On Cesàro-Volterra Method in Orthotropic Saint-Venant Beam
Source abstract
Abstract
The linearly elastic and orthotropic Saint-Venant beam model, with a spatially constant Poisson tensor and fiberwise homogeneous elastic moduli, is investigated by a coordinate-free approach. A careful reasoning reveals that the elastic strain, fulfilling the whole set of differential conditions of integrability and a differential condition imposed by equilibrium, is defined on the whole ambient space in which the beam is immersed. At this stage the shape of the beam cross-section is inessential and Cesàro-Volterra formula provides the general integral of the differential conditions of kinematic compatibility. The cross-section geometrical shape comes into play only when differential and boundary equilibrium conditions are imposed to evaluate the warping displacement field. The treatment of an orthotropic Saint-Venant beam is applied to investigate about the locations of the shear and twist centres. It is shown that the position of the shear centre can be expressed in terms of the sole cross-section twist warping. The advantage with respect to treatments in the literature is that the solution of a single Neumann-like problem is required.
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Research context
OpenAlex topics: Composite Structure Analysis and Optimization · Nonlocal and gradient elasticity in micro/nano structures · Elasticity and Material Modeling
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References (44)
Show source-provided references
1. Barretta, R., Barretta, A.: Shear stresses in elastic beams: an intrinsic approach. Eur. J. Mech. A, Solids 29, 400–409 (2010)10.1016/j.euromechsol.2009.10.008 ↗
2. Barretta, R.: On the relative position of twist and shear centres in the orthotropic and fiberwise homogeneous Saint-Venant beam theory. Int. J. Solids Struct. 49, 3038–3046 (2012)10.1016/j.ijsolstr.2012.06.003 ↗
3. Beltrami, E.: Osservazioni sulla nota precedente. Atti Acad. Lincei Rend. 1(5), 141–142 (1886). = Opere 4, 190–223
4. Cesàro, E.: Sulle formule del Volterra, fondamentali nella teoria delle distorsioni elastiche. Rend. Napoli 12(3a), 311–321 (1906)
5. Ciarlet, P.G., Gratie, L., Mardare, C.: A generalization of the classical Cesàro-Volterra path integral formula. C. R. Acad. Sci. Paris, Ser. I 347, 577–582 (2009)10.1016/j.crma.2009.03.007 ↗
6. Ciarlet, P.G., Gratie, L., Mardare, C.: A Cesàro-Volterra formula with litte regularity. J. Math. Pures Appl. 93, 41–60 (2010)10.1016/j.matpur.2009.05.011 ↗
7. Cicala, P.: Il centro di taglio nei solidi cilindrici. Atti Accad. Sci Torino, Ser. Fis. 70, 356–371 (1935)
8. Clebsh, A.: Theorie der Elasticität fester Körper. Teubner, Leipzig (1862)
9. Dong, S.B., Kosmatka, J.B., Lin, H.C.: On Saint-Venant’s problem for an inhomogeneous, anisotropic cylinder, part I: methodology for Saint-Venant solutions. J. Appl. Mech. 68, 376–381 (2001)10.1115/1.1363598 ↗
10. Ecsedi, I.: A formulation of the centre of twist and shear for nonhomogeneous beam. Mech. Res. Commun. 27(4), 407–411 (2000)10.1016/S0093-6413(00)00111-7 ↗
11. El Fatmi, R., Zenzri, H.: A numerical method for the exact elastic beam theory: applications to homogeneous and composite beams. Int. J. Solids Struct. 41, 2521–2537 (2004)10.1016/j.ijsolstr.2003.12.011 ↗
12. Fichera, G.: Existence Theorems in Elasticity. Encyclopædia of Physics, vol. VIa/2. Springer, New York (1972)
13. Fichera, G.: Remarks on Saint-Venant’s principle. In: I.N. Vekua 70th Anniversary Volume, Moskow (1977)
14. Fosdick, R., Royer-Carfagni, G.: A Stokes theorem for second-order tensor fields and its implications in continuum mechanics. Int. J. Non-Linear Mech. 40, 381–386 (2005)10.1016/j.ijnonlinmec.2004.07.006 ↗
15. Gurtin, M.E.: A generalization of the Beltrami stress functions in continuum mechanics. Arch. Ration. Mech. Anal. 13, 321–329 (1963)10.1007/BF01262700 ↗
16. Gurtin, M.E.: The Linear Theory of Elasticity. Encyclopædia of Physics, vol. VIa/2. Springer, New York (1972)
17. Kirchhoff, G.: Über die Gleichungen des Gleichgewicht und die Bewegung eines elastischen Körpers bei nicht unendlicht Vershiebungen seiner Theile. Sitzgsber. Akad. Wiss. Wien 9, 762–773 (1852)
18. Kosmatka, J.B., Lin, H.C., Dong, S.B.: On Saint-Venant’s problem for an inhomogeneous, anisotropic cylinder, part II: cross sectional properties. J. Appl. Mech. 68, 382–391 (2001)10.1115/1.1365152 ↗
19. Ladevéze, P., Simmonds, J.: New concepts for linear beam theory with arbitrary geometry and loading. Eur. J. Mech. A, Solids 17(3), 377–402 (1998)10.1016/S0997-7538(98)80051-X ↗
20. Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco (1963)
21. Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity, third, revised and augmented edn., Moscow, Leningrad (1949). Translated from the russian by J.R.M. Radok, P. Noordhoff Ltd Groningen, Holland (1953)
22. Novozhilov, V.V.: Theory of Elasticity. Pergamon, London (1961)
23. Romano, G.: Scienza delle Costruzioni, Tomo II. Hevelius, Benevento (2003)
24. Romano, G.: Continuum mechanics on manifolds. Lecture Notes, University of Naples Federico II (2007). http://wpage.unina.it/romano
25. Romano, G., Barretta, R., Diaco, M.: On the general form of the law of dynamics. Int. J. Non-Linear Mech. 44(6), 689–695 (2009)10.1016/j.ijnonlinmec.2009.02.011 ↗
26. Romano, G., Barretta, R., Barretta, A.: On Maupertuis principle in dynamics. Rep. Math. Phys. 63(3), 331–346 (2009)10.1016/S0034-4877(09)00015-9 ↗
27. Romano, G., Barretta, R., Diaco, M.: On continuum dynamics. J. Math. Phys. 50, 102903–1–26 (2009)10.1063/1.3215979 ↗
28. Romano, G., Barretta, R., Diaco, M.: Algorithmic tangent stiffness in elastoplasticity and elastoviscoplasticity: a geometric insight. Mech. Res. Commun. 37(3), 289–292 (2010)10.1016/j.mechrescom.2010.02.002 ↗
29. Romano, G., Diaco, M., Barretta, R.: Variational formulation of the first principle of continuum thermodynamics. Contin. Mech. Thermodyn. 22(3), 177–187 (2010)10.1007/s00161-009-0119-z ↗
30. Romano, G., Barretta, R.: Covariant hypo-elasticity. Eur. J. Mech. A, Solids 30(6), 1012–1023 (2011)10.1016/j.euromechsol.2011.05.005 ↗
31. Romano, G., Barretta, A., Barretta, R.: On torsion and shear of Saint-Venant beams. Eur. J. Mech. A, Solids 35, 47–60 (2012)10.1016/j.euromechsol.2012.01.007 ↗
32. Romano, G., Barretta, R.: On Euler’s stretching formula in continuum mechanics. Acta Mech. 224, 211–230 (2013)10.1007/s00707-012-0746-5 ↗
33. Romano, G., Barretta, R.: Geometric constitutive theory and frame invariance. Int. J. Non-Linear Mech. 51, 75–86 (2013). doi: 10.1016/j.ijnonlinmec.2012.12.00610.1016/j.ijnonlinmec.2012.12.006 ↗
34. de Saint-Venant, A.J.C.B.: Mémoire sur la torsion des prismes. Mém. Acad. Sci. Inst. Fr. 14, 233–560 (1856)
35. de Saint-Venant, A.J.C.B.: Mémoire sur la flexion des prismes. J. Math. Pures Appl. Ser. 2 1, 89–189 (1856)
36. Sokolnikoff, I.S.: Mathematical Theory of Elasticity. McGraw-Hill, New York (1956)
37. Solomon, L.: Élasticité Lineaire. Masson, Paris (1968)
38. Stephen, N.G., Maltbaek, J.C.: The relationship between the centres of flexure and twist. Int. J. Mech. Sci. 21(6), 373–377 (1979)10.1016/0020-7403(79)90017-1 ↗
39. Synge, J.L.: The problem of Saint Venant for a cylinder with free sides. Q. Appl. Math. II, 307–317 (1945)10.1090/qam/12037 ↗
40. Timoshenko, S., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1951)
41. Toupin, R.A.: Saint-Venant’s principle. Arch. Ration. Mech. Anal. 17, 85–112 (1964)10.1007/BF00253050 ↗
42. Trefftz, E.: Über den Schubmittelpunkt in einem durch eine Einzellast gebogenen Balken. Z. Angew. Math. Mech. 15, 220–225 (1935)10.1002/zamm.19350150405 ↗
43. Volterra, V.: Sur l’équilibre des corps élastiques multiplement connexes. Ann. Ecole Norm. 24, 401–517 (1907)
44. Weinstein, A.: The center of shear and the center of twist. Q. Appl. Math. 5, 97–99 (1947)10.1090/qam/20437 ↗