Analysis of Masonry Vaulted Systems: The Barrel Vaults

Transcription

1 Structural Analsis of istorical Constructions, ew Delhi 6 P.B. ourenço, P. oca, C. odena, S. Agrawal (ds.) Analsis of asonr Vaulted Sstems: The Barrel Vaults A.Baratta and O.Corbi Universit of ales Federico II, Deartment of Structural ngineering, aoli, Italia ABSTACT: In the aer one resents a general theoretical treatment of the roblem of analsis of masonr vaults. The shell reresenting the mid-surface of the vault is referred to in order to imose equilibrium conditions in case of a shell element of general form; due to the behaviour of vaults, membrane admissible stress fields are considered for inferring equilibrium and subsequent geometrical simlifications are introduced when referring to the case of indefinite barrel vaults loaded b forces acting in the lane of their cross-sections, which are demonstrated (and confirmed) to behave according to the model of a series of indeendent arches. TOTICA ST UP OF T POB. eneral aroach The surface of a shell of general form ma be defined b the equation f(, ). A tical element ABCD of the mid-surface of such a shell, defined b two meridians and two arallels, and its rojection on the lane A'B'C'D' are shown in Fig.. The incination of the element to the horiontal is measured b the angles and. The comonents of alied load er unit area in the lane are denoted b,,. The shell element is considered to be in a membrane state of stress under the forces,, having rojections in the Iane,,, resectivel. eferring to Fig., one can write: d d d d ds, ds, da ds ds () ds ds ds da da i i d d i d d d d d d d d i d d i i, i,, d ()

2 8 Structural Analsis of istorical Constructions where tan, tan. Figure : A shell of general shae f(, ) and its rojection on the lane. The sides of BC and AD the element are given b intersection with the lane, whilst the sides AB and CD result from intersection with the lane. For the equilibrium of the element in the and directions, therefore one has d d d d d d d d (3) d d d d d d d d (4) where ( ), ( ), ( ), ( ) and ( ) denote quantities evaluated at the oints,,,,, resectivel, of the element ABCD. Additionall, for equilibrium along the -direction one has

3 A.Baratta and O.Corbi 8 sin sin sin sin d d d d d sin d sin d sin d sin d d d d d d (5). The case of barrel vaults As regards to barrel vaults, one can secialie the general equilibrium roblem as follows. First of all one should consider that, since the vault geometricall derives b the translation along a directri of a generating arch curve, in this case, the meridian lines coincide with the generatri in their shaes; if one considers a rectilinear directri, the vault arallels are horiontal and rectilinear as well. B considering a barrel vault characteried b a horiontal directri arallel to the ais ( ), the meridian curves of the shell are contained in lanes arallel to the lane (Fig. ). Figure : Shell reresenting the mid-surface of a barrel vault with horiontal directri. The surface of the shell reresenting the mid-surface of the vault ma be defined b the equation f(). Because of the vault geometr, one has that

4 8 Structural Analsis of istorical Constructions, ds tan, tan d, ds d d, da ds ds d d d d (6) oreover in absence of horiontal loads and if the vertical load is not deendent on "", as it haens when the vault is subject to onl vertical loads due to the self-weight (i.e. ( ) ), and, additionall, assuming that the vault has an indefinite length in the direction, one has,,, ( ) (7) Therefore, for the equilibrium of the element in the direction one has d d d (8) where ( ), ( ), ( ), ( ) and ( ) denote quantities evaluated at the oints,,,,, resectivel, of the element ABCD. After some algebraic develoment, q. (8) turns into d d d d d d (9) whilst the equilibrium of the element in the direction is immediatel satisfied since it reduces to an identit. Additionall, for equilibrium along the -direction one has d d d d sin dsin () which, after some algebraic develoment,

5 A.Baratta and O.Corbi ( ) d d d d tan tan sin sin sin () Therefore, equilibrium conditions are reduced to the two equations relevant to the and directions, i.e. () Since from the equilibrium in the direction one has (3) after erforming the differentiation of the second of q. () one has (4) ence, rovided that a solution of the equilibrium in the form of q. (4) ma be obtained for the rojected forces, the actual forces ma then be readil determined from the relationshis q. (7). After introducing a stress function Ψ() such to immediatel satisf the equilibrium condition in the direction, and such that ( ) ψ (5) Then the third equilibrium equation for the barrel vault with horiontal ais arallel to and 83

6 84 Structural Analsis of istorical Constructions indefinite length turns into ψ ( ) (6) The solution of the roblem is thus reduced to the determination of stress function Ψ. Assuming that the directri curve of the vault is a circular arch (Fig. 3) of radius, with constant thickness "s" and unit weight γ, one has ( ) γ s γ s (7) ( ) t with t /. Figure 3 : Barrel vault with horiontal directri and circular arch generatri. Since the vault is undefined in the direction <> and the load is not deendent on "", one uts ψ ( ) b c, ( ) (8) So, q. (6) turns into ψ ( ) ( ) γ s ( ) (9) Considering that t, and that consequentl d d d dt d () dt d dt the revious equation can be written in function of "t" rather than of ""

7 A.Baratta and O.Corbi 85 d ( t) γ s () dt ( ) whose solution is arcsin t C () () t γ s () t with C and constants to be determined b the following boundar conditions (Fig. 4) t t ( ) γs ( C) o ( t) γs t arcsin( t) t C (3) o and are arbitrar ordinates, conditioned b the fact that (t) should be contained in the interior of the rofile of the vault. (t) s o C Figure 4 : Cross section of a barrel vault with circular arch generatri. The first of q. (3) ields C C o γ s (4) while the second of q. (3) ields

8 86 Structural Analsis of istorical Constructions γ s γ s γ s t γ s t γ s t arcsin ( t ) ( ) arcsin arcsin ( t ) ( t ) t arcsin o γ s γ s ( t ) t t t arcsin t t o γ s γ s γ s o ( t ) t γ s o o γ s (5) After this result, it is ossible to calculate the internal forces, and, (6) It is also ossible to realie that the equilibrium solution allows the structure to behave as a sequence of identical indeendent arches. In the case when the barrel vault is not indefinite in its length but two tmans are resent at its etremities, one can still consider the vault without an shear deformabilit. On the other side the tmans can be assumed to be rigid in their lanes, ehibiting no resistance out of such lanes, thus requiring that the normal membrane comonent is ero at the vault etremities. As a consequence of the assumed vault undeformabilit with regards to membrane tangential stress, the solidarit with tmans requires that the end arch-stris adherent to the tmans satisf the constraint condition ε over the whole contact surface (the transverse rofile of the vault), whence also at the etremities. As regards to the tangential membrane stress, in this secial case, it attains its maimum values on the tmans, where it cannot be neglected, thus requiring that the stress function Ψ is chosen in such a manner to be able to reroduce even this stress term. COCUSIOS In the aer a first aroach for the treatment of masonr vault analsis is outlined with secific reference to barrel vaults. Actuall the geometr of barrel vault allows to simlif the general roblem in the form as it is set u in the aer with regards to the element identified b the cut along two adjacent meridians and arallel in the mid-surface shell of a vault of general shae, which is rett comle. The general theor, referring to a membrane state of stress, coules the satial roblem to a lane roblem, which reresents the counterart of the real roblem rojected on a lane in all its geometrical and stress features. The general aroach, when alied to barrel vaults loaded b forces uniquel acting in the lane of their cross-sections (which is the usual case), confirms the basic behavior of this vault tolog, which acts as a series of indeendent arches. ACKOWDTS This aer is suorted b grants of Italian inistr of Universit esearch (IU) within a

9 A. Baratta and O. Corbi 87 financed PI roject. FCS Baratta, A., 984. Il materiale non reagente a traione come modello er il calcolo delle tensioni nelle areti murarie. J. of estauro, 75/76, , Ital. Baratta, A., 99. Statics and reliabilit of masonr structures, in eliabilit Problems: eneral Princiles and Alications in echanics of Solids and Structures, F.Casciati & J.B.oberts ds, CIS, Udine, Ital. Baratta, A. and Corbi, O., 5. On variational aroaches in T continua, J. of Solids and Structures, 4, Baratta, A. and Voiello,., 988. Teoria delle areti in muratura a blocchi: un modello discretiato di calcolo, in Franco Jossa e la sua oera. d. iannini, aoli, Ital. eman J., 977. quilibrium of shell structures, Oford Universit Press. eman, J.,966. The stone skeleton. J. of Solids and Structures., Ugural, A.C., 999. Stresses in lates and shells, craw-ill.

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