delft university of technology university of minho · tu-delft report no. 03.21.1.31.07 tno-bouw...

Delft University of Technology University of Minho Faculty of Civil Engineering Department of Civil Engineering

AN ANISOTROPIC MACRO-MODEL FOR MASONRY PLATES AND SHELLS: IMPLEMENTATION AND VALIDATION

Author : P. B. LOURENÇO Date : February 1997

TU-DELFT report no. 03.21.1.31.07 TNO-BOUW report no. 97-NM-R0564 TNO Building and Construction Research Computational Mechanics

Summary

An anisotropic model for the analysis of masonry plates and shells is proposed. The

“three-dimensional” model is an extension of a previously developed plane model

which incorporates independent yield criteria for tension and compression. This

composite model is implemented in modern computational plasticity concepts, utilising

local and global Newton-Raphson methods, implicit Euler backward return mapping

algorithms and consistent tangent operators. It contains different yield strengths along

the two material axes, both in tension and compression. The energy based regularisation

resorts then to four different fracture energies.

To validate the model a comparison is made between experimental and numerical

results, in the case of tests available in the literature in masonry panels subjected to out-

of-plane loading and in masonry wall elements subjected to combined actions.

Reasonable agreement was found in both cases.

Acknowledgements

The financial support by the Netherlands Technology Foundation (STW) under grant

DCT-33.3052 (project leader Dr. J.G. Rots) is gratefully acknowledged.

The calculations have been carried out with the Finite Element Package DIANA of TNO

Building and Construction Research on a Silicon Graphics Indigo R4000 workstation of

the Delft University of Technology.

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CONTENTS 1. INTRODUCTION.............................................................................................. 1 1.1 About anisotropic continuum models.......................................................... 1 1.2 Contents....................................................................................................... 3 2. MASONRY OUT-OF-PLANE BEHAVIOUR.................................................. 5 2.1 Flexural tensile strength - Uniaxial behaviour............................................. 6 2.2 Flexural tensile strength - Biaxial behaviour............................................... 9 2.3 The influence of normal compression......................................................... 14 2.4 Size effect and the randomness of the materials.......................................... 14 2.5 Size effect and the concept of flexural strength........................................... 16 3. THE ANISOTROPIC CONTINUUM MODEL FOR MASONRY PLATES AND SHELLS................................................................

19 3.1 Plates, shells and finite elements................................................................ 19 3.2 Tension - A Rankine Type Criterion........................................................... 22 3.3 Compression - A Hill Type Criterion.......................................................... 24 3.4 Orientation of the material axes................................................................... 25 3.5 Inelastic behaviour of the model.................................................................. 26 3.6 Non-linear analysis and thickness integration............................................. 29 3.7 Comparison with experimental data of masonry strength........................... 30 4. VALIDATION.................................................................................................... 33 4.1 McMaster University hollow concrete block panels................................... 33 4.1.1 Experimental versus numerical failure loads..................................... 65 4.1.2 The influence of the aspect ratio of the panels................................... 66 4.1.3 The influence of in-plane normal pressure......................................... 66 4.1.4 The influence of supports: simply supported on four versus three edges..................................... 67 4.1.5 Is yield-line design safe?..................................................................... 68 4.1.6 Mesh dependency of the solution....................................................... 69 4.2 University of Plymouth solid brick clay masonry panels............................ 77 4.3 ETH Zurich wall elements under eccentric loading.................................... 91 5. CONCLUSIONS................................................................................................ 95 BIBLIOGRAPHY...................................................................................................... 97

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1. Introduction This is the fifth report resulting of grant DCT 33.3052, from the Netherlands Technology Foundation (STW), “Numerical models for masonry structures and stacked construction”. Previous results of this project can be found in the Delft University of Technology Reports nos. 03.21.1.31.35 (1996), 03.21.1.31.27 (1995), 03.21.1.31.02 (1995), 03.21.22.0.01 (1994). This report deals with a macro-model for the out-of-plane behaviour of masonry structures. The composite plasticity model proposed in Lourenço (1996) for plane stress conditions is extended to shell structures. Modern algorithmic plasticity concepts - including implicit Euler backward return mapping schemes and consistent tangent operators for all regimes of the model - are utilised to combine anisotropic elastic behaviour with anisotropic plastic behaviour. The proposed yield criterion combines the advantages of modern plasticity concepts with a powerful representation of anisotropic material behaviour, which incorporates different hardening/softening behaviour along each material axis. The model is thus capable of reproducing independent (in the sense of completely diverse) elastic and inelastic behaviour along a prescribed set of material axes. The energy-based regularisation technique, which is employed to obtain objective results with respect to mesh refinement, resorts then to four different fracture energies. The numerical implementation and performance of the model is evaluated by means of a comparison between numerical results and experimental results for the case of masonry panels with out-of-plane loading and the case of masonry wall elements subjected to combined actions. It is stressed that the field of application of continuum models are large structures, subjected to loads and boundary conditions such that the state of stress and strain across a macro-length can be assumed to be uniform. In reality, the material in non-homogeneous and a close material representation is only possible if the units and joints are modelled separately. A macro-modelling strategy represents a compromise between efficiency and accuracy. 1.1 About anisotropic continuum models It is known that most structural materials exhibit some degree of anisotropy. Materials, such as timber, are naturally anisotropic, cold-worked metal sheets manifest predominant-directional properties which are intensified with increasing degree of plastic deformation. Other materials are anisotropic due to the manufacturing process such as plywood, reinforced concrete, masonry and most laminated composites. The difficulties in accurately modelling the behaviour of anisotropic materials are, usually, quite strong. This is due, not only, to the fact that comprehensive experimental results (including pre- and post-peak behaviour) are generally lacking, but also to intrinsic difficulties in the formulation of anisotropic inelastic behaviour. To describe the failure behaviour of anisotropic composites a criterion is needed which is able to describe the complex phenomena that govern failure in this type of materials. Criteria such as those of Hill (1948), Hoffman (1967) and Tsai-Wu (1971) have been defined with the aim of meeting this requirement. These anisotropic plasticity models have been

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proposed both from purely theoretical and experimental standpoints as failure criteria. But only a few numerical implementations and calculations have actually been carried out. Examples are given by the work of de Borst and Feenstra (1990) and Schellekens and de Borst (1990) which fully treated the implementation, in modern plasticity algorithmic concepts, of, respectively, an elastic-perfectly-plastic Hill yield criterion and an elastic-perfectly-plastic Hoffman yield criterion. In principle, inelastic behaviour could be simulated with the fraction model of Besseling (1958) but not much effort has been done in this direction. In fact, one of the serious problems that remains associated to the application of the above criteria is the description of inelastic behaviour. Besides the fraction model, other attempts can be found in the work of Owen and Figueiras (1983), which included material-axis-dependent hardening in the Hill criterion, Swan and Cakmak (1994), which included linear tensorial hardening in the Hill yield criterion, and Li et al (1994), which included linear hardening in a modified (pressure dependent) Von Mises criterion to fit either the uniaxial tensile or compressive behaviour. Nevertheless, all these approaches to model inelastic behaviour are relatively crude. In this article, the approach of Owen and Figueiras (1983) is extended to include softening behaviour and independent fracture energies along each material axis. The other problem associated with criteria such as Hill, Hoffman and Tsai-Wu is the poor representation of materials with a large difference between uniaxial compressive strength and uniaxial tensile strength, Lourenço (1996). Due to the smoothness of these criteria, unacceptable overestimation of strength can be found in the tension-compression regime. Figure 1 illustrates the problem of aiming to represent a plane stress isotropic no-tension material resorting to paraboloid yield criteria.

No-tension

Hoffman

fc σ1 σ2

fc

Figure 1.1 - Plane stress representation of the Hoffman criterion and a no-tension

material (fc is the uniaxial compressive strength)

To obtain a better representation of materials with considerable magnitude differences between compressive and tensile strengths, individual yield criteria are considered, according to different failure mechanisms, one in tension and the other in compression. The former is associated with a localised fracture process, denoted by cracking of the material, and, the latter, is associated with a more distributed fracture process which is usually termed crushing of the material. Again, the material model is developed for shells, which represents an extension of the work previously carried out for plane stress, Lourenço (1990).

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It is noted that a representation of an anisotropic yield criterion solely in terms of principal stresses is not possible. For shells, which is the case of the present article, a graphical representation in terms of the full stress vector in a predefined set of material axes is necessary. Anisotropic material behaviour, including a Hill type criterion for compression and a Rankine type criterion for tension, is proposed. This represents an extension of conventional formulations for isotropic quasi-brittle materials to describe anisotropic behaviour. In particular, it is an extension of the work of Feenstra and de Borst (1995), who utilised this approach for concrete with a Rankine and a Drucker-Prager criterion. 1.2 Contents Chapter 2 presents a brief review of masonry out-of-plane behaviour, including available experimental results and critical issues in using these results for numerical purposes. Chapter 3 describes the formulation of the model, including the mathematics and the necessary material parameters. Chapters 4 contains the validation of the model, in which experimental results available from the literature are compared with the results of the developed model.

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2. Masonry out-of-plane behaviour The out-of-plane behaviour is, in essence, a consequence of the in-plane behaviour. Recently, in the realm of a material description suitable for numerical purposes, significant improvements have been achieved in the characterisation of the in-plane behaviour of masonry, see Lourenço (1996) for a review. If the behaviour under plane stress conditions of a specific combination of mortar and units is described in a way suitable for the use of advanced numerical methods, then, the analysis of masonry plates and shells offers little difficulties. There has been much work done about the flexural tensile strength of masonry along the two principal material directions. Nevertheless, available experimental results are of limited interest for numerical purposes due to the lack of communication between numerical and experimental experts. In particular, moment-curvature diagrams are normally lacking and very little work was carried out in the following key-points: • Size effect: Consequences of the variability of material properties and of softening

behaviour (real tensile strength vs. measured flexural strength); • Strength for uniaxial testing at different angles with the bed joints or the interaction

relationships for biaxial flexural strength. It has been (and it still is) common practice to perform bending tests to describe structures subjected to out-of-plane forces. This is mostly done due to practical reasons, as bending tests are relatively easy to carry out, e.g. three-point bending, four-point bending or the bond-wrench test proposed by Hughes and Zsembery (1980), see Figure 2.1. No discussion is given here about different test set-ups and the reader is referred to van der Pluijm (1996) for a modern discussion on measuring flexural bond strength. A comprehensive review about flexural properties of masonry is given in Lawrence (1983), including the adequacy of testing methods and the reliability of results. Particular attention is given there to the influence in the results of secondary effects (such as non-ideal supports), lack of displacement and strain data, variability of properties, workmanship and age of testing. Clearly, the experimental work carried out in this area has not been oriented towards numerical analysis. It is, therefore, urgent that the experimental work reaches a level similar to that already obtained for the in-plane behaviour. A brief review, oriented towards numerical analysis, of experimental out-of-plane behaviour of masonry is presented next. The objective is to select the most adequate experimental results available in the literature for the validation of the numerical tools to be proposed in this report. Finally, it is noted that, if one is considering the development of modern tools for the analysis of masonry plates and shells, describing the flexural compressive inelastic behaviour is also important. This point might become critical in several structures subjected to high compressive stresses, such as pier-wall connections, curved masonry shells or masonry panels with boundaries that preclude any rotation. Nevertheless, the subject of out-of-plane compressive failure is almost virgin in the literature, both from experimental and numerical standpoints.

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FFF

(a) (b)

F

Joint undertestLever arm

(c)

Figure 2.1 - Examples of tests to obtain the tensile flexural strength of masonry: (a) three-point bending, (b) four-point bending and (c) bond-wrench test

2.1 Flexural tensile strength - Uniaxial behaviour The flexural strength of masonry has been mostly investigated in relation to the resistance of wall panels to wind loads. The flexural strength is, of course, different for bending perpendicular or parallel to the bed joints, see Figure 2.2, being normally several times larger when bending leads to failure in a plane perpendicular to the bed joints. In general, experimental results have been concerned solely with measurements of the flexural strength and mostly for the case when the plane of failure occurs parallel to the bed joints, while elastic and inelastic properties have usually been ignored.

p

p

p p

(a) (b)

Figure 2.2 - Four-point bending test in two different directions: (a) plane of failure parallel to the bed joints - vertical bending or bending normal to the bed joints - and (b) plane of failure perpendicular to the bed joints - horizontal bending or bending

parallel to the bed joints For bending leading to failure in a plane parallel to the bed joints (“vertical bending”), failure is generally caused by the relatively low tensile bond strength between the bed joints and the unit, see Figure 2.3a. In masonry with low strength units and greater tensile bond strength between the bed joints and the unit, e.g. high-strength mortar and units with numerous small perforations, which produce a dowel effect, failure may

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occur as a result of stresses exceeding the unit tensile strength. In any case, the typical moment-curvature relation is linear up to 70%-85% of the failure load, Lawrence (1983) and van der Pluijm et al (1995), see Figure 2.4a. Beyond this level micro-cracking starts to occur, with unloading (“softening”) of the extreme tensile fibres. For bending leading to failure in a plane perpendicular to the bed joints (“horizontal bending”), two different types of failure are common, depending on the relative strength of joints and units, see Figure 2.3b. In the first type of failure cracks zigzag through head and bed joints. The post-peak response of the specimen is governed by the fracture energy of the head joints and the friction behaviour of bed joints. In the second type of failure cracks run almost vertically through the units and head joints. The post-peak response is governed by the fracture energy of the units and head joints. In both types of failure, the typical moment-curvature relation indicates a sudden decrease of stiffness before non-linear behaviour starts to occur, see Figure 2.4b. This kind of behaviour was originally noticed by Ryder (1963). Lawrence (1983) attributed this behaviour to the gradual decrease of stiffness of the head joints, i.e. the softening behaviour, which has been confirmed by van der Pluijm et al (1995). It is noted that the above holds true only in the general case of units with a tensile strength substantially larger than the tensile strength of the head joints.

(a) (b1) (b2) Figure 2.3 - Possible failure modes for masonry subjected to bending along the material

axes. Failure in a plane parallel to the bed joints: (a) “debonding”. Failure in a plane perpendicular to the bed joints: (b1) “toothed” and (b2) “splitting”

M M

κκ (a) (b)

Figure 2.4 - Typical moment-curvature (M-κ) diagrams for masonry in bending such that failure occurs in a plane (a) parallel and (b) perpendicular to the bed joints

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The ratio of flexural strength along the two material axes varies significantly as can be seen in Figure 2.5a. Nevertheless, a definite trend seems to be distinguished in the orthogonal strength ratio, which decreases markedly with an increasing flexural bond strength, see Figure 2.5b and also Baker (1979). It is noted that, a large scatter occurs and such a simplified assumption is hardly acceptable. The scatter is partly caused by: • Different specimen and load configurations used to determine flexural strengths; • Variability of the masonry constituents and workmanship. Ideally, each point should

be established by averaging a convenient number of tests; • Different flexural strengths of the units used. Since failure in bending parallel to the

bed joints often occurs by unit breaking after degradation of the head joints strength, this parameter is important. This cannot be distinguished by such an empirical curve but a theoretical relationship that takes into account the flexural strength of the units can be found in Baker (1979).

Mftx

yx

Mfty

fftx [N/mm2]

ffty [N/mm2]

0.4 1.2

0.0

3.0

2.0

1.0

1.0

Satti 1:1/:6Satti 1:1/4:3

B.C.R.A. 1:1:6

Lawrence & MorganLawrenceB.C.R.A. 1:1/4:3

0.80.0 0.6

1:5 1:3 1:2

0.2

(a)

)1975(Lawrence,17.2

ftyfty

ftx

fff

=

f

fftx

fty

ffty [N/mm2]

0.4 1.2

0.0

12.0

8.0

4.0

1.0 0.8 0.0 0.6 0.2

(b)

Figure 2.5 - Flexural tensile strength along the two material axes, Hendry (1990): (a) strength along x-axis vs. strength along y-axis; (b) strength ratio vs. strength along y-axis

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2.2 Flexural tensile strength - Biaxial behaviour There is relatively little known about the flexural strength at different angles to the bed joints or about the interaction relationships for biaxial flexural tension. Losberg and Johansson (1969) present, probably, the first paper about four-point bending tests with masonry beams featuring different orientations with respect to the bed joints. This is a very complete article and a remarkable work for such early period. Satti and Hendry (1973) reported that the flexural strength at 45 degrees to the bed joints was approximately one half to a third of the strength in horizontal bending. In contrast, Cajdert and Losberg (1973) found that the diagonal flexural strength was higher than both the vertical and horizontal strengths. However, these early works were based on very few tests and there was a wide scatter of results. Baker (1979) carried out the first detailed experimental program in biaxial bending. Using single joint tests he found evidence of an elliptical interaction between the horizontal flexural strength fftx and the vertical flexural strength ffty, see Figure 2.6, for the case where both moments produce tension on the same face of the specimen.

fftx

ffty

Mftx

Mfty

Figure 2.6 - Failure criterion in biaxial bending according to Baker (1979)

This work, although of much interest two decades ago, is of limited interest today. Firstly, it is highly debatable that the adopted test set-up returns the flexural strength values of masonry. Secondly, Lawrence and Cao (1988), already showed that minor differences between an elliptical and rectangular (no-interaction) criterion can be expected in masonry panels. Thirdly, being masonry an anisotropic material, its out-of-plane behaviour cannot be described only in terms of principal moments. A complete out-of-plane description must be established in terms of, either, the three moment-components (Mxx, Myy and Mxy), or, the principal moments and one angle θ, which measures the orientation of the principal axes with respect to the material axes. It is noted that, in the general case of shell analysis, the material behaviour must be described by five stress components (the usual assumption of thin structures assumes the out-of-plane normal stress to remain zero) and a yield criterion established solely in terms of principal moments is inadequate. More recently, researchers tried to gather the necessary experimental insight in biaxial bending, Gazzola and Drysdale (1986), Guggisberg and Thürlimann (1990) and van der Pluijm et al (1995). Gazzola and Drysdale (1986) and van der Pluijm et al (1995) performed similar tests, resorting to four-point bending masonry beams loaded in different orientations with respect to the material axes, but only van der Pluijm et al

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(1995) was able to measure and report moment-curvature diagrams. Guggisberg and Thürlimann (1990) resorted to a more complex test set-up capable of applying a general combination of bending moments. Figure 2.7 illustrates possible flexural tensile failure modes of masonry subject to out-of-plane loading. The experimental failure patterns of Gazzola and Drysdale (1986), in hollow concrete block masonry, are summarised in Figure 2.8. Five tests were carried out, for each orientation of the principal bending moment θ. Failure occurred, generally, in a combination of tension and shear, except for some cracking through the units observed for bending parallel to the bed joints. The average and standard deviation of the five tests, for each orientation of loading, are represented in Figure 2.9. It is observed that, in this particular testing program, the flexural strength for bending parallel to the bed joints is 2.5 times larger than the flexural strength for bending normal to the bed joints.

(a) debonding (b) stepped (c) toothed (d) splitting (e) straight Figure 2.7 - Possible flexure failure modes

Bed joint M α

α = 90ºα = 75ºα = 45ºα = 15ºα = 0º

Figure 2.8 - Failure patterns in masonry wallettes, Gazzola and Drysdale (1986)

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fft [N/mm2]

Bed joint orientation θ [°]906015 4530 75 0

1.20

0.80

0.40

0.00

Figure 2.9 - Flexural strength for different orientations of loading, Gazzola and

Drysdale (1986) The results from van der Pluijm et al (1995), in solid brick clay masonry and calcium silicate block masonry, are summarised in Figure 2.10 and Figure 2.11 by the average and standard deviation of the tests, for each orientation of loading. Twelve tests were carried out, for each orientation of the principal bending moment θ. Failure occurred, generally, in a combination of tension and shear in the head and bed joints, except for loading making an angle of 70 degrees with the bed joints, where cracking through the units was also observed. It can be seen that, in this particular testing program, the flexural strength for bending parallel to the bed joints is approximately 4.0 times larger than the flexural strength for bending normal to the bed joints.

fft [N/mm2]


3.0

2.0

1.0

0.0

Figure 2.10 - Flexural strength for different orientations of loading in solid brick clay

masonry, van der Pluijm et al (1995)

12

fft [N/mm2]


2.50

1.67

0.83

0.00

Figure 2.11 - Flexural strength for different orientations of loading in solid block

calcium silicate masonry, van der Pluijm et al (1995) Guggisberg and Thürlimann (1990) resorted to a specially designed test rig to load masonry panels with a vertical normal force plus normal and twisting bending moments, see Figure 2.12 and Figure 2.13. The rig was designed to produce insignificant geometrical restraints along the boundaries. The whole rig was freely suspended and the load application system is statically determined.

Figure 2.12 - Experimental test set-up from Guggisberg and Thürlimann (1990)

The normal force has been kept constant during the test and the moments were increased proportionally from zero until failure. Figure 2.14 shows qualitative images of the cracking patterns and modes of failure (test Q9 was not successful), with a good representation in perspective for the tests without twisting moments that helps understanding masonry behaviour in flexure.

13

mx

mx

nx

nx

my

2mxy

2mxy

my

2mxy

2mxy

1200

2100

Figure 2.13 - Geometry and loading of test specimens,

Guggisberg and Thürlimann (1990)

Q3Q2Q1

Q6Q5Q4

Q10Q8Q7

Q13Q12Q11

Figure 2.14 - Cracking patterns and failure modes for test specimens, after Thürlimann and Guggisberg (1988)

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2.3 The influence of normal compression An interesting (and obvious) consequence of the arrangement of units and mortar was reported by Baker (1979). This author observed that a moderate in-plane compressive stress acting normal to the bed joints, increased, not only, the maximum moment for bending perpendicular to the bed joints, but also, the maximum moment for bending parallel to the bed joints. This is, of course, only true if a “toothed” type of failure is obtained. If a ”straight” type of failure, through head joints and units, is obtained then failure is determined by the unit flexural strength and no increase should be expected. In case of a “toothed” type of failure, the increase of the bending moment is a consequence of the strength increase in the joints which fail in shear. As given by the Coulomb friction law, higher normal compressive stresses lead to higher shear strength. This characteristic of masonry behaviour also occurs in plane stress situations. If one adopts a micro-modelling strategy, in which units and joints are modelled separately, this result comes naturally from the analysis, see e.g. CUR (1994). As the objective of the present study is a macro-model, the described behaviour can hardly be included in the model and seems to be the price to pay for adopting a simplified modelling strategy.

2.4 Size effect and the randomness of the materials As a result of many studies, it has been shown that brittle failure of concrete structures exhibits a size effect. For a long time, the size effect has been explained statistically as a consequence of the randomness of the material, particularly by the fact that in a larger structure it is more likely to encounter a material point of smaller strength. Various existing test data were interpreted in terms of Weibull weakest-link theory. Later, it was proposed by Bazant (1984) that whenever the failure does not occur at initiation of cracking, which represents most situations, the size effect should properly be explained by energy release caused by macrocrack growth, and that the randomness of strength plays only a negligible role. Presently, the role of the softening behaviour is accepted by all the scientific community to explain size effect. In the next section, the direct consequence of softening in the flexural strength of masonry will be discussed. Now, we address the problem of randomness of strength in masonry structures which seems to play a crucial role in size effect, Lawrence (1983). This effect, which is due to a much higher coefficient of variation, seems to have been firstly taken in consideration for masonry structures by Baker and Franken (1976), with respect to the flexural strength of brickwork. Of course, this is not only a problem for the flexural strength but in all failures related to the tensile strength (size effect due to “crushing” is more complex and will not be discussed here). Remarkably, the subject received a lot of publicity in relation to flexural strength and very little with relation to tensile strength! This seems to confirm a certain incipient knowledge of masonry structures. From the test results, e.g. Lawrence (1983) and van der Pluijm (1996), it is essential to treat a masonry beam as being composed of a number of discrete joints, each with a different strength. This is due to the high variability in masonry properties and the softening behaviour which leads to localisation of deformation in a single joint and unloading of the rest of the specimen. Because failure of each single joint leads to failure of the whole beam, a weakest-link theory has to be adopted. The theory of order

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statistics will not be reviewed here but, for a given distribution of joint strengths (which must be approximately normal), the mean and variance of the beam strength can be obtained from the mean and average of the joint strength, from the following relations:

×=

×−=

jointftbeamft

jointftjointftbeamft

k

kff

,2,

,1,,

σσ

σ (2.1)

where the values of k1 and k2 depend on the number of joints and can be found in e.g. Mosteller and Rourke (1973), see Table 2.1.

Table 2.1 - Weakest-link theory coefficients, Mosteller and Rourke (1973)

No. of joints

2

3

4

5

6

k1 0.56 0.85 1.03 1.16 1.27 k2 0.826 0.748 0.696 0.669 0.645

Figure 2.15 shows clay brick specimens tested in bending perpendicular to the bed joints, van der Pluijm (1996). Table 2.2 shows the results obtained in the different tests, which demonstrate the need of a correction. The mean flexural strength, in a single joint, is ≈ 0.66 N/mm2 and, in the wallettes, is ≈ 0.56 N/mm2 (85%). The standard deviation, in a single joint, is ≈ 0.20 N/mm2 and, in the wallettes, is ≈ 0.16 N/mm2 (80%). The weakest-link theory gives a ratio of 70% (four joints in pure flexure), both for the mean and the standard deviation. A cause for this outcome is not perfectly clear but it is questionable to use eq. (2.1), with the parameters of Table 2.1, in this case because the pier has a width of one unit and the wallettes have a larger width. Due to the manufacturing and curing processes different statistical distribution of strengths will be found in the pier joint and the wallette joint. A better agreement between the weakest-link theory and tests can be found in Lawrence (1983).

(a) (b) (c) Figure 2.15 - Clay brick specimens: (a) normal wallette, (b) double width wallette and

(c) stacked bonded pier, van der Pluijm (1996)

Table 2.2 - Average strength and coefficient of variation for flexural strength value according to three different specimens, van der Pluijm (1996)

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single joint (pier)

single joint (bond wrench)

normal wallettes

double width wallettes

fft [N/mm2] 0.64 0.68 0.57 0.56 c.v. [%] 30 29 29 27

Finally, it is noted that masonry modelling has been ill-treated for a long time, at least if one compares it other fields like concrete or soil mechanics. The author believes that, before trying to incorporate some length scale in the models, it is useful to gain more experience with the use of the complex masonry material models and the tests needed to calibrate them. The randomness of material properties will, therefore, be ignored in the present study. This procedure can find some justification in the fact that a large masonry structure is normally subjected to quite non-uniform stress conditions. Therefore, collapse will result from failure in a large number of joints and the panel failure mode is heavily constrained by the boundary conditions. This means that averaging effects are likely to occur, i.e. randomness of properties becomes less important.

2.5 Size effect and the concept of flexural strength Two arguments have been presented in favour of using bending tests (three-point or four-point bending) to obtain the flexural strength of a material. Firstly, the tests are relatively easy and inexpensive to perform and, secondly, if one is dealing with plates and shells it seems natural to directly characterise the bending tests behaviour. Nevertheless, one question that arises in practice is the relation between tensile and flexural strengths. In the following, it is assumed for simplicity that only tensile inelastic behaviour occurs in the structure. For a cross section of a beam or plate in pure bending, see Figure 2.16, the elementary linear elastic beam theory yields a tensile flexural strength

fM

bhft

u=6

2 (2.2)

where fft is the flexural tensile strength, Mu is the ultimate bending moment, and b, h are the dimensions of the cross section of the beam.

M M

fft

b

h

Assumed stressdistribution

Figure 2.16 - Beam under pure bending. Linear elastic theory

But this value is not the real uniaxial tensile strength. If the beam is subjected to increasing load, at a certain stage the tensile strength of the extreme fibre will be reached. The stress in this fibre starts to follow the descending branch, the micro-crack propagates upwards and the neutral axis of the cross section shifts towards the fibres in

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compression. Although micro-cracking is occurring, the bending moment can still be increased until the peak moment is reached. Only at the ultimate stage, a fully developed crack occurs. The internal stress distribution in the cross section through all the process is shown in Figure 2.17.

Linear ElasticPeakUltimate

σ−

σ+

Figure 2.17 - Normal stress distribution in a cross-section subjected to pure bending

The conclusion that the ultimate moment by itself is not enough to determine the constitutive behaviour of masonry is crucial. In particular, it means that most of the experimental out-of-plane results in masonry specimens are of limited interest for the purpose of numerical modelling. A complete material characterisation derived from bending tests has to include the ultimate moment (directly related to the flexural tensile strength) and the moment-curvature diagram. These “quantities” permit a description at constitutive level based on the (real) tensile strength and the fracture energy. In the following, the real uniaxial tensile strength ft is kept constant (equal to 3 N/mm2), while the fracture energy is varied from a very low value to a very high value. If the fracture energy is zero, failure occurs once the extreme fibre reaches the maximum tensile stress, i.e. a linear distribution of stresses is obtained at collapse. The correspondent ultimate moment, denoted by Mref (reference), can be directly obtained from eq. (2.2). Figure 2.18 shows the significant influence of the fracture energy in the moment-curvature diagrams. As the value of the ultimate bending moment can, in theory, be magnified by a factor three, it does not provide a reliable basis to estimate both the real uniaxial strength and the fracture energy. In masonry specimens, very little research has been carried out regarding this aspect but, it is obvious, that the influence of the descending branch of the stress-crack width diagram diminishes with increasing height h of the cross-section. As an example, for concrete, the Model Code 90, CEB (1991), provides the following expression

f fh h

h ht ft=

+

1 5

1 1 50

0 7

00 7

. ( / )

. ( / )

.

. (2.3)

where ft is the uniaxial tensile strength, h0 equals 100 mm and the height of the cross section h is expressed in mm.

18

M

Mref

κ [rad × 10-3]

2.5

2.0

1.5

1.0

0.5

0.0

0.4 0.6 1.00.80.20.0

Gf = 0.2

Gf = 10

Gf = 0.05

Gf = 0.025

Gf = 0.010 Gf = 0.005

(a)

Assymptot for Gf = +∞

Assymptot for Gf = 0

M

Mref

3

2

1

10-3 102101

Stress distributionin the cross section

Stress distributionin the cross section10010-110-2

(b) Figure 2.18 - Influence of the fracture energy, measured in N.mm/mm2, on

(a) moment-curvature diagram and (b) value of maximum bending moment. The uniaxial strength is kept constant.

19

3. The anisotropic continuum model for masonry plates and shells In this study, we address the problem of modelling laminar masonry structures (plates and shells with one dimension substantially smaller than the other two dimensions). A typical hypothesis in this type of elements is “zero normal stress”. This hypothesis states that the normal stress component perpendicular to the plane of the structure equals zero, and simplifies material modelling to a great extent. While, conceptually, it is relatively straightforward to include the three-dimensional behaviour in the model, one should keep in mind that the complexity of an anisotropic material model might preclude its numerical implementation and its use in practice, due to a large number of material parameters. The approach followed here is, basically, an engineering approach where a compromise is sought between accuracy and simplification. 3.1 Plates, shells and finite elements Plates and shells are but a particular form of a three-dimensional solid, the treatment of which presents no theoretical difficulties. Owing to the symmetry of the stress and strain tensors, it is normal to collect the tensors components, see Figure 3.1, in vectors as

{ } { }

{ } { }

=

=

Txzyzxyzyx

Txzyzxyzyx

γγγεεεε

τττσσσσ

,,,,,

,,,,, (3.1)

z

τyz

x

τzy τxz

τzx τxy τyx

σz

σy

σx y

Figure 3.1 - Stress components in a three-dimensional body

However, the thickness of plates and shells (denoted in the following as t) is small when compared with other dimensions, and complete three-dimensional numerical treatment would be, in general, not only costly but in addition could lead to serious equation conditioning problems. To ease the solution several classical assumptions regarding the behaviour of such structures were introduced. Clearly such assumptions resulted in a series of approximations, which led, basically, to a thin plate theory and a thick plate theory. Here, no discussion is given regarding the approximations involved or the range of validity of the theories. For a comprehensive review of plate and shell bending approximations and the finite element implementation the reader is referred to Zienkiewicz and Taylor (1991).

20

Here, the finite element adopted to reduce the degrees-of-freedom in a complete three-dimensional analysis, is the curved shell element degenerated from a three-dimensional formulation. This element, originally proposed by Ahmad et al (1970) for the linear analysis of moderately thick shells, has been extensively used for the geometrical and non-linear analysis of shell structures. Typical characteristics of this element are the two hypotheses on which the degeneration is based: “straight normals” and “zero normal stress”. The first hypothesis assumes that the normals to the mid-plane of the element remain straight after deformation, but not necessarily perpendicular to the mid-plane. The second hypothesis, already introduced above, states that the normal stress component perpendicular to the mid-plane equals zero, and the element formulation has been obtained ignoring the strain energy resulting from this stress. Assuming that the local z-axis represents the normal to the mid-plane, the five stress components left are σx, σy, τxy, τyz and τxz, see Figure 3.2. Five degrees of freedom are defined for each element node: three translations and two rotations, see Figure 3.3. The definition of the independent translations and rotations includes the influence of shear deformation. The rotations are not coupled to the gradient of the mid-plane. In this study, two by two Gauss integration in the plane and seven-point Simpson integration in the thickness direction are used.

(a)

τzx τzy

(b)σy

τxy

σx

Figure 3.2 - Thin shells: (a) layered shell with five stress components; (b) layer, essentially, in plane stress conditions

21

uz

φx φy

uy ux

t

b

Figure 3.3 - Curved shell element (applicable for thin shells with t τ > τ > τ = 03 2 1 0

Hill type Rankine type

Figure 3.4 - Adopted plane stress composite yield criterion with iso-shear stress lines,

Lourenço (1996) It is noted that the model is implemented in the DIANA finite element package. In this package, there is no difference between the formulation for a complete three-dimensional stress-state and the formulation for shells, which is quite convenient for a general purpose finite element program. Owing to the expansion/compression mechanism described in de Borst (1991), the normal stress component σz is iterated to zero in the case of shell analysis. Therefore, in the present study, all the formulation will be given in the full six components stress-strain space. It is noted that, in the following, only aspects concerned with shell developments will be considered. Details about the computational implementation of the model, resorting to a modern implementation of the plasticity theory, can be found in Lourenço (1995) and will not be reproduced here.

22

3.2 Tension - A Rankine Type Criterion For modelling tensile behaviour, it will be assumed that cracks, at each integration point, always arise normal to the mid-surface of the element. This assumption means that each layer of the shell element is considered to be in plane stress and the additional stresses from the shell formulation (τyz and τxz) will be ignored. Of course, the assumption entails some approximation as diagonal “shear” cracks in the thickness direction are replaced by cracks stepwise normal to the mid-surface. Nevertheless, this is a widely used simplification in thin shell analysis which, for the case of anisotropic shells, becomes particularly attractive because the input data are heavily reduced. In particular, the material behaviour along the z-axis (normal to the element mid-plane) and the contribution to failure of the two additional shear stresses do not have to be described. An adequate formulation of the Rankine criterion is given by a single function, which is governed by the first principal stress and one yield value σt that describes the softening behaviour of the material as, see Feenstra and de Borst (1995),

)(22

22

1 ttxyyxyxf κστ

σσσσ−+

−+

+= (3.2)

where the scalar κt controls the amount of softening. This expression can be rewritten as

22

1 2))(())((

2))(())((

xyttyttxttyttxf τ

κσσκσσκσσκσσ+

−−−+

−+−= (3.3)

where coupling exists between the stress components and the yield value. Setting forth a Rankine type criterion for an anisotropic material, with different tensile strengths along the x, y directions, is now straightforward if eq. (3.3) is modified to

22

1 2))(())((

2))(())((

xyttyyttxxttyyttxxf τα

κσσκσσκσσκσσ+

−−−+

−+−=

(3.4) where the parameter α, which controls the shear stress contribution to failure, reads

ατ

=f ftx ty

u t,2

(3.5)

Here, ftx, fty and τu,t are, respectively, the uniaxial tensile strengths in the x, y directions and the pure shear strength. Note that the material axes are now fixed with respect to a specific frame of reference. Thus, it shall be assumed that all stresses and strains for the elastoplastic algorithm are given in the material reference axes, see also section 3.4.

23

Eq. (3.4) can be recast in a matrix form as f T t

T1

12

12

12= +( { } [ ]{ }) { } { }ξ ξ π ξP (3.6)

where the projection matrix [Pt] reads

−

−

=

00000000000000200000000000000000

][P

21

21

21

21

αt (3.7)

the projection vector {π} reads { } { }π = 1 0 0 0 0 0 T (3.8) the reduced stress vector {ξ} reads { } { } { }ξ σ η= − (3.9) the stress vector {σ}, as given in eq. (3.1), reads { }σ σ σ σ τ τ τ= { }x y z xy yz xz

T (3.10) and the back stress vector {η} reads {η} = { ( ) ( ) }σ κ σ κtx t ty t

T0 0 0 0 (3.11) Exponential tensile softening is considered for both equivalent stress-equivalent strain diagrams, with different fracture energies (Gfx and Gfy) for each yield value, which read

σ κ σ κtx txtx

fx

t ty tyty

fy

tfh f

Gf

h f

G= − = −exp( ) exp( )and (3.12)

where the standard equivalent length h is related to the element size, Bazant and Oh (1983). A non-associated plastic potential g1 g T g

T1

12

12

12= +( { [ ]{ )ξ} ξ} {π} {ξ}P (3.13)

is considered, where the projection matrix [Pg] represents the original Rankine plastic flow, i.e. α = 1 in eq. (3.7). The inelastic behaviour is described by a strain softening hypothesis given by the maximum principal plastic strain ε t

p. as

24

κ εε ε

ε ε γtp x

pyp

xp

yp

xyp= =

++ − +⋅

⋅ ⋅⋅ ⋅⋅ ⋅

11

22 2

2( ) ( ) (3.14)

which reduces to the particularly simple expression κ λt t=

⋅ ⋅ (3.15) 3.3 Compression - A Hill Type Criterion In case of crushing it is physically appealing and it results quite simple to include the contribution of the additional stresses from the shell formulation (τyz and τxz) in the failure criterion. The simplest yield criterion that features different compressive strengths along the two material axes is a rotated centred ellipsoid in the full stress space. The expression for such a quadric can be written as

f cy c

cx c

x x ycx c

cy c

y xy yz xz cx c cy c22 2 2 2 2 0= + + + + + − =

σ κ

σ κσ βσ σ

σ κ

σ κσ γ τ τ τ σ κ σ κ

( )

( )

( )

( )( ) ( ) ( ) (3.16)

where σ κcx c( ) and σ κcy c( ) are, respectively, the yield values along the material axes x and y. The β and γ values are additional material parameters that determine the shape of the yield criterion. The parameter β controls the coupling between the normal stress values, i.e. rotates the yield criterion around the shear stress axis, and must be obtained from one additional experimental test, e.g. biaxial compression with a unit ratio between principal stresses. The parameter γ , which controls the shear stresses contributions to failure, can be obtained from

γτ

=f fcx cy

u c,2

(3.17)

where fcx , fcy and τu,c are, respectively, the uniaxial compressive strengths in the x, y directions and a fictitious pure shear in compression. For the purpose of numerical implementation, it is convenient to recast this yield criterion in a matrix form as f T c c c2 12

12= −( {σ} σ})[P ]{ σ κ( ) (3.18)

where the projection matrix [Pc] reads

25

=

γγ

γ

κσκσ

β

βκσ

κσ

200000020000002000000000

0000)()(

2

0000)()(

2

][P ccyccx

ccx

ccy

c (3.19)

the yield value σc is given by σ κ σ κ σ κc c cx c cy c( ) ( ) ( )= (3.20) and the scalar κc controls the amount of hardening and softening. The inelastic law adopted comprehends parabolic hardening followed by parabolic/exponential softening for both equivalent stress-equivalent strain diagrams, with different compressive fracture energies (Gfcx and Gfcy) along the material axes, see Lourenço (1996). The problem of mesh objectivity of the analyses with strain softening materials is a well debated issue, at least for tensile behaviour, and the stress-strain diagram must be adjusted according to an equivalent length h to provide an objective energy dissipation, see Feenstra and de Borst (1996). An associated flow rule and a work-like hardening/softening hypothesis are considered. This yields

κσ

σ ε λcc

T pc= =

⋅ ⋅ ⋅1 { } { } (3.21)

3.4 Orientation of the material axes For the sake of simplicity, the formulation of the plasticity model was presented based on the assumption that the principal axes of anisotropy coincided with the frame of reference (local or global) for stresses and strains in finite element computations. Since this is not necessarily the case, such non-alignment effects must be taken into account. Two different approaches can be followed. In the first approach, with each call to the plasticity model, stresses, strains and, finally, consistent tangent stiffness matrices must be rotated into and out of the material frame of reference, respectively, as pre- and post-processing. In the second approach, before the analysis begins, the elastic stiffness matrix [D], the projection matrices [Pt], [Pg] and [Pc], and the projection vectors {π} and {η} must be rotated from the material frame of reference into the global frame of reference at each quadrature point, eliminating the need of all subsequent rotation operations. The drawback of the latter approach is that the matrices then lose their sparse nature, resulting in less clear algorithms. For this reason, the plasticity model is implemented employing the former option.

26

3.5 Inelastic behaviour of the model The behaviour of the model in uniaxial tension and compression along the material axes has been discussed in detail in Lourenço (1996). A single element test with 100 × 100 × 100 mm3 has been selected for this purpose. The material properties given in Table 3.1 and Table 3.2 are assumed, in which all the material parameters in the y direction are multiplied by a factor 0.5 (except the fracture energy which is multiplied by a factor 0.3) to simulate severe orthotropic behaviour. Here, E represents the Young’s modulus, ν represents the Poisson’s ratio, G represents the shear modulus and κp represents the equivalent plastic strain at peak compressive strength, see Lourenço (1996).

Table 3.1 - Material properties for single element test. Elastic properties

Ex Ey Ez νxy νyz νxz Gxy Gyz Gxz 10000 5000 7500 0.2 0.2 0.2 3000 3000 3000 N/mm2 N/mm2 N/mm2 - - - N/mm2 N/mm2 N/mm2

Table 3.2 - Material properties for single element test. Inelastic properties

Tension regime ftx fty α Gfx Gfy 1.0 0.5 1.0 0.02 0.006

N/mm2 N/mm2 - Nmm/mm2 Nmm/mm2

Compression regime fmx fmy β γ Gfcx Gfcy κp

10.0 5.0 -1.0 3.0 5.0 1.5 5 × 10-4 N/mm2 N/mm2 - - Nmm/mm2 Nmm/mm2 -

The responses under pure uniaxial tension and compression are illustrated in Figure 3.5, which demonstrates that completely different behaviour along the two material axes can be reproduced.

27

σ [N/mm2]

ε [10-4]

1.0

0.8

0.6

0.4

y direction x direction

0.2

0.04.03.02.01.00.0

(a)

σ [N/mm2]

ε [10-3]

10.0

8.0

6.0

4.0


2.0

0.0 10.0 7.5 5.0 2.50.0

(b)

Figure 3.5 - Possible in-plane behaviour of the model along the material axes: (a) uniaxial tension; (b) uniaxial compression

The anisotropic out-of-plane behaviour of the model will be now discussed. Firstly, the element is loaded in pure flexure. Figure 3.6 shows the moment-curvature response along the two material axes. For both responses only the tensile criterion becomes active. For bending in the x direction, the maximum compressive stress is around 2.0 N/mm2, which is far below the compressive strength of the material. This is, probably, a typical situation of masonry out-of-plane behaviour.

28

2.5

2.0

1.5

1.0

0.5

0.06.05.04.03.02.01.00.0

M [kN.mm]

κ [10-4]


Figure 3.6 - Possible out-of-plane behaviour of the model along the material axes.

Pure bending In the case of confined masonry subjected to high vertical loads or masonry shells confined by the boundary conditions, the influence of the in-plane compressive behaviour might become relevant. To assess the model out-of-plane behaviour under inelastic compressive behaviour, the element is now subjected to a uniform pre-compression σx of increasing value, which is kept constant during the analysis. Figure 3.7 shows the response for bending in the x direction, for different values of the normal pre-compression σx. As expected, higher pre-compression leads, up to a certain limit, to higher maximum bending moment and increasing ductility. For σx = 0.4 fmx, the ductility of the response is reduced due to crushing in the compression zone and, for

σx = 0.2 fmx

σx = 0.4 fmx

σx = 0.1 fmx

σx = 0.05 fmx

σx = 0.0 fmx

60.0 50.0 40.0 30.0 20.0 10.0 0.0 0.0

4.0

σx = 0.8 fmx

8.0

12.0

M [kN.mm]

κ [10-4] Figure 3.7 - Possible out-of-plane behaviour of the model along the

material axes. Bending with normal compression

29

σx = 0.8 fmx, both the ductility and the maximum bending moment are heavily reduced due to early crushing of the compression zone. The critical importance of the inelastic compressive behaviour for high in-plane compression stresses (the cases of 0.4 fmx and 0.8 fmx) is illustrated in Figure 3.8, where the moment-curvature response of the element, with and without compressive inelastic behaviour is shown.

σx = 0.2 fmx

σx = 0.4 fmx

σx = 0.1 fmx σx = 0.05 fmx

σx = 0.0 fmx

60.0 50.0 40.0 30.0 20.0 10.0 0.0 0.0

10.0

σx = 0.8 fmx30.0

20.0

40.0

M [kN.mm]

κ [10-4]

fmx = 10 N/mm2 fmx = + ∞

Figure 3.8 - Possible out-of-plane behaviour of the model along the material axes. Bending with normal compression. Influence of compressive softening

3.6 Non-linear analysis and thickness integration To finalise the reader’s attention is driven to the fact that the a reasonably large number of integration points in the thickness direction must be provided for non-linear analysis and softening behaviour. In the case of the degenerated shell element presented before, the use of more than three integration points in the thickness direction is useless in the case of a linear elastic analysis. The question is: And what about non-linear analysis? Figure 3.9 shows a comparison of the results for pure bending of Section 3.5, using (a) one regular element with seven-point Simpson integration in the thickness direction and (b) one layered element with ten layers and two-point Simpson integration in the thickness direction. Option a represents a typical selection for non-linear analysis and option b represents the “exact” solution. Clearly, the zigzag response of the regular element (with seven-point Simpson integration) for this analysis is unacceptable. Nevertheless, for large structures, the local oscillating response is likely to be irrelevant because it is smoothed by the global behaviour. This is confirmed by the analysis of Chapter 4, where this element with the given integration rule was used without perturbations of the global response.

30

2.5

2.0

1.5

1.0

0.5

0.06.05.04.03.02.01.00.0

M [kN.mm]

κ [10-4]

Layered element (10 × 2-point integration)

Regular element (seven-point integration)

x direction

y direction

Figure 3.9 - Comparison of different element types for pure bending

3.7 Comparison with experimental data of masonry strength The ability of the proposed model to represent the strength of different masonry types subjected to in-plane stresses has been demonstrated in detail by Lourenço (1996). Here, for the sake of completeness, the model will be compared with experimental results of masonry subjected to out-of-plane loading. It is noted that, as stressed before, out-of-plane behaviour is a direct consequence of in-plane behaviour and this comparison is seen by the author as redundant. In all of the experimental data reported next, only the tension regime is active, which means that no comparison can be made for the compression regime (again, for in-plane tests, available data has been compared in Lourenço (1996)). The experimental data available from different authors is reported in terms of maximum bending moments or flexural tensile strengths. The adopted methodology to fit the experimental data was to use the tensile strengths along the two material axes and calculate the shear strength, i.e. the α parameter in eq. (3.4), with the least squares method. The problem that remains is how to determine the uniaxial tensile strengths from ultimate bending moments. This problem has been addressed in Section 2.5, where it has been concluded that the ultimate bending moment is a result of the uniaxial tensile strength and the fracture energy. As these parameters cannot be explicitly determined from the available experimental results, in terms of bending moments, eq. (2.3) has been adopted to calculate the uniaxial tensile strength along each material axis. From this value, each fracture energy is calculated, by an trial-and-error procedure, so that the experimental value for the ultimate bending moment along each material axis is retrieved. The most complete set of strength data for masonry subjected to out-of-plane loading seems to be given by Gazzola and Drysdale (1986), who tested 25 wallettes of hollow concrete block masonry, with different dimensions and with the bed joints making a variable angle θ with the direction of loading, in four-point bending. The

31

comparison between experimental values and the model is given in Figure 3.9, which shows the average and standard deviation of the tests, for each orientation of loading.

fft [N/mm2]

Numerical

Experimental

906015 4530 75 0

1.20

0.80

0.40

0.00

Bed joint orientation θ [°]

Figure 3.10 - Comparison between the plasticity tension model and experimental results from Gazzola and Drysdale (1986), (h = 150 mm).

Material parameters: ftx = 0.94 N/mm2; fty = 0.32 N/mm2; α = 0.4; Gfx = 0.9 N.mm/mm2; Gfy = 0.06 N.mm/mm2

A smaller, but more complex, testing program of masonry panels subjected to out-of-plane loading was carried by Guggisberg and Thürlimann (1990). The thirteen panels of hollow clay brick masonry, denoted by Q1 to Q 13, with dimensions 2100 × 1200 × 150 mm3, were, initially, loaded with a vertical normal force nx. Afterwards, normal bending moments, mx and my, and twisting moments mxy were applied proportionally up to failure, while the normal force was kept constant. The test results, the proposed plasticity model results and the ratio between experimental and predicted failure are given in Table 3.3 and Figure 3.10. Panel Q9 was not successfully tested and was not reported in Guggisberg and Thürlimann (1990). The plasticity model seems to be able to reproduce also the strength behaviour of this experimental programme with good accuracy.

Table 3.3 - Comparison between plasticity tension model and experimental results from Guggisberg and Thürlimann (1990) , (h = 150 mm).

32

Material parameters: ftx = 0.20 N/mm2; fty = 0.005 N/mm2; α = 1.0; Gfx = 0.014 N.mm/mm2; Gfy = 0.001 N.mm/mm2

Test Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 ny [kN/m] -267 -67 -67 -67 -67 -267 -20 -67

mx [kN.m/m] 12.56 4.88 3.95 0.08 1.35 15.54 0.53 -4.00 my [kN.m/m] 1.77 0.59 1.75 1.88 1.95 1.29 1.83 1.44

Ratio 1.05 1.04 1.07 0.97 0.94 0.98 1.00 1.22

Test Q10 Q11 Q12 Q13 ny [kN/m] -67 -67 -67 -67

mx [kN.m/m] 0.00 0.00 1.02 1.20 my [kN.m/m] 0.00 1.25 0.49 1.35 mxy [kN.m/m] -3.51 -2.50 -1.87 -2.50

Ratio 0.89 0.83 1.07 0.77

��

��

��

��

��

��

��

��

��

��

��

��

0.80

Q13Q12Q11Q10Q8Q7Q6Q5Q4Q3Q2Q1

Test

Ratio

1.40

1.20

1.00

0.60

0.40

0.20

0.00

Figure 3.11 - Comparison between the plasticity tension model and experimental results

from Guggisberg and Thürlimann (1990)

33

4. Validation The use of the anisotropic continuum model for the analysis of masonry plates and shells is validated next by a comparison with experimental results available in the literature. Traditionally, experiments in masonry panels loaded with constant normal pressure, applied via air-bags, have been adopted by the masonry community as the most common out-of-plane test. 4.1 McMaster University hollow concrete block panels The first series of panels analysed consists of hollow concrete block masonry, see Figure 2.9 on page 11. The tests were carried out by Gazzola et al (1985) and are denoted by W. The inelastic properties of the composite material are obtained from Gazolla et al (1985), see Table 4.1 and Table 4.2. Only tensile inelastic behaviour is considered in the analysis. The elastic properties were not available and had to be estimated. The Young’s modulus in the vertical direction has been calculated from the masonry compressive strength according to Hendry (1990), the elastic orthotropy is assumed to equal two and the adopted Poisson’s ratio is 0.2. All the panels from the experiments, WI, WII, WIII, WP1 and WF, see Figure 4.1, are analysed with the composite plasticity model. All the panels were loaded until failure with increasing out-of-plane uniform pressure p. The only panel with in-plane action was WP1. This panel was loaded, previously to the application of the out-of-plane loading, with an in-plane confining pressure of 0.2 N/mm2, which was kept constant until failure of the specimen due to the pressure p. For each configuration, three different tests have been carried out and the results reported in Gazzola et al (1985) represent the average of the tests.

Table 4.1 - Material properties for McMaster University panels

Ex Ey Ez νxy νyz νxz Gxy Gyz Gxz 5000 10000 10000 0.2 0.2 0.2 3125 3125 3125

N/mm2 N/mm2 N/mm2 - - - N/mm2 N/mm2 N/mm2

Table 4.2 - Material properties for McMaster University panels

Tension regime ftx fty α Gfx Gfy

0.94 0.32 0.4 0.9 0.06 N/mm2 N/mm2 - Nmm/mm2 Nmm/mm2

For the numerical analyses eight-noded degenerated shell elements with two by two in-plane Gauss integration and seven-point Simpson through thickness integration are used. Due to the double symmetry of the structure only one quarter (or one half) of the structure needs to be modelled. Nevertheless, for easiness in the post-processing and to obtain more clear pictures, the full structure is modelled. For this purpose, a regular mesh of five by five rectangular elements is used in all the analyses. The analyses are

34

carried out with arc-length control enhanced with line searches, aiming at a globally convergent algorithm.

h = 150 mm

simply supported

z y x

p

(a)

Panel WI 2800

3400

Panel WII

5000

Panel WIII

5800

Panel WP1(in-plane action also)

5000

2800Panel WF

5000

Dimensions in mm

2800

28002800

(b)

Figure 4.1 - Panels with constant out-of-plane pressure p, Gazzola et al (1985): (a) out-of-plane loading; (b) geometry and in-plane loading.

The first panel analysed here, denoted by WI, has dimensions 3400 × 2800 × 150 mm3. The panel is simply supported in the four edges. Figures 4.2 to 4.7 illustrate the behaviour of the panel for different load stages.

35

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.2 - Panel WI. Results of the analysis at a pressure p equal to 2.6 N/mm2 (25% pre-peak): (a) total and (b) incremental deformed mesh; (c) principal moments; cracks in the (d) bottom and (e) top; plastic strain at the (f) bottom and (g) top face

36

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.3 - Panel WI. Results of the analysis at a pressure p equal to 5.2 N/mm2 (50% pre-peak): (a) total and (b) incremental deformed mesh; (c) principal moments;

cracks in the (d) bottom and (e) top face; plastic strain at the (f) bottom and (g) top face

37

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.4 - Panel WI. Results of the analysis at a pressure p equal to 7.8 N/mm2 (75% pre-peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


38

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.5 - Panel WI. Results of the analysis at a pressure p equal to 10.4 N/mm2 (peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


39

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.6 - Panel WI. Results of the analysis at a pressure p equal to 5.2 N/mm2 (50% post-peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


40

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.7 - Panel WI. Results of the analysis at a pressure p equal to 3.2 N/mm2 (ultimate): (a) total and (b) incremental deformed mesh; (c) principal moments;


41

It can be observed that the response is typical of out-of-plane loaded panels, with cracking starting to occur in the bottom of the panel, Figures 4.2 to 4.4. Clearly, predominant cracking occurs in the shorter span direction, which corresponds to higher bending moments and lower tensile strength. With increasing loading, cracking in the bottom progresses towards the supports and cracking in the top starts to appear, Figure 4.5. The principal bending moments at this stage have rotated significantly, cf. Figure 4.2, and the maximum bending moments occur know in the direction of the longer span. This is not only due to softening in the direction of the shorter span but also to the anisotropy of the material, with higher strength in the longer span direction. After peak load, the cracking pattern indicates a yield-line type of collapse with marked softening lines, Figure 4.6 and Figure 4.7. It is noted that the softening lines are not identified at peak but only at ultimate stage. This seems to confirm the idea that yield line design for a quasi-brittle material like masonry is not adequate. At ultimate stage, the principal moments distribution shows complete degradation of strength in the corners of the panel and, along the shorter span, in the centre of the panel. For panel WI, the calculated load-displacement diagram for a displacement d at the centre of the specimen and the experimental collapse load are given in Figure 4.8. A ductile behaviour is observed and good agreement is found in terms of failure load.

Experimental failure load

Displacement d at centre point [mm]

10.0 8.06.0 4.02.0 0.0

p [N/mm2]

0.0

4.0

8.0

12.0

Figure 4.8 - Panels WI. Load-displacement diagram and experimental failure load

(average of three tests). The second panel analysed here, denoted by WII, has dimensions 5000 × 2800 × 150 mm3. The panel is simply supported in the four edges. Except for the width (5000 mm), the panel is equal to panel WI. Figures 4.9 to 4.13 illustrate the behaviour of the panel for different load stages.

42

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.9 - Panel WII. Results of the analysis at a pressure p equal to 1.6 N/mm2 (25% pre-peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


43

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.10 - Panel WII. Results of the analysis at a pressure p equal to 3.3 N/mm2 (50% pre-peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


44

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.11 - Panel WII. Results of the analysis at a pressure p equal to 6.6 N/mm2 (peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


45

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.12 - Panel WII. Results of the analysis at a pressure p equal to 3.3 N/mm2 (50% post-peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


46

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.13 - Panel WII. Results of the analysis at a pressure p equal to 1.8 N/mm2 (ultimate): (a) total and (b) incremental deformed mesh; (c) principal moments;


47

Again, the response is typical of out-of-plane loaded panels, with cracking starting to occur in the bottom, Figures 4.9 to 4.10. It is now even more salient that predominant cracking occurs in the shorter span direction, which corresponds to higher bending moments and lower tensile strength. This behaviour remains until peak load, with, practically, no diagonal cracking near the supports. At peak, only a centre crack parallel to the longer span crosses completely the panel, Figure 4.11. After peak load, cracks start rapidly to progress towards the corners of the panel and a yield-line type of collapse with clearly marked softening lines is finally obtained, Figure 4.12 and Figure 4.13. The regions with zero bending moments identify the softening lines and, again, a secondary load path is formed with the slab arching through the long span. This example, is even more critical in demonstrating that a yield line type of analysis might not be adequate for masonry panels because, at peak, softening lines are not formed. The yield-line type of failure is retrieved, after peak, for substantially lower loads. For panel WII, the calculated load-displacement diagram for a displacement d at the centre of the specimen and the experimental collapse load are given in Figure 4.14. A ductile behaviour is observed and good agreement is found in terms of failure load. A lower failure load is obtained because the width/height ratio of panel WII is larger that the width/height ratio of panel WI.

Experimental failure load

Displacement d at centre point [mm]

10.0 8.06.0 4.02.0 0.0

p [N/mm2]

0.0

2.0

4.0

6.0

8.0

Figure 4.14 - Panels WII. Load-displacement diagram and experimental failure load

(average of three tests). The third panel analysed here, denoted by WIII, has dimensions 5800 × 2800 × 150 mm3. The panel is simply supported in the four edges. Except for the width (5800 mm), the panel is equal to panel WI and panel WII. Figures 4.15 to 4.20 illustrate the behaviour of the panel for different load stages.

48

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.15 - Panel WIII. Results of the analysis at a pressure p equal to 1.8 N/mm2 (33% pre-peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


49

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.16 - Panel WIII. Results of the analysis at a pressure p equal to 3.0 N/mm2 (first-peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


50

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.17 - Panel WIII. Results of the analysis at a pressure p equal to 3.2 N/mm2 (50% pre-peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


51

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.18 - Panel WIII. Results of the analysis at a pressure p equal to 5.5 N/mm2 (peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


52

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.19 - Panel WIII. Results of the analysis at a pressure p equal to 3.5 N/mm2 (67% post-peak): (a) total and (b) incremental deformed mesh; (c) principal moments;


53

(a) (b)

(c)

(d) (e)

(f) (g)

Figure 4.20 - Panel WIII. Results of the analysis at a pressure p equal to 2.4 N/mm2 (ultimate): (a) total and (b) incremental deformed mesh; (c) principal moments;

cracks in the (d) bottom and (e) top face; plastic strain at the (f) bottom and (g) top face The response is similar to the response of panels WI and WII. Cracking starts to occur in the

delft university of technology university of minho · tu-delft report no. 03.21.1.31.07 tno-bouw...

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