A joint fatigue–creep deterioration model for masonry with acoustic emission based damage assessment

Construction and Building Materials 43 (2013) 575–588

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

A joint fatigue–creep deterioration model for masonry with acoustic emission based damage assessment Adrienn Tomor a,1, Els Verstrynge b,⇑ a b

Faculty of Environment and Technology, University of the West of England, Frenchay Campus, Coldharbour Lane, Bristol BS16 1QY, UK Department of Civil Engineering, Catholic University of Leuven, Kasteelpark Arenberg 40, B-3001 Heverlee, Belgium

h i g h l i g h t s  Development of SN and ST type failure prediction models for fatigue and creep in masonry, based on experimental results.  Characterisation of stages of the damage accumulation process with acoustic emission techniques.  A joint deterioration model for fatigue and creep in masonry.

a r t i c l e

i n f o

Article history: Received 29 October 2012 Received in revised form 14 February 2013 Accepted 26 February 2013 Available online 3 April 2013 Keywords: Fatigue Creep Masonry Acoustic emission technique Failure prediction Joint deterioration model

a b s t r a c t The paper investigates the long-term fatigue and creep deterioration processes in historical brick masonry. Based on two independent laboratory test series, the relationship between stress level and life expectancy was considered for fatigue and creep loading in the form of SN type models. The process of deterioration was investigated with the help of acoustic emission technique to identify stages and characteristics of the damage accumulation process. Based on the test data and acoustic emission results, a joint SN type deterioration model was proposed to incorporate the static, fatigue and creep deterioration mechanisms. A mathematical relationship was proposed for the joint fatigue–creep model and good agreement was found between the test data and the proposed model. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Over a long period of time fatigue and creep phenomena can lead to changes in strength and material properties, cause gradual deterioration or even collapse. While historical masonry structures are generally expected to have relatively low stress levels with little consequence on their long-term performance, increasing age and higher performance demands can result in increasing stress levels and accelerated long-term deterioration. Fatigue and creep are both time-dependent progressive deterioration phenomena. Fatigue is time-dependent deterioration under long-term cyclic loading and creep occurs under constant loading. Fatigue may be relevant, for example, to masonry arch bridges with heavily increased traffic loading or to tall structures subjected to wind loading. While compressive fatigue loading is rarely seen on masonry arch bridges, fatigue shear failure can be ⇑ Corresponding author. Tel.: +32 (0) 16 321987; fax: +32 (0) 16 321976. E-mail addresses: [email protected] (A. Tomor), [email protected] (E. Verstrynge). 1 Tel.: +44 117 328 3516. 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.02.045

relatively often observed in the form of ring separation (see Fig. 1). Although the present paper deals with fatigue compression only, the basic principles behind the two phenomena are very similar and the proposed theory can be adapted for both compression and shear. Creep damage can occur in tall masonry towers with high compressive stresses due to their large self-weight [1,2]. Creep may be observed as a series of vertical cracks through the vertical mortar joints and bricks. Fig. 2 shows an example of creep cracks at the base of a bell tower (Saint-Willibrordus church, Meldert, Belgium). The cracks were monitored for just 3 weeks, during which highly unstable crack growth was observed and creep damage suspected. The church was subsequently closed, but the bell tower collapsed before strengthening measures could be put in place [2]. Both fatigue and creep failure develops through time-dependent deformation, during which micro cracks coalesce into macro cracks and can lead to unstable crack development and sudden failure. Damage development may occur over a long period of time (even under relatively low stress levels) and may be difficult to assess as it is not necessarily related to sudden changes in the loading

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Fig. 1. Fatigue shear failure in the form of ring separation in a masonry arch.

conditions. Environmental effects (e.g. moisture saturation) and material deterioration may also interact with the creep and fatigue processes and reduce the residual strength over time [3,4]. Fatigue and creep are long-term, stochastic damage mechanisms with non-linear damage propagation. They are sensitive to external influences and small changes in the masonry properties, which make accurate modelling and failure prediction difficult. Fatigue and creep in masonry can be modelled by empirical formulae or by more advanced constitutive models, e.g. based on rheological models and/or damage mechanics. Empirical models are based on regression analysis of experimental data and can predict fatigue and creep development for a specific set of data. Constitutive

models are more general, but require a large number of input parameters and the awareness of scatter on these input values. Rheological models allow the effects of changes in humidity, temperature and ageing to be taken into account [5]. Advanced creep models for masonry have been developed and experimentally validated by Papa and Taliercio [6], Binda and Anzani [1,7], Verstrynge et al. [8], Pina-Henriques and Lourenço [9]. Modelling the combined effects of creep deformation, humidity and age at loading was addressed by Van Zijl [10] and Choi [11], while Ferretti and Bazant [12] also included the effects of carbonation. Advanced modelling of fatigue deterioration in masonry has not been possible to date due to the complexity of the issue and the limited amount of available test data. Material testing under fatigue has been carried out by Roberts et al. [13,14], Abrams et al. [15], Brencich et al. [16,17], Ronca et al. [18], Tomor and Wang [19] under laboratory conditions and reviewed by Wang et al. [20]. Large-scale fatigue tests have been carried out by Melbourne et al. [21,22] on masonry arches and basic principles for assessing the fatigue capacity of masonry have been proposed by also Melbourne et al. [23]. Creep testing of masonry has been performed by Forth et al. [24,25], Lenczner [26], Binda and Anzani [7,27], Ignoul et al. [28] and Verstrynge et al. [2,29]. While fatigue and creep deterioration have so far been studied separately for masonry, their joint interaction has not been considered before. The current paper investigates the long-term fatigue and creep deterioration processes through two sets of experimental test series (obtained by two independent research groups) and brings them together into a joint fatigue–creep deterioration model. Only basic fatigue and creep are considered in the current research, focussing on stress induced damage, leaving external environmental effects, such as weathering or moisture ingress outside the scope of this paper. To help identify the relevant characteristics of fatigue and creep deterioration processes, Acoustic Emission (AE) monitoring has been applied during both laboratory test series. Acoustic emission has previously been used for studying fatigue deterioration in masonry by Melbourne and Tomor [30], De Santis and Tomor [31] and Masera et al. [32] under laboratory conditions and by Tomor and Melbourne [33], Shigeishi et al. [34], Carpinteri and Invernizzi [35,36] on vaults and masonry arch bridges. To study creep deterioration in masonry, acoustic emission monitoring has been used by Verstrynge et al. [37,38] and by Carpinteri and Lacidogna [39]. 2. Experimental approach 2.1. Setup for fatigue testing

Fig. 2. Creep damage at the base of the bell tower of Saint-Willibrordus church, Belgium, a few weeks before collapse.

Brick masonry prisms (Fig. 3) have been built and tested at the University of the West of England (UWE, Bristol, UK) under long-term fatigue loading to help develop understanding of the performance of masonry arch bridges under long-term traffic loading. The prisms were built using solid 213  100  65 mm3 Wienerberger Warnham Red Terracotta moulded bricks with 22.6 N/mm2 strength and 2127 kg/ m3 density. 1:1:6 cement:lime:sand mortar was used by volume with NHL3.5 lime and 3 mm sharp washed sand. Joint thickness was 8 mm and specimens were cured for a minimum of 6 month before testing. The average compressive strength of prisms (SAv) was 10.9 N/mm2 (1.0 N/mm2 standard deviation (SD) and 9.3% coefficient of variation (CV)), see Table 1. Prisms were tested under compression using a 250 kN actuator at 0.15 kN/s loading rate. Layers of 3 mm plywood and 30 mm thick steel plates were placed on top and bottom of the specimens for load distribution. 10 prisms were tested under static loading to identify the average compressive strength (SAv). Long-term cyclic loading was subsequently applied to further specimens at 2 Hz frequency between a minimum stress level (SMin) and a maximum stress level (SMax), that were defined as percentage of the average static strength (SAv), see Fig. 4. The minimum stress is intended to represent the dead load of a bridge due to its self-weight and the maximum stress the variable live loads induced by passing traffic. The minimum stress (or base load) was applied either at 10% or 30% of the average static strength (SAv) during the tests.

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Fig. 3. Prisms with instrumentation (a), before (b) and after (c) testing.

Table 1 Overview of material characteristics of bricks and masonry for fatigue tests. Values are indicated with average strength (SAv), standard deviation (SD) and coefficient of variation (CV).

Clay bricks Prisms

Number of specimens

Compressive strength (SAv) (N/mm2) (SD; CV)

Remarks

70 10

22.6 (2.3; 10.2%) 10.9 (1.0; 9.3%)

Wienerberger Warnham Red solid

Stress SMax

SMin Number of cycles Fig. 4. Loading pattern for fatigue tests.

2.2. Setup for creep testing A similar setup was used in the Reyntjens Laboratory at the Catholic University of Leuven (KU Leuven, Belgium) to test the creep behaviour of masonry under uniaxial compressive loading. Relatively low-strength solid 188  88  48 mm3 Terca ‘Spanish Red’ clay bricks and 1:2.5 lime:sand mortar was used by volume with hydrated lime and 2 mm sand. Both mortar type and composition were chosen to be representative of historical masonry in Western Europe. Strength characteristics of the masonry components and full scale specimens are listed in Table 2. Masonry columns with overall dimensions 188  188  600 mm3 (length  width  height) were built with 10 layers of bricks (two bricks per layer) and 10 mm mortar joints as shown in Fig. 5. Each specimen was constructed on a concrete tile and a similar concrete tile or a steel plate was placed on top of the columns after 1.5 months. Mortar samples and masonry specimens were stored at a temperature of 20 °C and relative humidity of 60 ± 5% for three months before testing. The lime mortar was fully carbonated by storing the mortar and masonry specimens in a carbonation chamber with high CO2 level for several weeks.

Displacement-controlled quasi-static compressive tests with 10 lm/s displacement velocity were performed for a set of 7 masonry test specimens to obtain the average compressive strength (SAv). Stress levels during subsequent creep tests were defined as a percentage of the average static strength. Accelerated Creep Tests (ACT) were performed following a stepwise loading pattern [7,40]. The load was first increased to a stress level below half the average compressive strength (SAv) and kept constant for a pre-defined period of time. The load was subsequently increased and kept constant for another period, see Fig. 6. The process was repeated until failure occurred. The magnitude of load increase was reduced when 70% of the average compressive strength was reached. The advantage of the accelerated creep test is that creep and acoustic emission rates could be observed and compared at different stress levels. If the specimen failed during a constant stress interval, a three-phase creep curve could also be observed. Accelerated creep tests were performed either in the short- or long-term: (a) for Short-Term Accelerated Creep Tests (short-term ACT) the duration of the loading steps (Dt) was 3 h and failure generally occurred within 24–36 h. (b) for Long-Term Accelerated Creep Tests (long-term ACT) the duration of the loading step (Dt) was 2 months and failure generally occurred within 1.5– 2 years for each specimen. (c) Additionally, 1-Step Creep Tests (1-SCT) were performed to provide data for damage accumulation at specific stress levels. During the 1-step tests the load was increased up to 80–90% of the average compressive strength within half an hour and kept constant until failure occurred. By maintaining the stress level for an extended period of time, the tertiary creep phase had more time to develop and the likelihood of failure during stress increase was reduced. A constant stress level of 80–90% of the average strength (SAv) was chosen based on experience, to allow the tertiary creep phase to develop within 3–24 h. Quasi-static compressive tests and short-term creep tests were performed in a Dartec hydraulic press (Fig. 5b) with 5000 kN capacity and an additional loadcell to limit the maximum load to 500 kN and increase accuracy. Long-term and 1-step

Table 2 Overview of material characteristics of bricks, mortar and masonry for creep tests. Values are indicated with average strength (SAv), standard deviation (SD) and coefficient of variation (CV).

Clay bricks Mortar Masonry

Number of specimens

Compressive strength (SAv) (N/mm2) (SD; CV)

Remarks

10 9 7

8.02 (1.13, 14.1%) 1.21 (0.19, 15.7%) 3.73 (0.47, 12.6%)

NBN EN 772–1:2000 Terca ‘Spanish Red’ clay bricks 1:2.5 lime:sand mortar

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Fig. 5. Test specimen, (a) short-term (b) long-term and (c) creep test setup.

Fig. 6. Loading pattern for stepwise short-term and long-term accelerated creep tests (ST-/LT-ACT) and for 1-Step Creep Tests (1-SCT).

creep tests were performed in individual steel frames for each specimen (Fig. 5c) due to the extended time requirements (up to 2 years). The latter setup closely resembled the loading and boundary conditions of the short-term test setup. The load was applied and increased manually using a hydraulic jack with an accumulator to compensate for relaxation and keep the loading constant. All creep tests were performed under load-control.

2.3. Instrumentation (strain and acoustic emission monitoring) During both test series Linear Variable Differential Transformers (LVDTs) were used to monitor deformation. During fatigue tests deformation was measured across two mortar joints. Under short-term creep tests axial and lateral deformation was constantly recorded on all four sides of the specimens using LVDTs and under long-term creep tests deformation was recorded periodically using a Demountable Mechanical DEMEC strain gauge. In addition to deformation monitoring, the Acoustic Emission (AE) technique was used during both test series to help identify damage development characteristics under fatigue and creep loading. The technique has the advantage that it records the structure’s response to loading in real time. Acoustic emission technique detects high-frequency transient elastic waves that are emitted by the material itself during crack growth [41,42]. Waves are recorded on the surface by piezoelectric sensors, pre-amplified, filtered and amplified before they are processed by the data logger. AE amplitude is detected in lV, converted to AE decibel by 1 dB = 20  log (Voltage (lV)/1 lV) [43] and energy is calculated as the area under the envelope of an AE hit. Background noise is eliminated through a minimum amplitude threshold. For each AE hit a number of parameters (e.g. amplitude, energy, duration, count, arrival time) and the waveform are recorded. If an AE hit is recorded by one or more sensors it is defined as an AE event. The amount of detected AE hits and energy is influenced by the applied hardware and software (e.g. sensor type and frequency range, applied filters, waveform sampling frequency, threshold level, hit duration discrimination time, etc.). Therefore, software defined parameters should be kept constant for subsequent tests and results obtained with different acquisition systems should be compared carefully and in relative terms. The amount of AE hits detected during a time interval is also sensitive to a number of setup-specific boundary conditions, such as quality of the coupling between sensor and test specimen, material density, speed of wave propagation, interference with surrounding test equipment, and presence of internal cracks and voids. Due to the variability of the boundary conditions, relative change in detection level is therefore better suited for defining damage accumulation instead of the absolute amount of hits/energy.

During the fatigue test series at the University of the West of England, acoustic emission monitoring was performed using a Physical Acoustics Micro-SAMOS system with eight-channel PCI-8 AE board connected to PC via PCMCIA port and AEwin™ software was used to process the data. IL40s voltage preamplifiers with 1–400 kHz frequency bandwidth and 40 dB gain were used. 150 kHz resonant R15 sensors with 50–200 kHz operation frequency were attached to opposite sides of the specimens by means of a thin layer of hot-melt glue, which has proven to be a good couplant for laboratory conditions. The AE system was calibrated using the standard method of pencil lead breaks [44] for every test to verify the sensitivity of the sensors. Threshold level was adjusted during each test to avoid saturation. AE monitoring was performed during each of the static tests and during some of the fatigue tests that failed within a relatively short time interval, as long-term tests exceeded the capacity of the data logger. During the creep test series at the KU Leuven, acoustic emission monitoring was performed using a 4-channel Vallen AMSY-5 system. 375 kHz resonance sensors with 250–700 kHz operation frequency were attached to opposite sides of the specimens by means of a thin metal plate that was glued to the surface and allowed easy re-mounting for periodic monitoring. Vacuum grease was used as a couplant between the sensor and the metal plate. The preamplifier gain was set to 34 dB with a fixed threshold level of 34.5 dB. High frequency noise was filtered by applying a low-pass filter at 500 kHz. Pencil lead breaks were used for system calibration [44] and could be detected up to 30 cm from the sensor, indicating that almost all damage sources could be detected by only two sensors in the middle of the specimen. During the quasi-static compression tests, short-term creep tests and 1-step creep tests, AE events were continuously monitored. For long-term creep tests, AE sensors were attached 1 day before stress increase and remained in position for 6 days after stress increase. If unstable or increasing damage accumulation was observed after 6 days, the monitoring period was extended. Additional AE monitoring was performed for 24 h halfway between two loading steps to identify if emission rate was stable. 2.4. Experimental setup: similarities and differences It is evident that the two test series for fatigue and creep deterioration were not part of the same research program and differed in the choice of materials and testing arrangements. However, despite of the differences, they had common goals and methodologies. Both fatigue and creep are time-dependent deterioration processes at stress levels below the maximum strength of the material. While fatigue is induced by variable stress levels, creep may be considered as fatigue loading with zero stress amplitude. It is therefore expected that common features may be observed between the two phenomena that can help develop greater understanding of the damage evolution process in masonry. Before discussing the test results in detail, differences between the two test series are summarised below.  The masonry composition, specimen geometry (stacked bricks and bonded bricks) and compressive strength of the fatigue and creep specimens were different (10.9 N/mm2 and 3.73 N/mm2 respectively). Therefore, relative stress (in % of SAv) rather than absolute stress will be used for comparison. Damage evolution during creep has been found to follow similar patterns for similar masonry types if compared in terms of relative stresses [2].  Strain was measured differently: axial strain across 2 bed joints during fatigue loading and lateral and axial strain over 1 and 3 or 5 bed joints during creep loading. Strain results will therefore not be directly compared in this paper.

A. Tomor, E. Verstrynge / Construction and Building Materials 43 (2013) 575–588  Different AE systems and sensors were used that are likely to influence the magnitude and properties of detected acoustic emission events. AE in masonry is however also affected by a series of further factors, such as system setup, inhomogeneities of the material, relative diffraction, reflection and attenuation of brick and mortar, etc. As direct quantitative comparison of the two test programmes would be unreliable, qualitative analysis will be used to identify changes in emission patterns and indicate damage evolution.

3. Deterioration process Deformation in masonry is a consequence of elastic and plastic strain, micro-cracking and pore collapse within the material (the latter phenomena producing acoustic emissions). Before deformation can be measured on the surface of the masonry, AE technique can generally capture emissions due to micro-crack development and crack growth. Based on the AE recordings under quasi-static compressive loading, fatigue loading and creep loading, characteristics of the deterioration processes were investigated to identify phases of damage development. Thereby, a parameter-based technique is adopted, analysing AE parameters such as amplitude, energy and event-counting to assess the damage accumulation within the masonry specimens. Other techniques, involving a more complex signal-based analysis were developed for concrete [41], but have not yet been applied on masonry due to the very complex fracture behaviour of masonry under compression and the heterogeneity of the material.

3.1. Deterioration under static loading It is generally accepted that macroscopic fracture in brittle materials under uniaxial compression is caused by nucleation, growth, interaction and coalescence of micro cracks [45,46]. Micro fracture originates at flaws in the material, such as grain boundaries, and the stress at the crack tip responsible for crack growth is of tensile nature. Consequently, cracks propagate in the direction parallel to the principal compressive stress. In masonry, the interaction between bricks and mortar, which have different stiffness properties, is assumed to contribute to the concentration of stresses and subsequent crack formation. In the general case where the brick units have a higher stiffness than the mortar, deformation of the mortar induces local tensile stress concentration in the adjacent bricks, while enclosed mortar joints are subjected to triaxial compressive stresses [47].

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With the help of AE technique, the process of micro crack formation was detected from the very onset during compressive tests on masonry prisms. Typical acoustic emission amplitude and average energy recordings, obtained under quasi-static compressive loading of prisms, are shown in Fig. 7a and b. Noticeable emission was recorded from almost the start of load application and clear increase was observed around 40% and 95% of the maximum stress (SMax) in both amplitude and average energy. The amplitude threshold was set to 40 dB at the beginning of the test and was progressively raised to avoid saturation (loss of linearity between input and output in the amplifier due to excessive signal drive). Although low-amplitude hits were not recorded under high loads, they were considered to be less significant in the presence of high intensity signals and have negligible effect on the energy output (shown on the log scale). Based on the AE recordings, three phases of the crack development can be identified during quasi-static compression (marked as ‘‘S-Phase’’ for ‘‘Static-Phase’’), such as:  S-Phase 1: relatively constant low-level emission is observed, likely to be associated with compaction and crack nucleation, while elastic strain is the dominant cause of deformation at this stage.  S-Phase 2: micro-crack development from around 40% of SMax, likely to be associated with development of vertical cracks, crack extension and further compaction of the mortar. Deformation of the mortar induces local stress concentration in the adjacent bricks with increasing plastic behaviour and development of micro-cracks in the bricks. S-Phase 2 appears to have a steeper initial section (a) with rapid amplitude and energy increase, followed by somewhat reduced amplitude and energy release (b), but the exact mechanisms in the two sub-phases are yet to be identified  S-Phase 3: very high energy release from around 95% of SMax, characterised by macro-cracking and bridging while the tensile strength of bricks is being reached and fast fracture occurs in the bricks. In order to gain better insight into the fracture development process of masonry components, half size bricks (105  100  65 mm3) and 100  100  100 mm3 (1:1:6) mortar cubes were subsequently tested under quasi-static compression (Fig. 8a and b). For bricks, the AE amplitude and average energy increased rap-

Fig. 7. Typical AE amplitude (a) and average energy (b) vs. stress during compression of a masonry prism.

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Fig. 8. Typical AE amplitude and average energy during compression of a brick (a and b) and mortar cube (c and d).

idly between 0% and 30% SMax (up to ca. 90–100 dB) and a very high emission level was maintained as the bricks disintegrated between 30% and 100% SMax. Elastic deformation and micro-cracking are likely to have taken place very early on during the loading process and macro-cracking is likely to occur from as early as 30% SMax due to the brittle nature of the bricks. For mortar, AE amplitude and energy remained relatively low (40–60 dB) up to ca. 95% SMax, followed by sudden increase and immediate failure (Fig. 8c and d). During compressive loading of prisms, changes in energy levels may therefore be associated with the following mechanisms:  S-Phase 1: low-level emission (40–60 dB and ca. 103 aJ, Fig. 7), associated with compression of the mortar joints.  S-Phase 2: medium-level (60–85 dB and 103–106 aJ), crack extension in the mortar joints is likely to be overridden by the micro-crack development in the bricks.  S-Phase 3: high-level emission (80–90 dB and 106–107 aJ), likely to be almost entirely associated with macro-cracking of bricks. When comparing the AE output for prisms, bricks and mortar cubes in terms of amplitude level, the micro-cracking and critical crack propagation seem to have initiated when ca. 60 dB was exceeded. 60 dB therefore appears to be an approximate critical limit for damage development for the applied test setup in all three cases presented.

3.2. Deterioration under fatigue loading 3.2.1. Test results under fatigue After the static loading tests, prisms were subjected to longterm cyclic loading to investigate the fatigue deterioration process. Sinusoidal cyclic loading was applied at 2 Hz frequency for a minimum of 3,000,000 load cycles unless failure occurred. Loading was however continued to several million cycles in most cases and was only stopped due to excessive time requirements. A typical example of acoustic emission recording during a fatigue test is shown in Fig. 9 for a loading range between 10% (=SMin) and 70% stress (=SMax). During the fatigue deterioration process three fatigue phases (‘‘F-Phase’’ for ‘‘Fatigue-Phase’’) have been distinguished:  F-Phase 1: (0–75% of the total number of cycles), relatively low, constant emission (40–50 dB amplitude and 10–102 aJ absolute energy);  F-Phase 2: (75–95% cycles), small increase in emission (50– 60 dB amplitude and 10–102 aJ absolute energy);  F-Phase 3: (95–100% cycles), rapid increase in emission and sudden failure. Brief warning period. While it is difficult to make specific assumptions about the fatigue deterioration process of prisms at this stage, it may be helpful

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Fig. 9. Typical AE amplitude (a) and average energy (b) vs. number of cycles during fatigue compression of a masonry prism.

to compare the fatigue recordings (Fig. 9) with the quasi-static compression tests for prisms (Fig. 7), bricks and mortar (Fig. 8). Any comparison should however be considered purely on a qualitative basis, as detected energy levels and wave amplitudes are dependent on a number of setup specific conditions. Under fatigue loading (Fig. 9) the very low emission in F-Phase 1 (40–50 dB, 10–102 aJ) and relatively low emission in F-Phase 2 (50–60 dB, 10–102 aJ) may be compared with the energy levels emitted during the first static compression Phase of a prism (SPhase 1, Fig. 7a) and static compression of a mortar cube (40– 60 dB, Fig. 8c) respectively. It is therefore likely, that F-Phase 1 and 2 are mostly associated with compaction of the mortar. Micro-cracking in bricks can be assumed to take part gradually as a very slow process with limited energy emission within F-phase 1 and 2. F-Phase 3 with sudden increase in emission (from 60 dB to up to 100 dB) is likely to encompass S-Phase 2b and 3 during static compression (60–100 dB, Fig. 7a). The high emission level indicates rapid macro-crack development in the bricks and is related to the high-energy emission under static compression of a brick from 30% SMax (80–100 dB, Fig. 8a). It can thus be assumed that once the mortar has been compacted and micro-cracking has initiated during F-Phase 1 and 2, marco-crack development takes place rapidly within the brick during F-Phase 3. In terms of critical amplitude, crack propagation appeared to have developed soon after 60 dB was reached, similarly to the static test results. In addition to the qualitative assessment, more work and dedicated waveform analysis will be necessary to allow emission levels to be defined as a function of the deterioration process. 3.2.2. SN curves under fatigue Fatigue behaviour is often described using analytical models which define the relationship between stress level (S) and number of cycles to failure (N). These models are referred to as SN curves and their parameters are estimated based on experimental results. The simplest model defines a linear relationship between the stress level and the logarithm of the number of cycles. Results of the static and fatigue compression tests are summarised for the relative stress (% of SAv) against the number of cycles at failure (N) in Fig. 10. During the fatigue tests the load was cycled between a minimum stress (SMin, 10% or 30% of SAv) and a maximum stress (SMax). Static test results are shown as failure at 1 cycle and tests that were stopped without failure are also included in the graph. For 10%

Fig. 10. SN curve for fatigue tests under compression: stress (S) vs. number of cycles to failure (N, log scale).

minimum stress the best fit curve is also indicated with upper and lower limits (dotted lines) defined by the coefficient of variation for the static test results (9.3%). At first inspection it is clearly visible that the maximum number of cycles increases for reduced stress levels. Also, the limited number of tests at 30% minimum stress (some overlap in the test data) show noticeably higher life expectancy compared to the 10% results. The higher the minimum stress, the shallower the SN trendline is expected to be due to the reduced stress variation. Comparing the test data at any particular stress level however shows large variations in the number of cycles. The scatter is not surprising, as fatigue deterioration is a stochastic process and the fatigue stress level is defined as a percentage of the average static strength (SAv) for a batch of samples. A selected fatigue stress level therefore does not necessarily represent the actual stress level for the particular specimen. Sensitivity to small variations in the material properties, loading and environmental conditions can result in relatively high scatter. A coefficient of variation up to around 20% was indicated by Schueremans [48] for the compressive strength of masonry specimens under controlled laboratory conditions and even higher percentage are expected under field conditions. Due to the natural variability of masonry, it is therefore inherently impossible to define the relationship between the stress level and life expectancy as a simple deterministic relationship. Probabilistic

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Table 3 Approximate life expectancy for 10% minimum stress (extrapolated from regression curve in Fig. 10). Maximum stress

Life expectancy

(%)

(Cycles)

Ratio

55 50 45

6  106 4  107 2  108

1/6.6 1 5

analysis is therefore necessary to take the aleatoric uncertainties into account and enable a practical relationship to be established between the range of stress levels and desired confidence level, as discussed by Verstrynge [2] for creep and Casas [49] for fatigue in masonry. In terms of practical application, SN curves may be used to indicate the life expectancy at any chosen stress level. Table 3 lists examples of the average life expectancy for 10% minimum stress, indicated with the help of the projected trendline in Fig. 10. As an example, at 50% maximum stress the trendline is indicating 4  107 cycles. If the stress level is reduced from 50% to 45%, the life expectancy increases fivefold (from 4  107 to 2  108). If the stress level is increased from 50% to 55%, the indicative life expectancy reduces almost sevenfold (from 4  107 to 6  106). The example illustrates the potentially enormous impact of the assumed stress level on the outcome of the fatigue assessment. Note, that the numbers are only indicative to demonstrate the methodology and cannot be taken as actual values. The maximum number of cycles during the testing program was ca. 6.6  106 cycles (at 2 Hz frequency) that took 5.5 weeks of constant testing. In terms of gathering long-term test data, it is unlikely that individual specimens could be physically tested over 6-month duration (3.1  107 cycles, indicated as boundary ‘A’ in Fig. 10) that would still be insufficient to prove, disprove or identify the existence of a long-term fatigue limit for masonry. The presented test data therefore does not support nor contradict the existence of a fatigue limit. While it may not be possible to identify a fatigue limit, in terms of practical application, a permissible limit state (PLS) ‘‘at which there is a loss of structural integrity that will measurably affect the ability of the bridge to carry its working loads for the expected life of the bridge’’ [23] would be of more interest. The permissible limit may be defined by the maximum possible number of cycles during the expected lifespan of a structure (e.g. number of vehicles over a bridge). If, for example, the expected life of a bridge is in the order of 300 years, under (non-realistic) continuous 2 Hz loading the maximum number of cycles would add up to ca. 2  1010 (indicated as boundary ‘B’ in Fig. 10). The related stress level for the maximum possible number of cycles would therefore indicate a permissible limit under which no fatigue deterioration is likely to take place in the structure/material. 3.2.3. Mathematical models for fatigue A limited number of models have been suggested to date for the SN relationship under fatigue loading of masonry. Roberts [13] proposed a lower bound fatigue limit based on a series of small-scale tests (on dry and wet masonry) Eq. (1):

ðDS  SMax Þ0:5 ¼ 0:7  0:05  logðNÞ SAv

ð1Þ

where DS is the stress range (DS = SMax  SMin), SMax the maximum stress, SAv the average compressive strength and N the number of cycles. The model suggests a linear relationship between the maximum stress level (SMax) and the logarithm of the number of cycles (log(N)) and can be used for modelling fatigue behaviour at

different base loads by re-calibrating the model parameters. The model is however based on experimental tests with limited duration (as are all fatigue and creep regression models) and log(N) does not become zero for SMax equalling the compressive strength if a non-zero base load is defined. The fatigue model given by Casas [49] Eq. (2) was developed for probabilistic analysis of masonry arch bridges and the parameters were based on statistical analysis of the experimental results by Roberts et al. [13]. Progressive fatigue deterioration is described according to the Weibull distribution, which is widely used for fatigue analysis of metals.

S ¼ A  NBð1RÞ

> 0:5

ð2Þ

where S is the ratio of the maximum stress to the average strength (S = SMax/SAv), N the number of cycles and R the ratio of the minimum stress to the maximum stress (R = SMin/SMax). The exponent of N depends on SMin, which enables the model to correctly represent the shallower slopes in case of higher base loads. An endurance limit of S > 0.5 is assumed. Taking a logarithm on both sides converts the model into:

logðSÞ ¼ logðAÞ  Bð1  RÞ  logðNÞ

ð3Þ

In contrast with the previous SN models, linear relationship is defined between log(S) and log(N). In Fig. 10 the analytical model by Casas is represented for 10% and 30% minimum stress (SMin) together with the experimental test results. The value of 1 is adopted for parameter A, as the failure time should be 1 cycle for values of SMax equalling the compressive strength. The value of B is set to 0.04 for the current test data. While it is relatively easy to achieve good agreement between the experimental and mathematical models for low numbers of cycles, the predicted relationship for high numbers of cycles becomes increasingly sensitive to the chosen model in the absence of experimental data. In order to improve the proposed model, the SN relationship will be revisited in Section 4 for a joint fatigue–creep model in the presence of creep data. 3.3. Deterioration under creep loading Parallel to the fatigue tests, creep deterioration of masonry was studied at the KU Leuven through a laboratory test series. Compressive creep occurs under constant long-term stress during which gradual increase in strain can be used to identify phases of creep deterioration (‘‘C-Phases’’) [1,2], such as:  C-Phase 1: primary creep phase with decreasing strain rate.  C-Phase 2: secondary creep phase with constant strain rate where damage development is related to the relative stress level.  C-Phase 3: tertiary creep phase with increasing strain rate and sudden failure. It is generally assumed that the tertiary creep phase occurs if the stress is high enough for the damage accumulation to become unstable. This implies that a stress limit value exists, which is called the ‘creep limit’ or ‘viscosity limit’ value below which only primary and secondary creep is observed. For concrete, a theoretical framework was presented by Rüsch [50] and adopted for masonry by Binda [1]. It has been suggested that the viscosity limit can be defined with the help of accelerated creep tests by analysing the strain rate (=slope of the strain evolution in time) in subsequent loading steps during the secondary creep phase. If the strain rate starts to increase during subsequent loading steps, the viscosity limit is likely to have been reached [2]. For the low-strength masonry specimens with air-hardening lime mortar the viscosity

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limit was found to be around 40–50% of the average strength (SAv) and for medium-strength masonry (with hydraulic lime mortar, hybrid lime-cement mortar or low-strength cement mortar) around 60–70% SAv [2,51].

3.3.1. Test results under creep In order to help identify the stages of creep deterioration in masonry described above, acoustic emission monitoring has been used for 1-step, short-term and long-term creep tests. A range of studies is available on acoustic emission monitoring of compressive creep in rocks [52,53] and concrete components under bending [54]. However, very limited information is available on AE monitoring of compressive creep in masonry. Fig. 11 shows a typical AE recording for a 1-step creep test at 88% of the average compressive strength. The evolution of the stress level during the test is indicated in Fig. 11b. In Fig. 11a, AE amplitude (left vertical axis) and cumulative number of AE events (right vertical axis) are shown against time. An acoustic emission event is defined as the detection of an AE hit at one or both of the AE sensors. The three phases of creep deterioration can clearly be distinguished. After initial stress increase, AE amplitude and the slope of the cumulative AE event curve decrease in C-Phase 1. This is followed by a lower, constant emission in C-Phase 2 and increase and failure of the specimen in C-Phase 3. Fig. 12 shows representative AE and strain recordings under a short-term accelerated creep test (short-term ACT) for loading intervals (DT) of 3 h. AE amplitude (left vertical axis) and cumulative number of AE events (right vertical axis) are shown against time. Fig. 12b presents the axial (negative) and lateral (positive) strains against time for the same test. Close relationship between

Fig. 12. Typical short-term accelerated creep test with stepwise load application (DT = 3 h) (see Fig. 6). (a): AE amplitude and cumulative AE event count vs. time. (b): axial (negative) and lateral (positive) strains vs. Time.

strain and acoustic emission rate can be observed in the two graphs. Under short-term accelerated creep tests, failure of the specimen often occurred during stress increase and a tertiary creep phase could not be observed. For 1-step creep tests and long-term accelerated creep tests (long-term ACT), failure mostly occurred during a constant loading phase and tertiary creep could be observed. Fig. 13 shows the AE event rate for three typical stress levels (41%, 82% and 95% of SAv) during a long-term accelerated creep test. The AE event rate is defined as the AE event count per time interval, in this case an interval of 1 min. An increase in the AE event rate within the secondary creep phase can be noticed when comparing Fig. 13a and b and the development of a distinct tertiary creep phase before failure can be observed during the final stress level (Fig. 13c). Within the creep test series, a clearly distinguishable 3-phased creep curve (ranging from 15 min to 65 days duration) was observed during 20 creep tests (short-term ACT, long-term ACT and 1-SCT). In terms of AE and strain recordings, the following common patterns can be observed during the short-term, long-term and 1-step creep tests:

Fig. 11. Typical 1-step creep test. (a): AE Amplitude and Cumulative AE event count vs. Time with indicated creep phases (C-Phase 1, 2, 3). (b): absolute and relative stress levels.

 After each stress increase, a primary creep phase with decreasing AE amplitude can be observed in all loading cases;  At relatively low stress levels (below the viscosity limit of 40– 50% SAv) strain and AE levels tend to diminish gradually to a minimum level during the secondary creep phase. Under higher stress levels a clear secondary creep phase can be observed (Figs. 13a and b respectively);

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Fig. 13. Typical long-term accelerated creep test with stepwise load application (DT = 2 months): AE event rate vs. time for 41% (a), 82% (b) and 95% of SAv (c).

 If creep failure occurs within the constant stress interval, tertiary creep can clearly be distinguished in the AE recordings (see Fig. 11a for the 1-step and Fig. 13c for the long-term creep tests). Figs. 11 and 12 presented AE amplitude levels for 1-step and short-term accelerated creep tests. During stress increase and in C-Phase 1, relatively high AE amplitudes can be observed. The lower AE amplitudes (50–60 dB) during C-Phase 2 can be related to compaction of the mortar and possibly a very slow micro-cracking process in the mortar and bricks, comparable with fatigue F-Phase 2. In C-Phase 3, an increase in AE amplitude accompanies macrocrack development within the bricks. As the specimen composition and AE acquisition systems differ for the creep and fatigue tests series, a quantitative comparison of the AE amplitude and energy emission levels will however not be made.

3.3.2. ST curves under creep Similarly to the SN curves for fatigue loading, the relative stress level (S in % of SAv) and time to failure (T) may be expressed as ST curves for creep loading. Fig. 14 shows the creep test results for the 20 specimens for which a tertiary creep phase has been observed. Time to failure (T) is defined as the time between stress increase and creep failure. There is a large scatter in the test results similarly to the fatigue tests that may be explained by similar reasons. A logarithmic trendline is indicated with upper and lower limits defined by the coefficient of variation (12.6%) of the static compression tests. It needs to be noted, that creep test results include not only the 1-step creep tests, but also the short- and long-term accelerated

Fig. 14. ST curve for compressive creep tests: stress (S) vs. time to failure (T, log scale) for specimens for which a tertiary creep phase was observed.

creep tests. For these latter tests, the load history is not taken into account and only the stress level and associated time between stress increase and failure in the last loading step is shown. The results therefore do not take into account damage accumulation in preceding loading steps, thus possibly underestimating the life expectancy. 3.3.3. Mathematical model for creep For creep loading, an analytical relationship between stress level and time to failure has been suggested by Verstrynge [2] as shown in Eq. (4).

T¼

ð1  ðA  S þ BÞÞnþ1 cðn þ 1Þ  Sn

ð4Þ

where T is time to failure and S is the ratio of the maximum stress to the average strength (S = SMax/SAv). Parameters A = 1.9, B = 0.9, c = 8.5  1011 and n = 8 were calibrated based on the creep test results reported in this paper, for which the calibration process can be found in [2]. The model is presented in Fig. 14 together with the experimental creep test results and good correspondence is observed. 3.3.4. AE–T curves under creep While Fig. 14 shows damage development in terms of stress level and time to failure, crack development may also be expressed in terms of AE activity against time. As higher stress levels are likely to induce higher intensity micro- and macro-crack development, strong correlation is expected to exist between stress level and AE intensity [37]. In this respect, the 3-phase creep curve and more specifically the relationship between the rate of deterioration during the secondary creep phase and stress level can be described with a Weibull distribution model. This statistical background for creep failure prediction based on AE detection was presented by Verstrynge et al. in [37]. Fig. 15 shows the AE event rate (number of AE events per minute) during the secondary creep phase against time to failure (T) for the same set of test results as Fig. 14. An improved relationship can be observed in the AE–T graph compared to the ST graph and can help identify the time to failure based on monitored AE event rate. In Fig. 15, values are presented on a double logarithmic scale that implies decreasing accuracy for increasing time to failure. Also, prediction accuracy may reduce for lower stress levels (longer times to failure) if the AE activity is of similar magnitude as the background noise. Although the values in the AE–T curve are only indicative of a specific masonry type and AE setup, tested under constant environmental conditions, it intends to demonstrate the general principle for prediction of creep failure in masonry based on AE detection.

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Fig. 15. AE–T curve for compressive creep tests: AE event rate (AE, log) during secondary creep phase vs. time to failure (T, log).

In terms of practical application, in situ stress levels are difficult to measure in historical structures. Overall stress levels may be computed and related to the time to failure with the help of ST graphs for the structural elements, local stress levels however may vary significantly from the overall stress levels (due to local imperfections, variations in the material properties, cracks, voids, etc.) and lead to localised failure. Local AE emission can however be easily monitored and the proposed AE–T relationship may be used to indicate the expected time to failure and severity of damage progress for the specific location. 4. Relationship between fatigue and creep Both fatigue and creep can be described as time-dependent deterioration processes as a function of stress. The current section attempts to draw together the relationship between stress level and time to failure for static, fatigue and creep loading discussed in the previous section. 4.1. Joint fatigue–creep deterioration model Fatigue loading can be expressed by the combination of a mean stress (SMean) and stress amplitude (SAmpl). Alternatively, fatigue

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loading may also be defined by a minimum stress (SMin), a maximum stress (SMax) and/or a stress variation (DS = SMax  SMin = 2  SAmpl). Using these same definitions, creep loading may be expressed by a mean stress with zero stress amplitude. A theoretical relationship between stress level and time or number of cycles to failure (ST or SN curves) is shown in Fig. 16, incorporating static, fatigue and creep loading. The mean stress (SMean) is represented on the vertical axis and stress amplitudes (SAmpl) are marked on individual curves. For example, for 60% mean stress, 40% amplitude represents failure under static loading, 30%, 20% and 10% amplitude represent fatigue loading (with 30%, 40% and 50% minimum stress (SMin) respectively) and 0% amplitude represents creep. For practical application, the stress level for masonry structures may be more commonly defined as the combination of a minimum stress (SMin) and stress variation (DS) for which the relevant (SMin and DS) curves are also included in the graph. The shape of the ST or SN curves are only indicative and may have greater curvature for lower stress levels. The curves may approach without intersecting a limit stress value as it tends to infinity, although the existence of a fatigue limit (FLS) has not been confirmed so far (see Section 3.2.2). An indicative Permissible Limit State (PLS) has been included in the graph instead (see Sections 3.2.2 and 3.3), but more work is required to identify its position and nature. During fatigue loading the stress was defined against the number of cycles, while during creep loading stress was defined against time. In order to collate the fatigue and creep test data into one graph, ‘time to failure’ needs to be converted into ‘cycles to failure’, or vice versa. Although various approaches may be taken for relating time to cycles, for the current model 1 s during creep will be equated to 2 cycles under fatigue loading at 2 Hz frequency. While it is expected that the frequency will have an effect on the rate of deterioration, further work is required to propose suitable conversion factors. In the meantime, a conversion factor of 1 will be used for 2 Hz frequency. It needs to be noted, that while static loading is considered as 1 cycle in theory, in practice load application under quasi-static loading took a certain period of time that would need to be taken into account when converting into time or cycles. However, in order to demonstrate the principles of the model, the duration of the quasistatic tests will be disregarded and static test results will be identified as failure at 1 cycle.

Fig. 16. Joint failure model indicating stress vs. time or cycles to failure (ST or SN curves) for static, fatigue and creep loading.

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4.2. Joint fatigue–creep mathematical model For relating the experimental test results to mathematical models, the curves indicating fatigue failure for a given base load (SMin) are simulated by means of the fatigue model adapted from Casas [49] as described in Section 3.2.3. The creep curve (upper SAmpl curve) is simulated by means of the creep model described in Section 3.3.3. While the fatigue model by Casas shows reasonable agreement with the fatigue test data at 10% minimum stress (Fig. 10), no adequate correlation has been found between the fatigue curves and the creep model for higher SMin values. For an amplitude value of zero, SMin, SMean and SMax become equal for a specific fatigue curve and the number of cycles to failure should correspond to the failure time indicated by the creep model at that specific stress level. In order to improve the relationship, the fatigue model by Casas Eq. (2) has been adapted and a correction factor (C) introduced Eq. (5).

S ¼ A  NBð1CRÞ

ð5Þ

where S is the ratio of the maximum stress to the average strength (S = SMax/SAv), N the number of cycles, R the ratio of the minimum stress to the maximum stress (R = SMin/SMax), parameter A is set to 1, parameter B is set to 0.04 and C is the correction factor. To achieve the best correlation with the current fatigue and creep test results, the value of 0.62 has been identified for parameter C. The correction factor allows the interaction between the creep and fatigue phenomena to be taken into account and the slope of the SN curves to be adjusted. This correction for higher stress levels might be explained by the interaction between creep and fatigue, as the influence of creep damage is expected to become more significant for higher stress levels. Consequently, the slope of the fatigue curve and the number of cycles to failure will be slightly reduced due to creep effects. In terms of fatigue and creep effect, creep deterioration is assumed to be the dominant cause of failure for higher base loads, while fatigue damage becomes more dominant for lower mean stresses. The experimental fatigue and creep test data, modified fatigue Eq. (5) and creep model Eq. (4) are jointly shown in Fig. 17. Good correlation is observed between the test data and proposed modified model. To relate the results of the fatigue and creep tests with their respective static tests, the static tests of the fatigue testing with 10% base load are included as fatigue tests with 1 cycle, 55% mean stress and 45% amplitude, while the static tests of the creep

test program are presented as creep tests at 100% mean stress with 0% amplitude. In an environment where ambient fluctuations cannot be controlled, fatigue and creep deterioration are likely to interact with other physical, physico-chemical and biological deterioration processes, making the prediction of life expectancy purely based on stress level increasingly unreliable. To the authors’ knowledge no extensive experimental test programs have been reported in the literature to date on the interaction between fatigue, creep and other deterioration phenomena. The influence of moisture ingress on creep failure of sandstone and the effects of moisture movement and salt crystallization on creep strain have however been studied [3,24]. Although the available test data is insufficient at this stage to fully quantify and validate the SN relationship, they are intended to be used to demonstrate the methodology and partially validate the proposed fatigue-creep model for masonry. 5. Conclusions Fatigue and creep deterioration are time-dependent mechanisms that can lead to failure over time even at relatively low stress levels. Fatigue and creep loading were considered for masonry in terms of deterioration process, life expectancy, SN or ST curves and mathematical models in order to develop a joint fatigue–creep deterioration model. Fatigue and creep deterioration were tested within two independent laboratory test series at two universities with slightly different test setups, however results were comparable in terms of relative stress and qualitative changes in the acoustic emission output. The acoustic emission technique has shown to be a valuable tool for damage detection during creep and fatigue testing. With the help of acoustic emission monitoring, stages of the deterioration process (micro-crack nucleation, growth, and coalescence into macro-cracks) were identified and characterised during static, fatigue and creep loading. Close relationship between AE signal characteristics (amplitude, energy and event count) and remaining life expectancy has been found during laboratory testing that can help predict the remaining service life during laboratory and field testing. Based on the fatigue test results, SN curves (stress vs. number of cycles) have been developed for a specific masonry type and indi-

90% SMin 80% SMin 70% SMin

60% SMin 50% SMin

45%SAmpl 90%Δ S

40% SMin 30% SMin 40%SAmpl 80%Δ S

20% SMin 35%SAmpl 70%Δ S

10% SMin 30%SAmpl 60%Δ S

25%SAmpl 50%Δ S

0% SMin 20%SAmpl 40%Δ S

0%SAmpl 0%Δ S 5%SAmpl 10%Δ S 10%SAmpl 20%Δ S 15%SAmpl 30%Δ S

Fig. 17. Joint failure model indicating stress vs. cycles to failure for static, fatigue and creep loading, with experimental data.

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cated very significant changes in life expectancy for relatively small changes in stress levels. A fatigue model was proposed and showed good correspondence with experimental data. Creep test results were summarised in an ST relationship (stress vs. time) and an existing creep model with good correlation with the experimental data was applied. An AE-T relationship (AE event rate vs. time to failure) was found to provide improved accuracy for estimating the time to failure and rate of damage propagation. A joint SN model was proposed to incorporate static, fatigue and creep loading in terms of stress against time or number of cycles to failure. Creep was considered as fatigue loading with zero amplitude and static loading as fatigue failure after one cycle. The proposed fatigue model was adapted to improve correlation with the creep model and both mathematical models were combined in a fatigue–creep deterioration graph. Good agreement was found between the test data and proposed model for the tested time span. No endurance limit was observed within the timeframe of the experimental tests. Further laboratory tests data are required to populate the proposed fatigue–creep model and allow a generic quantitative model to be developed for practical use. Also, further work is required to quantify the effects of external influences and fluctuating environmental conditions on creep and fatigue deterioration and on AE emission levels to enable practical life-cycle prediction. Acknowledgements The authors gratefully acknowledge the financial support provided by the Engineering and Physical Research Sciences Council (EPSRC) for the research project at UWE and by the Research Foundation – Flanders (FWO) for the postdoctoral research of E. Verstrynge. References [1] Binda L, editor. Learning from failure – long-term behaviour of heavy masonry structures. Advances in architecture, vol. 23. Southampton: WIT Press; 2008. [2] Verstrynge E. Long-term behaviour of monumental masonry constructions: modelling and probabilistic evaluation. PhD thesis. civil engineering department, K.U.Leuven: Leuven; 2010. [3] Verstrynge E, Konings S, Wevers M. The influence of moisture on creep behaviour of sandstone assessed by means of acoustic emission. In: Jasienko J, editor. 8th International conference on structural analysis of historical constructions. Wroclaw, Poland; 2012. p. 2573–2581. [4] Ferretti D, Bazant ZP. Stability of ancient masonry towers: moisture diffusion, carbonation and size effect. Cement Conc Res 2006;36(7):1379–88. [5] Bazant ZP. Prediction of concrete creep and shrinkage: past, present and future. Nucl Eng Des 2001;203(1):27–38. [6] Papa E, Taliercio A. A visco-damage model for brittle materials under monotonic and sustained stresses. Int J Numer Anal Meth Geomech 2005;29(3):287–310. [7] Binda L, Anzani A. The time-dependent behaviour of masonry prisms: an interpretation. Masonry Soc J 1993;11(2):570–87. [8] Verstrynge E, Schueremans L, Van Gemert D, Hendriks MAN. Modelling and analysis of time-dependent behaviour of historical masonry under high stress levels. Eng Struct 2011;33(1):210–7. [9] Pina-Henriques J, Lourenço PB. Testing and modelling of masonry creep and damage in uniaxial compression. Struct Stud, Repairs Maint Heritage Archit VIII 2003;16:151–60. [10] van Zijl GPAG. Computational modelling of masonry creep and shrinkage. PhD thesis. Delft University of technology: Delft; 2000. [11] Choi KK, Lissel SL, Taha MMR. Rheological modelling of masonry creep. Can J Civil Eng 2007;34(11):1506–17. [12] Ferretti D, Bazant ZP. Stability of ancient masonry towers: Stress redistribution due to drying, carbonation, and creep. Cement Conc Res 2006;36(7):1389–98. [13] Roberts TM, Hughes TG, Dandamudi VR, Bell B. Quasi-static and high cycle fatigue strength of brick masonry. Constr Build Mater 2006;20:603–14. [14] Roberts TM, Hughes TG, Dandamudi VR. Progressive damage to masonry arch bridges caused by repeated traffic loading. RCNG 144 Final, report; 2004. [15] Abrams DP, Noland JL, Atkinson RH. Response of clay-unit masonry to repeated compressive forces. In: 7th International brick masonry conference: melbourne; 1985. p. 567–576. [16] Brencich A, Corradi C, Gambarotta L. Eccentrically loaded brickwork: theoretical and experimental results. Eng Struct 2008;30(12):3629–43.

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