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

1281

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,



( )

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

1283

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"



( )( )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

1285

( )

( )

( )

( )

( ) ( ) ⎥⎦⎤

⎢⎣⎡ −−−

−=→

−=⎥⎦⎤

⎢⎣⎡ −−−→

=++−−−→

=++−−−→

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−+−

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


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.

analysis of masonry vaulted systems: the barrel vaults · 2006-10-18 · by considering a barrel...

Documents