SAC2014 9 th Internatonal Conference on Structural Analyss of storcal Constructons F. Peña & M. Chávez (eds.) Mexco Cty, Mexco, 14 17 October 2014 LIMIT ANALYSIS IN LARGE DISPLACEMENTS OF MASONRY ARCES SUBJECTED TO VERTICAL AND ORIZONTAL LOADS N. Cavalagl, V. Gusella and L. Severn Department of Cvl and Envronmental Engneerng, Unversty of Peruga, va G. Durant, 93, 06125, Peruga, Italy {ncola.cavalagl, vttoro.gusella}@unpg.t, laurasevern@strutture.unpg.t eywords: masonry arch, large dsplacements, lmt analyss, horzontal loads Abstract. Ths work ams to analyze the behavor of masonry arches at collapse. The study fts nto the frame of lmt analyss referrng to eyman s theory. The collapse mechansm s of rotatonal type, snce t shows the relatve rotaton of the masonry voussors around the socalled hnges, whch occur at the edge of the thckness. In partcular, two types of arches are analyzed, the crcular and ponted one. The loadng system conssts n vertcal and horzontal loads, whch refer respectvely to the self-weght and to the sesmc actons. In a frst step the collapse mechansm, the correspondng horzontal load multpler and the horzontal thrusts at abutments are determned, n the condton of rgd abutments, as functons of geometrcal features of the structure. Moreover, the mnmum thckness of the arch s determned as functon of the horzontal load multpler. Fnally, the crcular arch wth elastc abutments s studed. The falure condtons are determned as functons of the flexbltes at the abutments. The mnmum thckness that the crcular arch should have to stand s determned as functon of the horzontal load multpler and the flexbltes at the abutments.
N. Cavalagl, V. Gusella and L. Severn 1 INTRODUCTION The masonry arch has been wdely studed durng the last decades, especally to mprove the understandng of hstorcal constructons, for ther conservaton and restoraton. The turnng pont n studes on the stablty of masonry buldngs occurred n the early sxtes, when Jacques eyman extended the lmt analyss, ntally developed for steel structures, to the so-called Stone Skeleton. The applcaton of the classcal approach of lmt analyss to the masonry arch [1] requres the defnton of ) equlbrum condton, ) resstance crteron, ) mechansm condton. The frst ) and the second ) correspond respectvely to the ndvduaton of a thrust lne n equlbrum wth external loads and anywhere contaned n the boundary of the arch. The thrd condton ) corresponds to a rotatonal mechansm, wth hnges that grow at the edge of the thckness, f the followng hypothess about the masonry are assumed [2]: masonry has no tensle strength, the compressve strength of masonry s nfnte, sldng falure does not occur. The stablty of masonry arches s consdered as a geometrc problem, namely a rght shape desgn s needed to acheve a safe state. eyman [1] gves the law of the mnmum thckness for the crcular arch subected ust to self-weght, as functon of the angle of embrace. The analyss on the mnmum thckness n the presence of the self-weght has been recently extended also to ponted arches [3]. The nfluence of fnte dsplacements at the abutments on the structural behavor s consdered n the works of Francos [4] and Ochsendorf [5]. The comprehenson of the response of the masonry arch to envronmental actons, as the earthquake or the yeldng of abutments, must be completed. The prevous work doesn t take nto account the effects of horzontal loads and abutments fnte dsplacements on the mnmum thckness, for crcular and ponted masonry arches. In ths paper the behavor of masonry arches at collapse s studed. In the frst part the crcular and ponted arch s consdered to be supported by rgd abutments and the mnmum thckness n the presence of both vertcal and horzontal loads s determned. In the second part the condton of elastc abutments for the crcular arch s analyzed; moreover, the nfluence of the flexblty of the pers on the mnmum thckness s evaluated by a numercal procedure n large dsplacements. 2 RIGID ABUTMENTS 2.1 Geometrcal descrpton The analyss s performed as a plane problem. The geometrc characterzaton depends on the type of the consdered masonry arch. Referrng to the crcular arch, the geometry s descrbed by assgnng the span l, the rse f and the thckness s, as shown n Fgure 1(a). It follows that the central angle subtended by the arch s unquely defned. Ponted arch needs an addtonal parameter; actually, the structure s determned by assgnng the span l, the rse f, the thckness s and the angle (Fgure 1(b)). Let us denote by d the out of plane depth. The analyss, both for the crcular and the ponted arch, s carred out by the dvson of the structure nto n voussors, whch are numbered from left to rght. The resultng n+1 onts are obtaned by radal cuts. The geometrcal descrpton of the structure, referrng to a system of Cartesan axes (z, y), requres the knowledge of z and y coordnates of the followng ponts: center of gravty G of each voussor, ntrados I and extrados S of the radal onts, geometrc centers P of the radal onts. The loadng system conssts of vertcal and lateral ponted loads, 2
Lmt analyss n large dsplacements of masonry arches subected to vertcal and horzontal loads whch represent respectvely the self-weght F of each voussor and the correspondng sesmc acton F s. The latter s proportonal to the vertcal load through a multpler k. Indeed t results: F A d (1) Fs m k F (2) where m represents the masonry s specfc weght and A the area of each voussor. Wthout loss of generalty t s assumed that the sesmc acton,.e. the force F s, s drected from left to rght. See [6] for further detal regardng the parametrc descrpton of the arches. Fgure 1: Geometry of crcular (a) and ponted (b) masonry arch. 2.2 Balance condtons and numercal procedure The purely rotatonal falure mechansm of an arch corresponds, n the presence of both vertcal and horzontal loads, to a four-hnge mechansm [4] (Fgure 2). It follows that at the collapse the arch s a statcally determned structure, whch can be solved by usng balance condtons. Due to the non-symmetrc nature of the problem, the poston of the hnges can t be mmedately defned. Therefore, let us assume a frst attempt confguraton of hnges and mpose the equlbrum n ths collapse condton. From the statc theorem of lmt analyss, ths equlbrated confguraton results correct only f the correspondng lne of thrust les nsde the masonry and passes through the assgned four hnges, namely f the resstance crteron for masonry s satsfed. If on the contrary the lne of thrust falls outsde the arch, the poston of the hnges must be changed and the equlbrum mposed agan; the soluton can be attaned by an teratve procedure. As suggested n [4], the best practcal choce s to move the hnges where the dstance between the center lne of the arch and the lne of thrust s maxmum. So a few steps are requred to get the rght confguraton. Let us denote by M, Q, T, U the four collapse hnges correspondng to the m, q, t, u onts and by V U, U the vertcal and horzontal reacton at hnge U; by takng the momentum balance about the remanng hnges one obtans: 3
N. Cavalagl, V. Gusella and L. Severn ntu ntu U ( yt yu ) VU ( zt zu ) F ( zt zg ) k ( ) 0 F yt yg 1 1 nqu nqu U ( yq yu ) VU ( zq zu ) F ( zq zg ) k ( ) 0 F yq yg 1 1 nmu nmu U ( ym yu ) VU ( zm zu ) F ( zm zg ) k F ( ) 0 ym yg 1 1 (3) where n TU, n QU, n MU refer respectvely to the number of voussors between the onts t, q, m and u. The expresson (3) s a determned system of equatons, whch can be solved n order to provde the reactons at hnge U and the load multpler k. Fgure 2: Generc four-hnge poston assocated to the collapse condton n the presence of vertcal and horzontal loads. The drawng of the lne of thrust needs that the coordnates of the centers of pressure are known n correspondence of each radal ont, namely t requres the knowledge of the eccentrcty of the normal force. ence, by usng expressons dependng on the poston of the ont, the resultant vertcal and horzontal forces are defned (Fgure 3(a)): n u U k F 1 nu V VU F 1 nu nu M VU ( zu zp ) ( ) ( ) ( ) U yu yp F zg z P k F yg y P 1 1 In the Equaton (4), f the th ont s at the left of hnge U, one has to use the upper sgn, otherwse the lower one. Thus, the normal force s known at each ont; by usng respectvely the upper and lower sgn for the case zp < zi and zp zi: (4) N cos V sn (5) where represents the angle between the lne perpendcular to the th ont and the horzontal one. 4
Lmt analyss n large dsplacements of masonry arches subected to vertcal and horzontal loads Fnally, the eccentrcty of the normal force s gven by: e M N (6) Fgure 3: Stress state of the th ont. In order to check f the lne of thrust, obtaned by lnkng the centers of pressure, s anywhere nsde the masonry, the followng condton must be verfed at each ont: s s e (7) 2 2 It should be notced that the sgn of equalty holds only n correspondence of the hnges M, Q, T and U. As mentoned above, f Equaton (7) s satsfed, then the poston of hnges dentfes the actual falure mechansm and the correspondng load multpler. Otherwse, necessarly the hnges have to be moved n order to ensure that the statc theorem s verfed at each ont. 2.3 Results for crcular arches In order to pont out the nfluence of geometry on the collapse of masonry arches, the relatonshp between the load multpler k, the open angle and the dmensonless thckness s/l s shown n Fgure 4. Each curve was obtaned by settng a constant value of the angle, progressvely ncreasng the parameter s/l and evaluatng the correspondng value of the collapse multpler k. Notce that for the same thckness, the multpler k ncreases wth the decreasng of the angle. Ths means that the more the arch s lowered, the greater wll be the resstance to horzontal actons,.e. the earthquake. In [1] eyman studed the effects of geometrcal propertes on the stablty of crcular masonry arches subected ust to self-weght. e found the mathematcal expresson and the graphc trend of the mnmum thckness that the arch should have to stand. In ths work the study s extended by consderng the presence of sesmc acton too, whch s quantfed through the load multpler k. Fgure 5 depcts graphcally the mnmum thckness for the crcular masonry arch n the presence of an assgned sesmc multpler. The relatonshp s expressed between the half angle of embrace /2 and the dmensonless thckness s/r, beng r the radus. By settng a constant value of k, namely by consderng the ndvdual curve, t results that the mnmum thckness ncreases wth the ncreasng of the angle of embrace. 5
N. Cavalagl, V. Gusella and L. Severn Fgure 4: Load multpler k of crcular arches as functon of geometry. Fgure 5: Mnmum thckness for crcular masonry arches wth vertcal and horzontal loads. Geometrcal features affects also the collapse horzontal thrust. In Fgure 6(a) and 6(b) the path of the abutments horzontal thrust s shown. Each curve, relatng to an assgned open angle, presents an upper and a lower branch, whch ntersect n a pont correspondng to the eyman s mnmum thrust. Ths state corresponds, as known, to the arch subected only to self-weght. The lower branch refers to the left abutment s thrust, whle the upper branch dentfes the rght one. As expected, due to the prevous assumpton about the drecton of horzontal loads, the path decreases n the frst case and ncreases n the latter. In Fgure 6 we denote by W the self-weght of the arch. 2.4 Results for ponted arches The same procedure of analyss s extended to ponted arches. The effects of geometry are shown by comparng the crcular arch, whch has span l and rse f0, wth ponted arches havng the same span and dfferent values of f and. We are consderng two types of ponted arch, whch dffer from the rato f / f0. 6
Lmt analyss n large dsplacements of masonry arches subected to vertcal and horzontal loads Fgure 6: Normalzed horzontal thrust for crcular arches as functon of the open angle and of dmensonless thckness s/l (a), the angle and of load multpler k (b) (lower branch = left abutment thrust, upper branch = rght abutment thrust). Fgure 7: Load multplers for ponted arches f / f 0 = 1.5 (a), f / f 0 = 2 (b). Fgure 7(a) refers to arches characterzed by the rato f / f0 = 1.5, whle Fgure 7(b) corresponds to a value f / f0 = 2. Each curve was determned by settng a constant value of the angle, progressvely ncreasng the parameter s/l and evaluatng the correspondng value of the collapse multpler k. Notce that Fgure 4 was obtaned for a rato f / f0 = 1,.e. for the crcular arch. Fgure 8 corresponds to Fgure 5, snce t gves the mnmum thckness for ponted masonry arches wth f / f0 = 1.5 (a) and f / f0 = 2 (b) at dfferent values of the load multpler k, as functon of the angle. As shown above for crcular arches, the graph of the abutment horzontal thrust s drawn n Fgure 9 and 10; notce that the trend of the left and rght thrust for ponted arches results smlar to that of crcular ones. 7
N. Cavalagl, V. Gusella and L. Severn Fgure 8: Mnmum thckness for ponted masonry arches f / f 0 = 1.5 (a), f / f 0 = 2 (b). Fgure 9: Normalzed horzontal thrust for ponted arches f / f 0 = 1.5 (a), f / f 0 = 2 (b). Fgure 10: Normalzed horzontal thrust for ponted arches f / f 0 = 1.5 (a), f / f 0 = 2 (b). 8
Lmt analyss n large dsplacements of masonry arches subected to vertcal and horzontal loads 3 ELASTIC ABUTMENTS If the abutments are perfectly rgd then the arch behaves as a rgd-plastc system, because an ncreasng horzontal load doesn t determne any dsplacement before the collapse. At falure, n correspondence to a four-hnge mechansm, undefned movements occur. On the contrary, f the structure has spreadng supports, some hnges form at the begnnng of the load hstory, when the self-weght starts actng after the removal of the rb: the arch s n the socalled mnmum thrust condton. eyman n [2] shows the shape of the lne of thrust and ndcates the arrangement of hnges. Let us denote by the horzontal relatve dsplacement of abutments, we assume: 0 for k 0 0 for k 0 Condtons (8) mean that the abutments start spreadng when horzontal loads are appled; nstead, when t results k=0 the condton of lttle dsplacements s assumed. A non-zero horzontal load causes fnte dsplacements of abutments and then the shft of statc hnges, whch correspond to the mnmum thrust condton. The multpler k can ncrease untl a lmt value, correspondng to the collapse of the structure by the growng of a four-hnge mechansm. The research of the collapse mechansm s carred out n accordance wth the statc theorem of lmt analyss, by gradually ncreasng the horzontal dsplacement: the hnges arrangement correspondng to an equlbrated lne of thrust that les nsde the masonry, s found at several values of δ. When ths s no longer possble, t means that the falure condton s reached. 3.1 nematc descrpton n fnte dsplacements The geometrcal descrpton of arches wth rgd abutments, referred to n 2.1, s vald also for arches wth elastc abutments n the mnmum thrust condton. The relatonshp between horzontal dsplacements and correspondng abutment reactons R (wth =A, B) s assumed to be elastc and t s gven by: (8) c c R A A A R B B B wth A B and havng denoted by c A and c B the two flexbltes at abutments. For fnte dsplacements, the knematc descrpton s made by referrng to the Cartesan axes (z, y ) havng orgn n the lower pont of the frst ont and modelng the arch as a system of two rgd lnks T and, both hnged at ts ends (Fgure 11(a)). The knowledge of z and y coordnates of the ponts of nterest n the knematc confguraton (G, I, P and C, referred to n 2.1) s needed n order to mpose the equlbrum condton n large dsplacements. 3.2 Balance condtons and numercal procedure By referrng to the rgd lnks T and, sx balance condtons can be wrtten, ncludng four translatonal and two rotatonal. In order to obtan a determned system, t s needed to add a further compatblty equaton, whch nvolves horzontal dsplacements at abutments. So one obtans: (9) 9
N. Cavalagl, V. Gusella and L. Severn nt VT V F 0 1 nt T k F 0 1 nt nt VT w1 T w2 F ( z ' z ' G ) k ( ' ' ) 0 F y y G 1 1 n V V F 0 1 n k F 0 1 n n V w3 w4 F ( z ' z ' G ) k F ( ' ' ) 0 y y G 1 1 ca( A Ag ) cb ( B Bg ) (10) where: V T, V, V and T, of sesmc acton at hnges T, and, n the presence of sesmc acton,, denote vertcal and horzontal reactons n the presence A and Ag and Bg B are the horzontal reactons at abutments are the horzontal reactons at abutments n the presence of the only self-weght, n T and n refers respectvely to the number of voussors between the hnges T- and -, w1 a1 cos 1, def, w2 a1 sn 1, def, w3 a2 cos 2, def, w4 a2 sn 2, def, wth respectvely a and, def ( = 1, 2) the length of the th rgd lnk and the angle between the lnk and the horzontal lne n large dsplacements. The equaton system (10) can be solved n order to gve the seven unknowns V T, V, V, T,, and k. Then, as done above for rgd abutments, the normal force and the moment actng at each ont are determned n order to evaluate the eccentrcty and the center of pressure. Last, the satsfacton of Equaton (7) s verfed, namely t s checked that the lne of thrust les everywhere nsde the boundary of the arch. At several ncreasng values of the dsplacement δ, the equlbrum condton for the three-hnged arch and the shape of the lne of thrust are determned, untl t s no longer possble to assure the respect of Equaton (7) because another hnge forms and leads to collapse. 3.3 Results for crcular arches The flexblty of abutments affects the collapse multpler. In order to underlne ths effect, each curve of Fgure 4 s redrawn by consderng three dfferent values of flexblty. In Fgure 12 one can observe, by settng c A = c B = c, a constant dmensonless thckness s/l and a constant angle of embrace, the decreasng of k wth the ncreasng of flexbltes. As expected, ths means that the arch wth spreadng supports s more vulnerable than the arch wth rgd ones. Fgure 13 depcts graphcally the mnmum thckness for crcular masonry arches n the presence of an assgned sesmc multpler and for assgned values of flexblty at abutments. The relatonshp s expressed, as done n Fgure 5 for rgd abutments, between the half angle of embrace /2 and the dmensonless thckness s/r. 10
Lmt analyss n large dsplacements of masonry arches subected to vertcal and horzontal loads Fgure 11: nematc descrpton n large dsplacements (a) and forces actng on the two rgd lnks (b) - three hnged arch. Fgure 12: Load multplers for crcular arches as functon of geometry and flexblty of abutments. Fgure 13: Mnmum thckness for crcular masonry arches wth vertcal and horzontal loads, at several values of flexblty - case k =0.2 (a), case k =0.3 (b). 11
N. Cavalagl, V. Gusella and L. Severn 4 CONCLUSIONS The method of lmt analyss has been appled n order to study the collapse condton of crcular and ponted masonry arches, n the presence of vertcal and horzontal loads. The analyss has been carred out by referrng to eyman s hypothess about the masonry and consderng the nfluence of the flexbltes of the abutments. As extenson of eyman s results n the state of rgd abutments, the mnmum thckness has been determned as functon of the horzontal load multpler and geometrc features: t results an ncreasng mnmum thckness wth the ncrease of the horzontal load multpler. The obtaned curves could be a useful tool to defne the thckness that the crcular or the ponted arch should have to wthstand an earthquake of assgned ntensty. Then, the mnmum thckness analyss n the presence of both vertcal and horzontal loads has been extended, for crcular arches, to the condton of elastc abutments. The results have been obtaned by mposng the equlbrum n large dsplacements. As could be expected, the analyss has shown that t s needed an ncreasng mnmum thckness wth the ncrease of the flexblty at the abutments and the obtaned curves descrbe ths relatonshp. REFERENCES [1] J. eyman, The safety of masonry arches. Internatonal Journal of Mechancal Scences, 11, 363-385, 1969. [2] J. eyman, The masonry arch. Ells orwood Ltd., Chnchester, 1982. [3] A. Romano, J. Ochsendorf, The Mechancs of Gothc Masonry Arches. Internatonal Journal of Archtectural ertage, 4(1), 59-82, 2010. [4] C. Francos, Lmt behavor of masonry arches n the presence of fnte dsplacements. Internatonal Journal of Mechancal Scences, 28, 463-471, 1986. [5] J. Ochsendorf, The Masonry Arch on Spreadng Supports. The Structural Engneer, 84(2), 29-36, 2006. [6] L. Severn, Anals lmte d arch n muratura n grand spostament, Master Thess, Unversty of Peruga, Italy, 2013 (n talan). 12