Construction and Building Materials 28 (2012) 372–381

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Construction and Building Materials

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Modelling the compressive mechanical behaviour of granite and sandstonehistorical building stones

Marco Ludovico-Marques a,⇑, Carlos Chastre b, Graça Vasconcelos c

a CICC, Barreiro School of Technology, Polytechnic Institute of Setúbal, Rua Américo da Silva Marinho, 2839-001 Lavradio, Portugalb UNIC, Department of Civil Engineering, FCT, Universidade Nova de Lisboa, Portugalc ISISE, Department of Civil Engineering, University of Minho, Portugal

a r t i c l e i n f o

Article history:Received 21 December 2010Received in revised form 24 August 2011Accepted 29 August 2011Available online 22 October 2011

Keywords:AnalyticalCompressionExperimentalNon-destructivePorosityPropertiesStone

0950-0618/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.conbuildmat.2011.08.083

⇑ Corresponding author. Tel.: +351 212064660; faxE-mail addresses: ludovicomarques@gmail.com,

o.ips.pt (M. Ludovico-Marques).

a b s t r a c t

Building stones, particularly sandstone and granite, are very important in the building elements of Por-tugal’s historical and cultural heritage. Experimental research, based on uniaxial compressive tests, wascarried out on selected representative samples of lithotypes of rocks used in historic built heritage, with aview to evaluating the compressive mechanical behaviour of different building stones. The resultsshowed that porosity plays a central role in the compressive behaviour of granites and sandstones. Asporosity can be evaluated in field conditions with non-destructive tests it was decided to derive an ana-lytical model to predict compressive behaviour based on the knowledge of porosity of the buildingstones. A cubic polynomial function was adopted to describe the pre-peak regime under compressionto implement the model. Furthermore, a statistical correlation between mechanical and porosity datahad to be defined. Good agreement between experimental and analytical compressive stress–strain dia-grams, from which the mechanical properties like compressive strength and modulus of elasticity can bederived, was achieved.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Built heritage such as castles, churches and palaces play animportant role in the cultural life of Portugal. In general, massivemasonry walls characterize the construction of these ancient con-structions and natural stone is the most widely applied material.The use of dimension stones in traditional constructions is closelyrelated to the distribution of rock outcrops. Granitic rocks are pre-dominant in the northern and central regions of Portugal, but italso possible to find them in some important monuments in thesouth. Sandstone is less widely distributed in Portugal, but itsuse is common in traditional buildings at regional level, particu-larly in the Western regions close to the sea (Peniche, Lourinhãand Silves). Fig. 1 shows some traditional buildings with graniteand sandstone loadbearing masonry.

Conservation, rehabilitation and strengthening of the built her-itage are clearly required by modern societies, meaning that appro-priate intervention techniques on materials and structures shouldbe available. The proper rehabilitation of ancient buildings shouldbe based on appropriate diagnosis and understanding of the exist-ing materials [1]. In addition, the principles of safeguarding archi-

ll rights reserved.

: +351 212075002.marco.marques@estbarreir-

tectural heritage according to the international charters of Athenscited by Venice [2] and Krakow [3] recommend that studies shouldbe carried out on the building stone with the lowest degree ofintrusion and fullest respect for their physical integrity. In fact,one of the main problems of diagnosis with respect to an ancientbuilding is the difficulty of removing material for mechanical andphysical characterization. The principle of minimum intrusionhas been broadly taken into account by the scientific community,which has been proposing alternative non-destructive techniquesto evaluate the mechanical and physical properties of constructionstone [4,5]. Ultrasonic pulse velocity (UPV) and the Schmidt ham-mer (rebound hammer) are two examples of simple and inexpen-sive solutions that can predict the elastic mechanical propertiesand the weathering state of building stones [6]. Porosity is a prop-erty that can be estimated also by a Schmidt hammer and by ultra-sonic pulse velocity [6–8].

The dependence of the compressive mechanical properties onthe physical properties of rocks has been reported by severalauthors [9–14]. This relation was also assessed for granite usinga set of statistical correlations between mechanical and physicalproperties [15]. In general, increasing porosity is associated withdecreasing compressive and tensile strength and a lower modulusof elasticity. This behaviour is to great extent related to the higherheterogeneity and presence of weak bonds such as pores, voids andmicrocracks in very porous rocks.

Fig. 1. Traditional buildings with loadbearing masonry: (a) vernacular masonry buildings of granites; (b) historical and vernacular construction in sandstones.

M. Ludovico-Marques et al. / Construction and Building Materials 28 (2012) 372–381 373

The dependence of the basic mechanical properties on the phys-ical properties (porosity) can make the mechanical evaluation ofexisting building stones in old masonry walls much easier.

Following this idea, a model is proposed that describes the com-pressive mechanical behaviour of distinct building stones based ontheir physical properties. The analytical model proposed simulatesthe mechanical compression behaviour of granite and sandstone interms of the stress–strain relation as a function of physical (poros-ity) and mechanical parameters (compressive strength and modu-lus of elasticity).

The implementation of this method involves a first phase ofexperimental investigation of the physical and mechanical proper-ties of the building stones under compressive loading (modulus ofelasticity and compressive strength). Once the model has been de-fined, it is intended to use it to predict the basic engineering prop-erties, based on porosity, which can be given by non-destructivetests.

The major significance of the proposed method is the possibilityof gathering enhanced information on the basic engineering prop-erties of ancient building stones without using destructive testing.It should be stressed that compressive strength and the modulus ofelasticity are the most important mechanical properties needed toestimate masonry’s compressive strength. In addition, these prop-erties have a major role in the numerical simulation of oldbuildings.

2. Selection of rock lithotypes

The analytical model was developed for sandstones and gran-ites based on the results of experimental work on the mechanicaland physical properties of granitic rocks and sandstones.

The granitic stones studied were mostly collected from thenorthern region of Portugal, i.e. from Afife (AF), Ponte de Lima(PTA), Mondim de Basto (MDB) and Gonça (GA). Mineralogical, tex-tural and structural characteristics were used to select granitetypes. In this paper only the results obtained for fine to mediumand medium granites are reported. The mean length of sectionsintercepted by a single circle was measured in order to evaluate

the grain size of the granitic types, in accordance with the princi-ples of the Hilliard single-circle procedure described in ASTME112-88 (1995) [16]. Four circles were studied for each granitic fa-cies and sections in the less weathered granitic types were consid-ered. Mean length of sections measured was about 0.5–0.6 mm inGA and AF and about 0.7–0.9 mm in MDB and PTA lithotypes. Thesmallest grain sizes were about 0.3 mm in GA, MDB and PTA lith-otypes, while the smallest grain size of 0.1 mm was recorded inAF granite.

The sandstones were collected in Atouguia da Baleia, in Peniche,a region in the centre of Portugal [17]. Four varieties, which arerepresentative of the two lithotypes in existing monuments, wereidentified. It should be noted that neither coeval quarries nor out-crops of similar materials to those used in the monuments could befound in areas near to Peniche. Thus, stone masonry walls were se-lected in the vicinity of the built heritage and some samples wereextracted from them, taking into account their similarity in termsof appearance, mineralogical composition, texture and structure, tothe stone in the monuments. Physical tests were also carried out todetermine porosity. The four varieties have similar porosity to thestone found in the monuments. Both lithotypes have the sameclassification according to Folk [18], i.e. they are classified as lithicarkose [17].

The lithotype designated A + B, which includes the varieties Aand B, has around 34–40% carbonates and 30–32% quartz, whereasthe lithotype C + M encompasses typology M which has about 20–21% carbonates and 45–51% quartz. The carbonate content in bothlithotypes is so significant such that they were designated as lithicarkose with carbonate cement. In this paper only the results ofvarieties A, B and M are shown.

Lithotype A + B exhibits macroscopically well defined lineationsand variety A has clearly visible laminations. Lineations were notdetected in variety M. However, in thin sections under a polarizingmicroscope, variety A exhibits one preferred orientation of micaminerals and variety B shows no preferred orientations, with linea-tions being randomly distributed. Thin sections of variety M showtwo preferred orientations of mica minerals. All these varietieshave about 4–6% mica minerals.

374 M. Ludovico-Marques et al. / Construction and Building Materials 28 (2012) 372–381

The average size of grains of quartz and feldspar in the sand-stone varieties A and B ranges from 0.1 to 0.13 mm, and in varietyM the average size is about 0.24 mm. Sandstones A and B are gen-erally fine-grained, whereas variety M sandstones are medium tofine to grained [17].

The smallest grain sizes in granite lithotypes GA, MDB and PTAare similar to the average size of grains in sandstone variety M.Granite lithotype AF has the smallest grain sizes of the four graniticlithotypes studied, which correspond to the average size of sand-stone lithotype A + B grains.

3. Experimental programme

3.1. Introduction

The experimental programme was carried out in the laboratory and involveduniaxial compression tests to obtain the stress–strain diagrams and the mechanicalengineering properties (compressive strength and modulus of elasticity), and poros-ity tests to obtain physical properties (porosity and density). In this section the de-tails of experimental testing are provided and experimental results are discussed.

3.2. Preparation of samples

The granite lithotypes selected in this study are part of a group that was sub-jected to extensive experimental research for the mechanical characterization ofdifferent types of granite which are typical of most historical and vernacular build-ings in the north of Portugal [8]. For the mechanical characterization of granites itwas decided to use cylindrical specimens with a diameter of 75 mm and a height todiameter ratio of approximately two. These measurements followed the recom-mendations of ISRM [19] so that representative samples of the studied granitescould be obtained. The granites selected exhibited no significant planar anisotropy.The direction of loading was always parallel to the rift plane. With respect to thesandstones, the samples of the three varieties mentioned were cut into prismaticspecimens in order to make best use of the scarce rock available. It was decidedto use prismatic specimens of 50 � 50 � 100 mm3, corresponding to a height tolength ratio of 2. The macroscopic laminations and lineations of lithotype A + Bwere aligned parallel to the axial length. As no macroscopic lineations were de-tected in variety M, the prismatic specimens were randomly cut [17].

Fig. 2. Testing equipment: (a) glass vessel; (b) specimen con

3.3. Study of physical properties

The evaluation of the physical properties of rocks can be a simple way to assesstheir quality and can assist with the interpretation of the results achieved bymechanical characterization [9]. Previous studies have shown that mechanicalproperties such as compressive strength and elastic modulus are dependent onporosity and density [12,20,21].

The porosity and density of the granites were determined according to themethod suggested by ISRM [19], while the porosity and density of the sandstoneswere obtained following the Recommendations of RILEM [22] and EN1936 [23].Both standards suggest using a vacuum to saturate the specimens. Fig. 2 showsthe experimental apparatus. The hydrostatic weighing was carried out after airvoids were filled with trapped water. The grain mass, Ms, is defined as the equilib-rium mass of the sample after oven drying at a temperature of 105 �C. The pore vol-umes accessible to water were then determined by using the Archimedes principleallowing to calculate porosity and real densities.

The authors carried out tests to determine other physical properties, such asbulk density. Additional tests were carried out on the sandstones to determinethe absorption of water at low pressure and by capillarity, as well as to determinethe mercury intrusion porosimetry [17].

The porosity tests were carried out on all specimens used in the mechanicalcharacterization to enable a direct correlation between porosity and mechanicalproperties. The average porosity obtained for granites and sandstones are pre-sented in Table 1. The values range from 0.42% (granite GA) to 5.23% (graniteMDB). The MDB porosity is rather high, indicating that this granite is consider-ably more weathered than the other granites studied, and this is denoted mac-roscopically by the change of colour and the rough surface. According toGoodman [9] the expected porosity in fresh granites is lower than 1% but theporosity of igneous rocks tends to rise to 20% or more as weathering degreeincreases.

The higher porosity of weathered granites can reach values near the lowerporosity of sound sandstones. The M sandstone samples exhibit the highest poros-ity of the varieties studied. In relation to the sandstones, there is a clear differencein the porosity of varieties A, B and M, with values ranging from 3.6% to 18.6%.

Bulk density average values obtained for granites range from 2524 kg/m3 (MDB)to 2660 kg/m3 (GA), reaching values of 2568 kg/m3 (AF) and 2652 kg/m3 (PTa) [8].Sandstones bulk density values range from 2179 kg/m3 (variety M) to 2510 kg/m3

(variety B) and 2594 kg/m3 (variety A) [17]. MDB bulk density average values aresimilar to variety B average values. Bulk density and porosity follow the same trend:these properties values of weathered granites can nearly reach the lowest values ofsound sandstones.

tainer; (c) deionised water reservoir; (d) vacuum pump.

Table 1Mean values of physical and mechanical properties of the sandstones and granites.

Rock Sample rc (MPa) eR n (%)

Sandstones AP1 102.3 0.00500 4.60AP5 105.2 0.0051 4.30AP6 104.0 0.00530 4.20AP9 120.3 0.0052 4.20AP11 136.2 0.00625 3.80AP13 135.7 0.00663 3.60BP3 95.0 0.00720 7.00BP13 105.3 0.00780 6.70MP1 18.7 0.00793 18.40MP2 20.0 0.00673 18.50MP3 24.5 0.00798 17.20MP5 17.9 0.00883 18.60MP6 17.6 0.00798 18.60

Granites GA3 125.3 0.00327 0.42GA5 120.8 0.00301 0.43GA1 136.2 0.00375 0.45GA4 137.1 0.00367 0.44GA2 135.9 0.00351 0.49GA9 135.4 0.00352 0.48PTa_l5 109.2 0.00430 1.10PTa_l4 111.2 0.00435 1.11PTa_l6 116.0 0.00446 1.11PTa_l3 116.9 0.00443 1.11AF_L13 68.9 0.00526 2.99AF_L12 57.7 0.00480 3.04AF_L8 67.1 0.00559 3.06AF_L1 66.7 0.00535 3.11AF_L2 66.1 0.00551 3.19AF_L11 63.1 0.00536 3.26MDB_L4 41.1 0.00596 4.77MDB_L51 39.1 0.00557 4.91MDB_L61 38.9 0.00582 4.95MDB_L5 39.1 0.00563 5.14MDB_L2 41.2 0.00619 5.19MDB_L71 38.7 0.00562 5.23

M. Ludovico-Marques et al. / Construction and Building Materials 28 (2012) 372–381 375

Weathering of sandstone is responsible for a greater increase in their porositythan in that of granite, with figures of up to 40% and nearly 50% being achievedin sandstone [10]. Those higher figures are very close to those reported by Tugruland Zarif [24] for weathered sandstones.

Ludovico-Marques [17] presented the pore size distribution of sandstone vari-eties B and M obtained by mercury intrusion porosimetry. Microporosity settledas the percentage of pores radii lower than 7.5 lm [25], is 80–85% in variety Band about 75% in variety M.

Several authors studied granites in the North of Portugal which generally showmicroporosity values higher than macroporosity values [26–29]. Microporosity ofthese granites is around 65–75%, but can reach 80% or more. Machado et al. [29] ob-tained pore size distribution of more weathered MDB granite samples (Lamaresgranite) by mercury intrusion porosimetry. Microporosity of MDB granite (Lamarestype) is around 80% and it is very similar to microporosity values of B sandstones.These results are consistent with those obtained from bulk densities.

3.4. Characterization of mechanical behaviour

3.4.1. Experimental procedures for monotonic uniaxial compression testsThe uniaxial compression tests on granites and sandstones were carried out in

two Portuguese Universities. The sandstones were tested at the Laboratory of Struc-tures of Universidade Nova de Lisboa, and the granites were tested at Laboratory ofStructures of University of Minho.

The uniaxial compression tests on the sandstones used a Seidner servo-con-trolled press, model 3000D, with load capacity up to 3000 kN and a piston strokeof 50 mm [17]. The tests were carried out under axial displacement control at a rateof 10 lm/s. One displacement transducer (LVDT) was attached at each side of thespecimen between plates of the testing machine. The average displacement wascalculated from the displacements measured in the four LVDTs. These displacementtransducers have 100 mm of stroke and 100 � 10�6 strain/mm of resolution.

In case of the granites the uniaxial compression tests used a stiff pre-stressedsteel frame so that stable response of granite after peak load could be obtained. Aset of preliminary tests showed that a very brittle behaviour characterized the gran-ites, particularly the fresh granites. The main aim of the extensive experimentalwork on uniaxial compression tests carried out by Vasconcelos [8] was to determinethe full behaviour under compression, for which the complete stress–strain dia-

grams are needed. The uniaxial compression tests were therefore carried out withcircumferential displacement control. For this, a special device using a pantographwas designed to measure the lateral deformations (see Fig. 3). This device is com-posed of a central ring that is attached locally to the specimen by means of threesteel pins. The expansion of the ring is made possible by the lateral spring. Thetwo rods attached to the central ring can move freely when the lateral displacementof the specimen increases, since they are connected through an axis. The controlLVDT is placed at the end of one of the rods and is able to measure the deviationbetween the two rods during the compression test. This device further allows theactual diametric displacement to be amplified by a factor of seven, which meansthat if the programmed velocity of the control LVDT is 2 lm/s the correspondinglateral increment measured in the specimen is approximately 0.3 lm/s. A set oftests was carried out with uniaxial and circumferential control on weathered gran-ites under uniaxial compression and it was concluded that, apart from the possibil-ity of obtaining the post-peak behaviour, no differences were found in using the twoseparate deformation controls [8]. The stress–strain diagrams were obtained fromthe average of three displacements measured by the three LVDTs placed betweenplates and spaced 120� apart. More details about the experimental procedurescan be found in Vasconcelos [8] and [17].

3.4.2. Experimental mechanical behaviour under uniaxial compressionThe mechanical behaviour of granular rocks in uniaxial compression can be de-

scribed through the stress–strain curves covering the following stages [30,31](Fig. 4): (i) pre-existing crack closure; (ii) linear elastic deformation; (iii) crack ini-tiation and stable crack growth; (iv) crack damage and unstable crack growth; (v)failure and post peak behaviour. Eberhardt et al. [30] defined the initial stage ofthe stress–strain curve as a nonlinear region corresponding to volumetric reductiondue to pre-existing microcracks and voids closure until the stress level rcc. Fromthis stage, the stress–strain diagram exhibits a linear stretch corresponding to elas-tic deformation until the microcracking stress level rci. This stress level correspondsto the onset of microcracking. Many previous studies have demonstrated thatmicrocracks induced during uniaxial tests are mainly tensile cracks [32–34]. Theshape of the stress–axial strain is not sensitive to this deformational mechanism.This is essentially because the compressive stress-parallel cracks do not changethe axial stiffness [35,36].

The unstable microcracking occurs for the crack damage stress level, rcd, and itis associated with the point of reversal in the total volumetric strain diagram. Thisstage is connected to the maximum compaction of the specimen and to the onset ofdilation since the increase in volume generated by the cracking process is largerthan the standard volumetric decrease due to the axial load. For this stage, a rapidand significant increase of the lateral strains is observed, as a result of the volumeincrease. The microcracking spreading is no longer independent, the local stressfields begin to interact and the previously formed microcracks tend to coalesce.After the peak load is reached, the material becomes weaker and the strain is con-centrated in the weaker elements (strain localization), which constitute the dam-aged zone [8].

After the peak load is reached, the compressive behaviour is characterized bymacrocracking growth as strain localization occurs. Macrocracks result from coales-cence of the microfractures developed until the peak load is reached. At this stage,the tensile or the shear fractures are fully formed and are visible to the naked eye.The strain location and the growth of the crack length are followed by a significantreduction in the load carrying capacity of the material, which often results in thebrittle failure of the stone. In general, it is hard to record the post-peak behaviourof high strength rocks because their very brittle nature results in an abrupt, suddenfailure. When it is recorded, however, the stress–strain diagram drops off abruptlyafter peak load and the softening branch presents distinct negative slopes, withsome of them even being positive (snap-back). The softening branch is muchsmoother in low-strength rocks. This trend can be seen in both sandstones andgranites. Figs. 5 and 6 show the stress–strain diagrams for both rocks under analy-sis. It is observed that the post-peak response of low strength sandstones can becaptured, whereas the stress–strain diagrams can only be recorded up to peak stressin the case of hard sandstones. Vasconcelos et al. [15] have shown that the post-peak behaviour of high strength granites can be recorded more easily if the circum-ferential displacement control is adopted in the compression tests. It is clear that, aswith sandstone, the post-peak behaviour of granite depends on its strength. Forhigh strength granites (GA, PTA) the post-peak shows a sharp decrease in stressfor increasing displacements, whereas for soft granites (AF and MDB) the post-peakis smooth. In GA granites there are even few abrupt failures after peak load isreached. This behaviour appears to be also related to the internal structure of rocksdescribed by the porosity, as it arises from the internal distribution and arrange-ment of the grains and internal microfissures, pores and voids.

The ‘post mortem’ evaluation of the failures of the tested specimens reveals thatthey appear to be related to weathering and consequently to the porosity level. Infresh granites cracks develop in the subparallel direction to loading, at an angleapproximately 10� below the axial longitudinal axis of the specimen. In weatheredand high porosity granites, however, the macrocracks occur within a shear band(Fig. 7), but in sandstone the double shear crack bands appear to be better definedin high porosity sandstones (Fig. 8). Clear double shear develops in the lithotype Msandstone specimens, whereas a more distributed subvertical cracking is more usu-

Fig. 3. System used for the uniaxial compression tests on granite samples under the circumferential displacement control.

376 M. Ludovico-Marques et al. / Construction and Building Materials 28 (2012) 372–381

ally observed in the lithotype A + B specimens (Fig. 8). This can be associated withthe existence of microcracks aligned according a preferential plane, as pointed outby Gupta and Rao [37].

The pre-peak behaviour is also dependent on the lithotype. Lower strengthrocks also generally have lower values of initial stiffness than high strength rocks(Figs. 5 and 6). Comparison of the compressive strength and porosity values (Ta-ble 1) makes it clear that the porosity regulates the behaviour of rocks in uniaxialcompression as it is the result of the distribution and arrangement of grains. It isshown that high porosity granites, which are associated essentially with moreweathered levels, have considerably lower stiffness and lower compressivestrength. The dependence of the compressive strength and stiffness of sandstoneson porosity follows the same trend as the granites.

The strain corresponding to peak stress increases in both the sandstones andthe granites as the compressive strength decreases, which is related to the lowerstiffness of low strength granites and sandstones. This means that this parameteris also dependent on porosity. Increasing values of porosity are associated withincreasing deformation at peak stress, as shown in Table 1, confirming that porosityplays a major role on the behaviour of rocks under compression.

4. Analytical modelling of compressive behaviour

4.1. Analytical model of Ludovico-Marques [17] for sandstones

Given the important role of porosity in the compressive behav-iour of rocks, it was decided to find an analytical model that makesit possible to describe compressive behaviour from the knowledgeof porosity.

Fig. 4. Stress–strain

The analytical model was developed for sandstone by Ludovico-Marques [17] and has also been used to predict the compressivebehaviour of granites. As mentioned in 1, according to Vasconceloset al. [15], the mechanical properties of homogeneous granite(without significant planar anisotropy) are also reasonably corre-lated with porosity.

The compressive stress, r, developed in granites and sandstonescan be determined by Eq. (1):

r ¼ f ðe=eRÞ � rc ð1Þ

where rc is the compressive strength of rocks and f(e/eR) is theshape function characterizing the pre-peak behaviour of the rocksunder study.

The shape function (f) is obtained by normalizing the compres-sive stress by the compressive strength, rc, and it is dependent onthe strain, e, normalized by the strain at peak strength (eR):

f ðe=eRÞ ¼rrc

ð2Þ

The shape function is calibrated based on the results of uniaxialcompression tests for both the types of rocks considered in thiswork. Thus, from the analytical expressions defined by Ludovico-Marques [17] for sandstone and taking into account the experi-mental stress–strain diagrams for granite, it can be seen that the

curve stages.

σc (MPa)

0

20

40

60

80

100

120

140

160

0.0 5.0 10.0 15.0

ε (x10-3)

B

A

M

Fig. 5. Stress–Strain diagrams representing the varieties of sandstones.

σc (MPa)

0

20

40

60

80

100

120

140

160

0.0 5.0 10.0 15.0

ε (x10-3)

GA

AF

PTa

MDB

Fig. 6. Stress–strain diagrams representing the varieties of granites.

M. Ludovico-Marques et al. / Construction and Building Materials 28 (2012) 372–381 377

pre-peak behaviour is nicely described by a cubic polynomial func-tion. Thus the shape function valid both for the sandstones andgranites is given by Eq. (3):

f ðe=eRÞ ¼ �ðe=eRÞ3 þ 1:47ðe=eRÞ2 þ 0:5ðe=eRÞ ð3Þ

The coefficient 1.47 that multiplies the square term in Eq. (3)tends to 1.5, so that the shape function tends to value 1 whenthe strain tends to eR.

Through direct substitution of Eq. (3) in Eq. (1), the compressivestress, r, is given by:

r ¼ ½�ðe=eRÞ3 þ 1:47ðe=eRÞ2 þ 0:5ðe=eRÞ� � rc ð4Þ

It should be stressed that the model can be easily applied toother types of rocks because their behaviour in the regime ofpre-peak is easily adjusted to a cubic polynomial function.

4.2. Correlation between porosity and compressive strength and strainat peak stress

To implement the analytical model intended to describe thecompressive behaviour of distinct types of rocks through porosity,statistical correlations need to be found between porosity and bothexperimental compressive strength and the strain at peak stress.Fig. 9 illustrates the variation of uniaxial compressive strengthwith porosity in the samples of sandstone and granite lithotypes.The regressions that best fit the experimental results listed in Ta-ble 1 are set forth as Eq. (5) for the sandstones and Eq. (6) forthe granites:

Fig. 7. Failure modes of granite specimens tested in monotonic uniaxial compres-sion: (a) fresh granites; (b) weathered granites.

378 M. Ludovico-Marques et al. / Construction and Building Materials 28 (2012) 372–381

rc ¼ 206:7e�0:129n ð5Þ

rc ¼ 148:8e�0:263n ð6Þ

In fact, there is a very significant correlation between the com-pressive strength of sandstones and granites with the porosity,which confirms that this parameter determines the behaviour ofrocks under compression as discussed previously. This result isconsistent with the correlations found between compressivestrength and porosity of stones by other authors [20,21]. It shouldbe noticed that porosity in granite may be hindered by high anisot-ropy, due to the fact that the compressive strength varies with thedirection of loading and porosity is not obviously a directionalproperty.

Another important parameter for the complete definition of theanalytical model is the strain at compressive strength, which canbe similarly correlated with the physical properties of materials.Fig. 10 shows the variation of strain at failure (eR) with porosityin the samples of sandstone and granite lithotypes. As previouslymentioned, there is a clear trend for the compressive strain at peakstress to increase as the porosity decreases, indicating that rockswith high porosity are more deformable.

Fig. 8. Failure modes of sandstone specimens tested in monotonic uniaxial compression.

As the porosity increases, so does the dispersion of strain valuesat failure (eR), as Fig. 10 illustrates. Nevertheless, the coefficient ofvariation corresponding to strain at failure (eR) obtained from a sta-tistical analysis of sandstone variety M data (Table 1) is smallerthan 10% and for the sandstone variety A is around 12%.

The regressions that best fit the experimental results listed inTable 1 are set forth as Eq. (7) for the sandstones and Eq. (8) forthe granites:

eR ¼ 0:0043n0:215 ð7Þ

eR ¼ 0:0041n0:214 ð8Þ

Taking all the sandstone and granite specimens, the relation be-tween strain at peak stress and porosity is given by the granites Eq.(9):

eR ¼ 0:0042n0:222 ð9Þ

The better correlation (R2 = 0.942) found between porosity andstrain at peak stress is given for the granites by Eq. (8). The corre-lation between porosity and strain at peak strain found for thesandstones is considerably worse. Thus, it was decided to considerfor the analytical model the correlation found, taking into consid-

(a) A variety. (b) B variety. (c) and (d) M variety, front and rear features of specimens.

σc = 206.7e-0.129 n

R2 = 0.987

σ c = 148.8 e-0.263 n

R2 = 0.989

0

20

40

60

80

100

120

140

160

0.0 5.0 10.0 15.0 20.0

sandstones

granites

n (%)

σc (MPa)

GRANITES

SANDSTONES

Fig. 9. Relationship between compressive strength (rc) and porosity (n) obtainedfrom sandstone and granite samples.

εR = 0.0043 n0.215

R2 = 0.597

εR = 0.0041 n0.214

R2 = 0.942

εR = 0.0042 n0.222

R2 = 0.898

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

0.0 5.0 10.0 15.0 20.0

SandstonesGranites

n (%)

ε

GRANITES

SANDSTONES

SANDSTONES & GRANITES

Best LineSandstones & Granites

Best LineSandstones

Best Line Granites

Fig. 10. Relationship between the strain at rupture (eR) and porosity (n) obtainedfrom sandstone and granite samples.

σc (MPa)

0

20

40

60

80

100

120

140

160

0.0 5.0 10.0 15.0

ε (x10-3)

ExperimentalAnalytical Model

BP3

AP11

MP5

Fig. 11. Analytical modelling of some experimental curves of sandstone samples.

M. Ludovico-Marques et al. / Construction and Building Materials 28 (2012) 372–381 379

eration the results of both rocks. Most values of strain at compres-sive strength (eR) calculated through Eq. (9) vary by less than 10%

of the experimental values obtained for the sandstone and granite.In the granites only specimen GA5 differs by 16%. In the sandstonesamples AP13, BP13 and MP2 differ by 16%, 18% and 19%,respectively.

It is clear that porosity, particularly the porosimetry distribu-tion, influences the compressive mechanical behaviour of thematerial. On one hand, the higher amount of pre-existing micro-cracks, pores and voids contributes to higher initial deformationscaused by their closure. The more porous microstructure has morevoids, which reduces the stiffness of the stone skeleton and con-tributes to the increase in the strain at peak stress. Additionally,rocks with more porous microstructure also show a more remark-able nonlinear behaviour in the pre-peak regime, which also helpsto increase the strain at peak stress [38].

The proposed model makes it possible to simulate the compres-sive mechanical behaviour of rocks in terms of stress–strain and soto find the compressive strength and the modulus of elasticitybased on the knowledge of porosity. The proposed method is ofmajor significance in terms of practical applications to the studyof stone masonry buildings for the estimation of elastic properties,when the extraction of rock core samples is forbidden. Notice thatthe estimation of the stone’s mechanical properties is very impor-tant when it is needed to assess stability based on numerical sim-ulation or simplified methods.

4.3. Comparison of experimental and analytical results

The analytical model for each stone is fully defined by substitut-ing the compressive strength and strain at peak stress by theexpressions that correlate them with porosity. So, in Eq (4), thecompressive strength (rc) must be replaced by Eqs. (5) or (6) for

σc (MPa)

0

20

40

60

80

100

120

140

160

0.0 5.0 10.0 15.0

ε (x10-3)

Experimental

Analytical

GA 9

AF L11

PTa L4

MDB L4

Fig. 12. Analytical modelling of some experimental curves of granite samples.

380 M. Ludovico-Marques et al. / Construction and Building Materials 28 (2012) 372–381

the sandstones and granites respectively, and the strain at com-pressive strength (eR) by Eq. (9).

The final expression enabling the definition of the stress–straindiagram through the porosity for sandstones is given by Eq. (10):

r ¼206:7e�0:129n � e0:0042n0:222

� �3þ 1:47

e0:0042 � n0:222

� �2�

þ0:5e

0:0042 � n0:222

� ��ð10Þ

In case of granites the stress–strain diagram can be obtained bythe following equation:

r ¼ 148:8e�0:263n � e0:0042n0:222

� �3þ 1:47

e0:0042 � n0:222

� �2�

þ0:5e

0:0042 � n0:222

� ��ð11Þ

The performance of the analytical expression is assessed bycomparing the experimental stress–strain diagrams with those ob-tained by Eqs. (10) and (11). From Figs. 11 and 12, where experi-mental and analytical stress–strain diagrams for the sandstonesand granites are compared, it can be seen that in general goodagreement is achieved between the experimental and analyticalresults. It is possible to observe that better agreement with respectto stiffness is achieved for the sandstones. But the analytical modelallows the proper estimation of the compressive strength of thegranites. This estimation is slightly better than in case ofsandstones.

5. Conclusions

This paper has given a general overview of the compressivebehaviour of two distinct types of masonry stones that are com-monly used in old masonry buildings of historical value or in ver-

nacular architecture. Besides the experimental details on theuniaxial compressive tests, a discussion of the main results hasbeen provided, viz. the stress–stain diagrams and the values ofthe key parameters characterizing the pre-peak behaviour suchas compressive strength and strain at peak stress. It has beenshown that the compressive behaviour is influenced largely byporosity, which is a property connected to the arrangement ofthe internal skeleton of stones. More porous rocks have clearlylower compressive strength and higher strain at peak stress. Thehigher porosity was also found to influence the rocks’ stiffness(modulus of elasticity). Stiffer rocks are associated with lowporosity.

Given this dependency, it was decided to derive an analyticalmodel to describe the compressive mechanical behaviour of sand-stones, and the model was then extended to granites. This modelwas defined by taking into account the general shape of the pre-peak stress–strain diagrams obtained in the experimental testsand by considering that it is well defined by a cubic polynomialfunction. The polynomial function is dependent on the compres-sive strength and on the strain normalized by the strain at peakstress. The final model for sandstones and granites was stated byconsidering the statistical correlations found between the com-pressive strength and strain at peak stress with porosity.

The performance of the analytical model was evaluated by com-paring the analytical stress–strain diagrams with the stress–straindiagrams obtained in uniaxial compressive tests. A good agree-ment between the analytical and experimental results was found,meaning that the compressive behaviour can be predicted wellwhen porosity is known. The major advantage of this procedureis that the mechanical properties can be estimated under compres-sion without requiring the destructive testing of samples.

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