MASONRY ORTHOTROPIC VAULTS IN HISTORICAL … › arch04 › BerbieriCarloniTommaso.pdfBENDING THEORY OF ORTHOTROPIC SHELL 4 TH INTERNATIONAL CONFERENCE ON ARCH BRIDGES MASONRY ORTHOTROPIC

MASONRY ORTHOTROPIC VAULTS IN MASONRY ORTHOTROPIC VAULTS IN HISTORICAL CONSTRUCTION:HISTORICAL CONSTRUCTION:THE HERRINGTHE HERRING--BONE PATTERNBONE PATTERN

Barbieri A.*, Carloni C.°, Di Tommaso A.*

4th International Conferenceon Arch Bridges17-19 November 2004Barcelona, Spain

*University IUAV of VenicedCaDorsoduro 220630123 Venice (Italy)

*University of BolognaDISTARTViale Risorgimento 240136 Bologna (Italy)

MASONRY ORTHOTROPIC VAULTS IN HISTORICAL CONSTRUCTION: THE HERRING-BONE PATTERNBarbieri A., Carloni C., Di Tommaso A.

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PRESENTATION SUMMARYPRESENTATION SUMMARY

q INTRODUCTION

q AIM OF RESEARCH

q HERRING-BONE MASONRY TECHNIQUE

q CONCLUSIONS

q BENDING THEORY OF ORTHOTROPIC SHELL

q FE MODEL METHOD

INTRODUCTIONINTRODUCTION4T

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In the masonry vaults the brick patterns could be different depending on the geometry of structural element. The aim of design criteria was to

prevent possible kinematical modifications, because they could not made dimensional design on the base of stresses criteria (not yet formulated).

The mason sensibility led to define some masonry building patterns more suitable than others because the brick pattern could contribute to reduce the support elements cost and to guarantee an increment in arch stability.

Hinge

INTRODUCTIONINTRODUCTION4T

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Nevertheless, masonry vaults cover wide rooms in many historical buildings. These structural elements were built using a material which is considered as “no tesion” material.

These structures need strengthening design due to loading increment and material damage. Their stability is evaluated considering collapse mechanisms and few data are available about their elastic behaviour under dead and live loadings.

Their technology is know even less, especially if the masonry has particular pattern.

AIM OF RESEARCHAIM OF RESEARCH4T

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To develop an analytical model in order to investigate the To develop an analytical model in order to investigate the structural behaviour of masonry barrel vaultsstructural behaviour of masonry barrel vaults

built with herringbuilt with herring--bone patternbone pattern

•These structural elements are analysed in the field of shell theory and linear elastic behaviour of material

•The masonry barrel vaults are modelled as bi-dimensional orthotropic elements

•The necessity to evaluate the effect of herring-bone pattern on the stresses configuration requires the introduction of flexural components in the equilibrium conditions

•The angle α of masonry configuration in herring-bone pattern is assumed as parametric element for the analysis.

HERRING HERRING -- BONE MASONRY TECHNIQUEBONE MASONRY TECHNIQUE4T

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Herringbone stonework, common in small parish

churches (Vienne-en-Bessin).

Herringbone pattern for paving.

Herringbone pattern for stone wall. Particular of stone disposition. Dossomaggiore Castle, sec. XIV

HERRING HERRING -- BONE MASONRY TECHNIQUEBONE MASONRY TECHNIQUE4T

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Masonry barrel and crossed vaults of Tolentini monastery (Venice).

BENDING THEORYBENDING THEORYOF ORTHOTROPIC SHELLOF ORTHOTROPIC SHELL

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q vault thickness is very small compared to curvature radius and length of vault;q strains in vault are enough small so that the terms of secondand over order could be neglected;q normal stress along vault thickness is enough small compared to other stresses, so that this could be neglected;q the normal to un-deformed vault surface is still normal in deformed configuration of vault.


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, ,

2, , , ,

2, , ,

0

10

12 0

x x x x

x x x x

x x x xx r

RN N p R

N RN M M p RR

N M RM M p RR

φ φ

φ φ φ φ φ φ φ

φ φ φ φ φφ

+ + =

+ + − + =

+ + + + =

;

;

;

;

;

,x x xuε =, z

u u

R Rφ φ

φε = −,

,x

x x

uu

Rφ

φ φγ = +

,x z xxuχ =,

2

zu

Rφφ

φχ =,z x

x

u

Rφ

φχ =

Strain components referring toDonnell formulation:

Equilibrium equations:

ij ijlk lkHσ ε= ( i,j,l,k=1,2,3)

Hooke’s law:


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;

;

;

;

;

Equilibrium equations:

213 66 13 32 13 32

11 , , 12 66 , 12 ,33 33 33

( ) ( ) ( ) 0x xx x x z xH H H H H H

H R u u H H u H uH R H Hφφ φ φ

− ⋅ ⋅ + ⋅ + − + ⋅ − − ⋅ =

2 2223 31 23 23

21 66 , 66 , 22 , 22 ,333 33 33

22 223 31 2312

66 , 22 ,33 33

1( ) ( ) ( )

12

1( ) ( ) 0

6 2 2

x x xx z

z xx z

H H H HtH H u H R u H u H u

H R H R H

H H HHt RH u H u p

R H R H t

φ φ φ φφ φφφ

φ φ φ

− + ⋅ + ⋅ ⋅ + ⋅ − ⋅ + ⋅ − + ⋅ +⋅

+ ⋅ − + ⋅ + ⋅ − + ⋅ + =⋅ ⋅

2 22 223 31 23 13 13 3212

21 , 22 , 11 , 66 ,33 33 33 33

2 22 223 23

22 , 22333 33

1( ) ( ) ( ) ( )

12 3 2 2

1( ) ( ) 0

12

x x z xxxx z xx

z z r

H H H H H HHt R tH u H u H u H u

H R H H R H

H Ht RH u H u p

R H R H t

φ φ φφ

φφφφ

⋅− ⋅ + ⋅ − ⋅ + ⋅ − ⋅ + ⋅ + − ⋅ +⋅ ⋅

+ ⋅ − ⋅ + ⋅ − + ⋅ + =⋅

These equilibrium equations can not be solved in an analytical form, hence a numerical procedure has been proposed.

F.E. MODEL METHODF.E. MODEL METHOD4T

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2

1 1

2

y

x x

y 1

2 1

2

Clay brick: 250 mm long, 60 mm high and 120 mm thick; Eb=4000 MPaMortar joint: 8 mm thick; Em=400 MPa.

1-1

2-2

3-3 z

x

y

a

IJ

KL

a

Mechanical characteristics of masonry E11=3290 MPaE22=1775 MPaνν12= 0.111νν21= 0.206G12=1188 MPa.


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-0,7

-0,65

-0,6

-0,55

-0,5

-0,45

-0,4

-0,35

0 1000 2000 3000 4000 5000 6000 7000

Section along generator axis [mm]

Rad

ial

dis

pla

cem

ent

[mm

]

L0FL30FL45FL60FL90F

-7,0E-05

-6,0E-05

-5,0E-05

-4,0E-05

-3,0E-05

-2,0E-05

-1,0E-05

0,0E+00

1,0E-05

2,0E-05

3,0E-05

4,0E-05

5,0E-05

6,0E-05

7,0E-05

0 1000 2000 3000 4000 5000 6000 7000

Sections along generator axis [mm]

Ro

tati

on

res

pec

t to

y a

xis

L0FL30FL45FL60FL90F


Radial displacement along generator axis at crown, for different orthotropic angle

Rotation along generator axis at crown, for different orthotropic angle


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-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0,00 0,31 0,63 0,94 1,26 1,57 1,88 2,20 2,51 2,83 3,14

Section along profile axis [rad]

Tan

gent

ial d

ispl

acem

ent [

mm

]

T0FT30FT45FT60FT90F

Tangential displacement in middle arch, for different orthotropic angle

-8,0E-04

-7,0E-04

-6,0E-04

-5,0E-04

-4,0E-04

-3,0E-04

-2,0E-04

-1,0E-04

0,0E+00

1,0E-04

2,0E-04

3,0E-04

4,0E-04

5,0E-04

6,0E-04

7,0E-04

8,0E-04

0,00 0,31 0,63 0,94 1,26 1,57 1,88 2,20 2,51 2,83 3,14

Sections along profile axis [rad]

Rota

tion r

espec

t to

z a

xis

T0FT30FT45FT60FT90F

Rotation along z axis in middle arch, for different orthotropic angle


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Orthotropy:0°

Orthotropy:45°Orthotropy:90°

Radial displacement in middle arch, for different orthotropic angle

-0,8

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,00 0,31 0,63 0,94 1,26 1,57 1,88 2,20 2,51 2,83 3,14

Section along profile axis [rad]

Rad

ial d

ispl

acem

ent [

mm

]

T0FT30FT45FT60FT90F

CONCLUSIONSCONCLUSIONS4T

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üAnalytical and numerical approach in order to investigate the structural behaviour of masonry barrel vaults built with herring-bone pattern is considered.

üThe displacements components along the middle arch and the generator axis defined by φ=0° are represented for different values of the inclination angle of the directions of orthotropy.

üWhen φ decreases the values of the displacements components decreases too, meaning that the barrel vault becomes more stiff.

üMoreover the more used pattern (i.e. φ=90°), seems to be the one with lower stiffness.

üResearch in progress: to analyse the limit analysis of these kind of vaults, in order to study the influence of the herring bone pattern on the collapse mechanism.

Documents

MASONRY ORTHOTROPIC VAULTS IN HISTORICAL … › arch04 › BerbieriCarloniTommaso.pdfBENDING THEORY OF ORTHOTROPIC SHELL 4 TH INTERNATIONAL CONFERENCE ON ARCH BRIDGES MASONRY ORTHOTROPIC