Analysis of Masonry Vaulted Systems: The Barrel Vaults · 2006-10-18 · By considering a barrel...
Transcript of Analysis of Masonry Vaulted Systems: The Barrel Vaults · 2006-10-18 · By considering a barrel...
Structural Analysis of Historical Constructions, New Delhi 2006P.B. Lourenço, P. Roca, C. Modena, S. Agrawal (Eds.)
1 THEORETICAL SET UP OF THE PROBLEM 1.1 General approach The surface of a shell of general form may be defined by the equation z = f(x, y). A typical element ABCD of the mid-surface of such a shell, defined by two meridians and two parallels, and its projection on the xy plane A'B'C'D' are shown in Fig. 1. The inc1ination of the element to the horizontal is measured by the angles ϕ and θ. The components of applied load per unit area in the xy plane are denoted by zyx ppp ,, . The shell element is considered to be in a membrane state of stress under the forces yxxyyx NNNN =,, having projections in the xy pIane
yxxyyx NNNN =,, , respectively. Referring to Fig. 1, one can write:
θϕ==
θ=
ϕ=
coscos,
cos,
cosdydxdsdsdAdydsdxds yxyx (1)
zyxippdydxpdydxpdApdAp
NNNdyNdyNdyNdsN
NNdxNdxNdxNdsN
NNdyNdyNdyNdsN
iiiizii
xyyxxyxyxyxyyxy
yyyyyxy
xxxxxyx
,,,coscoscoscos
coscoscos
coscos
coscoscos
coscos
coscoscos
=θϕ=→=θϕ
→=
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
==→=θ
θ→=θ
θϕ=→=
ϕθ→=θ
ϕθ=→=
θϕ→=ϕ
(2)
Analysis of Masonry Vaulted Systems: The Barrel Vaults
A.Baratta and O.Corbi University of Naples Federico II, Department of Structural Engineering, Napoli, Italia
ABSTRACT: In the paper one presents a general theoretical treatment of the problem of analysis of masonry vaults. The shell representing the mid-surface of the vault is referred to in order to impose 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 simplifications are introduced when referring to the case of indefinite barrel vaults loaded by forces acting in the plane of their cross-sections, which are demonstrated (and confirmed) to behave according to the model of a series of independent arches.
where .tan,tanyz
xz
∂∂=θ
∂∂=ϕ
Figure 1 : A shell of general shape z = f(x, y) and its projection on the xy plane. The sides of BC and AD the element are given by intersection with the plane xz, whilst the sides AB and CD result from
intersection with the plane yz. For the equilibrium of the element in the x and y directions, therefore one has
0dydxpdxdyy
NNdxN
dydxx
NNdyN
GGx
HH
xyxy
MMyx
LL
xx
EEx
=θϕ
+⎥⎥⎦
⎤
⎢⎢⎣
⎡
ϕϕ⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂
++ϕ
ϕ−+
+⎥⎦
⎤⎢⎣
⎡θ
ϕ⎟⎠
⎞⎜⎝
⎛∂
∂++
θϕ−
coscoscoscos
coscos
coscos
coscos
(3)
0dydxpdxdyy
NNdxN
dydxx
NN
dyN
GGy
HH
yy
MMy
LL
yxxy
EEyx
=θϕ
+⎥⎥⎦
⎤
⎢⎢⎣
⎡
ϕθ⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂
++ϕ
θ−+
+⎥⎥⎦
⎤
⎢⎢⎣
⎡
θθ⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂
++θ
θ−
coscoscoscos
coscos
coscos
coscos
(4)
where (⋅)E, (⋅)H, (⋅)L, (⋅)G and (⋅)M denote quantities evaluated at the points E,H,L,M,G, respectively, of the element ABCD. Additionally, for equilibrium along the z-direction one has
1280 Structural Analysis of Historical Constructions
A.Baratta and O.Corbi
0dydxpdxdyy
NNdxN
dydxx
NNdyN
dxdyy
NNdxN
dydxx
NNdyN
GGz
HH
yy
MMy
LL
xyxy
EEyx
HH
xyxy
MMyx
LL
xx
EEx
=θϕ
+⎥⎥⎦
⎤
ϕθ⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂
++⎢⎣
⎡ϕ
θ−+
+⎥⎥⎦
⎤
θθ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂++⎢
⎣
⎡θ
θ−+
+⎥⎥⎦
⎤
ϕϕ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂++⎢
⎣
⎡ϕ
ϕ−+
+⎥⎦
⎤⎢⎣
⎡θ
ϕ⎟⎠
⎞⎜⎝
⎛∂
∂++
θϕ−
coscoscossin
cossin
cossin
cossin
cossin
cossin
cossin
cossin
(5)
1.2 The case of barrel vaults
As regards to barrel vaults, one can specialize the general equilibrium problem as follows. First of all one should consider that, since the vault geometrically derives by the translation
along a directrix of a generating arch curve, in this case, the meridian lines coincide with the generatrix in their shapes; if one considers a rectilinear directrix, the vault parallels are horizontal and rectilinear as well.
By considering a barrel vault characterized by a horizontal directrix parallel to the y axis (θ = 0), the meridian curves of the shell are contained in planes parallel to the xz plane (Fig. 2).
Figure 2 : Shell representing the mid-surface of a barrel vault with horizontal directrix. The surface of the shell representing the mid-surface of the vault may be defined by the
equation z = f(x). Because of the vault geometry, one has that
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dydxdydxdsdsdAdydydsdxds
0yz
xz0
yxyx ϕ=
θϕ===
θ=
ϕ=
=∂∂=θ
∂∂=ϕ=θ
coscoscos,
cos,
cos
tan,tan, (6)
Moreover in absence of horizontal loads and if the vertical load is not dependent on "y", as it
happens when the vault is subject to only vertical loads due to the self-weight (i.e. ( ) 0xpp zz ≥= ), and, additionally, assuming that the vault has an indefinite length in the
direction y, one has
( )
ϕ=
ϕθ=
≥=ϕ=θϕ=
===→===
==→==
coscoscos
,coscoscos
,,
xxx
zzzzz
yxxyyyxxyy
yxyx
NNN
0xppppp
0NNN0NNN
0p0p0p0p
(7)
Therefore, for the equilibrium of the element in the x direction one has
0dydxx
NNdyN
LL
xx
EEx =⎥
⎦
⎤⎢⎣
⎡θ
ϕ⎟⎠
⎞⎜⎝
⎛∂
∂++
θϕ−
coscos
coscos (8)
where (⋅)E, (⋅)H, (⋅)L, (⋅)G and (⋅)M denote quantities evaluated at the points E,H,L,M,G, respectively, of the element ABCD.
After some algebraic development, Eq. (8) turns into
0x
N0dydx
dx
Ndxx
NN
0dydxx
NNN
xE
Ex
L
Lxx
L
Lxx
E
Ex
=∂
∂→=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡θϕ
−θϕ
⎟⎠
⎞⎜⎝
⎛∂
∂+
→
=⎥⎦
⎤⎢⎣
⎡θϕ
⎟⎠
⎞⎜⎝
⎛∂
∂++
θϕ
−
coscos
coscos
coscos
coscos
(9)
whilst the equilibrium of the element in the y direction is immediately satisfied since it reduces to an identity. Additionally, for equilibrium along the z-direction one has
0dydxpdydxx
NNdyNGG
zL
Lx
xE
Ex =θϕ
+⎥⎦
⎤⎢⎣
⎡θ
ϕ⎟⎠⎞
⎜⎝⎛
∂∂++
θϕ−
coscoscossin
cossin (10)
which, after some algebraic development,
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A.Baratta and O.Corbi
( )
0pxzN
x
0pNx
011pNx
011pNx
0dydxpdx
Ndxx
NN
zx
zx
GGzx
GGzx
GGz
E
Ex
L
Lxx
=+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂→
=+ϕ∂∂→
=θϕ
+⎟⎠⎞
⎜⎝⎛ ϕ
θϕ
∂∂→
=θϕ
+⎟⎠⎞
⎜⎝⎛
θϕ
∂∂→
=θϕ
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡θϕ
−θϕ
⎟⎠
⎞⎜⎝
⎛∂
∂+
tan
coscostan
coscos
coscoscossin
coscoscossin
cossin
(11)
Therefore, equilibrium conditions are reduced to the two equations relevant to the x and z
directions, i.e.
⎪⎪⎩
⎪⎪⎨
⎧
=+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=∂
∂
0pxzN
x
0x
N
zx
x
(12)
Since from the equilibrium in the x direction one has
0xz
xN x =
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂ (13)
after performing the differentiation of the second of Eq. (12) one has
z2
2
xzx
2
2
x px
zN0pxz
xN
xzN −=
∂∂→=+
∂∂
∂∂
+∂∂ (14)
Hence, provided that a solution of the equilibrium in the form of Eq. (14) may be obtained for the projected forces, the actual forces may then be readily determined from the relationships Eq. (7).
After introducing a stress function Ψ(y) such to immediately satisfy the equilibrium condition in the z direction, and such that
( )2
2
x yyN
∂ψ∂= (15)
Then the third equilibrium equation for the barrel vault with horizontal axis parallel to x and
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indefinite length turns into
( )z2
2
2
2
px
zy
y −=∂∂
∂ψ∂ (16)
The solution of the problem is thus reduced to the determination of stress function Ψ. Assuming that the directrix curve of the vault is a circular arch (Fig. 3) of radius R, with
constant thickness "s" and unit weight γ, one has
( )( ) 22zz
t1
1sRx1
1sxpp−
γ=−
γ== (17)
with t = x/R.
x
y
z
Figure 3 : Barrel vault with horizontal directrix and circular arch generatrix. Since the vault is undefined in the direction <y> and the load is not dependent on "y", one
puts
( ) ( )xzzcby2
yHy2
=++=ψ , (18)
So, Eq. (16) turns into
( ) ( )( )2z2
2
2
2
2
2
Rx1
1sxpx
zHx
zy
y
−γ−=−=
∂∂=
∂∂
∂ψ∂ (19)
Considering that x =Rt, and that consequently
dtd
R1
dxdt
dtd
dxd == (20)
the previous equation can be written in function of "t" rather than of "x"
1284 Structural Analysis of Historical Constructions
A.Baratta and O.Corbi
( )( )2
2
z
2
2
2
Rx1
1H
RstRpH
Rdt
zd
−γ−=−= (21)
whose solution is
( ) ( ) ⎥⎦⎤
⎢⎣⎡ +−+γ−= Ct1t
HRstz 2
2arcsin (22)
with C and H constants to be determined by the following boundary conditions (Fig. 4)
( ) ( )
( ) ( ) 12111
2
11
11
o
2
zCt1ttHRstz
Rxtxx
zC1HRs0z0t0x
=⎥⎦⎤
⎢⎣⎡ +−+γ−=⇒=⇒=
=+γ−=⇒=⇒=
arcsin (23)
zo and z1 are arbitrary ordinates, conditioned by the fact that z(t) should be contained in the
interior of the profile of the vault.
x
z
x
z
CR
L
s
z1
zo
x1
z(t)
Figure 4 : Cross section of a barrel vault with circular arch generatrix. The first of Eq. (23) yields C
⎟⎟⎠
⎞⎜⎜⎝
⎛γ
+−=s
zRH1C o
2 (24)
while the second of Eq. (23) yields H
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( )
( )
( )
( )
( ) ( ) ⎥⎦⎤
⎢⎣⎡ −−−
−=→
−=⎥⎦⎤
⎢⎣⎡ −−−→
=++−−−→
=++−−−→
=⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+−−+−
2111
1
2
12111
2
1
221
2
11
2
12
2221
2
11
2
122111
2
1arcsin1
1arcsin1
1arcsin
1arcsin
11arcsin
tttzz
RsH
zztttHRs
zzHRst
HRstt
HRs
zs
zRH
HRs
HRst
HRstt
HRs
zs
zRHttt
HRs
o
o
o
o
o
γ
γ
γγγ
γγγγγ
γγ
(25)
After this result, it is possible to calculate the internal forces 0NN0N xyyx ==≤ , and
0NN0N xyyx ==≤ ,
ϕ=
ϕ=
coscosHNN x
x (26)
It is also possible to realize that the equilibrium solution allows the structure to behave as a
sequence of identical independent arches. In the case when the barrel vault is not indefinite in its length but two tympans are present at
its extremities, one can still consider the vault without any shear deformability. On the other side the tympans can be assumed to be rigid in their planes, exhibiting no resistance out of such planes, thus requiring that the normal membrane component Ny is zero at the vault extremities. As a consequence of the assumed vault undeformability with regards to membrane tangential stress, the solidarity with tympans requires that the end arch-strips adherent to the tympans satisfy the constraint condition εx=0 over the whole contact surface (the transverse profile of the vault), whence also Nx=0 at the extremities.
As regards to the tangential membrane stress Nxy, in this special case, it attains its maximum values on the tympans, where it cannot be neglected, thus requiring that the stress function Ψ is chosen in such a manner to be able to reproduce even this stress term.
CONCLUSIONS
In the paper a first approach for the treatment of masonry vault analysis is outlined with specific reference to barrel vaults. Actually the geometry of barrel vault allows to simplify the general problem in the form as it is set up in the paper with regards to the element identified by the cut along two adjacent meridians and parallel in the mid-surface shell of a vault of general shape, which is pretty complex. The general theory, referring to a membrane state of stress, couples the spatial problem to a plane problem, which represents the counterpart of the real problem projected on a plane in all its geometrical and stress features. The general approach, when applied to barrel vaults loaded by forces uniquely acting in the plane of their cross-sections (which is the usual case), confirms the basic behavior of this vault typology, which acts as a series of independent arches.
ACKNOWLEDGEMENTS
This paper is supported by grants of Italian Ministry of University Research (MIUR) within a
1286 Structural Analysis of Historical Constructions
A. Baratta and O. Corbi 1287
financed PRIN project.
REFERENCES
Baratta, A., 1984. Il materiale non reagente a trazione come modello per il calcolo delle tensioni nelle pareti murarie. J. of Restauro, 75/76, p. 53-77, Italy.
Baratta, A., 1991. Statics and reliability of masonry structures, in Reliability Problems: General Principles and Applications in Mechanics of Solids and Structures, F.Casciati & J.B.Roberts Eds, CISM, Udine, Italy.
Baratta, A. and Corbi, O., 2005. On variational approaches in NRT continua, J. of Solids and Structures, 42, p. 5307-5321.
Baratta, A. and Voiello, G., 1988. Teoria delle pareti in muratura a blocchi: un modello discretizzato di calcolo, in “Franco Jossa e la sua opera“. Ed. Giannini, Napoli, Italy.
Heyman J., 1977. Equilibrium of shell structures, Oxford University Press. Heyman, J.,1966. The stone skeleton. J. of Solids and Structures. 2, p. 269-279. Ugural, A.C., 1999. Stresses in plates and shells, McGraw-Hill.
1288 Structural Analysis of Historical Constructions