Concrete behaviour in multiaxial compression : … BEHA VIOUR IN MULTIAXIAL COMPRESSION Experimental Research Proefschrift ter verkrjging van de graad van doctor aan de Technische

Concrete behaviour in multiaxial compression :experimental researchvan Geel, H.J.G.M.

DOI:10.6100/IR515170

Published: 01/01/1998

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Citation for published version (APA):Geel, van, H. J. G. M. (1998). Concrete behaviour in multiaxial compression : experimental research Eindhoven:Technische Universiteit Eindhoven DOI: 10.6100/IR515170

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CONCRETE BEHAVIOUR IN MULTIAXIAL COMPRESSION

EXPERIMENT AL RESEARCH ERIK VAN GEEL

48

CONCRETE BEHA VIOUR IN MULTIAXIAL COMPRESSION

Experimental Research

Proefschrift

ter verkrjging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de

Rector Magnificus, prof.dr. M. Rem, voor een commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen op donderdag 2 juli 1998 om 16.00 uur

door

ERIK VAN GEEL

geboren te Roermond

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir. H.S. Rutten en prof.dr. -ing. H.-W. Reinhardt

ISBN 90-6814-548-7

© 1998, Technische Universiteit Eindhoven, Faculteit Bouwkunde, Capaciteitsgroep Constructief Ontwerpen

"Sit back and relax Crawl into the maze

Within your mind Leave it all behind

Now allow yourself to Slide into a world

As real as you want it to be

Actual Fantasy"

Ayreon

ACKNOWLEDGEMENTS

The author wishes to thank the following persons and organisations:

Prof. dr. ir. H.S. Rutten, prof. dr.-ing. H.-W. Reinhardt, dr. ir. J.G.M. van Mier, ir. H.J. Fijneman and especially ir. J.P.W. Bongers for their support and discussions throughout the research project

The Dutch Technology Foundation (STW) for financial and scientific support of the project

Colleagues at the Department of Structural Design and the Pieter van Musschenbroek Laboratory, in particular mr. Sip Overdijk and ing. Martien Ceelen, for their assistance

My friends on the World Wide Web

My family and friends

Nancy

Contents

1. Introduction 3 1.1. Why is multiax.ial concrete behaviour investigated? 3 1.2. Aim of research 3 1.3. Method of working 3 1.4. Preview of contents 4

2. Questions at hand 6 2.1. Does localisation of deformations occur in multiaxial 6

compression as observed in uniax.ial compression? 2.2. What is the influence of the test environment on the 11

observed multiax.ial response? 2.3. Is concrete behaviour significantly influenced by its 16

damage history? 2.4. Is the effect of multiax.ial loading similar for other types of 19

concrete?

3. Testing technique 21 3 .1. Loading apparatus 21 3.2. Test path control 23 3.3. Measuririg loads and deformations 26 3.4. U1traviolet photographical crack pattem recognition 27 3 .5. Data handling 28 3.6. Materials 29 3.7. Specimen preparation 29 3.8. Tensile tests 31

4. Experimental results 34 4.1. Uniaxial compression 34 4.2. Uniax.ial tension 44 4.3. Biaxial compression - Plane stress 46 4.4. Biaxial compression - Plane strain 48 4.5. Triax.ial compression 53

1

5. InDuence of test environment 63 5.1. Boundary friction 63 5.2. Loading apparatus stiffness 71 5.3. Layout ofloading platens 80

6. Loealisation of deformations in macroscopie cracks 84 6.1. Macroscopie crack development 85 6.2. Post-peak continuum behaviour and size effect 89 6.3. Two-dimensional analytical model for shear cracks in 100

plane strain compression 6.4. Some general considerations about macroscopie cracks 115

7. InDuenee of damage from previous loading 120 7 .1. Plane strain test program 122 7.2. Pre-loading experiments 126 7.3. Loading path dependency due to damage at the 134

mesoscopic level 7.4. Macroscopie interpretation of the influence of damage 140

induced at the mesoscopic level 7.5. Loading path dependency due to combined meso- and 150

macroscopie damage 7.6. Finalremarks 151

8. Summary and conclusions 152

References 156

List of symbols 164

Samenvatting 166

2

Chapter 1: Introduction

1. lntroduction

1.1. Why is multiaxial concrete behaviour investigated?

In a large number of concrete structures and structural elements multiaxial compressive stress states occur. As is generally known, these stress states in general have an advantageous effect on the behaviour of the material, compared to the behaviour under uniaxial loading: the load bearing capacity increases and larger deformations are allowable. Unfortunately, this effect has not yet been implemented in the building codes, because there are too many unknowns to describe it in a useful quantitative manner. Therefore insight in the behaviour of concrete under this type of loading is important with respect to safety and deformational capacity of several kinds of concrete structures. At Eindhoven University of Technology (EUT) a research project was started in the early eighties to gain more insight in the failure behaviour of concrete in multiaxial compression. The research described in this thesis is to be regarded as a continuation of the work done earlier at EUT. It deals with several aspects of concrete behaviour in multiaxial compression that were not investigated before or from a different point of view, and intends to answer some remaining questions on the subject.

1.2. Aim of the research

The aim of the present research is to obtain a better understanding of the failure processes of concrete subjected to multiaxial compression. From previous research by Vonk [1992] it is well known how concrete failure takes place in uniaxial compression and how the failure process is influenced by extemal factors like the stiffuess of the loading apparatus. In multiaxial compression however different failure modes are observed (Van Mier [1984]). The present thesis therefore deals with the processes that lead to failure of concrete in multiaxial compression and the factors influencing these processes. The final goal of the research program is to incorporate concrete behaviour under multiaxial loading in building codes and to provide the structural engineering practice with a utility to handle this behaviour. This utility will be the finite element program 'FEMMASSE' in which the recent research results, both experimental and numerical, will be implemented in the very near future. The incorporation of multiaxial concrete behaviour in structural

3

Chapter 1: lntroduction

designs will not only optimise the use of the material, but also result in more effective designs.

1.3. Method of working

A large number of uniaxial, biaxial, plane strain and triaxial compression tests on concrete specimens has been carried out. On the macroscopie level stress-deformation behaviour and macro-crack patterns have been recorded. Mesoscopic cracking, i.e. cracking through mortar and aggregates and at the aggregate-cement paste interfaces, has been determined using the vacuumimpregnation-technique, developed previously at EUT. This experimental research is carried out in close cooperation with numerical research by J.P.W. Bongers. A complete overview of multiaxial concrete behaviour can only be obtained by a research program incorporating both numerical and experimental investigations. Whereas experimental results provide the required input data for numerical modelling, insight obtained from numerical calculations can specify the emphasis of the experimental test program. Numerical models at both macroscopie and mesoscopic level are under development. Initially these models are developed at the twodimensional level. Tuis bas led to an emphasis on plane strain experiments, which are quite close to a 2D situation.

1.4. Preview of contents

In Chapter 2 some background information on the subject is given, taken from a literature study carried out in an early stage of the research project. From this survey the goals for this research have been determined.

Chapter 3 describes the loading apparatus used in the present experimental research. Since this apparatus was modified during the present research, descriptions of both loading apparatus configurations are shown. Furthermore specifications of test control, measuring technique and specimen composition and preparation are given.

Test results from specimens subjected to monotonie loading are shown in Chapter 4. Because a large number of different types of compressive tests have been carried out on the same type of concrete, these results form a solid basis for numerical simulations. Furthermore, these results will be used as a resource for determining influences of test environment and loading path on the behaviour as

4

Chapter 1: Introduction

will be described in the following chapters. In this chapter stress-deformation behaviour is emphasised.

The influence of the test environment can be found in Chapter 5. Changes in the amount of friction at the loading platen-specimen interfaces and in the layout of the loading platens have been investigated. Deformation measurements at the loading frame have been carried out to ascertain the origin of the stimulation of rotations of the loaded specimen boundaries in uniaxial compression.

Chapter 6 deals with localisation of deformations in macroscopie cracks after peak stress has been reached. The evolution of these cracks has been examined in multiaxial compression. An important aspect in this is the determination of the contribution of the 'uncracked' specimen continuum parts to the post-peak behaviour. A simple model has been developed to describe macroscopie crack behaviour. Some general considerations are presented with respect to this behaviour.

In Chapter 7 the influence of damage from previous loadings on the stressdeformation behaviour is described. Results on specimens loaded via different stress paths and specimens subjected to several types of pre-loadings are presented. The particular differences observed between the several test results are described qualitatively. The observed macroscopie stress-deformation behaviour is qualitatively linked to two types of damage at the mesoscopic scale level.

In Chapter 8 the research at hand is summarized and the most important conclusions are outlined.

5

Chapter 2: Questions at hand

2. Questions at hand

At an early stage of the present research project, an extensive literature study was carried out in which all attainable studies with respect to the short-term behaviour of concrete under compression [Van Geel, 1995a] were searched and analysed. From this study, a number of questions arose concerning multiaxial behaviour of concrete. These questions and their origin are described in this chapter.

2.1. Does localisation of deformations occur in multiaxial compression as observed in uniaxial compremon ?

Concrete is a composite material, consisting of cement, gravel, sand and water. After hardening of this composite material, the material can be regarded as a mortar matrix including distributed aggregates. Tuis composite exhibits a nonlinear stress-strain relation, even at very low loads, and after reaching the peak load a descending branch, called softening (see for example Ziegeldorf [1984]), develops, as is shown in Figure 1.

cr Peak stress

So~ening curve

"-

Figure 1: Stress-strain behaviour of concrete in axial direction under uniaxial compression

6


It has been found that the nonlinear pre-peak behaviour is caused by a progressive growth (Spooner [1975,1976], Carrasquillo [1980], Vonk [1992]) of three different types of mesoscopic cracks: • Cracks running through the mortar matrix; • Cracks running through the aggregates; • Cracks at the bond between mortar and aggregates, see Blakey [1955], Hsu et al. [1963], Slate and Olsefski [1963], Sturman et al. [1965], Krishnaswamy [1968], Stroeven [1973], Mazars [1981], Loo [1995]. To some extent micro- and mesocracks (mostly bond cracks) already exist before any mechanica! load is applied to the concrete ([Slate and Olsefski [1963], Shah and Slate [1965], Krishnaswamy [1968], Dhir and Sangha [1974]). These cracks are mostly due to influences of temperature and moisture gradients within the material after casting. Cracks running through the aggregates are found when the difference in aggregate and matrix properties decreases, for example, in high-strength concrete (Carrasquillo [1980]). It is observed that the interface between aggregate and mortar has a very different structure (Scrivener and Pratt [ 1987], Mindess and Shah [1988]) and represents the weakest link in normal-strength concrete (note: in the case of other types of concrete this interface strength may be higher than for example the aggregate strength. In genera} it can be said that the ratio between aggregate, bond and cement paste strength determinates the properties of a certain type of concrete, see for example Vervuurt [ 1995]. At about 70 to 90 percent of peak stress these mesocracks join and start to form continuous crack patterns, accompanied by a transition from volume compaction to volume dilatation (Hsu et al. [1963], Krishnaswamy (1968]). The long-term strength of concrete is often related to this stress level, called 'critica! stress level'. At higher stress levels instable crack growth occurs (Slate and Hover [1984]). At peak stress these crack patterns are developed so far, that no additional load can be applied to the material. The mesoscopic crack pattems finally grow into macroscopie cracks which dominate the behaviour after peak stress (Kotsovos [1983], Van Mier [1984, 1986], Torrenti et al. [1986, 1989, 1991], Shah and Sankar [1987], Vonk [1992]). The softening behaviour is then governed on the one hand by the formation of these macrocracks itself and on the other hand by aggregate interlock and frictional characteristics of these cracks (Walraven [1980], Vonk [1992]). Figure 2 shows results from Vonk's numerical model demonstrating this progressive failure. Deformations tend to localise within these macrocracks while the more or less uncracked concrete parts show unloading. Therefore it is more suitable to describe the post-peak behaviour of concrete in terms of

7


(a)

'(d)

Figure 2: Progressive failure of concrete under uniaxial compression (/rom a to d,for increasing axial deformation) [Vonk, 1992]

displacements rather than strains (Van Mier [1984, 1986], Torrenti [1986, 1989], Vonk [1992]). Van Mier [1984], Van Mier et al. [1997] showed that this localisation is independent of the size of the test specimen, see Figure 3 (left), resulting in almost identical post-peak stress-displacement curves for different specimen sizes despite the differences in fracture modes observed (see also Jansen and Shah [1997]). However, according to Vonk (1992] the uncracked continuum parts still contribute to the post-peak load hearing capacity, leading to less brittle softening (stress-displacement) curves for larger specimens as shown in Figure 3 (right). This contradiction in experimental results will be discussed later on. When subjected to multiaxial compression, the formation of meso- and macroscopie crack pattems appears to be similar to that in the case of uniaxial compression (Launay et al. [1970], Kotsovos and Newman [1977], Newman [1979]). Dependent on the amount of lateral confinement, in multiaxial tests usually a more nonlinear relation exists between stress and

8


strain. At peak stress very large strains can occur, indicating a highly deformed specimen. When the level of confinement increases, peak stress and strain increase, according to Krishnaswamy [1968] caused by a 'slowdown' ofmicrocracking due to this confinement (see also Robinson [1967]).

1.0 ·::-,,

,....., ...!...

0.8

.. " \\~···································"············

~ ~ 0.6

~ ~ 0.4 ~

0.2

~ \ ..................... " ..................... ~ \\ """""" "\

~· , ... '~~ ....... ".

''-~..:·••l'u~«"••u•"u•• • ...... ......

~-------

0 0 -0.2 -0.4 -0.6

Post-peak inelastic disptacement [mm]

1.0

0.8 ::!:: "" <Il 0.6 ~ "" 0

-~ ..!! ~

0.4

0.2

0 0 -0.2 -0.4 -0.6 -0.8 -1.0

Post-peak inelastic displacement [mm]

Figure 3: Observed post-peak behaviour for different specimen sizes in uniaxial compression. Top: results by Van Mier ( [1984]), bottom: results by

Vonk {1992]).

9


Like in uniaxial compressive tests Van Mier observed the formation of localised macrocracks. Two clearly different fracture modes were distinguished by Van Mier as shown in Figure 4: • Planar failure mode: A pronounced shear band fracture mode ( a clear

localisation of deformations ), occurring when a preferential direction of failure was present (two different confining stresses or 'plain-strain' tests in which the deformation in one direction is completely prevented). Tuis means the presence of one large positive (i.e. non-compressive) deformation (in this case in the 3-direction in Figure 4, right).

• Cylindrical failure mode: A more distributed fracture mode, which is the result of mutually crossing shear bands, occurring in stress regions near the compressive meridian ( equal lateral confining pressures ). Uniaxial compression is also a load case in this region. Tuis means the presence of two large positive deformations (in this case the 2- and 3-direction in Figure 4, left). When applying equal lateral stresses, ductility increases with increasing confinement. Tuis is commonly referred to as the 'brittleductile transition' (Jamet et al. [1984], Willam et al. [1986], Smith et al. [1989], lmran and Pantazopoulou [1996] ).

Cylindrical mode Planarmode

Figure 4: Multiaxial compression tests on concrete cubes (Van Mier [1984])

10


The planar mode failure is a more brittle failure mode, whereas the cylindrical mode is a more ductile failure mode. In both modes shear bands develop in planes where in one direction a large compressive deformation and in another direction a large positive deformation occurs. In Figure 4 these are the 1-2 and 1-3 planes in the left figure and the 1-3 plane in the right figure. No multiaxial experiments have been carried out so far, in which localisation of deformations after peak stress is examined including a variation in specimen size. Because of the larger deformations observed in multiaxial compression, this might clarify the question raised above, whether there indeed is a contribution of uncracked specimen parts to the post-peak load hearing capacity of concrete or not.

2.2. What is the intluence of the test environment on the observed multiaxial response?

After initiation of the macroscopie cracks the material can no longer be regarded as a continuum. It becomes a structure consisting of uncracked elements divided by local fracture planes. It is clear that the behaviour of this structure is a structural response rather than a material characteristic (Van Mier [1984], Kotsovos [1983], Torrenti [1986], Vonk [1992], see also Sture and Ko [1978] and Bieniawski et al. [1967,1969] in the case of fractured rock). For example, consider a uniaxial compression test where the residual load hearing capacity at the end of the softening branch appears to be the result of the presence of a more or less intact specimen core (Van Mier [1984]). Furthermore, when interpreting test results one should bear in mind that the load application system -at least those parts directly connected to the specimen- is in fact part of this structure. Different load application systems influence the behaviour observed differently.

Layout of the loading apparatus Researchers at TU Munich (Linse [1978], Winkler [1985]) studied the influence of the layout of the testing device exhaustively and developed a 'multi-part' loading apparatus. This is an apparatus in which the three loading axes are able to move independently of each other. Hence additional shear stresses at the specimen surfaces ( especially in the softening regime) are minimised and non-symmetrie deformations of the specimen are avoided. These are significant problems encountered in a 'one-part' loading apparatus with fixed axes. See for example Linse [1978], Van Mier [1984] and Winkler

11


[1985]. Likewise differences in stress-deformation measurement can be expected when comparing results from a triaxial loading device, in which the lateral stress is equal in all lateral directions, and the above mentioned apparatuses. An example of Winkler' s results is shown in Figure 5.

Friction between loading platen and specimen surface It is known that frictional stresses between loading platens and specimen boundaries play a large role in the observed behaviour of concrete. In a uniaxial compression test large frictional stresses prevent the specimen to deform laterally and result in triaxial stress states near the loading platens. Increasing boundary friction therefore results in higher peak strains and stresses and less brittle softening behaviour, see for example Kotsovos [1983], Vonk (1992], Mansur et al. [1995] and Choi et al. (1996]. Therefore when analysing experimental data one should be aware of the boundary conditions present in the experiments. Dry steel loading platens, as used in standard compressive cube tests, introduce large frictional stresses at the specimen boundaries. Because the material behaviour becomes more complex (and thus more difficult to interpret) with increasing boundary friction, these loading platens are nowadays mostly used for comparative reasons. E.g. in the cooperative research program by Van Mier et al. [1997] (for uniaxial experiments using dry steel platens see e.g. Weigler and Becker [1963], Krishnaswamy [1968], Torrenti et al. [1989, 1993b], Taerwe [1991]). To eliminate or at least reduce the amount of boundary friction, different loading platens have been developed. The two most applied types are: • loading platens with intermediate layers, like chalk (sometimes with

aluminium foil) by Kobayashi and Koyanagi [1972], thin metal sheets by Erdei [1980], teflon sheets with hearing grease by Vonk [1992], multiple layers (grease, teflon and aluminium sheets) by Murakami et al. [1994], etc.

• Steel 'brush' platens. These platens, developed by Hilsdorf in [1965], consist of a large number of small steel rods assembled in a package. These loading platens have been used in multiaxial compression tests by for example Nelissen [1970, 1972], Kupfer [1973] and Van Mier [1984]. A disadvantage of these loading platens is that due to bending of the steel rods the resistance against lateral deformation increases with this deformation and on that account stresses are introduced that can be considerable when compared to stresses due to friction.

The effect of boundary friction on peak stress and strain is found to bè similar in the case of multiaxial loading (Gerstle et al. [1978]).

12


---aLnhLUg

-80

t

-80

Hl-'

t a,

-eo -eo

-80 -80

Ml-' Hl-'

OM5 Lo f'oo.3

5000 " ....

5pa11wu•9•vt•hl1LhLa cr,:o2:cr1 " t: t:o

IJl"' L• l'oa.2

IDH5 Ln l'aa.3

Figure 5: Biaxial experiments on concrete in a one-part (solid lines) and a multi-part (dashed lines) loading apparatus (left: dry steel loading platens,

right: brush platens) (Winkler [1985])

13


cr,

i 111 i ___.,..___ -î t t t î Figure 6: lnjluence of boundary friction decreases with increasing height in

uniaxial compression (afler Van Vliet and Van Mier {1996])

From previous research it is known that this influence ofboundary friction in uniaxial compression decreases with increasing specimen height 01 onk [1992] Mansur et al. [1995], Choi et al. [1996], Van Mier et al. [1997]). With increasing height the specimen volume, which is not influenced by local triaxial stress states, increases (see Figure 6). In the case of cylindrical specimens, the specimen height needed for the fäilure zone to form unobstructed by frictional stresses is found to be about three times the specimen diameter (Rokugo and Koyanagi [1992], Markeset [1993], Jansen and Shah [1997]). Besides the friction introduced by the loading platens the rigidity of the platens needs to be considered. lnfinitely rigid loading platens would theoretically lead to a constant axial displacement at the specimen surface. On the contrary, infinitely flexible platens would result in a constant axial stress distribution over the boundary.

Non-uniform deformations From the point of view of homogeneous continuum theory, a uniform distribution of stress or strain in a specimen is thought, resulting in a 'material' behaviour of the specimen. Surely, in a specimen subjected to uniaxial loading non-uniform deformations develop, particularly after peak

14


stress as a result of the localisation of deformations during the softening process. This nonuniformity of deformations occurs both in tensile loading (Willam et al. [1986], Van Mier [1986], Hordijk et al. [1987], Hordijk and Reinhardt [1989]) and compressive loading (Vonk [1992]). Vonk [1992] has shown that in the case of uniaxial compression, non-uniformity of deformations is related not only to the heterogeneity of the material but also to eccentric loading, imperfect contact between loading platen and specimen, the stiffness of the loading apparatus, the specimen dimensions and the slope of the softening curve. In Figure 7 the influence of an initial angle between loading platen and specimen surface is shown as an example. Both in uniaxial tension and compression the occurrence of non-uniform deformations is mainly characterised by an irregular shape of the nomina} stress-deformation curve.

-50

1 -«>

-30

1 i

-20

-10

0 0.0 - 0.2 -0.4

(a)

- tpo=-QOOO

- - - 9'0=0.001

-- ço"=0.002 . . . . . tpo=-Q.003

·- ,,,"-0.()()4

-0.6 -o.e -1.0

1.0()"""'~---..,

i: 1 oa5

o.eo ,___....._ ___ __, 0.000 0.001 Q.002 Q.003 Q.004

Initia! IOtatiol\ .....

(b)

Figure 7: Irifluence of an initia/ angle between specimen surface and loading platen on the nomina/ stress-deformation curve (a) and on peak stress (b)

from computations by Vonk {1992]

Unfortunately, when trying to eliminate boundary friction by using loading platens with low friction, the stabilising effect of the boundary restraint decreases and redistribution of stresses within the specimen becomes more difficult, thereby stimulating localisation of deformations and consequently the development of non-uniform deformations (Choi et al. [1996]). Vonk [1989] showed that in multiaxial compressive experiments the influence of nonuniform deformations is much smaller because the specimen

15


is completely enclosed by loading platens. In this way rotation of the specimen boundaries is prevented to a large extent.

2.3. Is concrete behaviour significantly inftuenced by its damage history?

In most multiaxial experiments loading is applied through proportional loading paths, i.e. constant ratios of stresses cr/cr/cr3 or displacements w/w/w3 • In practice, however, the occurrence of other loading paths is very likely. Furthermore, the highly non-linear behaviour observed in multiaxial compression tests on concrete raised the thought that this behaviour might very likely be dependent on the loading path followed, in other words on the amount of damage inflicted to the specimen during its load history. Therefore several researchers have examined the influence of this damage history on the observed stress-deformation behaviour. The influence of the damage history can be examined in many different ways as will be shown in this section, like subjecting specimens to cyclic loading or to different loading paths in stress or deformation space (stress and deformation paths). Because for a long time research into the behaviour of concrete under multiaxial compression was focussed on ultimate load hearing capacity, most comparisons between different stress or deformation paths were made emphasising the effect on peak stress only. See for example Kobayashi and Koyanagi [1972] (Figure 8) and Taylor and Patel [1974] for biaxial compression stress paths. In both studies it was concluded that no significant influence of the followed stress path on peak stress occurred. Recently, Torrenti [1993a] carried out a similar biaxial stress path comparison, using two different loading devices. A small influence on the strain levels at failure was observed, though the difference between the two applied loading systems appeared to be a factor of major influence. Triaxial stress path comparisons have been carried out by Gerstle et al. [1978], Kotsovos [1979], Smith et al. [1989] and recently by Imran and Pantazopoulou [1996], Figure 9. Again, only small differences have been observed between the behaviour following different stress paths. Kotsovos ascribes the differences, observed at stresses larger than 0.8 times the uniaxial compressive strength, to microcracking under hydrostatic loading. The fact that no differences are found between the 'envelope curve' of cyclic loading and the stress-strain curve of monotonie loading in both uniaxial (Spooner and Dougill [1979]) and multiaxial (Van Mier [1984]) experiments also indicates that no significant influence of the loading path exists.

16

400

:z 6

N u..

200


200 400 F 1 [kN]

Figure 8: Biaxial loading paths examined by Kobayashi and Koyanagi [1972]

However, it is shown ·by Van Mier [1984, 1986] that concrete behaviour is dependent on the damage history in some cases where macroscopie cracks are already present. Tuis has been shown by carrying out experiments in which the major principal stress direction is rotated after reaching peak stress. Tuis is depicted in Figure 10 and Figure 11. It appears that in the case of the 'cylindrical mode rotation test' . the macrocracks initiated during the first loading are situated in such directions that they can propagate more easily under the subsequent loading. Altogether it can be said that concrete stress-deformation behaviour in multiaxial compression seems to be quite insensitive to variations in loading paths. So far a significant influence of the damage history is only observed in situations, where peak stress is surpassed. The remaining question is: is prepeak concrete behaviour always more or less independent of the loading history?

17


.1 mJ----------iU 11 ~------

~ : u - - - -

Confilling pressure IMPa]

'::=======================:

A>Öal strain [oio]

Muftistep - Monotonie

4"4 U.I IU: 2U

Confining pressure IMP•]

--Mul!lstep Mon-i<

Allialstrain (o/o)

Figure 9: Comparison of 'multistep' and monotonie stress path by Imran and Pantazopoulou {1996]

<r2 =-1MPa

<r 3 = - 1 MPa

0 1 = -1 MPa

d(A w3)/dt = C d(A w3)/d! = C

Second loading L--'-ll__-"Y'

Second loading

Planar mode rotation test Cylindrical mode rotation test

Figure JO: Two types ofrotation tests carried out by Van Mier [1984]

18


-70 -D Cylindrical mode rotatlon test

8 6 4 2 0 ·2 ·4 -6 -8 -10 ·12 4 2 0 -2 -4 -6 -8 -10 -12 strain t 1,t3 1•1-1 strain E1.E3 l°lool

Figure 11: Stress-displacement results of Van Mier's rotation tests (Van Mier, {1984})

2.4. Is the effect of multia:xial loading similar for other types of concrete ?

Concrete types that are more homogeneous show a more linear uniaxial stress-deformation relation before peak stress, because the properties of mortar, aggregate and aggregate-cement paste interfaces are more alike. An example of this is high-strength concrete, where due to the addition of e.g. microsilica the properties of particularly the aggregate-cement paste interface are improved, resulting in an increase in peak stress. However, due to the increased homogeneity only a small redistribution of stresses is possible within the specimen, which results in unstable cracking soon after the first macrocracks are formed. Less possible fracture surfaces have been present and the final fracture surface is smoother. High-strength concrete therefore is also more brittle than normal-strength concrete. See for example Wischers [1978], Carrasquillo [1980], Dahl [1992] and Taerwe [1993]. In some uniaxial tests on high-strength concrete even snap-back behaviour is observed (Rokugo et al. [1986], Glavind and Stang [1991], Taerwe [1991,1993]). Dahl's results are shown in Figure 12. Until now no experiments have been carried out in which the influence of concrete composition is extended to the post-peak region. From multiaxial tests up to peak stress the results in genera! indicate that the (relative) failure surface in the stress space is hardly influenced by the concrete composition. See for example Mills and Zimmermann [1970], Hobbs [1971], Kobayashi and Koyanagi [1972], Kupfer [1973] and Linse and Stegbauer [1976].

19


Systematic research into the influence of several composition parameters on the softening response will be necessary. In this thesis a first effort will be made by examining . the behaviour of a high-strength concrete under multiaxial compressive loading.

120

(ii' a.. ~ 80

40

0 0.002 0.004 0.006

Strain [o/oo] Figure 12: The effect of improving the aggregate-cement paste bond on pre

peak behaviour, peak stress and brittleness (after Dahl {1992})

20

Chapter 3: Testing technique

3. Testing technique

As described in the previous chapter, the test environment can significantly influence measurements on specimens subjected to compressive loads. It is not only material behaviour that is measured but in fact a response that is the result of both material and test environment characteristics. Thus it is important to know how these measurements are acquired to be able to evaluate the influence of the test environment on the observations. In this chapter the loading equipment and the measuring technique in the present research are presented. In addition, the way in which specimens are prepared before testing and the test results are handled during and after testing are shown.

3.1. Loading apparatus

In the present researcb, two configurations of loading apparatus are used. The first configuration consists of three independent loading axes, which are able to apply 'true triaxial' stress states on prismatic specimens. Bach axis has a compressive capacity of 2000 kN and each axis is hung in its own steel frame by means of steel cables. Tuis is the loading apparatus used at Eindhoven University of Technology until 1996. For a detailed description of this apparatus see Van Mier [1984], Vonk [1992] and Van Geel [1994a,1995b]. The second configuration is similar to the first, but in this case the vertical axis has been replaced by an axis with a compressive capacity of 5000 kN. In this configuration the vertical axis is placed upon air springs to allow this axis to translate in vertical direction, as shown in Figure 13. In both configurations the horizontal axes are identical to the vertical axis in the first configuration. One loading axis (Figure 13) consists of: • A hydraulic cylinder. Within the cylinder an internal L VDT is mounted

with a range of ±100 mm. A servo valve is mounted upon the cylinder to control the movement of the cylinder' s piston;

• A steel frame upon which the cylinder is mounted, consisting of two steel platens connected by four steel rods. In the case of loading apparatus configuration 2, the steel rods of one horizontal axis pass through holes in the rods of the vertical (5000 kN) axis;

• A hinge between the cylinder and the loading platens. The hinges are always fastened (with steel bolts) during testing and only loosened to position the loading platens on the specimen;

21


Hydraulic Cylinder

Endplaten

Piston

Hinge

Loading platens

Specimen

Steel rods

Load cel!

Endplaten

Air springs

Figure 13: Layout of a single loading axis. Left: 5000 kN axis, right: 2000 kN axis

1 metre

• Loading platens. To be able to examine the effect of different specimen size and shape, three different types of loading platens are used, shown in Figure 14. Using these loading platens, prismatic specimens with a crosssection of l 00 by 100 mm2 can be tested with specimen heights of 50, 100 and 200 mm. All loading platens are hardened ( approximately 50 Rockwell) and all surfaces in contact with the specimen surface are polished (Ra 0.05 - 0.12 µm). In most experiments, the loading platens are covered with a very thin layer of hearing grease (Molykote BR2 Plus) and a 0.05 mm teflon layer to reduce friction between loading platens and specimen (see Vonk [1989,1992]);

• A load cell, consisting of a steel cylinder upon which four strain gauges are mounted in a full Wheatstone bridge configuration;

22


• In loading apparatus configuration 2: four air springs on which the vertical axis is placed. These air springs allow the vertical loading axis to translate in its axis direction to prevent nonsymmetrical deformations in the case of multiaxial testing (see e.g. Van Mier [1984]). Tuis means that the translation of this axis should equal 0.5 times the axial compressive deformation of the specimen (several millimeters). Within the range of specimens tested in this research, this translation is feasible.

Figure 14: Shape of the loading platens used to test different specimen heights multiaxially

3.2. Test path control

Manual test path control In manual test control (see [Van Geel, [1995b]) the major principal stress direction (master axis) is always in deformation control. A function generator (MTS 418.91 MicroProfiler) is used to generate a constant displacement rate. In the present tests this rate is either -1 µmis (uniaxial and biaxial tests) or -2 µmis (multiaxial tests). This signal is further processed by a Schenck S59 Servo-controller. In the case of multiaxial testing one or two so-called slave axes are in function. The forces applied by these slave axes are then dependent on the force measured in the master axis direction. A slave axis is always in force control, adjusted by dedicated electronics: using potentiometers the ratio AF.1av/Mmaster can be controlled manually. Disadvantage of this manual test control is that the time span needed for changes in stress path is too large to avoid serious deviations ftom the desired path. Therefore the automated test path control as described in the next section has been developed.

23


Test path control software During the present research a computer program has been developed which allows for fully automated tests in uniaxial, biaxial, plane strain and triaxial compression. This program allows the user to define stress paths, in which one axis (master axis) is in deformation control and -in the case of multiaxial testing- one or two axes are in force control (slave axes). The loading rate can be specified per part of the stress path, that consists of a number of linear branches in stress space. For a detailed description of the computer program see [Van Geel, 1996]. About half of the experiments has been controlled using this software. The former half has been carried out 'manually', as described in the previous section.

Figure 15: Test control panel during a multiaxial test

24


Extreme/y britt/e behaviour While defonnation-controlled tests in general are a good way to measure the post-peak response of concrete specimens subjected to compressive stresses, this control method fails in the case of very brittle post-peak behaviour or tests in which so-called snap-back behaviour is observed, like uniaxial tests on high-strength concrete specimens or biaxial compression tests. To be able to measure the post-peak behaviour of concrete in these tests, an alternative test control is used as proposed by Rokugo et al. [1986]. The feedback signal in these experiments consists of both deformation and force signals of the major principal stress direction, as indicated in Figure 16. In this manner the force-defonnation graph is in fact translated over a varying distance, proportional to the force signal, to overcome the problems of a very steep descending branch or snap-back behaviour. The factor y as shown in Figure 16 detennines the degree of translation.

Control parameter: w' = w-F tan Y

Axial deformation (w)

w' Figure 16: Alternative test contra/ used in uniaxial tests on high-strength

concrete specimens and biaxia/ compressive tests

25


3.3. Measuring loads and deformations

Generally the specimen deformations are measured using Hottinger WK20 LVDTs with a range of±lO mm. The LVDTs are fixed to the loading platens using aluminium frames. In Figure 17 (left) the position of these L VDTs in a uniaxial experiment is shown. Four L VDTs are used to measure the axial deformation, while two additional L VDTs are used for both measurement and test control.

•o~LVDT Strain gauge

0 0

o• ~Control LVDT Clip gauge

Figure 17: Measured deformations in a uniaxial experiment

26


Figure 18 shows the placement of the L VDTs (both measurement and test control) in the case of multiaxial testing. While in uniaxial tests strain gauges and clip gauges (see e.g. Van Mier [1984]) are used to measure lateral deformations, as shown in figure 17 (right), in multiaxial experiments all deformations are measured using the L VDTs. Loads are measured using the load cells mentioned earlier in this chapter. In the case of manual test path control, all load and deformation values are read using a HBM UPM 60 data logger, that scans the measuring channels every two seconds and stores the data on a Tulip PC286. In the case of automated test path control, the test software not only controls the test but also writes the actual values of loads and deformations to a measurement file every two seconds.

•

• ....._Control LVDT

Figure 18: Measured deformations using LVDTs in a multiaxial experiment

3.4. Ultraviolet photographic crack pattem recognition

At the end of the test, specimens tested multiaxially are taken out of the loading apparatus and are impregnated with an epoxy resin containing fluorescent powder, which fills cracks in contact with the outer specimen surfaces. After hardening of the resin the specimens are sawn into slices. By photographing these slices using ultraviolet light the failure modes of the specimens become clearly visible. Tuis technique bas been developed at Eindhoven University ofTechnology by Vonk and Goudswaard [1989,1992]. To obtain information about the evolution of cracks, series of tests have been carried out in which similar experiments are halted at different stages of postpeak loading. This has been done for both plane strain and triaxial tests as indicated in Figure 19.

27


-b

Abortion

+--Unloading

W1 Figure 19: Abortion of experiments to examine crack evolution

3.5. Data handling

The measured deforrnations have to be corrected for various reasons: • initial setting of the loading platens on the specimen surface due to non

flatness of the specimen, non-parallellness of specimen and loading platen;

• deforrnations of the loading platens and compression of the teflon/grease layer between loading platen and specimen.

In most cases, a quadratic regression analysis is applied to obtain the 'final' load-displacement diagrams. It is found in the plane strain test program (see Chapter 4) that the initial settings influence the measured deforrnations up to stresses of about -15 MPa (Van Geel [1995b]). Due to these settings regression of the initial part of the stress-deforrnation curves can be quite difficult in tests with small lateral stresses. Therefore specimens used in the triaxial experiments (with lateral stresses cr2=cr3) have been pre-loaded hydrostatically up to -10 MPa to avoid most of the difficulties caused by initia! settings due to compression of the teflon/grease layer.

28


Table 1: Concrete materials (weight percentages)

Normal Strength High Strength 1

Concrete Concrete PC Cement Type B 15.8% 21.2% Max. aggregate size 8mm 8mm

(Rounded river 2ravel) Size S-8mm 17.0% 15.5%

distrib. 3-Smm 5.7% 5.1% 2-4mm 15.3% 14.0% 1-2mm 11.5% 10.5%

0.5-1 mm 11.5% 10.5% 0.2-0.63 mm 9.8% 9.0% quartz sand 5.3% 4.9%

Admixtures None Microsilica 1.5% Water/cement ratio 0.5 0.35 plus 0.5%

superplasticizer

3.6. Materials

Two different materials have been used in the present experimental research. Both a normal-strength (standard compressive cube strength 55 MPa) and a high-strength (standard compressive cube strength 80 MPa) concrete are used, both with a maximum aggregate size of 8 mm. See Table 1.

3. 7. Specimen preparation

In Table 2 the followed specimen preparation procedure is shown. The specimens are cast in prismatic moulds of 150*150*450 mm3

, stiffened by extra partitions. From these prisms the specimens are sawn with an oversize of 2 mm in every direction, necessary for grinding the surfaces. In uniaxial testing the final cross-section of the specimens is 97*97 mm2

( equal to the cross-section of the loading platens ), in multiaxial experiments this crosssection is 100* 100 mm2 to avoid contact between the loading platens. Three different specimen heights have been applied (axial stress direction): 50, 100 and200mm.

29

Time: 1st day

2ndday

After 2 days

After 28 days


Table 2: Specimen preparation

Action: Casting of 3 prisms and 5 standard cubes (150 mm). Compaction by means of a vibration table (5 kHz) for about 45 (prisms) or 30 (cubes) seconds. Prisms and cubes covered with plastic, together with two bowls of water. Removing the bowls and covering the speci with wet clothes.

After 8 weeks and Testing before 10 weeks

'

0 0.01 mm

Il 0.01 mm A

100.0 mm

' ' '

1.Smm -' ' : '

----r-----1~ - - - - - - 3

Figure 20: Size and tolerances of specimens (multiaxial testing)

30


In Figure 20 the accuracy of grinding the specimens is shown. In the case of the first multiaxial test series, bevels were grinded at the specimen corners to avoid large stress concentrations at the specimens corners. However, comparison between experiments on specimens both with and without bevels showed that this precaution was not necessary. Therefore, most multiaxial experiments have been carried out on specimens without bevels.

3.8. Tensile tests

Tensile tests are carried out using a tensile loading apparatus which has been developed recently at TU Eindhoven and which is capable of keeping the two loading platens parallel during a test.

Loading apparatus This apparatus is originally built to perform tensile tests on masonry (brickjoint bond) by Van der Pluijm [1997]. The parallelism of the loading platens is guaranteed by a parallelograrn-construction, which is described by Van der Pluijm [1997]. See Figure 21 and Figure 22. The ends of the specimen are glued to steel loading platens. The lower steel platen is bolted into a fixed position, the upper loading platen is fixed to the loading frame.

Figure 21: Tensile loading apparatus used in the present research (after Van der Pluijm [1997})

31


Specimens In six 100 mm normal-strength concrete cubes a notch of 5 mm width is sawn at middle height of the specimens. The remaining cross-section at the notch is 57*57 mm2

• See Figure 23.

Measurement and test control Four L VDTs (Sangamo SM3, full range 3 mm) are mounted over the notch as indicated in Figure 23. The average signal of these LVDTs is taken as the control parameter. The tests are controlled by a Schenck S59 servocontroller. All tests are displacement-controlled at a loading rate of 1.2 µm per minute. In the lower part of the descending branch this loading rate is increased to save time.

Figure 22: Tensile loading apparatus used in the present research (after Van der Pluijm [ 1997})

32


Control L VDT

Figure 23: Tensile test specimen and LVDT positions

33

Chapter 4: Experimental results

4. Experimental results

In this chapter the results from monotonie experiments on both a normal- and a high-strength concrete are presented. The emphasis is laid on the normalstrength concrete test results, because initially the numerical modelling at EUT focusses on this type of concrete and because the high-strength concrete experiments provide only the basis for further research into the effect of concrete composition. Experiments carried out on both types of concrete are: uniaxial, biaxial, plane strain and triaxial compression. All of these experiments are carried out using a teflon intermediate layer between specimen and loading platens to reduce the effect of boundary friction on the observed behaviour. To provide additional parameters for numerical modelling, some tensile experiments are carried out on normal strength concrete specimens. All these different types of experiments have been carried out on the sarne type of concrete under identical laboratory conditions. Hence, these test results provide a solid basis for numerical simulations. The influence of the test environment and boundary conditions, which will be described in chapter 5, and the influence of darnage from previous loadings, in chapter 7, will be easier to evaluate because of the availability of these 'standard' experimental results.

4.1. Uniaxial compr~ion

As mentioned before, building codes generally do not take the advantageous effects of .multiaxial compressive stress states into account. Even more, the uniaxial stress-strain diagram is radically schematised, as is shown in Figure 23 in the case of Dutch building code 'VBC 1995' [1995]. Tuis type of diagram is assumed to be valid in nearly every situation regardless of boundary conditions, though it is known that stress-strain behaviour of concrete is dependent on the present boundary conditions too. In the case of compressive experiments on concrete specimens these boundary conditions include the layout and stiffuess of the loading apparatus, friction between loading platen and specimen and the size of the specimen. A bi-linear diagram is defined by two strain levels and a maximum compressive stress. Besides the fact that these strain levels are presumed to be identical for all types of concrete, the uniaxial strength is underestimated and the concrete' s softening behaviour is disregarded. When comparing the

34


diagram from Figure 23 with, for example, the stress-strain curves shown in Figure 9, it should be clear that a better understanding of concrete behaviour under multiaxial stresses can both improve design and reduce costs of concrete structures.

f'b

1.75 3.5 E [o/oo]

Figure 23: A stress-strain diagram/or concrete in building code 'VBC 1995' [1995]

However, there still is no international standard test procedure, not even for uniaxial compression, mainly because of differences between existing testing machines. In order to establish such a standard test method, RILEM Committee 148SSC (Strain Softening of Concrete) has set up a round robin test in which initially two structural aspects are emphasized: boundary friction between loading platen and specimen and specimen size. The tests described in this paragraph are part of this round robin test. The results from all laboratories are gathered in [Van Mier et al., 1997]. lndividual results can be found in [Bascoul et al" 1994, Choi et al., 1994, Dasenbrock et al., 1995, Gobbi and Ferrara, 1995, König et al., 1994, Lange-Kornbak and Karihaloo, 1994, Markeset, 1995, Van Geel, 1994a, 1994b, Van Vliet and Van Mier, 1995,1996, Zissopoulos et al., 1994].

Test program Uniaxial compressive softening tests on two types of concrete have been carried out, varying the type of loading platen and the specimen size. In total 37 specimens are tested, 18 with polished steel loading platens and 19 with

35


loading platens with a teflon layer. The tests carried out without teflon layer will be discussed in Chapter 5. Half of the specimens are made of normalstrength concrete, the other half of high-strength concrete. The specimen height is varied from 50 mm to 200 mm. In Table 3 an overview of all tests is presented indicating the loading platens used in the tests, the specimen size and the concrete type. As can be seen in Table 3 every variation was carried out in triplicate. From the test results it is found that the influence of differences in casting batches or position of the specimens in the casting prisms is negligible (Van Geel [1994a]).

Table 3: Layout of experiments

Normal-stren2th concrete Hi2h-stren1 tb concrete Specimen Rough Teflon Rough Teflon hei2ht platens platens platens platens 50mm 3 tests 3 tests 3 tests 3 tests lOOmm 3 tests 3 tests 3 tests 3 tests 200mm 3 tests 3 tests 3 tests 4 tests

4.1.1. Normal-strength concrete results

In Figure 24 and Figure 25 the axial stress-strain and stress-displacement curves are presented for all normal-strength concrete specimens loaded with teflon platens. These figures clearly show the difference between pre- and post-peak behaviour: before peak-stress different specimen sizes exhibit almost identical stress-strain behaviour while after the top different specimen sizes exhibit almost identical stress-displacement behaviour. It appears that peak stress and strain show a slight increase with decreasing specimen height. It is suspected that this increase is due to the still present boundary friction. It is known (Bazant [ 1984]) that during softening deformations tend to localise in the smallest volume possible, because in this way the smallest amount of energy is required for failure. Localisation of deformations is most pronounced in tests on large specimens resulting in just a few macrocracks, while smaller specimens show a more distributed crack pattern. See figure Figure 26.

36


-50

-40

l -30 ! ' ,, :Il (tj,S~fl .... ,, .... ,,,

b -20 ,,,

" ~

0 -5 -10 ·15 -20 ·25

e 1 [o/oo]

Figure 24: Axial stress-strain curves/or normal strength concrete loaded with teflon intermediate layer (3x3 experiments)

-50

-40

.... -30 as a.. . !. ' .... • b -20 .

1

' : 1

-10 ' I '·

I I

IJ ,~

. ,, ,,, ~

i• 1

-0.25 -0.50 -0.75 -1.00 -1.25 -1.50

W 1 (mm]

Figure 25: Axial stress-displacement curves for normal strength concrete loaded with teflon intermediate layer (3x3 experiments)

37


Figure 26: Typical macrocrack patterns /or concrete specimens in uniaxial compression

Closer inspection of the post-peak stress-displacement curves (Figure 27) reveals that the fracture energy per area increases with increasing specimen height, similar to the findings of Vonk [1992]. In Figure 27 elastic unloading of the continuum at peak stress is taken into account. Post-peak load hearing capacity appears to be not merely a local process. A contribution of the (microcracked) continuum part of the specimen seems to be present. Tuis continuum contribution increases with increasing specimen height. Tuis will be discussed further in Chapter 6.

Lateral deformations In Figure 28 the lateral deformations measured by the clip gauges mounted at the specimen si des are plotted for one 100 mm specimen. The figure clearly shows the typical initial volume compaction and the large volume expansion after peak stress. Measurements of lateral deformations of all other uniaxial experiments can be found in [Van Geel, 1994a]. In general these measurements show graphs similar to Figure 28, but for smaller specimens the scatter in clip gauge measurements is smaller than for higher specimens. Tuis can be explained by the difference in failure modes: smaller specimens show a more distributed crack pattem and therefore a more uniform lateral displacement (see also Van Mier [1984]).

38

1.00

0.80

0.60

0.40

0.20


"' "' 1 • '

' \\ \ "~,

'' \\

.--.-+------~

0.00 -0.20 -0.40 -0.60 -0.80 -1.00

wl, postpeak (inelastic) [mm]

Figure 2 7: Post-peak stress-dis placement curves for normal strength concrete loaded with teflon intermediate layer (3x3 experiments)

-50 , 6

, -40 ,

, 1

4 ...... -30 ,'E til Q.

1 .s !. , C') ...- N'

l'.J 1

-20 , 3::

2 , -10 ,

, 0 0

0.0 -0.4 -0.8 -1.2 -1.6 w

1 [mm]

Figure 28: All lateral versus axial displacements of one uniaxial experiment

39


Furthermore, in the lateral deformation measurements of the higher specimens large differences occur between clip gauges that cross the localised (macro)cracks (large lateral deformation) and clip gauges that do not (small lateral deformation). The majority of lateral versus axial deformation measurements in Figure 28 (the lines with a maximum lateral deformation of about 4 mm in this graph) consists of measurements at different height positions, as shown in Figure 1 7, and is found to be located in a very small zone. The closeness of these lateral deformations prove the efficacy of the friction reducing effect of the teflon intermediate layer. Similar graphs were obtained for all other uniaxial experiments (Van Geel [1994a]).

4.1.2. High-strength concrete results.

Again both axial stress-strain and stress-deformation results are shown for all specimen sizes, in Figure 30 and Figure 31. Like in the case of normalstrength concrete, in the pre-peak region similar stress-strain curves are observed (though it is not quite clear what causes the somewhat lower Young's modulus of the small specimens) and similar stress-displacement curves in the softening regime. As expected peak stress increases and the post-peak response becomes more brittle. High (200 mm) specimens are so brittle that the softening curve cannot be completely obtained in displacement con trol. Therefore the altemative test control system (using a combination ofload and displacement signal as feed back signal as explained in section 3.2. and Figure 16) bas been applied in the case of high-strength concrete specimens loaded with teflon loading platens. This method appears to give satisfactory results, though the number and size of 'control loops' in the stressdisplacement curves are larger than in displacement control. Like in the tests performed by Rokugo et al. [ 1986], notable control loops start with significant cracking, just before peak stress. An example of these control loops is shown in Figure 29. In Figure 32 the difference between normal test control ( displacement control) and the altemative test control is shown for the large high-strength concrete specimens. The stress-displacement curves are identical up to peak stress. In the softening branch only the altemative test control provides a stable descending curve.

40


Figure 29: Control loops around peak stress as observed in a uniaxial compressive experiment using the alternative test control

-80

-60

..... 1'11 a. !. -40 .... b

-20 ", ,,

0

'' 1

' 1

1'' '' '

11 \ \ \

l~-ttt---+-~' ~\ '1

li \

-8

E 1 [o/oo]

-12 -16

Figure 30: Axial stress-strain curves for high-strength concrete loaded with teflon intermediate layer (3x3 experiments)

41


-80

1 1 /1' -60 ,11 "'U

'i' i;,,. r I, J, .:' 1" 1 .... " ':/ CIS ' Q. Il\ 11

1. i: 1 -40 1 \~" \j .... Il it 1:) Il I.', ,,

I '', ,\ Il ,, 1 . ,, 1 '\

1 1. ·20 ; I

,, ' 1 ·

,, I

\ ,, ,, ' , , . l1

' \

0

0.0 -0.4 -0.8 -1.2 -1.6 -2.0 w 1 [mm]

Figure 31: Axial stress-displacement curves for high-strength concrete loaded with teflon intermediate layer (3x3 experiments)

-80

-60

'i !. -40 .... b

-20

0.0 -0.4

'• ' 1

Displacement controlled tests

' 'r-Tests using alternative ' 1 control signal ' 1

' ' 1 1

'

w 1 [mm] -0.8 -1.2

Figure 32: Experimental results of 200 mm high-strength concrete specimens

42


Lateral deformations Figure 33 shows all lateral deformations measured by clip gauges in one uniaxial experiment carried out on a high-strength concrete cube (100 mm). The difference with a similar experiment on a normal-strength specimen (Figure 28) is clear. Whereas lateral deformations start to increase rapidly right after peak stress in the case of normal-strength concrete, this increase only starts when the remaining load-bearing capacity is very small in the case of high-strength concrete. Tuis is a reflection of the differences in the fracture process of both types of concrete: in the case of high-strength concrete, shear cracks can develop much easier because of the increased homogeneity of the material, which limits the redistribution of stresses within the specimen. From Figure 33 it follows that the formation of these shear cracks requires only small lateral deformations, indicating small openings of these cracks. Tuis can be explained by the fact that these cracks are smoother than those in normal-strength specimens and do not have to open that far to develop around aggregates.

-80 , 3

J ~I

1 1

-60 , I 1

1 1

J 2 , 1 - 'Ë CIS

tl. ie 1 :E -40 .... 1 - ,

f') ..... ti /

b 1

J ;i: 1

1 1

1 " -20 ,

J

0 0 - - -- - - --0.0 -0.4 -0.8 -1.2 -1.6

w 1

[mm]

Figure 33: All lateral versus axial displacements in a single uniaxial experiment on a high-strength concrete specimen

43


Comparison with normal-strength concrete Qualitatively speaking, the macroscopie behaviour of high-strength concrete does not differ significantly from the normal-strength concrete behaviour. Though the difference is large between the way macrocracks develop and thus between the softening curves of both concretes, no difference in crack pattems can be detected. The large brittleness of high-strength concrete makes a stable test control more difficult, leading to a larger variation in test results compared to normal strength concrete and to a larger scatter in lateral deformation measurements.

4.2. Uniaxial tension

In Figure 34 the stress-deformation curves are shown from six tensile experiments. One test could not be kept stable in the descending branch. Two other tests show much larger stresses in the post-peak region. As a result, the fracture energies of these tests are much higher. From the other test results, it is concluded that the tensile fracture energy is about 70 N/m.

'ii' tL

4

3

!. 2 ....

1

Tension softening aecording to Equation 1

0 40 80 120 160 w 1 [µm]

Figu.re 34: Tensile test results and tension softening model (Eq.l) [1991]

The following equation by Reinhardt et al. [1986] is often used for implementation of the tensile post-peak behaviour in numerical models.

44


(Eq.1)

with: c1=3, c2=6.93 and wc=5.14*G1 !ft. From the experiments carried out it is found that the fracture energy Gr is about 70 N/m. The tensile strength ~ is taken as the average of all tests (2.96 MPa). From this it follows that wc=121.6 µm. The analytica! curve according to Equation 1 is also shown in Figure 34.

4

3

.... Average stress-displacement curve

1

0 20 40 60 80 100 w 1 [ µm]

Figu,re 35: Displacement measured at the specimen corners in a tensile test (top) and the tortuous fracture surface as observed in a tensile test (bottom)

45


From the four L VDT measurements per experiment it follows that fracture always starts at one specimen side and that cracking at the other specimen side only starts at a further stage in the softening curve. See Figure 35. The fracture surfaces of all specimens are very tortuous. This is also shown in Figure 35.

4.3. Biaxial compression - Plane stress

Some plane stress compressive experiments have been carried out on normalstrength concrete specimens in the range cr/cr1 = 0.1 to cr/cr1 = 1.0 as shown in Figure 36. At increasing ratios of lateral and axial stress specimens fail quite explosively in the out-of-plane (3-)direction. Even using the altemative test control described in Chapter 3, in some cases no complete descending branch could be measured. A possible solution in this case might be to select the out-of-plane deformation as a feedback signal.

1.50

1.25

1.00 ,~ ...

-0.25

... Present biaxial test results • Biaxial test results by Van Mier [1984]

_______ } Biaxial test results by Kupfer [1973]

! -·-·-·-·- Biaxial test resu lts by Liu [ 19 7 2] Biaxial test results by TUM in [Gerstle, 1978]

"""""" .. " Biaxial test results by BAM in [Gerstle, 1978]

0.25

Biaxial test results by CU in [Gerstle, 1978]

0.50 0. 75 1.00

cr2/ cr · · 1 un1ax1a

1.25

[-] 1.50

Figure 36: Biaxial test results compared to previous research

46


Striking is the observed drop in peak stress at crzfcr1 = 1.0 in Figure 36. Only at this stress ratio a large number of splitting cracks is observed, while at lower stress ratios a small number of shear cracks develops in the 1-3 plane as indicated in Figure 37. The fact that the crack pattem observed at stress ratio cr2/cr1 = 1.0 is deviant follows from the fact that in this case there is no difference anymore between the 1-3 and the 2-3 direction, causing the similar splitting crack pattems in both planes. The observed drop in peak stress at this stress ratio is larger than found in previous research (Figure 36). Tuis might be caused by the amount of friction at the boundaries between specimen and teflon loading platens, which is lower than in the case of other end conditions. In fact, an entirely different stress path is followed locally when the amount of boundary friction is reduced. After all, the lower the frictional stresses at the boundary, the lower the triaxial stress state introduced locally into the specimen. Therefore peak stress and ductility decrease with decreasing boundary friction. In Figure 38 it can be seen that even at low biaxial stress ratios post-peak deformations in the third direction increase very rapidly compared to uniaxial experiments. The main characteristic ofbiaxial experiments is that on the one hand the presence of a confining stress causes peak stress to increase compared to uniaxial experiments, while on the other hand failure is stimulated in the out-of-plane direction exactly because of the presence of this confining stress in combination with the absence of a confining stress in the out-of-plane direction. The latter dominates at large ratios of crzfcr1 and cancels out most of the advantageous effect of the confining stress.

Figure 37: Biaxialfailure modes. Left: at stress ratios cr/cr1 < 1.0. Right: at stress ratio cr /cr / = 1. 0

47

8 6


4 2 w 1,2,3 [mm]

'ii -60 a.. !.

0 -2

Figure 38: Stress-deformation curve fora biaxial test with stress ratio a ja 1

= 0.05

4.4. Biaxial compression - Plane strain

Like in the case of uniaxial compression, plane strain experiments (w2=0) have been carried out on both normal-strength concrete using the three different specimen heights of 50, 100 and 200 mm (with a cross-section of 1OOx100 mm2) and on high-strength concrete using only specimens with a height of 100 mm.

Normal-strength concrete Per specimen size five different proportional stress paths have been carried out with stress ratios in the range 0.05 s;; cr/cr1 ~ 0.5. The displacement w2 in the intermediate direction is kept zero. It should be noted that a compressive stress of about -15 to -20 MPa is required to complete initial settings due to compression of the teflon/grease intermediate layer and imperfect specimen geometry (Van Geel [1995b]). In plane strain experiments this stress level is often not or only in an advanced stage of the test reached in both the intermediate and particularly the minor principle stress direction. Hence curve fitting of the initial part of stress-deformation curves in these directions is rather susceptible to errors. Figure 39 shows the stress-deformation response of 100 mm cubes loaded with constant ratios cr/cr1 ranging from 0.05 to 0.5. Tuis figure shows that at

48


higher stress states larger nonlinear deformations occur resulting in larger peak defonnations and that the descending branch is similar in these experiments. Another distinct feature of multiaxial concrete behaviour is evident from Figure 39; initially only volume compaction occurs, while during a later stage of the experiment, dependent on the amount of lateral confinement, also volume expansion might occur. The stress in the intermediate direction (axis 2) is the result of the Poisson dilatation caused by the minor and major principal stress. Unloading in the intennediate direction in the post-peak region indicates unloading of the continuum parts of the specimen during softening (Van Mier [1984]). Tuis unloading has also been shown by strain gauge measurements (see Chapter 6).

6 5 4 3 2 w 3 [mm]

'i' D. ::& ....

N

b

O' 1 [MPa]

-200

1

-80

-40

0

\ 1 1

0

0

O' 3 I O' l = 0.15

-1 -2 -3 -4 -5 -6 w 1 [mm]

-1 -2 -3 -4 -5 -6 w 1 [mm]

Figure 39: Stress-deformation curves from plane strain tests carried out on 100 mm normal-strength concrete cubes

49


When comparing the results from Figure 39 with the plane strain results by Van Mier [ 1984, 1986] the influence of the boundary conditions is clear: because steel brush platens introduce less frictional stresses in the pre-peak region, pre-peak deformations are larger when using teflon platens. In the softening regime however the tests using teflon platens show more brittle behaviour and lower residual stresses. In all plane strain tests, regardless specimen size or stress path, a similar macroscopie failure mode is observed. A specimen is always divided into several -more or less- uncracked blocks by shear cracks at an angle of about 24 degrees with the major principal stress direction. See Figure 40. The ultraviolet photographs in Figure 40 are made using the vacuumimpregnation-technique mentioned in Chapter 3.

Figu,re 40: Macroscopie crack patterns for different specimen sizes observed in plane strain experiments

In the case of 200 mm high specimens only one shear crack develops. The angle from one specimen corner to the corner diagonally opposite is 27 degrees. Tuis means that -if at both specimen corners shear cracks start to develop at an angle of 24 degrees- the two macrocrack parts will not join in the specimen center. At the specimen centre additional energy is required to join the two macrocrack parts into one shear crack. Tuis phenomenon has been observed in most of the 200 mm experiments. Therefore the last part of the softening curve in these tests is less steep than in the case of 100 mm cubes. See Figure 41. In Figure 41 it is also clear that in the case of small specimens the descending curve is less steep during the entire softening process, especially in the lateral

50


(3-)direction. This can be explained by the fact that in this case several macroscopie cracks are formed. Because each of these cracks has to find its way around the aggregates, the total lateral displacement required for the formation of these cracks is larger. Note that in this figure the 100 mm specimen exhibits a higher peak stress, like all specimens taken from the casting batch from which this specimen is taken. This is probably due to a better compaction of this batch after casting compared to the other batches.

8 6 4 2 w 3 [mm]

'ii' a. i!.

N

t>

cr 1 [MPa]

-120

0

-40

-20

0

0

\ \ \

-2 -4 -6 w 1 [mm]

-2 -4 -6 w 1 [mm]

Figure 41: Stress-deformation curve/or plane strain test (cr/cr1 = 0.10) on normal-strength concrete specimen with h = 50, 100 and 200 mm.

While in the case of 50 and 100 mm high specimens frequently a distinct residual stress level is observed, in all 200 mm specimen experiments the stress appears to drop to zero quite fast. The absence of a residual stress may be the result of the fact that only in the case of 200 mm specimens no 'uncracked' specimen parts are blocked between the loading platens. It is

51


believed that residual stresses are not only the result of residual frictional capacity of the shear cracks, but also a result of the obstruction of shear displacements between · (partly) intact specimen parts that are blocked between the loading platens.

High-strength concrete Figure 42 shows plane strain stress-deformation curves for three 100 mm high-strength concrete cubes. The differences in behaviour with normalstrength concrete are manifest. Peak stress increases enormously due to the improved bond between the several constituents of the material. Pre-peak deformations are smaller than in the case of normal strength concrete in the major principle stress direction. The material has become more homogeneous. Tuis is also the cause of the increase in brittleness.

6 4 2 w 3 [mm]

'äi' Q.

i! N

b

cr 1 [MPaJ

-320

0

-80

-40

0

0

- cr 3

1 cr 1 = 0.15

;------- ---- cr 3

/ cr 1

= 0.10

-2 -4 -6 w 1 [mm]

, .. · - \

-2 -4 -6 w 1 [mm]

Figure 42: Stress-deformation curves /rom plane strain tests carried out on 100 mm high-strength concrete cubes

52


Especially noteworthy is the absence of residual stresses at the end of the descending branch. Though the failure mode in this case is exactly the same V-like crack pattem as observed in normal-strength 100 mm cubes (Figure 44 ), no residual stresses are observed. This confirms the assumption that the combination of specimen and loading apparatus geometry and residual load hearing capacity of shear cracks is the origin of residual stresses.

Figure 43: Macroscopie crack pattern of a 100 mm high-strength concrete cube subjected to plane strain compression

4.5. Triaxial compression

Triaxial compressive experiments have been carried out on both normal- and high-strength 100 mm cubical specimens. In these tests always constant lateral stresses cr2=cr3 have been imposed on the specimen. The cylindrical failure mode is not only essentially different, less localised, from the planar failure mode observed in plane strain experiments, hut also the failure mode changes with increasing lateral confinement (see section 2.1 ). Therefore the constant lateral stresses have been varied in the range -1 to -100 MPa for both types of concrete. An additional test with cr2=cr3=-195 MPa has been carried out in the case of normal-strength concrete.

53


cr 2 =cr 3 =-195MPa

cr 2 = cr 3 = -25 MPa ~--=-

- -- '-

6 4 2 0 ·2 -4 -6 -8 -10 w 2 = w 3 [mm] w 1 [mm]

Figu.re 44: Axial stress versus axial and lateral displacements in triaxial experiments on normal-strength concrete (dashed lines represent a uniaxial

experiment from §4. J)

Normal-strength concrete The axial stress minus lateral stress is plotted against axial and lateral displacement in Figure 44. Note that in this graph the vertical axis shows the axial minus the lateral stress instead of just the axial stress, because in this way the curves for the experiments with small confining stress are more clear, and that the lateral displacement actually plotted is the average lateral displacement (w2+w3)/2. The results are very similar to those by Jamet [1984] and Smith et al. [1989] though the confining stress level where no descending branch can be detected anymore is found to be higher in the present research. From this graph it is obvious what huge influence only a small confining stress has on the post-peak load-bearing capacity; already at a lateral stress of -1 MPa a large increase of ductility is observed.

54


When considering the term 'brittle-ductile transition' it can of course be argued whether there is an actual transition to ductile behaviour. The designation 'brittle-to-less brittle transition' would be more appropriate. In the case of the -100 MPa lateral stress experiment, for example, it is most likely that a peak stress would be found if the maximum attainable force and displacement of the loading apparatus would be higher. Even at stress states very close to pure hydrostatic loading this could be expected. A preferential direction for failure still exists there. Only in the case of purely hydrostatic loading pure compaction of the specimen would occur (see Bazant et al.(1986]). A certain amount of compaction is present in all compressive experiments, as can be seen in the stress-displacement curves shown in this chapter, where the axial displacements at least in the first part of the ascending branch are far greater than the lateral displacements. The origin of that initial volume compaction will be discussed further in Chapter 7. Similar to the findings by Jamet et al. (1984], in these triaxial tests a quite localised failure mode has been observed at low lateral stresses (up to -10 MPa). Normally, in one direction (most of the time the direction of casting) a crack pattern resembling that of plane strain experiments is observed, only with less distinct shear cracks. In the other direction generally a larger number of shear cracks is found. Tuis is shown in Figure 45 ( a and b ). At higher lateral stress levels, hardly any damage can be detected with the naked eye, except for the tremendous change in the shape of the specimens. Figure 45 (c) shows a photograph of a specimen loaded with confining stresses of -25 MPa, where no clear macroscopie failure mode can be detected, hut a lot of damage is present at the mesoscopic level.

Lateral deformations As mentioned, at lower confining stresses differences exist between the crack pattems in both lateral directions. In Figure 46 the lateral versus axial displacement plots are shown for two triaxial experiments, one loaded with lateral stresses cr2=cr3=-10 MPa, the other with cr2=cr3=-25 MPa. The graph on the left is representative of the lateral deformations in all triaxial tests with low confining stresses and shows that there indeed is a preferential direction for failure, although the lateral stresses in the 2- and 3-direction are equal. The graph on the right shows that at higher confining stress no differences exist anymore between the displacements in the two lateral directions. Tuis is due to the stabilising effect of the relatively high triaxial stress state, identical to that in the case of increased boundary friction (Vonk [1992]), and suggests

55


that the failure process at higher triaxial stress states indeed differs from that at lower stress states.

: a) b)

Figure 45: Crack patterns observed in triaxial experiments on normalstrength concrete: a) specimen loaded with a2=cr3=-3 MPa, view of plane formed by axial stress direction and casting direction, b) specimen loaded with a 2=a3=-3 MPa, view of plane perpendicular to a, c) specimen loaded with cr2=a3=-25 MPa

Mean 'volumetrie displacement' curves are shown in Figure 47, showing the increasing level of specimen compaction with higher confining stress. The curve for the experiment carried out with confining stresses of -50 MPa demonstrates that even at the end of the softening curve no volume expansion has occurred. This reveals that volume expansion and localisation of deformations into macroscopie cracks are not always typical characteristics of failure under compressive stresses. Apparently, the formation of macroscopie cracks and dilatant behaviour are in this test not possible due to the high confining pressures and the softening curve is probably the result of shearing and compaction at the mesoscopic level.

56

b


-100 "

" -BO "

" 1

-60 " 'ë' iE ....

" C') 1N' -40 " ;:

" -20 "

"

2 I /

/ /

/ I

I I

I I

/ /

W3 / /

I /

0 - 0 -1--=-~--....---,.---.-----,,..--....---,-------,

-160 " 4

..!

-120 " 3

..!

'Ê 1E

-80 " .... "'!. 2 1N

..! ;: 1

-40 "

..!

1

0

0

I

I

' I

I

-2 -4 w 1 [mm]

"~" ....... _" ... ,.,,. " ........ ;' ~. ___ _ # ... "_"_"

,' /

-2

/ /

/ /

I I

-4

/ I

I I

w 1 [mm]

w2 / I

I

/ W3 I

I

, I

-8

Figu.re 46: Lateral versus axial displacement in triaxial experiments on normal-strength concrete. Top: cr2=cr3=-10 MPa, Bottom: cr2=cr3=-25 MPa.

57


cr 2 = cr 3 • -3 MPa

w 1 [mm]

-8

\ Cl:I

Q.. :! ~ • Il

C")

t>

tf Cl:I .1! Il g .... :i; •

" Il ..., C") b

t>

Il N

t>

cr 2 = cr 3 = -1 MPa

2 1 0 -1 -2 (w 1 + w 2 + w 3 )13 [mm]

Figure 47: Axial displacement versus mean 'volumetrie displacement' in triaxial tests on normal-strength concrete

High-strength concrete The differences between high-strength and nonnal-strength concrete are similar to those observed in plane strain experiments. Figure 48 shows that ascending branch exhibits less defonnation, that peak stress is higher and that the post-peak behaviour is more brittle. Localised failure modes similar to those found in the case of nonnal-strength concrete have been observed for the present high-strength concrete specimens loaded with confining stresses up to -50 MPa, though the failure modes are in general more evidently unidirectional, i.e. a clear localised V-pattern in one lateral direction and hardly any macroscopie damage in the other lateral direction.

Lateral deformations Figure 49 expresses the fact that failure takes place in one direction more clearly than in the case of nonnal-strength concrete. When only small confining stresses are present, macroscopie failure takes place almost completely in one lateral direction only (upper graph in Figure 49). Tuis is

58


due to the fact that the brittleness of the material causes the first macroscopie crack pattern initiated to develop further immediately. With increasing lateral stresses, the failure mode is more symmetrie with regard to both lateral directions, see the lower graph in Figure 49 where initially both lateral deformations are rather close. At higher levels of confining stress the lateral deformations are equal, similar to those for normal-strength concrete in Figure 46. The level of confinement where no volume expansion is observed anymore is higher compared to normal-strength concrete (Figure 50), because in this case at larger confining stresses still macroscopie localisation can take place because of the brittleness of the concrete.

-400

cr 2 = cr 3 = -100 MPa

cr 2 = cr 3 = -25 MPa

cr 2 = cr 3 = -10 MPa ----=--~--cr-A2 = cr 3 = -3 MPa

~-- =cr =-1 MPa

6 4 2 0 -2 -4 -6 -8 -10

w 2 = w 3 [mm] w 1

[mm]

Figure 48: Axial stress versus axial and lateral displacements in triaxial experiments on high-strength concrete (dashed lines represent a uniaxial

experiment/rom §4.1)

59


·160 ï 10

J " 1 I 1

w2 8 -120 ï

1

J ' ';' 1 'ë' 6 ' Q.

\

::.i 1 .5. - -80 ï C"') ,..... N' b

1 ' J ~ 4 '•

~-. 1 "". -

" ... "" ... -." ...... ,._ .... " .... """" .... -"~ -40 " 2

J

W3 0 0 -- -- -- -- -- -- --

0 ·2 -4 -6 w 1 [mm]

-300 " 6 ~ """-" ... " ... " ................. " ,

, ·~ .. w2 / •,. 1 ' .....

I -.. .J ' -..

I I I

I

-200 1 I

1 4 ...... 'ë' :. !.. .s

" .... C")

b N' • -100 .J 2 -------w3

1 /

/

1 / - /

1 / /

0 - 0

0 -2 -4 -6 -8 -10 w 1 [mm]

Figure 49: Lateral versus axial displacement in triaxial experiments on high-strength concrete. Top: cr2=cr3=-JO MPa, Bottom: cr2=cr3=-50 MPa.

60


w 1

[mm]

-8 n:s a.. n:s :i

a.. 0 :i ID

1

ID Il N O' 2 = O' 3 = -10 MPa 1 <"')

-6 Il b (")

Il b N Il b

N

b -4

2 1 0 ~

(w 1 + w 2 + w 3 )/3 [mm]

n:s a.. :i 0 0

"""' 1

Il (")

b Il

N

b

-2

Figure 50: Axial displacement versus mean 'volumetrie displacement' in triaxial tests on high-strength concrete

Final remarks In figure 52 the increase of peak stress with increasing confining stress is shown for the present triax.ial experiments, both for · normal- and highstrength concrete. It can be seen that this increase is larger for the highstrength concrete. Tuis is probably due to the improved qualities of the aggregate-cement paste interfaces of the high-strength concrete compared to the normal-strength concrete, which seems to become more significant at higher compressive stresses. Furthermore results from similar experiments on normal-strength concrete cylinders by Newman [1979] and Jamet [1984] are plotted in this figure. Compared to the present normal-strength results, the increase in strength with increasing lateral stress is much larger in the case of cylinder tests. However, it is not clear whether these differences are caused by the differences in

61


specimen (and loading apparatus) geometry, or by the boundary frictions present in the different loading systems.

4 0

... 0 .... ....

-; ·;c

3 ');

ftS ï: 0 ::s .Jli ftS Cl> CL

t) ... /

0 - v 0 Normal-strength concrete results , ~ ftS 2 Cl> 0 --v High-strength concrete results CL

t) ... , ... Results by Jamet [1984) }!'

" 0 Results by Newman [1979]

1

0 10 20 30 40

O' 2

= O' 3

[MPa]

Figure 51: Comparison of peak stress es in triaxial tests between the present experiments and results by Jamet [1984] and Newman [ 1979]

62

Chapter 5: Influence of test environment

5. Influence of test environment

As described in Chapter 2, concrete behaviour as observed in laboratory measurements is always influenced by the test environment. In the previous chapter it already has been outlined how the layout of the loading apparatus in combination with the specimen dimensions can have a significant effect on residual stress levels. Tuis chapter deals with other effects caused by the loading conditions. Section 5.1. deals with the influence ofboundary friction, in section 5 .2. the occurrence of non uniform deformations in uniaxial testing will be linked to the stiffness of the loading apparatus and in section 5.3. the influence of the prismatic layout of the loading platens will be discussed.

5.1. Boundary friction

When increasing the coefficient of friction of the surfaces of the loading platens in contact with the specimen, sliding between specimen and loading platen becomes more restricted due to the frictional stresses at the specimen boundaries. In the case of uniaxial compression, this leads to the occurrence of triaxial stress states in regions near the loading platens and because of that to the formation of the well-known hourglass failure mode. It is clear from Figure 44 that even small lateral stresses have a significant influence on the observed stress-deformation behaviour. To obtain insight in concrete's failure behaviour with and without frictional boundary stresses, it is therefore important to reduce these stresses as much as possible. In the present research, two different loading platens have been used: dry steel platens and the same steel platens with a grease/teflon intermediate layer (see Chapter 3). Figure 52 shows frictional characteristics of similar loading platens used by Vonk at EUT (Vonk [1992]). Frictional coefficients were determined by sliding a concrete cube between two loading platens at constant values of norrnal stress. In the figure coefficients of friction are shown measured at the first peak: in the curve of the coefficient of friction versus sliding distance, which is caused by stick-slip behaviour, and at a sliding distance of 1.5 mm.

Uniaxial compression Figure 54 and Figure 55 show the measured axial stress-displacement curves for specimens loaded in uniaxial compression with dry steel loading platens. When comparing these graphs to those of specimens loaded with a teflon intermediate layer (Figure 25 and Figure 31) the effect of increased boundary

63

Chapter 5: lnfluence of test environment

0.20

À. ' ' ' Dry platen c:

' ---.t: 0 0.15 '&.... first peak :;::::;

u --" ·c ·-_______ ..__ --- -- --· Dry platen "+-- --·-"+-- w = 1.5 mm 0 ....... 0.10 c: --+- Teflon platen <IJ ·o first peak !E

<IJ Teflon platen 0 ----u 0.05 ·- w = 1.5 mm

1

·-·------ --+

·--------·--- -- ---tl 1 1 1 1

0 -10 -20 -30 -40

Axial stress [MPa] Figure 52: Coefficients of friction/or dry steel platens and teflon

platens (after Vonk {1992)).

Figure 53: Hourglassfailure mode/or two normal-strength concrete specimens loaded with dry steel platens

64

Chapter 5: Injluence of test environment

friction is obvious; both peak stress and displacements are much higher in the case of dry steel platens and the post~peak behaviour is more ductile due to the formation of the hourglass failure mode. This failure mode has even been observed in the case of the 200 mm high specimens (Figure 53). This confirms the observations by Rokugo and Koyanagi [1992], Markeset [1993] and Jansen and Shah [1997], mentioned in Chapter 2, that the height/width ratio of the specimen, required for failure to be unobstructed by boundary friction, should be higher than the present value of 2. At this ratio of height and width, however, the differences with teflon platens experiments are already much smaller than in the case of 50 or 100 mm high specimens, primarily with regard to peak stress.

-80

.·~~ . ' ~ '" 1" f

f ~ \ I~

', \. -60

IJ I ~ 111

.... Cl D. i! -40 -b

-20

0.00 -0.50 -1.00 -1.50 -2.00 -2.50 w 1 [mm]

Figure 54: Stress-displacement curves for normal-strength concrete specimens loaded uniaxially with dry steel platens (3x3 experiments)

65

Chapter 5: lnjluence of test environment

-100

- -60 &U

I a.. :E \ ......

..... I b ' 40 /IJ

• m ' 1 g

~ 1 ' • '-... 1 •

-20 , ,,' ' ..__ -~

0.00 -0.50 -1.00 -1.50 -2.00 -2.50 w 1 [mm]

Figu,re 55: Stress-displacement curves /or high-strength concrete specimens loaded uniaxially with dry steel platens (3x3 experiments)

The observed influence of boundary friction on stress-displacement behaviour and macroscopie cracking (the formation of the hourglass failure mode) has been found to be similar in the case of normal- and high-strength concrete.

Lateral deformations In general, lateral deformations measured in experiments using dry steel loading platens are in the same order of magnitude as those measured using teflon loading platens. In the case of dry steel platens however a large scatter in lateral deformations is already observed in tests on medium sized specimens (100 mm cubes). See Figure 56. Tuis is the result of the fact that sometimes large pieces of concrete are splitting off the specimen.

66


-60 ï 5

J I " 4 1

1 \

-40 1 1

...... 1,..... 3 l t1$ iE

0... ie :E .J -

1 ..... C") 1

~ N' b

1 3:: 2

-20 ' j

1 ...

0.0 -0.4 ..0.8 -1.2 -1.6 -2.0 w 1 [mm]

Figure 56: Lateral deformations versus axial deformation in a uniaxial experiment on a normal-strength 100 mm cube loaded with dry steel platens

Plane strain compression Experiments using dry steel platen have also been carried out in the plane strain domain. Note that in the intermediate (w2=0) direction, teflon loading platens are used to preserve the more or less two-dimensional character of these tests. Tuis bas been done for the sake of two-dimensional (plane strain) numerical modelling. Stress-displacement results from these tests are shown in Figure 57. Striking is the closeness of the individual curves in the pre-peak region while differences in the post-peak region are quite large. The latter is due to the fact that initially the V-like macroscopie crack pattem is initiated hut cannot evolve due to the friction at the specimen boundaries; at some stage a crack pattem resembling the hourglass failure mode is activated. Tuis interaction between two failure modes that strive to develop simultaneously causes the large scatter in post-peak results.

67

8. 6


4 2 w 3 [mm]

::

cr 1

[MPa] -160

0

-80

!. -40 N

b

0

-2

-2

-4 -6 -8

w 1 [mm]

-4 -6 w 1 [mm]

-10

-10

Figure 57: Plane strain experiments on 100 mm normal-strength concrete cubes carried out with dry steel p/atens

Figure 58 shows the macroscopie failure pattems of two specimens tested in plane strain compression using dry steel platens. The initiation of the Vpattem is clear in both photographs, while the regions influenced by boundary friction also can be distinguished very well. Although in the experiments, both with 5% and 10% confining stress, peak stress and deformation are larger compared to the teflon loading platens tests, this difference seems to be smaller in the case of cr/cr,=0.10. Tuis might be explained by the fact that the introduction of additional triaxial stress states by boundary friction bas a relatively smaller influence when a specimen is already subjected toa larger triaxial stress state.

68


Figure 58: Ultraviolet photographs of normal-strength concrete specimens loaded in plane strain compression (left: cr/cr1=0.05, right: cr/a1=0.10)

using dry steel platens

Triaxial compression Triaxial experiments using dry steel platens in all axis directions have been carried out on normal-strength concrete 100 mm cubes. Overall, the same tendency can be observed as in the triaxial experiments with teflon loading platens: at higher confining stresses the behaviour becomes more ductile, see Figure 59. However, because of the influence of the boundary friction the behaviour is already less brittle at low confining stresses. Macroscopie failure modes at low confining stresses are similar to those observed in plane strain tests using dry steel platens, being a composition of both a localised failure mode and the hourglass crack pattem (Figure 60). However, in these triaxial tests this failure mode is found in both lateral directions and the hourglass failure mode appears to be more dominant in this case than in the plane strain experiments. The relatively smaller influence of boundary friction at larger triaxial stress states becomes very clear when looking at Figure 61. At a confining stress level of -25 MPa, the differences in both peak stress and displacement become relatively small for the two types of loading platens. Like in the case of teflon platen experiments, at this level of lateral stress no macroscopie failure can be detected with the naked eye but a lot of damage on the mesoscopic level can be identified.

69


-140

er 2

=er 3

= -25 MPa

er =cr 3 =-to MPa

-20-

6 4 2 0 -2 -4 -6 -8 -10 w 2 =w 3 [mm] w 1 [mm]

Figure 59: Triaxial experiments on normal-strength concrete 100 mm cubes using dry steel platens

Figure 60: Ultraviolet photograph of a 100 mm cubic specimen subjected to triaxial compression (er2=cr3=-JO MPa) using dry steel platens

70

6


-140

-•........•. ·······.., :. -120", !.

-· .~ "" ..

4

.. -..-

2 w

2 =w

3 [mm]

C")

1:::)

.... '

-80 1 ,

-20

0

' '

-2 -4 w 1 [mm]

-6 -8

Figure 61: Comparison oftriaxial experiments on 100 mm normal-strength concrete cubes using teflon (solid lines) and dry steel (dashed lines) loading

platens

52.Loadingapparatusstiffness

Because localisation of deformations leads to the least energy-demanding failure mode, the formation of a partial failure mechanism is stimulated, sometimes resulting in a significant nonuniformity of deformations (rotation of the loaded boundaries) in uniaxial compression. In some uniaxial tests (section 4.1) a large non-uniformity of deformations is observed from the different L VDT-measurements. The nonuniformity increases when using teflon instead of dry loading platens or when decreasing the size of the specimen. Following the analytica! model by Vonk [1992] it is found that the critica! rotational stif:fness is smaller than the rotational stiffness of the

71


loading apparatus only for large (200 mm high) specimens (when comparing normal-strength concrete teflon tests among themselves), indicating rotational instabilities for the smaller specimens. In the case of high-strength concrete specimens deformations are even more non-uniform. Vonk [1989] showed that non-uniformity of deformations is limited in multiaxial testing due to the fact that the specimen is enclosed by loading platens in all directions. As a result the present investigation is limited to the case of uniaxial compression. A large non-uniformity of deformations is observed for all uniaxial teflon platen tests, with the exception of large normal strength concrete specimens, indicating a lack of bending stiffuess of the loading apparatus. Uniaxial testing of very brittle concretes (like the high-strength concrete tests in the present research) seems to be beyond the reach of the loading apparatus. It still can be questioned to what extend the observed non-uniform deformations influence the average axial stress-displacement curves. According to Vonk [1992] the significance of non-uniformity of deformations (i.e. rotation of the loaded boundaries) is dependent on: 1. The eccentricity e of the compressive load; 2. The initial rotation cp0 of the loading platen; 3. The rotational stiffhess C of the loading apparatus; 4. The cross-section b*d of the specimen; 5. The combination of stress cr and stress-gradient dcrldw. Tuis combination

becomes more critica! for: 5a) more brittle types of concrete; 5b) specimens of greater height.

However, argument 5b is in contradiction with the present test results, in which a decreasing non-uniformity is observed with increasing specimen height. See Figure 62. Obviously, the rotational stiffhess of the loading frame is only sufficient for higher specimens. Therefore, some additional measurements have been carried out to detect the cause of this increase of nonuniformity of deformations with decreasing specimen height.

Measurements A description of the loading apparatus (in this case the 3 x 2000 kN configuration) is given in chapter 3. To detect the 'weakest rotational link' in the loading frame, all parts of this frame which can possibly give cause to rotations ( except for the loading platens, because the bending stiffhess of these platens can be calculated easily) have been mounted with L VDTs or strain gauges, as indicated in Figure 63:

72


• Three L VDTs measuring the axial deformation of the specimen at three different positions;

• Four L VDTs measuring the deformations of the (fixed) hinge; • Three L VDTs measuring the deformations of the piston (relative to the

steel frame); • Twenty-four 120 mm strain gauges measuring the deformations of end

platens and steel rods; One additional measurement was carried out to measure the deformations of the load cell using four L VDTs.

-50 -50

-40 -40

'ii' -30 'ii' -30 ll. ll.

i!!. i!!. ... ... b -20 b -20

-10 -10

0 0 0.0 -0.4 -0.8 -1.2 -1.6 0.0 -0.4 -0.8 -1.2

w 1 [mm] w 1 [mm]

-50

-40

'iii' -30 ll. i!!. ... b -20

-10

0 0.0 -0.4 -0.8 -1.2 -1.6

w 1 [mm]

Figure 62: LVDT measurements in axial direction for three normal-strength concrete specimens with h= 50, 100 and 200 mm loaded in uniaxial

compression (dashed line average curve)

73

Test results


LVDTsat4 positions

2 strain gauges --~ per steel rod

4 strain gauges per endplaten ---

surface

4 strain gauges on load cell

Figure 63: Measurement of loading axis deformations

In Figure 64 through Figure 69 the results from a uniaxial compression test on a normal-strength concrete 100 mm cube are shown. Note that initial settings are not corrected in these graphs. From Figure 64 the rotation of the loading platens after peak stress is obvious. Ata certain post-peak stress the axial displacement at one si de of the specimen is larger than the displacement at the other side, where even unloading takes place temporarily.

74

-50

-40

-30

b -20

-10


0.00

1 1 1

1

1 \

LVDT positions:

2•

"

-1.00 -2.00 , LVDT displacement [mm]

-3.00

Figure 64: LVDT measurements of axial disp/acement at three different positions around a 100 mm normal-strength concrete cube (solid lines) and average stress-disp/acement curve measured by two control-LVDTs (dashed

line)

The defonnations of the hinge (Figure 65) are found to be relatively small compared to the specimen defonnations and are found to be quite uniform. It can be concluded that these defonnations are no cause for rotations of the loading platen. The defonnations of the piston (Figure 66) however show a non-unifonnity of defonnations already in the ascending branch of the stress-defonnation curve, which is in the order of magnitude of the non-unifonnity of the specimen L VDT measurements. From Figure 67 through Figure 69 it can be concluded that, though some defonnations occur in the steel frame, these defonnations -as could be expected- are small compared to the specimen displacements and certainly are no cause for any rotation.

75


"iii' -30 D. !. .... t> ·20

-10

~------1,3,4,2 (left to right)

1 1

• • 1

'

1 1

1

'

, " ' ' . ' \

1

' ' \ ' 1

' " •

1

' 1

' ' " " "

L VDT posltions around hlnge:

" """ ....... ______ _ o---------~-~----~

0.0 -0.4 -0.8 ·1.2 -1.8 w 1 [mm)

Figure 65: LVDT measurements of hinge deformations in a uniaxial experiment (4 solid lines) and average stress-(specimen) displacement curve

( dashed line) -50

-30 LVDT positions around piston:

0 -20

-10

0.00 0.50 1.00 1.50 2.00 2.50 LVDT displacement [mm]

Figure 66: LVDT measurements of piston deformations in a uniaxial experiment (4 solid lines)

76

'ii' Q.

i!. .... t:>


~·] -40 '

-30

-20

-10

-0.02 o.oo

123 4

0.02 0.04 Strain [o/oo]

5 6 7 8

3 ~6 ~

Cross section of steel rods with straln gauge posltions

0.06 0.08

Figu.re 67: Strains measured on steel tension rods of the loadingframe

4,8

during a uniaxial experiment

-50 1,5

2,6

--40 CIS Q. :E -.... b

-30

3,7

...!L

17;8 3,41

1,2

View on upper endplaten with strain gauge positions

(1,3,5,7 on top surface, 2,4,6,8 on bottom surface)

-0.40 -0.20 0.00 0.20 0.40 Strain [o/oo]

Figu.re 68: Strains measured on the upper steel endplaten of the loading frame during a uniaxial experiment

77


-50 3,7

1,5

0

'iii' a.. ~ .... b

2,6 4,8

View on lower endplaten with straln gauge poeltlons

(1,3,5,7 on top surface, 2,4,6,8 on bottom surface)

-0.20 -0.10 0.00 0.10 0.20 Strain [o/oo]

Figure 69: Strains measured on the lower steel endplaten of the loading frame during a uniaxial experiment

From the above findings it seems that rotation of the piston, which takes place already in an early stage of the experiment, further stimulates failure to initiate at a particular side of the specimen (from the measurements it follows that this is the left-front side of the specimen). During the descending branch of the stress-displacement curve the rotation of the piston (Figure 66) is in the same order of magnitude as the rotation of the loaded boundaries (Figure 64). Therefore the rotational stiffness of the loading apparatus is found to be not only dependent on the dimensions of the loading platens and the piston but, moreover, on the position of the piston within its housing. Since a larger part of the piston is outside the piston covering with decreasing specimen height, the rotational stiffness of the loading apparatus is related to the height of the specimen. Therefore the relation between specimen height and nonuniform deformations is different from that proposed by Vonk' s modelling of the test environment influences. These measurements explain why more uniform deformations have been observed in experiments on large (200 mm) specimens. In this case a smaller section of the piston is outside the housing so that it's bending length is smaller. Tuis factor seems more significant than the fact that the combination of stress and stress-gradient becomes more critica! with increasing specimen height.

78


-60

LVDT Positlons:

e2 -40

'ii' 1• .3 Q.

!. .... t;) .4

-20

0.0 -0.4 -0.8 -1.2 w 1 {mm]

Figure 70: Additional test to measure the deformations of the load cel/ during a uniaxial experiment. The thick grey line represents the average load-disp/acement diagram as measured by two contro/-LVDTs, lines 1-4

represent the LVDT measurements around the specimen. -60

... b

LVDT Positions:

•2

··•·· .4 0.00 0.04 0.08 0.12 0.16 0.20

w load cell [mm]

Figure 71: Additional test to measure the deformations of the laad cel/ during a uniaxial experiment. Lines 1-4 represent the LVDT measurements

around the /oad cel/.

79


Figure 70 and Figure 71 show results from an additional measurement in which the deformations of the load cell are measured at four positions. In these figures it can be seen that a rotation takes place in an early stage of the experiment not only in the piston but also in the load cell. The magnitude of the rotation of the load cell might also be large enough to stimulate the occurrence of non-uniform deformations. However, the quite uniform deformations when loading 200 mm specimens, where the possibility to rotate is prevented for the piston, shows that the influence of the load cell rotations is small compared to the influence of rotations of the piston. Even when performing uniaxial pilot experiments with the loading axis of 5000 kN, where the rotational stiffness of the piston is a factor 10.5 greater and the load cell also bas a larger stiffness, this phenomenon has been observed, indicating that merely increasing the stiffness of the loading frame is not sufficient to prevent the heterogeneous structure of the material from causing the occurrence of non-uniform deformations.

5.3. Layout ofloading platens

As mentioned briefly in Chapter 2, different test results may be obtained from experiments carried out using different layouts of the loading apparatus. Previous investigations · into the influence of the loading apparatus lay out were restricted to the manner in which the three loading axes were connected, or test results from 'true triaxial' (i.e. three independent loading axes) loading apparatuses were compared to those from triaxial loading devices. In this research emphasis is laid on the influence of the layout of the loading platens instead of the apparatus as a whole. From previous research (Hansen et al. [1962], Sigvaldason [1966]) it is known that already differences exist between results from uniaxial experiments on cylindrical and prismatic specimens, having equal height and cross-sectional area. Tuis already indicates the importance of the combination of specimen and loading platens geometries. In the multiaxial experiments, described in this thesis, it has been observed that macroscopie cracks often initiate at the specimen corners i.e. at singularities on the bounding surfaces of the specimen. In multiaxial testing it is necessary for the loading platens to be smaller than the specimen to avoid contact between the loading platens and the transfer of stresses via these platens instead of via the specimen. To prevent the loading platens from piercing into the specimen and hence introducing additional stress concentrations, the specimen corners have been bevelled as shown in Figure 20. Because the loading platens are

80


smaller than the specimen shear displacements may occur at the specimen corners. It is questioned to what degree the localised V -shaped failure mode (the shear crack angle) in plane strain compressive experiments, for example the 100 mm cube in Figure 40, is determined by this loading platens layout instead of actual structural behaviour. For the investigation of this question additional loading platens layouts have been manufactured, as shown in Figure 72a through Figure 72c. The additional layouts prevent shear displacements at the specimen corners. Singularities are introduced away from the corners. Figure 72d shows the original loading platens layout in a plane strain test (from here on denoted as 'common plane strain test'). With these additional loading platens, plane strain experiments with stress ratios cr/cr1=0.05 and cr/cr1=0.15 have been carried out. In the intermediate principle stress direction (w2=0), the regular loading platens with a cross-section of 97*97 mm2 have been used.

Figure 72: Different loading platen geometries used to investigate the inj/uence of the loading apparatus lay-out and the macroscopie crack

patterns observed

81


Like the original loading platens, the additional loading platens have been manufactured extremely accurate with respect to parallelism and orthogonality. Furthermore they have been hardened and polished as the regular loading platens (see Chapter 3). A teflon intermediate layer with hearing grease has not only been applied between specimen and loading platens, but also between the original and the supplementary loading platens. Results from these experiments are shown in Figure 73, where the designations a through d correspond with the labelling of the loading platens layouts shown in Figure 72. The stress-displacement results from the experiment with symmetrie (additional) loading platens layout (labelled a in Figure 72) are found to be very close to those from the common plane strain test with 5% confining stress ( d). Also a similar crack pattern as in the common experiment has been observed as shown in Figure 72. Though the possibility for concrete parts to shear off at the specimen corners is ruled out, shear displacements are now possible at the center of the lower specimen surface as shown in this figure. Therefore the failure mode is identical to the regular failure mode, though macroscopie cracks in this case iniate at the singularity at the tip of the V, instead of at both ends as observed in the experiments before. In the case of layouts b and c the formation of an asymmetrie failure mode is enforced, starting again at the places of singularities on the bounding surface of the specimen. While in the experiments with 5% confining stress the differences are not very large, in the case of 15% confinement large deviations from the common test results are observed. Smaller pre-peak deformations, a decrease in peak stress and a less brittle softening branch are found. Obviously it is in these situations easier to attain the mesoscopic 'continuum' damage needed for initiation of macroscopie failure than in the common test, but more energy is required for further development of the macroscopie cracks. The brittleness of the descending branch of layout b is closer to the brittleness in a common test than the one observed in layout c, because in case b the macroscopie shear crack angle can be identical to the common situation (Figure 72). The softening parts of the curves of these two experiments make very clear that in fact the combined stress-deformation behaviour of the material and the structure ( consisting of both 'uncracked' concrete parts divided by shear cracks and several loading blocks) is measured. Due to rotation of the additional loading platens at large displacements in layout c, the stress level even increases at the end of the descending branch.

82

6


c)----

b)-~-r-:.,,.

4 2 w

3 [mm]

';' ll. :& -N

t>

-160

0 -2 -4 -6

-60 w 1 [mm]

-40

-20

0

0 ·2 -4 -6 w 1 [mm]

Figure 7 3: Results from plane strain experiments with different loading platen geometries according to Figure 72

The results from these experiments show clearly that the layout of the loading platens has a significant effect on the observed macroscopie behaviour at higher multiaxial stress states and that results from common experiments (i.e. experiments with the loading platens layout as described in Chapter 3) may overestimate peak stress and deformations. Tuis again shows the importance of understanding the influence of boundary conditions on concrete behaviour under compressive stresses and incorporating this influence when extrapolating experimental results to other situations. But it also shows the necessity for numerical models to be able to simulate concrete behaviour based on sufficient experimental verification for many different boundary conditions, because the actual failure behaviour of concrete is very dependent on the boundary conditions.

83

Chapter 6: Localisation of deformations in macroscopie cracks

6. Localisation of deformations in macroscopie cracks

Near the peak of the stress-displacement curve of compressive experiments a macroscopie failure mode is initiated. The development and growth of these macroscopie cracks govems the mechanica! behaviour in the softening region to a great extent. Macroscopie crack evolution in triaxial experiments has already been determined by Hallbauer et al. [1973] in the case of quartzite specimens and Sture and Ko [1978] for rock specimens, see Figure 74. Sture and Ko showed that during the descending branch the rock specimen consists of relatively intact specimen pieces divided by sbear zones and that these zones in particular determine the post-peak stress-displacement behaviour. In section 6.1. the formation of macroscopie cracks in concrete specimens in multiaxial compression will be discussed.

PEAK STREJ«;TH, ENVEl.OPE

@ o- r .

/ @ ~t--n ® 0-

AXlAL STRRAIN

Figure 74: Schematic description of macroscopie crack evolution in a rock specimen subjected to triaxial compression (Sture and Ko [1978])

According to Van Mier et al. [1997], a perfect localisation of post-peak deformations in these shear cracks takes place in uniaxial compression. However, the uniaxial test results by Vonk [1992] indicated that the more or less uncracked 'continuum' parts of the specimen still contribute to the load

84


hearing capacity in the softening region. In the present research a technique has been developed to measure the continuum deformations in multiaxial compression. Because of the large nonlinear pre-peak deformations observed in multiaxial compression, the alleged contribution of continuum parts to post-peak load-bearing capacity is easier to detect than in uniaxial experiments. Tuis will be outlined in section 6.2. Section 6.3. shows a simple two-dimensional analytical model that describes the mechanica! behaviour of macroscopie shear cracks in plane strain compression. This model shows that concrete's post-peak behaviour might very well be described numerically in terms of shear crack displacements. In section 6.4. some genera} considerations are made with respect to the behaviour of macroscopie cracks in genera!.

6.1. Macroscopie crack development

-120 a)

b)

6 5 4 3 2 1 0 •1 -2 -3 -4 -5 -6 w 3 [mm] w 1 [mm]

-40 'ii' ll. :E -20 .....

N

t? 0

0 -1 -2 -3 -4 -5 -6 w 1 [mm]

Figure 75: Six plane strain experiments an 200 mm specimens laaded with a/cr1=0.10 to detect the evolution ofmacroscopic cracks

85


Both in the plane strain and the triaxial test program, experiments have been carried out to obtain insight in the development of macroscopie crack pattems. To achieve this insight several tests have been carried out, following an identical stress path, that have been interrupted at different stages of postpeak: loading. This was already shown schematically in Figure 19. Figure 75 shows the stress-displacement results of six experiments, carried out on 200 mm high normal-strength concrete prisms, loaded in plane strain configuration with cr/cr1= 0.10. One experiment was interrupted just hefore peak: stress, while the other five tests were interrupted at different levels of post-peak: stress in the major compressive direction. Tests were interrupted by unloading the specimen, i.e. applying a positive displacement in the major principle stress direction .

a)

. 0

b) c)

e) f)

Figure 76: Evolution ofmacroscopic cracks in a plane strain experiment. The labels a through f correspond with those in Figure 75

86


In Figure 76 photographs are shown from cross-sections of the six specimens tested. These photographs have been made using the technique described in section 3.4. From these results it appears that the localisation of deformations starts right after peak stress has been passed (photograph b ). At that stage macroscopie shear cracks are initiated at the specimen corners (see also Chapter 5). Note that in quite some cases not only those cracks are initiated that will eventually form the final macroscopie crack pattern, but also other cracks that might form this pattern equally well (photograph b, lower right corner). Tuis has also been observed in other experiments. Such an example is shown for a 100 mm cube in Figure 77, where the V -shaped crack pattem is in fact initiated in all four specimen corners but only one of the two possible V-shapes is finally formed completely.

Figure 77: Ultraviolet photograph of a 100 mm cube loaded in a plane strain test with o/o1=0.10

During the steep part of the descending branch of the stress-displacement curve the macroscopie cracks start to extend towards the center of the specimen (photographs c and d in Figure 76). When the two parts of the macroscopie crack have joined (photographs e and f), the deformational behaviour of the now completely formed macroscopie cracks largely determine the measured stress-displacement behaviour. The stressdeformation curves corresponding with photographs e and f clearly show the increased ductility due to the necessary bridging between the two macroscopie cracks.

87

6


5 4 3 2 w 2 =w 3 [mm]

;f' -80 i!.

C")

b ....

1

)

0 -1 -2 -3 -4 -5 w 1 [mm]

e)

-6

Figu.re 78: Five triaxial experiments on 100 mm cubes loaded with cr2=cr3= -3 MPa to detect the evolution of macroscopie cracks

. . ."' . .

•

a)

~ ... " "'. . . ' ~ ,:.-

•

.l .•. b)

e)

Figu.re 79: Evolution of macroscopie cracks in a triaxial experiment. The labels a through e correspond with those in Figu.re 78 (c almost identical tob

andd)

88


Similar experiments have been carried out on a 100 mm cube in triaxial compression. Five specimens were tested with constant lateral stresses

-3 MPa. Figure 78 and Figure 79 show results from these experiments. Note that the photographs in Figure 79 have been taken from the plane formed by the axial stress direction and the casting direction ( compare to Figure 45). Again, the macroscopie cracks are found to initiate at the specimen corners (photograph b and d in Figure 79). In plane strain experiments on 200 mm prisms it was found (Figure 76) that the two macroscopie cracks, that eventually join into the final shear band, are initiated at opposite specimen edges. In the case of both plane strain and triaxial experiments on 100 mm cubes two macroscopie cracks start at the same specimen side, growing towards the middle of the opposite specimen edge where they join (photograph e in Figure 79). It should be mentioned that this occurs in both lateral directions in triaxial experiments. Besides that, the crack pattem observed in these triaxial tests is not as localised as found in plane strain experiments ( compare Figure 79e with Figure 77). It should be mentioned here that in triaxial experiments both lateral stresses are kept constant, while in plane strain experiments in genera! the minor principle stress is proportional to the major principal stress.

6.2. Post-peak continuum behaviour and size effect

Using the same loading apparatus, Van Mier (using steel brush loading platens, [1984]) and Vonk (using teflon loading platens, [1992]) observed quite different behaviour in the post-peak region when testing specimens of different height under uniaxial loading. Tuis has already been outlined in section 2.1. Whereas Van Mier found identical post-peak stress-displacement curves for different specimen heights, indicating a perfect localisation of deformations in macroscopie shear bands, in the experiments carried out by Vonk the more or less uncracked specimen blocks between these shear zones seemed to contribute to the post-peak load-bearing capacity, expressed by a decreasing brittleness of the softening branch with increasing specimen height (and toa lesser extent also with increasing specimen width). Figure 80 shows a comparison of the observed post-peak stress-displacement behaviour, as observed by Vonk, with the present uniaxial test results, as described in Chapter 4. The grey areas representing the results from the current research are the zones bounded by the individual lines in Figure 27. In this graph unloading of the continuum is taken into account by assuming linear elastic unloading at peak stress.

89


It is striking that again results obtained from the same loading apparatus are different from each other. However, in this case the differences are even more surprising because the loacling platens and testing technique used in the present research are almost identical to those used by Vonk. The only known clifference between Vonk' s test setup and the present setup is the way in which the loading platens are hardened, but it seems most unlikely that the clifferences observed originate from this relatively small difference. Besides differences in test setup, differences in concrete mix or specimen preparation might be the origin of the differences observed. However, differences in concrete mix are considered to be quite small (see Vonk [1992] and Chapter 3). Making statements with regard to clifferences due to the applied specimen preparation would be mere speculation. Systematic experimental investigations into the influence of varying specimen composition and preparation on concrete behaviour should be subject of future research.

1.0

-1 ...... 0.8 rlJ

m b Ctl

& 1.6 ~ -:= -! i::ii= 0.4

0.2

0 -0.2 -0.4 -0.6 -0.8 -1.0

Post-peak inelastic displacement [mm]

Figure 80: Comparison of uniaxial post-peak stress versus inelastic displacement curves by Vonk (lines) with the present results (grey areas)

90


In the case of the experiments by Vonk it is clear from Figure 80 that the brittleness of the post-peak curve decreases with increasing specimen height. In the present test results almost equal slopes of the curves for the different heights are oberved. The largest difference between the different specimen sizes is observed right after peak stress, where stress levels are higher at a certain post-peak displacement for higher specimens (the shapes of the curves for the different specimen heights at relative stress levels lower than approximately 0.9 are almost equal). Though in the cooperative research by Van Mier et al. [1997] it is concluded that a perfect localisation of deforrnations takes place in the softening region, a sirnilar trend as observed here can be detected in the results from tests with teflon platens by Van Vliet and Van Mier [ 1996], as shown in Figure 81.

1.0 \

Cf} t\ Cf}

0.8 t'., (IJ

J:l '\• \ .\

Cl.l '\' h/d "'C

\ ,\ '\' 0.5 (IJ 0.6 \ ·\

N '\' ·- \' 1.0 - '\ c..s \ ' e ' \ 2.0 \.

0 0.4 s::

0.2 -~:::..-_-::...-_~--- -----

0 250 500 750 1000 post-peak deformation [µm]

Figure 81: Uniaxial post-peak stress versus displacement results after Van Vliet and Van Mier [1996]

Therefore experiments have been carried out, in which it was attempted to measure the behaviour of the continuum parts of the specimen after peak

91


stress in multiaxial compression, in order to clarify the alleged post-peak load-bearing contribution of these continuum parts. Figure 82 shows the measured post-peak response in plane strain experiments (cr/cr1=0.15) on three normal-strength concrete specimens with varying height. In this figure no continuum behaviour bas been accounted for. Tuis graph, which is representative for all plane strain experiments, shows that large differences are observed between the individual post-peak curves. But there does not seem to be a direct link between the path of these curves and the . specimen height, because the highest stress at a certain post-peak deformation is observed in the case of the specimen with a height of 100 mm. The fact that in the case of the batch, from which this 100 mm specimen was taken, peak stress was also found to be higher (see Figure 41) than in the other experiments, revealed the possibility of a better compaction after casting of this batch. Therefore the suspicion arises that the differences observed in Figure 82 might also be caused by differences in specimen quality (i.e. differences in casting batches) instead of differences in size.

'i"" .... .x: .,. CD a.

b --b

1.0

0.8

0.6

0.4

Post-peak response in plane strain experiments wlth 15% lateral confinement

' '

0.2 -+----~---,...-----.----,---~-----,

0 -2 -3

w 1, postpeak [mm]

Figure 82: Measured post-peak stress-displacement curves in plane strain compressive experiments on three specimens with varying height

92


To determine the behaviour of the more or less intact continuum specimen parts after peak stress, specimens have been manufactured containing grooves on those surfaces that are in contact with the loading platens in the intermediate principal stress direction (w2=0 direction), as shown in Figure 83. The width and depth of the grooves is 7 and 4 mm, respectively.

Figure 83: Specimen with grooves in order to determine the post-peak continuum behaviour by means of strain gauges.

In these grooves strain gauges have been applied ( eight per groove) to measure the post-peak continuum behaviour with 24 strain gauges in both the major (axis 1-direction) and minor (axis 3-direction) principal stress directions. The measuring length of these strain gauges is 6 mm, the maximum strain 2%, in the case of these 100 mm cubes corresponding with a maximum axial specimen displacement of 2 mm. Similar specimens were fabricated with heights of 50 and 200 mm. Because the results obtained from these experiments were similar to those from tests on 100 mm cubes (Van Geel [1995b]), only the cube results will be discussed here. Figure 84 shows a comparison between the observed macroscopie stressdisplacement curves for specimens tested in plane strain compression, both with and without the grooves sawn into the specimen surfaces. The closeness of the results in this graph, and in those obtained from specimens with other heights (Van Geel [1995b]), is reason to believe that not only a qualitative but also a quantitative comparison between these specimens is justified. Note that the displacements in the experiment with 15% confinement are too large for the strain gauges to measure. Therefore here only the results from the experiment with 5% confining stress are discussed.

93


-200 'ii' ll. :iiE .._.

- --~160 .... \ b

·• I

I

/

- - _ O' 3 I O' l = 0.05

r 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6

w 3 [mm] w 1 [mm]

-60 'ii' ll. -40 i!.

N -20 b

0

0 -1 -2 -3 -4 -5 -6 w 1 Imm]

Figure 84: Macroscopie stress-displacement curves obtained /rom experiments in plane strain compression on specimens both with and without

grooves sawn in specimen surfaces.

Figure 85 and Figure 86 show the results in axial direction from the strain gauge measurements in the case of the plane strain experiment on a 100 mm cube with cr/cr1= 0.05. Note that in these graphs the strains have been mul tip lied by a factor l 00 to facilitate comparison with the macroscopie stress-displacement curve. Figure 85 shows the strain gauge measurements at their position on the specimen surface. The observed macroscopie crack pattem bas been sketched roughly in this figure. It is obvious from this graph that, while measured strains in those strain gauges that are in the localised shear zone remain increasing (until they fail), the strain gauges active on specimen parts free from macroscopie cracks show steep post-peak. unloading behaviour.

94


0 ·2 0 ·2 0 ·2

LVDI-measurement:

0 ·2

.... \? 1 ...

w 1 [mm]

·80n, -40 1 \_

0

Figure 85: Strain gauge measurements in axial direction on a 100 mm cube in plane strain compression. Note that the strain readings have been multiplied with a factor 100 to compare them to the macroscopie specimen displacement (lower right corner).

95


-80

.... "' 0.. !. -b

-20

1

1 \ ' ' ", ...

0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 w 1 [mm]

Figure 86: All strain gauge measurements in axial direction /rom a plane strain test on a 100 mm cube in plane strain compression gathered (dashed

line represents the macroscopie (L VDT) stress-displacement curve).

The observed crack pattem is sketched in this graph. Though a large scatter is observed in the local pre-peak stress-strain curves, as shown in Figure 86, the measured local post-peak responses appear to be quite uniform. The scatter in the pre-peak region might be caused by the fact that the measuring length of the gauges (6 mm) is even smaller than the maximum aggregate size (8 mm) or may be even solely because the strain gauge measurements are surface measurements. The fact that the strain gauge readings in general show smaller deformations than the average macroscopie measurement might also originate here from. It was already observed by Van Mier [1984] in the case of uniaxial compression that surface strain measurements showed smaller deformations than L VDT measurements between the loading platens. However, Van Mier attributed this toa curvature of the applied steel brush loading platens.

96


In Figure 87 the continuum and local behaviour, as derived from the strain gauge measurements, are shown for this particular experiment. Note that to derive the pre-peak continuum behaviour only those strain gauge readings have been used that were close to the overall stress-displacement curve as measured by the L VDT' s. The shape of the post-peak curve has been determined by using all strain gauge measurements showing unloading behaviour. Figure 87, like the other strain gauge measurements not shown here (Van Geel [1995b]), shows that the continuum parts of the specimen only start unloading after a certain post-peak deformation.

-80

-60

-cv ll..

i!. -40 T'"

b

-20

Continuum contribution derived from strain gauge measurements

Measured stress-deformation curve

Local macrocrack contribution derived from strain gauge measurements

Continuum behaviour derived from strain gauge measurements

0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 w

1 [mm]

Figure 87: Derived post-peak continuum behaviour from the strain gauge readings

In Figure 87 the light grey area represents the continuum behaviour, accompanied by isolated damage at the mesoscopic level only, whereas the dark grey area represents the localised macroscopie (shear) crack behaviour. Damage at the mesoscopic level means inelastic deformations within the cement paste and at the aggregate-cement paste interfaces (and exceptionally within the aggregates ). Numerical simulations at the mesoscopic level by Bongers [1998] indicate that irreversible damage within the cement paste is mostly damage at interfacial zones. This means that, when aggregates do not fail, most continuum damage is concentrated at aggregate-cement paste

97


interfaces. Hence, when the quality of these interfaces is improved, which seems to be the case for the 100 mm specimen in Figure 41 and Figure 82, the maximum load hearing capacity of the specimen increases. Right after peak stress macroscopie cracks are initiated, hut are not covering the entire cross section of the specimen yet (see e.g. Figure 76b). Tuis means that at this stage of the stress-displacement curve part of the load is still being transferred by the continuum. Combination of Figure 41 and Figure 82 supports the thought that the extent of this continuum contrihution is also related to the casting and compactation quality of the concrete of the specimen. Similar measurements and derivations have been carried out in the minor principal stress (3-) direction, with similar results (Van Geel [ 1995b ]). The derived post-peak stress-displacement curves for the localised cracks in the major principal stress direction from the findings herefore are shown in Figure 88 and Figure 89 for several specimen heights, with cr/cr1= 0.05 and cr/cr1= 0.10, respectively. lt is evident that the differences in the initial part of the softening curve, as visible in Figure 82, disappear completely when taking the measured continuum post-peak behaviour into account. Therefore it is concluded that those differences are to a large extent due to the fact that the continuum parts of the specimen do not unload immediately at peak stress hut remain contributing to the load-bearing capacity until unloading of these parts starts at a certain level of post-peak displacement. As already mentioned, the extent of this continuum contribution is thought to be highly related to the properties of the aggregate-cement paste interfaces, thus also to the degree of compaction of the material after casting. When taking this continuum behaviour into account, as is done in Figure 88 and Figure 89, a perfect localisation of deformations occurs. In the case of uniaxial compression, Figure 80, differences in concrete casting quality do not reveal themselves as strongly as in multiaxial testing because the continuum strains are much smaller and hardly nonlinear. The differences in the uniaxial post-peak stress-displacement curves seem to be merely the result of not taking the actual post-peak continuum behaviour into account, hut assuming linear unloading at peak stress. Though the continued loading of the continuum right after peak stress is identical for all specimen sizes in terms of stress-strain behaviour, this is not the case when converted to stress-displacement behaviour. Therefore higher specimens show higher stresses at displacements right after the peak.

98


1.0

0.8 50 mm specimen

'i"' 100 mm specimen .... 0.6 .Il:

lll

!. b .... OA

b

0.2

0.00 -OAO -0.80 -1.20 -1.60

w 1, local [mm]

Figure 88: Post-peak stress-deformation curves for plane strain experiments (a/a1= 0.05) when actual continuum unloading behaviour is considered

1.0~

0.8

"';"' .... .Il: 0.6

l b

- 0.4

0.2

. '\ \ ~\ f,

' \ ', \ ', \" 1 ,, ',

\\'-,

'\ '·,, \,

50 mm specimen

200 mm specimens

0 ~ ~ ~ 4 ~

w 1, local [mm]

Figure 89: Post-peak stress-deformation curves for plane strain experiments (a/a1= 0.10) when actual continuum unloading behaviour is considered

99


6.3. Two-dimensional analytical model for shear cracks in plane strain compression

An analytical model is developed, based on a recent model by Murakami and Ohtani (1994], which is described in subsection 6.3.1. In subsection 6.3.2. the adaptions are outlined that have been made to the Murakami model. Subsection 6.3.3. shows some model results compared to laboratory experiments and conclusions drawn from the model calculations. The model is used to: • obtain a better insight in the post-peak behaviour of concrete and the

processes that dominate the observed behaviour during that stage; • investigate the sensitivity of concrete's post-peak behaviour to changes in

stress path; • investigate the possibility of modelling macroscopie shear cracks

numerically using interface elements.

6.3.J. Murakami model

A simple two-dimensional analytical model for post-peak behaviour of concrete has been developed by Murakami and Ohtani [1994], based on experiments, in which always a localised shear crack developed that could be interpreted as a 2D-ftacture, similar to the present plane strain experiments. In the model the continuum parts of the specimen are assumed to be rigid bodies, all deformations are assumed to take place within the macroscopie shear cracks. A Mohr-Coulomb failure criterion is assumed for these shear cracks, defined by a cohesion C and a friction coefficient µ at both peak stress and a 'final stable stage' (residual stress level). This is shown in Figure 90. The parameters C and µ are assumed to be linearly dependent on a damage index a, which itself is dependent on the crack shear deformation, as shown in Figure 91, together with the assumed distribution of shear strain along the axis perpendicular to the shear crack. Note that in the figure B50 is the mimimum width of the shear band (corresponding to zero confinement cr1=0, note that Murakami et al. denote the major compressive stress with cr3 and the

confining stress with cr1 in Figure 90!) and mYt is the averaged shear strain over the shear band width B5.The tangential displacement öt within the shear band is given by:

(Eq.2)

100


Mohr-Coulomb's fracture criterion

't' = cpeak + µpeak O' n

Residual strength criterion

't, = cfigal + µfinalO'n

cpeak ~c~ciinal f.\eaic $; µ < JJmai

Figure 90: Fai/ure and residual strength criterionfor macroscopie shear cracks according to Murakami and Ohtani [1994]

in which B. is the shear band width (Figure 91) and ~ the nondimensional width of the shear band i.e. B/Bso- From Equation 2 it follows that a is a function of the shear displacement at. At residual stress µ is larger than at peak: stress, because ~eak may be dominated by friction between cement paste and aggregates in mesoscopic cracks, whereas µfinai may be the result of interlocking between two sliding surfaces (separation of crack surfaces, crushing of material). From the experiments, in which the lateral compressive stress cr 1 is constant, Murak:ami and Ohtani concluded that the residual strength is mainly dominated by friction. The angle of the shear band is assumed to be constant during the fracture process:

(Eq. 3)

101


The ratio between normal and tangential displacements within the shear band, on/01 , during the entire fracture process is assumed to be equal to the ratio ön,finai/81,final and only dependent on the lateral compressive stress cr1•

:c B, ~ Shearband

1.0

0 1.0 2.0

Bso mÎt final = 3.6 mm

\ "'".1. 3.0 4.0

Figure 91: Assumption of shear strain distribution and the damage parameter a as a fanction of the shear strain and shear band width

according to Murakami and Ohtani (1994]

6.3.2. Modified model

In the modified analytica! model the following assumptions are made:

Shear crack angle The shear crack angle 0 is assurned to be constant: 25 degrees. Tuis shear crack angle is based on the observed angle in the present plane strain experiments.

Failure criterion From three plane strain experiments on 100 mm cubes with different confining stress cr3 (0.05, 0.10 and 0.15 times the major compressive stress cr1

as shown in Figure 39) in which also a 2D fracture proces takes place, it is found that the Mohr-Coulomb failure criterion is only applicable for low compressive stress levels. At higher stress levels Murakarni's model largely overestimates both cr3 and cr1• See Figure 92. Note: to obtain these results Murakarni's parameters were used, except the uniaxial compressive strength which was taken as 37.5 MPa in order to fit the test with cr/cr1= 0.10 (while the actual compressive strength of the concrete used was about 44 MPa).

102


-200

0'1 [MPa]

-100

0

' '

2

" "

' "

\ '-

-- -- -Calculations according to Murakami's model [1994)

--Present test results

" \

\

\

\

\

\

\

" " ___

4 6 8

W 1 [mm]

Figure 92: Comparison of Murakami 's model, using a linear Mohr-Coulomb failure surface with present plane strain test resu/ts (with conjining stresses

cr/cr1= 0.05, 0.10 and 0.15 /rom bottom to top respectively)

Therefore a curvilinear failure criterion has been assumed for the shear crack. Tuis criterion is defined at peak and residual stress. During softening the failure surface transforms from the state at peak stress to the state at residual stress. The following failure surface is proposed by Bongers [1994]:

(Eq. 4)

in which a(;;::O) is related to the angle of friction, b(;;::l) is a measure for the curvature of the failure surface and ~ is the tensile strength of the material. In Figure 93 and Figure 94 the fits of Equation 4 to the three plane strain test results mentioned are shown for both peak and residual stress. These fits have been obtained graphically, not numerically. It will be shown that these fits suffice for the purposes of the model.

103


-140

-120

-100 êi ~

-80 ~ - ~ 0 / -60 ~ / --~

/ / -40 / -----\-

/ I \ -20

I I

-200 -150 -100 -50 0

cr0 [MPal

Peak stress: a=0.2, b=l.4,J;=4 Nlmm2 (Equation 4) Figure 93: Bongers' curvilinear failure criterion used in the present model (solid line) and Mohr 's stress cire/es at peak stress for three plane strain

experiments

-50

-40 ";' ~

-30 ~ ...... 0

1:-..lf

--- -..: -20 /

/ -10 I /-\-

1

-60 -50 -40 -30 -20 -10 0

cr0 (MPa]

Residual stress: a=0.54, b=J.19,J;=O Nlmm2 (Equation 4) Figure 94: Bongers' curvilinear criterion at residual stress used in the

present model (solid line) and Mohr 's stress cire/es at residual stress for three plane strain experiments

104


Softening parameter The transformation of Bongers' criterion from peak to residual stress level is defined by a softening parameter a, similar to the damage parameter used by Murakami and Ohtani. The failure criterion parameters are chosen linearly dependent on this damage parameter:

a = aa peak +(1-a )a final

b = abpeak +(1-a )bfinal

ft = aft,peak +(1-a )!t,fmal

with 0 :c:;; a :c:;; 1 ( apeat = 1, afl1101=0).

(Eq. 5)

Unlike in the Murakami model, this parameter a is chosen dependent on both the shear and normal displacements within the macroscopie crack (w1 and wn respectively ):

From Equations 4 and 5 and the following equilibrium equations:

't t =-(er l -er 3)sin8 cose

er n =cr1 sin2 e +cr3 cos2 e

(Eq. 6)

(Eq. 7)

in which 0 is the angle of the shear band, as well as the values for a, b and ~ from Figure 93 and Figure 94, the following implicit relation between a and cr1 can be derived:

J

. (4a -er 1(sin2 e + rcos2 e ))0.2la+l.l9 -cr1(1-r)sm0 cose = -034a +054 (Eq. 8)

in which ris the stress ratio cr/cr1. Equation 8 is used to evaluate the three plane strain test results to find the relationship between a and the shear crack displacements. From this evaluation it appeared that the softening parameter a in all cases is the same function of K, as can be seen in Figure 95.

105


0 2 4 6

K = w n + w t [mm]

Figure 9 5: Softening parameter a. as a function of total macroscopie crack displacements for three different plane strain tests and curve fit (thick line)

The following curve fit is used:

(-4soo( ;J 5

J K < 25mm: a = e

(Eq. 9) K:;:: 25mm:

In this case it proved to be easier to use these two functions instead of trying to find one function describing the entire curve ofFigure 95.

Coupling of stress es and displacements In the Murakami model the ratio between the shear band displacements on/ot (in the present model denoted as w/w1) is assumed to be equal to the ratio On,finalöt,final during the entire fracture process and dependent only on the lateral compressive stress. Though -using this assumption- good results were obtained with the Murakami model, simulating experiments with constant

106


lateral stress, this relation between macroscopie stresses and displacements appeared to be insufficient in the case of varying confining stress. Therefore in the modified model a more detailed relationship is formulated, in which the ratio w/w1 is directly related to the ratio cr/cr1• Tuis assumption seems valid, because the dilatant behaviour of the macro-crack seems mainly dependent on the ratio of normal and shear stress acting in the crack zone. A coupling between wn and w1 in three stages is assumed during the softening proces, depending on the stress ratio r=cr/cr1:

Stage 1:

wn s (0.20+r):

1

Stae;e 2:

wn > (0.20+r)andw, s (l.6-2r):

Awn 1 ---Awt 1 + lO*r

Awn = 1.1 Aw,

~ wn = wn,end ofstagel + l.l(w, - wt,end ofstagei>

Stage 3:

w, > (1.6- 2r): Awn = 1-3r Aw,

~ w n = wn, end of stage2 + (1- 3r )( w, - w,. end of stage2)

(Eq. 10)

(Eq. 11)

(Eq. 12)

in which ris again the stress ratio cr/cr1• These relations between shear and normal displacement are shown in Figure 96.

Calculation of stresses and deformations The transformation from stresses and deformations within the shear crack to specimen displacements can be obtained from a simple geometrical calculation, see Figure 97. From the geometry of the cracked specimen, the following relationships can be derived:

107


3 ~ { ~

/ <::)' / / ()" t:::JJ' )

" /

// / ()•

""' / 6 ~/ " ó / ' / 6 " '

I 6 ".J; / ' "?: / / /

6 "> ó 7 /

2

-E E ....... -==

1

0

0 1 2 3 w n [mm]

Figu,re 96: Shear versus nonna/ displacement observed in three plane strain experiments (solid lines) and Equations 9-11 used in the present model

(dashed lines)

Wn = -W1 sin0 -W3 COS0

wt = w1 cos0 - w3 sin0 (Eq.13)

in which 0 is the angle of the shear band. In Equation 13 the nurnber of macroscopie cracks (n) as shown in Figure 97 is not taken into account because here the total crack displacements are calculated. Similar equations for the specimen and shear crack stresses have already been shown in Equation 7.

108


6.3.3. Model results

In Figure 98 the results from the calculations with cr/cr1=0.05, 0.10 and 0.15 are shown, compared to the experimental results on 100 mm cubes. Of course, since these three experiments have been used to determine the model parameters, the calculation results fit the experimental curves quite well.

0'1

~

--+W3/n

Figure 97: Shear crack and specimen stresses and deformations (n designates the number of macroscopie shear cracks, in the case of 100 mm

cubes in plane strain experiments n=2)

However, the validity of the model assumptions, in particular the relationship between wn and w1 depending on the stress ratio cr3/cr1, in fact depending on the normal and shear stresses acting at the macroscopie crack (this will be discussed further in the next section), can be confirmed by tests with other (e.g. constant) confining stresses. As will be outlined in more detail in the next chapter, besides proportional tests experiments have been carried out with non-proportional post-peak stress ratio's. In Figure 99 through Figure 103 some model calculations are shown for loading paths with other post-peak stress ratio's, compared to experimental results. Only in two cases significant deviations from the experimental results are found: in Figure 101 and Figure 102 a higher peak stress is found in the model calculation.

109


4 2 w3 [mm]

-200

0 -2 -4 w1 [mm]

Figu,re 98: Calculated (dashed lines) and measured (solid lines) stressdisplacement curves for three proportional plane strain experiments on 100

mm cubes

But the observed deviations in peak stress can be explained easily. As already mentioned in Chapter 4 and in the previous section the casting batch, from which the three proportional tests on 100 mm cubes were taken, appeared to be of a quality different from the other batches, resulting in higher peak stresses than in the case of the other batches, see also Figure 41. Because the failure criterion of the modified model has been fit to those proportional tests, all peak stresses calculated in Figure 99 through Figure 103 are overestimated. The fact that this only leads to significant differences between model and experimental results in the case of constant confining stress can be explained using Figure 105. Tuis figure qualitatively shows the shape of the mean failure surface of concrete as observed in plane strain compression (see

110

Chapter 6: Loca/isation of deformations in macroscopie cracks

Van GeeJ [1995b]), as well as some possible test results corresponding to that mean failure surface. Furthermore, a statistica} distribution of the test results is shown. Suppose that some specimens show peak stresses that are larger than those corresponding to the mean failure surface, like the three proportional tests used to fit the model. Then it is obvious that the observed difference in peak stress in the major principal stress direction between these tests and tests with a 'mean' peak stress is dependent on the stress path follow ed. Figure 105 shows why for experiments with constant confining stress (like in Figure 101 and Figure 102), denoted by 'stress path 1' in the figure, this difference is larger than for experiments with a post-peak stress ratio of dcr3=-dcr1 (like in Figure 100 and Figure 103), labelled 'stress path 2' in the figure.

-120

00

-80

'ëi ~

-60 ~ ......... -b -40

-20

3 2 1 0 -1 -2 -3 w3 [mm] w 1 [mm]

Figure 99: Experimental (so/id lines) and model results (dashed lines) fora 100 mm cube loaded with post-peak stress ratio dcr3=-0.25*dcr1 (this means

that during the softening process cr 3 increases) starting at cr 1=-105 MPa

111


-100

-80 ~ b

-60

3 2 1 0 W3 [mm)

Figure 100: Experimental (solid lines) and model results (dashed lines) fora 100 mm cube loaded with post-peak stress ratio do 3=-do1 ( again o 3

increases during the softeningprocess) starting at 01=~96 MPa

' ,....., ''

-80 <1S t:I... :E

-60 ....... -"

-40

-20

t--~~-t-~~-r~~~t--~~-;---~~-+-~~~1

3 2

w3 [mm] 1 0 -1 -3

Figure 101: Experimental (solid lines) and model results (dashed lines) fora 100 mm cube loaded with post-peak constant lateral stress of o1=-13 MPa,

starting at 0 1=-113 MPa

112


3 2 1 w3 [mm]

-160

"-140

-120

-100

-80

-60

-40

-20 1

0 -1 -2 -3 w1 [mm]

Figure 102: Experimental (solid lines) and model results (dashed lines) fora 100 mm cube loaded with post-peak constant lateral stress of cr 3=-20 MPa,

starting at cr1=-l 35 MPa

-140 ." ......

-120

-100

-80

-60

-40

-20

3 2 1 0 -1 -2 -3 W3 [mm) w 1 [mm]

Figure 103: Experimental (solid lines) and model results (dashed lines) fora 100 mm cube loaded with post-peak stress ratio dcr3=-dcr1 (again cr3

increases during the softening process) starting at cr1=-134 MPa

113


Probability distribution of test results '

Mean failure surface

(Acr1)1 cr 1 Figure 104: Two-dimensional representation of the injluence of differences

in specimen strength due to the quality of specimen preparation when following different stress paths

The model appears to describe the experimental results quite well. In the first place this means that the assumed behaviour of macroscopie shear cracks, being composed of a varying failure surface dependent on the normal and tangential displacements in a shear crack, leads to a satisfactory description of concrete's post-peak behaviour. When comparing the shapes of the (w1,wn)-curves used in the model (Figure 96) to the evolution of macroscopie cracks in Figure 76 and Figure 79, it can be seen that in the model it is also assumed that there is a dominance of 'sliding' (shear) displacements during the final stage of the softening curve (stage 3 in the model) and a relatively larger importance of normal crack displacements (opening of shear cracks) during the steep part of the softening curve (stage 2). The validity of these assumptions is confirmed by the calculation results in Figure 99 through Figure 103, where different (post-peak) stress paths are described with satisfactory accuracy. Comparing the results of the analytica! model with the experimental results, hearing in mind the origin of the deviations found, it can be said that the simple way of modelling shear cracks as is done here indicates that discrete modelling of macroscopie shear cracks in finite element models is an acceptable approximation. Within the range of experiments at hand, it is shown that different post-peak stress paths can be simulated by the model without adjustment of model parameters, indicating

114


that a similar fracture mechanism occurs independent of the loading path. Furthermore, the model shows that this post-peak failure mechanism is highly insensitive to variations in the followed pre-peak loading path, even though in some of the simulated experiments large inelastic displacements occur. Tuis will be discussed further in the next chapter. One should bear in mind, considering the results from section 6.1., that in fact the macroscopie cracks are not formed completely until an advanced stage of softening. In the model these cracks are assumed to have been completely formed already at peak stress. The model actually describes the behaviour of the completely developed macroscopie cracks instead of the behaviour as a result of both formation and development of these cracks, hut this simplification of observed behaviour seems to be valid when comparing experimental and analytical results. In a sophisticated numerical model several conditions should be taken into account that have been neglected in this section for the sake of the simplicity of the model. In this chapter, 'material behaviour' is determined from plane strain experiments with teflon loading platens and the results are only compared for this type of tests. In a numerical model, discrete crack elements should be able to describe macroscopie crack behaviour regardless of loading and boundary conditions. Secondly, the unloading behaviour of the continuum as shown in section 6.2. bas not been taken into account in the model. All post-peak specimen deformations have been assumed to take place within the macroscopie cracks. However, incorporating post-peak continuum behaviour would mostly influence the shape of stage 1 in the (w1,wn)-curves (Figure 96). This would mean only a slight adaption of the modified model. Finally, the analytica! model only considers the case of 100 mm cubical specimens. This means that only one crack pattem and one macroscopie crack length have been taken into account.

6.4. Some general considerattons about macroscopie cracks

In Figure 105 a qualitative overview is given of shear versus normal displacement curves of macroscopie cracks. The curves at cr/t1 = -0.76, -0.61 and -0.47 are the curves of the three plane strain compression tests used to fit the analytica! model in the previous section. According to this diagram the shear- versus normal displacement curve is highly dependent on the ratio of normal and shear stress, if this ratio is relatively small (about lcr n I t 11 < 1 ). As the modulus of the ratio of normal

and shear stress increases, the magnitude of the displacements in both

115


directions decreases and differences in the shear- versus normal displacement curve become smaller and smaller. In the compression!shear region this is due to an increasing normal compressive stress acting on a crack surface, obstructing the development of both opening and shear displacements within the crack. In the tensile/shear region this is due to an acceleration of macroscopie tensile failure. Furthermore, differences in the tension/shearregion are thought to be smaller than differences in the compressionlshear region. Tuis can for example be observed when comparing uniaxial and biaxial tensile tests on concrete specimens: in the shape of the descending branch of these two tests hardly any differences can be noticed.

~

3

2

1

Compression 0 0

t

1 CJn/'Ct= +et:)

Compression/shear

Shear/T en si on

Tension

Figure 105: Qualitative representation of the relation between shear and normal displacements within a macroscopie crack under varying stress states

As can be distinguished in Figure 39, the post-peak stress-displacement curves are somewhat more brittle for specimens loaded with higher confining stresses. In general, brittleness is directly related to the homogeneity of the

116


material. It seems that a larger lateral (macroscopie) confining stress results in a more homogeneous mesoscopic stress distribution (see also Bongers [1998b]). Furthermore, if the assumption is made that dilatancy is merely the result of the fact that macroscopie shear cracks have to go around the aggregates, this behaviour implicates that at higher confining stresses the macroscopie cracks have to be less tortuous. At a high level of confinement, this can only be achieved when these macroscopie cracks do not go around the aggregates but go through them. With increasing confining stress, crack propagation through the aggregates instead of around them could be a less energy-demanding failure mode, dependent on the size, mutual distance and strength (related to the strength of the bond zone and cement paste) of the aggregate grains (Bongers [ l 998b ]). Macroscopie cracking through the aggregates has been observed to occur at higher confining stress levels in the present experiments. In Figure 106 the difference between cracking at low and high confining stress is depicted schematically.

ooo 0

Figure 106 : Different crack mechanisms at different levels of confining stress. Left: at low confining stress macroscopie cracks generally jind their

way around the aggregates, right: at high confining stress macroscopie cracks more often run through the aggregates

Figure 105 shows that with increasing confining stress smaller normal deformations occur. Tuis means that macroscopie cracks in tests with higher confinement have to be less open. Note that the macroscopie cracks in an experiment with higher confining stress as shown in Figure 107 are also clearly less wide than in the case of a test with lower confining stress, see Figure 77.

117

Chapter 6: Loca/isation of deformations in macroscopie cracks

Figure 107: Ultriaviolet photograph of a specimen loaded in plane strain conjiguration with cr/a1=0.15

In Figure 109 close-up photographs are shown of specimens loaded with higher confining stresses, clearly showing fracture through the aggregates. Figure 108 indicates at what locations on the specimen surfaces these photographs were taken.

1,5,6

2,3 Figure 108: Indication of the locations of the photographs shown in Figure 109 on 100 mm cubes loaded in plane strain conjiguration with cr /cr / =0.15

118


Figure 109: Examples of cracks running through the aggregates in plane strain experiments with cr /cr / =0.15

119

Chapter 7: lnjluence of damage /rom previous loading

7. Influence of damage from previous loading

In section 2.3. it was already outlined that it is not yet clear whether damage inflicted to the concrete specimens in the pre-peak region of compression tests has a significant influence on the material' s stress-deformation behaviour or not. In this context the term 'damage' compromises all irreversible deformations at mesoscopic level within a specimen, e.g. crushing and cracking within the aggregate-cement paste interfaces, cracks through the cement paste or through the aggregates, wbich result in the nonlinear pre-peak stress-deformation behaviour. On the macroscopie scale 'damage' consists of irreversible deformations within macroscopie cracks in the post-peak stage of loading. An increase in damage in general therefore means an increase of the irreversible deformations within those zones and cracks, be it at the meso- or macroscopie level due to, for example, continued shear or tensile action effects in the zones of the cracks. A classification of types of damage occurring in multiaxial compression tests is developed in this chapter. In the past most researchers have examined the influence of damage evolution on macroscopie concrete behaviour by following different stress paths up to peak stress and then comparing differences in the peak stress values corresponding to the different types of stress paths. An example of this was already shown in Figure 8. However, it is also important to know if and how damage, already inflicted to the material in an earlier stage of loading, influences its deformational behaviour and, especially with respect to the safety of structures, its post-peak load hearing capacity. Therefore not only the effect of the load history on peak stress should be examined, but the effect on the stress-deformation behaviour as a whole. The behaviour of a material is said to be 'loading path dependent' in the present research if the macroscopie stress-deformation behaviour of the material changes significantly when the load history of the specimen alters. In this case 'load history' refers to the loading states, i.e. (quasi)static stresses or deformations, the material bas been subjected to during it's lifetime. Dynamic loadings and changes in material behaviour due to shrinkage, creep and different temperatures are not included here. Loading path dependent behaviour is closely related with physical and geometrical nonlinearities. Because of the large pre-peak nonlinear deformations occurring in multiaxially loaded concrete, loading path dependent behaviour is likely to occur in concrete even before macroscopie cracks have been initiated.

120

Chapter 7: Influence of damage from previous loading

Load histories can be defined in stress space or in deformation (strain or displacement) space. For several reasons, it is easier to define loading paths in stress space. Firstly, absolute values of stresses are much easier to interpret than those of deformations. A positive stress for example is definitely a tensile loading, while a positive specimen deformation (be it displacement or strain) in a certain direction might be accompanied by a compressive stress in that direction. Besides, after peak stress deformations tend to localize in a limited number of macroscopic cracks as shown in the previous chapter. The specimen then can be regarded as a discontinuum, whose behaviour is govemed by displacements instead of strains, as is the case before peak stress is reached. This means that stress-deformation behaviour is dependent on the specimen size in the pre-peak region if it is regarded in terms of displacements, and in the post-peak region if regarded in terms of strains. Therefore the easiest way to detect loading path dependent behaviour is to carry out different loading paths in stress space and then to compare the deformations of the specimens at loading path intersections in stress space. In the present research this is done in an exhaustive plane strain test program which will be discussed in section 7.1., where (per specimen size) displacements at stress path intersections have been compared. This is shown schematically in Figure 110.

,WA =Wa?

\ )

" ....... /Stress Path B

·-------~ O'l

Figure 110: Two-dimensional representation of comparing deformations at a stress path intersection

121

Chapter 7: Injluence of damage from previous loading

In section 7 .2. a different approach to unveil loading path dependency is chosen: specimens are loaded in uniaxial, plane strain and triaxial compression after they first have been subjected to different types of preloading. It will be shown that due to these pre-loadings significant loading path dependent behaviour can occur. In section 7.3. results from the experiments carried out are considered more closely and the influence of damage inflicted to the material at the mesoscopic level, in particular cracking at the aggregate-cement paste interfaces, on stress-deformation behaviour is emphasised. Two different mesoscopic damage mechanisms, softening and strengthening of aggregate-cement paste interfaces, appear to be the main causes of loading path dependency observed. In section 7.4. a macroscopie interpretation of these mesoscopic darnage mechanisms is undertaken. Lastly, in section 7.5. the influence of combinations of meso- and macroscopie darnage, i.e. macroscopie cracks, is analysed.

7.1. Plane strain test program

The above idea of comparing deformations at stress path intersections has been carried out in a plane strain test program. In this program proportional tests are carried out at stress ratios (cr/cr1) of 5, 10, 15, 25 and 50%. Furthermore the test program consists of two 'random', two 'deviatoric' and two '25%' tests, which will be described further on. Identical stress paths have been followed per specimen size. This is shown in Figure 111 in the case of 100 mm cubes. In this figure the stress path intersections where both pre- and post-peak deformations have been compared are denoted by 'A' and 'B'. Pre-peak deformations have been compared at all other path intersections. Figure 112 shows stress-displacement results for three different stress paths (100 mm cubes). In the 'deviatoric' test initially a stress path is followed according to cr3=cr1• From a certain stress point -defined in advance- this stress ratio is changed in dcr3=-dcr1, which is maintained until the end of the test. As can be seen in Figure 112, the steep softening branch as observed in the proportional tests disappears in a 'deviatoric' test. The failure behaviour observed has changed from a very brittle one into a quite ductile one, due to the stress ratio imposed on the specimen. Tuis is a result of the increase in confining stress in axis 3 when peak stress bas been passed and cr1 decreases, hindering the formation of the final macro-crack pattem. Though the major and minor principal stresses (cr1 and cr3) were kept equal very precisely in the

122


first stage of these 'deviatoric' tests, in all tests the displacement in the 3-direction was not exactly equal to the displacement in the 1-direction. lt is not clear whether this is a result of the loading conditions (the orthogonality, parallellism and the positions of the loading platens were checked regularly) or of the way in which the specimens were casted or prepared. In the intermediate stress direction (axis 2) no descending branch bas been observed at all in 'deviatoric' tests. The experiments denoted with '25% tests' are very similar to these 'deviatoric' tests: in these tests initially a stress path is followed according to cr3=0.25*cr1• From a certain stress point -defined in advance- this stress ratio is changed in dcr3=-0.25*dcr1, which is maintained until the end of the test.

-100

l !. -50

C")

t?

Proportional tests

'Devlatoric' tests

'Random' tests

'25%'test

o~,~~~~===-r--,-----.--------.---~ 0 -40 -80 -120 -160 -200

O' 1 [MPa]

Figure 111: Plane strain stress path test program

The 'random' test stress paths consist of several linear branches, in which either cr1 or cr3 is kept constant. Before reaching the failure surface, cr3 is kept constant until the end of the test. As expected, the steepness of the descending branch of the stress-displacement curve lies somewhere between that of the proportional and the deviatoric test (in the random tests cr3 is kept constant after the peak while in the proportional and deviatoric tests this stress decreases respectively increases).

123


In Figure 112, stress path intersection 'B' (see Figure 111) is indicated in the case of 100 mm cubical specimens. In this graph no '25%' test is shown because this type of test was not carried out at stress path intersection B. Similar graphs have been produced for stress point A, also for the other specimen sizes. See [Van Geel, 1995b] for more details. From these graphical comparisons one could conclude that there is not much diff erence between the deformation results of the different stress paths at the examined stress path intersections.

ca Q.

:& ...... et?. T"'

b

4 2

-200

-160

-120

-80

0 -2 w 1,3 [mm]

B

Proportional test

- 'Deviatoric' test

'Random' test

-4 -6

Figure 112: Comparison of different stress paths at a stress path intersection

124

Chapter 7: Injluence of damage /rom previous loading

Only some proportional paths showed larger deformations than the nonproportional paths. The reason for the deviation of these proportional tests might be found in the essential difference between these tests and the others: the proportional test is the only test in which from the start of the test positive deformations in the 3-axis occur. In the other tests always the initially developed negative deformations have to be counterbalanced first, in order to reach the positive deformations in the 3-direction necessary to form the macroscopie crack pattem. However, when comparing the peak stresses of the deviating proportional paths, originating from the same casting batch (see also Figure 41 ), it is found that these tests show a larger peak stress, accompanied by larger deformations than expected. It is therefore suspected that the observed deviations are caused primarily by differences in the quality of specimen preparation instead ofloading path dependent behaviour.

-6

-4

2

4

6

6 4

• .·o

. . •

0

2

- - - - - Line x = y

•

•

Wu [mm]

(proportional tests)

o w3. 50mm specimens a w1, 50mm specimens

• w3, 100mm specimens • w1, 100mm specimens

• w3, 200mm specimens • w1. 200mm specimens

0 -2 -4 -6

Figure 113: Comparison of displacements of proportional and 'random' plane strain tests at all stress path intersections (Van Geel [1995b])

Figure 113 shows a comparison of measured displacements, compared for two different stress paths at every stress path intersection. In this graph the displacement of one stress path is displayed on the horizontal axis, the displacement of the other stress path on the vertical axis. If no loading path

125

Chapter 7: Injluence of damage /rom previous loading

dependency would exist at all (and if no material variations would exist at all), all data points would be situated on the line x=y. Thus, the distance between the data points and the line x=y provides an indication of the phenomenon of loading path dependency. These comparisons have been carried out for all types of tests and for all specimen sizes [Van Geel, 1995b ]. Again, the only trend that might exist is that the proportional tests show larger displacements in both directions when compared to other stress paths. However, this trend is not very clear to distinguish and could even be rejected when comparing the displacement w1 of the proportional and the •random' paths in Figure 113. Deviations between several stress paths might very likely originate from variations in material properties, like peak strength, due to the variability (heterogeneity) of concrete and from variations in specimen preparation. Furthermore, the large running-in effects due to the presence of the teflon/grease intermediate layer might have their effects on the material response (e.g. the stress in the intermediate direction). Finally some difficulties are encountered when handling the measured data, like curvefitting initia} parts of stress-displacement curves. When considering all these factors that can have a significant influence on the observed displacements, it is remarkable that the differences in Figure 113 are as small as they are. From these test results the occurrence of significant loading path dependent behaviour of concrete can therefore not be confirmed. As will be shown in the next section, more extreme types of loading paths have to be applied to reveal a behaviour as such.

7 :1.. Pre-loading experiments

7.2.1. lnfluence of macroscopie damage

To gain more clarity about whether or not significant loading path dependency occurs, some experiments are carried out in which a pre-loading is applied. From Van Mier's rotation tests it is clear that loading path dependency does exist: peak stress can decrease significantly if a specimen is pre-loaded into the post-peak region in one direction and consequently loaded in a direction perpendicular to the previous loading direction. Tuis loading path dependency is then the result of the presence of macroscopie cracks and occurs only when these cracks are oriented in a favourable position for further growth under the subsequent loading. Some multiaxial tests have been carried out in the present research to confirm these observations. Figure 114 shows a plane strain rotation experiment (w2=0) in

126


which initially axis 3 is the major principal stress direction and the specimen is loaded with cr/cr3 = -0.05/-1. Right after peak stress, the major principal stress direction is rotated to axis 1 and the specimen is subsequently loaded with cr/cr1 = -0.05/-L It can be seen in this figure that the macroscopie behaviour after rotation of the principal loading axis has barely changed. A new V-shaped macroscopie crack pattern is formed in a direction perpendicular to the direction of the V-pattern initiated during the first loading. Tuis confirms the observation by Van Mier, that macroscopie cracks, located unfavourably for further growth under subsequent loading, do not affect the macroscopie stress-deformation behaviour significantly. Even in a triaxial rotation experiment with cr2 = cr3 = -10 MPa as shown in Figure 115, pre-loaded with axis 2 as principal loading axis (cr1 = cr3 = -10 MPa), the stress-deformation behaviour is hardly influenced by the preloading. Tuis means that even when the macroscopie cracks (initiated before rotation of the loading axes) are located favourably for further growth, the material bas to be subjected to loading further into the softening region than is done in this triaxial test to have a significant influence on the behaviour after rotation. Likewise, the response in plane strain experiments (no rotation of loading axes) seems hardly affected by macroscopie damage inflicted in the postpeak regime in a uniaxial test (Figure 117).

7.22. Intluence of mesosoopic damage

lt may be clear that even when significant macroscopie cracking bas occurred within a specimen, this severe damage might not influence the behaviour of the specimen in a consecutive loading. Whereas macroscopie cracks have a distinct orientation with respect to the loading directions and therefore can be of small influence in some loading cases, concrete's size-independent prepeak crushing and cracking at the mesoscopic level is arranged more distributedly throughout the specimen and might therefore -if the extent of this mesoscopic damage is large enough- influence the material behaviour regardless of the loading case. F ollowing this line of thought a number of experiments has been carried out in which the specimens are pre-loaded hydrostatically, so that relatively large pre-peak deformations occur and thus a substantial amount of mesoscopic damage is introduced in the specimen. Figure 116 shows the results of uniaxial experiments pre-loaded triaxially up to three different lèvels of hydrostatic stress.

127

2

Chapter 7: /nfluence of damage /rom previous loading

4

-60

-200 l

1 \

j

0 -4 w 1,3 [mm]

Regular plane strain test (CJ 3 t cr 1 = 0.15) Plane strain rotation test

0 -2 -4 w 1 [mm]

-6

Figure 114: Result of a plane strain rotation test compared toa regular plane strain test with cr/cr1= 0.15

128

'ëi' ll. :E .....

""!. -b

2

'E' .5.

CW)

N' ~

2

Chapter 7: Jnjluence of damage from previous loading

-120

-80 J ,

/

-40

f

0

4

'

/ I -I -,,.-

,,< -t'

Regular triaxial test (cr lat= -10 MPa)

Triaxial rotation test

,,.. ,.... -......... -

• w 2 - cr 2 (before rotation)

w 1 - cr 1 (after rotation)

-2 -4 -6 w 1,2 [mm]

/

//

,/

r'

-2 -4 -8 w 1 [mm]

Figure 115: Result of a triaxial rotation test compared to a triaxial test with cr2 = cr3 = -10 MPa

129

Chapter 7: lnfluence of damage from previous loading

The effect of increasing the level of hydrostatic pre-loading on uniaxial peak stress is a very clear example of loading path dependent behaviour, though not very surprising: due to the in some cases relatively large triaxial preloading the specimen is damaged quite severely at the mesoscopic level. During subsequent uniaxial loading macroscopie failure is reached at an earlier stage of loading because continuous cracks are easily formed via the mesoscopic cracks created during pre-loading. The decrease of initial stiffness with increase of pre-loading can be explained by the closure again of the crushed zones at the mesoscopic level, created during the first loading and opened during unloading. It is very striking that the difference between the experiments with preloading levels of -92 and -132 MPa is quite large, while the difference between the experiments with levels of -132 and -192 MPa is much smaller. Obviously most damage is induced within the former range of hydrostatic stress. Tuis damage rnight consist of the formation of shear and tensile cracks at the mesoscopic level, while the damage induced at higher hydrostatic stress might be characterised by further development of mesoscopic cracks already formed before.

-40

'Il"" -30 b

-20

-10

Regular uniaxial test

Uniaxlal test after hydrostatlc pre ... oading up to -92 MPa

Uniaxial test after hydrostatlc pre-loadlng up to -132 MPa

Uniaxial test after hydrostatic pre-loadlng up to -192 MPa

0 --+--------.-------__._,..--------1

0.00 -0.50 -1.00 "1.so w

1 [mm]

Figure 116: Uniaxial experiments after triaxial pre-loading

130

6

.... CIS

Q.

:E -..... b


-160

-. ~" \

4 2 0 -2 -4 w 3 [mm] w 1 [mm]

Regular proportional plane strain tests

·· .. " '•,

-6

Plane strain test with er 3 I er 1 = 0.05 after uniaxial pre-loading 1

Plane strain test with er 3 Ier 1 = 0.15 after uniaxial pre-loading 2

-40 - 0 End of uniaxial pre-loading 1 (cr 1 = -39 MPa)

I \ End of uniaxial pre-loading 2 (cr 1 = -25 MPa]

-20 -

' I

0 1 1

0.0 -0.5 -1.0 w 1 [mm]

Figure 117: Plane strain tests after uniaxial pre-loading

131

2


Regular proportional plane &train test (cr 3 I cr 1 = 0.05)

Plane strain test (cr 3

1 er 1

= 0.05) after plane straln pre-loading up to cr 3

= cr 1 = -92 MPa

Plane strain test (Cl' 3

Ier 1 = 0.05) after plane strain pre-loading up to er 3

= er 1

= -132 MPa

Plane straln test (er 3 1 cr 1 = 0.05) after plane &train pre-loading up to cr 3

= cr 1 = -192 MPa

1 w 3 [mm)

-20

-200 l '

l'

1

-160 l •

0

.... b

, ,

-1

w 1 [mm]

-2

'" • 1

w 1 [mm]

-3

Figure 118: Plane strain tests after plane strain pre-loading compared to a regular plane strain 5% proportional test

132

Chapter 7: Jnjluence of damage /rom previous /oading

Figure 118 shows the results of three plane strain tests with stress ratio cr/cr1= -0.05/-1, pre-loaded similarly to the uniaxial tests discussed above. Note that for practical reasons the pre-loading in this case has in fact not been hydrostatically but plane strain with stress ratio cr3=cr1• This figure shows that the effect of pre-loading is completely different from that in the uniaxial case. Instead of a decrease an increase in peak stress is observed. The stiffness during the reloading branch of the stress-deformation curve is also greater than the initial stiffness. Similar results have been obtained for triaxial tests with lateral confinements of cr2=a3= -3 and-10 MPa (an example is shown in Figure 119) with varying levels of hydrostatic pre-loading.

-150

-100 'ii' a.. !. -b

-50

0

Regular triaxial test with er 2 = er 3 = -3 MPa

Triaxial test with er 2 = er 3 = -3 MPa

after hydrostatic pre-loading up to -132 MPa

/

/ ,

/

'1 /,

!. ,' ' 1

"-....

-2 w 1 [mm]

-4 -6

Figure 119: Axia/ stress-displacement curve of a triaxial test with a2=a3= -3 MPa after hydrostatic pre-loading compared to a similar triaxial test without

pre-loading

133


Clearly the damage induced during pre-loading does not have the negative effect during consecutive plane strain loading as observed in consecutive uniaxial loading. This increase in peak stress cannot be explained by the higher intermediate principle stress at the start of reloading (see Figure 118, cr2-w1 diagram) because this higher stress would merely accelerate failure in the minor principal stress direction. Furthermore it seems logical to think that the negative deformations, that initially develop in the minor principle stress direction during pre-loading, can only have their effect on the magnitude of deformations at peak stress instead of on the peak stresses themselves. Therefore it is concluded that besides the way in which damage at the mesoscopic level causes the reduction in peak stress in the case of uniaxial loading after pre-loading, there must be a different phenomenon with an advantageous effect in the case of a multiaxial loading-unloading-reloading cycle as shown in the experiments.

7.3. Loading path dependency due to damage at the mesoscopic level

From the experiments described in the preceding sections and the results obtained previously by Van Mier [1984] it appears that two types of loading path dependency can be distinguished in the case of concrete behaviour: 1. Loading path dependency related to macroscopie damage. This is the case

when a specimen is loaded until localisation of deformations is initiated and hence a macroscopie failure crack pattern has been formed. More generally this means loading beyond peak stress, like in the described rotation experiments. If the macroscopie cracks are situated in such directions that they can develop easily under further loading and if they already have developed sufficiently, the macroscopie stress-deformation behaviour will be dependent on the already present amount of damage by the macrocracks. This type of loading path dependency bas been described earlier in section 7 .2.1.

2. Loading path dependency related to mesoscopic damage. If no macroscopie failure pattem has been initiated (pre-peak loading), an other type of loading path dependency can occur. This dependency is then caused by distributed damage at the mesoscopic level This type of loading path dependency is only observed if a specimen has been subjected to relatively high multiaxial compressive stresses. Tuis has been shown in section 7 .2.2.

In the following this loading path dependency related to mesoscopic damage will be discussed.

134

Chapter 7: lnjluence of damage from previous loading

Strengthening effect due to mesoscopic damage Mesoscopic loading path dependency occurs, if a specimen is subjected to a relatively large multiaxial compressive stress state. From both plane strain and triaxial tests pre-loaded multiaxially (pre-peak pre-loading), it is found that the peak stress increases with increasing levels of compressive preloading. Obviously, a mesoscopic strengthening effect develops when applying high multiaxial compressive pre-peak loads. It has generally been accepted, that the aggregate-cement paste interface forms the weakest link in concrete specimens for low and medium strengths. Larbi [ 1991] showed that the interfacial zone around aggregates bas a thickness of about 50 µm and occupies quite a large part of the total cement paste volume (30 to 50%). Bourdette et al. (1995] furthermore showed that this transition zone is very porous. Therefore it seems incorrect to assume that this interface is a surface that is only damaged under tensile or shear loading: one could assume that the transition zone volume can be deformed also under compressive loading.

Figure 120: UV-Photographs of 100 mm specimens tested in plane strain configuration at lower (left, 5%) and higher (right, 15%) confining stress.

Figure 120 shows two photographs taken from 100 mm cubes, loaded (plane strain) proportionally according to cr3/cr1=0.10 (left) and cr/cr1=0.15 (right). These photographs are taken at the end of the softening branch. Besides the macroscopie crack patterns the difference in amount of mesoscopic damage between both tests is clearly visible. It should be mentioned that in these pictures only cracks in contact with the outer specimen surfaces are visible. Closer inspection of the mesoscopic cracks showed, that the aggregatecement paste interfaces perpendicular to the direction of major principal

135

Chapter 7: lnjluence of damage /rom previous loading

stress are often heavily damaged, especially in the zone around the macrocracks (shear band). From Figure 120 it is obvious that this amount of damage is larger in the case of higher lateral stress. Figure 121 shows an example of a damaged aggregate-cement paste interface, taken from Figure 120. It is postulated that interfaces, as shown in Figure 121, are damaged during loading due to a compressive load ('crushing') and tend to open during unloading.

Figure 121: Microscope photograph of compressive interface damage

This crushing effect might account for: • larger nonlinearities at higher multiaxial compressive stress levels; • the observed initial plastic volume compaction in compressive

experiments; • a strengthening effect leading to a loading path dependency related to

mesoscopic damage only. It is believed that this type of damage is the primary cause of the strengthening behaviour observed in multiaxial compressive experiments on concrete. Therefore this effect can be regarded as mesoscopic hardening. Closure and collapse of the pore system, indicated by Bazant et al. [ 1986] to be the major cause of nonlinearities in hydrostatic tests, seems to take place primarily within the aggregate-cement paste transition zone. Considering the stress-deformation behaviour of multiaxial compressive tests, the crushing of the transition zone seems to start at a compressive stress of about -30 to -40 MPa. This might also explain why in uniaxial compressive tests no significant hardening behaviour is observed.

136


In the case of the pre-loaded plane strain tests mentioned in the previous section, crushing appears to have a positive influence on the further load hearing capacity of a specimen. According to Bongers [1998b] this strengthening effect is also a result of the decreasing distance between the aggregates due to crushing. Tuis means that at contact areas of continuous mesoscopic cracks, as shown in Figure 122, the deviation from the ideal crack angle of such a crack increases, resulting in an increase of energy needed for further development of the crack.

Figure 122: Contact area of a continuous crack and regions of compressive crushing in aggregate-cement paste interfaces (Bongers [1998b])

Softening effect due to mesoscopic damage Whereas triaxial pre-loadings have an advantegeous influence on peak stress in the case of plane strain and triaxial reloading, in the case of uniaxial reloading a decrease in load hearing capacity due to the same preloading is observed. Besides the profitable hardening effect caused by crushing of the aggregate-cement paste interfaces, as described above, some tensile and shear damage at the mesoscopic level is induced under high multiaxial compressive stresses, i.e. softening of aggregate-cement paste interfaces. Tuis mesoscopic softening was already implemented in the mesoscopic numerical model by Vonk [1992], which showed that implementation of this softening at the mesoscopic level suffices to describe macroscopie softening of concrete in uniaxial compression quite well.

137

Chapter 7: lnjluence of damage from previous loading

Combined mesoscopic hardening and softening Suppose that most of the nonlinear pre-peak deformations are concentrated at the aggregate-cement paste interfacial zones and that (mesoscopic) hardening and softening of the failure surface of these interfaces occurs as indicated in Figure 123, as proposed by Bongers [1994]. Then the stress-deformation behaviour of concrete in a monotonie multiaxial test can be explained as follows. During the first stage of loading no nonlinear behaviour occurs. Tuis means that at the aggregate-cement paste interfaces no significant crushing, shear and/or tensile damage has taken place yet and that in this stage of loading the 'elastic' properties of the material constituents determine the stress-deformation behaviour. With increased loading inelastic deformations occur, in combination with the occurrence of inelastic compaction indicating a compressive failure of the interfaces. At peak stress the combination mesoscopic hardening/softening of all interfaces has reached a critica! stage, resulting in the maximum load at the macroscopie level. Right after peak stress mesoscopic softening behaviour dominates in the aggregate-cement paste interfaces and this mesoscopic softening damage has become so large that continuous macroscopie cracks are initiated. Tuis results in a localisation of deformations in macroscopie cracks and softening behaviour at the macroscopie level.

... " ....... " ... " ... """""""" ___ .., _____ "_"""" __ --.... ,.,,--

'

Figure 123: Hardening and softening of the failure surface of an aggregatecement paste interface (after Bongers {1994])

A similar line of thought has been described in detail by Bongers [1998b]. See Figure 124, where a monotonie multiaxial compression test is analysed by dividing the stress-deformation behaviour in four stages:

138


:······· ··········:········-200 ················ ................................. .

4 2 wlateral [mm]

0

II 111

-2 -4 waxial [mm]

Figure 124: Stages in a multiaxial compression test (Bongers [1998b])

-6

Stage 1: Elastic stage, where the influence of growth of already present and the formation of new mesoscopic cracks is negligible.

Stage II: Inelastic hardening stage, where crushing of the interfacial bond zones at the mesoscopic level results in macroscopie hardening behaviour accompanied by inelastic volume compaction.

Stage 111: Around peak stress, where mesoscopic cracks are formed into continuous cracks and still a strengthening effect occurs due to crack arrest, which are highly dependent on the relative strength and size of the aggregates and their mutual distance.

Stage IV: Softening stage, where macroscopie cracks determine the macroscopie stress-deformation behaviour and gradually loose their load hearing capacity.

A numerical model developed by Bongers [1998], in which both mesoscopic hardening and softening mechanisms as described in this section have been incorporated, appears to be able to describe the behaviour of concrete in multiaxial compression very well.

139


In the case of hydrostatic preloading, hardening behaviour dominates and the effect of the crushing of interfaces results in an inflation of the failure surface of these crushed interfaces. After unloading and re-loading in, for example, a proportional plane strain test this leads to amore linear pre-peak behaviour, because it takes longer to reach the failure surface where nonlinear hardening behaviour is taken up again, and a higher peak stress. In uniaxial loading most interface loading paths are close to the axis 0'0 =O (zero stress normal to the interface), i.e. compression/shear and tension/shear, as a result hardening is hardly observed in uniaxial tests and softening dominates strongly. This is also the reason why Vonk's numerical model [1992] was able to simulate uniaxial compression tests very well without taking mesoscopic hardening ( crushing) into account. When loading a specimen uniaxially after triaxial pre-loading, part of the softening has already taken place resulting in a decrease of peak stress. In Table 4 the types of damage in concrete under multiaxial compressive stresses are summarised.

Table 4: Types of damage occurring in multiaxial compression

MACROSCOPIC DAMAGE

After peak stress macroscopie cracks are formed, which gradually loose their load hearing capacity

MESOSCOPIC DAMAGE

Due to crushing of the Due to tensile and shear aggregate-cement paste cracking m the interfaces under a aggregate-cement paste compressive load a interfaces and in the strengthening effect cement paste itself a occurs at the softening effect occurs mesosco ic level at the mesosco ic level

7 A. Macroscopie interpretatlon of the influence of damage induced at the mesoscopic level

Because in the pre-peak region the stress-deformation behaviour of concrete has been found to be to a large extent independent of the specimen size, the mesoscopic damage induced to the material during this stage of loading can be regarded as damage distributed equally well over the entire specimen volume. This means, when considering the effecs of mesoscopic hardening and softening at the macroscopie level, that the observed nonlinear macroscopie behaviour in the pre-peak region might be expressed in terms of

140

Chapter 7: Influence of damage /rom previous loading

an integration over the entire volume of the mesoscopic deforrnations due to mesoscopic hardening and softening. In this section therefore a macroscopie interpretation of these two types of mesoscopic damage and the way in which they affect the macroscopie behaviour is outlined.

Levels of equal total mesoscopic hardening and softening damage Because crushing of the aggregate-cement paste interfaces perpendicular to a compressive load is the main cause of hardening behaviour, the hardening effect in a certain direction is closely related to the compressive stress in that direction. Furtherrnore, the presence of confining stresses obstructs the development of mesoscopic softening damage and meanwhile increases the amount of crushing of interfacial zones. Therefore a certain level of total hardening damage (i.e. the total of the hardening darnage in all bond zones within the volume) is reached at a lower axial stress when the amount of confining stress increases. Close to the hydrostatic axis crushing is accomplished most easily. The assurnption of levels of equal total hardening damage in axial direction as shown in Figure 125 (two-dimensional representation) seems justified, as an approximate qualitative indication of hardening damage. Note that in this figure these lines are only drawn in the region cr1>cr3, because crushing of the aggregate-cement paste interfaces in fact is directional dependent: closer to the hydrostatic axis mesoscopic hardening darnage will be distributed more equally around the aggregates, while in the region closer the the cr1-axis most crushing will take place in the interfaces perpendicular to the cr 1-direction. Whereas it is easy to imagine that levels of equal total hardening are closely related to the axial stress, it is also imaginable that levels of equal total mesoscopic softening damage are more closely correlated with the ratio of lateral and axial stress. As mentioned before, if the confining stress decreases, lateral expansion of the material becomes easier and hence a certain amount of shear and tensile cracking at the mesoscopic level can be accomplished at a lower level of axial stress. Close to peak stress mesoscopic softening damage increases rapidly. Even under pure hydrostatic compression some softening damage might occur due to the heterogeneous structure of concrete, which causes the levels of equal total softening darnage to be approximately located in two-dimensional stress space as indicated in Figure 126.

141


onset of mesoscopic hardening

increase of mesoscopic , .. · · hydrostatic axis

hardeningdamage .-~ ( crushing) /

Figure 125: Qualitative representation oflines of equal total mesoscopic hardening damage in axial direction (2D case)

onset of mesoscopic softening

increase of mesoscopic softening damage

(•~g

, , , ·, hydrostatic axis

.-·-·-·-·-·-"c.;;ventional plane strain ·-·-·--·- failure surface

Figure 12 6: Qualitative representation of lines of equal total mesoscopic softening damage in axial direction (2D case)

142


Measured macroscopie quantities representative of total mesoscopic hardening and softening When. for the sake of simplity of macroscopie interpretation, the correlation between the damage induced at the mesoscopic level with its location (as a result of the direction and magnitude of the stresses acting on the material) is disregarded, it can be conceived that mesoscopic hardening is closely related to a volume change of the material. See Figure 127 (left), where a schematic representation is shown of an aggregate embedded in a cement paste volume.

Cement paste Interfacial zone

Aggregate

Initial situation

l

Interface crushing due to volume change

1 .................................... . .

. . : .................................. :

Interface shearing and opening due to shape change

Figure 12 7: Idealised concrete volume element and two types of mesoscopic damage related to changes in the elements' volume and shape

143


If the volume of the material decreases without a change in shape of the original volume (and if it is assumed that all nonlinear deformations take place within the aggregate-cement paste interface) crushing of the bond zones without shear and tensile cracking within these zones is mostly responsible for this pure volume change. Likewise, if the shape of this volume element changes, as is shown in Figure 127 (right), this is mainly a result of shear and tensile deformations within the bond zone. However, the latter deformations may be accompanied by a very · small amount of interface crushing at the upper and lower surfaces of the aggregate, depending on the ratio between lateral and axial stress. From all different plane strain stress experiments carried out on 1 OOmm normal strength concrete cubes, following the stress paths as shown in Figure 111, the inelastic hydrostatic and deviatoric strains have been determined, as well as the energy required to achieve these nonlinear deformations. Following the above line of thought, the inelastic hydrostatic strain and energy can be regarded as a measure of mesoscopic hardening damage and the inelastic deviatoric strain and energy as a measure of mesoscopic softening damage. In Figure 128 and Figure 129 levels of equal hydrostatic and deviatoric strains and energies are plotted in plane strain stress space. The qualitative levels of equal total hardening and softening damage as presented in the previous subsection are also drawn in these figures. The depiction of damage levels in Figure 125, Figure 126, Figure 128 and Figure 129 indicate that at every point in the stress space a different combination of mesoscopic hardening and softening damage exists. The fact that every point in the stress space corresponds with a different combination of the two types of damage is thought now to be the main cause of loading path dependent behaviour, as observed in the experiments in the previous sections. How this loading path dependent behaviour could be related to the two types of damage will be outlined in the following where the influence of mesoscopic hardening and softening damage on failure stress and loading path followed is considered.

Failure surface Normally the failure surface for concrete is determined by carrying out a series of multiaxial compressive tests in which always the same type of stress path is followed, e.g. proportional stresses. The line or small zone through the measured peak stresses then is considered to be the authentic failure line or surf ace of the material.

144


But the results from the experiments in which a pre~peak pre-loading bas been applied indicate that the failure surface is not the conventionally assumed statie shape in stress space, hut that it is a shape that evolves constantly with increasing damage at the mesoscopic level.

E 0, inelastic [-] -0.001

-0.001

-0.002 60 -0.002

-0.002

50 -0.003

ro -0.004

c. ! -0.004

~ 40-i ''

-0.004

0 ' ..

-0.005 ~ ' ' \

{-..> : <: ~: ': ',~ ~~ ~ \ -0.005 3 ... :>N;~t··

-0.006

·0.007

20 ~ -0.007

~ -0.007

-0.008 -110 -100 -90 -80 -70 -60 -60 -40 -30 -20

-0.009

-0.009 O' O [MPa]

Y 0, inelastic M 0.012

0.011

0.010

5 0.009

ro 0.008 0.

~ 4 0.007

0 ~

0.006

30 '

0.005

0.004

20 ~ 0.003 ~

0.002

-110 -100 -90 -80 -70 -60 -50 -40 -30 -20 0.001

'o.ooo O' O [MPa]

Figure 128: Inelastic hydrostatic and deviatoric strains as measured in plane strain experiments on 1 OOmm cubes following different stress paths up to peak stress. The upper boundary of the shaded surf aces corresponds with the conventional plane strain failure surface.

145

'iii' a..


Hydrostatic inelastic damage energy [MPa]

60

50

30

20

-110 -100 .go .ao .70 -eo .50 -40 -so -20

O' O [MPa]

Deviatoric inelastic damage energy [MPa] :f _ ' ' ' ' ' ' ' ::a: 40~ -· "-· ~0 I ____ " ••

""'

20

-110 • 100 -90 -80 -70 -60 -50 -40 ·30 ·20

O' O [MPa]

0.35

0.30

0.25

0.15

0.10

0.05

0.00

0.10

0.05

0.00

Figure 129: lnelastic hydrostatic and deviatoric damage energies as measured in plane strain experiments on 1 OOmm cubes following different stress paths up to peak stress. The upper boundary of the shaded surfaces corresponds with the conventional plane strain failure surface.

From the test results discussed and the measures of total mesoscopic hardening and softening introduced it is deduced that increasing mesoscopic softening damage leads to a shrinkage of the macroscopie failure surface,

146


mostly in those regions where mesoscopic tensile and shear cracking can develop relatively easily (like uniaxial compression tests after a triaxial preloading), while increasing mesoscopic hardening damage leads to an expansion of the failure surface, especially in the regions where mesoscopic tensile and shear cracking are obstructed, i.e. the presence of lateral stresses, like plain strain tests after a triaxial pre-loading (these are the regions below and above the 'rotation point' in Figure 130 respectively). The shape of the failure surface then always is a function of the highest levels of mesoscopic hardening and softening reached in a particular case. Tuis is qualitatively shown in Figure 130.

-ns ll. :E -

Increasing mesoscopic softening and hardening damage during preloading

@ ® expansion of the macroscopie failure surface due to mesoscopic hardening -------------------------

_,,,,,,. ,.,.@ __ } ® _,,,,,,. -- -_,,,,,,. -

'rotation point' ~...::;:::... --

-- ,;::::r- .......

shrinkage of the macroscopie failure surface due to mesoscopic softening

er 1 [MPa]

Figure 130: Movingfailure surface as a fanction of mesoscopic hardening and softening levels (2D case), at stress point B higher levels of mesoscopic

hardening and softening damage are reached

Figure 131 shows that from this point of view the failure surface, as determined by for example three 'deviatoric' plane strain tests, is in fact a line connecting points on three different failure surfaces, each corresponding with a specific pair of values of measures of mesoscopic hardening and softening darnage. It means that in principle to each cross section of lines of equal mesoscopic hardening and softening darnage (see Figure 125 and Figure 126) in the zone where peak stresses are attained, a particular failure line (or surface) belongs.

147

Chapter 7: /nfluence of damage from previous loading

However, the concept of the 'moving' failure surface can only be valid if the rotation point, indicated in Figure 130, is situated above the cr3=0 axis. lf this would not be the case, mesoscopic hardening would dominate even in uniaxial experiments. But it is easy to conceive why this rotation point can not be situated below the cr3=0 axis: in this region (at least) in one direction a tensile stress occurs. Tuis means that mesoscopic softening dominates strongly and the amount of crushing is negligible. Therefore here only a shrinkage of the failure surface can occur at the macroscopie level. Though the rotation point most likely is not a fixed point (or line in the 3D case) in stress space either, it can never be situated outside the domain of multiaxial compression.

CJ' 1 [MPa]

Figure 131: Apparent failure surf ace composed of three different failure surfaces corresponding with different levels of hardening (2D case). With increasing levels ofmesoscopic damage the macroscopicfailure surface

develops from state A to C (corresponding with stress paths A to C)

Loading path dependency As described above the failure surface expands or shrinks dependent upon the combination of the amount of mesoscopic hardening and softening attained in a particular case. Using this concept, the behaviour in the pre-loading tests described in section 7.2.2. can be evaluated as shown in Figure 132 and Figure 133. In the case of plane strain reloading, the level of mesoscopic hardening reached causes the macroscopie failure surface to expand resulting in a higher peak stress (Figure 132). In the case of uniaxial reloading, the

148


level reached of mesoscopic softening of identical pre-loading becomes the dominant factor, i.e. the macroscopie failure surface now shrinks and thus leads toa lower peak stress (Figure 133).

C")

b

er 1 [MPa]

Figure 132: Higher peak stress as a result of mesoscopic hardening during preloading, 'A ' denotes a regular proportional plane strain stress path and

the failure surface at which peak stress is reached, 'B ' denotes a plane strain pre-loading stress path and the failure surf ace at which peak stress is

reached

.... ns Q. :E .....

C")

b @ ·········································®

er 1 [MPa]

Figure 133: Lower peak stress as a result ofmesoscopic softening during preloading, 'A ' denotes a regular uniaxial compression, 'B' denotes a

uniaxial test after a plane strain pre-loading. 'A ' and 'B ' also denote the failure surfaces at which peak stress is reached in each case.

149


7.S. Loading path d.ependency due to combined meso-- and macroscopie damage

Whereas the macroscopie failure surface can shrink and expand due to mesoscopic damage in the pre-peak region as outlined above, in the post-peak region the failure surface only can shrink. The analytica} model in Chapter 6 explicitely showed this. In this section it will be shown that this post-peak shrinkage of the failure surface may be influenced in certain cases by mesoscopic damage inflicted before peak stress. As can be seen in Figure 11, the second peak in a triaxial rotation experiment can be significantly lower than the first peak, whereas the shape of the descending branch does not seem to change. Because in the case of this experiment (Figure 11) relatively low triaxial stresses occur, it is concluded that in the case of this rotation test no significant damage at the mesoscopic level is involved. Tuis can also be concluded from the rotation tests as a part of the research project at hand, where loading before rotation is continued untiljust after the peak and almost no decrease in peak stress after rotation is found. Tuis means that the loading path dependency found in Figure 11 is a result of macroscopie (local) damage instead of mesoscopic, distributed damage. The decrease in peak stress in Figure 11 can be explained easily. If a stress path reaches the failure surface this failure surface will start to shrink, due to the gradual loss of load hearing capacity of the macroscopie cracks. Refer also to the analytical model in Chapter 6. Tuis shrinkage of the macroscopie failure surface is accompanied by a descending branch in the stressdeformation diagram. Tuis macroscopie softening continues until the residual stress surface, i.e. the 'final failure surface', is reached. The analytica! shear crack model described before shows how this shrinkage advances with increasing macroscopie crack deformations in the case of plane strain tests. This shrinkage is relatively insensitive to mesoscopic damage, because the model is able to describe macroscopie softening behaviour both at low and high stress levels (i.e. low and high levels of mesoscopic damage) and because the post-peak curves of proportional tests both at low and high stress levels are almost identical. Only in the case of high multiaxial pre-loading (Figure 115) damage at the mesoscopic level increases the brittleness of the macroscopie descending branch. One should bear in mind that at least in the first part of the descending branch macroscopie cracks are formed by growing through mesoscopic cracks already present. It is therefore logica! that the behaviour of macroscopie cracks changes if the mesoscopic cracks already present are developed further. This signifies that the behaviour of

150


macroscopie cracks in extreme cases can also be influenced by damage at the mesoscopic level. If macroscopie cracks are already present in a specimen, shrinkage of the macroscopie failure surface towards the residual surface only occurs in those regions of stress space where the already present macrocracks are situated in a favourable position for further crack growth (note that this is obviously true fora regular proportional test). This is for example not true in the case of a plane strain rotation test, in which a V -shaped macrocrack pattem has to be formed perpendi.cular to the existing V-shaped macrocrack pattem (Figure 11 ). The only possible loading path dependency that might occur in this type of test would be due to mesoscopic damage, e.g. in the case of a plane strain rotation test with relatively high confining stress before rotation.

7.6. Final remark.s.

The actual quantitative influence of mesoscopic hardening and softening damage as outlined in this chapter can not be gained from experimental research in which only macroscopie quantities are determined, i.e. forces and deformations. Without this lack of ascertainment it would be improper to regard the concept of mesoscopic hardening and softening levels outlined before as 'material behaviour'. The concept carne out of the process of interpretation and understanding of the phenomena controlling loading path dependent behaviour. It proved to be a convenient conceptual tool to comprehend macroscopie concrete behaviour. Nevertheless, the numerical model by Bongers [1998] -a quantitative model- supports the ideas presented here on the influence of damage at the mesoscopic level, inflicted in the material in the pre-peak region.

151

Chapter 8: Summary and conclusions

8. Summary and Conclusions

Background of this research The present research is part of an exhaustive investigation into the mechanica} behaviour of concrete in compressive loading. It aims at answering some questions (described in Chapter 2) that were not answered in the research projects carried out earlier at Eindhoven University of Technology (Van Mier [1984), Vonk [1992)). While in this thesis the results are described from tests on a high-strength concrete, emphasis is still laid on the behaviour of a normal-strength gravel concrete under multiaxial stresses.

New laboratory tools Because of the long history of concrete research at EUT, both the 'true triaxial' loading apparatus and the measuring equipment were already available at the start of the present research project. However, the application of high multiaxial stress states and the use of a high-strength concrete were the reason to increase the capacity of the loading apparatus. One of the 2MN loading axes therefore has been replaced by a 5MN axis. The desire to apply different multiaxial stress paths and compare them at different points in stress space made a more accurate test control method necessary. A software program has been developed that can handle all compressive stress paths.

'Standard' test results In the case of the normal-strength concrete a large number of experiments with various loading conditions (uniaxial, biaxial, plane strain and triaxial compression) is carried out. In all these tests loading platens with a teflon/grease intermediate layer is used as applied before by Vonk [1992]. The experimental results obtained from the experiments are a solid base for numerical simulations since the laboratory conditions have been identical for all tests. Similar experiments have been carried out on a high-strength concrete. In general, disregarding the quantitative differences, the failure behaviour of both types of concrete is very similar. Some uniaxial tension on the normal-strength concrete have been carried out to provide further data for numerical simulations.

Increase ofboundary friction Uniaxial and multiaxial compression tests have been carried out using both teflon loading platens and dry steel loading platens, both on normal- and high-strength concrete. The results confirm many observations done in the

152


past: peak stress and deformations increase with increasing lateral boundary restraint, and the softening branch of the stress-deformation curve becomes less steep. The influence of additional triaxial stresses in (parts of) the specimen introduced by boundary friction is smaller at higher multiaxial stress states because the magnitude of the frictional stresses becomes relatively smaller.

Non-uniformity of deformations As observed by Vonk [1992] in uniaxial compression non-uniform deformations occur. Due to the heterogeneous nature of concrete, specimens tend to fail according to a partial failure mechanism causing a rotation of the loaded specimen boundaries. Vonk showed that this rotation is highly dependent on the rotational stiffhess of the loading apparatus. In the present research uniaxial experiments have been carried out in which not only the specimen deformations but also the deformations of the entire loading axis have been measured. From the test results it is concluded that, while a rotation of the load cell has been observed too, non-uniform deformations appear to be stimulated further mainly by a rotation of the piston. To minimise the influence of this rotation, the distance between specimen and hydraulic cylinder should be kept as small as possible.

Formation and behaviour ofmacroscopic cracks It is known from previous research that after peak stress a localisation of deformations in macroscopie cracks occurs in multiaxial compression. Both in plane strain and triaxial compression tests the development of macroscopie cracks has been recorded using a vacuum impregnation technique (Vonk [1992]). It bas been shown that macroscopie cracks start to form at specimen corners right after peak stress. The geometry of the loading apparatus is found to be of major importance for the initiation of the cracks and has a significant influence on the observed stress-displacement diagrams. During the steep part of the softening branch the cracks grow to form a complete failure pattem. During this stage opening of the cracks dominates the stressdisplacement behaviour. After the final failure pattem is completely formed, a curvature of the stress-displacement diagram towards the residual stress level is observed. In this final stage of the descending branch, sliding of the crack surfaces dominates the observed macroscopie behaviour. Tuis has been showh both experimentally and using an analytica} model for macroscopie crack behaviour. The model results indicate that a numerical modelling of macroscopie cracks in concrete using discrete interface elements can give acceptable results.

153


With higher multiaxial stress states macroscopie cracks are found to be less wide and tend to run through aggregates more often instead of around them.

Post-peak continuum contribution While in the pre-peak region almost identical stress-strain relations are found for specimens with different geometries, in the post-peak region -due to localisation of deformations- almost identical stress-deformation curves are observed. In previous research contradictive results were found in the case of uniaxial compression: while some (e.g. Van Mier [1984]) observed a complete localisation of deformations in macroscopie cracks, others (e.g. Vonk [1992]) concluded that the more or less uncracked continuum specimen parts still contributed to the post-peak load hearing capacity. By carrying out multiaxial experiments in which the post-peak continuum deformations have been measured, it bas been shown that indeed the continuum specimen parts still contribute to the post-peak load hearing capacity. The extent of this contribution seems closely related to the casting quality of the concrete, in particular the quality of the aggregate-cement paste interfaces.

Damage from previous loading From Van Mier's rotation tests [1984] it is known that damage at the macroscopie level can have a significant influence on the stress-deformation behaviour of concrete. A large number of multiaxial stress paths has been carried out to detect wether concrete behaviour is dependent on damage induced at the mesoscopic level, i.e. damage induced in the pre-peak region. When applying multiaxial pre-peak pre-loadings to a specimen, a significant influence of the accumulated mesoscopic damage is observed. Surprisingly, even an increase of peak stress can be observed with increasing damage. But in the case of stress paths with monotonically increasing loads it is found that the influence of variations in these loading paths is not significant. Tuis means that there is no need to take the influence of load histories into account explicitly in the case of regular structural engineering applications.

Mesoscopic hardening effect In Vonk's numerical (mesoscopic) model for concrete in uniaxial compression [1992] the macroscopie stress-deformation behaviour is a result of mesoscopic softening laws at the aggregate-cement paste interfaces. The model appeared to give satisfactory results in the case of uniaxial compression. However, the present pre-loading experiments and numerical results by Bongers [1998] indicate that besides mesoscopic softening another phenomenon plays a role in multiaxial compression. In this thesis a

154


mesoscopic hardening effect as a result of the crushing of aggregate-cement paste interfaces is introduced. Mesoscopic softening and hardening appear to be the two factors determining concrete's macroscopie mechanical behaviour under multiaxial stresses. The two phenomena have been linked to measured macroscopie quantities and test results have been explained on the basis of this new insight.

Further research As mentioned before, at present a mesoscopic numerical model is being developed at EUT by ir. J.P.W. Bongers. The model parameters have been determined from the experiments described in this thesis. The incorporation of both mesoscopic softening and hardening in the model appears to lead to promising results. Furthermore, it seems that Bonger' s model is sui tab Ie for other types of concrete too by adapting the model parameters. A comprehensive experimental research into the influence of the composition of concrete on the behaviour in compression as a next step in the research program therefore seems appropriate. In the very near future Bonger's results will be implemented in the finite element code 'FEMMASSE' and will therefore become available to structural engineers in practice.

155

Re/er ene es

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163

List of symbols

List of symbols

The following list shows the syrnbols used in this thesis. Note that the section where the first use of the syrnbol occurs is given for those syrnbols that are applicable only in a specific part ofthis thesis.

Symbol Description

Bs shear band width in Murakami 's model (section 6.3.2.)

E Young's modulus Gr fracture energy

a, b coefficient determining the shape of Banger' s failure surface (section 6.3.2.)

c1, c2 coefficient determining the shape of the tension softening model by Reinhardt et al. (section 4.2.)

d specimen width r b (uniaxial) compressive design strength according

to VBC 1995 ( peak stress in uniaxial compression tests using

teflon loading platens (in this research program) ~ peak stress in uniaxial tensile tests (in this

research program) h specimen height n number of macroscopie shear cracks in a

specimen (section 6.3.2.) r ratio of cr3/cr1 (section 6.3.2.)

w1, w2, w3 specimen displacements in directions 1, 2 and 3 respectively

wn normal shear crack displacement wt tangential shear crack displacement

a softening parameter (section 6.3.2.) ~ non-dimensional shear band width in

Murakami's model (section 6.3.2.) ot tangential displacement within a shear band in

Murakami's model (section 6.3.2.) Eo hydrostatic strain

164

Unit

mm

MPa N/mm

mm MPa

MPa

MPa

mm

mm

mm mm

mm

List of symbols

E1, E2, E3 specimen strains in directions 1, 2 and 3 respectively

Yo deviatoric strain

mÎt averaged shear strain over shear band width in MPa Murakami's model (section 6.3.2.)

K summation of normal and tangential shear crack mm displacements (section 6.3.2.)

e shear crack angle rad

cro hydrostatic stress MPa

cr', cr2, cr3 specimen stresses in directions 1, 2 and 3 MPa respectively

crn normal stress acting on a shear crack MPa

to deviatoric stress MPa tl tangential stress acting on a shear crack MPa

165

Samenvatting

Samenvatting

Achtergrond van dit onderzoek (hoofdstuk 2) Het onderhavige onderzoek maakt deel uit van een uitgebreide studie naar het mechanische gedrag van beton onder drukbelastingen. Het heeft tot doel enkele vragen te beantwoorden die in eerdere onderzoeksprojecten uitgevoerd aan de TU Eindhoven onbeantwoord zijn gebleven. (Van Mier [1984], Vonk [1992]). Hoewel in dit proefschrift ook resultaten zijn beschreven van proeven op een hoge( re )-sterkte beton, ligt de nadruk nog steeds op normale-sterkte grindbeton onder meerassige spanningen.

Nieuwe laboratorium apparatuur (hoofdstuk 3) Door de lange geschiedenis van betononderzoek aan de TUE, waren zowel de drie-assige proefopstelling als ook de meetapparatuur reeds beschikbaar bij de aanvang van het onderhavige onderzoeksproject. De toepassing van hoge meerassige spanningstoestanden en het gebruik van een hoge-sterkte beton zijn echter de redenen geweest om de capaciteit van de belastingapparatuur te vergroten. Eén van de 2 MN belasting-assen is daartoe vervangen door een 5MN as. De wens om verschillende meerassige spanningspaden toe te passen en deze onderling te vergelijken ter plaatse van van te voren gedefinieerde punten in de spanningsruimte maakten een nauwkeuriger sturing van de proefopstelling noodzakelijk. Een computerprogramma is ontwikkeld dat alle (druk-)belastingpaden geautomatiseerd kan laten uitvoeren.

'Standaard' proefresultaten (hoofdstuk 4) In het geval van het normale-sterkte beton is een groot aantal proeven uitgevoerd in diverse belastingsituaties ( éénassige, twee-assige, 'plane strain' (vlakke vermormingstoestand) en drie-assige druk). In al deze proeven zijn belastingplaten gebruikt met een teflon/smeemiddel tussenlaag zoals eerder toegepast door Vonk [1992]. De experimentele resultaten verkregen uit deze experimenten vormen een solide basis voor numerieke simulaties omdat deze alle zijn uitgevoerd onder identieke laboratorium-omstandigheden. Soortgelijke proeven zijn uitgevoerd op een hoge-sterkte beton. In het algemeen, afgezien van de kwantitatieve verschillen, blijkt het bezwijkgedrag van de twee beproefde betonsoorten hetzelfde te zijn. Enkele éénassige trekproeven op het normale-sterkte beton zijn uitgevoerd om de diversiteit aan gegevens voor numerieke simulaties te vergroten.

166

Samenvatting

Wrijving aan de proefstukranden (hoofdstuk 5) Eén- en meerassige drukproeven zijn uitgevoerd met stalen belastingplaten zowel met als zonder teflon tussenlaag, op beide betonsoorten. De resultaten bevestigen een groot aantal proefresultaten uit het verleden: de bezwijksterkte en -vervormingen nemen toe als de laterale wrijving aan de proefstukranden toeneemt, en de dalende tak van de spanning-vervormingscurve wordt minder steil. De invloed van additionele drie-assige spanningen in (delen van) het proefstuk, geïntroduceerd door deze wrijving, is kleiner in geval van hogere meerassige spanningstoestanden omdat de grootte van deze wrijvingsspanningen relatief kleiner wordt.

Niet-uniforme vervormingen Zoals door Vonk [1992] reeds geobserveerd treden in éénassige drukproeven niet-uniforme vervormingen op. Ten gevolge van de heterogene structuur van beton hebben proefstukken de neiging om via een partieel (onvolledig) bezwijkmechanisme te bezwijken, wat resulteert in een rotatie van de belaste proefstukranden. Vonk heeft laten zien dat deze rotatie in hoge mate afhankelijk is van de rotatiestijfheid van de proefopstelling. In het onderhavige onderzoek zijn proeven uitgevoerd waarin niet alleen de proefstukvervormingen maar ook de vervormingen van het gehele belastingframe zijn gemeten. Uit de resultaten wordt geconcludeerd dat, hoewel ook een rotatie van de krachtmeetdoos is gemeten, niet-uniforme vervormingen met name gestimuleerd worden door een rotatie van de drukcylinder. Om de invloed hiervan te minimaliseren, dient de afstand tussen proefstuk en hydraulische cylinder zo klein mogelijk te worden gehouden.

Vorming en gedrag van macroscopische scheuren (hoofdstuk 6) Uit voorgaand onderzoek is bekend dat na de top in het spanningvervormings-diagram de vervormingen localiseren in macroscopische scheuren. Zowel in plane strain als drie-assige drukproeven is de ontwikkeling van het macroscopische scheurpatroon vastgelegd met behulp van een vacuum-impregnatie techniek (Vonk [1992]). Er is gebleken dat macroscopische scheuren direct na de top aan de proefstukranden worden geïnitieerd. De geometrie van de proefopstelling blijkt van grote invloed te zijn op de initiatie van macroscopische scheuren en heeft een significante invloed op het waargenomen spanning-verplaatsings-diagram. Gedurende de meest steile tak van de softening-curve groeien de scheuren uit tot een volledig bezwijkpatroon. In deze fase domineert het openen van de scheuren het spanning-verplaatsings-diagram. Nadat het uiteindelijke bezwijkpatroon

167

Samenvatting

is gevormd buigt de spanning-verplaatsings-curve geleidelijk af naar het restspanning-niveau. In deze laatste fase van de dalende tak domineert het over elkaar glijden van scheuroppervlakken het waargenomen macroscopische gedrag. Dit is zowel experimenteel als met behulp van een analytisch model voor het gedrag van macroscopische scheuren aangetoond. De modelresultaten geven aan dat een numerieke modellering van macroscopische scheuren door middel van discrete interface elementen tot acceptabele resultaten kan leiden. Bij hogere meerassige spanningstoestanden blijken de macroscopische scheuren minder ver te openen en de neiging te vertonen om door het toeslagmateriaal te groeien in plaats van er om heen.

Bijdrage van continuum delen na de top Terwijl in het pre-peak gebied bijna identieke spanning-rek relaties worden gevonden voor verschillende proefstuk afmetingen, worden na de top -ten gevolge van de localisatie van vervormingen- bijna identieke spanningverplaatsings-relaties waargenomen. In voorgaand onderzoek werden tegenstrijdige resultaten verkregen in het geval van éénassige druk: sommigen (bijv. Van Mier [1984]) observeerden een volledige localisatie van vervormingen in macroscopische scheuren, terwijl anderen (bijv. Vonk [1992]) concludeerden dat de -min of meer- ongescheurde continuum proefstukdelen na de top nog steeds bijdragen aan de draagcapaciteit. Door meerassige drukproeven uit te voeren waarin de post•peak continuum vervormingen na de top worden gemeten, is duidelijk gemaakt dat de continuum proefstukdelen na de top inderdaad nog aan bijdrage leveren aan de totale draagkracht. De grootte van deze bijdrage lijkt sterk gerelateerd aan de kwaliteit van de proefstukvervaardiging, met name de kwaliteit van de hechtvlakken tussen cement-matrix en toeslagkorrels.

Schade uit voorgaande belastingen (hoofdstuk 7) Uit Van Mier's rotatie proeven [1984] is bekend dat schade op het macroscopische niveau een significante invloed kan hebben op het spanningvervormings-gedrag van beton. Een groot aantal meerassige spanningspaden is uitgevoerd om te onderzoeken of betongedrag al dan niet afhankelijk is van schade toegebracht op het mesoscopische niveau c.q. schade toegebracht voor de top. Wanneer meerassige pre-peak voorbelastingen worden uitgeoefend wordt een significante invloed waargenomen van de opgebouwde mesoscopische schade. Er kan zelfs een toename van de bezwijkspanning worden waargenomen bij een toenemende hoeveelheid toegebrachte schade. Maar in het geval van belastingpaden met monotoon

168

Samenvatting

stijgende belastingen wordt gevonden dat de invloed van variaties in deze belastingpaden niet significant is. Dit betekent dat er geen noodzaak bestaat om in de reguliere constructieve praktijk-toepassingen rekening te houden met de invloed van belastinghistories.

Mesoscopisch verstevigings-ejfect In Vonk's numerieke (mesoscopische) model voor beton onder éénassige drukbelastingen [1992] is de macroscopische spanning-vervormings-curve het resultaat van mesoscopische softening-wetten die gelden ter plaatse van de hechtvlakken tussen cement-matrix en toeslagkorrels. Het model bleek in het geval van éénassige druk bevredigende resultaten te geven. Echter, de onderhavige voorbelasting-proeven en numerieke resultaten verkregen door Bongers [1998] geven aan dat naast mesoscopische softening een ander fenomeen een rol speelt in het geval van meerassige druk. In dit proefschrift wordt een mesoscopisch verstevigings-effect ('hardening') ten gevolge van verbrijzeling van genoemde hechtvlakken geïntroduceerd. Mesoscopische softening en hardening blijken de twee factoren te zijn die het macroscopische mechanische gedrag van beton onder meerassige druk bepalen. De twee fenomenen zijn gerelateerd aan gemeten macroscopische grootheden en proefresultaten zijn verklaard op basis van dit nieuwe inzicht.

Verder onderzoek Zoals al eerder vermeldt wordt op dit moment aan de TU Eindhoven een mesoscopisch numeriek model ontwikkeld door ir. J.P.W. Bongers. De modelparameters zijn bepaald op basis van de experimentele resultaten beschreven in dit proefschrift. De verwerking van zowel mesoscopische softening als hardening in het model lijken tot veelbelovende resultaten te leiden. Daarnaast lijkt Bonger's model geschikt om het gedrag van meerdere betonsoorten te beschrijven door aanpassing van de modelparameters. Een uitgebreid experimenteel onderzoek naar de invloed van de betonsamenstelling als eerstvolgende stap in het onderzoeksprogramma lijkt daarom de meest zinvolle. In de nabije toekomst zullen Bonger' s resultaten worden geïmplementeerd in het eindige elementen pakket 'FEMMASSE' en daarmee toegankelijk worden voor de contructie-praktijk.

169

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nr37 Design Research in the Netherlands editors: Prof. dr R.M.Oxman, Prof. dr ir. M.F.Th. Bax, lr H.H. Achten

nr38 Communication in the Building Indus try Bauke de Vries

nr 39 Optimaal dimensioneren van gelaste plaatliggers

nr40 Huisvesting en overwinning van armoede dr.ir. P.H. Thung en dr.ir. P. Beekman (red.)

nr41 Urban Habitat: The environment oftomorrow George G. van der Meulen, Peter A. Erkelens

nr42 A typology of joints John C.M. Olie

nr43 Modeling constraints-based chokes for leisure mobility planning Marcus P. Stemerding

nr44 Activity-based travel demand modeling D. Ettema

nr45 Wind-induced pressure fluctuations on building facades Chris Geurts

nr46 Generic Representations Henri Achten

nr47 Johann Santini Aichel Dirk De Meyer

Documents

Concrete behaviour in multiaxial compression : … BEHA VIOUR IN MULTIAXIAL COMPRESSION Experimental Research Proefschrift ter verkrjging van de graad van doctor aan de Technische