To date, there have been conspicuous advances in simulating the response of ancient masonry structures, mainly with the aim of determining the ultimate vertical loads and capacity with respect the lateral seismic actions. For example, among other studies, modeling criteria for ancient bridges may be found in Laterza et al. (2017b) and D'Amato et al. (2017), while for ancient churches they are reported in Pelà et al. (2009); Formisano and Marzo (2017); Betti et al. (2018); D'Amato et al. (2018), Formisano et al. (2018); Fuentes et al. (2019); Ramirez et al. (2019), and Lopez et al. (2019), and they are discussed for towers in Shakya et al. (2016); Bartoli et al. (2018) and Sarhosis et al. (2018). Models of general historical buildings are discussed in Caprili et al. (2017) and Milani et al. (2018), while detailed study on in-situ tests may be found, among the others, in Krstevska et al. (2010); Bartoli et al. (2013), and Luchin et al. (2018).

Nowadays, the study of ancient masonry structures' responses is a relevant topic since most of them are still in service without any kind of limitation.

By contrast, in the scientific community there is not yet the same level of interest for a correct understanding of fatigue effects and the prediction of the remaining service life of masonry elements under cyclic compressive loads. Among ancient masonry structures, it is known that arch bridges are affected by fatigue problems. This is due to the fact that they are currently subjected to higher and more frequent cyclic loads due to population growth, resulting in premature cracking and deterioration. As such, the available fatigue strength, that is the maximum stress acting in cyclic conditions, is significantly lower than the one obtained under quasi static loading conditions. Therefore, rather than the ultimate carrying capacity, it is important to know the actual fatigue strength, starting from which useful indications on the remaining service life with a possible traffic load limitation may be established. Moreover, the deterioration of materials and cyclic action may accelerate masonry deterioration and reduce carrying capacity. Clark (1994) and Roberts et al. (2006) performed cyclic tests on brick masonry columns and concluded that the stress limit of dry specimens was about 50% of the static compressive strength, whereas Choo and Hogg (1995) and Schueremans and Van Gemert (2001) instead suggested limiting the applied load to < 50% of the ultimate vertical load. Furthermore, in Melbourne et al. (2004) the cyclic vertical load capacity of multi-ring masonry arches varied between 37 and 57% of the static load carrying capacity. This significant reduction was due to a separation of rings, which was provoked by a shear failure mode among bricks and mortar joints instead of a four-hinge failure mechanism. Finally, in Melbourne et al. (2007), a unitary assessment procedure (named SMART procedure) was proposed for evaluating fatigue performance of masonry arches, involving the application of the fatigue model reported in Roberts et al. (2006).

Stress-life curves for masonry elements, indicated as S-N curves, are usually established in a limited number of experimental tests and, very often, are not adequate to reproduce the elements of ancient masonry. Moreover, in many of the actual design codes—among which are (Ministerial Decree D.M. 14/01/2008 (NTC-08), 2008), with the related (Italian Design Code Instructions (NTC-08 Instructions), 2009), and Eurocode 3 (EC3, 2003)—appropriate indications for evaluating the fatigue strength of masonry elements are still missing, contrarily to steel elements. Nevertheless, as highlighted from laboratory tests carried out on prototype models cast in a reduced scale, cycling loads related to in-service conditions may provoke fatigue failure for a vertical load significantly lower than the one related to the ultimate condition (Melbourne et al., 2004).

In this study, an updated review is provided for the main stress-life curve models available in the literature for estimating the fatigue strength of masonry elements. In particular, the models proposed by Ronca et al. (2004), Roberts et al. (2006), Casas (2009), Tomor and Verstrynge (2013), and Koltsida et al. (2018a) are considered. At first, they are separately described and shown. Then, the considered models are applied to a case study, an ancient masonry arch bridge. In particular, the main arch fatigue strength is assessed by considering the numerical simulations for repeated vertical loads reported in a previous study (Laterza et al., 2017a).

Koltsida et al. (2018b) investigated deterioration of the elastic modulus of masonry during compressive cyclic loading. This study was conducted starting from similar studies regarding concrete specimens, as reported in Crumley and Kennedy (1977); Holmen (1982); Cachim et al. (2002); Mu and Shah (2005); Breitenbucher and Ibuk (2006); Zanuy et al. (2011);Vicente et al. (2014).

Specifically, Crumley and Kennedy (1977) in their tests concluded that the elastic modulus decreased by about 40% over the concrete usable life, and that a remarkable reduction of elastic modulus arose often at 75% of the fatigue life. In Holmen (1982) the elastic modulus of concrete cylinders consisted of three different phases: a first rapid decrease from 0 to about 10% of the number of failure cycles, a constant decrease from 10 up to 80% of the number of failure cycles, and then a sudden decrease until the specimen's fatigue failure. These reductions resulted from increasing the number of cycles up to the failure. In addition, during the performed tests it was found that the absorbed energy at failure was the same for static and fatigue loads with different intensities. Mu and Shah (2005) performed experimental fatigue tests on the concrete cylinders to evaluate the damage evolution in the case of biaxial fatigue loading, compression and torsion. The results showed that the evolution of cracks in the material may be divided into two phases: the first phase was characterized by a deceleration of the crack, and the second one by a sudden acceleration. The authors proposed the following relationship:

where k, N and N_f are the elastic modulus, cycle and fatigue life, respectively. It should be noted that this relationship is independent of the fatigue load range. A different approach for predicting the elastic modulus of concrete under repeated compressive loads was proposed by Zanuy et al. (2011). In accordance with this model, the maximum strain (ε_max) and the elastic modulus (E) were directly related to the number of cycles:

The deterioration influence on Equation (7) and Equation (8) depends on the maximum and the minimum stresses (σ_max/f_c, σ_min/f_c). In particular, the authors defined three different deterioration stages. In Stage 1, concrete deterioration was due to micro-cracks forming at the aggregate-paste interface. This stage covered approximately 10–15% of the fatigue life. In Stage 2, micro-cracks grew steadily, with a constant reduction of the elastic modulus. This stage covered about 80–90% of the fatigue life. In Stage 3, micro-cracks converged to form a macro-crack causing specimen failure. The expressions proposed to determine elastic modulus in the transition from Stage 1 to Stage 2 and the rate of modulus decline in fatigue Stage 2 were as follows (Zanuy et al., 2011):

where E_c is the static modulus of deformation.

Starting from the previous studies, a few research groups have proposed similar formulations by considering, instead of concrete, the masonry material. Among these groups, it is worth mentioning the studies conducted by (Alshebani and Sinha, 2001). In this work, the authors concluded that deterioration of the elastic modulus began at 20% of the compressive load capacity. Moreover, the strength and stiffness deterioration depended on the number and intensity of cyclic loads, and in particular the degradation increased as the load and number of cycles increased.

As mentioned above, Koltsida et al. (2018b) recently proposed a different formulation to describe the evolution of the stiffness degradation under cyclic loading. The tests highlighted that until 95% of the fatigue life, the variation of the elastic modulus remained constant, while beyond the 95% mark it rapidly decreased until failure. To describe the evolution of the elastic modulus during cyclic loading, a linear model was proposed as described by Equation 11 and Equation 12, where E/E₀ represents the ratio between the elastic modulus and initial elastic modulus, and N/N_f is the ratio between the number of cycles and the number of cycles to failure. In Equation 12 the coefficient a is the gradient coefficient, while b is the intercept coefficient of the linear equation:

In Koltsida et al. (2018b) the following function was found as the best fit curve of the maximum induced stress:

By substituting the previous Equation (13) into Equation (11), the following relationship describing the reduction of the elastic modulus under cycling loading was proposed:

In Figure 5 the comparisons among the stress–life curves considered in this work are reported and illustrated in the semi-logarithmic plane Log N- S. In the comparisons the following values of the ratio R = S_min/S_max are considered: 0.2, 0.4, 0.6, and 0.8. Moreover, the curves proposed by Casas (2009) and Koltsida et al. (2018a) are drawn by referring to the 5% of failure probability, as considered by the EC3 and NTC-08 for the material nominal compressive strength. Also, Figure 5 reports the Tomor and Verstrynge (2013) stress life curves, and the ones proposed by Ronca et al. (2004), related to the ratio R equal to 0.8, since 0.8 is approximately the average value of the imposed R in all the performed tests. In all the graphs, the fatigue endurance limit S = 0.5 is also highlighted. It is important to outline that all the considered models may predict a low number of cycles at failure, even lower than the stress level corresponding to the classic endurance limit. From the comparisons, it is possible to highlight that when the stress ratio R decreases, i.e., the stress fluctuation between the minimum and maximum compressive stress increases, the number of cycles to failure decreases for any value of ratio S = S_max/S_u. Moreover, it can be observed that for R equal to 0.6 and 0.8, the fatigue strength becomes significantly higher than the endurance limit, very high also for a number of cycles. Finally, the curves proposed by Tomor and Verstrynge (2013) indicated very different values for the number of cycles to failure compared to the other curves, especially for high values of ratio S, where the effect of the plastic deformations are more influent.

In this paper fatigue assessment related to a case study is discussed. In particular, the results related to the main and secondary arches of an ancient masonry arch bridge are shown.

The case study, “Cavone Bridge,” was built in Italy before the Second World War and is currently still open to the traffic. It takes its name from the river it crosses and consists of seven arches of brick masonry. The bridge is 140 m in length and 5.6 m wide, and it is composed of three main arches with span lengths of 22 m and three secondary arches with span lengths of 10 m. The arches are supported by two piers of which 14 m are outside the riverbed. The piers have a total height from the foundation plane of about 24 m.

The bridge serves a provincial road according to the Italian Transport Classification (1992). The bridge was subjected to some in situ tests to identify the typology and the thickness of all its elements and components. The tests showed that the piers, abutments and spandrel walls consist of an external leaf of regular stone blocks. The piers have a core of cohesive backfill, while the bridge deck is formed by an incoherent backfill that has the function of spreading the traffic loads to the supporting arches. More details about numerical investigations carried out on this bridge for fatigue and seismic performance may be found performance in Laterza et al. (2017a,b) and D'Amato et al. (2017). In particular, in this section the numerical investigations illustrated in Laterza et al. (2017a) regarding the most unfavorable section for fatigue assessment verifications are discussed.

As for the load scheme, in this study Fatigue Model 3, which has two axles with a load of 120 kN/axle, is used. In accordance with Eurocode 1 (EC1, 2003) the considered fatigue model is more appropriate for typical heavy traffic on European main roads or motorways. In addition, a Traffic Category 2 has been assumed, resulting in 5 × 10⁵ passages for the year (N_cat). Since there is an absence of a relative procedure for masonry elements, in this study the fatigue assessment for steel elements has been applied, as proposed in the Italian Design Code Instructions (NTC-08 Instructions) (2009) and EC3 (2003), while applying fatigue stress-strain curves for masonry.

In accordance with the Italian Design Code Instructions (NTC-08 Instructions) (2009), two different values for the masonry compressive strength associated with Knowledge Level 1 (KL1) and Knowledge Level 3 (KL3) have been considered for the arches (NTC-08). Precisely, the compressive strength of masonry is assumed equal to 2.4 MPa for KL1 and 3.2 MPa for KL3. The strength is further reduced by a confidence factor FC = 1.35 for KL1 and FC = 1 for KL3.

Figure 6 reports the comparisons of the fatigue curves considered in this study, plotted for main and secondary arches. It should be noted that, in accordance with the NTC-08 and EC3 methods, all the curves are factorized, i.e., the fatigue strength is divided by γ_Mf (fatigue strength partial factor), assumed in this case equal to 1.35 by supposing a high consequence of arch failure due to the fatigue strength achievement. Meanwhile, the stress range Δσ_i, due to the stresses fluctuation resulting from the transit of the load along the arch, is amplified by γ_Ff = 1, where γ_Ff is the partial factor for equivalent constant amplitude stress range Δσ_i.

As far as the stress fluctuation is concerned, three different sections (2 haunches section, and 1 key section) for each arch have been numerically investigated in Laterza et al. (2017a) by means of FEM models. However, in this study only the most unfavorable section for fatigue assessment in terms of Δσ_i fluctuation is considered, which is the haunch section for the main arches and the key for the secondary ones. For these sections the ratio S_min/S_max results in value equal to 0.868 for the main arches and to 0.872 for the secondary ones, where S_min and S_max are the minimum and maximum axial stresses. In Tables 3, 4, the numerical results are summarized for the main and secondary arches, reporting the ratios S_max/S_u (maximum stress S_max over the ultimate strength in compression S_u) for both KL1 and KL3.

The fatigue performance is evaluated through the calculation of the residual service life, expressing the residual time (in year) before the failure to fatigue. This is given by the following ratio:

where N is the cycle number to failure obtained from the stress-life curve and N_cat is the number of passages per year assumed for the category of the bridge (N_cat = 5 × 10⁵ passages/year).

It should be noted that for KL3, by referring to the main arches, and for KL1 and KL3 in the case of secondary ones, the cycles' number to failure N are always very high (greater than 10⁸) so that, substantially, the residual life may be considered infinite.

On the other hand, the fatigue assessment results drastically change if the KL1 is considered for main arches. In this case, the models of Tomor and Verstrynge (2013) and Ronca et al. (2004) are not applicable, since the stress level S_max/S_u = 0.790 falls beyond the stress-life curves for any value of logN. As for the model proposed by Casas (2009) and Koltsida et al. (2018a), the residual life is < 1 year. Meanwhile, according to the Roberts et al. (2006) model, the residual life is about 3.5 years.

By comparing the chosen models, it is possible to conclude that the models proposed by Koltsida et al. (2018a), Tomor and Verstrynge (2013) and Casas (2009) actually represent the most complete models, since they have been derived from experimental results considered with a probabilistic approach. Differently from the Ronca et al. (2004) model (derived by imposing a small variation of the alternating loads), both the Roberts et al. (2006) and Casas (2009) models estimate, at a number of cycles that is not too high, fatigue strength ratios (S_max/S_u) significantly lower than 0.5, which is the value usually indicated as the fatigue endurance limit. This demonstrates the importance of an appropriate evaluation of the fatigue strength that may lead, if simplified methods are applied (such as the endurance limit criterion), to an overestimation of the actual fatigue strength.

In this paper, a critical review has been made and we have applied some fatigue models to a case study, considering also the deterioration of the masonry due to the effect of the cyclic loads.

In particular, different fatigue curves have been considered for evaluating the damage accumulation due to traffic load, in compliance with the procedure proposed by the EC3 (2003) and NTC-08 (2008). Since in the examined codes no clear indication is reported for masonry elements, the fatigue performance approach has been applied similarly to the one proposed for steel elements, with masonry fatigue curves.

Different stress-life curves have been considered in accordance with the models proposed by Ronca et al. (2004), Roberts et al. (2006), Casas (2009), Tomor and Verstrynge (2013) and Koltsida et al. (2018a), which are also capable of estimating the degradation of the material due to fatigue.

By comparing the chosen models, it possible to conclude that the models proposed by Casas (2009), Tomor and Verstrynge (2013), and Koltsida et al. (2018a) actually represent the most complete models, since they have been derived from the experimental results considered with a probabilistic approach. It must be remarked, however, that they have been derived from laboratory tests performed on masonry specimens having compressive strength values higher than the usual ones encountered in existing masonry. To this aim, particular attention should be also paid to the influence on the fatigue capacity of the cyclic load frequency. As a matter of fact, the fatigue models examined in this study have been proposed for cycled loads having frequencies (more than 1 Hz) higher than the ones associated with traffic loads indicated in the considered codes (for example, in the case analyzed the traffic load frequency results equal to 0.015 Hz). A first approach has been adopted to evaluate the degradation of the material due to the fatigue effects by using the recent model proposed by Koltsida et al. (2018b).

Finally, new laboratory tests focused on ancient masonry specimens will permit researchers to study in a more in-depth manner the fatigue behavior and the evolution of the material degradation of masonry, which have not yet been fully detailed.

The datasets generated for this study are available on request to the corresponding author.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.