Abstract
In this paper, we introduce a phenomenological model approximating the behaviour of masonry structures, which is based on a low-tension elastic–brittle (EB) assumption with evolutionary tensile behaviour. The EB model is conceived by embedding a decaying tensile strength in the material behaviour, and it is able to achieve good agreement with the real behaviour of masonry. Since the model is quite sophisticated, non-holonomic, and the EB solution depends—amongst other things—on the loading path, it is worthwhile to investigate the relationships with more manageable and stable models rather than searching for unreliable solutions that depend on poorly predictable data. Namely, whereas it is quite clear and largely agreed upon that structural models widely applied in engineering (like perfectly plastic or no-tension models or other ones) are well-conditioned problems, the same does not apply to brittle structures. In this case, exact solutions are hard to be found and are scarcely attractive from the engineering point of view since they also depend on the load history and on unverifiable variables such as the local tensile strength. In view of these considerations, in this paper it is proved that stress fields in tensioned EB problems can be approached by highly stable solutions, on the upper and lower sides of the relevant complementary energy, and that the approximation gets closer as the limit tensile strength of the brittle material becomes lower.
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Abbreviations
- CE:
-
Complementary energy
- MCE:
-
Minimum complementary energy
- EB:
-
Elastic–brittle
- EL:
-
Purely elastic
- NT:
-
No-tension
- PL:
-
Elastic–plastic
- VWP:
-
Virtual work principle
- O :
-
Axis origin
- s :
-
Curvilinear abscissa
- \(\ell \) :
-
Length of the model mid-line
- \(V, S_{1}\) :
-
Body volume and constrained part of the body surface
- \(\theta \) :
-
Direction tangent to the barycentre line
- e(s):
-
Eccentricity
- \(h'(s), h''(s)\) :
-
Distances of the upper and lower profiles of the arch from the cross-sectional barycentre
- n(s):
-
Neutral axis
- \(A_{r}(s)\) :
-
Resistant part of the cross section
- \(G_{r}(s)\) :
-
Barycentre of \(A_{r}(s)\)
- C(s):
-
Solicitation centre
- \(e_{r}\) :
-
Distance of C(s) from \(G_{r}(s)\)
- \(d_{Gr}(s)\) :
-
Distance of \(G_{r}(s)\) from n(s)
- \(\rho \) :
-
Material density
- E :
-
Elastic modulus in compression of the masonry
- g, c :
-
Gravity acceleration, fraction of the gravity acceleration
- \(a_\mathrm{max}\) :
-
Maximum horizontal ground acceleration
- \(\tau \), t:
-
Parameter governing the loading process, and one single value
- \(\mathbf{p }\) :
-
Surface loads
- \(\mathbf{G}\) :
-
Body forces
- \(\mathbf{u}\) :
-
Imposed displacement field on \(S_{1}\)
- \(X_{i}~~ (i = 1,2,3)\) :
-
Unknown static variables (static redundancies)
- \(X_{1}\), \(X_{2}\), \(X_{3}\) :
-
Thrust force, support force, and bending moment redundancies
- \(\mathbf{S}_{0}(s)\), \(\mathbf{S}_{i}(s)\) :
-
Stress resultant vectors for the isostatic schemes under the applied loads and the i-th unit redundancy \(X_{i}\)
- N(s), T(s), M(s):
-
Normal force, shear force, and bending moment
- \(N_{0}(s)\), \(T_{0}(s)\), \(M_{0}(s)\) :
-
Stress resultants referred to the isostatic scheme under the applied loads
- \(N_{i}(s)\), \(T_{i}(s)\), \(M_{i}(s)\) :
-
Stress resultants referred to the isostatic scheme under the i-th unit redundancy \(X_{i}\)
- \(N_{d}\), \(T_{d}\), \(M_{d}\) :
-
Values of the static redundancies on the basis of the leftward abutment of the portal arch
- \(u_{d}\), \(v_{d}\), \(\phi _{d}\) :
-
Settlements of the foundation basis of the leftward abutment of the portal arch
- \(\sigma \) :
-
Statically admissible stress
- \(\sigma '_{o}\), \(\varepsilon '_{o}\) :
-
Tensile yield stress and strain
- \(\sigma _{\theta }\) :
-
Stress component along \(\theta \)
- \(\varepsilon \), \(\varepsilon _\mathrm{e}\), \(\varepsilon _\mathrm{f}\) :
-
Strain and relevant elastic and fracture components
- \(\varepsilon _\mathrm{EB}\), \(\varepsilon _\mathrm{fEB}\) :
-
EB strain and fracture component
- \({\mathcal {C}}(X_{1}\), \(X_{2}\), \(X_{3})\) :
-
Convex functional over the convex set (\(X_{1}\), \(X_{2}\), \(X_{3})\)
- \({\mathcal {C}}\) :
-
Complementary energy functional
- \(D_\mathrm{EB}\), \(\sigma _\mathrm{EB}\), \({\mathcal {C}}_\mathrm{EB}\) :
-
EB admissible domain, solution stress and complementary energy
- \(D_\mathrm{EL,} \sigma _\mathrm{EL}\), \({\mathcal {C}}_\mathrm{EL}\) :
-
EL admissible domain, solution stress and complementary energy
- \(D_\mathrm{NT}\), \(\sigma _\mathrm{NT}\), \({\mathcal {C}}_\mathrm{NT}\) :
-
NT admissible domain, solution stress and complementary energy
- \(D_\mathrm{PL}\), \(\sigma _\mathrm{PL}\), \({\mathcal {C}}_\mathrm{PL}\) :
-
PL admissible domain, solution stress and complementary energy
- \(\mathbf{T}\), \(\mathbf{T}_\mathrm{NT}\) :
-
Reactions in equilibrium with any NT statically admissible stress \(\sigma \) and with the stress solution \(\sigma _\mathrm{NT}\)
References
Heyman, J.: The stone skeleton. J. Solids Struct. 2, 249–279 (1966)
Bazant, Z.P., Li, Y.N.: Stability of cohesive crack model: part I: energy principles. J. Appl. Mech. 62(12), 959–964 (1995)
Kooharian, A.: Limit analysis of voussoir (segmental) and concrete arches. J. Am. Concr. Inst. 24, 317–328 (1952)
Heyman, J., Pippard, A.J.S.: The estimation of the strength of masonry arches. Proc. Inst. Civ. Eng. 69(4), 921–937 (1980). doi:10.1680/iicep.1980.2177
Khludnev A.M., Kovtunenko V.A.: Analysis of cracks in solids. Advances in Fracture Mechanics, pp. 386. Computational Mechanics (2000). ISBN 978-1853126253
Andreu, A., Gil, L., Roca, P.: Computational analysis of masonry structures with a funicular model. J. Eng. Mech. 133(4), 473–480 (2007)
Anthoine, A.: Homogenization of periodic masonry, plane stress, generalized plane strain or 3D modeling. J. Commun. Numer. Methods Eng. 13(5), 319–326 (1997)
Baratta, A., Corbi, I., Corbi, O.: Stress analysis of masonry structures: arches, walls, and vaults. In: D’Ayala, Fodde E. (eds.) Structural Analysis of Historic Constructions: Preserving Safety and Significance, pp. 321–329. CRC Press (2008). ISBN 978-0-415-46872-5
Baratta, A., Corbi, I., Corbi, O.: Analytical formulation of generalized incremental theorems for 2D no-tension solids. J. Acta Mech. 226(9), 2849–2859 (2015). doi:10.1007/s00707-015-1350-2
Baratta, A., Corbi, I., Corbi, O.: Stability of evolutionary brittle-tension 2D solids with heterogeneous resistance. J. Comput. Struct. (2015). doi:10.1016/j.compstruc.2015.10.004
Baratta, A., Corbi, I., Corbi, O., Rinaldis, D.: Experimental survey on seismic response of masonry models. In: Proceedings of the 6th International Conference on Structural Analysis of Historic Constructions: Preserving Safety and Significance, SAHC08, Bath, 2–4 July 2008, 8, pp. 799–807, (2008). ISBN 0415468728;978-041546872-5
Baratta, A., Corbi, I.: Equilibrium models for helicoidal laterally supported staircases. J. Comput. Struct. (2013). doi:10.1016/j.compstruc.2012.11.007. ISSN 00457949
Baratta, A., Corbi, O.: Heterogeneously resistant elastic–brittle solids under multi-axial stress: fundamental postulates and bounding theorems. J. Acta Mech. 226(6), 2077–2087 (2015). doi:10.1007/s00707-015-1299-1
Baratta, A., Corbi, O.: Contribution of the fill to the static behaviour of arched masonry structures: theoretical formulation. J. Acta Mech. 225(1), 53–66 (2014). doi:10.1007/s00707-013-0935-x
Drosopoulos, G.A., Stavroulakis, G.E., Massalas, C.V.: Limit analysis of a single span masonry bridge with unilateral frictional contact interfaces. J. Eng. Struct. 28(13), 1864–1873 (2006)
Fanning, P.J., Boothby, T.E.: Three-dimensional modelling and full-scale testing of stone arch bridges. J. Comput. Struct. 79(29–30), 2645–2662 (2001)
Foti, D., Diaferio, M., Giannoccaro, N.I., Mongelli, M.: Ambient vibration testing. dynamic identification and model updating of a historic tower. NDT E Int. 47, 88–95 (2012).doi:10.1016/j.ndteint.2011.11.009. ISSN:0963-8695
Furtmüller, T., Adam, C.: Numerical modeling of the in-plane behaviour of historical brick masonry walls. J. Acta Mech. 221(1–2), 65–77 (2011). doi:10.1007/s00707-011-0493-z
Füssl, J., Lackner, R., Eberhardsteiner, J., Mang, H.A.: Failure modes and effective strength of two-phase materials determined by means of numerical limit analysis. J. Acta Mech. 195(1–4), 185–202 (2008). doi:10.1007/s00707-007-0550-9
Pietruszczak, S., Ushaksaraei, R.: Description of inelastic behaviour of structural masonry. Int. J. Solids Struct. 40(15), 4003–4019 (2003). doi:10.1016/S0020-7683(03)00174-4
Vintzileou, E.: Testing historic masonry elements and/or building models. J. Geotech. Geol. Earthq. Eng. 34, 267–307 (2014)
Baratta, A., Corbi, O.: Closed-form solutions for FRP strengthening of masonry vaults. J. Comput. Struct. 147, 244–249 (2015). doi:10.1016/j.compstruc.2014.09.007
Baratta, A., Corbi, I., Corbi, O.: Bounds on the Elastic Brittle solution in bodies reinforced with FRP/FRCM composite provisions. J. Compos. Part B Eng. 68, 230–236 (2015). doi:10.1016/j.compositesb.2014.07.027
Elmalich, D., Rabinovitch, O.: Nonlinear analysis of masonry arches strengthened with composite materials. J. Eng. Mech. 136(8), 996–1005 (2010). doi:10.1061/(ASCE)EM.1943-7889.0000140
Baratta, A., Corbi, I., Corbi, O.: Algorithm design of an hybrid system embedding influence of soil for dynamic vibration control. J. Soil Dyn. Earthq. Eng. 74, 79–88 (2015)
Corbi, I., Corbi, O.: Macro-mechanical modelling of pseudo-elasticity in shape memory alloys for structural applications. J. Acta Mech. (2016). doi:10.1007/s00707-016-1624-3
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Baratta, A., Corbi, I. & Corbi, O. Theorems for masonry solids with brittle time-decaying tensile limit strength. Acta Mech 228, 837–849 (2017). https://doi.org/10.1007/s00707-016-1722-2
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DOI: https://doi.org/10.1007/s00707-016-1722-2