollowand silicate specimen using SPHB technique. In general ca. 20 cylindrical specimenswere tested, in order to perform further dynamic analysis of whole masonry wall. Theshapes of these elements were obtained using jet machining system. The diameter wasabout 12 mm, while the thickness was in range 8 to 13 mm. The test were performedfor solid and silicate materials in general. The outcomes allow for prediction of DIFsunder constant rate of the strain fixed to ca. 600 s1, see Figure 2.6. This constantboundary value is able to found using small mass projectile in SPHB system, becauseof the fact that all specimens are brittle. The laboratory place2 and the specimensare presented in Figure 2.2. Author presents an exampled results data for solid brickspecimen during the dynamic test, see Figure 2.3. This graph shows two curves i.e.the rate of the strain (black) and true stress (grey) in true strain function in both.This is a lucid study, which shows the strong dependency between the results andspecimen quality. The vertical arrow shows the region necessary for alignment of bothextreme specimen surfaces. This inaccuracy is caused by incorrect machining the ini-tial material. Nevertheless, the result are clear, and proves the dynamic increase effect.

Additionally, there is used the high speed camera to catch the fracture steps duringthe process. The used frame rate was about 20k (20 000) frames per second, and theresults are showed in Figure 2.5. Nevertheless, partial of experimental results utilizedin this work are taken mainly from [51, 102, 91, 92] and further used by the authorto approximate the behavior of masonry. The framework of this approximation isexplained and discussed in Chapter 4 in detail.

The average stress evolution during quasi-static compression made by the authoris presented in Figure 2.4. They represents the dynamic increasing factors in time

2Institute of the Structural Engineering.


Figure 2.2: SPHB laboratory and the tested specimens

Figure 2.3: Finall output data: SPHB test for solid brick unit, = 600 s1


Figure 2.4: Stress evolution in time function for solid (dashed), hollow (dark grey), andsilicate (light grey) bricks during quasi-static compression; d is tensor of strain rates

Figure 2.5: Fracture mode of solid and silicate brick under rapid compressive loading

function, for three types of brick unit. These curves rise rapidly reaching the limitvalues. After the failure curves stopped rising and decreased simultaneously. Completeresults data are presented in legend of Figure 2.4. It is well established that equivalentvalue of the strain rate d comes directly from tensorial measure d, and it means thescalar measure. There is equivalent measure to which denotes for the small strains.Furthermore, d can be reduced to the ratio of boundary velocity and unit dimension.The peak value of the force divided by the initial area of masonry unit shows thecritical stress. Furthermore, the failure mode corresponds to the critical strain. Finally,the maximum impulse value is obtained as an area beneath the curves, before failureoccurs. Furthermore, the main results of critical stresses were generated based ondifferent authors and information from Figure 2.6. The general conclusion is that thedynamic increasing factors (DIFs) due to different values of the loading velocities are


Figure 2.6: Critical stresses from experiments for different state, material and strain rate

significantly different for tension and compression.Based on the study conducted in the area of concrete [75, 77] the same comparable

values of dynamic increasing factor exist for tension and compression under strain rateof = 102 s1 and = 103 s1, respectively. Such big discrepancies clearly imply thatthe description of material must be combined from two independent states, i.e. thetension and compression.

Let us focus on the strain rate of the process. The simplest way to consider dynamicsis recommended by standard approaches. In this case, however, some simplificationsare used by introducing the safety factors. In order to provide safety these factors mustbe set to, or be greater than one. For example the Unified Facilities Criteria (UFC)standards for concrete [140, 139] distinguish the different loading states, see Figure 2.6,and additionally the type of the stresses, see Table 2.3. Furthermore, the first columnrepresents the dynamic- to the static yield ratios, while the second one the ratios ofthe dynamic- to the static ultimate strength of reinforcing bars in both. In the thirdone columns, as for far design- as for close-in ranges, the bold values show the ratio ofthe dynamic- to the static ultimate strength of concrete.

Furthermore, the UFC safety factors are designed particularly for explosive loading.In that sense the stand-off distance is also taken into consideration. This featureincreases the standard of designing in agreement with UFC rules [140]. Furthermore,these standards provide the outline for basic properties such as global compressivestrength value for concrete masonry units (CMU) ranging from 9.3 to 12.4 MPa, andthe modulus of elasticity equals to Em = 1000 fm. Any kind of increasing factors are


Table 2.3: Dynamic increase factors for design of reinforced concrete [139],[140],[141]

Type of Stress Far Design Range Close-In Design RangeBending 1.17 / 1.05 1.19 1.23 / 1.05 1.25Diagonal Tension 1.00 / - 1.00 1.10 / 1.00 1.00Direct Shear 1.10 / 1.00 1.10 1.10 / 1.00 1.10Bond 1.17 / 1.05 1.00 1.23 / 1.05 1.00Compression 1.10 / - 1.12 1.13 / - 1.16

not taken into account particularly for bricks.Similar approach is presented by Comite Euro-International du Beton (CEB). It

recommends independent utilization of the safety factors, TDIF for tension and CDIFfor compression. All of the presented approaches are gathered and presented in Fig-ure 2.6. The dashed curves represent the CEB approach. It is important to mentionthat current research [65, 75] show that the CEB values, especially in tension, are un-derestimated for strain rates above 100 s1. On the other hand, value = 1000 s1

is assumed as maximal and all potentially greater values are matched to it preventingoverestimation of the results.

The masonry structures under blast action are not verified despite frequent utiliza-tion of this material in constructing walls, see Figure 2.7. In other civil engineeringstructures masonry may be used in reinforced concrete frames or frontal elevations.During unexpected incidents like explosions, the masonry fragmentation influencesdirectly the safety of personnel inside the building. Due to lack of information re-garding masonry under explosive loading in the technical literature, author assumedthis material as a two-phases composite. The first phase i.e. mortar behaves similaras a concrete, and the second one i.e. brick unit, is described with the similar equa-tions hence, different material properties. The crucial problem of realistic and reliabledescription of the behavior of masonry under blast phenomenon requires extensiveresearch and sophisticated numerical simulations. Chapter 4 addresses this issue byproviding innovatory solution to the stated problem.

Among large variety of equations describing various material behaviors for quasi-static loading there are also those describing brittle behavior. One of them is theformula proposed by Geers [98], as presented in Equation 2.1, which is directly derivedfrom Burzyski (1928) approach [23].

eqv =k 1

2k(1 2)I1 +1

2k

(k 11 2 I1

)2+

6k

(1 )2 J2 (2.1)

Equation 2.1 introduces two invariants: I1 and J2 which represent the first and the sec-ond invariants of stress and deviatoric stress tensors, respectively. Let us assume, thatAij is the second-order stress tensor. A unique property of the second-order tensors isthe possibility to uncouple it into two parts. The first part is the isotropic component


Figure 2.7: Masonry usage in typical construction structures

AOik, and the second one is the stress deviator ADij , as presented in Equation 2.2.

AOik =1

3Appij , and ADij = Aij

1

3Appij , (2.2)

where, App is the sum of all diagonal terms of tensor Aij . Here, I1 = Aii, andJ2 = det(ADij). The formula presented by Geers in Equation 2.1, was initially used toobtain the strength properties for concrete. In this case, however, parameter k is of thecrucial importance, which is responsible for shape of a failure surface in the principalstresses space. Completing the description of Equation 2.1 represents the Poissonratio.

Based on provided discussion, the author presents the evolution of parameter k fordifferent data, in aspect of the strain rate function. In order to perform analysis ofmasonry for blast and explosion this kind of study must be performed independentlyfor each phase of the composite i.e. mortar and unit. The detailed description of suchresearch and obtained results are presented in Chapter 4.

Let us move our consideration into the mortar which is equivalent to concrete. Theexperimental studies provide data of critical values for uniaxial stresses which fromobvious reasons are different for tension and compression. The graphical representationof the above formula is presented by Jankowiak [65] for concrete under quasi-staticloading.


Figure 2.8: Impulse vs. loading under dynamic process

Considering the strength of materials in aspect of rapid dynamics, very importantfactor is the time of loading action. It is therefore called impulse. The loading pressuresI,II,III, may be significantly different, however, on the other hand the impulse givesthe same values. This situation is presented in Figure 2.8. This approach allowsutilization of different limits for the strength values under the same loading but fordifferent loading time action. The strength of material increases while the load durationdecreases. It is directly linked to the strain rate. The higher the d the higher theultimate strength, which could exists in the material point. Hence in order to maximizethe material resistance the load duration in this point should be decreased. Withrespect to Figure 2.8, the area under the pressure-impulse (P-I) curve, where pressuremeans exactly the force, represent the safe region. For selected combination of pressureand impulse values, in some material points, the pressure value is above the limitingcurve and hence the critical strength is reached.

While introducing the impulse into Burzynski equations, it is assumed that thestructures can carry any kind of loading. The crucial question that needs to beanswered is about the duration of occurrence of such a loading. In that sense thedynamic strength of structural elements must be considered at least in aspect of forceand impulse.

Similar approach was realized through cumulative criterion which is utilized forconcrete and introduced by Campbell [25], Klepaczko [75], and numerically adopted in[65] by Jankowiak.

This criterion exists also in the US codes, which deal with the blast resistance ofglass, doors, windows [109], as well as other references. Besides, the pressure-impulserelation is introduced to improve the public safety, see Figure 3.33 in addition.

2.3. ENGINEERING DESIGN OF MASONRY UNDER EXPLOSION 23

Figure 2.9: Behavior of non-reinforced masonry exposed to blast action

2.3 Engineering design of masonry under explosion

During last years masonry gained a lot of popularity particularly as a building mate-rial for walls in civil engineering applications. Base on the UFC [140] the masonry isvery unique material with fireproof properties, good acoustical and thermal insulation,structural mass and resistance to flying debris. Due to circumstances of its utilizationit may also be exposed to exterior and interior blasts. Nevertheless, it is shown by au-thor, that the approach to the description of the masonry behavior under explosion asa complete structure is not taken into consideration in the standard manner. Further-more, properly designed masonry walls can provide economical resistance to relativelylow blast over pressures. Meaning of the term: low blast can be extended, whenmasonry is properly designed. In order to increase the blast resistance of masonry thereduction of tensile cracking along with the introduction e.g. the steel reinforcementmust be provided.

Let us concentrate on non-reinforced masonry wall. The UFC presents the studyon the singular example of brick wall under lateral dynamic pressure loading. Thisresistance is a function of three features as follows: wall deflection x, the compressivestrength of the mortar fm, and the equivalent of the supports stiffness h = h

h.These parameters are presented in Figure 2.9.

For the symmetrically supported wall, the blast action causes the cracks in thecentre of the wall. As a result two parts of the wall move as two separate yet rigidbodies. Due to explosive action the m point of the wall, where the crack appearsundergoes lateral motion x. This motion occurs while the point o moves towards the


upper support. This condition corresponds to the deflection value denoted by xc whichis derived from the geometry of the wall, according to Equation 2.3, as follows:

(t xc)2 = L2 (

h

2+

h h2

)2(2.3)

Furthermore, the value L could be derived from the initial state for the unloaded wall.It is worth mentioning that there exists also the maximum deflection value xmax, whencompressive stress f m occurs in points m and o. This value can be computed, seeEquation 2.5, as a function of fm corresponding to the strain m. These parametersare shown together in Figure 2.9. Due to presented approach the resistance of thewall ru to the lateral force, firstly introduced in Equation 2.4, appears as a functionof displacement x. First, the resistance is a linear function, starting from the xc. Themaximum resistance occurs when x reaches xmax, and the above mentioned value ofthe resistance function decreases according to Equation 2.8.

ru =8 Mu

h2(2.4)

xmax xct xc =

f mfm

=f m

E m , (2.5)

where E stands for the elasticity modulus of the mortar. Additionally, the unit strainin the wall caused by the shortening will be described by Equation 2.6, while theinterpretation of the Mu presents Equation 2.7.

m =L h/2

L(2.6)

Mu = 0.25 f m (t x) (2.7)

ru =2

h2f m (t x)2 (2.8)

The presented solution comes directly from the UFC [140], where more detaileddescription can be found. In agreement with the presented equations the resistancefunction for one type of solid unit was calculated. For practical design the blast re-sistance of the masonry could be also determined from the [63]. Where, the ultimateresistance is the function of the wall deflection, compressive strength fd, and the sup-ports stiffness of the masonry. Author tries to show the sensitivity of engineeringconsiderations in function of input parameters. Due to this assumption the studies areconducted for different values of four separate features, which are: hight, thickness ofthe wall, support stiffness, and the compressive strength of the mortar. The resultsare obtained using developed Matlab procedure, and are presented in Figure 2.10.

Initial resistance for typical properties of brick unit wall is represented by the blackand continuous curve. In this case, this curve represents the one-layer solid unit wall.

2.3. ENGINEERING DESIGN OF MASONRY UNDER EXPLOSION 25

Figure 2.10: Elastic range of resistance function of masonry wall for different properties

It is 2 meters high and 0.125 meters thick. The support stiffness is fixed at h andequals to 0.01 m. The fm parameter is fixed to be 10 MPa. All other curves plottedin Figure 2.10 represent the sensitivity of the resistance function and changes of allinitial data. These changes are multiplied by fixed factors of 0.9 and 1.1 of initialparameters. Thickness of the wall is the most sensitive parameter. During the increaseof thickness by 10% the wall resistance increases more than 100%. On the other hand,slight changes of the wall resistance cause the increase of mortar strength.

Main damage criteria for any kind of brick walls, which are presented in UFC,concern the angle of the support rotation. This support rotation critical angle is in therange from 0.5 to 1.0 degree [141]. Author presented obtained results using the rigidboundaries and the non-reinforced wall. This approach allowed for the amount decreaseof parameters during the preliminary computations. However, the UFC approach isfully verified with the use of FEM in the Chapter 4.

Despite that one has to take into account the different boundaries, the resistant isthe function on the stiffness of the supports.

Furthermore, simply supported wall, without the top boundary against verticalmotion must be checked as a simply supported beam, and the maximum moment isdetermined via the rupture value function of the mortar. The above considerations arecrucial for the further description of the masonry behavior, particularly while employ-ing FEM approach.

Chapter 3

Explosion and blast waves action onstructures

3.1 Introduction

In order to provide a credibly performing numerical model addressing structures fail-ure, it is necessary to properly define the loading conditions. The research presentedin this work is mainly focused on the topic of external pressure generated by the blast.The topics such as perforation of the structure which results from the internal ex-plosion are not included in details. Nevertheless, in order to understand the furtherresearch on masonry failure and damage evolution imposed by stress wave action insidethe brick structure it is particularly required to introduce to explosion phenomenonand blast wave propagation in the surrounding air. The intense study of these effectsallowed author to obtain the alternative loading schemes, which come from the blastand are further used in Chapter 4, as a primary objective of the dissertation. Theconclusions regarding the structure loading under stress wave are presented in thisChapter. Furthermore, this study is extended to consider the energy and detonatedproduct which are severely affecting the surrounding air. The general aspects of ex-plosions are discussed, however, the theory of ignition and detonation phases are notaddressed in detail. The gas dynamics, chemistry of gaseous detonations and analy-sis of reaction mechanisms including comprehensive set of reaction rates attemptingto represent all chemical processes within a given system are described in details in[30, 34, 123, 124, 125].

27

28 CHAPTER 3. EXPLOSION AND BLAST WAVES ACTION

3.2 Explosion and blast wave features

In order to understand further description regarding the structures safety under theblast wave action, there is necessity for description and definition of basic and relevantconcepts.

Definition 3.1. Explosion describes a rapid phenomenon of physical, chemical ornuclear conservation where the change of potential energy to mechanical and thermalwork is inseparable. This work is carried out by expanded gases, which before were incompressed state, or were formed during the phenomenon [121].

Expanding the provided definition, the explosion concept can be described as thesudden release of large amounts of energy within a limited space during the detonationprocess of the charge. In this case the detonation means a type of reaction generated asa result of explosion, i.e. a high intensity shock wave. During such process the gener-ated wave begins to travel with the initial velocity of 103104 meters per second. Thegood example of physical conversion is a fast disruption of tank infield with gas. Explo-sions are also accompanied by chemical phenomenon. Their duration is calculated inmicroseconds and the usual cause is the exothermic reaction in charges, which initiallywere in solid or liquid state, or were already initiated in the gas mixture. The lastconcept given in the definition, i.e. nuclear conservation concerns the high influenceof thermal and nuclear radiation. The thermal radiation involves the wide range ofthe electromagnetic spectrum including infrared, visible and ultraviolet light. The nu-clear radiation can be described with the initial ionizing and further residual radiationconsisting mainly of neutrons and gamma rays emitted within the first minute afterdetonation. It is noteworthy, that the power of nuclear explosion is a few order higherthan the conventional condensed explosion. This reaction is measured in nanosecondswhich generates huge amount of energy in addition. This view is permanent since 1938,where the first nuclear fission was performed by Hahn and Strassmann [52]. The shortdraft shows the instruction scheme for all explosive processes:

Explosive + Heat source = Energy + Detonation products

In most of the cases the heat source is delivered typically to condense or liquid explo-sives and is starting the process of transmutation or conversion. The right hand siteof the above draft is the product of detonation which is important for the obstacleloading. The author neglects any kind of nuclear loading in the further considera-tion and focuses only on the conventional phenomenon. In the conventional chargeterm, meaning thermodynamically metastable configuration there are exist rapid andself-sustaining processes. There are initiated by heat, electrical or mechanical influ-ences.The high-condensed gases or vapours are connected directly with the processwhich are capable of performing a mechanical work. They are so-called detonationproducts. These products are highly compressed at the initial step. Right after thedetonation there is rapid change of pressure, e.g. 10 GPa, on the medium boundaries

3.2. EXPLOSION AND BLAST WAVE FEATURES 29

(charge-surrounding). The detonation products are called high pressure carriers baseon Wodarczyk [147].

The important property for each explosive is its exothermic. This feature is re-sponsible for the self-sustainability independently on external factors, like e.g. at-mospheric conditions. The general parameter connected with exothermic property isspecific heat or capacity. This term means precisely the amount of energy formed dur-ing explosion of 1kg of any explosive material. This term rapidly deals with the timeof conversion, and there are fundamentals for quality of the explosion. The specificheat for the conventional condensed charges equals up to 10 MJ per kilogram. Theexact values for the different charges are obviously well known, and are presented byvarious authors in tabular forms, e.g. see [57, 58, 147]. Furthermore, author collectedthis data in Appendix A, see Tables A.4 and A.5.

Due to the fact that the primary aim of this work deals with the structural en-gineering response the considerations are reduced only to one explosive charge, i.e.the well known pure TNT (trinitrotoluene) compound. The properties of any othercompounds are recalculate if necessary, based on the TNT equivalence. This termis usually stated in the units of kilotons or even megatons, particularly for nuclearexplosions. The equivalent base is that the explosion of one ton of TNT is assumed torelease 4.5 MJ of energy. For any other material the properties are translated includ-ing specific heat ratios. The energy released during the explosion is expressed in termof the mass of trinitrotoluene, which would release the same amount of energy whenexploded.

There are couple of general processes accompanying the first milliseconds of explo-sion. The first one is visual fireball; the expansion rapidly compresses the surroundingair and as a result produces a powerful blast waves. The radius of the bright fireball isof a crucial importance during the field measurements and must be taken into accountto neglect the overexposure of the visual recording. The second process is the rapidtemperature changes which is transmitted on the front of the pressure wave. On theother hand the density of the front of the shock wave is also important. It can reachthe values exceeding 2500 kg/m3 having the properties similar to a solid body [147].However, for the most examples the density of the air increases to be 8 times higherthen initially.

Definition 3.2. Shock Wave is a continuously propagated pressure wave in the sur-rounding medium which may be air, water or earth and it can be initiated by theexpansion of hot gases produced by e.g. a nuclear explosion [121].

The most important shock wave parameters were published in 1870 by Rankineand Hugoniot [114]. For loading purpose of any structural obstacle very important ispressure evolution in space and time. The typical scheme for some point in the freespace is presented in Figure 3.1. This point is separated from the charge center witha particular distance called stand off distance. Furthermore, the blast pressure


Figure 3.1: Typical pressure-time history of an airblast in free point of air space

consists of two significantly different phases called: positive and negative. When theprocess is initiated, following the explosion at the time of arrival tA , the pressuresuddenly increases to a peak value PSO (or pSO) which exceeds the ambient pressureequals to P0. Hence, the pressure decays to P0 in time t0, and again reaches PSOpressure in order to finally reach again the barometric value, at time t0 . The sum oftimes of over and under pressures is called the time of duration T. The value of PSO isusually referred to as the peak side on overpressure or incident peak overpressureand remains in accord with the established rules [140, 141].

Definition 3.3. Overpressure is a transient pressure, usually expressed in Pascals,exceeding the ambient pressure, manifested in the shock wave from an explosion. Thepeak overpressure is the maximum value of the overpressure at a given location and isgenerally experienced at the instant the shock wave reaches that location.

The integral form of positive pressure is called positive impulse iS and iS is anegative impulse. These values are usually important both for further failure analysisof any obstacle. Furthermore, the impulse of overpressure could be more dangerousthan the underpressure especially for typical civil engineering structures. On the otherhand, the experimental tests which allow to measure the real pressure in the certaindistance from the ignition point are relatively rare and very costly. In addition to that,

3.3. THERMAL EFFECT 31

they often end with the damage of the gauges and cannot be repeated under the sameconditions.

Nevertheless, many authors based on numerous experiments, e.g. [19, 81, 71, 130,138], have elaborated some simplifications for predicting the pressure values, whichmay be crucial for rapid assessment of blast loading effect. Part of these values arecollected in Figure 3.15 in the further part of this Chapter.

3.3 Thermal effect

The energy value released during explosion is proportional to the difference between thetotal binding energy contained within the initial and unstable system and the energycontained within the final and stable state. This net energy release is often describedas the heat of explosion. The idea of the thermal energy comes directly form thenuclear explosion analysis, where the increasing heat effect is crucial for the damagephenomenon. Let us consider the following definition:

Definition 3.4. Thermal Energy The energy emitted from the fireball as thermalradiation. It is usually expressed in terms of calories per square meter and describesthe total amount of thermal energy received per the unit area at a specified distancefrom the explosion.

In addition to that, the pressure-thermal ratio strongly depends on the type ofexplosion. Basing on the literature research [35] these dependencies are gathered inFigure 3.2. Furthermore, several authors investigate the results of the thermal effect onthe engineering structure after typical (mixture and condensed) charge explosions. In1963 Mader [97] presented the results of his research regarding the thermal propertiesfor different condensed explosives. The author presented the simplified method forpredicting the heat value as a function of charge mass and the distance. Nguyen et al.[105] analyzed the impact of the heat action due to the fire on masonry barriers. Theyobtained critical displacement values for hollow burnt-clay bricks under 103 Celsiusdegree. Leiva et al. [83] obtained experimentally mechanical properties of mortarcylinder specimens. The loading conditions were introduced by heating the mortar upto 103 Celsius degree.

In papers presented above, the authors have analyzed long duration heating whichis not directly comparable with the heat effect during explosion. In this case the gradi-ents of temperature field are extremely high. Due to lack of any real experimental datafor failure of masonry under rapid thermal radiation release and staying in the agree-ment with Figure 3.2, the reliable consideration of this effect is neglected by authorin dissertation. Furthermore, the thermal wave transmits considerably longer throughthe medium than the pressure wave and usually reaches the obstacle after the failure,see [117, 121]. The interpretation of the last sentence is graphically represented inFigure 3.3 where shock against heat fronts are compared with respect to time and


Figure 3.2: Influence types after explosion [35]

distance function. These fronts intersect at a finite time t = t1 and at some distancer = r1. The presented result is obtained for a Taylor-Sedov solution of point explosion[134, 126], for a sharply fronted spherical wave, which moves supersonically, i.e. con-siderably faster than typical hydrodynamic flow.Furthermore, there is a slight influence under thermal wave propagation for struc-ture failure in comparison to the rapid pressure changes. For further consideration ofconventional explosives the thermal effect is ignored.

3.4 Basic concepts of thermodynamics for explosion

One of the basic laws of thermodynamics provides the conservation of energy whichrequires that energy must be released in case of converting system to another one withgreater stability and lover amount of entire energy. When we consider non-nuclearcharges, the molecules of commonly used explosives are considered to be in highlyenergetic and unstable state. In case of reaction of such compound the products ofgreater stability are formed consequentially releasing the energy. For the conventionalexplosives the energy is released through rapid and violent chemical or physical reac-tion.

Further investigation considering blasts requires brief introduction to the basic ex-plosion process and commonly used terminology. Let us first consider the detonation.

Definition 3.5. Detonation is a reaction on explosive where the high intensity shock

3.4. BASIC CONCEPTS OF THERMODYNAMICS FOR EXPLOSION 33

Figure 3.3: Trajectories of shock and heat fronts for strong explosion [134]

wave is produced [58].

In general, there are two types of detonation. The non-stationary detonation wherethe explosion propagates with various velocities and the classic detonation with the con-stant velocity (velocity D range is from 2100 up to 8000 ms1 for condensed explosives[147]). During this research author concentrates on the primary effects of explosionhence chemical details of detonation are neglected.

3.4.1 Introduction and governing equations

The methods for deriving the value of the blast pressure fall under three differentcategories [34]. According to the conducted studies and literature reviews the mostaccurate are the first principle methods, which represent the numerical solution ofthe system of partial differential equations. They usually engage Computational FluidDynamics (CFD) models. Hence, the fluid represents the medium (usually air orliquid) throughout the propagation of the blast pressure. There are three fundamentalequations governing the flow of the fluid [159]. These are therefore called Navier-Stokesequations:

Conservation of the massThe mass stays constant within the given system, however, the density can vary


across the fluid field. In this work we consider as density and ui as velocity:

t+

xi(ui) = 0. (3.1)

Conservation of momentumThe equation describing this law is the direct result of the Newtons second law.The change rate of the forces acting on the fluid III is equal to the change rateof the momentum with respect to time I and distance II :{

tui

}I+

{

xi(uiuj)

}II{ijxi

+p

xi fi

}III= 0. (3.2)

Conservation of energyThis law obviously implies that the total energy of the system remains constant.While considering fluids the form of energy is mainly expressed by pressure,however, there are also other types of energies such as kinetic k or heat energyqH which in case of explosions are usually driven by the chemical reactions.Mechanical energy fiui could be expressed by work being done by externalforces.

tE +

xi(uiH)

xi

(kT

xi

)+

xi(ijui) fiui qH = 0 (3.3)

Where H is heat content (enthalpy), see Equation 3.4.

H = E +p

, (3.4)

where E is a total energy per unit mass Equation 3.5, and there is a sum of in-trinsic energy e = e(T, p). Furthermore, it depends on the temperature, pressureof the fluid, and the kinetic energy k.

E = e+1

2ui

2. (3.5)

Solving the above equations for the entire domain is very tedious and difficult. Dueto the pressure, the velocity and density fields vary spatially, hence the entire domainmust be discretized into equations that apply to smaller volumes of fluid. This kind ofapproach is commonly called control volumes.

The system of equations 3.1, 3.2, and 3.3 are well known Navier-Stokes equations.Let us assume that there exists a particular case of above equations which correspondsto the the inviscid fluid. That directly implies that the viscosity is equal to zero, hencethe shear stresses ij and heat conduction do not exist. For such case we are consider-ing the Euler equations. After the explosion, the shock wave propagates through themedium in completely inviscid style [21, 34]. Beyond the fuel mixture, especially for


condensed charges, there is no combustion during this propagation. For this particularsituation the Euler equations seems to be a most suitable approximation of the prob-lem.

The solution of the one-dimensional Euler equation Equation 3.6 is a solid basisfor further investigation and analysis of one-, two-, and three-dimensional systems ofspherically-, radially-symmetrical, and spatial implementation of the explosion phe-nomena, respectively. An interesting approach of solution is performed by Wada andLiu [143] (1997). They based their analysis on the flux splitting scheme utilizing the im-proved Advection Upstream Splitting Method (AUSMDV). However, their approachrequire additional time integration for producing a second-order implementation inspace and time, following MUSCL-Hancock [138].

U

t+F

x= 0 (3.6)

Relatively large number of sophisticated codes for analyzing blast effects, see Table 3.7,utilize this method to obtain solution. In case of spherically symmetrical analysis i.e.one-dimensional case, the first-order accuracy is sufficient only if the space discretiza-tion, i.e. between the explosion center and the nearest obstacle surface, is fine enoughto get demanded peak pressure. In such case, the partial differentiation equation Equa-tion 3.7 remains almost unchanged, apart from an extra term S(U). For each timestep U is being updated and hence the S term is recalculated and added to the entireflux term.

U

t+F

x+ S(U) = 0 (3.7)

The detailed parts of Equation 3.7 present as follows:

U =

u(e+ V

2

2

)F =

uu2 + pu(e+ V

2

2

)+ pu

S = 2r

uu2u(e+ V

2

2

)pu

(3.8)where the parameters such as source energy are available to be released as a heat e,detonation velocity is denoted as V , and initial density as . The initial density repre-sents the feature which is constant for each type of condensed explosive. The symbol rreflects the radial distance from the charge center and u is the one component velocity.

The typical scheme for solving one-dimensional problem can be presented as fol-lows. We assume that the space must be discretised into radial parts r, and eachpart is equivalent to the scaled distance having constant charge weight. It shouldconsist of roughly fifty computational cells through the thickness of the charge in or-der to produce very accurate blast parameters. The initial density, internal energy,and the speed of the detonation wave are defined as initial conditions. Furthermore,the replacing radius of the charge is calculated based on the mass of the charge W ,explained further as in Equation 3.34, in order to obtain the starting time t0. Hence,the time is the ratio of charge radius and detonation velocity. The total number of


computational cells, presented above and represented by integer number, comes fromdividing the largest radius rmax by r. The first computational cell is filled by ex-plosive properties, whereas the remaining cells are filled by the surrounding mediumi.e. the air. All other necessary parameters are calculated base on specific heat of theair at constant volume accounting for the specific heat ratios. These properties arediscussed at a later stage in details. The calculations are performed as long as eitherthe initially specified time is reached or the velocity in the final computational cellis greater then zero. The obtained data are used as an input data to the two- andthree-dimensional approach. This algorithm is widely utilized in many sophisticatedindustrial implementation such as the Air3d code, developed at Cranfield Universityin the UK.

The radially symmetrical solution of Euler equations, requires implementation oftwo-dimensional problem. It basically relies on the symmetry conditions about thevertical axis (coordinate h). Now, the coordinate r is a radial distance from the axis ofsymmetry. The existing velocity has radial u(r, h) as well as the axial v(r, h) compo-nents. Hence, the Euler form is extended with a geometric source term G and presentsas follows:

U

t+F(U)

r+G(U)

h+ S(U) = 0, (3.9)

and the separate elements there are:

U =

u

v

(e+ V

2

2

) , F =

u

u2 + p

uv

u(e+ V

2

2

)+ pu

,

G =

v

vu

v2 + p

v(e+ V

2

2

)+ pv

, S = 1r

u

u2

uv

u(e+ V

2

2

)pu

.(3.10)

U

t+F(U)

x+G(U)

y+H(U)

z= 0 (3.11)


U =

u

v

w

(e+ V

2

2

)

, F =

u

u2 + p

uv

uw

u(e+ V

2

2

)+ pu

,

G =

v

vu

v2 + p

vw

v(e+ V

2

2

)+ pv

, H =

w

wu

wv

w2 + p

w(e+ V

2

2

)pw

.(3.12)

Very interesting study on the blast pressure evolution was performed and presentedby Krzewiski [78]. Author evaluates parameters which are of a significant importanceto the front of the shock wave for spherical charges. These are: blast front velocity,pressure and density. If one assumes that the differential equation system is similar tothe ones presented in Equations 3.1, 3.2, and 3.3 than the blast model in gas mediumcan be described as presented in Equation 3.13:

t + u

r +

ur + ( 1) ur = 0

ut + u

ur +

1pr = 0

pt + u

pr + p

(ur +

1r u

)= 0

c

= rrc(t)

(3.13)

where the first two formulas represent the mathematical formulation of mass and mo-mentum conservation laws (as previously), the third part is responsible for the adi-abatic conditions and the last one for the density distribution in disturbed medium.Furthermore, the boundary conditions for above equations are presented for the explo-sive centre Equations 3.14, and 3.15 for the wave front. In such setup, crucial role isplayed by the starting condition stated in Equation 3.16.

u (0, t) = 0 (3.14)u (r0, t) = uc = f1 (r0) = 1 (t)

p (r0, t) = pc = f2 (r0) = 2 (t)

(r0, t) = 0 = f3 (r0) = 3 (t)

(3.15)

r (0) = 0 (3.16)

These formulae are directly connected to the dynamic agreement conditions Equa-tion 3.17 [147].

uc =(

1 10)

dr0dt

pc p1 = 1uc dr0dtE0 E1 + pc+p12

(10 11

)= 0

(3.17)


Figure 3.4: Blast wave front in different time instants

The last part of the first two lines of equation Equation 3.17 are representing the samevalue i.e. the velocity of the front of the wave D, see Equation 3.18, which can bederived directly from Figure 3.4a.

D =dr0dt

(3.18)

Figure 3.4b shows a small part of the cylinder representing subsequent positions ofthe disturbed front and describing the material state for times t and t+ dt. The frontof the wave is going to move from the section I, in time t, to the section I in time dt.During this process some of the particles are translated to a new state pc and c undervelocity uc.

This move is formed as a consequence of the rapid energy release Q0. The frontof the wave travels with velocity D in time dt. Afterward, it arrives to section II,following path Ddt. At this moment particles move from the surface I to I, followingpath ucdt. In the framework of the conservation of mass law the explosive mass, withinthe range I-II and I-II should remind the same as presented in Equation 3.19.

0FDdt = cF (D uc) dt (3.19)after mutual reduction of area F the uc can be expressed as follows:

uc =

(1 0

c

), (3.20)

also including [147]:pc p1 = 0ucD. (3.21)

The last element of Equation 3.17 is an integral form of the energy conservation law,which is necessary to start the explosion phenomenon. Hence, we obtain the system


of equations, i.e. Equation 3.13 and 3.17, with boundary conditions presented in3.14, 3.15 and 3.16. In order to make this formulation complete one has to considerthe general medium equation (Equation 3.22) along with the internal energy descriptionof gas Equation 3.23.

p = RT (3.22)

E = cT (3.23)

Where R represents a gas constant and all other variables can be derived directly fromFigure 3.4. For further consideration it is necessary to introduce the representation ofthe specific heat c variable. This important thermodynamic parameter characterisesthe ratio between heat absorbed by the unit mass and temperature increase. In agree-ment with the first law of thermodynamics, the change

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Sielicki Piotr W Masonry Failure Under Unusual Impulse Loading (1)

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