UNIVERSITY OF NAPLES FEDERICO II Faculty of Engineering. PH.D. PROGRAMME in MATERIALS and STRUCTURES COORDINATOR PROF. DOMENICO ACIERNO XXII CYCLE

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1 UNIVERSITY OF NAPLES FEDERICO II Faculty of Engineering PH.D. PROGRAMME in MATERIALS and STRUCTURES COORDINATOR PROF. DOMENICO ACIERNO XXII CYCLE DOMENICO ASPRONE PH.D. THESIS Advanced analysis and modeling of strategic infrastructures subjected to extreme loads TUTOR PROF. GAETANO MANFREDI CO-TUTOR DR. ANDREA PROTA

2 ACKNOWLEDGEMENTS These few lines are to show my gratitude to those people that make my Ph.D possible. I want to particularly thank Prof. Nanni for his scientific coordination of the SAS and the Tenza projects, which included many of the activities presented in this work. Prof. Nanni always encourages me enthusiastically contributing to my increasing love for research. I am also particularly grateful to Prof. Auricchio whose scientific guidance is valuable, precious and fundamental for my work in particle methods. I really thank him for the time he spends with me and his patience. I am also grateful to Prof. Cadoni that conducted with me all the activities of my work involving dynamic characterization of materials. Through these years Ezio has really become a good friend, beyond our scientific interest and it is always a pleasure for me to work with him. A devoted gratitude is toward Dr. Jalayer that conducted with me the work on the multi-hazard assessment of structures. Fatima is a friend and a brilliant researcher whom I have learnt from a lot. The deepest gratitude is toward Prof. Manfredi, Prof. Cosenza and Dr. Prota that gave me the opportunity of working with them, making me proud to belong to a unique team. Their scientific guidance and teachings are precious for me. I am grateful to Prof. Manfredi and Dr. Prota particularly for their generous devotion they show every day and for the patience they had with me when I was not convinced to start this adventure. Thank you.

3 3 Index INDEX INDEX...3 INTRODUCTION...7 Chapter I ANALYSIS OF STRATEGIC INFRASTRUCTURES IN CASE OF BLAST...10 MULTI-HAZARD ASSESSMENT/DESIGN...10 BLAST LOADING...14 BLAST FRAGILITY...16 Using simulation-based reliability methods for risk assessment...16 Closed-form solution for the local dynamic analysis...17 Kinematic plastic analysis on damaged structure...19 Calculating the blast fragility implementing the MC simulation...23 NUMERICAL EXAMPLE...24 Structural model description...24 Characterization of the uncertainties...28 Characterization of the parameters defining the local dynamic analysis...30 Blast fragility...32 Seismic fragility...36 Discussion on the case study...37

4 4 Index Appendix 1.A: Analysis of un-damped free vibration for fixed-end beam flexure...41 Appendix 1.B: Analysis of Impact loading for fixed-end beam...43 Appendix 1.C: Linear Programming problem...46 Appendix 1.D: The Coefficient of the Variation (COV) of the Mean of a Bernoulli Variable...46 REFERENCES...47 Chapter II DYNAMIC PROPERTIES OF CONCRETE...50 INTRODUCTION TO THE FRAMEWORK ACTIVITIES...50 The Tenza Bridge...52 Traditional assessment and characterization...53 STATIC CHARACTERIZATION OF CONCRETE...54 DYNAMIC TESTS ON CONCRETE...55 Introduction to the dynamic properties of concrete...55 DynaMat facilities...56 Set-up for high strain rate tests...56 Test results...62 ASSESSMENT OF EXPERIMENTAL RESULTS...67 REFERENCES...73 Chapter III DYNAMIC PROPERTIES OF REINFORCING STEEL...75 INTRODUCTION TO FRAMEWORK ACTIVITIES...75 EXPERIMENTAL ACTIVITY...76 Dynamic tests set-up...78 Tests results...81 ANALYSIS OF TEST RESULTS...90 REFERENCES...96

5 5 Index Chapter IV DYNAMIC PROPERTIES OF GFRP...98 INTRODUCTION TO FRAMEWORK ACTIVITIES...98 STATIC CHARACTERIZATION OF GFRP...99 DYNAMIC EXPERIMENTAL ACTIVITIES ON GFRP Experimental set-up Discussion of tests result NUMERICAL ASSESSMENT OF RESULTS REFERENCES Chapter V DYNAMIC PROPERTIES OF A NATURAL STONE INTRODUCTION TO THE FRAMEWORK ACTIVITIES STATIC CHARACTERIZATION OF NYT DYNAMIC TESTS ON NYT Medium strain-rate tests High strain-rate tests ASSESSMENT OF EXPERIMENTAL RESULTS REFERENCES Chapter VI PARTICLE METHODS FOR DYNAMIC ANALYSIS INTRODUCTION TO PARTICLE METHODS CLASSICAL SPH APPROACH APPROXIMATION OF DERIVATIVES Original formulation Chen and Beraun's formulation RKPM method FPM formulation Modified FPM formulation A NOVEL SECOND-ORDER FPM FORMULATION NUMERICAL TESTS...160

6 6 Index Derivative test Elastic static test Elastic vibration test ASSESSMENT OF NUMERICAL RESULTS REFERENCES Chapter VII DEVELOPMENT OF A BLAST PROTECTION BARRIER INTRODUCTION TO FRAMEWORK ACTIVITIES GEOMETRY AND MECHANICAL PROPERTIES OF THE BARRIER ASSESSMENT OF THE ELECTROMAGNETIC DISTURBANCE BLAST TESTS EXPERIMENTAL-THEORETICAL COMPARISONS Appendix 7.A: An example of the proposed numerical procedure REFERENCES CONCLUSIONS...202

7 7 Introduction INTRODUCTION Recent terrorist acts have contributed to change the design approach to critical infrastructures; in fact, malicious disruptions, blasts, or impacts have unfortunately become part of the possible load scenarios that could act on constructed facilities during their life spans. Hence, a sustainable design aims to ensure the satisfactory performance of the structure during its entire lifetime considering all the possible critical actions, which the structure could be subjected to, including severe dynamic load conditions. The evaluation of the actions on the structure in case of such events is fundamental but represents a critical concern, since uncertainty related to loads definition is often quite high, especially for blast actions. Furthermore, structural response in case of such severe dynamic actions represents a critical issue, since both mechanical properties of materials and dynamic behavior of structural elements under severe dynamic loads can be very different from that exhibited under static actions. Moreover, numerical procedures used to simulate high dynamic loading conditions on structures can suffer of lack of accuracy, due to the rate and the intensity of deformations occurring on structural elements. Hence specific investigations become necessary for all these concerns. In particular, the present work addresses the assessment and design of strategic structures which are to be subjected to multiple hazards during its lifetime, including severe dynamic events, especially blast. At this aim, the most critical

8 8 Introduction issues related to the assessment and design of strategic infrastructures potentially subjected to high dynamic conditions, are discussed and analyzed. Given the uncertainty involved in characterizing the load conditions, it seems inevitable to address the design based on a probabilistic framework. The design can be addressed by limiting the probability of failure below a certain de-minimis risk level that is deemed acceptable by the society. Inevitably, evaluation of the probability of failure requires taking into account possible actions or hazards that the structure could be subjected to; in other words, it needs to be evaluated based on a multi-hazard approach. In details, a multihazard framework is proposed and implemented for a strategic reinforced concrete buildings subjected to both seismic and blast hazard. The methodology is described in Chapter I and applied to a case study. Then, a deep investigation is presented on mechanical properties of construction materials in case of dynamic loading conditions. In particular, the strain rate sensitiveness of such material is investigated through a wide experimental activity conducted at Dynamat Laboatory at University of Lugano, Switzerland. In details, results of research activities are presented for: concrete, in Chapter II, steel for concrete internal reinforcement, in Chapter III, Neapolitan yellow tuff, a natural stone widely used in Neapolitan area for masonry structures, Chapter IV, GFRP (glass fiber reinforced polymer), in Chapter V. A further critical issue related to numerical simulations in case of high dynamic loading conditions on structures. In fact, to address dynamic loading conditions on structural elements, a variety of numerical methods have been recently proposed in the literature; the objective is to address advanced mechanical problems, such as those involving rapid deformations, high intensity forces, large displacement fields. In many of these cases, in fact, classical finite element methods (FEM) suffer from mesh distortion, numerical spurious errors and, above all, mesh sensitiveness. Hence, to overcome such issues, a number of numerical methods, belonging to the family of the so-called meshless techniques, have been widely investigated and applied. The objective of

9 9 Introduction employing these methods is to avoid the introduction of a mesh for the continuum, preferring a particle discretization, with the goal of obtaining an easier treatment of large and rapid displacements. Recently, a number of researchers have tried to extend meshless methods also to solid mechanics problems. Among the several meshless numerical methods proposed, particle methods and in particular Smoothed Particle Hydrodynamics (SPH) has been widely implemented and investigated. A revision of the most common SPH methods is presented in Chapter VI and a rigorous analysis of the error is conducted, focusing on 1D problems. A novel second-order accurate formulation is also proposed. A further issue is addressed in Chapter VII and is related to protection interventions to be introduced in structural design to minimize disruptive effects in case of malicious blast actions and guarantee the safety of the occupants. In particular, a GFRP porous barrier is developed as fencing structure to prevent malicious disruptions, provide a standoff distance in case of blast actions, and reduce the consequences of an impact. The proposed barrier provides protection through two contributions. First, its geometrical and mechanical characteristics ensure protection against intrusions and blast loads. Second, its shape provides a disruption of the blast shock wave, adding additional protection for structures and facilities located beyond it. The efficacy of the proposed barrier under blast loads is presented by showing the results of the blast tests conducted on full-size specimens with a focus on the reduction of the blast shock wave induced by the barrier. A simplified model is also proposed to predict the reduction of the blast pressure due to the porous barrier, providing a procedure to design the geometrical characteristics of the barrier.

10 10 Chapter I Analysis of strategic infrastructures in case of blast Chapter I ANALYSIS OF STRATEGIC INFRASTRUCTURES IN CASE OF BLAST MULTI-HAZARD ASSESSMENT/DESIGN Given the uncertainty involved in characterizing critical events possibly occurring on strategic structures during their life-time, it seems inevitable to address the design based on a probabilistic framework. In lieu of economical cost considerations, one can address the sustainable design by limiting the probability of failure below a certain de-minimis risk level that is deemed acceptable by the society. The probability of failure can be measured by the mean annual frequency that the structural response exceeds a certain limit threshold, which is identified based on the design performance objectives. Inevitably, evaluation of the probability of failure requires taking into account possible actions or hazards that the structure could be subjected to; in other words, it needs to be evaluated based on a multi-hazard approach. This study aims to evaluate the probability of failure following a multi-hazard approach. In particular, it considers the case of a strategic structure, which is to be subjected to both earthquake and blast actions during its lifetime. The attention is focused on the structural collapse as the limit threshold for which the mean annual frequency of exceedance is calculated. By structural collapse, it is intended the loss of ability to withstand gravity loads. The multi-hazard

11 11 Chapter I Analysis of strategic infrastructures in case of blast approach enables consideration of both seismic and blast actions in the form of seismic and blast fragilities, respectively. The seismic fragility, defined as the probability of structural collapse given that an earthquake of interest has taken place in the seismic zone around the structure, can be calculated by following a non-linear static analysis (pushover) approach. The blast fragility, defined as the probability of collapse given that a blast event has taken place in the structure, is evaluated using an advanced simulation method. It is assumed that a possible blast scenario is identified by the quantity of the explosive and the location of the center of the blast within or close to the structure. For each possible blast scenario realization, generated by the simulation, the stability of the structure is verified by performing a plastic limit analysis on the damaged structure. Finally, the mean annual frequency of collapse can be evaluated and benchmarked against the acceptable risk threshold. As a case study, the blast and seismic fragilities of a generic four-storey RC building located in the seismic zone are calculated and implemented in the framework of a multihazard procedure, leading to the evaluation of the annual risk of collapse. The probability-based multi-hazard design of a structure is performed taking into account all possible events that could potentially cause significant damage. In particular, for the limit state of collapse, the probability of collapse can be written as [1]: P ( C) P( C A) P( A) (1.1) = A where A stands for a critical event, such as, Earthquake, Blast, etc. Formally, A can be written as the logical union of the potential critical events, that is: A Eartquake + Wind+ Gas Explosion + Blast + (1.2) Equation (1.1) is written using the total probability theorem assuming that the critical events A are mutually exclusive (i.e., they cannot happen at the same time) and collectively exhaustive (i.e., all of the potential A are considered). Obviously, the events that contribute to A vary based on the type, location, and function of the structure to be designed or assessed. That is, depending on the

12 12 Chapter I Analysis of strategic infrastructures in case of blast particulars of each problem, some of the terms in A might be dominant with respect to the others. The de minimis risk ν dm, which defines that risk below which society normally does not impose any regulatory guidance, is in the order of 10-7 /year [2]. Therefore, if the annual risk of occurrence of any critical event A is considerably less than the de minimis level, it could be omitted from the critical events considered in Equation (1.2). Therefore, the multi-hazard acceptance criteria can be written as following: P( C) = P( C A) P( A) ν (1.3) A dm The above-mentioned criteria could be used both for probability-based design and assessment of structures for limit state of collapse. Considering a particular case in which the critical events are earthquake (EQ) and blast, the design/assessment criterion can be written as: ν C = P( C EQ) ν EQ + P( C Blast) ν Blast ν dm (1.4) where ν C stands for the annual rate of collapse and ν EQ and ν Blast stand for the annual rates of occurrence of earthquake and blast events of significance, respectively. P(C EQ) and P(C Blast) represent seismic and blast fragilities. In this case, considering earthquake and blast hazards as mutually exclusive implies assuming that just one of them can induce structural collapse. Note that ν C is a rate of exceedance and not a probability; however, for very rare events, the probability of exceedance is approximately equal to the annual rate of exceedance. The annual rate of an earthquake event of interest can be calculated using probabilistic seismic hazard analysis (PSHA) for the site of the project. On the other hand, estimation of the annual rate of the occurrence of a blast event caused by terrorist attack cannot be easily quantified and defined analytically. However, estimation of ν Blast is not an engineering problem since it depends on socio-political considerations and how strategically vulnerable the structure is against such events. Therefore, it is highly questionable whether blast event can be characterized as an aleatory stochastic process. However, in order to facilitate calculations, it is assumed here that ν Blast can be quantified numerically. In order to gain a qualitative understanding of relative importance

13 13 Chapter I Analysis of strategic infrastructures in case of blast of blast compared to earthquake, one can normalize the design/assessment criterion with respect to seismic risk: ν Blast ν dm P( C EQ) + P( C Blast) (1.5) ν ν EQ EQ Alternatively, in cases where ν Blast cannot be identified, one could perform a scenario-based calculation of the probability of collapse and compare it against an acceptable threshold that is larger than de minimis level (e.g., in the order of 10-2 ). The probabilistic multi-hazard design can be considered as an iterative procedure, in which, starting from a proposed structural design, one can calculate the seismic and blast fragilities and check the design criteria stated in Equation (1.5) for a pre-determined ratio of blast risk to seismic risk. The iteration continues until the design criterion is satisfied. However, it should be taken into account that the so-called de minimis risk is a qualitative threshold based on the design needs. Moreover, it is important to note that the structural resistance to seismic and blast actions are correlated; that is, the structural design choices aimed at seismic risk mitigation are likely to improve the structure s performance also in the case of blast. It is interesting to note that the first generation of probability-based limit states design criteria are based on the reliability analysis of individual members and components under a number of distinct load combinations. Each of these load combinations includes the maximum value over time of one of the loads plus the arbitrary-point-in-time values of other loads [3, 4]. For each load combination, the reliability of element/component (calculated using first order reliability methods) was compared against an acceptable threshold (e.g., safety index of about 4 which corresponds to a de minimis level of about 10-5 ). Alternatively, in the multihazard approach, the reliability of the structure is calculated from Equation (1.1) considering all possible critical load configurations that could lead to collapse. Finally, the annual rate of collapse can be calculated from Equation (1.4) or Equation (1.5) and compared with the adopted de minimis threshold. It should be noted that employing the multi-hazard formulation makes it possible to consider the rehabilitation strategies with respect to both blast and

14 14 Chapter I Analysis of strategic infrastructures in case of blast earthquake. Considering that structural collapse cases induced by earthquake and blast are initiated by similar local failure mechanisms [5], the multi-hazard approach is especially efficient since the corresponding risk reduction interventions are similar or at the very least complementary (i.e. composite wrapping of columns, steel bracing installations). BLAST LOADING An explosion induces mainly a quick and significant increase of pressure in the medium where it occurs, i.e. air or water. Such overpressure propagates as a wave, the so called blast wave, and is characterized by its speed, duration and intensity. These parameters are fundamental in order to evaluate the actions that an explosion can induce in the structural elements in its vicinity. The numerical values of these parameters depend on several aspects, such as, type and the amount of the exploding mass, distance of the target of interest from the explosion, geometry of the target, type of reflecting surfaces (e.g., the ground in case of external explosions or walls or slabs in case of closed-in explosions). In the past decades, several investigations have been performed on such aspects and they have provided reliable numerical procedures for quantification of the overpressure time-histories. In the case of blast explosion, the induced overpressure follows a trend over time similar to that shown in Figure 1.1, where a positive decaying phase is followed by a weaker negative phase. The duration of the blast wave depends on the amount of the exploding charge and its distance from the target; however, the phenomenon is very quick and can last up to 10-2 s.

15 15 Chapter I Analysis of strategic infrastructures in case of blast P [Mpa] t [s] Figure 1.1 Blast overpressure in air generated by 1kg of TNT at 2m from the charge The primary effect of a blast explosion on civil structures is due to such rapid and intense action, able to induce severe local structural damages. In fact, the applied loads are so fast that they are unable to activate the global vibration modes of the structure, since the inertia corresponding to such modes has no sufficient time to react. Therefore, in case of an RC framed structure, the blastinduced overpressures hit directly the single frame elements, which behave as independent structures, and can be modeled as fixed ends elements [6]. An indirect effect of blast explosion on civil structure is progressive collapse. The progressive collapse can be defined as a mechanism involving a large part of a structure, triggered by local less-extensive damage in the structure. In fact, a blast explosion occurring within or near an RC framed building can cause the loss of one or more single frame elements. Having lost some elements, the whole structure can become unstable, failing under the present vertical loads. That is, the structure can eventually develop a global mechanism, which is widely referred to as the progressive collapse mechanism [7, 8, 9]. Design

16 16 Chapter I Analysis of strategic infrastructures in case of blast and/or assessment of structures accounting for such failure mechanism can follow a direct approach or an indirect approach [10]. In the first case, resistance to progressive collapse is pursued guaranteeing minimum levels of strength, continuity and ductility, whereas in the direct approach progressive collapse scenarios are directly analyzed. Actually, the progressive collapse mechanism is most often identified as the actual cause of blast-induced damage [11] and it is already the subject of wide research related to the protection of critical infrastructures [8, 9, 11]. The design approach to such failure mechanism follows BLAST FRAGILITY Using simulation-based reliability methods for risk assessment The blast fragility denoted by P ( C Blast), in the context of this work, can be defined as the conditional probability for the event of progressive collapse given that a blast event takes place near or inside the strategic structure in question. Consider that real vector θ represents the uncertain quantities of interest, related to structural modeling and loading conditions. Let the positive real number p (θ ) represent the probability density function (PDF) for the vector θ. The ( C Blast) P P can be written as follows: ( C Blast) I C Blast ( θ ) p( θ ) θ C Blast (1.6) where I ( ) is an index function which is equal to unity in the case where θ leads to blast-induced progressive collapse and zero otherwise. Here, the probability of progressive collapse P ( C Blast) is calculated using standard Monte Carlo (MC) simulation for generating N samples θ sim i from PDF p (θ ). The event of progressive collapse is identified by the ratio index λ θ ) which is the factor by which the gravity loads should be multiplied in C ( i = dθ

17 17 Chapter I Analysis of strategic infrastructures in case of blast order to create a global collapse mechanism. In case it assumes a value less than unity, the event of progressive collapse is actually activated, since the acting loads are sufficient to determine instability of the structure. Moreover, the uncertain quantities of interest here are the amount of explosive and its position with respect to the structure. Obviously, any other uncertain quantity such as those related to structural modeling can be added to vector of uncertain parameters θ. For each simulation realizationθ i, the following two steps are performed: 1. A local dynamic analysis is performed on the column elements affected by the blast in order to verify whether they can resist the explosion and keep their vertical load carrying capacity. 2. After identifying the damaged columns to be removed, a kinematic plastic analysis is performed on the damaged structure in order to evaluate the progressive collapse index λ C ( θ i ) and to control whether the structure is able to carry the gravity loads in its post-explosion state. Closed-form solution for the local dynamic analysis As mentioned in a previous section, the blast action can be modeled by a quick decay pressure time-history curve. This curve can be approximated by a triangular shape identified by two parameters, namely, the initial peak pressure p 0 and the duration t plus of the positive phase. These parameters, which depend on the amount of explosive and the distance from the charge, can be evaluated according to empirical formulas available in literature (e.g., [6, 12]). Since the blast-induced action is very rapid and consequently the structural inertia does not have sufficient time to respond, the individual elements react to it as if they were fixed-end elements. Moreover, for the same reason, the structural damping can be ignored. For each simulation realization, the step 1 described above is conducted, performing the dynamic analysis of an un-damped distributed-mass fixed-end beam subject to triangular impact loading, for all the columns on the same floor as the explosion. Moreover, for the sake of simplicity in calculations, it is

18 18 Chapter I Analysis of strategic infrastructures in case of blast assumed that blast action is constant across the length of the columns. Consider that the beam in question has constant EI and constant distributed mass m, the free vibration equation of motion for this system can be written as [13]: 4 2 (1.7) v( x, t) EI + m 4 x where v ( x, t) is the displacement of the beam. Assuming that this beam responds primarily according to its first mode of vibration, its displacement response v ( x, t) can be represented as that of a generalized SDOF system: v( x, t) = φ( x) Y ( t where (x) v( x, t) = 0 2 t ) φ is the normalized first mode-shape of the beam and ) (1.8) Y (t is the displacement time-history of an SDOF system with the same period of vibration as the fixed-end beam. The period of vibration and the first modeshape of the beam can be calculated by imposing the boundary conditions to Equation (1.7) and using the separation of variables in Equation (1.8). It can be shown (see Appendix 1.A) that the period of vibration of a fixed end beam with the above-mentioned properties is equal to: L T = 2π m EI (1.9) The first mode-shape of the beam is plotted in Figure Again using the variable separation in Equation (1.8), it can be shown that the equation of motion of the beam in response to impact loading is identical to that of an SDOF system with period of vibration equal to T and displacement response equal to Y (t). An analytic closed-form solution can be found for an undamped Single Degree Of Freedom (SDOF) system subject to triangular impulse loading (see Apendix B). It turns out that the impulse duration is much smaller than the natural period of vibration of the SDOF system; therefore, the maximum response will most likely be in the free-vibration response phase. The free-vibration response of an SDOF oscillator with angular frequency ω = 2π / T can be derived from following:

19 19 Chapter I Analysis of strategic infrastructures in case of blast Y ( t plus) (1.10) Y ( t) = Y ( t plus)cosωt + sinωt ω Where Y t ) and Y t ) are the displacement and velocity initial ( plus ( plus conditions for the free-vibration response evaluated at the end of the triangular impulse loading. Thus, the maximum response will be equal to: 2 (1.11) Taking into account the displacement profile of the beam shown in Figure 1.12, it can be deducted that the maximum bending moment and shear will take place at the fixed ends and will be calculated as following (see Appendix 1.B for details): M V ρ = max max Y ( t plus ) 4.73 = 1.26 L 4.73 = 1.24 L + ω Y ( t ) 2 plus 3 2 EIρ EIρ (1.12) In order to verify whether the individual column can resist the explosion, the maximum blast-induced bending moment and shear M max and Vmax are compared against the ultimate bending and shear capacity of the element at its ends. Kinematic plastic analysis on damaged structure After identifying and removing the damaged elements, it should be verified whether the damaged structure can withstand the applied vertical loads. This is essentially a global stability analysis of the damaged structure. A possible approach to perform such analysis would be to conduct a plastic limit analysis. A plastic limit analysis involves finding the load factor on the applied loads for which the following effects occur: 1. Equilibrium conditions are satisfied.

20 20 Chapter I Analysis of strategic infrastructures in case of blast 2. A sufficient number of plastic hinges are formed in the structure in order to transform the whole structure or a part of it into a collapse mechanism. Assuming that: (a) the non-linear behavior in the structure is concentrated at the element ends (b) the member ends are capable of developing their fully plastic moment (i.e., the fragile failure modes such as axial and shear failure or the ultimate rotational failure do not take place before the member has developed its plastic bending capacity). The plastic limit analysis is herein performed by finding the smallest kinematically admissible load for which the above conditions (1 and 2) are satisfied, employing the principle of virtual work. A kinematically admissible load corresponds to a mechanism in which both the external work e done by the applied forces on virtual deformations and the internal work u done by the ultimate moments M on the virtual rotations θ are positive. p It can be shown that the collapse mechanism can be described as a linear combination of the independent mechanisms that can be activated in the structure. The number of these independent mechanisms denoted here by m is equal to the difference between the number of possible plastic hinge locations, s, and the degree of indeterminateness in the structure, i [14]. Therefore, the collapse load factor λ C for the structure can be defined as: s (1.13) M pjθ j u j= 1 λc = = m e t e i= 1 In which, u is the internal work done during the formation of the mechanism leading to collapse and e is the external work done by the applied loads due to the formation of the collapse mechanism. The plastic rotation at the hinge location j can be calculated as i i θ = m t j i i= 1 θ ij, which corresponds to a linear combination of plastic rotation contributions from each independent

21 21 Chapter I Analysis of strategic infrastructures in case of blast mechanism i, θ, and the scale factor that reflects the relative contribution of ij the corresponding independent mechanism, t i. In the same manner, the external work e can be described as a linear combination of the external work e done by the independent mechanisms again weighted through the i corresponding scale factor t i. It has been shown [15] that the procedure for finding the smallest kinematically admissible load can be defined as a linear optimization programming with the objective of minimizing the load factor λ C. Formally, the linear programming problem can be outlined as following: Minimize subject to both u = θ = j s j= 1 m i M i pj θ t θ and ij j e = m i= 1 t e i i (1.14) It can be recognized that this linear programming problem could be resolved by employing a simplex algorithm [16] in which the basic variables are represented by θ and t (see Appendix 1.C). However, it should be ensured j i that the basic variables in the simplex algorithm are non-negative. On the other hand, θ and t j i can be negative since the plastic hinges can be formed in both directions of bending and also independent mechanisms can contribute negatively to the definition of the collapse mechanism. In order to overcome the problem of basic variables with negative values, the following changes of variables are performed. The plastic rotation θ can be defined as: θ + = θ θ j (1.15) j j + In which θ and θ, depending on the assumed convention, are defined as: j j j

22 22 Chapter I Analysis of strategic infrastructures in case of blast θ = θ + j θ = θ j j j if θ 0 and j if θ < 0 and j θ = 0 if θ < 0 + j θ = 0 if θ 0 j (1.16) Moreover, the corresponding plastic moment capacities in the positive and + negative direction are denoted by M and M, respectively. As it regards the pj pj contribution of the independent mechanisms, the following change of variables needs to be made: t = t t i i (1.17) In which t i is a non-negative quantity and t is an arbitrarily large positive quantity. Substituting from Equations (1.16) and (1.17), the linear programming problem in Equation (1.14) can be represented in the following form: Minimize subject to both u = j s j= 1 j pj + θ + θ + + M θ + M θ m i j ( t t) θ = 0 i pj j ij j j and m i= 1 ( t t) e i i = e (1.18) The input of the optimization problem consists of e, i + M and pj M pj. Since the final objective is to minimize the load factor λ C (which is the ratio of internal to external work), the external work e is introduced into the linear programming problem as an arbitrary positive constant. As a consequence, the + generated output θ and θ represent only the relative values of the plastic j j rotation in the hinges. For example, in the particular case of a RC framed structure, the independent mechanisms are classified as follows [15] (Figure 1.2): (a) the soft-story mechanisms in which the plastic hinges at both ends of all the columns within a given storey are activated, (b) the beam mechanisms in which (at least) three hinges are formed in given beam, and (c) the joint mechanisms in which the end hinges of all the frame elements converging into a given joint are activated.

23 23 Chapter I Analysis of strategic infrastructures in case of blast (a) storey mechanism (b) beam mechanism (c) joint mechanism Figure 1.2 Principal mechanisms, as defined in [15] Once the linear optimization problem described in Equation (1.18) is solved the minimum load factor λ C leading to structural collapse is obtained. In static loading problems, aλ C less than or equal to unity indicates that the structure is already unstable under the applied loads. On the other hand, as it regards an instantaneous dynamic loading the threshold should be moved to 2. In case of progressive collapse problem it has been shown that 2 is probably conservative and the actual value of λ causing instability in the structure is between 1 and C 2 [17]. Calculating the blast fragility implementing the MC simulation As mentioned in a previous section, the blast fragility is defined as the probability of progressive collapse event given that a blast event takes place inside or in the vicinity of the structure in question. The progressive collapse event can be characterized by a Bernoulli-type variable that is equal to unity in the event of progressive collapse and equal to zero otherwise. Using the kinematic plastic limit analysis described in the previous section, the Bernoulli collapse variable denoted by I ( θ ) can be determined as a function the C Blast collapse load factor λ C :

24 24 Chapter I Analysis of strategic infrastructures in case of blast I I C Blast C Blast ( θ ) = 0 ( θ ) = 1 (1.19) Where λ is the threshold value for the load factor indicating the onset of C, th progressive collapse varying between 1 and 2. The MC procedure can be used to generate N realizations of the uncertain vector θ i according to its sim if λ > λ C C if λ λ C, th C, th Probability Density Function (PDF) p (θ ). Finally, the conditional probability of progressive collapse in Equation (1.6) can be solved numerically as the expected value of the Bernoulli collapse index variable I ( ) : θ C Blast Nsim (1.20) IC Blast( θ i ) i= P( C Blast) 1 N sim It can be shown (see Appendix 1.D) that the coefficient of variation of the conditional progressive collapse probability can be calculated as follows: C. O. V. P( C Blast) = 1 P( C Blast) N P( C Blast) sim (1.21) NUMERICAL EXAMPLE A possible application of the methodology described in the previous section can refer to the calculation of the mean annual risk for progressive collapse of a generic RC framed building. A numerical example is here presented about such application; the characteristics of the case-study structure are outlined in the following. Structural model description The case-study building is a generic five-story RC framed structure. The structural model is illustrated in Figure 1.3, presenting a plan of the generic storey; column sections are all 60x30 at the first and the second floor and

25 25 Chapter I Analysis of strategic infrastructures in case of blast 50x30 at other floors, whereas two types of beam are present, Type A and Type B, whose width and height are 0.30m per 0.50m and 0.80m per 0.24m, respectively; the floors are supposed to be one-way joist slabs, 0.24m thick. Type A Type A Type A Type A Type A Type A Type A Type A Type B Type B Type A Type B Type A/Type B* Type A Type A/Type B* Type B Type A Type B Type B Type A Type A Type A y Type A Type A Type A Type A Type A x Figure 1.3 Storey view (dimensions in m) *this frames represent both storey beams (Type B) and stair knee beams (Type A) Figure 1.4 shows a 3D view of the model. Each storey is 3.00m high, except the second one, which is 4.00m high. The non-linear behavior in the sections is assumed to be only flexural and is modeled based on the concentrated plasticity concept. It is assumed that the plastic moment in the hinge sections is equal to the ultimate moment capacity in the sections which is calculated using the Mander [18] model for concrete and elastic-plastic model for steel rebar. Materials parameters and RC sections properties are outlined in Table 1.1 and, Table 1.2 respectively.

26 26 Chapter I Analysis of strategic infrastructures in case of blast Concrete strength [MPa] Table 1.1 Materials parameters Concrete strain corresponding to maximum stress Concrete ultimate strain [MPa] Steel yielding design stress [MPa] Steel Young modulus [GPa]

27 27 Chapter I Analysis of strategic infrastructures in case of blast element type Columns beams Table 1.2 RC sections properties Section x direction 1 st and 2 nd floor, entire length + 3 rd floor, bottom section columns 1, 6, 7, 8, 9, 10, 11, 12, 13, 18 y direction 1 st and 2 nd floor, entire length + 3 rd floor, bottom section columns 1, 6, 7, 8, 9, 10, 11, 12, 13, 18 x direction 1 st and 2 nd floor, entire length + 3 rd floor, bottom section columns 2, 3, 4, 5, 14, 15, 16, 17 y direction 1 st and 2 nd floor, entire length + 3 rd floor, bottom section columns 2, 3, 4, 5, 14, 15, 16, 17 x and y directions 3 rd floor, top section + 4 th and 5 th floor, entire length all columns 50 x 30 beams positive bending moment, midspan section + negative bending moment, ends sections 24 x 100 beams positive bending moment, midspan section + negative bending moment, ends sections Tensile rebar [mm 2 ] Compression rebar [mm 2 ] Stirrups [mm] φ8/10 at the end sections φ8/10 at the end sections φ8/10 at the end sections φ8/10 at the end sections φ8/10 at the end sections Not specified Not specified Rebar cover [mm]

28 28 Chapter I Analysis of strategic infrastructures in case of blast 5 th floor 4 th floor 3 rd floor 2 nd floor 1 st floor Assumed protection fence placed at 10 m from the structure Figure 1.4 3D model view Characterization of the uncertainties As mentioned in the methodology, the uncertain quantities of interest in this study are the amount of explosive W and its position with respect to a fixed point within the structure denoted by R. Formally, the vector of uncertain parameters contains two uncertain quantities: θ = { W, R}. The following assumptions are made in order to determine the possible values of θ : The access to the structure is allowed to people at each floor whereas at the first floor, representing an underground garage, the access is

29 29 Chapter I Analysis of strategic infrastructures in case of blast permitted to cars; at the second floor, corresponding to the ground level, a fence system is present at 10 meters from the structural perimeters, providing a stand-off distance for cars and trucks. Consequently, a backpack bomb can occur from the second to the fifth floor of the structure, a car bomb can take place a the first floor and, in addition a car bomb or a truck bomb can occur at a variable point, at 10 meters from the structural perimeter, in correspondence to the second floor. For each simulation realization, the center of explosion is initially evaluated, assuming that with a probability of 30% and 70% explosion occurs within the structure or outside, respectively. Once explosion realization occurs inside the structure, with the same probability it can take place at one of the 5 floors of the building. Then the amount of explosive is defined assuming that it can vary between 15 and 35 kg of equivalent TNT (simulating a backpack bomb) if the explosion takes place within the structure from the second to the fifth floor and between 200 and 500 kg of equivalent TNT (simulating a car bomb) if the explosion occurs at the first floor, corresponding to the underground level; furthermore, in case the explosion occurs outside, at the ground level, the amount of TNT has the 10% of probability to vary between and kg of equivalent TNT (simulating a truck bomb) and the remaining probability to vary between 200 and 500 kg of equivalent TNT. All uncertain quantities are assumed to be uniformly distributed (i.e., the possible values for the uncertain quantity are all equally likely). The process in determining the realization of θ vector is clarified in Figure 1.5.

30 30 Chapter I Analysis of strategic infrastructures in case of blast Blast scenario 30% 70% Explosion takes place inside the structure 20% 20% 20% 20% 20% Explosion takes place outside a 10 m standoff distance from the structure 10% 90% 1 st floor: Car bomb W=200 kg 500 kg 2 nd floor: Backpack bomb W=15 kg 35 kg 3 rd floor: Backpack bomb W=15 kg 35 kg 4 th floor: Backpack bomb W=15 kg 35 kg 5 th floor: Backpack bomb W=15 kg 35 kg Truck bomb W=15,000kg - 25,000kg Car bomb W=200 kg - 500kg Figure 1.5 Blast realization logic tree Characterization of the parameters defining the local dynamic analysis In case of inside explosion, it is assumed that only the columns on the same floor as that of the explosion are affected by it. This assumption is supported by the fact that the columns on the other floors are sheltered from the blast wave by the floor slab system [6]. Therefore, in case of outside explosion, only the external columns directly viewable from the charge location are affected by the explosion, since the internal ones are sheltered by the perimeter walls [6]. Then, for each of the columns hit by the explosion at the distance r from center the center of the charge, given the amount of explosive W, the reduced 3 3 distance Z = r / W is calculated. Once Z = r / W is calculated, a triangular impulse loading is considered acting on the columns (Figure 1.6), whose parameters p o (maximum initial pressure) and t plus (duration of the impulse) can be obtained based on the semi-empirical formulas available in the literature [12].

31 31 Chapter I Analysis of strategic infrastructures in case of blast (t) p0 p(t) p0 t t Figure 1.6 Blast impulse loading It is further assumed that the intensity of the impact loading is uniform across the column height. Furthermore, since such load generally acts in a direction that is not parallel to local axes of the column it is divided into two components and, applying the procedure described in Appendix 1.B, the maxima for bending moment and shear force are evaluated. These values are then used to verify whether the column fails; in particular, in case of bending moment a convex domain criterion is employed, using the following formulation [19]: α α M M (1.22) x y + < 1 M xu M yu where M x, M xu, M y and M yu represents the acting and the ultimate bending moment on both local axes directions, respectively, and α is a parameter defining the convex domain shape, equal to 1.5 in the present case [19]. On the contrary, in order to verify if a shear failure occurs, the acting forces in the two directions are compared separately with their respectively ultimate shears.

32 32 Chapter I Analysis of strategic infrastructures in case of blast Blast fragility A standard MC simulation technique was used to generate 500 blast scenario realizations, assuming that the structure was subjected to its gravity loads and to the 30% of the live loads (equal to 2.0 kn/m 2 ). For each of these realizations, the collapse load factor λ C was calculated. The cumulative distribution function for the load factor denoted by P( λ λc Blast) is plotted for possible values of λ C in Figure 1.7. The threshold value, identifying progressive collapse region, is λ Cth, = [ 1, 2], as marked in Figure 1.7. However, considering a conservative value equal to 2, it can be observed that probability P ( C Blast) that a blast event leads to progressive collapse of the case-study structure is around On the contrary, the value λ C = 4.80 corresponds to the case that none of the columns is eliminated due to the blast; in other words, it is the load factor corresponding to the original structure. This explains why the probability that a blast event leads to a collapse load factor load less than λc 4.80 is equal to unity.

33 33 Chapter I Analysis of strategic infrastructures in case of blast P(λ <= λ C Blast) λ C Figure 1.7 Blast fragility or the cumulative probability distribution for the collapse load factor given blast In order to gain further insight about the simulation results, the blast scenarios leading to progressive collapse, identified by λc 2, are plotted in Figure 1.8 and Figure 1.9.

34 34 Chapter I Analysis of strategic infrastructures in case of blast storey # Figure 1.8 The blast scenarios that lead to progressive collapse in the structure

35 35 Chapter I Analysis of strategic infrastructures in case of blast 6 Back-Pack Car Truck Weight of explosive [kg TNT] Figure 1.9 Weight of explosive in the blast scenarios that lead to progressive collapse in the structure Figure 1.8 illustrates the location of the blast charges that lead to progressive collapse on the structure s original geometry together with the histogram for the storey in which the explosion takes place. This kind of plot is very helpful for identifying the critical zones within which, an explosion could most likely lead to progressive collapse. It can be observed that the collapse scenarios take place predominantly on the ground level (second storey) at selected stand-off distance of 10m from the structure. Figure 1.9 shows the histograms for the quantity of the explosive, distinguishing the three blast categories (backpack bomb, car bomb and truck bomb) which seems to indicate that the event of progressive collapse is not very sensitive to the quantity of explosive, within each blast category (the histograms seem to imply uniform distribution with respect to the quantity of explosive). This is to be expected since the induced blast pressure depends on the explosive quantity by the cubic root, as indicated in [6, 12].

36 36 Chapter I Analysis of strategic infrastructures in case of blast Seismic fragility The seismic fragility for the case-study structure is calculated in two steps. In the first step, a non-linear static analysis is performed on the structure model using SAP2000 (version 10) software based on the concentrated flexural plasticity concept. The pushover curve or roof displacement versus the base shear for the analyzed structure is evaluated and the point at which the first element in the structure reaches its ultimate rotation capacity is determined, according to European seismic guideline [20]. The equivalent elastic-perfectly plastic SDOF system corresponding to the above-mentioned pushover curve is then approximated using a procedure recommended in [20] and is illustrated in Figure The point on the figure that is marked with d MAX corresponds to the first instance when ultimate rotation capacity takes place in the structure. In the second step a suite of 50 ground motion accelerations are applied to the equivalent elastic-plastic SDOF system based on the incremental dynamic analysis or the multiple-stripe analysis procedures [21, 22] Equivalent SDOF System F y 1400 Force [kn] d y d max Displacement [meters] Figure 1.10 The pushover curve for the equivalent SDOF system and the simple elastic-plastic curve fitted to it

37 37 Chapter I Analysis of strategic infrastructures in case of blast Based on these non-linear analysis procedures, the suite of ground motion records are scaled to increasing levels of spectral acceleration and applied to the structure. At each spectral acceleration level, the probability of structural failure was estimated with the ratio of number of records that cause maximum displacement in the equivalent SDOF system that are great than d MAX, to the total number of records (i.e., 50). The result is plotted in Figure 1.11, showing the probability of failure at each spectral acceleration level versus spectral acceleration. Figure 1.11 The seismic fragility curve or the probability of collapse for a given spectral acceleration Discussion on the case study In order to calculate the seismic risk, the fragility should be integrated with the hazard for spectral acceleration at a period close to the fundamental period of the structure. Here the annual rate of exceeding spectral acceleration at T1 = 0.50 has been extracted from Italian National Institute of Geophysics and

38 38 Chapter I Analysis of strategic infrastructures in case of blast Volcanology (INGV) database [23] assuming that the structure in question is located at Naples, Italy. Consequently, the seismic contribution to the total risk 5 (the first term in Equation (1.4)) is calculated and is equal to Therefore, the annual risk of collapse, to compare with the de minimis threshold, can be calculated from Equation (1.4) as follows: 5 ν = ν (1.23) C Blast where the value of P ( C Blast), evaluated with the presented procedure and equal to 0.20, is substituted. As it can be observed from Equation (1.23), also the blast fragility needs to be multiplied by the annual rate ν Blast that a significant blast event takes place. However, as mentioned before, this rate is difficult to evaluate as an engineering quantity and it depends more on the socio-political circumstances and the strategic importance of the structure. Nevertheless, the blast contribution to ν C, in the present case equal to 0.20 ν Blast, can be compared with the seismic contribution (in the present case 5 equal to ) in order to assess its relative importance and determine whether blast poses a significant risk of collapse in the investigated structure. For instance, in case of a non-strategic structure ν Blast can be in the order of 10-7 [1], making blast contribution to the annual risk of collapse negligible, compared with that of earthquake. Alternatively, in case of a strategic structure ν Blast can be as large as 10-4 ; in such case, blast hazard will be the dominant term in Equation (1.4) for calculating the annual risk of collapse. It should be noted that the terms blast fragility and earthquake fragility plotted in Figure 1.7 and Figure 1.11 are calculated differently, although both are referred to as fragility. Blast fragility is defined as the probability of progressive collapse given that a significant blast event has taken place and it needs to be multiplied by the annual rate of significant blast event taking place in order to yield the mean annual risk of collapse. On the other hand, seismic fragility is defined as the probability of structural collapse given a specific value of spectral acceleration and it needs to be integrated with the annual rate

39 39 Chapter I Analysis of strategic infrastructures in case of blast of exceeding spectral acceleration in order to yield the mean annual risk of collapse. Following observations and outcomes can be made: The results for the presented case-study seem to justify the choice of a MC simulation procedure for calculating the probability of progressive collapse. That is, given that a blast event takes place, the probability of progressive collapse is found to be around 20% which is within the range of probabilities calculated efficiently with MC simulation. For example, the 500 realizations generated by conducting the MC simulation herein lead to a reasonably low coefficient of variation in the failure probability estimate (equal to 0.09). The methodology presented herein exploits the particular characteristics of the blast action and its effect on the structure in order to achieve maximum efficiency in the calculations. More specifically, the use of plastic limit analysis (formulated as a linear programming problem) instead of common 3-D finite element analysis renders the calculations significantly more rapid and thereby feasible for implementation within a simulation procedure. Moreover, the derivation of an analytic closedform solution for the problem of dynamic impulse facilitates the damage analysis of individual structural elements for each simulation realization. The efficiency and rigor of the presented methodology make it particularly useful as a design and/or retrofit tool for strategic structures. More specifically, the outcome of the MC simulations can be used to mark the location of critical blast scenarios on the structural geometry and identify the risk-prone areas. An example of a simple and effective prevention strategy would be to limit or to deny the access to critical zones within the structure, once they are identified using the presented procedure.

40 40 Chapter I Analysis of strategic infrastructures in case of blast Once the annual rate of blast ν Blast is known, the blast fragility P( C Blast) evaluated herein can be used to determine the annual risk of collapse ν c (Equation (1.4)). Moreover, it should noted that the methodology presented herein for assessment of a case-study RC structure can be extended in order to evaluate the vulnerability of a class of structures, located in a seismic zone, against blast-induced progressive collapse (i.e. masonry buildings, steel-frame buildings, RC bridges).

41 41 Chapter I Analysis of strategic infrastructures in case of blast Appendix 1.A: Analysis of un-damped free vibration for fixedend beam flexure Consider that the beam mass and stiffness properties are constant across its length and are represented by EI and m. The free-vibration equation of motion can be written as [13]: 4 2 vxt (, ) vxt (, ) (1.24) EI + m = x t One form of solution for this equation can be obtained by separation of variables, namely: vxt (, ) = φ( xyt ) ( ) (1.25) Where φ ( x) is the free-vibration shape and Yt () is the time-dependent amplitude. Thus, Equation (1.25) yields two ordinary differential equations: Yt φ 2 () + ω Yt () = 0 iv 4 ( x) a φ( x) = 0 (1.26) (1.27) In which: 4 2 aei ω m (1.28) The second equation can be solved by introducing a solution in the form: φ( x) Gexp( st) (1.29) which leads to the following solution:

42 42 Chapter I Analysis of strategic infrastructures in case of blast φ( x) A cos ax + A sin ax + A cosh ax + A sinh ax (1.30) These constants can be derived by satisfying the known boundary conditions at the ends of the beam. For the case of fixed end beam, the following displacement and slope restraints need to be satisfied at the beam boundaries: φ(0) = φ( L) = 0 φ (0) = φ ( L) = 0 (1.31) Which leads to: A + A = 0, A + A = 0 A A(cos al cosh al) cos al cos al sin al+ sin al, sin al sinh al sin al sinh al cos al cos al 1 2 = = (1.32) The last Equation is solved numerically leading to the following solution: EI (1.33) al = 4.73, ω = ( ) L m cos al cos al cos al cos al φ = A1 (cos ax sin ax cosh ax + sinh ax) sin al sinh al sin al sinh al Where al = 7.73 is the smallest positive value that satisfies equation (1.33) and ω is the circular frequency for the first mode-shape. The first mode-shape φ 1 can be obtained by substituting the first-mode frequency ω in Equation (1.33) (Figure 1.12).

43 43 Chapter I Analysis of strategic infrastructures in case of blast Figure The first-mode shape for the fixed-end beam x/l Appendix 1.B: Analysis of Impact loading for fixed-end beam Assuming that the coefficients A of the fixed-end beam first mode-shape are set so that the φ 1 = 1 at x/ L= 0.50[13]: v(0.5, t) = Y( t) (1.34) and vxt (, ) = φ v(0.5, t) (1.35) 1 Assuming that the impulse loading is approximated by a triangular time-decay function, characterized by two parameters p 0 and t plus, and distributed uniformly over the length of the beam, the equation of motion can be written as:

44 44 Chapter I Analysis of strategic infrastructures in case of blast 4 2 vxt (, ) vxt (, ) t EI + m = p (1 ) t t EI plus x t tplus vxt (, ) vxt (, ) + m = 0 t > t 4 2 x t 4 2 plus (1.36) Which is divided in two phases: the first phase ( t t plus ) is the impact forcedvibration part and the second phase is a free vibration response. In order to calculate the forced-vibration time-history response, Equation (1.36) is calculated at x/ L= 0.50, using Equations (1.34) and (1.35): t EI Yt myt mω Yt myt p (1.37) L 2 ( ) + ( ) = ( ) + ( ) = 0(1 ) t tplus tplus It can be demonstrated that the particular solution to the equation of motion (1.37) is: ( ) p (1 t Y ) t t p t 0 = 2 plus (1.38) mω tplus Combining this solution with the free vibration response and evaluating its constants so that they satisfy the zero initial boundary conditions, the general solution to the differential equation of motion is obtained as following: p 0 1 t Yt ( ) = Yg( t) + Yp( t) = sin t cos t 1 t t 2 ω ω + plus mω ωtplus t (1.39) plus The displacement and velocity at the end of the forced-vibration phase t t ) will serve as initial conditions for the free vibration phase ( t > tplus ): ( plus

45 45 Chapter I Analysis of strategic infrastructures in case of blast p sinωt 0 plus Yt ( plus ) = cos 2 ωtplus mω ωt plus (1.40) p Yt ( plus ) = cosωt sin 2 plus + ω ωtplus mω tplus t plus It turns out that the impulse duration is much smaller than the natural period of vibration of the SDOF system; therefore, the maximum response will most likely be in the free-vibration response phase. That is, the response Y (t) will be of the following form: Y ( t) = Y( t (1.41) Where Y t ) and Y t ) are the displacement and velocity initial ( plus ( plus conditions for the free-vibration response evaluated at the end of the triangular impulse loading and shown in Equation (1.40). Thus, the maximum response will be equal to: (1.42) Furthermore, the bending moment and the shear in the beam can be calculated as: M = EIν ( x, t) = EI Y( t) φ ( x) (1.43) V = EIν ( x, t) = EI Y ( t) φ ( x) It can be shown (by calculating the second and the third derivative of φ ) that the maxima take place at the beam ends and is equal to: M V ρ = max max Y ( t plus plus 4.73 = 1.26 L 4.73 = 1.24 L Y ( t plus ) )cosωt + sinωt ω ) + ω Y ( t ) 2 plus 3 2 EIρ EIρ 2 (1.44) (1.45)

46 46 Chapter I Analysis of strategic infrastructures in case of blast Where ρ is the maximum value for Yt ( ) calculated in Equation (1.42). Appendix 1.C: Linear Programming problem A linear programming problem can be expressed as following [14]: Minimize subject to c T x A x = b x 0 (1.46) where x is the vector of variables to be solved, A is a matrix of known coefficients and c and b are vectors of known coefficients. c T x represents the objective function whereas the equation A x = b represents the constraint condition; the problem is defined standing the further condition that x 0. Appendix 1.D: The Coefficient of the Variation (COV) of the Mean of a Bernoulli Variable Suppose that the variable I can assume only two possible values{ 0,1} and that the parameter p [ 0,1] is used to describe the probability that I is equal to unity. Therefore, I will be equal to zero with the probability 1 p. The expected value and the variance for variable I can be calculated as: 2 E[ I] = E[ I ] = p 1+ (1 p) 0 = p Var[ I] = E[ I ] E[ I] = p p = p(1 p) Therefore, the expected value and the variance of the mean of a the variable I over N observations can be calculated as: (1.47)

47 47 Chapter I Analysis of strategic infrastructures in case of blast N E[ Ii ] i= 1 Np E[ I] = = = p N N 1 Var[ I] = p(1 p) N Finally the COV of the expected value for I is calculated as: σ[ I] COV I = = E[ I] 1 p(1 p) N = p 1 p Np (1.48) (1.49) REFERENCES 1 Ellingwood, B. R., Mitigating Risk from Abnormal Loads and Progressive Collapse, Journal of Performance of Constructed Facilities, Vol. 20, No 4, November 2006, pp Pate-Cornell, E., Quantitative safety goals for risk management of industrial facilities, Structural Safety Vol 13 No 3, 1994, pp Galambos, T. V., Ellingwood, B., MacGregor, J. G. and Cornell, C. A., Probability based Load Criteria: Assessment of Current Design Practice, Journal of Structural Division, Proceeding of the American Society of Civil Engineers, Vol 108, No ST5, May 1982, pp Ellingwood, B., MacGregor, J. G., Galambos, T. V. and Cornell, C. A., Probability based Load Criteria: Load Factors and Load Combinations, Journal of Structural Division, Proceeding of the American Society of Civil Engineers, Vol 108, No ST5, May 1982, pp NIST/GSA Workshop on Application of Seismic Rehabilitation Technologies to Mitigate Blast-Induced Progressive Collapse, Eds Nicholas J. Carino and H.S. Lew, September 10, 2001, Oakland, CA

48 48 Chapter I Analysis of strategic infrastructures in case of blast 6 Departments of the Army, the Navy and the Air Force - USA, TM Structures to resist the effects of accidental explosions, November 1990, 1796 pp. 7 Allen, D. E., Schriever, W. R., Progressive Collapse, abnormal loads and building codes, Division of Building Research Council, Québec, ASCE/Structural Engineering Institute (SEI), Minimum design loads for buildings and other structures, ASCE/SEI 7, Reston, Va, General Services Administration (GSA), Progressive collapse analysis and design guidelines for new federal office buildings and major modernization projects GSA, Washington, D.C., Ellingwood, B. R. and Leyendecker, E. V., Approaches for Design against Progressive Collapse Journal of Structural Division, Vol. 104, No. 3, 1978, pp National Research Council (NRC), Protecting people and buildings from terrorism, Committee for Oversight and Assessment of Blast-effects and Related Research, National Academy Press, Washington, D.C., Henrych, J, The Dynamics of Explosion and its Use, Elsevier, 1979, pp Clough, R. W. and Penzien, J, Dynamics of Structures, McGraw-Hill, 1993, pp Baker, J. and Heyman J., Plastic Design of Frames: Fundamentals, Cambridge University Press, 1969, pp Grierson, D. E. and Gladwell, G. M. L., Collapse Load Analysis using Linear Programming, Journal of Structural Division, Proceeding of the American Society of Civil Engineers, Vol 97, No ST5, May 1971, pp Dantzing, G. B. and Thapa, M. N., Linear Programming: 1: Introduction, Springer, 1997, pp Ruth, P., Marchand, K. A. and Williamson, E. B. Static Equivalency in Progressive Collapse Alternate Path Analysis: Reducing Conservatism While Retaining Structural Integrity, Journal of Performance of Constructed Facilities, Vol. 20, No 4, November 2006, pp

49 49 Chapter I Analysis of strategic infrastructures in case of blast 18 Mander, J.B., Priestley, J.N. and Park, R. Theoretical Stress-Strain Model for Confined Concrete Journal of Structural Engineering, Vol 114, No. 8, August 1988, pp Bresler, B. Design Criteria for reinforced columns under axial load and biaxial bending, Journal of the American Concrete Institute, Farmington Hills, Mi, November 1960, pp EN 1988 Eurocode 8 - Design of structures for earthquake resistance, CEN, Vamvatsikos, D and Cornell, C. A., Incremental Dynamic Analysis, Earthquake Engineering and Structural Dynamics, Vol. 21m No. 3, 2002, pp Jalayer, F., Direct probabilistic seismic analysis: implementing non-linear dynamic assessments PhD Thesis, Department of Civil Environment Engineering, Stanford University, 2003, pp Progetto INGV-DPC S1, Proseguimento della assistenza al DPC per il completamento e la gestione della mappa di pericolosità sismica prevista dall Ordinanza PCM 3274 e progettazione di ulteriori sviluppi., 2007,

50 50 Chapter II Dynamic properties of concrete Chapter II DYNAMIC PROPERTIES OF CONCRETE INTRODUCTION TO THE FRAMEWORK ACTIVITIES The activities presented in this Chapter are part of a wide research project, namely the Tenza project. The objective of the Tenza project is to study the effect of high dynamic loads on a reinforced concrete (RC) arch bridge, called Tenza, in southern Italy, part of the abandoned path of the Salerno-Reggio Calabria highway. The bridge was built in the 1960s and retrofitted in the 1990s. Four types of materials can be identified in the Tenza Bridge structure: the original concrete used in the 1960s when the bridge was built; this accounts for most of the concrete in the structure; the concrete used to strengthen the piers and arches; the original lightly ribbed steel used for the reinforcement of the original concrete; the new ribbed steel used as reinforcement of the more recent RC portions. In the first phase of the project, the structure was characterized through static analysis under gravity and live loads, and a complete seismic assessment. The Finite Element Method (FEM) model used to perform these analyses was validated by comparing the numerical vibration modes with the results of a vibrodyne test [1]. The objective of the second phase of the project was to

51 51 Chapter II Dynamic properties of concrete perform an assessment of the structure under severe dynamic loads, through numerical analysis and in situ tests. For this purpose complete dynamic characterization of the materials was performed. High strain-rate failure tests were conducted at the DynaMat Laboratory of the University of Applied Sciences of Southern Switzerland, using three modified Hopkinson bars. Stress strain relationships under different strain rates were evaluated under tensile loads for the materials of the bridge. The obtained results are fundamental to define the influence of dynamic loads on the constitutive behavior of materials aged in a real structure and will provide a legal point of reference to be used in structural analyses accounting for strain-rate effects. Indeed, knowledge of the dynamic behavior of both structure and materials is essential to perform a complete assessment of the bridge under impact or blast loads. In particular, under such load conditions two different types of failure can be distinguished [2]: Local failure; Global failure. The former can be due to an impact or an explosion occurring close to structural elements; the characteristics of this failure depend on the dynamic properties and ductility of the element concerned and its constituent materials. By contrast, global failure occurs after local failure and it is related to the ability of the structure to withstand the loss of elements without activating progressive collapse. It depends on global ductility properties of the structure and on the quality and frequency of connections between its elements. Obviously the more severe the local failure, the more likely global failure is to occur. Local failure can be distinguished into local failure of materials and local failure of structural elements. The first type of failure occurs when the explosion is so close to the structure that the consequent shock wave in the air, impacting on the surface of the element, causes a high field of compression and propagates a tensile wave inside the material. Hence, these stresses can cause the concrete to crack and consequent projection of debris. By contrast, the

52 52 Chapter II Dynamic properties of concrete second type can occur when the failure of one or more sections within the element is activated [3]. Instead, global failure takes place after severe damage to one or more structural elements, which can lead to progressive collapse of the structure. Hence, the possibility of this failure mechanism is linked to the capacity of the structure to redistribute loads on other structural elements and it depends on redundancy of the elements and ductility of connections [4]. Various experimental activities have been carried out on the tensile dynamic behavior of concrete in recent decades [5], but very few investigations have been performed on existing concrete taken from a real structure. [6], mainly examining low strain rate ranges induced by seismic loading conditions. The main significance of this research is therefore related to the evaluation of the tensile dynamic properties of an existing concrete under high strain rates, to confirm what expected according to available literature. Comparisons were thus performed between the obtained strength data and the existing tensile strength DIF (Dynamic Increase Factor) strain rate formulations, where DIF is defined as the ratio of the dynamic value of a mechanical parameter over its corresponding static value. The results confirmed that the available relationships provide good predictions of the mechanical behavior of concrete under dynamic loads, even if it is taken from an existing structure. The Tenza Bridge The Tenza bridge (see Figure 2.1) in southern Italy was built in the 1960s as part of the Salerno-Reggio Calabria highway and was open to traffic until a few years ago. Indeed, recently ANAS, the Italian road agency which owns the bridge, planned to change the geometry of the route, since it no longer respects current safety standards. Therefore, the bridge was closed to traffic as it belonged to a replaced section of the highway.

53 53 Chapter II Dynamic properties of concrete Figure 2.1 Tenza bridge view. The bridge structure of the Tenza viaduct consists of three different structures: a central superior way arch bridge and two approach viaducts. The arch structure is 120 m long and 40 m deep, while each arch approach ramp is about 30 m long. The bridge deck consists of a ribbed solid slab and is supported by multiple wall piers of different heights. Each individual pier is made of two RC columns; the external columns are connected over their entire height by an RC wall. In the 1990s the structure was strengthened by RC jacketing of the piers and widening the arch section. Traditional assessment and characterization The aim of the first phase of the project was to obtain detailed knowledge of bridge properties and structural behavior prior to a traditional investigation. First of all the actual bridge geometry was evaluated by analyzing the original drawings and acquiring data with a 3D Laser Scanner survey; a static characterization of materials was then conducted. After this phase, an FEM

54 54 Chapter II Dynamic properties of concrete model was built. Subsequently modal analysis was performed and validated through a vibrodyne in situ test, to ensure the reliability of the FEM model. Finally, static analysis under gravity and seismic loads was conducted to perform a traditional assessment of the bridge. Detailed results of this phase may be found in [1]. STATIC CHARACTERIZATION OF CONCRETE The aim of this test activity was to obtain a complete quasi-static characterization of the concrete of the Tenza bridge. As already explained, two types of concrete can be identified in the structure: the concrete used in the 1960s when the bridge was built and the concrete used for piers and arch strengthening. For both of them several specimens were taken from the real structure and compression tests were performed. Concrete cores were collected in order to achieve a representative characterization of the whole structure and then accounting for the spatial homogeneity of mechanical properties of concrete. With this aim, specimens were taken from arch and piers in regions close to the base and midspan of the bridge, and from the upper deck, at three different points of the carriageway. The specimens were cylinders with a diameter of either 75 mm (2.95 in.) or 100 mm (3.94 in.) and variable length. Compression test data were processed using the following European Standard relationship [5]: R cub = Rmeasured * D ( ) λ (2.1) where: R cub = compressive cubic strength; D = 2.3 for vertical extraction and D = 2.5 for horizontal extraction; λ = Height/Diameter

55 55 Chapter II Dynamic properties of concrete Multiplying cubic strength by a coefficient of 0.83, the value of the cylindrical strength was then calculated. The processed data (Table 2.1) show that the original concrete, especially in the deck, has a higher average compressive cylindrical strength compared to that of the strengthening concrete. This is probably due to long curing of about 40 years of the old concrete. Extraction zone of specimens Table 2.1 Static compression tests results Average cubic strength, [MPa] Average cylindrical strength, [MPa] Deck (original concrete) Arch (original concrete) Arch (strengthening concrete) Piers (original concrete) Piers (strengthening concrete) DYNAMIC TESTS ON CONCRETE Introduction to the dynamic properties of concrete The differences between dynamic and static properties of concrete are amply illustrated in the literature [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. It is accepted that, to evaluate correctly the behavior of concrete under extreme dynamic loads, dynamic mechanical properties have to be necessarily investigated. The scientific data available show that, under high strain rates, concrete can exhibit, both in compression and in tension: an increase in failure strength [6]; an increase in ultimate stress [6]; an increase in Young s modulus [7]; a different evolution of cracks, that do not develop through a local mechanism, like in a static range, but start and grow at the same time in several locations [8]. Moreover, an increase in strength is exhibited, both in compression and in tension, for flexural loads [6, 9].

56 56 Chapter II Dynamic properties of concrete These characteristics are explained by several technical codes or instructions, so as to guide the engineer to predict properly the real behavior of a structure under extreme loads. CEB information bulletin no. 187 [10] gives formulations to evaluate the dynamic properties of concrete by updating static properties. DIF strain rate relationships are suggested for compressive and tensile failure stresses, compressive and tensile ultimate strains and Young s modulus. TM [3] is a technical document that gives specific guidelines for structures subjected to blast loads. It has a very direct approach to structural design and gives detailed indications for several design situations of different structures under explosions. The issue of the dynamic properties of concrete is considered by suggesting different DIF for failure strength in several load conditions. In this case, the DIF is not a function of strain rate value, as expressed by CEB formulations, but it just depends on the distance from the blast source, distinguishing between far and close-in explosions. DynaMat facilities In order to determine the mechanical properties of concrete under high loading rates a dynamic test activity was conducted at the DynaMat Laboratory of the University of Applied Sciences of Southern Switzerland (SUPSI) of Lugano. The laboratory is equipped with four Modified Hopkinson Bar (MHB) [11] apparatuses 15 m in length ( s-1) with diameters of 60 mm for concretes and rocks, 10 and 12 mm for metals, ceramics, glass, and 20 mm for polymers, FRP, mortars etc.; a Hydro-Pneumatic Machine ( s-1) and two universal machines for quasi-static tests. Set-up for high strain rate tests Concrete specimens were cylindrical, with both diameter and height of 60 mm. The system used to perform the tests consists of two circular aluminium bars, with a length of 3 m and a diameter of 60 mm. The specimens were positioned between the two bars and glued to them using an epoxy resin (see Figures 2.2, 2.3). During the test each bar was equipped with semiconductor strain gauges in order to obtain measurements of the incident, reflected and transmitted

57 57 Chapter II Dynamic properties of concrete pulses acting on the cross section of the specimens during the test. The concrete specimen was also equipped with a strain gauge. 1. hydraulic actuator; 2. high strength steel bar for energy storage (3m); 3. blocking device; 4. input bar; 5. strain gauges to measure incident and reflected pulses; 6. specimen; 7. strain gauges to measure transmitted pulses; 8. output bar. Figure 2.2 Experimental set-up

58 58 Chapter II Dynamic properties of concrete Figure 2.3 Concrete specimen in a Hopkinson bar The phases of an MHB tensile test can be then summarized as follows: A hydraulic actuator, of maximum loading capacity of 1 MN, pulls a highstrength Maraging steel bar with a length of 3 m and a diameter of 35.8 mm; the pretension stored in this bar is resisted by the blocking device (Figure 2.2) linked to the first of the aluminium bars, called input bar. Then the fragile bolt in the blocking device fails, giving rise to a tensile mechanical pulse of 1200 ms duration, with a linear loading rate during the rise time, propagating along the input bar and then transmitted, through the concrete specimen, to the second aluminium bar, called the output bar. Then pulses acting on the specimen determine its failure (Figure 2.4).

59 59 Chapter II Dynamic properties of concrete 8000 incident pulse Input pulse Output pulse 6000 pulse [N] 4000 reflected pulse 2000 transmitted pulse time [s] Figure 2.4 Acquired signal measurement by input and output bar strain-gauge stations Aluminium was chosen as the bar material because of its acoustic impedance, which is not far from that of plain concrete. This minimizes the constraint to transverse deformation of the concrete specimen that depends on the ratio between the Poisson and Young modulus. The strain gauge station on the input bar measures the incident pulses e I and the reflected pulses e R. The strain gauge station on the output bar measures the pulses e T transmitted through the specimen. From the measurement of the reflected and transmitted pulse the stress and strain history is obtained using the formulation of the Hopkinson theory [12]: σ A A 0 () t = E ε () t 0 T (2.2)

60 60 Chapter II Dynamic properties of concrete 2 C t 0 ε () t = R ()dt t L ε 0 2C ( t) = L ε 0 ε R () t (2.3) (2.4) where: E 0 is the elastic modulus of the bars; A 0 their cross-sectional area; A is the specimen cross-sectional area; L is the specimen length; C 0 is the sound velocity of the bar material. The tests were carried out using an AlazarTech device for data acquisition, which has six ATS330 channels. The system is designed to provide high precision (12 bit) waveform acquisition and analysis capabilities with a maximum 50 MS/s real-time sampling rate, 25 MHz full power bandwidth and acquisition memory of 128K samples per channel. Each individual board has its own independent trigger circuitry and time base, such that record acquisition can be carried out with different time windows. The data acquisition system receives signals from the Pacific Instrument 6120 Programmable Transducer Amplifiers. This is an expandable signal conditioning system which allows the user to configure each individual signal conditioner from any host computer with IEEE-488 or RS-232 capability. Individual channels may be addressed, or an all-channel command will address all channels simultaneously for fast initial set-up. Figure 2.5 presents the logical architecture of the experimental measurements and recordings. During the experimental activities the specimens were divided into 4 categories: The original arch concrete; The arch strengthening concrete; The pier original concrete; The pier strengthening concrete.

61 61 Chapter II Dynamic properties of concrete For each, a static tensile test and several dynamic tensile tests were performed, as presented in table 2.2. Table 2.2 Results of dynamic tensile tests Extraction zone of specimens Arch (original concrete) Arch (strengthening concrete) Piers (original concrete) Piers (strengthening concrete)* Strain rate dε/dt, [s -1 ] f ct, [MPa] ε u, [με] Fracture Energy, [J/m 2 ] Quasi static Soft impact Hard impact Quasi static Hard impact Quasi static Soft impact Hard impact Quasi static Soft impact Hard impact Hard impact f ct = fracture stress; ε u = ultimate strain * the tests were carried out on small specimens h/d=1 d=20mm

62 62 Chapter II Dynamic properties of concrete Figure 2.5 Logical architecture of the experimental measurements Test results Direct data acquisition is represented by two strain time curves provided by strain gauge sensors positioned on input and output bars. Knowing the elastic properties of bars, they can be related to pulses moving through the system composed by an input bar, specimen and output bar. Figure 2.4 shows an example of the transformation of acquired data into forces acting on the system. Moreover, using strain time curves read by sensors on input and output bars through equations (2.2) and (2.3), stress-time and strain time curves can be evaluated for concrete specimens. Then, using time as a common parameter of the two last curves, stress-strain relationships can be processed for the specimen at the strain rate given by equation (2.4). Importantly, during such tests, the strain rate has no constant value. Hence, definition of the strain rate value for each test is not immediate. Frequently, in the literature the maximum value of strain rate is chosen as the strain rate level.

63 63 Chapter II Dynamic properties of concrete Here a different approach was preferred, defining the strain rate level as the strain rate obtained at the time when there was maximum stress. An example of stress-strain curves is shown in Figure 2.6 while Figure 2.7 depicts different results in terms of stress-time curves for the same strain rate and the same concrete type, allowing appreciating the results dispersion. Figure 2.8 shows a concrete specimen after failure stress [MPa] specimen_20mm (0.79 in.) strain Piers new concrete specimen tested at a strain rate of 52 s -1 : fracture strain: ε u =654 με; fracture stress: f ct =5.43 MPa Figure Example of tensile stress vs. strain acquisition

64 64 Chapter II Dynamic properties of concrete 8 stress [MPa] Test 1 Test 2 Test 3 Test 4 Test time [s] Figure 2.7- Comparison between stress-time curves for concrete specimens of the same extraction zone at the same strain rate (10 s -1 ) Figure 2.8- Concrete specimen after failure

65 65 Chapter II Dynamic properties of concrete The main results are shown in Figures 2.9. They represent the main tensile stress-strain curves of arch and piers concrete at different strain rates. It can be observed that tensile strength, ultimate strain and total deformation energy (evaluated as the area under the curves) increase as the strain rate increases. These data are summarized in Table 2.2. Interestingly, for the concrete of original portions, even though the specimens were taken from a forty-year-old existing structure, the observed trend is consistent with the results described in the literature [6, 8, 10, 13, 14, 15, 16, 17]. 8 stress [MPa] specimen_60mm strain-rate = 1 s strain Figure 2.9a- Stress-strain curves of arch original concrete at a strain rate of 1 s -1

66 66 Chapter II Dynamic properties of concrete 8 7 specimen_60mm strain - rate = 10 s - 1 stress [MPa] strain Figure 2.9b- Stress- strain curves of arch original concrete at a strain rate of 10 s specimen_60mm ( 2.36 in. ) strain - rate = 10 s stress [MPa] strain Figure 2.9c- Stress-strain curves of pier original concrete at a strain rate of 10 s -1.

67 67 Chapter II Dynamic properties of concrete 15 specimen_20mm strain-rate = 50 s stress [MPa] strain Figure 2.9d- Stress-strain curves of pier strengthening concrete at a strain rate of 50 ASSESSMENT OF EXPERIMENTAL RESULTS To appreciate correctly the influence of the strain rate on the mechanical properties of concrete the results in terms of failure strength were elaborated to obtain the DIF., thereby obtaining the curves in Figures 2.10.

68 68 Chapter II Dynamic properties of concrete Arch (original concrete) CEB Malvar Experimental data DIF Strain rate Figure 2.10a- DIF for failure stress of arch original concrete Arch (strengthening concrete) CEB Malvar Experimental data DIF Strain rate Figure 2.10b- Failure stress DIF for arch strengthening concrete

69 69 Chapter II Dynamic properties of concrete Piers (original concrete) CEB Malvar Experimental data 3 DIF Strain rate Figure 2.10c- Failure stress DIF for pier original concrete 10 8 Piers (strengthening concrete) CEB Malvar Experimental data 6 DIF Strain rate Figure 2.10d- Failure stress DIF for pier strengthening concrete

70 70 Chapter II Dynamic properties of concrete Moreover, in order to verify that the obtained data were consistent with results described in the literature, a comparison with existing theories was performed. As points of reference, the CEB [10] and Malvar expressions [13] were considered. Both these formulations seek to predict dynamic tensile strength, expressing the DIF as a function of the strain rate; the CEB formulation expresses the DIF as: 1.016δ f d ε DIF = = for ε 30s fs ε f d ε DIF = = η for ε > 30s fs ε (2.5) (2.6) where f d f s ε is the dynamic strength; is the static strength; is the strain rate; ε 0 is a constant equal to s -1 and has the meaning of the static strain rate; and log η = 7.11 δ 2.33, 1 δ = f cs where f0, f cs is the static compression strength; f 0 is a constant equal to 10 MPa. This formulation gives the DIF as a bilinear function of the strain rate in the logarithmic scale, and presents a slope variation at 30 s -1. The following Malvar formulation, instead, expresses the DIF in a similar way but fixes the reference static strain rate at 10-6 s -1 and moves the slope variation to 1 s -1.

71 71 Chapter II Dynamic properties of concrete M f d ε DIF = = for ε 1s fs ε 0M f d ε DIF = = β for ε > 1s fs ε 0M δ (2.7) (2.8) where fd, fs,ε have already been expressed above; and ε 0M is a constant equal to 10-6 s-1; log β 6δ 2 = M 1 δ M = f 1+ 8 cs where f0, f cs, f 0 have already been defined above; By plotting these relationships against the experimental results (Figures 2.10) the correspondence with the data obtained can then be appreciated. This shows that, as Malvar formulation suggests modifying CEB relationship, the slope variation is significant at 1 s -1, even though experimental data are not available for strain rates higher than 10 s -1. In any event, the CEB expression often underestimates or, at the very least, slightly overestimates the experimental results while the Malvar expression sometimes predicts a higher DIF than the test value. Hence the CEB expression appears more suitable for designing calculations where more conservative values should be considered. The data are summarized in Table 2.3

72 72 Chapter II Dynamic properties of concrete Table 2.3 Experimental and numerical DIF of the tensile strength Extraction zone of specimens Arch (original concrete) Arch (strengthening concrete) Piers (original concrete) Piers (strengthening concrete) Strain rate dε/dt, [s -1 ] Experimental values DIF of the tensile strength CEB expression* Malvar expression* Soft impact (+25%) 1.63 (+33%) Hard impact Hard impact (-1%) 3.51 (+110%) (-14%) 3.67 (+86%) Soft impact (+21%) 1.65 (+30%) Hard 10 impact (-31%) 3.55 (+46%) Soft impact (-17%) 2.95 (+46%) Hard 50 impact (+5%) 6.37 (+212%) Hard 70 impact (-40%) 7.12 (+78%) * the terms in parentheses represent the percent differences between the numerical and the experimental values The conclusions may be summarized as follows: our tests confirmed that the concrete of an old structure is also strain rate sensitive, as is widely pointed out concerning new concrete; both CEB and Malvar DIF strain rate formulations for tensile strength predict the actual behavior of concrete well; the CEB expression provides slight underestimates, while the Malvar expression overestimates the experimental data in many cases; therefore the CEB expression appears sounder for the purpose of design assessment; although both Malvar and CEB expressions give the DIF strain rate relationship as a bilinear function in logarithmic scale, they consider different positions for the slope variation; based on the tests performed

73 73 Chapter II Dynamic properties of concrete the variation in gradient occurs at 1 s -1, as predicted by the Malvar formulation; These results could provide a basic tool for engineers involved in assessing RC structures with respect to extreme loads. REFERENCES 1 Asprone, D., Cosenza, E., Manfredi, G., Occhiuzzi, A., Prota, A. and Devitofranceschi, A. Caratterizzazione dinamica di strutture da ponte: il progetto Tenza. National Conference Proceedings Sperimentazione su Materiali e Strutture December 2006, Italy. pp ; 2 Winget, D. G., Marchand, K. A. and Williamson, E. B. Analysis and Design of Critical Bridges Subjected to Blast Loads Journal of Structural Engineering, Vol. 131, No. 8, August 2005 pp Departments of the Army, the Navy and the Air Force - USA, TM Structures to resist the effects of accidental explosions, November 1990, 1796 pp. 4 NIST/GSA Workshop on Application of Seismic Rehabilitation Technologies to Mitigate Blast-Induced Progressive Collapse September 2001 Oakland, CA 5 Fu, H.C., Erki, M.A. and Seckin, M. Review of effects of loading rate on concrete in compression, Journal of Structural Engineering vol 117, 1991, pp Harris, D.W., Mohorovic, C.E., Dolen, T.P. Dynamic Properties of Mass Concrete Obtained from Dam Cores ACI Materials Journal, Vol.97, 2000, pp EN , (British Standard 1881) Testing Concrete, part 120 Methods for determination of the compressive strength of concrete cores, EN :2004 Eurocode 2. Design of concrete structures. General rules and rules for buildings, 2004, 230 pp.

74 74 Chapter II Dynamic properties of concrete 9 Cadoni, E., Solomos, G., Berra, M. and Albertini, C. High strain-rate behaviour of plain concrete subjected to tensile and compressive loading Proc. of 3rd Int. Conf. on Construction Materials: Performances, Innovations and Structural implication August Vancouver, Canada. 10 Cadoni, E., Labibes, K., Berra, M., Giangrasso, M. and Albertini, C., High strain-rate tensile behaviour of concrete, Magazine of Concrete Research, vol. 52, No.5, Oct. 2000, pp Comité Euro-International du Béton. Concrete structures under impact and impulsive loading CEB Bulletin 187, Lausanne, Switzerland, Malvar L.J. and Ross C.A. Review of strain-rate effects for concrete in tension ACI Materials Journal vol. 95, 1998, pp Cadoni, E., Albertini, C., Labibes, K. and Solomos, G. Behavior of plain concrete subjected to tensile loading at high strain-rate. Proc. of 4th Int. Conf. on Fracture Mechanics of Concrete and Concrete Structures FRAMCOS-4, Cachan, France, 2001 pp Cadoni, E., Albertini, C., Solomos, G. Analysis of the concrete behavior in tension at high strain-rate by a modified Hopkinson bar in support of impact resistant structural design, Journal de Physique, Vol.3, 2006, pp Albertini, C., Cadoni, E. & Labibes, K. Mechanical characterization and fracture process of concrete at high strain-rates in proc. of 2 nd Int. Conf. on Concrete under Severe Conditions, - CONSEC 98, Tromsø, Norway, 1998, pp Toutlemonde, F., Boulay, C. and Rossi, P., High strain-rate tensile behaviour of concrete: significant parameters Proc. of 2nd Int. Conf. on Fracture Mechanics of Concrete and Concrete Structures FRAMCOS-2, 2005 Freiburg, Germany 17 van Doormal, J.C.A.M., Weerheijn, J. and Sluys, L.J. Experimental and numerical determination of the dynamic fracture energy of concrete Journal de physique vol. 4, 1994, pp

75 75 Chapter III Dynamic properties of reinforcing steel Chapter III DYNAMIC PROPERTIES OF REINFORCING STEEL INTRODUCTION TO FRAMEWORK ACTIVITIES The activities presented in this Chapter were conducted, within the Tenza project, briefly described in previous paragraphs. In particular, in this Chapter the results of the dynamic characterization of the reinforcing steel in the structural elements of the bridge are presented and discussed. Reinforcing steel role in RC behavior under dynamic conditions is fundamental. In fact, energy dissipation mechanisms are essential under severe dynamic loading patterns and ductility contribution provided by steel needs to be correctly investigated. Therefore, analyzing how steel mechanical properties change from static to dynamic conditions represents an essential topic in order to correctly evaluate the plasticity capabilities of RC members under extreme dynamic loads. In fact, as experienced with many other metallic materials, dynamic mechanical behavior of steel presents significant differences if compared with mechanical behavior exhibited under static load conditions. This is due to several phenomena involved in steel strain-rate sensitiveness, but mainly the reason of such differences stands in the dynamic dislocations evolution, affecting the microscopic scale [1, 2, 3]. Available scientific data outline that, as strain rate

76 76 Chapter III Dynamic properties of reinforcing steel increases, the following changes in mechanical properties of reinforcing steel can be noticed [1, 2, 3, 4, 5]: an increase of yielding stress f y ; an increase of ultimate tensile stress f t ; an increase of the ultimate tensile strain ε t ; On the contrary no changes are experienced in terms of Young modulus. Unfortunately, available literature focuses the attention on dynamic properties of several steel alloys, mainly for industrial applications [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], whereas few data are available on reinforcing steel properties [4, 5, 17]. Therefore, further investigations on reinforcing steel appear necessary to evaluate the real dynamic behavior of RC members, in case of impact or blast loads. EXPERIMENTAL ACTIVITY In the present activity dynamic tensile failure tests were conducted on steel from Tenza bridge rebar and are here presented. Steel bars of 18mm of diameter were collected at the basis of the arch and specimens were obtained using automatic lathe. The geometry of the specimens is depicted in Figure 3.1.

77 77 Chapter III Dynamic properties of reinforcing steel Figure 3.1 Specimen geometry (dimensions in mm). In particular 10 dynamic tensile failure tests were carried out, with a strain rate ranging about from 150 s -1 to 600 s -1 ; such values are typically experienced in case of impact or blast loading. To conduct these tests a Modified Hopkinson Bar, available at DynaMat Laboratory [18] of the University of Applied Sciences of Southern Switzerland, was used. Moreover, also 3 static tensile failure tests were conducted on the same types of specimens in order to compare the dynamic and static results; to carry out these tests a universal electro-mechanical device was used and a speed displacement control of mm/s was applied (Figure 3.2).

78 78 Chapter III Dynamic properties of reinforcing steel Figure 3.2 Quasi-static test. Dynamic tests set-up The tests were conducted using the Hopkinson Bar equipment already described in previous Chapter. The steel specimen, characterized by a diameter of 3mm, is screwed to the input and output bars. Both input and output bars are instrumented with semiconductor strain gauges, measuring incident, reflected and transmitted pulses, acting on the cross section of the specimen. On the basis of the record of ε I, ε R and ε T, presented in Figure 3.3 in terms of acquired signal, it is then possible to calculate the engineering stress, engineering strain and strain-rate curves, using equations (2.2), (2.3) and (2.4).

79 79 Chapter III Dynamic properties of reinforcing steel 15 output pulse input pulse 10 Pulse [V] 5 ε I ε R ε T time [s] Figure 3.3 Measurements obtained by Modified Hopkinson Bar Then, the engineering stress vs. strain curves in tension have been transformed in true stress vs. strain curves by the application of the following relationships: ( ε ) σ true = σ 1+ (3.1) ε = ln 1 true ( + ε) (3.2) True stress vs. true strain relationships provide interesting data in order to predict the behavior of the investigated steel in case of stress conditions different from those induced by uniaxial tensile loading. In fact, it must be outlined that engineering stress does not represent the actual stress in the material during the tensile test, since it is calculated as the ratio between the applied load and the initial area, which differs from the real area during the test. Also engineering strain is inadequate to describe the actual strain of the

80 80 Chapter III Dynamic properties of reinforcing steel material, in particular in the plastic range, where large deformations occur. Then, true stress vs. true strain curves are more informative in studying plastic behavior and are useful in conducting research on material and in developing constitutive law for FE codes. On the contrary, the engineering curves provide satisfactory information for structural design. However true stress vs. true strain relationships can be considered significant up to the maximum stress point where necking begins; after this point, localization and fracture propagation governs the flow curve, which is no more representative of homogeneous mechanical properties of the materials. An image was also acquired after the test in order to observe the failure mechanism in the necking zone. Figure 3.4 depicts the specimen during the test, while Figure 3.5 presents a microscopic view of the specimen after the test, revealing the fracture zone. Figure 3.4 Specimen during the high strain-rate test.

81 81 Chapter III Dynamic properties of reinforcing steel Figure 3.5- Photo of the specimen after failure. Tests results Results of static tensile tests on steel specimens are presented in Table 3.1, while Figure 3.6 presents the obtained stress-strain relationships for the 3 tests. specimen Table 3.1 Static tensile failure tests results* f y, [MPa] f y,ave, [MPa] f t, [MPa] f t,ave, [MPa] ε t, [%] # # [4] [18] 9.9 # *The suffix ave indicates the average value; in square parentheses the standard deviation is presented ε t,ave, [%] 10 [1]

82 82 Chapter III Dynamic properties of reinforcing steel eng. stress [MPa] eng. strain Figure 3.6- Quasi static stress strain relationships. A mean yielding stress of 388 MPa was obtained, in agreement with available original design documents, declaring for the characteristic yielding stress of the concrete reinforcing rebar a value of 320 MPa. An average ultimate tensile stress of 708 MPa was also obtained. Data from static tensile failure tests conducted on steel bars are also available. Such tests revealed an average yielding stress and an average ultimate tensile stress of 400 MPa and 593 MPa, respectively. Comparing these data with those obtained on steel specimens reveals some differences, especially for ultimate tensile stress. This is probably due to some disturb introduced in the steel during the lathe phase. Using the Hopkinson bar apparatus 10 failure tests were performed. Table 3.2 provides the main results from such tests. It must be outlined that the exact strain rate level cannot be defined before the test starts but it can be evaluated only after the test.

83 83 Chapter III Dynamic properties of reinforcing steel specimen ε, [s -1 ] Table 3.2 Dynamic tensile failure tests results* ε,ave, [s -1 ] f y,dyn, [MPa] f y,dyn,ave, [MPa] f t,dyn [MPa] f t,dyn,ave [MPa] ε t,dyn, [%] # # #6 172 [18] 533 [43] 809 [43] 9 (Group #7 184 A) ε t,dyn,ave, [%] 11 [1] # # # [55] [8] # [10] (Group B) # [1] # *The suffix ave indicates the average value; in square parentheses the standard deviation is presented Therefore, depending on strain rate value reached during the tests, specimens are divided into two groups, Group A and Group B, characterized by an average strain rate of 174 s -1 and 562 s -1, respectively. Hence, in Group A tests an average yielding stress of 547 MPa and an average ultimate stress of 789 MPa were obtained, while in Group B, characterized by higher stress values, an average yielding stress of 626 MPa and an average ultimate stress of 830 MPa were obtained. Moreover an average ultimate strain of 11% and 15% were obtained for Group A and Group B, respectively. Figures 3.7 and 3.8 represent the stress vs. strain curves of Group A and Group B, respectively.

84 84 Chapter III Dynamic properties of reinforcing steel eng. stress [MPa] eng. strain Figure 3.7- Stress strain relationships from Group A tests (ε,ave, = 174 s.1 )

85 85 Chapter III Dynamic properties of reinforcing steel eng. stress [MPa] eng. strain Figure 3.8- Stress strain relationships from Group B tests (ε,ave, = 562 s.1 ) Such data were then processed in terms of Dynamic Increase Factor (DIF), defined as the ratio of the dynamic to static value, for each of the analyzed properties. In order to perform such process the mean values from Table 3.1 were used as reference static value. Table 3.3 reports the experimental DIF for yielding stress, ultimate tensile stress and ultimate tensile strain, evaluated for each specimen. It can be observed that DIF grows as strain-rate increases, for all of the 3 considered properties. Moreover yielding stress exhibits higher values of DIF, if compared with those related to ultimate tensile stress.

86 86 Chapter III Dynamic properties of reinforcing steel Table 3.3 Experimental Dynamic Increase Factors* specimen ε, [s -1 ] DIF,f y DIF,f y,ave DIF,f u DIF,f u,ave DIF,ε u DIF,ε u,ave # Group A # # [0.11] 1.14 [0.06] 0.9 # [0.1] # # Group B # # [0.03] 1.17 [0.01] 1.6 # [0.1] # *The suffix ave indicates the average value; in square parentheses the standard deviation is presented Figure 3.9, 3.10 and 3.11 present the true stress vs. true strain curves for the three strain-rate regimes.

87 87 Chapter III Dynamic properties of reinforcing steel 1200 true stress [MPa] true strain Figure 3.9- True Stress versus strain curves in static tests 1200 true stress [MPa] true strain Figure True Stress versus strain curves from Group A tests (ε,ave, = 174 s.1 )

88 88 Chapter III Dynamic properties of reinforcing steel 1200 true stress [MPa] true strain Figure True Stress versus strain curves of Group B tests (ε,ave, = 562 s.1 ) Moreover, by the microscopy measurements the reduction of the area (RA) at fracture obtained was obtained as: (A A ) RA = 0 F (3.3) A 0 where A 0 is the initial cross-sectional area and A F is the cross-sectional area of the fractured section. The fracture strain ε t,fracture and the fracture stress s t,fracture were then calculated measuring the outside radius of the cross section of the neck and the radius of curvature at the neck according to Bridgeman correction [19] which considers the triaxiality in the necking zone. Table 3.4 and Table 3.5 report the main data from the true stress vs. true strain curves.

89 89 Chapter III Dynamic properties of reinforcing steel Table 3.4 True stress-true strain elaborations (a)* Specimen ε, [s -1 ] σ,t,true [MPa] σ,t,true,ave, [MPa] ε t,true, [%] # ε t,true,ave, [%] # # [55] 13 # [3] # # # # [10] 18 # [1] # *The suffix ave indicates the average value; in square parentheses the standard deviation is presented 18

90 90 Chapter III Dynamic properties of reinforcing steel Specimen Table 3.5 True stress-true strain elaborations (b)* ε, [s - 1 ] RA [%] RA ave [%] ε t,fracture, [%] ε t,fracture,ave [%] σ,t,fracture, [MPa] σ,t,fracture,ave, [MPa] # # # [5] [9] 1538 # [89] # # # # [1] [9] 1474 # [141] # *The suffix ave indicates the average value; in square parentheses the standard deviation is presented ANALYSIS OF TEST RESULTS The obtained data were compared with theoretical prediction of the dynamic properties of the investigated steel, conducted using existing relationships available in literature. In particular 2 different formulations were considered: CEB information bulletin no. 187 formulation [4]; Malvar formulation [5]. CEB bulletin presents several formulations for different steel types, providing DIF for yielding stress and ultimate stress. In case of hot rolled reinforcing steel the following expressions are provided: f y, dyn 6.0 ε DIF f y = = 1+ ln (3.4) f f ε y y 0

91 91 Chapter III Dynamic properties of reinforcing steel fu, dyn 7.0 ε DIF f u = = 1 + ln (3.5) f u fu ε 0 where DIF f y is the DIF for the yielding stress; DIF fu is the DIF for the ultimate stress; f y,dyn is the dynamic yielding stress; f y, is the static yielding stress; f t,dyn is the dynamic ultimate tensile stress; f u, is the static ultimate tensile stress; ε is the strain rate; ε 0 is a constant equal to s -1 and has the meaning strain rate at quasi-static condition. On the contrary, Malvar work, presenting a review of available studies investigating strain-rate sensitiveness of reinforcing steel, proposes the following formulations: DIF DIF f y fu f = f f = f y, dyn y u, dyn u ε = 10 ε = f y f y (3.6) (3.7) where f y, is requested ksi. Using such formulations, data in Table 3.6 were obtained. A comparison between the numerically evaluated DIFs and those obtained from the experimental results reveals that CEB expression slightly underestimates yielding stress DIF, while Malvar formulation weakly overestimates it. On the contrary, both expressions well fit experimental DIFs for the ultimate stress.

92 92 Chapter III Dynamic properties of reinforcing steel Table 3.6 Comparison between experimental and numerical DIFs* specimens ε ave [s -1 ] Experimental CEB formulations* Malvar formulations* DIF,f y,ave DIF,f u,ave DIF,f y,ave DIF,f u,ave DIF,f y,ave DIF,f u,ave Group A Group B *The suffix ave indicates the average value; in square parentheses the standard deviation is presented The results are then plotted in Figures 3.12 and 3.13, where these comparisons are well appreciable. It must be outlined that both in case of CEB and Malvar formulation, background data, used to build the numerical expressions, are from tensile failure tests, with strain-rate values up to 10 s -1.

93 93 Chapter III Dynamic properties of reinforcing steel 1.8 DIFfy 1.6 Experimental data CEB formulation Malvar formulation 1.4 DIF Strain rate Figure DIF for yielding stress

94 94 Chapter III Dynamic properties of reinforcing steel 1.8 DIFfu 1.6 Experimental data CEB formulation Malvar formulation 1.4 DIF Strain rate Figure DIF for ultimate stress Then, the incongruence between experimental data of the conducted tests and the numerical evaluations could be addressed to this aspect. The obtained results allow concluding the following: The investigated reinforcing steel was found to be strain-rate sensitive, in terms of yielding stress, ultimate stress and ultimate strain.

95 95 Chapter III Dynamic properties of reinforcing steel As strain-rate increases, yielding stress increases more the ultimate stress; in fact yielding stress assumes a maximum DIF value of 1.62 for a strain-rate of 629 s -1, while ultimate stress reaches a DIF of 1.17, with the same strain-rate level. Both CEB and Malvar numerical prediction sufficiently fit the experimental data; in particular CEB expression underestimates the yielding stress with a maximum percent difference of -24%, while Malvar expression overestimates it, presenting a maximum percent difference of +38%. Furthermore, in case of ultimate stress, both expressions reproduce the experimental data even more reliably, with percent differences of about 2%. In the present case small specimen were used; actually some differences were experienced between static tests performed on such specimens and steel bar specimens, revealing that some disturb was probably introduced in the preparation of the dynamic samples via automatic lathe working. However the DIF were obtained processing dynamic and static tests performed on the same sample type. However it must be outlined that the most significant differences were obtained in terms of ultimate stress, whereas yielding stress, more useful for design calculations, was just slightly affected by such dissimilarities. Nevertheless further investigations on this issue appear necessary, in order to verify the reliability of the obtained data. Finally it can be observed that the experimental results represent interesting data about very high strain-rate behavior of reinforcing steel that could be very useful to assess further numerical formulations, providing dynamic mechanical properties of reinforcing steel. However, to do this, additional tests appear crucial, also investigating more recent types of reinforcing steel, in order to provide reliable design formulations, predicting dynamic mechanical properties of reinforcing steel under severe dynamic load conditions.

96 96 Chapter III Dynamic properties of reinforcing steel REFERENCES 1 Woei Shyan Lee, Chen-Yang Liu, The effect of temperature and strain-rate on the dynamic flow behaviour of different steels, Materials Science and Engineering, A 426, 2006, pp Mainstone, R.J. Properties of materials at high rates of straining or loading, part 4, State-of-the-Art report on Impact loading of Structures, Materials and Structures, vol. 8, No. 44, pp Uenishi, A. and Teodosiu, A., Constitutive modelling of the high strain-rate behaviour of interstitial-free steel, International Journal of Plasticity, vol. 20, 2004, pp Comité Euro-International du Béton. «Concrete structures under impact and impulsive loading CEB Bulletin 187, Lausanne, Switzerland, Malvar, L.J., Review of static and dynamic properties of steel reinforcing bars, ACI Materials Journal, vol. 95, 1998, pp Lee, W.-S., Lin, C.-F., Chen, B.-T., Tensile properties and microstructural aspects of 304L stainless steel weldments as a function of strain rate and temperature, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 219, 2005, pp Couque, H., Leung, C.P., Hudak Jr., S.J., Effect of planar size and dynamic loading rate on initiation and propagation toughness of a moderate-toughness steel Engineering Fracture Mechanics, vol. 47, 1994, pp Drar, H., On predicting the temperature and strain rate dependences of the fracture toughness of plain carbon steels Materials Characterization, vol. 31, 1993, pp Wang, G.Z., Wang, Y.L., Effects of loading rate, notch geometry and loading mode on the local cleavage fracture stress of a C-Mn steel, International Journal of Fracture, vol. 146, 2007, pp

97 97 Chapter III Dynamic properties of reinforcing steel 10 Yang, Z., Kim, C.-B., Cho, N., Beom, H.G., Ren, B., The static and dynamic fracture test of X70 pipeline steel, Key Engineering Materials, vol , 2007, pp Uenishi, A., Takahashi, M., Kuriyama, Y., High-strength steel sheets offering high impact energy-absorbing capacity Nippon Steel Technical Report, vol. 81, 2000, pp Oliver, S., Jones, T.B., Fourlaris, G., Dual phase versus TRIP strip steels: Comparison of dynamic properties for automotive crash performance, Materials Science and Technology, Vol. 23, 2007, pp Choi, I.D., Bruce, D.M., Kim, S.J., Lee, C.G., Park, S.H., Matlock, D.K., Speer, J.G., Deformation behavior of low carbon TRIP sheet steels at high strain rates, ISIJ International, vol. 42, 2002, pp Bleck, W., Schael, I., Determination of crash-relevant material parameters by dynamic tensile tests Steel Research, vol. 71, 2000, pp Mihalikova, M., Janek, J., Influence of the loading and strain rates on the strength properties and formability of higher-strength sheet, Metalurgija, vol. 46, 2007, pp Buršák, M., Mamuzić, I., The influence of the loading rate on the mechanical properties of drawing steel sheet, Metalurgija, vol. 45, 2006, pp Filiatrault, A, Holleran, M., Stress-strain behavior of reinforcing steel and concrete under seismic strain rates and low temperatures, Materials and Structures, Vol. 34, 2001, pp Cadoni, E., Amaro, W., Dotta, M., Albertini, C., Giorgetti P. DynaMat: Laboratory for mechanical characterization of materials at high strain-rate in Structural concrete in Switzerland, FIB-CH, ISBN , 2006, pp Bridgeman P., Studies in large plastic flow and fracture. Cambridge, MA: Harvard University Press; 1964.

98 98 Chapter IV Dynamic properties of GFRP Chapter IV DYNAMIC PROPERTIES OF GFRP INTRODUCTION TO FRAMEWORK ACTIVITIES In 2007 AMRA, an Italian research center based at University of Naples Federico II, coordinated a research project, funded by European Commission, namely SAS (Security of Airport Structures) project, focusing on the design of a discontinuous barrier to protect airport infrastructures from disruptive malicious actions, such as blast or intrusions. The barrier was essentially composed by tubular elements vertically installed into a concrete base. To achieve both transparency to the electromagnetic waves used in radiocommunications and high mechanical properties, GFRP (glass fibers reinforced polymer) was preferred to steel as component materials of the pipe elements. The designed barrier was subjected to a detailed test activity, investigating mechanical properties and radio-transparency of prototypes of the structure. Further discussions about blast behavior of the barrier are presented in Chapter VII. Within such activities, a dynamic characterization of GFRP used in pipe elements was carried out at DynaMat laboratory of the University of Applied Sciences of Southern Switzerland, conducting dynamic tensile failure tests on GFRP samples at selected strain rates. This chapter describes results from such phase, whose main objective was to investigate dynamic behavior of the

99 99 Chapter IV Dynamic properties of GFRP examined composite, in order to account for its strain rate sensitiveness in the definition of constitutive relationships to be implemented in numerical models. Indeed, mechanical properties of composite materials are generally affected by strain rate dependence, as shown in [1, 2, 3, 4]. In particular, initial Young modulus and failure stress increase as strain rate becomes higher [4, 5], under both compression and tensile loading conditions. Specifically, focusing on GFRP, available literature describes in details the effects of strain rate under compressive loading [6, 7, 8, 9]. In particular, it is shown that mechanical behavior of composites in compression is strongly controlled by resin properties, which appear to be highly strain rate dependent. Consequently, a significant increase is generally experienced in initial Young modulus, failure stress and failure strain, under compression. On the other hand, under tensile loading, GFRP also exhibits sensitiveness to rate of loading, as described in [10, 11, 12]. In particular, also in this case, an increase in failure stress and initial Young modulus is significant, whereas failure strain decreases as strain rate increases [13]. STATIC CHARACTERIZATION OF GFRP Tested GFRP was used in pipe elements of the SAS barrier; it is a pultruded polyester composite reinforced with unidirectional E-glass fibers; volume fraction of fibers is about 60%. Static tensile failure tests were preliminarily conducted to characterize the main mechanical properties of the investigated composite. These tests were carried out on prismatic samples, 3mm thick, 15mm wide and 300mm high. 6 tests were carried out, according to ASTM specifications [14]; a universal machine MTS 810 was used, with a load capacity of 500 kn. The displacement controlled test were conducted at a testing speed of 2 mm/min, corresponding to a strain rate of 10-4 s -1. Figure 4.1 depicts a specimen during and after the test.

100 100 Chapter IV Dynamic properties of GFRP (a) (b) Figure 4.1 Specimen during (a) and after (b) static test All specimens exhibited a linear behavior up to failure. Table 4.1 reports main mechanical properties obtained from such tests; an average failure stress of MPa and a Young modulus of 51.3 GPa are evidenced. Table 4.1 Experimental static mechanical properties of tested GFRP Failure stress [MPa] Failure strain Young modulus [GPa] Average Standard deviation

101 101 Chapter IV Dynamic properties of GFRP DYNAMIC EXPERIMENTAL ACTIVITIES ON GFRP As above mentioned, the experimental activities consisted of uniaxial tensile failure tests conducted at controlled strain rate on GFRP specimens, loaded along fibers direction. In particular, 4 strain rate nominal values were considered: 1 s -1, 10 s -1, 400 s -1 and 700 s -1. The first and the second strain rate levels can be considered as intermediate strain rates, as those possibly induced by severe earthquakes or low impacts; such strain rates were obtained using a Hydro-Pneumatic Machine (HPM). On the contrary, the other two values can be considered as representative of to high strain rate loading, as those caused by strong impacts or blast actions; such strain rate tests were conducted via a Modified Hopkinson Bar (MHB) apparatus, already described in previous chapters. Experimental set-up The tests were conducted on prismatic notched specimens. Weakened cross section was 0.5mm thick and about 3.5-4mm wide. The geometry of the specimen is depicted in Figure 4.2. A specific gripping system was also developed, both for HPM and MHB tests, to avoid slipping of samples from testing devices mm 1 mm 0.5 mm Figure 4.2 Geometry of specimens used in dynamic tests

102 102 Chapter IV Dynamic properties of GFRP The system is composed by two clamps made up of two wedge-shaped elements and a thermally aged Maraging steel support. Further details about testing set-up using HPM and MHB devices at Dynamat laboratory are presented in [15] and [16], respectively. Figure 4.3 show HPM set-up, whereas Figure 4.4 depicts a specimen before MHB test. sample (a) (b) Figure 4.3 HPM set-up: the testing device (a); specimen in its testing position (b) sample Figure 4.4 Specimen in its testing position in MHB

103 103 Chapter IV Dynamic properties of GFRP Discussion of tests result Several tests were conducted at each strain rate but many of them were affected by undesired failure or slipping of the specimen at the gripping zones. Hence, only three tensile failure tests are presented for each considered strain rate level (below referred to as Group A, Group B, Group C and Group D for target strain rate of about 1 s -1, 10 s -1, 400 s -1 and 700 s -1, respectively). Each specimen presented a sharp-cut failure, as depicted in Figure 4.5. Figure 4.5 Specimen after failure In case of intermediate strain rate tests, performed via HPM, acquired data consist just of stress-time curves, from which failure stress and failure time can be obtained. Indeed, recorded strain values are affected by spurious deformations occurring in dynamometric elastic bars and consequently cannot be considered reliable; the same applies to initial Young modulus. On the contrary, in case of high strain rate tests, conducted via MHB apparatus, acquired data consist of stress-strain curves and stress-time curves, from which failure stress, failure strain, initial Young modulus and failure time can be obtained. In these cases, specimens stress-strain relationships present a slightly non-linear behavior. Figures 4.6 presents stress-time curves for each group, whereas Figures 4.7 and 4.8 report stress-strain curves for Group C and Group

104 104 Chapter IV Dynamic properties of GFRP D, respectively. In Figure 4.6 it can be observed that, in case of intermediate strain rate tests, initial stress is not equal to zero but, before test starts, specimen is in equilibrium with a given tensile action, causing an initial stress not higher than the 20% of the failure stress # #2 #3 stress [MPa] Group A time [s] (a)

105 105 Chapter IV Dynamic properties of GFRP 1500 # #5 #6 stress [MPa] Group B time [s] (b)

106 106 Chapter IV Dynamic properties of GFRP 1500 # #8 #9 stress [MPa] Group C time [s] (c)

107 107 Chapter IV Dynamic properties of GFRP 1500 # #11 #12 stress [MPa] Group D time [s] (d) Figures 4.6 Stress-time curves at strain rate values of 1 s -1 (a), 10 s -1 (b) 400 s -1 (c) and 700 s -1 (d)

108 108 Chapter IV Dynamic properties of GFRP Stress [MPa] # 7 # 8 # Strain Figure 4.7 Stress-strain curves at strain rate values of 400 s -1

109 109 Chapter IV Dynamic properties of GFRP Stress [MPa] # 10 # 11 # Strain Figure 4.8 Stress-strain curves at strain rate values of 700 s -1 Results of the conducted tests are summarized in Tables 4.2 and 4.3.

110 110 Chapter IV Dynamic properties of GFRP Group A Specimen Table 4.2 Results of dynamic HPM tests Strain-rate [s -1 ] Failure stress [MPa] Failure stress DIF Loading rate [MPa/s] Failure time [µs] Average values Standard deviation [0.1] [54] [0.08] [1085] [15639] Group B Average values Standard deviation [0.2] [98] [0.13] [13549] [799]

111 111 Chapter IV Dynamic properties of GFRP Group C Specimen Strain-rate [s -1 ] Table 4.3 Results of dynamic MHB tests Failure stress [MPa] Failure stress DIF Failure strain Failure Strain DIF Initial Young modulus [MPa] Initial Young modulus DIF Loading rate [MPa/s] Failure time [µs] Average values Standard deviation [33] [39] [0.05] [0.0065] [0.43] [34.0] [0.66] [ ] [37] Group D Average values Standard deviation [44] [92] [0.12] [0.0053] [0.35] [15.4] [0.30] [ ] [3] It is pointed out that, both in HPM and in MHB tests, strain rate values cannot be exactly set before the test starts, but it can be only assessed at the end of it. Hence, for each test, the exact strain rate value is reported. Furthermore, since many papers in literature refer to loading rate rather then strain rate, in order to make results available for comparison also with data from other articles, corresponding values of loading rate are also reported. Obtained data were processed to determine Dynamic Increase Factor (DIF) for the main parameters; DIF is defined as the ratio between the values in dynamic regime and its corresponding value under static conditions; such evaluation was performed for failure stress, failure strain and initial Young modulus, and allowed for a comparison to reference values obtained in static tests. The outcomes are presented in Tables 4.2 and 4.3. An increasing trend with strain rate is evident for failure stress and initial Young modulus; it is pointed out that, in case of 700 s -1, DIF reaches average values equal to 1.90 and 2.78, for failure stress and initial Young modulus, respectively. Figure 4.9 depicts DIFstrain rate relationships for average failure stress. On the contrary, failure strain

112 112 Chapter IV Dynamic properties of GFRP presents large variability, which seems unaffected by clear strain rate sensitiveness. However, focusing on high strain rate tests, conducted via MHB apparatus, it can be observed that failure strain decreases moving from 400 s -1 to 700 s -1, which appears consistent with indications from available literature, predicting a reduction of about 20% from quasi-static conditions to very high strain rate levels [13] Failure stress 1.6 DIF strain-rate [s -1 ] Figure 4.9 DIF-strain rate relationship for failure stress NUMERICAL ASSESSMENT OF RESULTS Tested specimens exhibited a non-linear behavior at high strain rate levels, as presented in stress-strain relationships depicted in Figure 4.7 and 4.8.

113 113 Chapter IV Dynamic properties of GFRP Therefore, a viscoplasticity model appears appropriate to reproduce experienced behavior. Viscoplasticity theory has been widely used to represent dynamic behavior of composites. Firstly, parametric power laws were developed describing effective stress-effective plastic strain relationships at each strain rate [17, 18]; such approaches need to characterize experimental parameters values for each investigated strain rate level. More recently a viscoplastic rate dependent constitutive model was developed by Thiruppukuzhi and Sun [19], considering effective plastic strain rate, depending on the angle between fibers and loading direction. The model also provides a failure criterion, which reproduces the effect of strain rate on ultimate stress. In particular, in this model, strain rate is directly accounted for and parameters characterization is not necessary for each strain rate of interest. Specifically, viscoplasticity is introduced through the following expression: p p α β ε = B( ε ) ( σ) (4.1) p p where ε is the effective plastic strain rate, ε is the effective plastic strain, σ is the effective stress and B, α and β are parameters experimentally evaluated. It is underlined that this model does not need any reference stress; in other words, viscoplasticity contributions to constitutive law needs to be computed at each loading rate, without a quasi-static reference response [19]. This way of considering strain rate sensitiveness differs from other approaches in which a static behavior is defined and then updated, in order to account for viscosity enhancements. In case of monotonic and uniaxial loading applied on fibers direction, effective stress and effective plastic strain reduce to: σ = 3/2 σ x (4.2) p p ε = 2/3 ε x p where σ x and ε x are the stress and the plastic strain along fibers direction. From equation (4.1) the following relationship can be derived:

114 114 Chapter IV Dynamic properties of GFRP p n ε = A( σ ) (4.3) where p m A = χε ( ) (4.4) and n, χ and m are parameters experimentally evaluated. In particular, strainrate dependence is controlled by χ and m, whereas n represents a rateindependent constant. p p The dependence of both ε on σ and A on ε are linear in a bi-logarithmic plot; hence, considering recorded stress-strain relationships for Group C and Group D tests, best fitting values of n, χ and m have been evaluated and are equal to 1.76, MPa -n and -0.65, respectively. Using these values, p analytical σ -ε curves have been evaluated; Figures 4.10 and 4.11 reports p experimental σ -ε curves for Group C and Group D, respectively, plotted in both cases against the numerical curve evaluated considering the average effective plastic strain rate obtained in each group test.

115 115 Chapter IV Dynamic properties of GFRP Effective stress [MPa] # 7 # 8 # 9 ave. numerical curve Effective plastic strain Figure 4.10 Effective stress effective plastic strain curves (Group C)

116 116 Chapter IV Dynamic properties of GFRP Effective stress [MPa] # 10 # 11 # 12 ave. numerical curve Effective plastic strain Figure 4.11 Effective stress effective plastic strain curves (Group D) In order to characterize strain rate sensitiveness of failure stress, rate-dependent failure criterion proposed by Thiruppukuzhi and Sun [19] was also considered. This failure criterion is based an the assumption that rate dependent behavior would be exclusively due to matrix contribution; in fact, strain rate enhancement is considered negligible in case of load applied along fiber direction, where mechanical behavior of composite is governed mainly by fiber contribution. In other words, only rate dependence of polymeric matrix is considered, which becomes significant only in cases of off-axis load direction. On the contrary, in the current case, being loading direction aligned with fibers orientation, matrix contribution can be considered negligible and failure enhancements can be attributed to glass fibers contribution.

117 117 Chapter IV Dynamic properties of GFRP However, using this failure criterion, imposing angle between fibers orientation and loading direction equal to zero, failure condition reduces to: σ x = kcr (4.5) where σ x represents the stress exerted along loading direction and k cr is the corresponding failure stress, equal to: k γ ε cr = kcr, stat ε stat (4.6) In equation (4.6) k cr, stat represents failure stress under quasi-static strain rate, ε is the strain rate level, ε stat is the quasi-static strain rate level, suggested equal to , and γ is an experimental parameter. Using experimental data from quasi-static tests and dynamic tests, the best fitting value of γ was determined to be equal to The corresponding curve is reported in Figure 4.12 against experimental data. The comparison shows that, for tested samples, this approach seems to not provide a close correspondence between experimental and numerical trend. In fact, it implies a linear dependence of k cr on ε, in logarithmic scale; actually, this is not true, as clearly shown in Figure 4.9, where failure stress DIF-ε curve exhibits a clear slope variation between intermediate and high strain rate. This behavior is very common among ceramic composite materials, as concrete [20, 21]. Actually, glass used as reinforcing material of GFRP presents a microstructure similar to that characterizing ceramic material. Considering that loading is applied along fibers orientation and hence mechanical behavior is mainly due to reinforcing fibers, this could represent a further proof that experienced strain rate sensitiveness might be attributed to glass fibers reinforcement.

118 118 Chapter IV Dynamic properties of GFRP Failure stress [MPa] g= Experimental values Thiruppukuzhi and Sun criterion Strain rate [s -1 ] Figure 4.12 Failure stress-strain rate relationship As a final word, the author would like to precise that actually data from uniaxial tensile tests have been employed to calibrate a three-dimensional constitutive law, as that from Thiruppukuzhi and Sun [19]. In fact, even if a plastic potential needs of multi-axial tests to be fully determined, Thiruppukuzhi and Sun model was at any rate used in order to make results available for comparisons with similar investigations conducted on different composites and elaborated with the same model. Specifically, a strain rate sensitiveness was evidenced for high strain rate levels, whereas, within intermediate strain rate range, mechanical properties did not present significant enhancements. In fact, as depicted in DIF-strain rates curves, both initial Young modulus and failure stress presented significant

119 119 Chapter IV Dynamic properties of GFRP increases for high strain rate values, equal to 2.88 times and 1.90 times the static values, respectively; on the contrary, at intermediate strain rate, failure stress exhibited a very slight increment, equal approximately to 5-15%. Furthermore, in order to reproduce acquired stress-strain curves at high strain rate, a viscoplastic model, accounting for effective stress and effective plastic strain, and a strain rate dependent failure criterion was considered, as proposed by Thiruppukuzhi and Sun. Calibrating model parameters with experimental results, a good agreement between numerical and experimental effective stresseffective plastic strain curves was obtained. On the contrary, the failure criterion, which presents a failure stress-strain rate linear relationship in a bilog plot, did not appear suitable to reproduce the slope variation occurring between intermediate and high strain rate. Hence, a different formulation appears necessary; in order to achieve this objective, further investigations are necessary at strain rate levels between 10 s -1 and 400 s -1, where the slope variation occurs. Finally, it is emphasized the importance of the presented experimental study, representing a fundamental step toward the investigation of structural behavior of composite structures, subjected to severe dynamic loads. Indeed, results from the presented activities can be used to define constitutive models of the mechanical behavior of composites that can be implemented within numerical analyses to predict structural response of composite elements subjected to dynamic loads. REFERENCES 1 Hsiao, H. M. and Daniel, I. M., Strain rate behavior of composite materials Composites Part B, Vol 29B, 1998, Harding, J., Effect of strain rate and specimen geometry on the compressive strength of woven glass-reinforced epoxy laminates, Composites, Vol. 24, No. 4, 1993, pp

120 120 Chapter IV Dynamic properties of GFRP 3 Welsh, L.M. and Harding, J., Effect of strain rate on the tensile failure of woven reinforced polyester resin composites, Journal de Physique, Vol. C5, 1985, pp Sierakowsky, R. L., Strain rate effects in composites Appl. Mech. Rev., Vol. 50, No. 11, 1997, pp Okoli, I., The effect of strain rate and failure modes on the failure energy of fiber reinforced composites, Composite Structures, Vol. 54, 2001, pp Gary, G, Zhao, H. Dynamic testing of fibre polymer matrix composites plates under in-plane compression, Composites Part A, Vol. 31, 2000, pp El-Habak, A. M. A. Compressive resistance of unidirectional GFRP under high rate of loading, Journal of Composites Technology and Research, Vol. 15, No. 4, 1993, pp Tay T. E., Ang H. G. and Shim V. P.W., An empirical strain ratedependent constitutive relationship for glass fibre reinforced epoxy and pure epoxy, Composite Structures, Vol. 33, 1995, pp Huang, Z., Nie, X. and Xia, Y., Effect of strain rate and temperature on the dynamic tensile properties of GFRP, Journal of Materials Science, Vol. 39, 2004, pp Harding, J. and Welsh, L.M., A tensile testing technique for fiber reinforced composites at impact rates of strain, Journal of Materials Science, Vol. 18, 1983, pp Makarov, G., Wang, W. and Shenoi, R.A., Deformation and fracture of unidirectional GFRP composites at high strain rate tension, Proc. Of 2nd International Conference on Composites Testing and Model Identification, September, , Bristol, UK, Paper No Newill, J. F., and Vinson, R., Some high strain rate effects on composites materials Proc. Of ICCM9, Vol. 5, 1993, pp Majzoobi, G.H., Saniee, F.F. and Bahrami, M., A tensile impact apparatus for characterization of fibrous composites at high strain

121 121 Chapter IV Dynamic properties of GFRP rates, Journal of Materials Processing Technology Vol , 2005, pp ASTM D 3039/D 3039M Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials, 2003, pp Asprone, D., Cadoni, E. and Prota, A., Experimental analysis on the tensile dynamic behavior of existing concrete under high strain-rates ACI Structural Journal vol.106, issue 1, January-February 2009, pp Asprone, D., Cadoni, E., Prota, A. and Manfredi, G. Investigation on dynamic behavior of a mediterranean natural stone under tensile loading International Journal of Rock Mechanics and Mining Sciences, Volume 46, Issue 3, April 2009, Pages Sun, C.T., Chen, J.L., A simple flow rule for characterizing nonlinear behavior of fiber composites Journal of Composite Materials, 1987, Vol. 21, pp Gates, T.S., Sun C.T., Elastic/viscoplastic constitutive model for fiber reinforced thermoplastic composites, American Institute of Aeronautics and Astronautics Journal, 1991, Vol. 29, No. 3, pp Thiruppukuzhi, S.V., Sun, C.T., Models for the strain-rate dependent behavior of polymer composites, Composites Science and Technology, 2001, Vol. 61, pp Comité Euro-International du Béton. CEB-FIP Model Code 1990 Redwood Books, Trowbridge, Wiltshire, UK, Malvar L.J. and Ross C.A. Review of strain-rate effects for concrete in tension ACI Materials Journal vol. 95, 1998, pp

122 122 Chapter V Dynamic properties of a natural stone Chapter V DYNAMIC PROPERTIES OF A NATURAL STONE INTRODUCTION TO THE FRAMEWORK ACTIVITIES Currently results from several research activities are available in literature, focusing on experimental investigation of dynamic mechanical properties of common construction materials; in particular concrete and steel, widely used in recent civil infrastructures, have been dealt with in several research activities [1, 2, 3, 4, 5, 6, 7]. However, especially in assessment of existing structures, such as historical buildings or monuments, a specific knowledge about dynamic properties of materials of less recent use, like masonry, becomes necessary. The present chapter discusses the results of an experimental activity conducted to investigate the dynamic properties at different strain-rate of a classical Mediterranean natural stone, namely Neapolitan Yellow Tuff (NYT). NYT is a zeolitized tuff, product of a huge eruption dated about 15,000 years ago in the Phlegrean Field volcanic area (Neapolitan area), that generated about 50 km 3 of material [8]. Mechanical behavior of NYT is similar to that exhibited by several natural stones of Mediterranean area that, generated by volcanic eruptions, present common characteristics and composition properties. Such stones were widely used in Mediterranean cities as construction stones (Figure 5.1 shows a NYT quarry), and consequently are present in numerous buildings

123 123 Chapter V Dynamic properties of a natural stone belonging to architectural heritage of historical cities but also in strategic structures (Figure 5.2). Focusing the attention on one of such stones is unavoidable, due to their wide variability, but the results from the present study can provide interesting data about their dynamic behavior. Figure 5.1 NYT quarry

124 124 Chapter V Dynamic properties of a natural stone Figure 5.2 Castel dell Ovo, a complete NYT building Due to its extensive employment, quasi-static mechanical properties of NYT are well known [9], but no data are available about its behavior under controlled dynamic conditions. Hence, the objective of the presented research is to assess the dynamic behavior of NYT stone, at different strain-rate, in order to verify its behavior under dynamic loading conditions, such as those induced by seismic or blast actions. NYT is a brittle material characterized by good mechanical properties and by a sufficient workability that make it an appreciable construction stone. As it could be expected by a natural stone, NYT is affected by a high variability of its mechanical characteristics, which may depend on the extraction location or on the depth of extraction. Table 5.1 reports mean compression strength of different varieties of NYT, as reported in [10]. It can be observed that a medium strength stone is characterized by a compression failure stress f c of

125 125 Chapter V Dynamic properties of a natural stone about 4 MPa. On the contrary, tensile failure stress f t is characterized by values of one order smaller than f c that, according to [11], can be evaluated as: t f (5.1) t f c = T p a p a where p a is the atmospheric pressure and T and t are two empirical parameters, that can be posed equal to and 0.805, respectively [12]; such expression provides values of f t that represents about the 9% of f c. Table 5.1 Compression strength of different yellow tuff varieties [10] Type Average values of compression strength f c [MPa] Very low compression strength tuff (< 2MPa) Low compression strength tuff (2-3MPa) Medium compression strength tuff (3-4MPa) Quite high compression strength tuff (4-5MPa) High compression strength tuff (5-7.5MPa) Very high compression strength tuff (>7.5MPa) Porosity and saturation represent two important parameters that can influence mechanical properties of NYT; in fact, as both increase mechanical properties degrade and in particular compression strength can be affected by strong reductions, up to 36% [10]. Young modulus E of NYT can vary between 800MPa and 3000MPa presenting the 80% of values over 1000MPa [10]. Furthermore, the specific weight g is

126 126 Chapter V Dynamic properties of a natural stone also affected by high variability, ranging between 14.6kN/m 3 and 17.5kN/m 3 [9]. Here a dynamic tensile characterization is conducted on a NYT variety, performing failure tensile tests at different strain-rates, ranging between 10-5 s -1, conducted in quasi-static loading conditions, to 50 s -1. Static tensile and compression tests were preliminarily performed in order to assess the mechanical properties of the investigated NYT variety. The obtained results are detailed in following paragraphs. It is outlined that tests were conducted on specimens under ordinary conditions of temperature and humidity; however to control such aspect, total porosity and saturation grade were also determined, obtaining values equal to and 2%, respectively. STATIC CHARACTERIZATION OF NYT A preliminary static characterization was conducted on the investigated NYT. 5 compression failure tests were performed on 60mm high cylindrical specimens with a diameter of 60mm and as many tests were conducted on cylindrical specimens of 20mm of diameter and height. Moreover 3 measurement of the Young modulus E was conducted on 3 specimens with of 60mm diameter and 110mm high. Results in Tables 5.2 and 5.3 were then obtained. It can be observed that, in case of specimens of 60mm of diameter an average compression failure stress of 3.57MPa was obtained revealing, according to characterization defined in [10], a Medium compression strength tuff. Moreover, results from smaller specimen tests (20mm of diameter) revealed a higher average compression failure stress, equal to 5.15MPa. This is probably caused by a size effect influencing the behavior of NYT, which is due to its inhomogeneity at a microscopic scale. In fact, it is widely accepted that a reduction in the size of the specimens, in case of not homogeneous materials, can determine an increase in experienced strength values.

127 127 Chapter V Dynamic properties of a natural stone Table 5.2 Results of compression failure tests Specimen f c [MPa] (Ø= 60mm) f c [MPa] (Ø= 20mm) average Standard Deviation Table 5.3 Measurements of Young modulus Specimen E [MPa] average 2407 Standard Deviation 271 Tensile failure tests were also performed on cylindrical specimens of 20mm of diameter and 20mm of height. In particular 6 tests were conducted, obtaining the results presented in Table 5.4. Figure 5.3 shows a specimen before and after the test. An average tensile failure stress of 0.68MPa was obtained, corresponding to the 13% of the average compression failure stress, obtained for the same dimension specimens. Such value is actually close to values indicated in available literature that, as reported above, suggests a ratio of about 9%.

128 128 Chapter V Dynamic properties of a natural stone Table 5.4 Results of static tensile failure tests (Ø= 20mm) Specimen f t [MPa] average 0.68 Standard Deviation 0.15 (a) (b) Figure 5.3 Specimen before (a) and after (b) static tensile failure test DYNAMIC TESTS ON NYT Tensile failure tests were conducted, in the DynaMat laboratory of the University of Applied Sciences of Southern Switzerland, on NYT specimens, investigating in particular medium strain-rate and high strain-rate. The former, commonly considered as ranging between 10-1 s -1 and 10 s -1, can be induced by severe seismic actions, while the latter, varying between 10s -1 and 10 2 s -1, can be due to impact or blast loading.

129 129 Chapter V Dynamic properties of a natural stone Two different set-up were used to conduct such tests. In particular, for medium strain-rate tests a Hydro-Pneumatic Machine (HPM) was employed, whereas a Modified Hopkinson Bar (MHB) apparatus was used to conduct high strain-rate tests. In both cases the specimens were cylindrical, with a diameter of 20mm and an height of 20mm. Medium strain-rate tests In medium strain-rate range 6 tests were conducted at a strain-rate level of 10-1 s - 1 and 7 tests at a strain-rate level of 5 s -1. The employed HPM is depicted in Figure 5.4, whereas Figure 5.5 presents its functioning scheme. At the beginning of the test, a sealed piston divides a cylindrical tank in two chambers; one chamber is filled with gas at high pressure (e.g. 150 bars), the other one is filled with water.

130 130 Chapter V Dynamic properties of a natural stone Figure 5.4 Hydro-Pneumatic Machine Figure 5.5 HPM functioning scheme At the beginning equal pressure is established in the water and gas chambers so that forces acting on the two piston faces are in equilibrium. The test starts when the second chamber discharge the water through a calibrated orifice, activated by a fast electro-valve. Then, the piston starts moving expelling out the gas through a sealed opening; the end of the piston shaft is connected to the steel specimen; The specimen is linked to the piston shaft and to one end of an elastic bar, whose other end is rigidly fixed to a supporting structure. When the piston shaft moves, the specimen is pulled at a fixed strain-rate level, depending on the velocity of the gas expelling from the chamber. The elastic bar is instrumented with a strain-gauge that provides, trough the elastic properties of the bar, the force acting on the specimen during the test. Two targets are attached on both ends of the specimen and their movements are measured by two contact-less displacement transducers.

131 131 Chapter V Dynamic properties of a natural stone It can be evidenced that a constant speed movement of the piston guarantees the constancy of the strain-rate during the test; this depends mainly by the constancy of the force exerted by the gas pressure on the piston face. A good result in this sense was obtained with small change of gas volume during the test in order to have small gas pressure decrease and consequently small piston force decrease. Furthermore, the load P resisted by the specimen is measured by the dynamometric elastic bar, whereas the specimen elongation ΔL is measured by the displacement transducers, sensing the displacement of the plates target fixed to both specimen ends. Such acquisitions allow obtaining the stress vs. strain relationship at the strain-rate level, achieved during the test. Figure 5.6 shows a NYT specimen during a HPM test. Results of the conducted tests are presented in Table 5.5. It can be observed that average tensile failure stresses of 0.86MPa and 1.19MPa were obtained for strain-rate of 10-1 s -1 and 5 s -1, respectively. Strain-rate [s -1 ] Table 5.5 Results of medium strain-rate tensile failure tests Specimen f t [MPa] f t average value [MPa] f t standard deviation [MPa]

132 132 Chapter V Dynamic properties of a natural stone Figure 5.6 A NYT specimen during medium strain-rate dynamic test High strain-rate tests In high strain-rate range 3 tests were conducted at both 20 s -1 and 50 s -1 strainrate level. To conduct such tests the MHB apparatus, already described in previous Chapters, was employed. Results of the conducted tests are reported in Table 5.6, revealing an average tensile failure stress of 1.71MPa and 1.97MPa for a strain-rate level of 20 s -1 and at 50 s -1, respectively. In Figure 5.7 two stress versus time curves at 20 and 50 s -1 are shown. It shows that, increasing strain-rate, specimens were able to withstand higher forces before achieving failure.

133 133 Chapter V Dynamic properties of a natural stone Strain-rate [s -1 ] Table 5.6 Results of high strain-rate tensile failure tests Specimen f t [MPa] f t average value [MPa] f t standard deviation [MPa] s s-1 NYT 2 stress [MPa] time [s] Figure 5.7 Stress vs. time curve for the two high strain-rates

134 134 Chapter V Dynamic properties of a natural stone It is outlined that in the presented cases stain-rate was calculated, according to equation presented in Chapter 2 (from (2.2) to (2.4)), in correspondence of the maximum stress [13]. In fact, contrarily to ductile materials, where plasticity makes strain-rate constant, in a brittle material, as NYT, the stain-rate varies over time. Furthermore, in order to make useful the obtained results in comparison with other experimental activities considering loading rate instead of strain-rate, the following considerations are necessary. Starting from the elastic properties of input and output bars, a corresponding loading-rate can be evaluated for each strain-rate; Table 5.7 reports such conversions values. Moreover, time necessary to achieve failure at the different strain-rates was also measured revealing an increment as strain rate decreases. Table 5.8 reports time of fracture for each strain rate. Table 5.7 Loading rates Strain-rate [s -1 ] Loading-rate [GPa/s] Table 5.8 Time of fracture Strain-rate [s -1 ] Time of fracture [s]

135 135 Chapter V Dynamic properties of a natural stone ASSESSMENT OF EXPERIMENTAL RESULTS Experimental data were processed to obtain stress vs. strain curves at different strain-rates. Figure 5.8 reports the curves obtained for each of the investigated strain-rate, compared with that from quasi-static loading conditions, indicated with a strain-rate of 10-5 s s s-1 1 s-1 stress [MPa] s-1 50 s-1 yellow tuff strain Figure 5.8 Average Stress vs. strain curves at different strain-rates From such figure it can be observed that as strain-rate increases ultimate failure stress increases whereas the corresponding strain slightly reduces its value. Moreover, Figure 5.9 depicts the strain histories measured by the input strain gauge during high strain-rate tests, distinguished into two curves (20 s -1 and at 50 s -1 of strain-rate). In this figure it can be noticed how different pre-loads, employed to obtain different strain-rates, generate different velocities of the

136 136 Chapter V Dynamic properties of a natural stone tests; this is observable in the slopes of the initial and final parts, representing the transmission and the reflection of the strain wave, respectively, which are higher as strain-rate increases input strain (20 s-1) input strain (50 s-1) NYT input strain ε I (50 s -1 ) ε R (50 s -1 ) ε R (20 s -1 ) ε I (20 s -1 ) time [s] Figure 5.9 Input strain history at different strain-rates In order to appreciate the improved strength under dynamic loading conditions the results of the average tensile failure stresses were processed in terms of Dynamic Increase Factor, DIF, f t, defined as the ratio of the dynamic values of the tensile failure stress over the static one. Hence, the results in Table 5.9 were obtained, revealing a maximum DIF, f t equal to 2.92, in case of strain-rate of 50s -1. Such data were then processed as depending on the strain-rate in logarithmic scale, obtaining the curve depicted in Figure 5.10.

137 137 Chapter V Dynamic properties of a natural stone Table 5.9 Experimental DIF for tensile failure stresses Strain-rate [s -1 ] Experimental DIF, f t DIF, ft Strain rate[s -1 ] 2.2 Figure 5.10 DIF,f t vs. strain-rate experimental data As it was expected, such figure reveals a behavior similar to that experienced in case of brittle ceramic materials, like concrete. In case of concrete, in fact,

138 138 Chapter V Dynamic properties of a natural stone DIF, f t dependence on strain-rate level in logarithmic scale presents two different slopes for medium strain-rate and high strain-rate and the variation of gradient is located at about 1s -1 [2, 3]. In the first case, the increase of strength is due to moisture content, involving viscosity mechanisms, whereas for high strain-rate inertia effects in cracking propagation become relevant, providing further resistance [2, 3, 14]. In the present case, a two segment trend-line was also drawn revealing a knee located at about 2.2 s -1 of strain-rate. Unfortunately, no specific formulations were found in literature for natural stones. However, given the similarities above described with concrete behavior and given that both concrete and NYT are brittle ceramic materials presenting a certain level of porosity a content of water, formulation provided in [2] was employed. Such relationship, namely CEB formulation, actually elaborated for concrete, allows evaluating the dynamic tensile strength at a given strain-rate level, starting from static mechanical properties. It allows predicting DIF, f t as: 1.016δ 1 DIF, f = ε t for ε 30s (5.2) ε ε DIF, f t = η for ε > 30s ε 0 1 (5.3) where ε is the strain-rate; ε 0 is a constant equal to s -1 and has the meaning of the static strain-rate; and logη = 7.11δ 2.33, 1 where δ =, f c f 0 f c is the static compression strength; f 0 is a constant equal to 10 MPa.

139 139 Chapter V Dynamic properties of a natural stone Then this relationship was employed trying to predict the experimental values of DIF, f t via a numerical elaboration; to do this the static compression strength was assumed equal to 3.57MPa, as obtained from the preliminary static tests. Results in Table 5.10 were found, while Figure 5.11 depicts the obtained curve on the experimental data. It can be observed that CEB formulation overestimates the experimental results, providing higher values of DIF, f t. This was actually expected since CEB formulation was calibrated for concrete, presenting higher static compression strength. Table 5.10 CEB formulation DIF for tensile failure stresses Strain-rate [s -1 ] Experimental DIF, f t

140 140 Chapter V Dynamic properties of a natural stone Experimental data CEB formulation DIF, ft Strain rate [s -1 ] Figure 5.11 DIF,f t vs. strain-rate compared with CEB formulation The conducted experimental analyses revealed that the investigated NYT, under high strain-rate conditions, is able to withstand to tensile forces about equal to three times those resisted in static conditions. Such results allow affirming that NYT under tensile loading conditions is strain-rate sensitive. Then, the author tried reproducing the obtained results in terms of DIF, f t, via CEB relationship for tensile strain-rate sensitiveness of concrete, given the similarities that NYT presents with concrete. However, CEB formulation applied to the investigated NYT strongly overestimates the experimental data.

141 141 Chapter V Dynamic properties of a natural stone Hence, it appears necessary that further mechanical dynamic characterization under tensile loading conditions needs to be conducted, also accounting for size effect, in order to obtain further data and calibrate a specific relationship for NYT. Furthermore, the influence of content of water on dynamic behavior of NYT needs to be investigated, given its importance in determining static mechanical properties of NYT. Hence, the author is currently working on replicating the conducted experimental activity on completely dry and completely wet NYT samples. As a final words, it is outlined that the conducted activities provides preliminary experimental data that can be helpful to assess design formulations, accounting for improved strength of a natural construction stone under dynamic loading conditions, as those induced by severe earthquakes, impacts or blast. REFERENCES 1 Mainstone, R. J., Properties of materials at high rates of straining loading, Matériaux et Constructions, Vol. 8, No. 44, 1975, pp Comité Euro-International du Béton. CEB-FIP Model Code 1990 Redwood Books, Trowbridge, Wiltshire, UK, Malvar L.J. and Ross C.A. Review of strain-rate effects for concrete in tension ACI Materials Journal vol. 95, 1998, pp Banthia, N., Mindess, S., Bentur, A. and Pigeon, M. Impact testing of concrete using a Drop-weight Impact Machine Experimental Mechanics, vol. 29, 1989, pp Lindholm, U.S. High strain-rate tests, in measurement of mechanical properties Techniques of Metals Research vol. 5, 1971, pp van Doormal, J.C.A.M., Weerheijn, J. and Sluys, L.J. Experimental and numerical determination of the dynamic fracture energy of concrete Journal de physique vol. 4, 1994, pp

142 142 Chapter V Dynamic properties of a natural stone 7 Malvar, L.J., Review of static and dynamic properties of steel reinforcing bars, ACI Materials Journal, vol. 95, 1998, pp Scarpati, C, Cole, P. and Perrotta, A., The Neapolitan yellow tuff. A large volume multiphase eruption from Campi Flegrei, Southern Italy, Bulletin of Vulkanology. Vol. 55, 1993, pp Rippa, F. and Vinale, F. Structures and mechanical behavior of a volcanic tuff Proceedings of the 5 th Congress of the International Society of Rock Mechanics, Melbourne, Dell Erba, L., Il tufo giallo napoletano, ed. Raffaele Pironti, Naples, Italy, P.V. Lade, Three parameter failure criterion for concrete, J. Engng. Mech. Div. A.S.C.E, Pellegrino, A. and Evangelista, A., Caratteristiche geotecniche di alcune rocce tenere italiane, Atti del 3 Ciclo di conferenze di Meccanica e Ingegneria delle Rocce, SGE ed., Torino, Novembre 1990, 13 Cadoni E., Labibes K., Berra M., Giangrasso M., Albertini C., High strain-rate tensile concrete behaviour, Magazine of Concrete Research, vol. 52, No.5, Oct. 2000, pp Rossi, P., van Mier, J.G.M. amd Toutlemonde, F. Is the dynamic behavior of concrete influenced by the presence of free water?, Fracture Mechanics of Concrete Structures (ed. Z.P. Bazant), Elsevier applied Science, London, 1992, pp

143 143 Chapter VI Particle methods for dynamic analysis Chapter VI PARTICLE METHODS FOR DYNAMIC ANALYSIS INTRODUCTION TO PARTICLE METHODS A variety of numerical methods have been recently proposed in the literature to address advanced mechanical problems, such as those involving rapid deformations, high intensity forces, large displacement fields. In many of these cases, in fact, classical finite element methods (FEM) suffer from mesh distortion, numerical spurious errors and, above all, mesh sensitiveness. Hence, to overcome such issues, a number of numerical methods, belonging to the family of the so-called meshless techniques, have been widely investigated and applied. The objective of employing these methods is to avoid the introduction of a mesh for the continuum, preferring a particle discretization, with the goal of obtaining an easier treatment of large and rapid displacements. Thus, meshless methods have been widely applied, mainly to fluid dynamics problems, where particle approaches appear to be more feasible. However, recently, a number of researchers have tried to extend meshless methods also to solid mechanics problems. Among the several meshless numerical methods proposed, particle methods and in particular Smoothed Particle Hydrodynamics (SPH) has been widely implemented and investigated. Historically SPH was

144 144 Chapter VI Particle methods for dynamic analysis introduced by Lucy [1] and Gingold and Monaghan [2] to treat astrophysics problems, and, then, a variety of formulations have been proposed to apply its principles to different problems, such as incompressible flows [3], elasticity [4], fracture of solids [5, 6] heat conduction [7]. Furthermore, in order to address a number of criticalities and issues, several improvements have been proposed: for instance, Swegle and co-workers [8] highlighted tension instability", fixing it through a viscosity-based procedure; Johnson and Beissel [9] proposed a method to improve strain calculation and Liu [10, 11] introduced the so-called Reproducing Kernel Particle Method (RKPM) to overcome deficiencies occurring on the boundary. Nowadays SPH represents itself a family of methods, given the large variety of formulations available in the literature stemmed from the original SPH approach. The main objective of the present chapter is to analyze some of the proposed particle formulations in a one dimensional setting, where a rigorous error analysis can be conducted. In particular, beside the original SPH approach, the following methods are focused on: Chen Beraun SPH [12], Finite Particle Method (FPM) [13, 14], First and second-order RKPM methods [10, 11, 15], Finally a novel enhanced SPH-like procedure, able to guarantee second-order accurate approximations is also proposed; it is called in the following a secondorder FPM. Accordingly, the chapter starts with a brief introduction to classical SPH methods, followed by a review of the methods listed above and by a presentation of the mainly proposed methodology. Then, all the numerical schemes are tested against three different 1D benchmarks, all dialing with the second derivative approximation, i.e.: Derivative test: second derivatives of given functions are evaluated via particle formulations and the results are compared with the exact solution. The order of convergence of the obtained solutions is determined as well. Elastic static test: the displacements of an elastic rod are determined via particle procedures, given boundary conditions and the body load (i.e.,

145 145 Chapter VI Particle methods for dynamic analysis the second derivative of the displacement field). Results are then compared with the exact solution and the order of convergence of the obtained solutions is determined as well. Elastic vibration test: again referring to an elastic rod, the eigenvalue problem is addressed, determining the errors related to the numerically obtained spectra. Finally, conclusions arising from the conducted investigations are presented and discussed. CLASSICAL SPH APPROACH The classical SPH approximation procedure [1, 2] takes its origin from the fact that, for a generic function B( x) defined in a continuous domain Ω, the value of the function in a generic point x i can be expressed using the following relationship Bx ( i ) = Bx ( ) δ i ( xd ) Ω (6.1) Ω where δ i ( x) is the Dirac's function centered in x i. A first approximation of the above relation can be introduced replacing the Dirac's delta with a smooth kernel function Wi ( x ), Bx ( ) B( xw ) ( xd ) Ω (6.2) i Ω i The approximation clearly depends on how the kernel function approximates the Dirac's delta and, so, in the original works [1, 2], Wi ( x ) was required to satisfy at least the following conditions: Wi ( x) 0, x Ω; Wi ( x) dω= 1; Ω

146 146 Chapter VI Particle methods for dynamic analysis Wi ( x ) regular enough, i.e., derivable many times with continuous derivatives; Wi ( x ) defined on a compact support. Kernel functions may assume different expressions, but typically Gaussian or spline functions are used. Gaussian functions are often preferred, even if they do not respect the last of the listed properties. However, for practical purposes they are negligible outside a compact region that, with abuse of notation, is referred to as support. The diameter measure of such a support is called smoothing length, and is denoted by the parameter h. As h approaches zero, the kernel approaches a Dirac's delta. A second approximation may now be introduced performing a discretization of the domain Ω, introducing a partition of Ω through a finite number N of subdomains ΔΩ, such that j N ΔΩ i= 1 j =Ω (6.3) A centroid x j, referred to as particle, may be associated to each subdomain ΔΩ j ; then, equation (6.2) is further approximated substituting the integration with a summation of the integrand function, computed at the centroids x j, and weighted using ΔΩ j, i.e. N Bx ( ) Bx ( ) W( x) ΔΩ i j i j j j= 1 (6.4) Observe that, with some notational abuse, both the subdomain and its measure is denoted with ΔΩ. This step can be interpreted as a numerical quadrature of equation (6.2). j

147 147 Chapter VI Particle methods for dynamic analysis APPROXIMATION OF DERIVATIVES To obtain an expression for the approximation of derivatives via the SPH approach, several procedures may be proposed. The following paragraphs examine and compare some of the most significant methods available in the literature, investigating in particular the corresponding error orders. For the sake of simplicity, only the 1D case is discussed. Original formulation Following the original formulation proposed by Gingold and Monaghan [2] and applying the first approximation step to the gradient of a generic function B( x ), the following relationship is obtained (6.5) Employing Green's formula yields (6.6) where is the boundary of and n is the outward normal. If is compactly supported in, the first term of the right-hand-side of equation (6.6) is null, leading to (6.7) Discretizing by means of numerical quadrature it yields (6.8)

148 148 Chapter VI Particle methods for dynamic analysis Accordingly, given the values, the quantities can be computed in a typical SPH fashion. Expression (6.7) is a reasonable approximation in the interior of ; indeed, it is exact at least on constant functions when is null (or negligible) on, which implies (6.9) However, close to the boundary, that is, when is not negligible on, typically such assumption fails, and approximations (6.7) and (6.8) deteriorate. To overcome this difficulty, many authors suggest to replace (6.7) by (6.10) Observe that (6.7) and (6.10) are equivalent when (6.9) holds, but (6.10) is always exact on constants. Discretizing (6.10), it yields (6.11) To obtain also the second derivative, this procedure can be reiterated, giving rise to the following expression (6.12) It is important to remark that, though being an improvement with respect to (6.8), expression (6.11) is still not fully satisfactory, since it could be proven that it does not converge when evaluated close to the boundary (as proven in the next subsection) and this motivated the development of other approaches. Chen and Beraun's formulation Chen and Beraun [12] proposed a generalized SPH formulation, directly derived by a Taylor's expansion of up to the first derivative term, i.e.

149 149 Chapter VI Particle methods for dynamic analysis (6.13) In fact, multiplying both sides of equation (6.13) by over yields and integrating (6.14) where is the approximation error (second-order, in this case) and is the -norm. From (6.14) it yields (6.15) Since, neglecting the second term in the right-hand-side of the previous equation results in an error of order. Notice that, if the second-order momentum of,, is null or negligible, as it typically happens far from the boundary, then and (6.15) turns out to be second-order accurate when e is neglected. This is not the case, however, in a neighborhood of the boundary. Then, discretizing by numerical quadrature, the following relationship holds

150 150 Chapter VI Particle methods for dynamic analysis (6.16) It can be observed that this expression is different from the original formulation (6.11), because it presents as denominator the static momentum of the derivative of the kernel function. Since the formulation is derived from Taylor's formula truncated at the first order, such a denominator can be considered as a correction factor, which allows to obtain a first-order approximation everywhere. Hence, adopting (6.11) instead of (6.16), it can be clearly obtained a method which is first-order where the missing denominator is close to 1 (i.e., far from the boundary) and not necessarily converging where the missing denominator is far from 1 (i.e., close to the boundary). Then, to obtain an expression for the second derivative, the procedure is reiterated starting from the Taylor's formula up to the second derivative, i.e. (6.17) Multiplying both sides of (6.17) by yields and integrating over the domain (6.18) where, in this case,. Then, from (6.18), it follows

151 151 Chapter VI Particle methods for dynamic analysis (6.19) Also in this case, as in the expression for the first derivative, neglecting the error term in the right-hand-side introduces an error. Far from the boundary, the term e increases again its order, becoming, since the third order momentum of vanishes; hence (6.19) becomes second-order accurate when e is neglected. Discretizing by numerical quadrature it can be finally obtained (6.20) It should be mentioned that the proposed procedure was also used to solve the tension instability issue [16]. RKPM method Introduced by Liu and coworkers [10, 11, 15], this method has been widely employed to correct SPH classical formulations improving accuracy (see, e.g., [17, 18, 19]). The key idea is to replace (6.2) by an improved representation

152 152 Chapter VI Particle methods for dynamic analysis (6.21) where is an improved Kernel function modified for each particle as (6.22) where is a classical Gaussian function and is a polynomial, i.e.: Starting from the Taylor's series expansion and projecting it against (6.23) yields (6.24) Then, it is imposed that the zeroth-order momentum (i.e., the integral) of is equal to one, and that higher-order momenta of are null. Such conditions allow to define an algebraic system whose unknowns are the coefficients of the polynomial in (6.23). Solving this system for each particle within the domain leads to determine the kernel functions. If only the first-order momentum of is forced to be zero, the error of the Taylor's series expansion in equation (6.24) is a second-order; hence, this approach is hereafter referred to as first-order RKPM. If, instead, both the firstand the second-order momenta of are forced to be zero, the term e in equation (6.24) represents a third-order error and the approach is referred to as second-order RKPM. Once corrected kernel functions are determined, the derivatives can be expressed, adopting a classical SPH approach, through the following equations

153 153 Chapter VI Particle methods for dynamic analysis (6.25) (6.26) which are completely similar to (6.11) and (6.12). It should be mentioned that Bonet and Kulasegaram also discussed the evaluation of the second derivatives through this approach in [20], involving the evaluation of the second derivatives of the Kernel functions. A direct formulation was derived, coming from a double application of the Taylor's series expansion. FPM formulation Batra and Zhang [13] and Liu et. al [14] independently proposed a different formulation to determine at the same time a function and its derivatives and after referred in the literature as Finite Particle Method (FPM). The basic idea is to project the Taylor's formula on a number of independent functions. This provides a linear algebraic system for each particle, whose unknowns are the approximations at of the function and its derivatives, up to the order of the Taylor's formula. Hence, in the 1D case, considering a series expansion up to the first-order and employing as projecting function a kernel function and its derivative, the following relationships are obtained (6.27)

154 154 Chapter VI Particle methods for dynamic analysis (6.28) where and represent the errors due to the series truncation. Then, defining (6.29) equations (6.27) and (6.28) can be re-arranged as (6.30) These equations represent a linear algebraic system which can be solved with respect to and. To assess the corresponding error, the following considerations can be made. For particles far from the boundary, the associated kernel functions are negligible on the domain boundary; hence, given that is a skew-symmetric function, its second-order momentum vanishes and it is obtained that. Being symmetric, it is also obtained that. Moreover, considering again the symmetry of and the skew-symmetry of, the terms and vanish and the system yields (6.31)

155 155 Chapter VI Particle methods for dynamic analysis Being and, the error introduced neglecting and for the evaluation of and its first derivative, equal to in both cases. On the other side, for particles close to the boundary, both functions and are not completely developed; thus it is obtained that and. Moreover, close to the boundary, terms and do not vanish; hence the error introduced on and neglecting and can be evaluated as (6.32) where and represent constant terms. Thus, the evaluation of is affected by an error of order inside the domain and of order close to the boundary. Trying to evaluate also the second derivative of the function B(x), the procedure can be in principle reiterated. Unfortunately, this operation causes a propagation of the error, making the formulation not converging. Modified FPM formulation To overcome the problems highlighted above and to have a converging solution also for the second derivative, a slightly different procedure is here proposed and discussed. In case the values of do not represent real unknowns, the system can be rearranged to introduce also the second derivatives. In fact, introducing a Taylor's expansion for up to the second order, multiplying

156 156 Chapter VI Particle methods for dynamic analysis once by and by and integrating, the following relationships are obtained (6.33) (6.34) where terms and represent the errors due to the series truncation. Then, defining (6.35) equations (6.33) and (6.34) can be re-arranged as (6.36) In this case, conducting a discussion similar to the previous case, it can be observed that, far from the boundary, given that is a symmetric function, its third-order momentum is null and, whereas

157 157 Chapter VI Particle methods for dynamic analysis. Moreover, due to the symmetry of and the skew-symmetry of, terms and vanish, yielding (6.37) Given that is proportional to and is proportional to, the error related to the first and second derivative evaluation is. Instead, close to the boundary, functions are not completely developed and it yields and. Terms and do not vanish and the error introduced neglecting and can be evaluated as (6.38) Hence, the maximum error related to first and second derivative evaluation is of the order of far from the boundary and of the order of h close to. A NOVEL SECOND-ORDER FPM FORMULATION Aiming at obtaining in the whole domain a second-order accurate estimate of the second derivative, a novel FPM-based procedure is here presented. Based on a modification of the kernel functions, this method is able to overcome the

158 158 Chapter VI Particle methods for dynamic analysis reduction of accuracy occurring close to the boundary. The basic idea to achieve this goal is to construct kernel functions satisfying particular conditions directly coming from the error evaluation procedure conducted in the previous section. Similarly to RKPM, it is defined: (6.39) where are the classical kernel functions and (6.40) are polynomials to be determined by the following conditions: (6.41) with, i.e. for each particle. Imposing the above conditions directly in the discrete setting, it is obtained (6.42) The choice of the polynomial (6.40) guarantees that the linear system given by (6.42) for the determination of the coefficients a, b, and c is nonsingular for all i. Instead, a quadratic polynomial would give a singular or close to singular system when is far from the boundary. Moreover, observe that the Gaussian kernel functions on an infinite

159 159 Chapter VI Particle methods for dynamic analysis domain naturally verify the previous conditions, therefore far from the boundary. So, only kernel functions close to the boundary are effectively modified. Then, as in the modified FPM formulation, projecting the Taylor's series expansion against the functions and, the following algebraic linear system in terms of the derivatives of is obtained for each particle: (6.43) where and are the first and second momenta of and are the first and second momenta of (see for example equations (6.35)); finally, and represent the errors due to the series truncation. As a consequence of conditions (6.41) (or (6.42)), it is obtained,, and, as usual,, for every i. Then the error on the evaluation of the first and second derivative of B is (6.44) It is then clear that the employed conditions allow to obtain at least a secondorder error everywhere in the domain.

160 160 Chapter VI Particle methods for dynamic analysis NUMERICAL TESTS To assess the performance of the discussed formulations, the following three different sets of numerical tests are performed: Derivative test: second derivatives of known functions are evaluated via particle formulations and the results are compared with the exact solution. The order of convergence of the obtained solutions is determined as well. Elastic static test: the displacements of an elastic rod are determined via particle procedures, given boundary conditions and body load; results are then compared with the exact solution and the order of convergence of the obtained solutions is determined as well. Elastic vibration test: again referring to an elastic rod, the eigenvalue problem is addressed, determining the errors related to numerically obtained spectra. Such tests have been conducted using the different SPH-based approaches discussed in previous sections, i.e., in particular: original formulation; Chen and Beraun's formulation; first and second-order RKPM; modified FPM; novel second-order FPM. All tests are carried out on the domain, with a uniformly spaced particle discretization. A Gaussian function, with h equal to the distance between two consecutive particles, is considered as the basic kernel function. Derivative test The aim of this first test is to evaluate the approximation, obtained through the discussed formulations, of the second derivative of given functions, comparing the results with the exact solutions. In particular the following two functions are considered:

161 161 Chapter VI Particle methods for dynamic analysis, whose second derivative is ;, whose second derivative is. Figure 6.1 Infinity norm error in the second derivative approximation vs. particles number, function To evaluate convergence orders, the infinity norm of the difference between the exact and the numerical solution has been computed for each investigated formulation. These values, for the exponential function, are plotted in logarithmic scale, as a function of the number of particles, in Figures 6.1; the slope of the obtained curves indicates the order of convergence. A similar behavior was obtained for the sinusoidal function. It can be observed that the original formulation and the RKPM formulations appear to be not converging. This happens because the infinity norm of the error is governed by the maximum error in the domain, which occurs on the boundary. On the contrary, in the FPM formulations and, for the sine function (but not in general) also for the Chen and Beraun's formulation, the infinity norm of the error decreases as the particle number increases. It is finally possible to see that modified FPM and (only in the case of the sine function, that is, when it converges) Chen and Beraun's method show a first order convergence, while the novel FPM approach shows as expected a second order convergence.

162 162 Chapter VI Particle methods for dynamic analysis Elastic static test The aim of this test is to address an elastic rod static problem (assumed to have unitary length and axial stiffness), determining the axial displacement u at particle positions given an axial body load f, and comparing the obtained results with the exact solution. The second order differential equation governing the problem is (6.45) with appropriate boundary conditions. In particular, it is considered a first case with and homogeneous Dirichlet boundary conditions, i.e.. The corresponding exact solution is. Moreover a second case is considered, with and a traction-free condition for (i.e., an homogeneous Neumann boundary condition). So, it is imposed and the corresponding exact solution is. Then, also in this case, the infinity norm of the difference between the exact and the numerical solution have been evaluated for each investigated formulation, in order to study the order of convergence for the different formulations. The results for the two studied problems are reported in Figures 6.2, for the exponential function; a similar behavior was obtained for the sinusoidal function. It can be observed that the FPM formulations present a second-order convergence, whereas original formulation and first-order RKPM procedure are characterized by a first-order convergence. Instead, second-order RKPM is not convergent at all, whereas the Chen and Beraun's formulation presents a second-order convergence in the first test case, which reduces to a first-order convergence in the second test case. It is finally highlighted the better performance, in terms of global error, of the novel FPM formulation with respect to the modified FPM in the second test case, where a Neumann boundary condition is imposed.

163 163 Chapter VI Particle methods for dynamic analysis Figure 6.2 Displacement infinity norm error vs. particle number, for exponential load imposing a Neumann boundary condition Elastic vibration test Also in this test it is referred to an elastic rod, which is assumed to have unitary axial stiffness and density, and Dirichlet boundary conditions: The governing equation of the corresponding dynamic problem is (6.46) (6.47) where X and x are the initial and the current configuration of the element, respectively. In a stationary problem the following equation holds (6.48) where is the vibration frequency of the rod; hence, it can be written (6.49) and the solution of this differential equation is given by

164 164 Chapter VI Particle methods for dynamic analysis (6.50) Thus, imposing boundary conditions, it may be found that where n is a positive integer. The discrete counterpart of (6.49) is (6.51) (6.52) where is the stiffness matrix associated to the second-order discrete derivative (note that the mass matrix in SPH-based approaches is the identity). Hence, the eigenvalues of K represent the numerical approximation of and the ratio between the numerically evaluated vibration frequencies and the corresponding exact ones can be computed for each vibration mode, giving rise to a normalized numerical spectrum. Figures 6.3 report the different normalized numerical spectra computed, employing 100 particles and using the different considered SPH-based methods as well as, for comparison reasons, using standard linear finite elements. It can be observed that the original and RKPM formulations strongly underestimate vibration frequencies, also for lowest modes, presenting a rapidly decreasing spectrum (the original and the Order RKPM formulation curves overlap in Figure 6.3). On the contrary, Chen and Beraun's formulation, the modified FPM and the second-order FPM show very similar results (the three curves overlap in Figure 6.3), approximating much better the overall spectrum and, in particular, predicting very well low frequencies.

165 165 Chapter VI Particle methods for dynamic analysis Figure 6.3 Normalized numerical spectra (100 particles) ASSESSMENT OF NUMERICAL RESULTS Here several particle procedures are investigated, with the goal of evaluating their performance in the approximation of derivatives and, as a consequence, in the approximation of static and dynamic elasticity problems. In particular, an analysis of the error has been conducted for a selection of methods, derived by means of Taylor's series expansion. The original SPH method, the Chen and Beraun's formulation, the RKPM procedures and a modified FPM formulation have been considered. The performed numerical tests reveal that the original and RKPM formulations appear to suffer of lack of accuracy in boundary zones, and perform poorly in frequency analysis. The modified FPM formulation, instead, presents second-order accuracy in approximating second derivatives inside the domain, but only first-order accuracy close to the boundary. A novel method is also illustrated with the objective of overcoming such boundary deficiencies, coupling the projection procedure of FPM and enhanced kernel functions inspired by RKPM. In this way, second-order accuracy in approximating second derivatives is maintained even close to the boundary. The conducted numerical tests confirm these conclusions and, in particular, the second derivative tests highlight the improved accuracy provided by the proposed formulation.

166 166 Chapter VI Particle methods for dynamic analysis REFERENCES 1 Lucy LB. A numerical approach to the testing of the fission hypothesis. Astronomical Journal 1977; 88: Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society 1977; 181: Monaghan JJ. Simulating free surface flows with SPH. Journal of Computational Physics 1994; 110: Libersky LD, Petschek AG. Smooth particle hydrodynamics with strength of materials. In Proceedings of the Next Free-Lagrange Method, Lecture Notes in Physics Trease HE, Fritts MJ, Crowley WP (eds). vol. 395, Springer-Verlag, Berlin, 1991; pp Benz W, Asphaug E. Impact simulations with fracture: I. methods and tests. Icarus, 107; Benz W, Asphaug E. Simulations of brittle solids using smooth particle hydrodynamics. Computer Physics Communications : Cleary PW, Monaghan JJ. Conduction modeling using smoothed particle hydrodynamics. Journal of Computational Physics 1999; 148: Swegle JW, Hicks DL, Attaway SW. Smoothed particle hydrodynamics stability analysis. Journal of Computational Physics 1995; 116: Johnson GR, Beissel SR. Normalised smoothing functions for SPH impact computations. International Journal for Numerical Methods in Engineering 1996; 39: Liu WK, Jun S, Li S, Zhang YF. Reproducing Kernel Particle Method. International Journal for Numerical Methods in Fluids 1995; 20: Liu WK, Jun S, Li S, Adee J, Belytschko T. Reproducing Kernel Particle Methods for structural dynamics. International Journal for Numerical Methods in Engineering 1995; 38:

167 167 Chapter VI Particle methods for dynamic analysis 12 Chen JK, Beraun JE. A generalized smoothed particle hydrodynamics method for nonlinear dynamic problems. Computer Methods in Applied Mechanics and Engineering 2000; 190: Batra RC, Zhang GM. Analysis of adiabatic shear bands in elastothermo-viscoplastic materials by modified smoothed-particle hydrodynamics (MSPH) method. Journal of Computational Physics 2004; 201: Liu MB, Xie WP, Liu GR. Modeling incompressible flows using a finite particle method. Applied Mathematical Modeling 2005; 29: Jun S, Liu WK, Belytschko T. Explicit Reproducing Kernel Particle Methods for large deformation problems. International Journal for Numerical Methods in Engineering 1998; 41: Chen JK, Beraun JE, Jih CJ. Improvement for tensile instability in smoothed particle hydrodynamics. Computational Mechanics 1999; 23: Bonet J, Kulasegaram S. A simplified approach to enhance the performance of smooth particle hydrodynamics methods. Applied Mathematics and Computation 2002; 126: Bonet J, Kulasegaram S. Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics (CSPH) methods. International Journal for Numerical Methods in Engineering 2001; 52: Vidal Y, Bonet J, Huerta A. Stabilized updated Lagrangian corrected SPH for explicit dynamic problems. International Journal for Numerical Methods in Engineering 2007; 69: Bonet J, Kulasegaram S. Finite increment gradient stabilization of point integrated meshless methods for elliptic equations Communications in Numerical Methods in Engineering 2000; 16:

168 168 Chapter VII Development of a blast protection barrier Chapter VII DEVELOPMENT OF A BLAST PROTECTION BARRIER INTRODUCTION TO FRAMEWORK ACTIVITIES Recent terrorist acts have contributed to change the design approach to critical infrastructures; in fact, malicious disruptions, blasts, or impacts have unfortunately become part of the possible load scenarios that could act on constructed facilities during their life spans. Consequently, specific protection interventions are introduced to minimize disruptive effects, guarantee the safety of the occupants of a facility and to the extent possible, maintain the functionality of a facility. Mitigation techniques used for other extreme events such as earthquakes could be adopted to resist blast loads [1], and fiber reinforced polymer (FRP) composites comprise a promising strengthening solution in the form of an externally bonded system [2]. Conversely, using FRP for the construction of barriers or similar security has not yet been explored [3] even though glass FRP (GFRP) porous barriers have been identified as a possible measure to prevent malicious disruptions, provide a standoff distance in case of blast actions, and reduce the consequences of an impact. In December 2006, the Italian research center AMRA, based at University Federico II of Naples, and its industry partners started the Security of Airport

169 169 Chapter VII Development of a blast protection barrier Structures project, already mentioned in Chapter IV, focusing on protecting airport infrastructures against man-made disruptions and eco-terrorist acts in particular. The end user partner of this project was ENAV, the Italian agency for air traffic control. The project had the financial support of the European Commission Directorate General Justice, Freedom and Security, through the 2006 European Programme for Critical Infrastructures Protection [4, 5]. According to the end-user requirements, the main objective of the research project was to develop and deploy a structural fencing system able to protect VHF Omnidirectional Range (VOR) stations against malicious actions consisting of intrusion and/or blast loads of relatively small explosive charges placed in the vicinity or in contact with the barrier. A critical feature of any protective barrier for this type of facility is radio transparency, which is necessary to avoid any disturbance to radio communications of specific frequencies. Hence, to achieve such goals, a discontinuous (porous) barrier composed by GFRP and precast concrete elements reinforced with GFRP bars was designed to take advantage of electromagnetic and mechanical properties of composites. The proposed barrier provides protection through two contributions. First, its geometrical and mechanical characteristics ensure protection against intrusions and blast loads. Second, its shape provides an attenuation of the blast shock wave, adding some level of additional protection for facilities located beyond it. The end-user specified the design charge for the blast action equal to 5 kg of commercial explosive, intended to represent the possible disruptive action of an eco-terrorist attacking radio-communication infrastructures. Given the amount of explosive as a fixed parameter, in addition to contact charge configuration employed in third test, the first two tests investigated two different stand-off distance configurations aiming at reproducing the cases of prevented direct access to the barrier. The objective of this work is to describe some results of the blast test activity conducted on full-size specimens of the proposed barrier and to discuss their effectiveness in mitigating blast shock waves. A simplified model is also proposed to predict the reduction of the peak pressure due to the porous barrier; an example of the computational procedure is described in Appendix 7.A.

170 170 Chapter VII Development of a blast protection barrier GEOMETRY AND MECHANICAL PROPERTIES OF THE BARRIER The barrier is composed of vertical GFRP pipe elements secured in a precast reinforced concrete pedestal base. Each individual pipe is not connected to the adjacent ones to avoid the domino effect in case of failure. Each pipe has an external diameter of 85 mm and a wall thickness of 5.5 mm. Pipes are spaced 65 mm on centers. The GFRP pipe wall is a typical pultruded product with an internal core reinforced with unidirectional E-glass fibers (to provide strength and stiffness in the longitudinal direction) and two external layers reinforced with a mat of E-glass chopped fibers (to facilitate manufacturing and allow for a resin rich surface). Both core fibers and mats are impregnated with a polyester resin. The concrete base and the GFRP pipes are 0.5 m and 2.5 m high, respectively. A prototype barrier is represented in Figure 7.1. The concrete base is made of interconnected precast segments (dry joint, male-female connection to provide shear interlock) having a thickness of 150 mm. Each segment has an inverted T cross-section, whose web and flange are 0.3 m and 0.7 m wide, respectively. The overall height and flange thickness are equal to 0.5 m and 0.2 m, respectively.

171 171 Chapter VII Development of a blast protection barrier Figure 7.1 Prototype of the barrier One pipe is installed in each precast concrete segment in a 0.3 m deep, 100 mm diameter hole centered on the web of the precast element. The interspace between pipe and concrete is filled with a no-shrink cementitious mortar. Two additional holes for tie anchors are present in the flanges of each segment. Two GFRP bars can be inserted through these holes and secured with cementitious mortar to create a deep foundation system. The diameter and depth of these GFRP tie bars have to be determined based on ground characteristics. The GFRP s mechanical behavior was investigated for a quality control process and to collect data that will be used in performance modeling (together with high-strain rate properties) not discussed herein. To this aim, coupon tests were conducted on flat GFRP specimens taken from the pipe to evaluate their direct tension properties [4]. Two typologies of specimens were tested: Integral, composed of the internal core and the external layers.

172 172 Chapter VII Development of a blast protection barrier Peeled, composed only of the internal core (after machining the surface layers). The former (integral) were 5.5 mm thick, while the latter (peeled) were about 3 mm thick. All the specimens were 300 mm long and 15 mm wide. Figures 7.2a and 7.2b show a typical specimen during and after the test. All the specimens exhibited a linear behavior up to failure. Table 7.1 shows the results derived from such tests. Figure 7.2a Specimen during test Figure 7.2b Specimen after failure Table 7.1 Summary of the results of tensile failure tests on GFRP specimens Integral specimens (5 samples) Peeled specimens (6 samples) Failure stress [MPa] (11.5%) (6.4%) Experimental ultimate strain (4.8%) (2.5%) Chord modulus of elasticity +, [GPa] 44.1 (3.5%) 51.3 (2.1%) Theoretical ultimate strain* Terms in parentheses represent the coefficient of variation + Accordingly to [4], chord modulus is evaluated as the ratio of stress increment over strain increment, corresponding to the strain interval from to * evaluated as the ratio of the failure stress over the chord modulus of elasticity

173 173 Chapter VII Development of a blast protection barrier Four points bending tests were also performed on five pipe specimens to investigate their flexural behavior. Figure 7.3 depicts a typical specimen during the test, whereas Figure 7.4a describes the test scheme. This type of test can be very simply undertaken and is complementary to the tensile test on coupons in terms of quality control. The midspan cross-section was instrumented with linear variable displacements transducers (LVDT) and with strain gauges on the top and bottom of the cross-section. A load cell was used to record the applied load. The data were processed to obtain moment-curvature relationships, and those slopes indicate the flexural stiffness of the tested pipes. Table 7.2 presents the values obtained for each of the performed tests. Figure 7.3 Pipe specimen in four points bending test configuration

174 174 Chapter VII Development of a blast protection barrier Table 7.2 Summary of the results of four-point bending tests on GFRP pipe specimens Flexural stiffness [knm 2 ] Global Young s modulus [GPa] Compression Young s modulus [GPa] specimen specimen specimen specimen specimen Average value 41.2 knm GPa 32.7 GPa Coefficient of variation 6.1% 6.1% 8.8% Note: The flexural stiffness was evaluated on the midspan cross-section, at a tensile strain of 0.003; the inertia modulus of the compression zone and tensile zone referred to neutral axis and the global inertia modulus referred to the center of the cross section were equal to mm 4, mm 4 and mm 4, respectively. Support GFRP pipe Load application points F Load cell A Figure 7.4a Four point bending test set-up (dimensions in m)

175 175 Chapter VII Development of a blast protection barrier Figure 7.4b Global shear failure occurring in four point bending test Dividing the experimental flexural stiffness by the moment of inertia of the cross section, equal to 10.7x10 6 mm 4, a global Young s modulus was obtained that combines both tensile and compression contributions. To separate them and determine the compression Young s modulus, the following equation was used: ( EI = E I + E I (7.1) ) exp tens tens comp comp In which (EI) exp is the experimental flexural stiffness E tens is the tensile Young s modulus obtained from the tensile tests performed on the integral specimens, equal to 44.1 GPa I tens and I comp represent the moment of inertia of the tensile and compression zone, respectively, related to neutral axis E is the compression Young s modulus to be evaluated comp

176 176 Chapter VII Development of a blast protection barrier The results shown in Table 7.2 reveal average global tension and compression Young s moduli of 38.4 and 32.7 GPa, respectively, in good agreement with values commonly exhibited by GFRP [5]. The ratio between the average compression Young s modulus and the average tensile Young s modulus is 0.74, in good agreement with literature data [6] reporting During the performed tests, a global shear failure was experienced. As depicted in Figure 7.4b, a longitudinal fracture developed in the shear critical zone and spread towards midspan at the location of the neutral axis line. The tests revealed an ultimate bending moment of 7 knm and a maximum tensile strain of The latter, using the tensile Young s modulus obtained from tensile failure tests, corresponds to a stress value of 282 MPa, equal to 44 percent of the tensile failure strength reported in Table 7.1. To complete the mechanical characterization, a dynamic tensile characterization activity was performed, where high strain-rate values of interest ranged from 1 s -1 to 700 s -1. The performed tests revealed an increase in tensile strength, which reached 88% in case of the highest strain rate. The Young modulus presented an even higher enhancement, increasing up to 178%, in case of the highest strain rate. A detailed discussion of these characteristics is reported elsewhere [7]. The results of the dynamic tensile characterization will be used to construct complete constitutive laws and to validate the dynamic behavior of the barrier system via sophisticated numerical models. ASSESSMENT OF THE ELECTROMAGNETIC DISTURBANCE To validate the designed solution, electromagnetic disturbance introduced by the barrier to radio electromagnetic waves was completely assessed. Numerical analyses describing the electromagnetic phenomena were carried out. Full-scale experimental tests were also performed on samples of the barrier in an anechoic chamber, simulating the barrier submerged into the electromagnetic field generated by the radio-communication antennas. The results of the numerical analyses were then compared with the data obtained in the experimental tests.

177 177 Chapter VII Development of a blast protection barrier These studies confirm that composites, and in particular GFRP (Glass Fiber Reinforced Polymers), result in low interference with electromagnetic fields, as could be argued and also expected from the available literature. In particular, the GRFP tubes seem to have minimal influence on the electromagnetic field, whereas the concrete basement of the barrier produced more considerable disturbance, even if not reinforced with steel rebar. The obtained data suggest that internal steel reinforcement would not have increased the disturbance introduced by the concrete elements. Thus the proposed barrier appears to fit the low disturbance requirements to fence air traffic control sites if the concrete basement is placed under the counterpoise level of the antennas, which currently occurs in most VOR sites. BLAST TESTS Three blast tests were performed on the three separate barriers to validate the system s capability to withstand blast-induced loads and reduce the shock wave propagating beyond the barrier. In these tests, a TNT charge was detonated at a specified distance from a specimen composed of 13 pipes, installed as described previously, to form a 2 m long barrier. Figure 7.5 shows a specimen before the test. The charge was placed at 1.5 m from the ground to avoid energy dissipation in the formation of the crater and consequent lack of accuracy in evaluating the quantity of energy propagating in air. In each test, 5 kg of commercial grade quarry explosive, corresponding to 4 kg of equivalent TNT explosive was used, and the distances between the charge and the barrier were set at 5 m, 3 m, and 0.5 m, respectively. The tests were filmed with highspeed cameras, and specimens were instrumented with six accelerometers, three contact pressure gauges and nine strain gauges, located on the central pipe, as depicted in Figure 7.6, reporting strain gauges labels. Eight free air pressure gauges were distributed around the barrier to measure the evolution of the pressure field induced by the explosion.

178 178 Chapter VII Development of a blast protection barrier Figure 7.5 Barrier specimen before the blast test 125.0cm charge 62.5cm cm Figure 7.6 Strain gauges set up

179 179 Chapter VII Development of a blast protection barrier Strain gauge records allow evaluating the structural response of the central pipe under blast action; such data are available for the first and second blast tests, since damage occurred on the strain gauges during the third test. A preliminary elaboration of strain data is presented, aiming at (a) evaluating the stress levels occurring on instrumented pipes during blast excitation and (b) performing a comparison with levels reached during mechanical static tests. Maximum absolute strains recorded by strain gauges 1-6 during the first and the second test are reported in Table 7.3. Maximum deformations were recorded at sensors 5 and 6, placed close to the base of the pipe; in particular, both sensors recorded the same maximum values of and , during the first and the second test, respectively. These values are much lower than the failure strain obtained in the laboratory for quasi static tensile tests (0.0150) (Table 7.1), quasi static four-point bending tests (0.0064) and high strain rate tensile tests (0.0124). Furthermore, the pipes were carefully inspected and no local damage was visually observed. This allows affirming that no structural damage occurred in the GFRP during the first and the second test. Table 7.3 Maximum absolute GFRP pipe strains during blast tests Strain Gauge First shot [m/m] Second shot [m/m] Furthermore, strain data were employed to derive the internal bending moment acting on the instrumented pipe during blast excitation. Onto this end, strain gauges pairs 1-2, 3-4, 5-6 (Figure 7.6) were used. Each of these pairs was placed at a different level and diametrically opposite sides of the pipe element (specifically 125.0, 62.5 and 2.5 cm from the concrete base, respectively),

180 180 Chapter VII Development of a blast protection barrier identifying a cross section. For each pair, the curvature χ was derived as shown below, assuming that a plane section remains plane under loading: ε1+ ε 2 χ = (7.2) h Then, using the average flexural stiffness (EI) exp obtained from four-point bending tests, the bending moment history was determined at each instrumented cross section as: M = χ( EI) exp (7.3) Figure 7.7 and Figure 7.8 report bending moment histories at instrumented cross sections for the first and the second test, respectively. In both cases, bending moment histories exhibited sinusoidal trends, clearly induced by oscillating displacements. Maximum values are equal to 0.65 knm and 1.81 knm in the first and the second blast respectively, occurring, in both cases, at the base of the pipe, as it could be expected by a cantilever beam behavior. As further proof of the absence of damage, computed moment values are below 7 knm, representing failure bending moment experienced during four-point bending tests. Both in the first and the second blast tests, oscillation started about 330 ms after explosion time. Considering that the blast pressure reduced to zero not more than 40 ms after explosion time, as recorded from contact pressure sensors, pipe displacements can be considered as elastic free vibrations, induced by an initial impulsive action.

181 181 Chapter VII Development of a blast protection barrier Section A - 2.5cm from the basis Section B cm from the basis Section C cm from the basis 1.00 bending moment [knm] time after explosion [ms] Section C Section B Section A Figure 7.7 Bending histories at instrumented cross sections First test

182 182 Chapter VII Development of a blast protection barrier 2.00E E+00 Section A - 2.5cm from the basis Section B cm from the basis Section C cm from the basis 1.00E+00 bending moment [knm] 5.00E E E E E E+00 time after explosion [ms] Section C Section B Section A Figure 7.8 Bending histories at instrumented cross sections Second test

183 183 Chapter VII Development of a blast protection barrier Figure 7.9 provides a schematic plan view of the test set up showing the location of the two air pressure gauges S1 and S6, placed at opposite sides with respect to the charge. Gauge S1 measures the free air pressure at a given distance while gauge S6 measures the free air pressure at the same distance from the charge, but behind the barrier. Hence, by comparing the pressure history acquired by these sensors, the interference of the barrier on the shock wave can be directly evaluated. With reference to Figure 7.9, the distance between gauge S6 and the center of the barrier (D T - D B ) was kept constant at 4m for all tests. For this reason, D T was set equal to 9m, 7m, and 4.5m for the first, second, and third blasts, respectively, with D B equal to 5m, 3m, and 0.5m. The length of the barrier test article was chosen recognizing that the blast wave beyond the barrier, recorded by rear sensors, is surely affected by diffractions occurring around the barrier. This effect is governed by the height of the barrier, a constant value in this program; hence, specimen width was selected so that the barrier boundaries, both horizontal and vertical be at the same approximate distance from the charge.

184 184 Chapter VII Development of a blast protection barrier Sensor S1 D T 1m 1m Blast D T D B Barrier Sensor S6 Figure 7.9 Pressure gauges set up After the first and second tests, the barrier was carefully inspected and no damage was observed, even in the concrete composite interface; the third blast determined the failure of five of the central pipe elements as depicted in Figure 7.10.

185 185 Chapter VII Development of a blast protection barrier Figure 7.10 Barrier specimen after the third test Figures 7.11 to 7.13 show the pressure-time curves obtained from gauges S1 and S6 in each test (data from gauge S1 in the second test are not available). All diagrams also show the peak pressure numerically estimated according to a numerical model [8] based on the amount of equivalent TNT and the distance of the sensor. This numerical model is currently used in the scientific community for free air explosions and is considered reliable. The acquisition of the pressure histories was triggered by the explosion. Therefore, the first portion of the acquired signals reporting zero values represents the delay time as the shock waves traveled to the sensors. As shown in the figures, such time decreases from the first to the third test, as the distance between the explosive and the sensor D T decreases. While the arrival time of the shock wave decreases as the distance D T decreases, the three diagrams show that acquired pressure values increase as distance D T decreases.

186 186 Chapter VII Development of a blast protection barrier 100 First shot (5 m) 75 Sensor S6 Sensor S1 Pressure (kpa) Peak pressure evaluated with Enrych formulation 8% 11% Time (μs) Figure 7.11 Acquired pressure histories: first blast

187 187 Chapter VII Development of a blast protection barrier 100 Second shot (3 m) Sensor S6 Peak pressure evaluated with Enrych formulation 47% Pressure (kpa) Time (μs) * data from sensor S1 not available Figure 7.12 Acquired pressure histories: second blast

188 188 Chapter VII Development of a blast protection barrier 100 Third shot (0.5 m) Pressure (kpa) % 36% Sensor S6 Sensor S1 Peak pressure evaluated with Enrych formulation Time (μs) Figure 7.13 Acquired pressure histories: third blast Focusing on the barrier s effect in the attenuation of the blast wave s peak pressure, a percentage reduction factor η exp, representing the mitigation effect, can be derived from Figures 7.11 to η exp was evaluated using the following equation: pnb,exp pb,exp ηexp = 100 (7.4) pnb,exp with p NB,exp being the peak pressure recorded by gauge S1 (or derived with the numerical model), and p B,exp being the peak pressure evaluated and recorded by gauge S6. As a general observation, the maximum reduction factor η exp was obtained for the second test, with a distance between the charge and the barrier of 3 m. This result likely occurs because in the first test (D B = 5 m), the blast

189 189 Chapter VII Development of a blast protection barrier wave had more space to grow and surmount the barrier without losing much energy. On the contrary, in the third test (D B = 0.5 m), the porosity of the barrier permits the blast wave to pass through the pipes more easily that in the previous tests. Additionally, the damage to the pipes in the third test could have limited the barrier mitigation effect on the blast wave; unfortunately, no strain gauge data are available for this test to provide an experimental confirmation of this explanation. During each blast test, no crater formation occurred, as desired for the reasons stated above. EXPERIMENTAL-THEORETICAL COMPARISONS The availability of these experimental data provides an opportunity to validate a simplified numerical procedure to predict the mitigation effects of a porous barrier. In cases of other potential targets requiring specific protection measures, such procedures could facilitate designing the optimum geometrical configuration of a porous barrier to limit the blast wave to an acceptable value. Available literature addresses the interaction of shock waves with porous barriers, with many papers focusing on grid configuration [9, 10, 11, 12] and stressing the importance of porosity in the fluid-dynamic interaction with shock waves. Other studies investigate the attenuation of different blast waves impact on a porous barrier [13, 14, 15]. Different geometrical configurations of the porous surface were also analyzed, and the interaction with a target placed beyond the barrier was discussed [16, 17, 18, 19]. However, differently from continuous barriers, no simple numerical model is available to predict the attenuation of a blast-induced shock wave due to a porous barrier. To this effect, Chapman et al. [20] conducted a blast test activity, investigating the protection offered by a continuous rigid barrier against a blast-induced shock wave and proposed a simple formulation providing the peak pressure reduction factor. A similar study was conducted by Zhou and Hao [21], who, after an experimental activity, proposed another numerical formulation. Based on available literature, the main parameters governing the wave abatement phenomenon can be identified as follows:

190 190 Chapter VII Development of a blast protection barrier equivalent TNT charge weight W height of the charge H C from the ground height of the barrier H B height of the target H T distance between the charge and the barrier D B distance between the charge and the target D T porosity of the barrier r computed as the ratio of the voids area over the total area The proposed procedure does not take into account the material properties of the barrier. This assumption is consistent with the hypothesis used in the analysis of continuous barriers [19, 20]; that is, the barrier is rigid during the first part of the shock wave when the peak pressure is moving throughout it. Since the barrier response is much slower than the shock wave time, as mentioned in above, the barrier behaves as a rigid body even if significant damage is inflicted, as occurred in the third test. However, this assumption may not be valid in the case of very weak barriers, not capable of withstanding the first part of the shock wave. For these cases, more sophisticated numerical models should be used. The procedure does not account for blast wavelength, but for the case at hand, this is not detrimental since the ratio blast wavelengthto-barrier porosity is very large and a small change in this ratio would not affect the procedure effectiveness. In particular, the blast wavelength at the barrier was about equal to 2 m in each of the conducted tests. For this reason, the proposed procedure remains valid for blast scenarios in which the ratio blast wavelength-to-barrier porosity is similar to that occurring in the investigated cases. Moreover, the procedure does not account for boundary effects, occurring at the lateral sides of the barrier; actually lateral boundaries should not provide significant effects, since, in the test set-up, the top boundary is as far as lateral boundaries from the detonation point and should provide similar interactions with blast waves. More sophisticated models could be developed to account for this effect and remain valid in a wider set of cases. The proposed simplified procedure allows for evaluating the reduction factor of the peak pressure at a given distance from the charge, because of the

191 191 Chapter VII Development of a blast protection barrier interposition of a porous barrier. This procedure, summarized through the steps listed below, is presented in Appendix 7.A using the data from the first blast test. Step 1 The peak pressure p 1, corresponding to the barrier (Figure 7.14), at the distance D B, due to the detonation of W, is evaluated according to the numerical formula [8], since it is a typical free air phenomenon. Barrier Charge p = + p p 1 1V 1S p B,num = p BV + p BS W W 1 W 2 Target D B D T Figure 7.14 Geometrical representation of the proposed model Step 2 The blast pressure p 1 is divided into two components, depending on the porosity of the barrier: p V ρ (7.5) 1 = p 1 p S = p (1 ) (7.6) 1 1 ρ The pressure p 1V and p 1S represent the impacting actions on the voids and the solid parts, respectively. Step 3