15 th Intenationa ick and ock Masony Confeence Foianópois azi 01 3D INTERACTION DOMAINS FOR UNREINFORCED MASONRY PANELS SUJECTED TO ECCENTRIC COMPRESSION AND SEAR Paisi, Fuvio 1 ; Augenti, Nicoa 1 PhD, Post-Doc, Univesity of Napes Fedeico II, Depatment of Stuctua Engineeing, fuvio.paisi@unina.it Pofesso, Univesity of Napes Fedeico II, Depatment of Stuctua Engineeing, augenti@unina.it Fexua stength of uneinfoced masony (URM) coss-sections is typicay pedicted by means of two-dimensiona (D) inteaction domains. Atenativey, thee-dimensiona (3D) inteaction domains can be used to incude the baance of the whoe masony pane in the imit state equations of an exteme section. In this pape 3D domains ae pesented to descibe the inteaction between shea foce, axia foce and its eccenticity fo pismatic URM panes with ectangua coss-section. Fo both eastic and utimate imit states, sectiona equiibium equations coesponding to a given axia stain diagam ae meged with those of the entie masony pane fo cacked and uncacked conditions sepaatey. Such domains ae defined in a dimensioness fomat in ode to be independent of the compessive stength of masony. Limit state equations wee deived fo five diffeent stess-stain eationships, to investigate the infuence of masony behaviou in uniaxia compession unde the assumption of zeo tensie stength. In this pape the genea fomuation is speciaised fo a stength-degading constitutive mode to emphasise the effects of stain softening. Finay, the 3D domains wee sectioned with panes coesponding to diffeent eves of axia foce and axia foce eccenticity in ode to deive D inteaction domains expessed espectivey in tems of shea foce vesus axia foce eccenticity and shea foce vesus axia foce. It is shown that any incease in the axia foce eccenticity causes a otation of the shea foce vesus axia foce domain, esuting in an aowabe axia foce owe than that associated with concentic compession. Keywods: Uneinfoced masony panes, fexua stength domains, eccentic compession, shea INTRODUCTION A geat amount of eseach has focused on the stength pediction of eccenticay oaded uneinfoced masony (URM) coss-sections. Eccentic oading is athe diffeent fom concentic oading because the atte is a staticay deteminate pobem whee compessive stength can be assumed as the peak axia foce divided by the goss sectiona aea. On the othe hand, eccentic oading is a staticay indeteminate pobem whee bending moment capacity can be detemined though integation of noma stesses coesponding to a pedefined axia stain distibution ove the coss-section. This appoach cas fo the assumption of a macoscopic constitutive mode of the entie masony, which actuay is an assembage of stones/bicks and dy/mota joints. The axia stain sectiona distibution can be assumed to be pana unde eccentic oading. In fact, expeimenta studies have vaidated the Navie
15 th Intenationa ick and ock Masony Confeence Foianópois azi 01 hypothesis fo URM coss-sections in the cases of soid cay bick masony (iozi 1998; encich & Gambaotta 005) and cacaenite masony (Cavaei et a. 005). Typicay, two-dimensiona (D) inteaction domains ae used to pedict bending moment capacity of URM coss-sections unde a given axia foce (Augenti & Paisi 010a). Atenativey, thee-dimensiona (3D) inteaction domains can be used to incude the baance of the whoe masony pane in the imit state equations of an exteme section. In this pape 3D domains ae pesented to descibe the inteaction between shea foce, axia foce, and its eccenticity fo pismatic URM panes with ectangua coss-section. Such domains ae defined fo eastic and utimate imit states sepaatey, incuding the effect of the tensie cacking suffeed by the pane. A defomation-based sectiona anaysis aowed to speciaise imit state equations fo five stess vesus stain eationships of masony, some of them incuding stain softening as we. 3D domains ae pesented in a dimensioness fomat in ode to be independent of the compessive stength of masony. Finay, D inteaction domains in the fom of shea foce vesus axia foce eccenticity and shea foce vesus axia foce ae aso poposed. They wee deived by sectioning the 3D domains with panes coesponding to diffeent eves of axia foce and eccenticity, espectivey. ANALYSIS PROCEDURE In this pape the 3D stength domains ae pesented in the (V/N m, N/N m, e/) dimensioness space and incude a actions appied to the pane. In the fomuation N and V denote the axia and shea foce espectivey, N m the aowabe axia foce coesponding to concentic compession, e the axia foce eccenticity, and and the height and the ength of the pane espectivey. This epesentation was fist poposed by Augenti (007) in ode to investigate the fexua stength of spande panes at eastic imit state (ELS) by using an easticpefecty pastic (EPP) constitutive mode fo masony. Such an appoach was extended to othe constitutive aws of masony and utimate imit state (ULS). Specia emphasis was given to the oe of stain softening. In ode to get 3D stength domains, sectiona equiibium equations coesponding to a given axia stain diagam wee meged with those of the entie URM pane fo cacked and uncacked conditions sepaatey. The poposed inteaction domains ae to be defined fo each exteme section in the case of non-zeo appied oads. Theoetica anaysis was caied out with efeence to the simpified scheme shown in Figue 1, whee the URM pane is subjected to eccentic compession and shea, as we as extena foces F, P, and Q. N F e V G P + Q V e f N Figue 1: Intena and extena foces acting on the masony pane
15 th Intenationa ick and ock Masony Confeence Foianópois azi 01 In the case of a spande pane, namey a hoizonta maco-eement in the fame of a macoeement ideaisation of an URM wa with openings (Paisi 010), F stands fo the faction of eathquake action tansmitted by the foo sab, P is the faction of gavity oads tansmitted by the foo sab, and Q is the sef-weight of the pane. The foowing symbos ae aso used beow: - N and N, which ae the axia foces ove the eft and ight exteme sections, espectivey; - V and V, which ae the shea foces the eft and ight exteme sections, espectivey; - e and e, which ae the eccenticities of the axia foces N and N, espectivey, with espect to the centoid of the pane; - f, which is the distance of the foce ine of F fom the owe exteme of the pane. Whist intena foces and extena actions ae nomaised to the maximum axia foce N m, eccenticities ae nomaised to the goss height of the pane. The foowing sign convention is used in the fomuation: - N > 0 fo compessive foces; - M > 0 fo cockwise bending moments; - V > 0 fo shea foces inducing cockwise bending moments aound the centoid of the pane; and - e > 0 fo compessive foces inducing cockwise bending moments. Opposed to D stength domains, the 3D domains pesented heein can gaphicay be epesented ony if the geomety of the pane is quantitativey defined. In ode to aow compaisons between fomuations pesented heein to that poposed by Augenti (007), the same dimensions ae assumed fo the masony pane, namey 90 140 60 cm. Fo each imit state of inteest, the peak esisting shea foce can be pedicted by equations esuting fom otationa equiibium of the pane with espect to the point whee the axia foce ove the section unde veification is appied. The eationship between the shea foce V ove the ight section, the axia foce N, and the eccenticities e and e is: 1 V = N ( e e ) ( P Q) F f e + + + + (1) which can be divided by the maximum axia foce and goss height of the section, eaching the dimensioness fomat: 1 V = N ( e + e ) + ( P+ Q) + F f e () Simiay, fo the eft section it tuns out to be: 1 V = N ( e e ) ( P Q) F f e + + + + 1 V = N ( e + e ) ( P+ Q) + F f + e (3) (4)
15 th Intenationa ick and ock Masony Confeence Foianópois azi 01 The impementation of these equiibium equations of the pane in those eated to a singe coss-section aowed to get 3D fexua stength domains V/N m N/N m e/ fo sevea stessstain eationships of inteest. Key diffeences between D and 3D domains ae that the atte ae not geneay the same fo the exteme sections of the pane and depend on the sign of the axia foce eccenticity. 3D LIMIT STRENGT DOMAINS Five diffeent stess-stain eationships wee assumed fo masony in ode to deive the coesponding 3D imit stength domains. In this pape imit state equations ae pesented fo the stess-stain eationship poposed by Augenti & Paisi (010b) fo masony subjected to uniaxia compession paae to mota bed joints. Reades ae efeed to Paisi (010) fo the equations based on constitutive modes without stength degadation, i.e. EPP and Euocode 6 (CEN 005) modes, and othe constitutive modes incuding stength degadation, i.e. those poposed by Tunšek & Čačovič (1971) and by Augenti & Paisi (010b) fo masony subjected to uniaxia compession othogona to mota bed joints. When modeing masony was with openings though thei maco-eement ideaisation, one coud assume not ony diffeent compessive stengths fo pies and spandes, namey the vetica and hoizonta maco-eements espectivey, but aso diffeent constitutive aws. This was the need that inspied the authos to deive two stess-stain eationships fo compessive oading diections paae and othogona to mota bed joints (Augenti & Paisi 010b). Since the poposed 3D stength domains wee deived to be mainy appied to spande panes, and then to pie panes, this study focuses on the fomuation deived fo the fome panes. Unde such assumptions, it was found that any given ectangua coss-section eaches ELS in cacked conditions if the nomaised axia foce N/N m fas in the inteva [0,0.64], and uncacked conditions if N/N m fas in the inteva [0.64,1]. In both conditions one can compute the neuta axis depth nomaised to the goss height (i.e., h = h/) though the foowing equations espectivey: N = 0.64h (5) 0.45 0.09 N = 1 + 3 h h (6) At ULS the coss-section is cacked if N/N m fas in the inteva [0,0.69] and uncacked if N/N m fas in the inteva [0.69, N im /N m ]. In such cases the nomaised neuta axis depth can espectivey be deived by the foowing equations: 0.7 3 4 5 N = h 0.7 + 0.34με 0.17 με + 0.03με 0.003με + 0.0001μ ε με (7) 3 3 3 0.8με 0.8με 0.54με 0.09με + 0.36με N = 1.64με 0.8με 0.36με + h h 3 0.09μ ε 0.7 3 4 5 + + h 0.7 0.48μ 3 ε 0.07 με + 0.1με 0.003με + 0.0001με h με (8)
15 th Intenationa ick and ock Masony Confeence Foianópois azi 01 whee μ ε is the avaiabe stain ductiity assumed fo masony in compession. The nomaised imit axia foce N im /N m depends on such a ductiity. Assuming the Augenti- Paisi constitutive mode, N im /N m anges fom 0.97 to 0.64 fo μ ε anging fom 1.75 to 6. One can assume, e.g., μ ε = 1.75 in design and up to μ ε = 6 in noninea assessment pocedues. As stated above, the poposed 3D stength domains depend on the exteme section to be assessed. The incusion of the equiibium equations eated to the whoe masony pane ed to equations that povide the maximum shea foce ove an exteme section at which coesponds the attainment of ELS o ULS at the opposite section. ence, the eft [ight] section of an URM pane is safe at ELS o ULS if the absoute vaue of shea demand on the ight [eft] section does not exceed the absoute vaue of shea capacity on the atte section. The absoute vaue is used because both the sign and magnitude of the shea capacity depend on the sign of the axia oad eccenticity on the section to be assessed. Fo the eft section of the URM pane at ELS in cacked conditions, the shea capacity can be pedicted as foows: 1 = N e + 0.57( N F) + F( f 1) + ( P+ Q) fo e > 0 1 3 = N e 0.57( N F) F f ( P Q) fo e 0 + + + + < (9) (10) which ae epaced in uncacked conditions by: M 1 = N e + ( h ) + F f + ( P+ Q) fo e > 0 M 1 = N e ( h ) + F f + ( P+ Q) fo e < 0 (11) (1) being: M 0.03 0.11 ( h ) = N e = + (13) 3 h h Simiay, the ight section attains ELS in cacked conditions if the shea demand eaches the vaue povided by one of the foowing equations: 1 = N e + 0.57( N + F) + F f ( P+ Q) fo e > 0 1 = N e + 0.57( N + F) + F( f 1) ( P+ Q) fo e < 0 (14) (15)
15 th Intenationa ick and ock Masony Confeence Foianópois azi 01 which ae epaced in uncacked conditions by: M 1 = N e + ( h ) + F f ( P+ Q) fo e > 0 M 1 = N e ( h ) + F f ( P+ Q) fo e < 0 (16) (17) In ULS equations the avaiabe stain ductiity pays a key oe. Fo the eft section in cacked conditions the foowing equations wee obtained: 1 3 = N e κμ ( N F) F( f 1) ( P Q) fo e 0 + + + + > 1 3 = N e κμ ( N F) F f ( P Q) fo e 0 + + + + < (18) (19) being: κ 0.13 0.35μ 0.17 μ + 0.08μ 0.0μ + 0.00μ 3 4 5 ε ε ε ε ε μ = 3 4 5 6 0.7 0.7 με 0.34με + 0.17 με 0.03με + 0.003με 0.001με (0) In uncacked conditions the neuta axis depth must aso be detemined though Eq. (6) o (8), depending on the imit state of inteest. Then, one can pedict the esisting shea foce on the ight section coesponding to the attainment of ELS on the eft section, as foows: M 1 = N e + ( h ) + F f + ( P+ Q) fo e > 0 M 1 = N e ( h ) + F f + ( P+ Q) fo e < 0 (1) () whee: 3 3 3 M 0.03με 0.0με + 0.09με 0.14με 0.05με 0.09με ( h ) = N e = + + + 3 h h h 0.07 0.17 3 4 + h 0.35 + + 0.16μ ε + 0.0με 0.03με + 0.001με + με με 0.13 + h 0.35 0.4μ 0.04μ + 0.06μ 0.00μ με 3 4 ε ε ε ε (3)
15 th Intenationa ick and ock Masony Confeence Foianópois azi 01 To assess the ight section at ULS in cacked conditions the foowing equations can be used: 1 3 = N e κμ ( N F) F f ( P Q) fo e 0 + + + + > 1 3 = N e κμ ( N F) F f ( P Q) fo e 0 + + + + < (4) (5) which ae epaced in uncacked conditions by: M 1 = N e + ( h ) + F f ( P+ Q) fo e > 0 M 1 = N e ( h ) + F f ( P+ Q) fo e < 0 (6) (7) whee M ( h ) can be computed though Eq. (3) poviding that the supescipt is epaced by. The 3D stength domains at ELS and ULS ae espectivey shown in Figues 3(a) and 3(b) fo an URM pane with dimensions 90 140 60 cm and no extena foces F, P, and Q. In fact, ony if the atte ae zeo the 3D stength domains ae the same fo both exteme sections. These domains have the same shape, but that eated to ULS povides owe shea capacity vaues because of the high compessive stength degadation of masony associated with μ ε = 6. On the contay, utimate shea capacity is highe than that at ELS if significanty owe vaues of avaiabe stain ductiity ae assumed fo masony (fo instance the typica vaue μ ε = 1.75 coesponding to the assumption of cacking and utimate axia stains equa to 0.00 and 0.0035 espectivey). (a) (b) Figue 3: V/N m N/N m e/ stength domains at (a) ELS and (b) ULS
15 th Intenationa ick and ock Masony Confeence Foianópois azi 01 D LIMIT STRENGT DOMAINS The paticua shape of the 3D stength domains can be investigated by means of D stength domains deived by sectioning the fome domains at diffeent eves of V/N m, N/N m, and e/. It is woth noting that ony if the atte is zeo the D stength domain of the URM pane defined in the (V/N m, N/N m ) pane is symmetica to the N/N m -axis (Figs. 4(a) and 4(b)) and the shea capacity vanishes at an aowabe axia foce coesponding to concentic oading. 0. 0. 0.1 0.1 V e /N m 0 0 0. 0.4 0.6 0.8 1 N/N m V u /N m 0 0 0. 0.4 0.6 0.8 1 N/N m -0.1-0. e/ = 0 e/ = 0. e/ = 0.4 e/ = -0. e/ = -0.4 (a) (b) Figue 4: V/N m N/N m stength domains at (a) ELS and (b) ULS -0.1-0. e/ = 0 e/ = 0. e/ = 0.4 e/ = -0. e/ = -0.4 Othewise, as the axia foce eccenticity inceases, the stength domain gaduay otates aound the coodinate system oigin. It foows that the V/N m N/N m stength domain is tuncated at deceasing axia foce vaues povided by the foowing equations: ( ) N = 0.87 1 e fo e > 0 (8) aw ( ) N = 0.87 1+ e fo e < 0 (9) aw in the case of cacked section at ULS, and: N aw 0.09 0.45 = 1+ 3 h h (30) in the case of uncacked section at ULS. Eq. (30) equies the detemination of the nomaised neuta axis depth by numeica soving the foowing equations: 0.11( h + 0. 4) ( h 0.53)( h 0.)( h + 0. 76) e = fo e > 0 (31) e 0.11( h + 0.4) ( h 0.53)( h 0.)( h + 0.76) = fo e < 0 (3) Fo a cacked section at ULS the aowabe axia foce is given by:
15 th Intenationa ick and ock Masony Confeence Foianópois azi 01 1 Naw = ( 1 e) foe > 0 (33) κ μ 1 Naw = ( 1+ e) fo e < 0 (34) κ μ wheeas in the case of uncacked section at ULS one can use the foowing equation: N aw 3 3 3 0.09με 0.09με + 0.36με = 1.64με 0.8με 0.36με + + 3 h h 3 0.8με 0.8με 0.54με + h 0.7 + h 0.7 0.48μ 0.07 μ + 0.1μ 0.003μ + 0.0001μ με 3 4 5 ε ε ε ε ε (35) whee h can be detemined as foows: e 4 5 a+ bh + ch + dh + f h = fo e > 0 (36) 3 4 g+ mh + nh + ph + h e 4 5 a+ bh + ch + dh + f h = fo e < 0 3 4 g+ mh + nh + ph + h (37) A coefficients a,..., depend nonineay on μ ε (Paisi 010). If the 3D stength domains at ELS and ULS ae sectioned at diffeent N/N m vaues, the D stength domains shown in Figues 5(a) and 5(b) can espectivey be obtained. As the axia foce inceases, the stength domain expeiences inceasing cockwise otation aound the coodinate system oigin, and educes up to degeneate in a point when the axia foce eaches the aowabe axia foce associated with zeo eccenticity (that is N im /N m ). In fact, as the axia foce eccenticity inceases, both the effective section and the esuting axia foce educe. V e /N m 0.0 0.15 0.10 0.05 0.00-0.5-0.4-0.3-0. -0.1 0 0.1 0. 0.3 0.4 0.5-0.05 e/ V u /N m 0.0 0.15 0.10 0.05 0.00-0.5-0.4-0.3-0. -0.1 0 0.1 0. 0.3 0.4 0.5-0.05 e/ N/Nm= 0.1 N/Nm= 0.3 N/Nm= 0.5 N/Nm= 0.7-0.10-0.15-0.0 N/Nm= 0.1 N/Nm= 0.3 N/Nm= 0.5 (a) (b) Figue 5: V/N m e/ stength domains at (a) ELS and (b) ULS -0.10-0.15-0.0
15 th Intenationa ick and ock Masony Confeence Foianópois azi 01 CONCLUSIONS 3D domains have been pesented to descibe the inteaction between shea foce, axia foce and its eccenticity fo pismatic URM panes with ectangua coss-section. Fo both ELS and ULS, sectiona equiibium equations have been meged with those of the entie masony pane fo cacked and uncacked conditions sepaatey. A detaied discussion has been made on 3D domains coesponding to a stength-degading constitutive mode of masony, to emphasise the effects of stain softening on fexua stength. New types of D inteaction domains, in the fom of shea foce vesus axia foce and shea foce vesus axia foce eccenticity, have been pesented as we. It has been found that any incease in the axia foce eccenticity causes a otation of the shea foce vesus axia foce domain, esuting in an aowabe axia foce owe than that associated with concentic compession. ACKNOWLEDGEMENTS This eseach was caied out in the famewok of the ReLUIS-DPC 010-013 poject (Line AT1-1.1 - Evauation of the neabiity of Masony uidings, istoica Centes and Cutua eitage ) funded by the Itaian Depatment of Civi Potection. REFERENCES Augenti, N. Resistenza dee fasce di piano di edifici in muatua soecitate da azioni sismiche, 1 th Itaian Confeence on Eathquake Engineeing, Pisa, Itay, 007, Pape No. 7 (in Itaian). Augenti, N., Paisi, F. Utimate fexua stength of uneinfoced masony spande panes, 8 th Intenationa Masony Confeence, Desden, Gemany, 010a, pp 1653-166. Augenti, N., Paisi, F. Constitutive modes fo tuff masony unde uniaxia compession, Jouna of Mateias in Civi Engineeing,, 11, 010b, pp 110-1111. iozi, L. Evauation of compessive stength of masony was by imit anaysis, Jouna of Stuctua Engineeing, 114, 10, 1998, pp 179-189. encich, A., Gambaotta, L. Mechanica esponse of soid cay bickwok unde eccentic oading. Pat I: Uneinfoced masony, Mateias and Stuctues, 38,, 005, pp 57-66. Cavaei, L., Faia, A., La Mendoa, L., Papia, M. Expeimenta and anaytica esponse of masony eements unde eccentic vetica oads, Engineeing Stuctues, 7, 8, 005, pp 1175-1184. CEN. Euocode 6: Design of masony stuctues - Pat 1-1: Genea ues fo einfoced and uneinfoced masony stuctues, EN 1996-1-1. Comité Euopéen de Nomaisation, usses, egium, 005. Paisi, F. Non-inea seismic anaysis of masony buidings. PhD Thesis, Univesity of Napes Fedeico II, Napes, Itay, 010, 336pp. Tunšek, V., Čačovič, F. Some expeimenta esuts on the stength of bick masony was, nd Intenationa ick & ock Masony Confeence, Stoke-on-Tent, UK, 1971, pp 149-156.