4. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству 22. април Суботица, СРБИЈА

Transcription

1 Савремена достигнућа у грађевинарству. април 016. Суботица, СРБИЈА A NOTE ON MILANKOVITCH'S THEORY OF THRUST LINE APPLIED TO GOTHIC MASONRY ARCHES Dimitrije Nikolić 1 UDK: DOI: /konferencijaGFS Summary: Over the last few years, scholars have revisited the classical issue of identifyg the limit equilibrium states of the symmetrical masonry arches of different shapes and correspondg theoretical mimum thickness, when subjected to self-weight. More than century ago, Serbian scholar Milut Milankovitch his remarkable work set the complete and correct theory of thrust le for the equilibrium of the arch of general shape, and computed the mimum thickness of semicircular arch. Although poted arches, beside circular and elliptical, are very common historic structures, particularly Gothic architecture, the lack of formation about their structural behaviour accordg to thrust le analysis is noticeable. In this paper, the characteristic elements of analysis, as well as computation, such as eccentricity, beg the measure of potedness, or the position of the application pots of relevant forces, which diverse the poted from semicircular arches, have been noticed. Furthermore, radial stereotomy, which assumes that jots between voussoirs are concurrent to the centre of the arch, considerg both complete and overcomplete arches, is employed. Accordgly, the analytical pressions of the arch rg area and its centroid are provided, and, after equilibrium i.e. static approach, pression for the thrust le has been derived. Hence, it represents the basis for the various computational analyses of the mechanical behaviour of Gothic masonry arches, as well as other types of circular based arches and vaults contag the poted crown. Keywords: poted arches, equilibrium analysis, Milut Milankovitch, thrust le theory 1. INTRODUCTION More than century ago, Serbian scholar Milut Milankovitch his remarkable, but for a long period nearly unknown work [1] (see [] as well) set the complete and correct theory for the equilibrium of the masonry arch of general shape. He upgraded and ceeded precedg researchers, maly architects, engeers and mathematicians, who, ce the 18 th century, order to predict and prevent possible collapse of vaulted masonry structures, have developed various models applied the stability and safety analysis (for more formation see [], [4]). Milankovitch considered the equilibrium of an arch on the assumptions about the material behaviour made from Couplet [5] onward: masonry has no tensile strength, it has fite compressive strength and slidg cannot 1 Dimitrije Nikolić, M.Arch., University of Novi Sad, Faculty of Technical Sciences, Department of Architecture, Trg Dositeja Obradovića 6, Novi Sad, Serbia, tel: , dima@uns.ac.rs ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (016) 191

2 4 th INTERNATIONAL CONFERENCE Contemporary achievements civil engeerg. April 016. Subotica, SERBIA occur []. Furthermore, he followed the thrust le concept troduced the 19 th century (e.g. [6]), where thrust le (Milankovitch: Druckkurven) represents the load path, beg the locus of the application pots of the resultant thrust forces developed at the jots (beds) between the voussoirs of the arch. Milankovitch troduced the computation the true location of the centre of the gravity of each ideal, generic voussoir, which was until then assumed to be located along the centrele of the arch. Hence, after remarkable mathematical elaboration, he was the first who gave the correct solution for the mimum thickness of semicircular arch of constant thickness subject to its own weight. In addition, he considered the different types of stereotomy, i.e. the direction of generic sections (jots), showg the multiplicity of limit thrust les (see [7]). Recently, few scholars have revisited the classical issue of identifyg the limit equilibrium states of the symmetrical masonry arches of different shapes and correspondg theoretical mimum thickness, when subjected to self-weight (e.g. [8], [9]). Although poted masonry arches, beside circular and elliptical, are very common historic structures, particularly Gothic architecture, their structural behaviour accordg to thrust le theory has not been researched sufficient detail. However, [10] the geometric framework is set, and the aim of the present paper is to provide a more detailed consideration on the le of thrust and its analytical pression. Hence, employg radial stereotomy (directions of the jots between voussoirs are concurrent to the centre of the arch, as used Italian poted arches), the characteristic elements of analysis, as well as computation, such as eccentricity, beg the measure of potedness, or the position of the application pots of relevant forces, which diverse the poted from semicircular arches, have been noticed, and are presented the followg section.. PARTICULAR GEOMETRIC PARAMETERS OF POINTED ARCH Due to the symmetry of the arch, the followg discussion only half-arch is considered. In Fig. 1 (a) relevant geometrical parameters are shown: R and t denote the mean radius and the thickness of the arch rg, respectively. The mimum value of thickness to radius ratio, t/r, represents the mimum possible thickness of the arch normalized by the radius. Further, the value e, which measures the deviation from the circular shape, is the horizontal distance between the circular axis centre O and the centre C of the poted arch. The angle α represents the angle of embrace, which is the complement of the sprgg angle, and arches can be complete (segmental) or overcomplete (horseshoe), if this angle is less or greater than 90, respectively. The substantial parameter of poted arch is its eccentricity, beg the measure of potedness, and followg [11] represents the ratio between e and the difference between R and e. Thus, arches can be slightly poted (drop, depressed or obtuse arch), or strongly poted (also known as lancet, acute or narrow poted arch). The angle φ is angular coordate measured from the vertical axis of the symmetry of the arch, which defes the generic section. Thrust le cannot be poted, the absence of concentrated loads [11], and therefore cannot pass through the trados at the crown, as opposed to the circular arch. Hence, the parameters of particular importance are the application pots B and S of the horizontal thrust H actg at the crown jot, and the reaction force R actg at the sprggs, ce, among the eccentricity by default, their position affects 19 CONFERENCE PROCEEDINGS INTERNATIONAL CONFERENCE (016)

3 Савремена достигнућа у грађевинарству. април 016. Суботица, СРБИЈА the position of thrust le through the arch [10] and therewith the value of horizontal thrust, changg the location of critical sections as well. Figure 1. (a) Geometric parameters of poted arch, (b) free-body diagram of the top portion of the arch with correspondg thrust forces actg on it, (c) force polygon [1] The rigid arch is determate to the third degree such that for any arch there is a family of possible equilibrium solutions, which can be visualized with les of thrust obtaed through graphical statics methods [11]. Accordgly, the force polygon presses graphically the equilibrium of the system; the les of action of the resultant thrust forces generate the funicular polygon, and the les of action of the weights of the voussoirs meet at the corners of the funicular polygon to satisfy moment equilibrium (see [7]).. DERIVATION OF THE EXPRESSION FOR THE THRUST LINE THROUGH GOTHIC ARCHES As poted out by Milankovitch [1], the plicit equation of the le of thrust could be directly obtaed without deducg the differential equation and then tegratg it, when it is possible to fd the analytical pression of the resultant load and its pot of application for a fite portion of the system []. Thus, along with the usual assumption that specific weight of the masonry and the depth of the arch have unit value, for the sake of simplicity, the problem of the stability of arch can be reduced to purely ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (016) 19

4 4 th INTERNATIONAL CONFERENCE Contemporary achievements civil engeerg. April 016. Subotica, SERBIA geometrical task. Namely, a self-weight of the arch or its portion is substituted by the area of the arch rg, limited by trados and trados curves as well as by the particular jots between voussoirs, and is applied the centre of gravity i.e. the centroid of the limited area. Accordgly, knowg the weight W beg the area of the half-arch as well as the centroid of the area, and assumg the application pots B and S of the forces H and F, respectively (see Fig. 1 (a) and (c)), from rotational equilibrium about the sprgg hge S, one can determe the value H of horizontal thrust, given by the followg pression: ( ) W H W x (1) cos( ) 0 whereas the weight W of the half-arch, after [1], is: 1 1 R e W ( ) e e R 1 e 1 R cos( ) e e Rt () R and the abscissa x W of its centre of gravity is given by: R 1 e R e 1 xw ( ) e R 1 1 e e e e cos( ) R R cos( ) er cos( ) R e 1 e e e e e Rt R R 1 cos( ) () Furthermore, two boundary conditions defg the positions of application pots B and S, i.e. the distance of the thrust le from the centre of the arch at the crown and sprggs, ρ 0 and ρ α respectively, are as follows: and R t / e d qc ( 0) (4) 0 R t / qs d ( ) e (5) whereas d=e cot(α); q c is the radial distance between the pot of application of the horizontal thrust and the crown at trados, and q s is the radial distance between the pot of application of the reaction force at the sprggs and the trados, as shown Fig. 1 (a). Further, the resultant thrust T at generic section at the angle φ together with its pot of application A is uniquely determed from the force and moment equilibrium, either graphically with the force polygon (see Fig. 1 (b) and (c)) or analytically by solvg equilibrium equations. Thus, from rotational equilibrium about pot A follows: 194 CONFERENCE PROCEEDINGS INTERNATIONAL CONFERENCE (016)

5 Савремена достигнућа у грађевинарству. април 016. Суботица, СРБИЈА ( ) x H0 cos( ) V V (6) whereas, accordance with Fig. 1 and after [1], the weight V(φ) of the fite portion of the arch up to generic section at the angle φ, represented by the area of correspondg arch rg, and its centre of mass x V(φ) i.e. the centroid of that area, are given by Eq. (7) and Eq. (8), respectively: 1 V ( ) d yr e d yr d yr e d yr e e e e 1 d y 1 e r 1 d y 1 e r cot( ) xr xr R (7) 1 d y 1 e r x V ( ) e e e d yr e d yr e 1 d y 1 e r 1 / R e e e d yr e cot( ) xr xr d yr e d yr d yr e d yr e e e e 1 d y 1 e r 1 d y 1 e r cot( ) xr xr R (8) In Eqs. (7) and (8), R =R+t/ and R =R-t/ are radii of trados and trados circle, respectively, and the quantities x r, x r, y r and y r are the abscissas and ordates of the trados and trados of the arch, as one can see Fig. 1 (a). Fally, from Eq. (6) one can solve distance ρ between the thrust le and the centre of the arch, obtag the followg closed form pression for the thrust le of poted arches (pressed Cartesian coordates and traced with dashed le Fig. 1 (a) and (b)): H0 VxV ( ) (9) H cos( ) V ( ) whereas the H, ρ 0, V and x V are given by Eqs. (1), (4), (7) and (8), respectively (substitutg these pressions to Eq. (9) one can obta panded pression for the le of thrust, not shown here because of the length of the solution). Hence, for each generic section, the distance of the thrust le, i.e. of the application pot of the resultant thrust force, from the trados and trados, can be computed, and the critical sections regardg the jots where the thrust le approaches closest to the arch boundary can be detected. 4. FINAL REMARKS AND CONCLUSIONS Serbian scholar Milut Milankovitch was the first who provided the complete theory of thrust le, with the correct mathematical elaboration concerng the true location of the centres of gravity of generic voussoir, for the equilibrium analysis of the arch of general ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (016) 195

6 4 th INTERNATIONAL CONFERENCE Contemporary achievements civil engeerg. April 016. Subotica, SERBIA shape. Recently, this theory has been revisited the analysis regardg elliptical arches. However, the application to poted arches has not been considered. Therefore, this paper, the characteristic elements of analysis, as well as computation, such as eccentricity, beg the measure of potedness or the position of the application pots of relevant forces, which diverse the poted from semicircular arches, have been noticed. Hence, after equilibrium i.e. static approach and employg radial stereotomy, concerng both complete and overcomplete poted arches, the analytical pression for the thrust le has been derived. Therefore, it represents the basis for the various computational analyses of the mechanical behaviour of Gothic masonry arches, as well as other types of circular based arches or vaults contag the poted part. Inspectg the different values of the eccentricity and embrace angle, critical sections with appropriate collapse modes can be identified, enablg the detection of the mimum thickness and correspondg thrust values. In addition, different types of ercised stereotomy, such as vertical or normal, rema to be considered. Moreover, regardg the further possible developments of the present research, appropriate iterative procedures and computer codes can be developed order to enable the tensive elaboration of the problem as well as to provide the complete sight to the matter. ACKNOWLEDGEMENTS The paper was done with the Project No. TR604 supported by the Mistry of Education, Science and Technological Development of the Republic of Serbia. REFERENCES [1] Milankovitch, M.: Theorie der Druckkurven. Zeitschrift für Mathematik und Physik, 1907, 55, pp [] Foce, F.: Milankovitch s Theorie der Druckkurven: Good mechanics for masonry architecture. Nus Netw. J., 007, 9, pp [] Heyman, J.: Coulomb s memoir on statics: An essay the history of civil engeerg. Cambridge university press, Cambridge, 197. [4] Benvenuto, E.: An troduction to the history of structural mechanics. Sprger- Verlag, New York, [5] Couplet, P.: De la poussée des voûtes. Histoire de l Académie Royale des Sciences, Paris, 179, 170, pp and pp [6] Moseley, H.: The Mechanical Prciples of Engeerg and Architecture. London, 184. [7] Makris, N., Alakis, H.: The effect of stereotomy on the shape of the thrust-le and the mimum thickness of semicircular masonry arches. Arch. Appl. Mech., 01, 8, pp [8] Alakis, H., Makris, N.: Mimum thickness of elliptical masonry arches. Acta Mech., 01, 4, pp CONFERENCE PROCEEDINGS INTERNATIONAL CONFERENCE (016)

7 Савремена достигнућа у грађевинарству. април 016. Суботица, СРБИЈА [9] Cocchetti G., Colasante G., Rizzi E.: On the Analysis of Mimum Thickness Circular Masonry Arches. Appl. Mech. Rev., 011, 64. [10] Nikolić, D.: Geometric Framework for the Equilibrium Analysis of Poted Arches Accordg to Milankovitch s Theory of Thrust Le. 4th ecaade Regional International Conference: Between models and performative capacities, Novi Sad, Serbia, 016. (accepted manuscript) [11] Romano, A., Ochsendorf, J. A.: The mechanics of gothic masonry arches. Int. J. Archit. Herit., 010, 4, pp [1] Nikolić, D.: Determation of the Centroid of Poted Arches Accordg to Radial Stereotomy. 4th International Conference on Geometry and Graphics mongeometrija 016, Belgrade, Serbia, 016. (accepted manuscript) БЕЛЕШКА О МИЛАНКОВИЋЕВОЈ ТЕОРИЈИ ПОТПОРНЕ ЛИНИЈЕ ПРИМЕЊЕНОЈ НА ГОТСКЕ МАСИВНЕ ЛУКОВЕ Резиме: Током последњих неколико година, научници поново обрађују класична питања одређивања стања граничне равнотеже симетричних масивних лукова различитих облика, те њихове одговарајуће минималне дебљине под сопственом тежином. Пре више од једног века, српски научник Милутин Миланковић је у свом изванредном раду поставио свеобухватну и исправну теорију потпорне линије намењену анализи равнотеже лука општег облика, те израчунао минималну дебљину полукружног лука. Иако су преломљени лукови, поред кружних и елиптичких, веома заступљени у историјским структурама, посебно у готској архитектури, приметан је недостатак информација о њиховом понашању према теорији потпорне линије. У овом раду су уочени карактеристични елементи анализе и прорачуна који разликују преломљене од полукружних лукова, као што су ексентрицитет као мера преломљености лука, или положај нападних тачака релевантних сила. Осим тога, примењена је радијална стереотомија, којом се претпоставља да су лежишне спојнице између сводних каменова конкурентне с центром лука, узимајући у обзир и сегментне и потковичасте лукове. На основу тога су дати аналитички изрази за површину и тежиште прстена лука, те је према статичкој анализи изведен израз за потпорну линију. Тиме је постављена основа за различите рачунарске анализе механичког понашања готских масивних лукова, као и других кружних лукова и сводова који садрже преломљено теме. Кључне речи: преломљени лукови, статичка анализа, Милутин Миланковић, теорија потпорне линије ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (016) 197