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Review Articles

On the Analysis of Minimum Thickness in Circular Masonry Arches

[+] Author and Article Information
Giuseppe Cocchetti

Facoltà di Ingegneria (Dalmine),
Dipartimento di Progettazione e Tecnologie,
Università degli Studi di Bergamo,
viale G. Marconi 5, I-24044 Dalmine (BG), Italy;
Dipartimento di Ingegneria Strutturale,
Politecnico di Milano,
piazza Leonardo da Vinci 32,
I-20133 Milano, Italy

Egidio Rizzi

e-mail: egidio.rizzi@unibg.it
Facoltà di Ingegneria (Dalmine),
Dipartimento di Progettazione e Tecnologie,
Università degli Studi di Bergamo,
viale G. Marconi 5, I-24044 Dalmine (BG), Italy

Throughout the paper, with the purpose of subtle academic comparisons, numerical results will be provided with 6 significative digits.

1Corresponding author.

Manuscript received March 16, 2011; final manuscript received May 21, 2012; published online October 1, 2012. Assoc. Editor: Panos Papadopoulos.

Appl. Mech. Rev 64(5), 050802 (Oct 01, 2012) (27 pages) doi:10.1115/1.4007417 History: Received March 16, 2011; Revised May 21, 2012

In this paper, the so-called Couplet–Heyman problem of finding the minimum thickness necessary for equilibrium of a circular masonry arch, with general opening angle, subjected only to its own weight is reexamined. Classical analytical solutions provided by J. Heyman are first rederived and explored in details. Such derivations make obviously use of equilibrium relations. These are complemented by a tangency condition of the resultant thrust force at the haunches' intrados. Later, given the same basic equilibrium conditions, the tangency condition is more correctly restated explicitly in terms of the true line of thrust, i.e., the locus of the centers of pressure of the resultant internal forces at each theoretical joint of the arch. Explicit solutions are obtained for the unknown position of the intrados hinge at the haunches, the minimum thickness to radius ratio and the nondimensional horizontal thrust. As expected from quoted Coulomb's observations, only the first of these three characteristics is perceptibly influenced, in engineering terms, by the analysis. This occurs more evidently at increasing opening angle of the arch, especially for over-complete arches. On the other hand, the systematic treatment presented here reveals the implications of an important conceptual difference, which appears to be relevant in the statics of masonry arches. Finally, similar trends are confirmed as well for a Milankovitch-type solution that accounts for the true self-weight distribution along the arch.

Copyright © 2011 by ASME
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References

Figures

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Fig. 1

Sketch of a symmetric circular arch subjected only to its own weight (of specific weight γ) with all characteristic parameters involved (d: out-of-plane depth)

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Fig. 2

Five-hinge rotational collapse mechanism of a symmetric circular arch

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Fig. 3

Statics and kinematics of a symmetric rotational collapse mechanism of a circular arch supporting only its own weight

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Fig. 4

Equilibrium of the upper portion of the arch and Heyman's tangency condition of the resultant thrust at the haunches

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Fig. 5

Heyman solution. Fit βfit of β as a function of A.

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Fig. 6

Heyman solution. Fit βfit of β as a function of α.

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Fig. 7

Heyman solution. Fit ηfit of η as a function of α.

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Fig. 8

Heyman solution. Fit hfit of h as a function of α.

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Fig. 9

Eccentricity of the line of thrust with respect to the geometrical centerline of the arch

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Fig. 10

Qualitative representation of the line of thrust at the critical condition of minimum arch thickness

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Fig. 11

CCR solution. Functional dependence of A, η, h on β.

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Fig. 12

CCR solution. Fit βfit = a (A − 2/3)1/b (2 − A)1/cof β as a function of A.

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Fig. 13

CCR solution. Fit βfit = a (A(α) − 2/3)1/b (2 − A(α))1/cof β as a function of α.

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Fig. 14

CCR solution. Fit ηfit of η as a function of α.

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Fig. 15

CCR solution. Fit hfit of h as a function of α.

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Fig. 16

Plot of the eccentricity of the line of thrust for CCR (line tangent to the intrados) and Heyman solutions (line going out of the arch thickness), α = π / 2

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Fig. 17

Plot of the eccentricity of the line of thrust for CCR (line tangent to the intrados) and Heyman solutions (line going out of the arch thickness), α = 7 π/9

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Fig. 18

Analytical plots of the line of thrust for CCR solution, for α = π/2 = 90 deg (A = π/2) and α = 7π/9 = 140 deg (A = 0.889347)

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Fig. 19

Comparison between Heyman, CCR and Milankovitch solutions in terms of the solution couples (β,η), (β, h), (η, h)

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Fig. 20

Comparison between Heyman, CCR, and Milankovitch solutions in terms of A

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Fig. 21

Comparison between Heyman, CCR, and Milankovitch solutions in terms of α

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Fig. 22

Comparison between Heyman, CCR, and Milankovitch solutions for the horizontal nondimensional thrust h∧ = ηh in terms of A

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Fig. 23

Comparison between Heyman, CCR, and Milankovitch solutions for the horizontal nondimensional thrust h∧ = ηh in terms of α

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Fig. 24

Milankovitch solution. Fit βfit M= a(A−(✓31))1/b (2−A)1/c of β as a function of A.

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Fig. 25

Milankovitch solution. Fit βfit M= a (A(α) − (✓31))1/b (2 − A(α))1/cof β as a function of α.

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Fig. 26

Milankovitch solution. Fit ηfit M of η as a function of α.

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Fig. 27

Milankovitch solution. Fit hfit M of h as a function of α.

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