Numerical analysis of Masonry Spandrels with Shallow Arches

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1 Numerial analysis of Masonry Spandrels with Shallow Arhes Sujith Mangalathu Rose Programme, UME Shool, IUSS Pavia, Italy. Eole Polytehnique Fédérale de Lausanne (EPFL), Switzerland Katrin Beyer Eole Polytehnique Fédérale de Lausanne (EPFL), Switzerland. SUMMARY: In unreinfored masonry (URM) walls the vertial piers are onneted by horizontal andrel elements. These andrel elements onsiderably affet the global wall behavior when subjeted to seismi loading. Modeling of masonry walls without fully understanding the effet of andrels an lead to erroneous results. Deite the importane of andrels, the oupling ation of andrels is often negleted in design odes mainly due to the sarity of experimental data on masonry andrels and the absene of proper mehanial models. For the realisti seismi design of new buildings and assessment of existing buildings, mehanial models for masonry andrels are required. In this study, numerial investigations are arried out based on the simplified miromodeling approah using the ATENA software pakage. In the simplified miro-model, eah brik is modeled as a separate unit and the mortar joints are represented by ontat elements. The results of the numerial analyses are verified against experimental data from masonry andrel tests. It is found that the results from numerial studies are in good agreement with the experimental values. The numerial model is then used to ondut a parametri study on the peak strength of the masonry andrels with shallow arhes. Keywords: Unreinfored masonry, andrel elements, simplified miro-modeling 1. INTRODUCTION In unreinfored masonry (URM) walls, the vertial piers are onneted by horizontal elements alled andrels. Reent earthquakes (e.g., L Aquila, 9, Fig. 1) have shown that the andrel elements are often the first elements to rak or fail in a URM building under seismi loading. In ite of its importane, the framing ation of the andrel elements is negleted and only the vertial pier elements are onsidered while alulating the strength and the stiffness of the URM walls (Beyer and Dazio, 1). Various researhers (Magenes, ; Beyer and Dazio, 11) have pointed out that the andrel elements onsiderably affet the fore-deformation harateristis of the URM walls. The main reasons for ignoring the ontribution of masonry andrels are the sarity of experimental data on masonry andrels under seismi loading and the resulting absene of validated mehanial models. Reent experiments (Gatteso et al. 8; Beyer and Dazio, 1) have revealed the foredeformation harateristis of masonry andrels. The objetive of the urrent study is to obtain deeper insights into the behavior of masonry andrels by numerial modeling. The numerial representation of masonry an be ahieved either by miro-modeling, i.e., modeling the onstituents separately, or by maro-modeling, i.e., modeling the struture as a ontinuum. In miromodeling the units are represented by ontinuum elements and the joints are modeled by interfae elements. Depending upon the level of auray and simpliity, miro-modeling an be lassified into two modeling approahes detailed miro-modeling and simplified miro-modeling. In detailed miro-modeling, joints are represented by mortar ontinuous elements and disontinuous interfae elements, while in simplified miro-modeling, joints are represented by disontinuum elements (Loureno, 1994).

2 a) b) Figure 1. a. Old URM building during the L Aquila earthquake on April 6th, 9, showing andrel failure; b. Detailed view of the andrel failure (Beyer and Dazio, 1), Photo b ourtesy of A. Dazio The maro-modeling approah (e.g., Gambarotta and Lagomarsino, 1997) models the whole struture as a ontinuum without any distintion between the units and joints. Maro-modeling is useful for studying the global reonse of the struture. Simplified miro-modeling is suitable for small strutural elements as their reonses an be losely represented using the knowledge of material properties. In the urrent study, simplified miro-modeling is adopted in order to apture the foredeformation harateristis of the masonry andrels from the onstituent properties. The urrent study fouses only on masonry andrel elements with shallow arhes and with onstant axial load on the andrel.. EXISTING PROVISIONS FEMA 36 guidelines (ATC, 1998) and the Italian ode OPCM 3431 (OPCM, 5) are, to our knowledge, the only standards that propose equations for the strength of masonry andrels. The urrent setion provides a brief summary of the FEMA and OPCM models. For a detailed review of the various mehanial models for andrels readers are referred to Beyer and Mangalathu (1). FEMA 36 addresses the peak and the residual strengths of masonry andrels. The flexural apaity of the andrel in FEMA 36 (Eq..1, Table 1) is assumed to be derived from the shear stresses in joints between briks that are pulled out due to the opening of a flexural rak at the interfae between the pier and the andrel. The peak shear strength apaity of the andrel in FEMA 36 is based on the model by Turnsek and Caovi (197). The Italian seismi design ode OPCM 3431 (OPCM, 5) provides guidelines for omputing the shear and flexural apaities of andrel elements in URM buildings (Table 1). It distinguishes between andrel elements for whih the axial fores are either known or unknown. If the axial fore in the andrel is known, the andrels are treated like piers for the omputation of shear strength assoiated with the flexural mehanism. If it is unknown, the flexural apaity of the andrel is omputed from a strut-and-tie mehanism by replaing the axial fore in the andrel, P (Eq..) with the minimum of H p and.4h t f hd, where H p is the tensile strength of the horizontal tension elements suh as steel ties or ring beams. The geometri parameters, height of the andrel (h ) and thikness of andrel (t ) are shown in Fig. and f hd is the design ompressive strength of the masonry in the horizontal diretion. Like FEMA 36, OPCM 3431 adopts the shear apaity model by Turnsek and Caovi (Eq..5), if the axial fore is known. If it is unknown, the shear apaity is omputed using Eq..3. OPCM 3431 also provides a shear apaity equation (Eq..4) based on a sliding shear mehanism in the ompression zone. It is, however, not lear whether the OPCM equations address the peak strength or the residual strength of the masonry andrels.

3 Table 1 provides a summary of the FEMA and OPCM models. The geometri parameters l, l b, l j, h b and h j in Table 1 are defined in Fig.. The variable f dt in Table 1 is the diagonal tensile strength of the masonry, the ohesion, the oeffiient of frition, p is the mean axial ompressive stress in the andrel (p = P /h t ), and h the height of the ompression zone. Table 1. Summary of Equations for Prediting the Spandrel Strength (Beyer and Mangalathu, 1) h Vp, fl h f p,tot 1) (.1) l 3 4 h h FEMA OPCM Turnsek and Caovi (197) Flexure Flexure Shear 1 Shear Shear with lb f p,tot f p,bj tb f fp, bj. 5pier h h fl P l h j p,sj b lb h b NB 1 where, NB = number of wythes V 1 (.) 1 Vs ht with red red 1 h j hb / l b l j (.3) P Vs ht red.4 P with h (.4).85 f t dt hd and red as in Shear 1 p V p,s fdt h t 1 with h (.5) f l 1) End moments are onverted to a shear fore assuming the andrel is subjeted to double bending a) b) Figure. a. General geometry of andrel; b. Geometry of briks and mortar (Beyer, 1) 3. EXPERIMENTAL STUDY Beyer and Dazio (1) tested four masonry andrels that featured either a timber lintel or a shallow masonry arh. The results of one of the andrel elements with a shallow arh were hosen to validate the numerial model. A brief summary of the experimental setup is given in the following setion; for a more detailed desription of the test see Beyer and Dazio (1). The test onsisted of quasi-stati yli loading of masonry andrels. A shemati diagram of the test setup is shown in Fig. 3. A drift was imposed on the two piers, whih defined the deformation demand on the andrel. The axial elongation of the andrel was restrained by the horizontal rods. The axial fore in the andrel was kept onstant throughout the test with the help of a eial load ontrol system. The mehanial

4 properties of the onstrution materials were determined by material tests whih were arried along with the quasi-stati yli tests. These properties are given in Fig. 3. Test unit pier [MPa] [MPa] µ [ - ] f m [MPa] TUC pier Mean vertial stress on piers Cohesion of mortar joint f m µ All dimensions are in mm Axial fore in the andrel (P ) Constant 8 kn Compressive strength of masonry Coeffiient of frition of mortar joint Figure 3. Test setup for andrel test and mehanial properties of the onstituent materials (Beyer and Dazio, 1) 4. NUMERICAL MODELING OF MASONRY A two-dimensional modeling approah was adopted in the urrent study to repliate the experimental test setup. The ommerial finite element pakage ATENA (Cervenka, 7) was used for the urrent study. Using the simplified miro-modeling approah eah brik was modeled as a separate unit with ontat properties assigned at the interfae.. Residual surfae =, f t = Initial surfae f t ( ) 1 ( ft ) 1 ( ft ) ft, ft ft, ft Figure 4. Failure surfae for interfae elements

5 The interfae was modeled aording to the failure surfae as shown in Fig. 4. The initial failure surfae follows Mohr-Coulomb frition law with an ellipsoidal failure surfae in the tension regime (Fig. 4). One the interfae reahes its maximum shear stress (), it loses its ohesion () and tensile strength (f t ). The residual failure surfae aounts only for the fritional strength. The tensile and shear softening was defined based on the frature energy assoiated with eah mode. 4.1 Desription of the model The briks were modeled as plane stress elasti isotropi elements with a mesh element size of 1 m. The steel beams on the top and bottom of the masonry walls were also modeled as plane stress elasti isotropi elements with a mesh size of 1 m. The meshes, both in the briks and the steel beams, were of quadrilateral shape (CCISoQuad) and a mesh sensitivity study revealed that reduing mesh size further had no effet on the global reonse. The interfae was modeled with D interfae elements with the interfae properties and the failure mehanism as given in Fig. 3 and 4. The tensile and shear softening of the interfae was modeled based on the frature energy assoiated with eah mode and a linear variation was assumed in the urrent study. The frature energy of the shear mode (G II f) was alulated from the experiments as.1 N/mm and the frature energy (G I f) for the axial mode was assumed to be 1/1 th of that for the shear mode, i.e., G I f =.1 G II f. a) b) Figure 5. a. Numerial model for masonry-arh andrel; b. Measurement of drift A typial representation of the numerial model reated in ATENA is shown in Fig. 5a. Although quasi-stati yli loading was applied in the experiments of Beyer and Dazio (1) the urrent numerial study is limited to monotoni loading, i.e., loading only in a single diretion, for the following two reasons: (1) The envelope of the fore deformation urve from the quasi stati yli loading is an approximate representation of the fore deformation urve from the monotoni loading. () As the objetive of the urrent study is to ondut a parametri analysis, it is desirable to redue the omputational effort. The omputational effort assoiated with monotoni loading is less in omparison to quasi-stati yli loading. The analysis of the numerial model was arried out in two steps. The axial loads on the piers and the andrels were applied first, followed by the dilaement at the end of the steel beams. The dilaement was applied in the upward diretion for the left steel beam and in the downward diretion for the right steel beam (Fig. 5a). Numerial analysis was arried out with the standard Newton-Raphson algorithm. The tangent stiffness was updated at eah step of the analysis. The number of iterations was restrited to with a dilaement error tolerane of.1. If the algorithm failed to onverge the number of iterations was inreased to Validation of the numerial model The auray of the finite element model (FEM) was verified through the omparison with the experimental data for TUC from Beyer and Dazio (1). The fore-deformation behavior obtained from the numerial model and the experiments are shown in Fig. 6. The positive and negative envelope of the quasi-stati yli loading from the experimental results is plotted in Fig. 6. The

6 numerial model is able to predit the peak strength as well as the fore deformation harateristis up to a drift of.34 %, the maximum drift for whih onvergene was obtained. Figure 7 shows the deformed shape of the test unit and the numerial model at a drift of. %. The numerial model is able to apture the failure pattern with adequate auray. 1 1 Test (envelope in the positive diretion of loading) Test (envelope in the negative diretion of loading) ATENA model Shear fore (V Pier rotation (, %) Figure 6. Comparison of the numerial model with the experimental results a) b) Figure 7. Deformed shape at a drift () of. %, a. Test eimen TUC (Beyer and Dazio, 1); b. Numerial model in ATENA (magnifiation fator of 1) 5. PARAMETRIC ANALYSIS AND COMPARISON WITH THE EXISTING MODELS In order to identify the parameters that have a signifiant influene on the peak strength of the masonry andrels, a parametri study was arried out by varying one parameter at a time from the numerial model used for experimental validation An initial sensitivity analysis revealed that the axial stress on the andrel (p ), ohesion (), and the ratio of the height of the andrel to the length of the andrel (h /l ) are the most signifiant parameters that affet the peak strength and the mode of failure assoiated with masonry andrels. A detailed parametri study was arried out to hek the variation of the peak strength with reet to the parameters suh as p,, and h /l. The results of the parametri studies are shown in Fig. 8 to 1. These figures also provide a omparison of the numerially determined peak strength values with those predited by the existing mehanial models (setion ). The diagonal tensile strength of the masonry f dt was not determined experimentally and a value of f dt =.1 MPa was assumed for all andrels. The horizontal strength f hd of the masonry was also not determined experimentally and as a first approximation, f hd was assumed to be equal to the vertial ompressive strength of the masonry f m (Beyer and Dazio, 1). The ombined height of a brik and a joint was 74 mm. The length of a brik and the width of a head joint were assumed to be 1 mm and 1 mm, reetively.

7 5.1 The effet of axial stress on the peak strength of the andrel A parametri study was arried out using the model desribed in setion 4 in order to asertain whether the peak strength is affeted by the axial stress in the andrel. The axial stress was varied from MPa to.5 MPa. The results of the parametri sensitivity analysis are shown in Fig. 8. Figure 8 shows that the peak strength of the andrel is not muh influened by the axial stress on the andrel, but determines the mode of failure assoiated with the masonry andrel, i.e., flexural mode of failure is assoiated with low axial stresses and diagonal shear failure with higher axial stresses. 1 Peak strength of the masonry andrel (V p Numerial model (flexure) OPCM (flexure) FEMA (flexure) Numerial model (shear) OPCM (shear 1) OPCM (shear ) Turnse and Caovi (shear) Axial stress on the andrel (p, MPa) Figure 8. Variation of peak strength with reet to the hange in the axial stress on the andrel Figure 8 shows that none of the existing models are able to predit the peak strength of the masonry andrels. OPCM models signifiantly underestimate the peak strength of the masonry andrels. Although the Turnsek and Caovi model signifiantly underestimate the peak strength of the masonry andrels, it is the only model whih an provide a fair estimate of the peak strength of the masonry andrels. It is, however, to be noted that the urrent omparison by Turnsek and Caovi model is based on an assumed diagonal tensile strength. The FEMA flexural model predits a onstant peak strength when the axial load is varied. This is expeted sine the influene of the axial load was not onsidered in the FEMA model 5. The effet of ohesion on the peak strength of the andrel In order to study the effet of ohesion () on the peak strength of the andrel, a parametri study was onduted for various values of ohesion (ranging from.1 MPa to.3 MPa). The diagonal shear mode was observed to be the failure mode for the seleted andrels. The variation of the peak strength with reet to the hange in ohesion is shown in Fig. 9a. Figure 9a shows that the peak strength is greatly affeted by ohesion and inreases with an inrease in ohesion. The OPCM model (shear ) and the Turnsek and Caovi model predit onstant peak strength irreetive of the value of ohesion. The model by OPCM (shear 1) is able to identify the trend in the variation of peak strength with reet to the hange in the ohesion, albeit with a onstant offset.

8 a) 1 b) 1 Peak strength of the masonry andrel (V p Cohesion (, MPa) Peak strength of the masonry andrel (V p Numerial model (shear) OPCM (shear 1) OPCM (shear ) Turnse and Caovi (shear) Height to length ratio of the andrel (h /l ) Figure 9. Variation of peak strength with reet to, a. Change in ohesion; b. Height to length ratio of the andrel 5.3. The effet of height to length ratio of the andrel on the peak strength of the andrel Figure 9b shows the results of a parametri study on the influene of the ratio h /l on the peak strength of the masonry andrel. The load-arrying apaity diminishes as the pier beomes more and more slender. OPCM model (shear ) is unable to apture the variation of the peak strength with reet to the h /l ratio although they are able to predit the peak strength values at low slenderness ratios. The Turnsek and Caovi shear model an identify the variation of the peak shear strength with reet to the hange in the h /l ratio. a) Shear fore (V p = MPa P =. MPa P =.4 MPa P =.7 MPa P =.9 MPa P =.11 MPa P =.14 MPa P =.16 MPa P =.18 MPa P =.1 MPa P =.3 MPa b) Shear fore (V h /l =.88 h /l =.8 h /l =.75 h /l =.69 h /l =.63 h /l =.57 h /l =.51 h /l =.44 h /l = Pier rotation (, %) Pier rotation (, %) Figure 1. Fore-deformation harateristis of the masonry andrel with reet to a) hange in axial stress on the andrel; b) height to length ratio of andrel

9 6. FORCE-DEFORMATION CHARACTERISTICS OF MASONRY SPANDRELS The fore-deformation harateristis of the masonry andrel for different axial stresses (p ) and height to length ratio of andrel (h /l ) are shown in Fig. 1. The shear fore in the andrel varies linearly with the rotation ( up to a point where the first raks form in the andrel, similar to what was notied from the experiments (Beyer and Dazio, 1). The stiffness of the andrel redues one it raks, but the shear arrying apaity inreases till the peak strength is reahed. The shear strength apaity redues rapidly beyond the peak strength. 7. CONCLUSIONS In ite of the importane of masonry andrels in the fore-deformation harateristis of masonry piers, they are rarely onsidered in designs due to the lak of experimental data and mehanial models. A detailed parametri study was arried out to identify the parameters that have a signifiant influene on the peak strength of the masonry andrels with shallow arhes. Parametri studies revealed that the ohesion () and the height to length ratio of the andrel (h /l ) are the most sensitive parameters affeting the peak strength of masonry andrels. The mode of failure (flexure or diagonal shear) of the masonry andrels is determined by the axial stress in the andrel. The urrent study shows that the existing models (FEMA and OPCM) signifiantly underestimate the peak strength of masonry andrels. The peak strength apaity of the andrel might be of interest for the assessment of a building before an earthquake when a visual inetion has shown that the andrels are still largely unraked and the few existing raks are small. Future work should therefore expand the existing mehanial models of masonry andrels suh that they an apture the variation of the peak strength with reet to ohesion and height to length ratio of the andrel. The urrent study fouses only on the peak strength of the masonry andrel with shallow arhes; residual strength, limit rotations are not addressed and are the subjet of ongoing studies. ACKNOWLEDGEMENTS The finanial support for the first author by the European Commission through a sholarship for the Erasmus Mundus Master ourse in Earthquake Engineering and Engineering Seismology is gratefully aknowledged. REFERENCES ATC (1998) FEMA36. Evaluation of earthquake damaged onrete and masonry wall buildings. Basi Proedures Manual, Applied Tehnology Counil, Washington DC, United States. Beyer, K. and Dazio, A. (11). Modeling of andrel elements in URM struture with RC slabs or ring beams. Proeedings of the Eleventh North Amerian Masonry Conferene, Minneapolis, United States Beyer, K. and Dazio, A. (1). Quasi-stati yli tests on masonry andrels. Aepted for publiation in Earthquake Spetra (artile available on Beyer, K. and Mangalathu, S. (1). Review of strength models for masonry andrels. Communiated to Bulletin of Earthquake Engineering. Beyer, K. (1). Peak and residual strength of masonry andrels. Aepted for publiation in Engineering Strutures, 41, Carvenka, V. (7). AtenaComputer program for nonlinear finite element analysis of reinfored onrete strutures. Theory and User Manual, Prague, Czeh Republi. Gambarotta, L. and Lagomarsino, S. (1997). Damage models for the seismi reonse of brik masonry shear walls, Part II: The ontinuum model and its appliations. Earthquake Engineering and Strutural Dynamis, 6:4, Gatteso, N., Clemente, I., Maorini, L. and Noe, S. (8). Experimental investigation of the behavior of andrels in anient masonry buildings. Proeedings of the 14th world onferene on earthquake engineering, Otober 1-17, 8. Beijing, China, Loureno, P. B. (1994). Analysis of masonry strutures with interfae elementstheory and appliations. TU Delft report no , Tehnial University Delft, The Netherlands.

10 Magenes, G. (). A method for pushover analysis in seismi assessment of masonry buildings. Proeedings of the 1th world onferene on earthquake engineering, January 3-February 4,. Aukland, New Zealand, No O.P.C.M 374, /3/3 Primi elementi in materia di riteri generali per la lassifiazione sismia del territorio nazionale e di normative tenihe per le ostruzioni in zona sismia (in Italian). O.P.C.M 3431/5, 9/5/5 Ulteriori modifihe ed integrazioni all OPCM 374/3 (in Italian). Turnsek, V. and Caovi, F. (197). Some experimental results on the strength of brik masonry walls. Proeedings of the nd international brik masonry onferene, Stoke on Trent, United Kingdom,