Dimension stone design – kerf anchorage in limestone and marble

Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos ice | proceedings Proceedings of the Institution of Civil Engineers Construction Materials 165 June 2012 Issue CM3 Pages 161–175 http://dx.doi.org/10.1680/coma.8.00052 Paper 800052 Received 04/11/2008 Accepted 08/03/2009 Keywords: anchors & anchorages/design methods & aids/rock mechanics ICE Publishing: All rights reserved Dimension stone design – kerf anchorage in limestone and marble 1 & Rui S. Camposinhos PhD Coordinator Professor at ISEP, Polytechnic of Porto, School of Engineering, Porto, Portugal 1 2 2 & Rui Pedro A. Camposinhos MSc Project Engineer, Stucky Atlãntico SA, Porto, Portugal Rui de Sousa Camposin hos Digitally signed by Rui de Sousa Camposinhos DN: c=PT, st=Porto, l=Porto, o=Instituto Superior de Engenharia do Porto, ou=501540709, [email protected], cn=Rui de Sousa Camposinhos, [email protected] Date: 2012.07.11 18:39:19 +01'00' This paper describes a study on kerf anchorage behaviour in three types of stones. Two limestone and various marble specimens were submitted to a series of tests – namely standard absorption, bulk specific gravity and flexural strength tests – to determine their physical and mechanical properties. The paper focuses on the results of tests carried out according to the ASTM Standard Test Method for Strength of Individual Stone Anchorages in Dimension Stone. These tests were performed on a stiff rail continuous anchorage system on both the front and rear kerf legs. Stress analysis and finite element method calculations were the basis for the proposed semi-empirical formula to estimate this anchorage system’s breaking load under the above-mentioned conditions. Through a general but practical example, it is shown that a design focused on the bending strength of stone panels, to the detriment of the kerf anchorage, is an unsafe and yet common practice. The paper emphasises that designing stone cladding systems with continuous kerf must take into account different effects in order to evaluate the effective stress in the critical region of the kerf geometry. Separate stress concentration factors are proposed to account for the kerf geometry and the specimens’ specific properties. 1. Introduction Stone wall systems have been employed to achieve a wide range of architectural styles, aesthetic effects and appearances. Generally, thin stone wall systems are used in all environments and can have various finishes. Granites have had a long history of durable service, and certain marbles have a long history of successful use. However, some marble types, particularly some white marbles of pure calcite, have been found not to be durable materials because of their susceptibility to deterioration from heating and cooling cycles. Travertine, limestone and sandstone have a good history when applied as thick stone wall elements, but their service history as thin stone wall elements is fairly limited, particularly their durability. Inadequate initial evaluation of material durability and panel strength resulted in varying degrees of distress in some claddings. Although early stone cladding was installed with little thought to structural analysis, current evaluation techniques should predict capability by checking characteristics that were probably ignored by the initial engineering. Nevertheless, most failures in stone wall systems can be attributed to the fastening system (Lewis, 2007). Panel cracking, displacement or other failures can occur at locations where anchors are inadequately or improperly connected to the stone. Poor construction is often the result of poor quality and tolerance control during panel fabrication and/or erection. Moreover, damage during handling operations can also result in panel cracking, some of which may not become evident for several years. Yet, it must be recognised that a lack of information on stone cladding dimensioning and design rules is one of the most common reasons, if not the main reason, for the majority of failures (Lammert and Hoigard, 2007; Lewis, 1995). This paper focuses on the study of kerf anchorage behaviour regarding three types of stones. Two limestone and various marble specimens were submitted to a series of tests – namely standard absorption, bulk specific gravity and flexural strength tests – to determine their physical and mechanical properties. The paper focuses mainly on the results of a series of tests carried out 161 Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos according to the ASTM Standard Test Method for Strength of Individual Stone Anchorages in Dimension Stone (ASTM, 2004). These tests were carried out in a stiff rail continuous anchorage system on both the front and rear kerf legs. The mechanical behaviour of structures, particularly stressdeformation behaviour, is greatly influenced by the respective materials. Based on that behaviour, most engineering materials can be categorised as brittle, ductile or quasi-brittle (Shah et al., 1995). The stress quickly drops to zero when a brittle material fractures, whereas large deformation at constant stress occurs when a ductile material yields. On the other hand, a quasi-brittle material is characterised by a gradual decrease in post-peak stress. Stress analysis and finite element method (FEM) calculations were the basis for the proposed semi-empirical formula to estimate this anchorage system’s breaking load under the above-mentioned conditions. Separate stress concentration factors are proposed to account for the kerf geometry and the specific properties of the stone specimens. 2. Design principles Stone wall systems are traditionally constructed as a curtain wall or veneer, in which no building loads are transferred to the stone panels. Typically, the stone wall system must bear all lateral loads, such as those caused by wind and earthquakes, as well as vertical loads resulting from its own weight. The thermal-related expansion and shrinkage of the stone units may generate localised pressure where they come in contact with one another. Although loads such as accidental impacts, construction handling and transportation must also be taken into account in the design, these factors are not covered by the present paper. Design is not restricted to dimension stone elements, since the fixing elements must also be designed accordingly. Although the wind load is normally used to determine the load case, earthquake loads must not be neglected in relevant seismic areas. Procedures to estimate stone slab dimensions based simply on permitted bending stress calculations are, in most cases, not adequate to ensure safety. It must be emphasised that, for some cladding systems, there is relevant information about design methods or approaches (Camposinhos and Camposinhos, 2009; Lammert and Hoigard, 2007; Moreiras et al., 2007; West, 2004), which exceed the traditional ‘bending at midspan’ stress calculations, often referred to as allowable stress design. 3. Kerf anchorage A kerf is a sawcut groove or slot in the edge of a stone panel. A kerf clip or kerf bar is a flat bar or thin plate configured to engage a sawcut slot in the stone edge. These profiles are fastened to a support frame or connected directly to the building structure by bolts or anchors, thus providing the essential mechanical connection between the stone and the structure. Aluminium extruded profile (continuous kerf anchors _ T shaped) Holes for fastening with expansion bolts or threaded studs Formed stainless steel profile _ discrete length kerf anchors _ L shaped Figure 1. Kerf anchor profiles in discrete and continuous length configurations 162 Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos Figure 1 illustrates typical kerf anchor configurations, normally in formed stainless steel or extruded aluminium profiles. Other metals may be used if properly protected against moisture and galvanic action. They may be continuous or discontinuous and are typically located in the top and bottom edges for easier access and alignment during installation. The structural capacity of this type of anchorage depends essentially on the combined shear and flexural strength of the stone’s fin or leg. (e) (f) (g) The main issues regarding assembling stone panels using kerf anchors revolve around considerations which are implicit in the design; they are numbered and illustrated in Figure 2. (h) (i) (a) (b) (c) (d) Placing a plastic or metal adjustment bearing shim slightly larger than the anchor’s fastened face [1]. Preventing the connection from slipping after vertical adjustment with a diagonally slotted washer plate, welded washer or serrated anchor and washer [2]. Attaching the anchor with a fastener to a back-up structure [3]. Avoiding a kerf anchor (brake-formed split-ear shape (j) down) or extruded shape that fits into a sawn kerf [4] misalignment to prevent prying on kerf due to inaccurate fabrication or installation. Maintaining clearance to avoid contact and weight transfer (unless designed to stack) [5], thus allowing for movement, creep, expansion, and fabrication and installation tolerances. Placing a plastic or stiff rubber bearing shim to level, separate or to prevent bearing of stone kerf fin on anchor radius [6]. Maintaining clearance to avoid point loading [7, 10] at the kerf at the top and bottom of stone. Placing backer rod or foam tape at proper depth [8] to prevent a three-sided sealant bond. Sizing joints [9] to allow for hardware, tolerances, clearances, appropriate movement and joint filler capability; ensuring clearance and thus avoid point loading at the top and bottom. Kerf filling must be continuous with sealant to top of kerf to prevent moisture accumulation. Minimising distance [12] to reduce eccentric weight on the anchor (ASTM, 2003; Camposinhos and Amaral, 2007). The core parameters affecting the anchorage capacity have been depicted (Lewis, 1995), namely and mainly in the following manner (Figure 3). [1] (a) [3] (b) [2] [7] [6] [8] (c) [5] [4] [9] [10] [12] Figure 2. Typical cross-section of kerfed stone cladding systems (d) Kerf slot width (tk) the distance across the saw cut varies due to the original saw blade thickness and the blade’s reliability of plane and rotation. The degree of wear resulting in a thinner blade which tends to veer. The thickness of the stone fin (tf) remaining on the panel edge, after kerf anchorage assembly determines its potential strength. The distance from the inside face of the kerf to the finished face varies according to the stone panel gauge after having been sawn and the stone panel thickness, in accordance with the previously mentioned factors. The depth of contact (D), namely, the depth to which the kerf clip’s leg contacts the stone within the stone kerf. The length to which the kerf clip’s leg is engaged, taking into account that the actual length is not necessarily the effective length of engagement. Note that the continuous kerf rail in the stone edge does not provide a continuous support for the stone unless the kerf rail is much stiffer than the stone panel. In fact, when the support rail’s stiffness (which depends on the rail itself, on its cross-section’s moment of inertia and modulus of elasticity) is less than the dimension stone’s stiffness (determined by the stone’s thickness and modulus of elasticity) along the span of the dimension stone, the entire kerf rail length will not act as a continuous support (Conroy and Hoigard, 2007; Lewis, 1995; West and Heinlin, 2000). 163 Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos tf The tests were helpful in gaining a better understanding of the rupture behaviour of a common kerf anchorage. The test results were the basis for calibrating a simple formula that can be used to estimate the breaking load of the stone at the kerf anchorage. 4.1 Stone type and specimens 4.1.1 Lithological description Two limestone and one marble specimens were studied (Figure 4). D (a) Hf C tk (b) Molianos, an oolitic limestone formed of shells and shell fragments, is practically non-crystalline in character and found in massive deposits. This limestone is characteristically freestone, without cleavage planes and has a remarkably uniform composition, texture and structure. It has a high internal elasticity and adaptability to extreme temperature changes. Molianos is known by various commercial names, such as Gasgogne Beige or Gasgogne Blue, Guido Limestone or Porto Velho. Lioz, a beige semi-crystalline limestone with coarse elements, bioclastic and calciclastic. It is a biosparite t (a) (b) Figure 3. Kerf slot width, fin and depth of rail contact 4. Case study In the present study kerf anchorage behaviour based on a number of mechanical tests was analysed using two sedimentary and one metamorphic Portuguese stones. This study aimed to investigate the relationship between flexural strength and breaking load at the kerf anchorage. Stiff rails were used to ensure a continuous support. 164 (c) Figure 4. Images of three of the specimens studied, all samples are 30 mm thick Construction Materials Volume 165 Issue CM3 Sample Molianos Lioz Estremoz Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos Type Water absorption (EN 13755) Volumic mass: kg/m3 Open porosity (EN 1936) Limestone Limestone Marble 3?1% 0?1% 0?07% 2456 2658 2770 6?2% 0?4% 0?2% Table 1. Water absorption, volumic mass and open porosity of the stones studied (c) – microsparite that has been slightly affected by metamorphism. This limestone is often used as dimension stone and is usually well consolidated and has minimum graining or bedding direction. Estremoz, a high quality, fine- to medium-grained marble. This cream-white stone has excellent mechanicalphysical properties and is aesthetically pleasing. These marbles have been quarried since the Roman period and throughout the Middle Ages. 4.1.2 Physical and mechanical properties Samples were tested both wet and dry. The wet samples were soaked in water at 22 ˚C for a minimum of 48 h and a maximum of 120 h. The dry samples were dried prior to testing in a humidity-controlled oven at 65 ˚C and gradually cooled to 22 ˚C. The relevant mean values of the physical characteristics of the samples are shown in Table 1. The stones were tested in accordance with EN 1926 (CEN, 2006) to determine their unconfined compressive strength (UCS). Forty-five cylindrical samples were tested in dried conditions. The test results are shown in Table 2. Flexural strength was determined with three-point load tests. These tests were carried out on 33 prism samples (50 mm 6 30 mm cross-section) and on 59 slab samples with a (200 mm 6 30 mm) cross-section in accordance with European Standard EN 12372 (CEN, 1999). Hence, all specimens had a total length of 180 mm and were placed on rollers with a 150 mm gap. The samples were tested in dried conditions with a gang-sawn finish on the lower surface. The bending strength per type of stone and the number of tests are shown in Table 3. Sample Molianos Lioz Estremoz UCS (average): MPa Quantity 92?81 88?90 85?91 15 15 15 Table 2. Average unconfined compressive strength of stones studied as per EN 1926 The last two columns in Table 3 indicate the average flexural strength value and, as the tested specimens had the same spanto-thickness ratio, these values were adopted in the study. 4.2 Anchorage strength tests Specific tests were performed on the slab specimens to evaluate the strength of the kerfed edges. All anchorage strength tests were performed according to the ASTM C 1354 test method (ASTM, 2004) using specimens with nominal dimensions of 400 mm 6 200 mm by 30 mm thick and with both faces sawn. All specimens were tested after being in dry storage for over 1 week, without any additional wetting or drying procedures. The test configuration is shown in Figure 5. The kerf anchor was installed in only one of the edges of the stone specimen and the opposite edge was placed over a roller support. The load was applied at a close distance from the kerf anchorage (ðtzHf Þ; Figure 3) to avoid interference in the failure mode. The actual load applied to the anchor, which was less than the total applied load, was calculated using simple statics principles. 4.3 Performing the tests The anchors were bolted to a cold-formed channel support which was attached to a heavy steel reaction frame using threaded rods. The testing machine has a load cell capacity of 10 kN. Load values were displayed on a digital readout with a peak value memory. The load was applied at a constant rate of approximately 0?02 kN/s. Precautions were taken to ensure parallel alignment between the loading line and the kerf bottom. The anchor knife and the kerf’s leg formed an angle slightly greater than 90 ˚ to ensure that the contact line was parallel to the kerf’s slot (Figure 6). 165 Construction Materials Volume 165 Issue CM3 Molianos Lioz Estremoz Total Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos Prisms FS: MPa c.v. Qty Slabs FS: MPa c.v. Qty FS average: (MPa) 12?64 14?63 20?29 – 9?3% 7?9% 11?5% 9 12 12 33 9?10 12?94 19?8 – 10?1% 9?5% 9?9% 29 30 21 59 10?16 13?62 19?96 – c.v. 19?21% 10?63% 10?42% FS, flexural strength; c.v., coefficient of variation; qty, number of tests. Table 3. Flexural strength of tested stone according to EN 12372 Tests were performed using continuous steel stiff rail anchors installed in the continuous kerf. The typical kerf slot dimensions were 7 mm wide and 26 mm deep in the stone edge. The anchor rail was fitted inside the kerf groove at a measured depth of 13?5 mm. As the upper and lower kerf legs were slightly different in thickness, each specimen’s dimensions were measured after and before every test. Similarly, measurements were made for the slot’s contact depth. Some authors (Conroy and Hoigard, 2007; Lewis, 1995) have reported both laboratory and in-service kerf failures of curtain wall systems with full-width aluminium extrusion kerf hardware. These failures typically occur in the form of multiple short kerf breaks starting at the outer panel edges and proceeding inward towards the panel centre. In order to avoid this type of failure and to ensure that the kerfed stone cladding panels form a plate supported along two edges, the stiff steel rails were used to prevent any deflection between the kerf legs (Figure 6). During testing, the authors found that the coldformed channel support did not undergo any vertical or lateral displacement. Furthermore, the contact was defined at a precise contact depth consisting of a support line between the kerf leg and the steel rail of the kerf anchorage, as mentioned before. 5. Results A total of 106 tests were performed to determine the pullout load failure in accordance with Equation 1, which is the formula given in ASTM C1354 (ASTM, 2004). 1. P~ F |A L With reference to Figure 7: A is the distance between the roller support and the acting load (F); and L is the span, namely the distance between roller and kerf supports. The breaking load test results are summarised in Table 4. The 106 tests were performed on 27 slab specimens, and generally four tests were performed for each panel, two legs for each slab, except for one Molianos slab for which only one test was performed. All tests resulted in kerf leg failure. The spalling angle for the same type of stone was very similar. The spall failure orientation, angle, a, in Figure 6 was measured for all of the 106 spalls at three separate locations. The average value obtained as well as the range values of angle a are shown in Table 5, along with the variation coefficient. The shape and configuration of various spalls are illustrated in Figure 8 for each type of stone. Figure 9 shows a comparison of side-by-side spall angles. 5.1 Figure 5. Test arrangement before knife edge adjustment 166 Analysis A considerable number of theories have been proposed relating uniaxial to biaxial or triaxial stress systems. For brittle or quasi-brittle materials, the maximum stress criterion, or Rankine criterion, has been applied. In this criterion, it is assumed that failure occurs in a multiaxial state of stress when Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos α C - clearence D - contact depth D 13.5 C 11.5 11.5 7.0 tk t 6.0 11.5 tf Hf 10.0 Figure 6. Typical dimensions in mm of anchor cut-off – kerf groove, legs and contact depth for kerf steel angle either a principal stress reaches the uniaxial tensile strength, sut or a principal compressive stress reaches the uniaxial compressive strength, suc . As for natural stone, suc is considerably greater than sut , and in the present case the principal compressive stress is thus neglected. The nominal principal tensile stress in the critical plane of the kerf’s leg (Figure 10) is given by: 2. 3. snom ~ sx ~ sx zsy 1 z | 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 sx {sy z4|t2xy 6|P|C ; sy ~0 B|t2f 4. txy ~t* P B|tf Note that the shear stress in this region is assumed to have an average value when compared with the exact formula. Substituting Equations (4) and (3) in Equation (2), for the maximum principal tensile stress, snom , we obtain: 5. P|C z snom ~3| B|t2f sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2    P|C 2 P 9| z B|tf B|t2f When comparing snom with the flexural strength, FS, for each type of stone, it is necessary to multiply the value from Equation (5) by an effective concentration factor, Keff. A A 6. F F P P L L Figure 7. Test arrangement and typical lengths: L 5 290 mm; A 5 232 mm snom |Keff ~FS By substituting Equation (6) in Equation (5), the value of Keff is obtained, which is the effective or global stress concentration factor at the critical failure line. 7. Keff ~ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i  ffi FS|B C 2 | 9C 2 zt2f {3C 2 C|Pu where Keff is the effective concentration factor; Pu is the breaking load; FS is the flexural strength; B is the slab width; C~Hf {D is the clearance distance between the contact point 167 Construction Materials Volume 165 Issue CM3 Stone Molianos Lioz Estremoz – Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos Type of stone Number of tests Average value (P): kN Figure 7 c.v. for P (%) Average value (F): kN Figure 7 Limestone Limestone Marble Total 34 36 36 106 1?65 2?55 2?48 – 10?8% 19?3% 13?4% – 2?04 3?19 3?09 – Table 4. Cell applied load, F, and breaking load, P, at the kerf anchor of the anchor and the bottom of the kerf; and tf is the leg width. These values are plotted in Figure 11 for the tested stone and replace, for each test, the breaking load value, Pu, the medium value of the flexural strength given in Table 3 and the corresponding specimen dimensions in Equation (7). The average values obtained for Keff were 2?12 for Lioz, 2?40 for Molianos and 3?22 for Estremoz. It must be pointed out that these concentration factors include, not only stress concentration due to the abrupt geometry change near the edges of the stone, but also magnification due to the intrinsic properties of quasibrittle materials, such as natural stones, caused by particles, grains or inclusions, or simply regions of microscopically irregular surfaces causing topological interference (Sauoma et al., 2003). 5.2 Geometrical stress concentration factors The presence of shoulders, grooves, holes, and so on also modify stress distributions, which are obtained through elementary stress design formulas and are based on the members having a constant section or a section with a gradual contour change, so that localised high stresses occur and are measured by a stress concentration factor. In this case the elastic ‘theoretical’ or ‘geometric’ stress concentration factor, Kt, can be defined as the ratio between the actual maximum stress and the nominal stress: 8. Kt ~ smax snom where Kt is the elastic ‘theoretical’ or ‘geometric’ stress concentration factor; smax is the maximum stress to be Stone Molianos Lioz Estremoz Type In some cases, a theoretical factor can be derived for Kt based on the theory of elasticity, or it may be obtained through a laboratory stress analysis experiment. The universal availability of powerful, effective computational capabilities, usually based on the finite element method (FEM), has altered the need for and use of stress concentration factors. Nevertheless, it is desirable to compare the values obtained through both methods. In this case (Figure 12), typical kerf cross-section geometry is illustrated through a macro photograph. Based on the photo-elastic tests of Leven and Hartman (1951) and of Wilson and White (1973), cited by Pilkey and Pilkey (2007) the corresponding Kt can be determined for the present case. A value of Kt : 1:58 can be obtained using chart 3.9 in the bibliography of Pilkey and Pilkey (2007). 5.3 Finite element approach A two-dimensional (2D) finite element model representing a kerf anchorage configuration was developed to determine the stone’s stress state in the vicinity of the anchor. The model uses a four-node formulation for membrane elements with an isoparametric formulation for the translational in-plane stiffness component. A linear elastic analysis was performed using a refined mesh in the anchor’s vicinity in order to capture the elastic state of stress at the radius fillet as shown in Figure 12. Three runs Spall angle a: degrees c.v. Min a: degrees Max a: degrees 15?8 23?9 44?5 12?5% 23?1% 10?1% 19?4 32?7 52?6 13?4 16?6 39?0 Limestone Limestone Marble Table 5. Spall angle a measured (Figure 6) 168 expected in the member at the critical section; and snom is the reference or nominal principal tensile stress at the critical section according to Equation (6). Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos Molianos Figure 10. Normal and shear stress at a critical kerf leg section 5.4 Comparison of peak stress values Table 6 shows the calculated peak-factored elastic stresses, using Equation (8), which are summarised and compared with the maximum principal tensile stress obtained with the FEM. Lioz Note that the authors confirmed that the stress values obtained in tests coincided with the stress obtained in the prism samples using the same FEM formulation as that was used for the kerfsupported specimens. Estremoz Figure 8. Generated spalls after testing using a continuous stiff rail in Molianos and Lioz limestone and in marble Estremoz specimens The values in columns (4) and (6) in Table 6 may be used to compute the corresponding average elastic concentration factor value (Table 7) when the FEM is used ðKt : 1:67Þ. This value is in agreement with the one used in Equation 8 ðKt : 1:58Þ. 5.5 were made taking into account the self weight of the panels and the average breaking load values for each type of stone, according to Table 4 (Figures 6 and 7). The principal stress values are mapped in Figures 13, 14 and 15 for each stone type by applying the average breaking load value from tests. (a) (b) (c) Figure 9. Generated spalls after testing from left to right: (a) Molianos; (b) Lioz and (c) Estremoz specimens Magnification factors Standard design methods for engineering structures and components under static loading are usually based on avoiding failure caused by yielding/plastic collapse or buckling. The loading resistance is determined based on conventional solid mechanics theories of stress analysis. Conventional design procedures to prevent fatigue failure are based on experimental results for particular geometric details and materials. None of these procedures are capable of taking into account the effects of severe stress concentrations or crack-like flaws. The presence of such flaws is more or less inevitable to some extent in practical manufacturing and is a characteristic of quasi-brittle materials such as natural stones. The strength of a material can, in the simplest terms, be viewed as the maximum stress which the material can sustain under given conditions. Theoretical strength calculations often overestimate the strength if they do not incorporate mechanisms to account for material defects such as cracks. To propagate material defects, the theoretical strength must be overcome only locally, within a ‘stress concentration’ produced by these flaws; hence the effective strength of the material is lowered. 169 Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos Stress concentration factors (Keff) 6 Lioz Molianos Estremoz 5 4 3 2 1 Kaverage = 3.22 (Estremoz) Kaverage = 2.40 (Molianos) Kaverage = 2.12 (Lioz) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Specimens Figure 11. Effective stress concentration factors for kerf anchorages in Molianos, Lioz and Estremoz Fracture mechanics methods are particularly useful in making fitness for purpose assessments of the effects of flaws. An important effect arises when the crack affects the net crosssection area either in the case of a through-thickness crack in a plate of finite width, or in the case of remaining ligaments between the crack front and a free surface for part-thickness cracks (Albrecht and Yamada, 1977; Sauoma et al., 2003; Shah et al., 1995). When a crack is located in a region of geometric stress concentration, there will be a further increase in the stress. The overall effect depends on the relative size of the crack and the stress concentration zone. In this situation, a stress intensity magnification factor (Mc) is required to represent the necessary amplification (Albrecht and Yamada, 1977; Rooke and Cartwright, 1976). 5.6 11.5 mm Effective stress concentration factor Accordingly, the effective concentration factor, Keff, obtained through Equations (6) and (7), can be broken down to isolate the stress concentration caused by the kerfed edge geometry 1.7 1.7 1.6 2.04 kN 7 mm 1.6 1.65 kN 7.0 MPa 11.5 mm Z X -5,00 -4,00 -3,00 -2,00 -1,00 0,00 1,00 2,00 3,00 4,00 5,00 6,00 7,00 8,00 Figure 12. Kerf geometry and dimension in mm of edge crosssection 170 Figure 13. Finite element method calculation of maximum principal stresses at bottom of kerf: Molianos limestone Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos 3.19 kN 5.7 Formula for a kerf design anchorage The ultimate break design load, Pud, must be greater than the wind design acting load: 2.55 kN 11.0 MPa 11. Z X -7,0 -5,6 -4,2 -2,8 -1,5 -0,1 1,3 2,7 4,1 5,5 6,8 8,2 9,6 11,0 Figure 14. Finite element method calculation of maximum principal stresses at bottom of kerf: Lioz limestone (Kt) and the stress concentration caused by the intrinsic brittle behaviour of each type of stone (Mc) using the following relationship: The magnification factor values, Mc, are thus evaluated for each type of stone (Table 8). B 2 where wSd is the wind design of dynamic wind pressure; L is the panel span; and B is the panel width. By combining Equations 5 and 6 with Equation 10 and, for variable C, to solve the clearance or distance between the anchor’s contact point and kerf bottom (Figure 10), the following equation is obtained to establish the minimum clearance between the kerf rail and the bottom of the edge groove: 12. Keff ~Kt |Mc 9. PSd §wSd |L| C§ 1 ð2|tf |stRd Þ2 {ðwSd |L|Keff Þ2 | 12 stRd |wSd |L|Keff Alternatively, it is possible to define the minimum fin thickness by solving Equation 12 for tf: It is interesting to compare the values obtained for Mc with the constant 1z tan ðaÞ (where a is the average spall angle) [Table 5 – spall angle a measured (Figure 6)] for each stone type (Table 9). 13. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The relationship points to an equation of the following type: where C is the clearance between the kerf rail and the bottom of the edge groove; Keff is effective stress concentration factor equal to Kt |ð1z tan aÞ; Kt is the elastic geometric stress concentration factor; stRd is the design value of the stone’s flexural tensile strength equal to MR=cM ; MR is the flexural strength of the stone; cM is a partial safety factor for the tensile strength of the stone; tf is the fin or leg thickness of the kerfed edge stone; wSd is the design value of the dynamic wind pressure; L is length or span of the supported dimension stone; and a is the spall failure angle of the kerfed edge stone. Mc ~1z tanðaÞ 10. which is like saying that the amplification factor is somehow related with the stone’s intrinsic and rheological characteristic. Nevertheless, this relationship is not consistent with any theoretical or other empirical formula. Consequently, emphasis should be placed on the need for further work using other examples of limestone and marble to check whether this relationship is merely coincidental in this case. 10.6 MPa Z X -7,0 -5,6 -4,2 -2,8 -1,5 -0,1 1,3 2,7 4,1 5,5 12|C|Keff |wSd |L|stRd zðwSd |L|Keff Þ2 2|stRd A comparison was made between the required thickness given by Equation 13, due to the breaking load at the kerf and due to the bending resistance for spans between 500 and 1400 mm. 3.09 kN 2.48 kN tf ~ 6,8 8,2 9,6 11,0 Figure 15. Finite element method calculation of maximum principal stresses at bottom of kerf: Estremoz marble The total required thickness at the edge is calculated assuming that the groove has a typical width of 7 mm, a clearance of C 5 15 mm, a flexural strength of FS 5 20 MPa, cM 5 4, Keff 5 2 and for a wind design value of wSd 5 2?0 kPa. Partial safety factors were used in this example. However, it is reasonable to assume that an allowable stress approach, with a global safety factor, would yield similar results. 171 Construction Materials Volume 165 Issue CM3 (1) Stone Molianos Lioz Estremoz Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos (2) (3) Type P: kN Figure 7 Limestone Limestone Marble (4) (5) smax FEM: MPa 1?65 2?55 2?48 (6) smax Equation 8: MPa 7?0 11?0 10?6 snom Equation 5: MPa 6?8 10?3 10?0 4?29 6?49 6?32 Table 6. Peak stresses obtained using Equation 8 and finite element method (FEM) for each stone type Stone Molianos Lioz Estremoz Type smax FEM: MPa snom Equation 5: MPa Kt Limestone Limestone Marble average 7?0 11?0 10?6 – 4?29 6?49 6?32 – 1?63 1?69 1?68 1?67 Table 7. Elastic concentration factors with smax from finite element method (FEM) (1) Stone Molianos Lioz Estremoz Type Limestone Limestone Marble (2) (3) Keff Kt Mc 2?12 2?40 3?22 1?63 1?69 1?68 1?30 1?42 1?91 Table 8. Stresses from FEM analysis versus peak values using Equation 6 for each stone type Stone Molianos Lioz Estremoz Type Spall angle a ½1z tanðaÞ Mc Limestone Limestone Marble 15?8 ˚ 23?9 ˚ 44?5 ˚ 1?28 1?44 1?98 1?30 1?42 1?91 Table 9. Stress magnification factors compared with the fin’s detachment spall angles 172 Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos 5.8 30 Slab thickness: mm 25 t(kerf) t(midspan) 20 15 10 5 500 600 700 Safety factors design Allowable stress design factors were applied based on the stone’s physical/mechanical properties determined by testing that particular stone. In this field, some further work and tests are still necessary in order to establish adequate partial safety factors based on reliability calibration. Until that time, global safety factors based on allowable stresses may be used. 800 900 1000 1100 Bending span: mm 1200 1300 1400 Figure 16. Comparison between calculated thickness for breaking load and bending strength For example, based on recommendations by the Marble Institute of America (2008) a global safety factor of 3 is usually used for granite panel stresses away from connections, and a safety factor of 4 is usually used for stresses in the granite at connections. Based on the said Institute’s recommendations, a safety factor of 5 is commonly used for marble panel stresses caused by wind loading. For travertine, limestone and quartzbased stone, a safety factor of 8 is recommended. The Indiana Limestone Institute Handbook (1989) recommends that a safety factor of 6 be used for Indiana Limestone. 6. It seems evident that the required panel thickness is consistently determined by support strength, tkerf rather than by bending at midspan due to flexural resistance (Figure 16). The authors are of the opinion that stone panel thickness design will be unsafe if it continues to emphasise bending strength over anchorage breaking load, as is common practice. Conclusion It has been shown that a design focused on the bending strength of stone panels, to the detriment of the kerf anchorage, is unsafe. Yet, normal practice, as mentioned above, does not take this into account. In designing stone cladding systems with continuous kerf, two different effects must be taken into account when evaluating Figure 17. From left to right, different metamorphism stages: lower degree of crystallization in Molianos and Lioz limestone and a higher level of metamorphism in the Estremoz marble 173 Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos the effective stress in the critical region (kerf’s leg): first, the kerf geometry and, second, the stone-specific properties. ASTM (2004) ASTM C 1354 – 96: Standard test method for The first factor (kerf geometry) will cause stress concentration near the critical region that can be determined through analytical or numerical methods. The second factor (stone properties) further amplifies the stress, typical of quasi-brittle stones, and can be determined only through laboratory testing. The computed magnification factor values (column 3 of Table 8) are directly related to the type of stone studied in addition to its rheological properties, mineralogical flaws and grain structure. The stress magnification factor depends on the observed spall angle, and it seems evident that this angle is somehow related to the type of stone. It can be observed – such as in the case of Molianos, an oolitic limestone which is practically non-crystalline and with a regular internal structure (Figure 17, left) – that the spall angle is low and, consequently, the magnification factor is also low. For the Lioz, a semi-crystalline limestone with coarse elements and slightly affected by metamorphism (Figure 17, centre), the spall angle increases slightly when compared with the previous stone and, consequently, the magnification factor also increases accordingly. Lastly, the Estremoz marble has a higher level of metamorphism leading to a higher degree of recrystallisation (Figure 17, right), and thus it was found to have a greater spall angle and magnification factor. The derived semi-empirical formulae make it possible to evaluate the thickness required for a particular type of stone with kerfed edges in a continuous stiff rail. It must be emphasised that the stress concentration factor is not a safety factor. Regardless of the applied design principles, the safety factors must be applied after computing the effective stresses, taking into account relevant situations of geometry and the degree of metamorphism for each type of stone. 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Shah SP, Swartz SE and Ouyang C (1995) Fracture Mechanics of Construction Materials Volume 165 Issue CM3 Dimension stone design – kerf anchorage in limestone and marble Camposinhos and Camposinhos Concrete: Applications of Fracture Mechanics to Concrete, Rock, and Other Quasi-brittle Materials. John Wiley, New York, NY, USA. West D (2004) Safety factors for design of thin granite cladding – a review of international practice. Discovering Stone 3(5): 44–46. West DG and Heinlin M (2000) Anchorage pullout strength in granite: design and fabrication influences. In ASTM STP 1394. Dimension Stone Cladding: Design, Constructionm, Evaluation, and Repair (Hoigard KR (ed.)). ASTM, West Conshohocken, PA, pp. 121–134. WHAT DO YOU THINK? To discuss this paper, please email up to 500 words to the editor at [email protected]. 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