Dynamic Behavior of Concrete and Seismic Engineering E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 487

Veröffentlichungsjahr: 2013

Beschreibung

While the static behavior of concrete has been the subject of numerous works, the same cannot be said for the dynamic behavior. This book sets out to remedy this situation: it begins by presenting the most frequently used experimental techniques in the study of the dynamic behavior of concrete, then continues by examining seismicity and seismic behavior, soil behavior, models of concrete structures subject to seismic activity, seismic calculation methods of structures, and paraseismic engineering.

Leseprobe

Table of Contents

Preface

Chapter 1. Dynamic Behavior of Concrete: Experimental Aspects

1.1. Introduction

1.2. Tests in which the transient rate has little influence

1.3. Tests with transient phase conditioned interpretations

1.4. Other tests

1.5. Synthesis of the experimental data on concrete and associated materials

1.6. Conclusion

1.7. Bibliography

Chapter 2. Dynamic Behavior of Concrete: Constitutive Models

2.1. Dynamics of concrete structures

2.2. Fast dynamics applied to concrete

2.3. Scabbing

2.4. Effect of a shock wave on the structure of materials

2.5. Modeling types

2.6. Models

2.7. Conclusion

2.8. Bibliography

Chapter 3. Seismic Ground Motion

3.1. Introduction

3.2. Measuring seismic motions

3.3. Quantitative characterization of seismic movements

3.4. Factors affecting seismic motions

3.5. Conclusions

3.6. Bibliography

Chapter 4. Soil Behavior: Dynamic Soil-Structure Interactions

Introduction

4.1. Behavior of soils under seismic loading

4.2. Modeling soil behavior

4.3. Linear soil-structure interactions

4.4. Non-linear soil-structure interactions

4.5. Bibliography

Chapter 5. Experimental Methods in Earthquake Engineering

Introduction

5.1. The pseudo-dynamic method

5.2. The conventional pseudo-dynamic method

5.3. Continuous pseudo-dynamic method

5.4. Final comments

5.5. Shaking table tests

5.6. Laws of similarity

5.7. Instrumentation

5.8. Loading

5.9. Conclusion

5.10. Bibliography

Chapter 6. Experiments on Large Structures

Introduction

6.1. Instrumentation

6.2. Dynamic loads

6.3. Data processing

6.4. Application to buildings

6.5. Bridge application

6.6. Application to large dams

6.7. Conclusion

6.8. Acknowledgements

6.9. Bibliography

Chapter 7. Models for Simulating the Seismic Response of Concrete Structures

7.1. Introduction

7.2. Different discretization families

7.3. Behavior laws for concrete

7.4. A few examples with their validation through experiments

7.5. Conclusions

7.6. Bibliography

Chapter 8. Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts

8.1. Introduction

8.2. The modal method

8.3. Criticism of the modal method

8.4. A few reminders about random processes

8.5. Improvements to the modal method

8.6. Direct calculation of the floor spectra

8.7. Creation of synthetic signals and direct numerical integration

8.8. Seismic analysis of non-linear behavior structures

8.9. Conclusion

8.10. Bibliography

Chapter 9. Engineering Know-How: Lessons from Earthquakes and Rules for Seismic Design

9.1. Introduction

9.2. Lessons from earthquakes

9.3. The aims of anti-seismic protection standards

9.4. General design

9.5. Behavior coefficients

9.6. Designing and dimensioning reinforced concrete structure elements

9.7. Conclusions

9.8. Bibliography

List of Authors

Index

First published in France in 2004 by Hermes Science/Lavoisier entitled: Comportement dynamique des bétons et génie parasismique © LAVOISIER, 2004 First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Jacky Mazars and Alain Millard to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Comportement dynamique des bétons et génie parasismique. English   Dynamic behavior of concrete and seismic engineering / edited by Jacky Mazars, Alain Millard.     p. cm.   Includes bibliographical references and index.   ISBN 978-1-84821-071-4   1. Earthquake resistant design. 2. Structural dynamics. 3. Concrete--Plastic properties. 4. Buildings,   Reinforced concrete--Earthquake effects. I. Mazars, Jacky. II. Millard, Alain.   TA658.44.C653 2009   624.1'762--dc22

2009004399

British Library Cataloguing-in-Publication Data

Preface

The static and quasi-static behaviors of concrete have been the subject of so many works that we often consider that they are quite well known and mastered as far as modeling with a view to structure calculations is concerned. However, the same is not true of concrete’s dynamic behavior, because of the complexity of the tests needed to reach pertinent loading rates.

The subject matter of Chapter 1 is divided into two parts: it presents the most widely used experimental techniques to study the dynamic behavior of concrete, drawing attention to the difficulties in interpreting the results of tests designed to identify its intrinsic parameters. It also offers a synthesis of properties that have been published in the literature dealing with concrete (chiefly its traction and simple compression strengths), as well as values for reinforced or fiber-reinforced composites. An extensive bibliography enables the reader to refer to the relevant original articles.

Dynamic loadings can generate non-linearities and a range of deteriorations in concrete (failure from bending and/or shear, traction, mechanical spalling, tearing, compression, compaction and hole perforation, etc.), all of which have to be carefully modeled to enable prediction of the behavior of a specific structure under a violent action. The variety of responses has generated several unique modeling approaches. Depending on the phenomenon under consideration, we use either the damage approach for cracking, the plasticity or viscoplasticity approach for shear, or the still volume-pressure influence approach for compaction. The theoretical contexts are discussed in Chapter 2, before the essential elements of several “conventional” models are described, along with their strengths and weaknesses.

In Chapter 3, the subject matter turns to the particular category of dynamic oscillations associated with earthquakes. As an introduction, Chapter 3 deals with the way seismic movement measurements – which generate the data used for structure reaction calculations – are made. Besides presenting the addresses of databases of signals measured in different countries, this chapter also introduces the concept of spectral representation, which plays a key role in engineering practice. A geophysical interpretation of seismic movements in connection with subjacent phenomena is proposed, which integrates the contributory effects of the site and the topography of the environment around the structure.

Though typical practice involves calculating the reaction of a structure submitted to an earthquake by considering its base to be totally embedded, the nature of some soils, coupled with the exceptional character of some structures (like dams and nuclear reactors), demands that the behavior of the structure is modeled in a particular environment. This problem is called structure-soil interaction, and forms the subject of Chapter 4. To solve this problem, it is necessary to have a model of the soil’s behavior under cyclic loading. Different models exist, depending on the nature and amplitude of the loading. After modeling, the interaction problem can either be treated by superposition, by considering the soil and the structure separately for linear cases, or globally for non-linear situations.

The difficulty of conducting structure tests on full-sized models led to the development of experimental methods employing scale models. Vibrating tables, which reproduce earthquakes on a small scale, were designed for this purpose. The subsequent development of fast and powerful computers gave birth to the pseudo-dynamic method, in which the purely dynamic effects of an earthquake are simulated using calculations. These complementary techniques both have their own advantages and disadvantages. The quality of the results they can produce depends mainly on the quality of model implementations, which are described in detail in Chapter 5.

Chapter 6 is concerned with experimental techniques on large structures. Experiments play an essential role in obtaining realistic data about a structure’s dynamic signature; mechanically-controlled vibration tests are not easy to implement, but they are a crucial source of information. This chapter shows how an excitation with rotating masses, coupled to adapted instrumentation and measurement processing, gives access to a vast amount of key information concerning natural modes, frequency damping, damage indications and coupling effects between the structure and its environment. This forms an invaluable database that model-makers need to calibrate their models, which is a pre-requisite of any realistic analyses of the seismic response of an existing structure.

Chapter 7 examines the structure-modeling field as applied to the seismic analysis of concrete buildings. The chapter focuses on approaches that allow engineers to simulate reactions to the application of an earthquake by exploring the non-linear field and collapse modes.

In this context, three model families are considered: global, semi-local and local models. The first rely on empirical behavior descriptions, gathering phenomena at the level of a single section or structural element with occasional brief discretization. The second type of model works out global laws from phenomenological local models, with discretization made at the multi-fiber or multi-layer beam level. The third type of model is more sophisticated and takes the responses of a building’s constituent materials and their bindings into account. These demand very thorough discretization, their main disadvantage being that they are very time-consuming to implement.

Validation analyses are proposed based on experiments carried out on shaking tables or reaction walls. The results show that that modeling has reached such a sophisticated stage of development that it allows complete experimental and experiment-feedback analyses, and is therefore ready for transfer to everyday engineering.

Chapter 8 introduces a quite promising analysis procedure: probability analysis. It is clear that the uncertain nature of seismic loading must be taken into account for the dimensioning of large structures. However, though sophisticated methods are used (seismic movement correlation, structure-soil interaction, behavioral non-linearity, etc.), the models remain “deterministic”.

This chapter shows that the determinist approach can sometimes lead to erroneous predictions, and that better control of phenomena makes it necessary to take into account the probabilistic character of the problem. Probabilistic seismic analysis is a new field of research that should lead to significant advances in paraseismic engineering.

Chapter 9 considers the craft aspect of engineering in the field of seismic building analysis. The subject of this chapter, experiment feedback and regulations, is important, as engineers are ultimately responsible for the safety of people as well as buildings, despite the fact that building science is not a totally exact science. Numerous theories can help analyze the behavior of structures, but the limits of the problems faced by engineers in that field remain “fuzzy” (action characterization, structure complexity, local behavior facts, soil-structure interaction, etc.). Regulations can provide a framework that makes the analysis concepts reliable. Experiment feedback gives an indication of those approaches that have worked and those that have not, and considering this when developing regulations obviously assists progress in safety control. This is the subject of this chapter, which also gives an excellent account of the spirit in which the paraseismic design of various concrete structures should be approached.

Jacky MAZARSAlain MILLARD

Chapter 1

Dynamic Behavior of Concrete: Experimental Aspects1

1.1. Introduction

1.1.1. Meaning of the word “dynamic”

As distinct from the term “static”, “dynamic” implies the influence of time. A test is said to be “quasi-static” when the effects of time are present, but can be neglected. For a structure test, and for any real test, the effects of time are typically expressed in two ways:

– by forces of inertia resulting from the not equal to zero acceleration to which the elements of structures are submitted;

– by the behavior of each elementary volume of the material depending on the evolution in time of the elementary mechanical values (stress and strain) and possibly of their time derivatives. This dependence is described by the generic name of “viscosity”.

This distinction is strictly linked to the notion of elementary volume underlying the definition of behavior. Actually, the fact that viscosity effects might be the manifestation of inertial microscopic phenomena cannot be excluded. This remark is important in the case of concrete, as considerations about material homogenity involve decimeter elementary volumes (but this is not the case with metals for which the elementary volume is sub-millimetric).

Thus, to be quite clear, we will consider the dynamic behavior aspect as limited to the description of the effects of time using elementary mechanical values and excluding inertia effects.

From general physical and thermodynamic considerations concerning behavior laws [MAN 67], we can deduce that the generalized mechanical variables Q (t) (stress) and q(t) (strain) can be related in the following way:

[1.1]

where H describes the loading history. This formulation highlights the fact that these values do not play a symmetric role. The instantaneous mechanical reaction depends on the geometric history, its current value, and the values of its higher time derivatives. Thus, it is not natural to consider stress velocity as a behavior variable.

If we limit our attention to formulations likely to be easily integrated into calculation codes, the relation expressed in equation [1.1] can be re-written in the following incremental form:

[1.2]

The values αi are internal parameters that take process history into account. Their evolution has to be described as a complement to the relationship in equation [1.2]. Their dependence on the history of the process explicitly results in their loading and unloading paths being different. The values playing a part in equation [1.2] are tensors. We can see the complexity of this relation. In most cases, the simplifications carried out involve discarding strain time derivatives higher than 1, and expressing the strain speed using a scalar value. Such simplifying assumptions are justified for two types of reasons. Firstly, programming laws into codes will be simplified by doing this. Secondly, an insufficient variety of dynamic tests is available to identify more parameters. For this reason, from this point onwards, we will refer to “strain velocity” without going back over the definition.

As far as strain velocity is concerned, it is standard practice to study its effects on long time scales revealed through creep. Even though creep tests can clarify the analysis of dynamic tests, we will not be considering them. The experimental aspects of creep tests have no dynamic aspects, as typical strain velocities implemented are around 10-10 s-1, as compared to “static” test standard velocities that range from 10-6 to 10-5 s-1, and the strain velocities reached during “hard” shocks on civil engineering structures, which are usually in the range of 0.1 and 10 s-1.

These elementary considerations understood, it appears that a critical factor in the experimental characterization of concrete behavior is discarding the inertia terms. The problem is more delicate with concrete (a brittle material) than it is with metals. As a matter of fact, the first manifestation of inertial effects on a sample submitted to dynamic loading is the transient response observed when the time taken by elastic waves to pass through the sample (the transfer time) is significant relative to the test’s time duration. When studying this problem, the pertinent time-dependent parameter is not the strain velocity (which, in any case, is not well-defined in the transient phase), but the loading time relative to the transfer time. If sufficient strain levels are reached in very short periods of time, the sample could fail before a homogenous stress and strain state, measurable as an average, could be reached. In fact, low amplitude traction strains (ranging from 100 to 150 x 10-6) lead to material failure. Test analysis is generally difficult. For common sized samples (centimeter scale), we cannot go beyond 1/s average strain velocities when conducting a quasi-static test analysis. This feature of brittle materials can be exploited advantageously, and is used in scabbing tests (see section 1.3.1).

This limitation is far less a problem with metals, where important local strains arise, but do not cause failure. Such a situation can only occur in concrete if particular conditions that guarantee mechanical field homogenity exist to prevent cracking. This is the case when tests are conducted in strong confinement (under which circumstances, concrete behavior is described by plasticity-type models). As far as metals and most polymers are concerned, it is also important to take thermo-mechanical coupling into account, due to the adiabatic feature of dynamic tests. This effect can only be neglected when failure occurs under low strain for which the dissipated heat remains low: with concrete, it can also be neglected in confinement tests, since we can presuppose a low thermo-mechanical coupling.

1.1.2. Reminders about dynamic experimentation

1.1.2.1. Specificity of dynamic tests

As far as statics and dynamics are concerned, it is reasonable to consider sample analysis in a separate section, along with the overall measures it involves (generally carried out on the peripheral part of the material). This is the second aspect mentioned in the introduction.

The first difficulties encountered in dynamic experimentation fall under the first category mentioned in the introduction. They are linked to transient effects inside the machine and the associated sensors: the balancing time of the machine and its sensor array (elastic waves moving back and forth several times) are not negligible relative to the length of the test. Thus, carrying out quality measures often requires a transient analysis of the response of the machine itself. Hence, in a real situation, characteristic testing times have to be compared with the acquisition chain and the sensor pass-band. If the acquisition frequency is not far higher than the frequency of transient signals, the observed result can be completely modified by the measuring chain, and even average values can be wrong.

1.1.2.2. Hopkinson bar test

For average strain rates in excess of 50/s, because the transient effect inside the test machine cannot be neglected, a way round the problem involves explicitly taking wave propagation phenomena into account, using a bar system. Whilst the transient analysis of three-dimensional structures is too complex to be taken into account efficiently, using “one-dimensional” bars makes it possible, as we will now explain.

1.1.2.2.1. A description of the bar test

To carry out a dynamic compression tests with Hopkinson bars [HOP 14] (also called the SHPB (Split Hopkinson Pressure Bar) system, or Kolsky bars [KOL 49]: named after the first person to use the system in its current configuration), a small sample is placed between two identical long bars with a high elastic limit relative to the tested material (Figure 1.1). Strain gages are glued to both bars. Due to a projectile, a compression longitudinal elastic wave is induced into the input bar. Part of this gets reflected at the sample-bar interface, whilst another part is transmitted to the sample before inducing a wave in the output bar.

Figure 1.1.Hopkinson bar assembly

The waves at points A and B are determined by measuring and recording the structurally-associated longitudinal strains. The need to know A, the incident wave induced by the impact separator, and the reflected wave B, which depends on the reaction of the sample, arises because we need to find the optimal position of the measuring point at the middle of the bar. On the other hand, considering the bar as one-dimensional does not allow us to place the strain gauge too near an end. A typical recording for a concrete sample compression test is shown in Figure 1.2.

Figure 1.2.Compression test on a concrete sample (40 mm diameter aluminum bars): basic waves

Next, the waves have to be carried to the contacts between the sample and the bar. Then we can calculate the stresses and displacements (by integrating the velocities, which are directly accessible) on the corresponding faces.

The particulate velocities at the input and output faces can be written respectively as:

[1.3]

The forces on the input and output faces are respectively:

[1.4]

Measures on the two opposite faces of the sample allow estimation of strain field homogenity by comparing the forces on each face (section 1.3.2, Figure 1.12). We note that for this test, the assumption of homogenity in mechanical fields is hazardous. As a consequence, the notion of average strain velocity is also hazardous. In section 1.3.2 we will see the best way to use the available measurements. Thus, we should stress that the Hopkinson Bar leads to overall values of loads and displacements on both sides of the sample. All mechanical quantities are obtained by making additional assumptions completely separate from the test facilities. These have been widely reported in the literature [NIC 80].

1.1.2.2.2. Limitations of the conventional system

Accurate analysis of wave transport

To carry out a precise virtual wave transport between the measured points and the sample (forward transport for the incident wave and backward transport for the others), the three-dimensional feature of the bars need to be considered, and the dispersal correction must be introduced using a signal treatment technique. This parameter corresponds to signal modification during transport. An accurate time calibration (to within a micro-second) is also necessary [ZHA 96]; it is especially important for measurement of small strains, and thus for brittle materials such as concrete.

Multi-axial characteristics of the test

The uniaxial characteristic of the test is also an approximation. Let us examine this aspect in the case of compression. Whenever the material presents a Poisson effect, the longitudinal strain comes with a lateral strain (as is the case in statics if the support conditions are well controlled), which is opposed by radial inertial effects. This causes an induced confinement. The confinement explains the obvious sensitivity of concrete to strain velocity that is universally observed in dynamic compression (see section 1.3.2).

Measurement duration

The proportionality between the mechanical values associated with a wave inside a bar, on which the Hopkinson bar technique is based [1.3]-[1.4], only applies to a wave propagating in a single direction, which requires measurement of the incident wave (propagating one way) separately from the reflected wave (which propagates in the other direction). This limits the measuring duration to

C being the propagation velocity and L the length of the input bar. ΔT is thus a function of the length of the bars. Consequently, for a behavior test, the total strain cannot exceed the product of the average strain velocity and ΔT. For instance, measuring duration will not exceed 400 µs (C≈ 5,000 m/s) for a 2 m long aluminum bar, and the total strain will be limited to 4% for a test with a 100 s-1 average strain velocity. Because of this limitation, even with concrete (for which high strains are unlikely to be reached), the conventional Hopkinson bar system will not allow tests at average strain velocities lower than 50 s-1. On the other hand, for reasons explained in section 1.2.1, traditional machines used without specific precautions do not give reliable results at lower velocities. Besides, their superior limit is not clearly established and is determined to an extent by the material being tested (the test piece). The machine must be used in a particular way; it varies between 1 s-1 and about 10 s-1. However, a recent experimental technique using bars [BUS 02] that covers this problem now exists.

1.1.2.2.3. Difficulties inherent to dynamic measurements

The dynamic test facilities have numerous limitations, especially for stresses other than simple compression or small strains. This limitation mostly affects low strength stressed materials (impedance adaptation and high strain problems) and brittle materials (low strain at failure).

The Hopkinson bar example illustrates the generic difficulties quite well. The very short loading times do not enable us to carry out multi-axial dynamic loadings easily, and it is not easy to synchronize loading with two (or three) orthogonal Hopkinson bars. If synchronization is tricky in dynamics, it is all the more so when piloting the test. Therefore, we cannot (for now) contemplate carrying out tests under controlled multi-axial loading (deviatoric, for example), as is required in a quasi-static mode. The need to control the loading and the difficulty in carrying out dynamic displacement measurements limits the potential tests to a very small number, which are described, along with their specific problems, in sections 1.2 and 1.3.

1.1.2.2.4. Compression tests with confinement

It is quite easy to superimpose quasi-static confinement on a dynamic compression test. A cell in which a gas pressure confinement can be maintained during the compression test is described in [GAR 99]. Some authors have proposed a bi-axial loading scheme, where the secondary static stress is applied using a jack [WEE 88]. For higher confinements (necessary if we want to study compaction of concrete, for example), a metal cylinder can be used [GAR 99]. In this case, confinement pressure is not studied, but can be measured during the test by assuming the (most often elastic) response of the confinement ring is known (as in an oedometric test). Another way to carry out high confinements involves using the “plate on plate” test developed to study the high-speed spherical behavior of metals. It is a plane strain-loading test, the inverse analysis of which is based on behavior modeling. High confinement there is associated with very high strain speeds.

1.1.2.2.5. Traction tests

A conventional traction test can be carried out with a Hopkinson bar [REI 86]. If we consider only global measures, the main difficulty is due to keeping the sample in contact with the bars. To avoid having to resort to assemblies leading to impedance failures, it is reasonable to glue the sample to the bars. Some authors [TED 93] have had the idea of using the Brazilian test again in dynamics. In this case we have to check that the conditions of strain homogenity are compatible with the assumptions. Finally, the spalling test [DIA 97] allows an accurate measurement of the average stress just before failure, but its interpretation is difficult as it is between the classical traction test and the fracture test (toughness measurement).

1.1.3. Identifying the behavior of concrete under fast dynamic loadings

When identifying the dynamic behavior of concrete, we are confronted with a series of typical problems for each high-speed behavior identification test. Some of these problems are increased by the nature of concrete, which is the reason why we prompt the reader to be very cautious when using experimentation signals or results.

Due to its structure in aggregates, where it is mixed with sand and hardened cement paste, concrete can be a highly heterogenous material, and the size of a representative sample is not always an easy thing to state. As far as statics is concerned, a 2 slenderness cylinder, over five times as big in diameter as the aggregates, is the lowest volume necessary to obtain stable properties representing the material in these tests, particularly as far as strength is concerned, otherwise “scale effects” will be observed. Such a constraint raises several types of problems:

– for standard concretes, in which the maximum size of aggregates ranges from 20 to 25 mm, the dimensions of test samples (diameter over 10 or 12 cm, mass over 5 kg) involve resorting to important energies, particularly for high speed tests, which involves sometimes tricky technological arrangements;

– to avoid this difficulty, tests are often carried out on micro-concrete, mortar or cement paste samples. Transposing these results to structure concrete requires a critical analysis, mainly because the volume fraction of cement paste (generally considered as the viscous element of the composite) is not always constant. In the same way, the propagation of waves disrupted by the module differs between cement pastes, and aggregates can also be different depending on the composition of the studied material;

– even if we managed to identify the intrinsic properties of the material on big enough samples, for many structures, the “representative material point” size is important compared with the dimensions of the smallest pieces (building shells about 20 cm, bridge webs from 30 to 45 cm). Furthermore, significant stress variations on the scale of the structure can be discerned over short distances of the same order of magnitude as the dimensions of the test sample. What is then involved is the application of continuous medium theory, which is based on the assumption – generally not well verified – that the material point is infinitely small compared to the structure;

– a problem (which occurs in statics too) that becomes crucial as far as the dynamic interpretation of tests is concerned is that the sample is not submitted to an homogenous state of strain and stress owing to its size, and has to be considered as a structure submitted to transient loading.

Because concrete is a brittle material (like most geomaterials, concrete can only withstand very weak extension strains and its apparent “failure” takes place for a compression strength about 10 to 20 times as strong as its traction strength), most of the time, in practice, while interpreting the tests, we must consider:

– that we are dealing with an elastic homogenous material (which implies the size precautions referred to above): the assumption is necessary for relatively low velocities or low strain levels, in continuity with the quasi-static field. It is not good enough to interpret the totality of a test when the speed increases, since the maximum stress is reached when localized cracking has been reached significantly on only a part of the structure;

– that beyond the stage corresponding to localized cracking, the test sample can be modeled as a cracked structure where damage concentrates in the crack area, which corresponds to fracture models;

– that beyond a stage corresponding to a distributed deterioration (which corresponds to the bonding material crumbling away), the material can be described by combining damage and plasticity models.

Hence, at the material failure of the sample, the interpretation of the tests requires different analysis models, regardless of whether we are mainly in a deviatoric behavior with a possible extension direction allowing localized cracking, or in a mainly “spherical” behavior, and depending on the stress peak being identifiable or not.

It is important to note that because of the weak growths withstood by the material at high velocities, experimental precautions have to be taken – especially because of transient effects, in dynamic experiments where limit conditions are difficult to control. In experiments where an “energetic” approach is privileged, this aspect is also important: the inertia of the test sample cannot always be neglected with regard to that of the test machine, and the energy dissipation through damage on the support, or through contact with the impact separator can be important compared with the energy supposedly dissipated by the “normal” cracking expected in bending.

Finally, a delicate feature of concrete is its porosity: it has such a tortuous network that water exchange times with the environment are quite long (about 10 years for the representative volumes considered above). We can consider the hydration state of the sample as constant during dynamic tests, which is not the case for shrinkage or plastic flow tests. However, important relative pore moisture and mechanical state coupling, together with frequent cracking due to the stress levels reached when desiccated, begins at the sample’s surface and/or their environment as soon as they are fabricated. In at least one stress and velocity field ([DAR 95] [TOU 95a]), researchers have shown that the partly water-saturated feature of the porous network explains the modification of apparent mechanical properties: these are generally called “velocity effects” in the literature.

In following sections (1.2 to 1.4), we will detail the arrangements, test type by test type, used to analyze the results and infer the indications and modifications required to calculate and understand the behavior of fast dynamic loading concrete structures. The actual and measured behaviors are summarized in a rational way in section 1.5.

1.2. Tests in which the transient rate has little influence

In this chapter, we will deal with behavior identification tests that, for reasons developed in section 1.1 can have a “quasi-static” interpretation.

Two test families can be distinguished. The first is derived from typical concrete characterization tests and emphasizes growth or cracking failures. This is called deviatoric behavior, and is the failure kind that is also, indirectly, the cause of collapse observed in compression and even in biaxial compression. The second test type corresponds to “volumic” behavior, which can seldom be observed in ordinary structures, except in relatively confined areas where specific reinforcement by the surrounding material ensures tri-axial confinement at high velocity: concrete areas directly submitted to impact and those close to an explosive charge or perforating projectile are examples.

Combining both types of information in statics enables a definition of failure or plasticity criteria closed on the hydrostatic compression axle, as opposed to the “intrinsic curves” (Coulomb criterion), the validity of which is preferentially ensured when an extension direction is possible.

1.2.1. Tests involving deviatoric behavior

1.2.1.1. High-speed press machines and traction tests

Because of the difficulties connected with carrying out dynamic tests, most authors use privileged uniaxial tests. Owing to the basic feature, traction behavior identification stands out, and has given rise to a great number of tests. In order to ensure continuity in the geometry of test samples, by controlling the loading application speed and considering its limited artifacts, a direct traction test on a cylindrical specimen has become essential. This is detailed in [HOR 87], [REI 82] and [TOU 95a]. With particular precautions, this test can actually be carried out on conventional servo-controlled machines with load build-up speeds ranging from about 0.05 MPa/s (which is the standard loading rate for standard identification tests) to about 10,000 times this load, with identification at still higher speeds of the order of 50 GPa/s possible on the same specimen type thanks to the modified Hopkinson bar (SHB).

The necessary precautions particularly involve:

– choosing to glue the specimen in place with centering and a rigid (without spherical pairs) mounting onto the press to limit looseness which is a source of interfering moments;

– choosing aluminum hard supports to limit the transversal strain divergence at both ends of the specimen;

– controlling the hydration state of the specimen [TOU 95a];

– choosing a not too important slenderness ratio (1 to 1.5) to limit potential bending;

– gauge extensometer or extensometers fixed in the middle of the specimen to avoid the deformations due to the glue joint;

– using specimens with adequately sized diameters considering the maximum size of the aggregates, and if possible core cylinders for better homogenity of the material and to avoid scaling effects [ROS 92a].

With a sufficient automatic control and oil flow unit, and potentially using a preload to carry out high velocity tests, we can consider that the load build-up speed is rather constant during the test. The propagation speed of the waves within concrete – about 4,000 m/s, the standard size of specimens (10 cm) – and the traction failure stress (4 to 10 MPa) limit the quasi-static interpretation of this kind of test, results typically showing a divergence about 10% between the specimen’s input stress and output stress.

The measurements typically carried out during this test are of the applied force as a function of time, and of the average longitudinal strain at the center of the specimen (extensometer gages or sensors for which we have to check that the inertia will stay weak and the fixing will be ensured during the test). Taking into account the small size and fixedness of the assembly, we can consider that there are no differences between the measured force and the force applied to the specimen, so we can assess traction uniaxial behavior by eliminating time. In such a test, the specimen behavior corresponds quite well to brittle elastic behavior up to localized cracking. Localization brings about loss of the homogenity of the strains, and an almost instantaneous decrease of the load.

Going through these tests, which implies expressing the maximum measured stress according to the “load build-up” parameter in a logarithmic diagram, typically allows us to define a traction rate effect corresponding to the strength relative increase.

1.2.1.2. High-speed press machines and compression tests

The second most conventional test that can be performed at high speed is the compression test. It enables us to define a compression “rate effect” from the measurement of the maximum strain reached [BIS 91]. The size of the test sample necessary to free oneself from the size effect and to ensure the correct strain level reached lead to strict constraints on press dimensions, unit power and the jack flow rate. For this reason, a great number of the tests described in the literature were carried out on mortar, cement pastes or micro-concrete [HAR 90]. As is the case in traction, it has proved possible to look for a size compatible with the higher speed test performed with Hopkinson bars [DAR 95].

As it is difficult to stop the jack when its speed has been stabilized, few test reports have included extensometer measurements [BIS 91], measuring the load obviously remains the main data. For standard size specimens (10 cm), considering the wave propagation speed and the maximum stress reached, the load build-up rate beyond which the sample cannot be deemed to be in a stationary process is about 10 times as important as it is in traction tests, which correspond to the strength ratio. When expressed in terms of strain rate, the threshold is about 10 instead of 1 s-1 [MAL 98].

The incidence of superfluous interference moments is generally less important than it is in traction tests; however, the precautions to be taken to avoid restricting transversal strains are as important as in statics, especially for specimens with slenderness ratios below 2. To this end, we can mention lubricating the faces or using aluminum. The quality of the stress transmission surfaces is essential to avoid premature concentration of stresses.

We note that as in statics, and even for specimens that are simply laid, the relative displacement (including interface crush and deformations at the ends of the specimens) cannot result in a reliable indication of the strains of concrete in its standard part, the error typically ranging from 30 to 100% [BOU 99]. As in statics, failure obtained in compression tests begins with transversal extensions. The traction rate effect results in an “inertial confinement”. However, the maximum stress is only reached when the cracks parallel to loading direction meet, allowing either buckling in the “small columns” formed inside the test body, or shear localization. As a consequence, interpreting the strength evolution, where load build-up speed is the only parameter, becomes complex.

We could not find any references to tests deriving from standard quasi-static identification of multi-axial behavior with prevailing deviatoric behavior (bi-traction, pure shear, bi-compression), at least not in areas where transient test characteristic can be neglected. As a matter of fact, the most frequent cases of dynamic multi-axial behavior identification use unidirectional loading with a Hopkinson bar [GAR 98], [LOU 94] and [WEE 92], while confinement or loading in the other direction is often “static”. These tests will be described in section 1.3. Such a situation can indeed be explained by the difficulty in controlling and synchronizing dynamic loadings, even at the “low” speeds reached by conventional presses or jacks. Furthermore, taking the properties of concrete into consideration, the regulations rarely take multi-axial behavior into account. As a result this lowers the validation of high velocity dynamic models adapted to concrete, in situations other than simple traction, uniaxial compression or compaction.

1.2.1.3. Tests with small plates or beams submitted to pressure loading

Considering the difficulty in carrying out dynamic loading with mechanical application of the loads, some authors perform controlled loadings on mini-structures (small rectangular plates, beams or small plates), using a pressure loading generated by an explosion. The purpose is then to identify the bending behavior, the bend-moment law being material information directly transposable to the calculated structure, taking into account the similar nature of the tested material and the geometric and energy similarities – called Hopkinson’s – on the load. Detailed experimentation of this kind will be described in [BAI 87] and [BAI 88]. The limitation on the energies that can be used in a laboratory forces the use of centimeter thick test elements, therefore generally of mortar (or possibly fiber-reinforced) rather than concrete.

The loading process links the level of the applied overpressure with its duration and load build-up speed. However, interpreting the trial remains simple in so far as the load build-up times can be considered as very short compared with the specific period of the structure. Thus, we have what is called a pulse loading: the overpressure time, which causes the structure to start vibrating, is very slightly ahead of the latter’s peculiar period, which is then in a free-vibration system. As the probable area of maximum strain and even failure is known, the relevant section can be instrumented in a preferential way. Therefore, we can measure the traction by bending the final strain and the final bend. Note that shortly before failure, the strain of the compressed side is slightly inferior to that of the opposite side (the start of non-linearity which could be representative of micro-cracking). Since the structure is undergoing free vibrations, the deformations should be linked to the stresses generated from the by-pulse loading, which implies that a dynamic analysis can be used to calculate the moments to link to the bends within the scope of behavior law identification. Nevertheless as long as we stay at moderate loading levels and deal with the behavior just before a brittle material fails, an elastic analysis is satisfactory. The divergence from elastic behavior can be identified “at a quasi-static speed”.

1.2.1.4. Shock tube tests on plates

The principle of a gas pressure by-pulse loading can also be applied by resorting to a uniform loading the value of which is controlled thanks to a tube used as a wave guide and called a shock tube. Using such a device is quite conventional for testing industrial equipment in the defense field. Using the device for structure elements was developed more recently [TOU 93]. Using explosives is limited and the loading profile as well as its spatial repartition is better controlled than open-air explosions. In so far as the conditions at limits can also be well controlled, we can directly access to the behavior of a bending plate, which represents “basic” data for the structure designer [KRA 93] or a simple basic situation to validate a behavior model [PON 95, SER 98a].

The innovation of this trial was that it generated loading by means of a well-controlled air shock wave (Figure 1.3). By using the closed tube, for the same plate with the same support conditions, it is possible to carry out quasi-static loadings by slowly inflating the whole tube. As an example¸ a 35 m long tube, 66.6 cm in diameter, was used to compile an important experimental database about concrete and reinforced concrete plate bending parameters [TOU 95a].

Figure 1.3.Skeleton diagram of the shock tube trial (from [TOU 95a])

Considering the inner diameters of the tube and the support area (82 cm), to preserve the cylindrical symmetry of the test, the test sample is a “thin plate” (thickness/span < 1/10) 900 mm in diameter and 8 cm high. It has dimensions compatible with the performances of the tube (allowing it to actually reach failure requires using a concrete with aggregates that are not too small, or realistically standard reinforcement (welded wire mesh), or fiber reinforcements). We can note the particular care taken to achieve limit conditions close to those for an ideal simple support, the circular slab being “pinched” between the humps of two massive guides, a thin rubber-steel sandwich (a 3 cm wide ring) allowing absorption of geometric defects and distribution of the clamping load. Its stiffness has been measured, and control of the displacement and acceleration on the supports during blasts enables analysis of the bending of the support slab under uniform loading on a driven reference line.

Figure 1.4.Shock tube failure trials for a plain or reinforced concrete slab (from [TOU 95a])

In addition to excellent loading control and a size adapted to a well controlled trial on “realistic” concrete, the advantages of this test are the realistic representativity (bending is obtained with maximum deformation speed typically ranging from 0.01 s-1 and 1 s-1, which corresponds quite well to the “hard” shock range) and geometric simplicity (radial symmetry is preserved up to cracking) which make it possible to validate a calculation model as well as for comparing various materials. The relative ease of interpretation stems from the fast loading building up (about 10 μs for a maximum deformation reached in about 1 ms) and from the absence of pressure gradients on the loaded face. We can consider that the plate is loaded instantly (vibration setting with a first deformation peak which is particularly intense compared to static loading), but with a bearing constant loading, which allows a stationary vibration rate to be set up before unloading. A “conventional” modal analysis enables access to local stresses and strains, at least until cracking starts.

In [TOU 95a], the details about the instrumentation implemented to characterize strains in test samples in these types of trials are presented. We have seen that in a series of plain or reinforced concrete plates with strength in the range of 35 to 120 MPa (Figure 1.4), we are able to show the progressive deterioration of the modal response (frequency drop, increasing damping), the appearance of deflection, plastification of the reinforcement, crack progression (which is sometimes delayed with regard to the maximum strain rate) and the collapse mode type (shear force/bending competition) the respective appearances of which can be justified by limit analysis-inspired calculations [TOU 95a].

1.2.2. Tests with prevailing spherical behavior

When loading has a strong tri-axial component, concrete undergoes a global reaction resembling that of a coherent material, even when it has failed on a microscopic scale, which is the case for confinements over 10% [GAR 99]. The models used to describe this are generally plastic models (not necessarily standard and usually coupled to deviatoric and spherical behaviors). In these cases, even high strain gradients do not bring about failure or localization, and the concrete sample can be analyzed as if composed of a homogenous material.

1.2.2.1. Slab-plate tests

[1.5]

As the time of the shock is known (by contact measurement for example), measuring the free rear face speed allows us to locate the moment when the wave arrives and to measure D. It also allows us to calculate u. Thus, one test establishes a relationship between P and V, and also between D and u: these are called “shock polar curves”. To deduce a strain-stress uniaxial relationship from them, we will have to make a hypothesis about the behavior model of the material.

For metals and high strength shocks, the elastic response is neglected, and we assume that the plastic behavior is purely deviatoric (without any volume variation). Strictly speaking, concrete behavior analysis should be different. Each test gives a point on a curve. The “Hugoniot curve” links pressure to material physical speed and the “ shock polar curve” links shock speed with material speed (objective measurements). There is therefore no direct way of converting this to mechanical values that geomechanical engineers are familiar with.

The plate-plate test is a relatively pure trial. However, it has to be interpreted, is difficult to implement, and only can only inform us about concrete compaction behavior at very high strain rates (above 105 s-1).

1.2.2.2. Hopkinson bar tests with strong confinement

This test was developed at the LMS in co-operation with LMT Cachan [GAR 99]. A cylindrical specimen is confined within a metal cylinder (Figure 1.5). It is loaded using a large diameter (80 mm) steel Hopkinson bar, which allows the use of test samples large enough in comparison with aggregate size to be adequately representative of the material.

Figure 1.5.Confined sample for Hopkinson bar test (from [GAR 99])

The complete collection and analysis of the signals recorded on the bars (described in section 1.1.2.2) allows the measurement of the forces and displacements applied on both faces of the sample.

When the input and output forces are equal (which is the case shown in Figure 1.6) and we can assume an homogenous state of stress and strain, the stresses, strains and axial strain speeds can be deduced.

Figure 1.6.Static and dynamic volume-pressure relationship

Figure 1.7.Static and dynamic volume-pressure relationship

The behavior law of the metallic ring is known. A thick enough ring to remain in the elastic field allows the application of strong confinements. Using a ring made of material that enters the plastic field (brass for example) will enable controlled confinement to be applied. Thus, measuring the transversal strain of the ring allows the confinement to be calculated, after which we can calculate the values that are usually dealt with in geomechanics. As an example, Figure 1.7 shows evolution of the volume-pressure relationship compared to the same relationship obtained using a static trial.

1.3. Tests with transient phase conditioned interpretations

1.3.1. Tests involving mainly traction behavior

1.3.1.1. Modified Hopkinson bar

As explained in section 1.2.1.1, traction behavior is essential for characterizing the failure of brittle geomaterials like concrete, which is why adapted tests have been designed to obtain this data for high speeds, and has been widely studied.

The design has been achieved, mainly thanks to modified Hopkinson bar configurations in which the specimen is glued between the input and output bars, where it is submitted to traction produced by a shock to a retaining shoulder at the end of the input bar. The main results with this technique were obtained on the device of the University of Technology in Delft [REI 86] and [ZIE 82] between 1980 and 1995. The tested specimens are typically core sampling specimens 74 mm in diameter (the same diameter as the bars), with a 1 to 1.5 slenderness. The duration and energy of the shock which generates the traction wave depends on the mass used, hydrostatic pressure and the number of dampers inserted between the masses whose fall is triggered and the lower input bar shoulder.

In practice, as we want the shock to be intense enough to cause specimen failure, and the loading build-up rate to be constant during the trial, the device allows loading rates ranging from 4 to 200 MN/s, about 100 to 1,000 above the rates reached with conventional press machines with similar specimen geometries.

The analysis of specimen loading uses the transient analysis described in section 1.1. The quality of glueing interfaces and the nature of the aluminum bars contributes to impedance compatibility between concrete and the loaded material, so an important part of the wave is transmitted to the specimen and the obstacles to transversal strains are limited. We have verified that the transmitted-wave signal gives a precise measurement of the average stress developed inside the sample – after conversion into stress and calibration in time.

Figure 1.8.Direct traction tests on Delft University’s Hopkinson bar, plain concrete and very high performance concrete

The simultaneous measurement of the strains on the specimen (Figure 1.8) is made possible either by extensometers gages glued to the sample [TOU 95a] or by pre-slotted fiber concrete (where the measurements concerns crack opening), by gages fixed directly on the sample [TOU 99b]. For the speeds considered, the time delay between stress and strain signals is about 220 μs, whilst the space difference is about 1 meter. The “suitable” loading time (from 0 to maximum load) ranges from 100 to 500 μs, and sampling is carried out at 250 kHz. The excellent stress-strain linearity obtained confirms the validity of the hypotheses. Nevertheless, considering the time to go through the specimen (about 25 μs, i.e. a difference about 1 MPa), the rates reached limit the interpretation as far as sample homogenity is concerned.

Figure 1.9.Ispra Centre device for big sample testing

To improve understanding of the mechanisms of traction failure and crack dynamic propagation, a specific device has been developed for effort transmission and measurement and is included in the large-scale dynamic test equipment (LDTF) at the European Research Centre Ispra [CAD 01]. To increase the capacity of the shock transmitted to the specimen at that installation (20 cm-edge cube), the shock is generated by the violent release of a tight cable. The device (a Hopkinson Bar Bundle (HBB)) consists of a prismatic Hopkinson bar beam, each bar being instrumented, which transmits the traction wave to the specimen. Potential helical reinforcements at both ends of the specimen are eliminated. It is possible to follow both the opening of a crack across the specimen and the loading transmission remaining in the not yet broken ligament, by applying a simplifying hypothesis of wave propagation and load transmission inside the breaking specimen.

Most of the significant results concerning high-speed traction concrete behavior detailed in section 1.5 were discovered using this installation (Figure 1.9) on quite large scales.

1.3.1.2. Hopkinson bar Brazilian test

The test is an expansion of the Brazilian test, whose traditional analysis is based on the assumption of brittle elastic behavior. We consider an elastic cylinder compressed perpendicularly to its generators: compression is applied along two diametric generators. A plane deformation elastic calculation shows that loading causes practically constant traction maximum stress along the cylinder axle, at right angles to the compression axle. We assume cylinder failure takes place when the strain reaches the ultimate value. Carrying out this test in quasi-statics is not obvious, as it requires strict respect for limit conditions and the ideal elastic model (stiff supports among others). Nonetheless, this trial is easy to carry out and gives a consistent order of magnitude for simple traction failure stress.

Extension to the dynamic situation is easy. Compression is applied using a Hopkinson bar. If we want to analyze the results in the standard way, we suppose that the situation is not too far from the quasi-static case. To do this, we have to assume that inertial effects can be neglected. They can be neglected before failure but, as is the case for simple compression, they cause an apparent increase in the maximum load after failure, so consequently it is important to detect failure by direct observation (using high-speed imaging), as it is for dynamic compression tests where localization of strains with block development does not necessarily lead to load drop immediately. We should also check that the mechanical fields are not too far away from the fields we would have in statics at the same applied force value. Thus, we have to verify that failure occurs at a time when input and output forces are quasi-equal. Such a situation will only happen when loading is slower than the homogenizing time (typically the time for the elastic waves to cover the diameter of the sample several times).

Achieving all these conditions simultaneously is difficult, but as we saw in section 1.1.2.2, the Hopkinson bar provides us with information about the loads and displacements applied to the sample all the time. Assuming this data is accurate, we can then carry out a numeric simulation of the test (assuming brittle elastic behavior), which gives a more precise assessment of failure strength [TED 93]. However, this hybrid approach (calculation-test association) is that of a structure trial, and is better suited to model validation than directly determining a behavior parameter.

1.3.1.3. Scabbing test

The scabbing test is a test with a fundamentally transient analysis. Actually it is based on analyzing wave propagation inside a bar made of the material itself. Concrete, a brittle material has a uniaxial compression strength that is clearly superior to its traction strength.

By using an assembly like the one in Figure 1.10, we induce a compression wave (propagating to the right in the figure), which is reflected at the free end as a traction wave [BRA 99], [DIA 97].

Figure 1.10.Scabbing test diagram

The compression pulse produced by the impactor is measured via a strain gauge glued to the bar. The elastic properties of the bar and the sample being known, we can infer the shape of the pulse induced inside the sample. We can also glue a gauge on the specimen to measure it directly. The compression wave thus produced has a lower amplitude stress than the compression concrete failure stress. The opposite amplitude reflected traction wave is sufficient to cause failure in the sample at a specific position. By applying the principle of elastic wave superposition, we can infer the stress value at the failure point. The analysis is easy because the pulse is short compared to the propagation time inside the sample. This is why we use short impactors and long specimens. Making specimens respecting homogenity conditions is therefore delicate.

The accuracy of the test analysis can be improved by additional information such as the failure instant, which can be obtained by high-speed imaging. In some cases, we can observe successive failures in the sample, analysis of which gives redundant measurements of failure stress.

This trial also gives accurate and reliable measurements of limit conditions, and the loading parameters are well-mastered. Fine interpretation still remains difficult as it is one-dimensional (as far as wave propagation is concerned), whereas failure has to propagate in the transverse direction. Moreover, the characteristic phenomenon is quite local. High strain gradients do not allow easy measurement of the strain rate characteristic of the test. This speed is usually taken as the strain time derivative near the failure point; for a one-dimensional wave, this derivative is proportional to the deformation spatial derivative.

For concrete, a very marked increase of failure stress with strain rate has been observed [BRA 99]. Between 1 and 100 s-1, failure stress can be multiplied by as much as a factor of 10. The physical interpretation of this result still has to be more closely examined.

1.3.2. Tests implementing compression behavior

1.3.2.1. Hopkinson bar trial

As explained in section 1.1.2.2, the Hopkinson bar allows an accurate measurement of the forces and displacements applied on a both faces of a sample, especially in compression. Particular precautions pointed out give access to the weak strain area in the case of concrete. Figure 1.11 shows an example of the forces measured on each face, as well as the rates applied to each face of the sample (Figure 1.12) and the associated displacements (Figure 1.13).

For this test, the specimen is initially 40 mm in both length and diameter. Its relative density is 2.25 kg/m3, with a largest aggregate diameter of 8 mm. It is loaded via an aluminum Hopkinson bar, 40 mm in diameter. The 1.3 m long impactor is projected with a speed of 14.5 m/s. When observing the speeds to be measured, we notice that the specimen absorbs little of the available energy, since the loading bar speed is roughly equal to the initial speed of the impactor at the end of the test, i.e. when the sample has failed. The induced displacements are very low, as the displacement associated with the force peaks is below 1 mm. The post-peak phase observed on the loads says a lot about the existence of inertial confinement.

Figure 1.11.Hopkinson bar compression test input and output loads

Figure 1.12.Hopkinson bar compression test input and output rates