Stress analysis of generally asymmetric non-prismatic beams subject to arbitrary loads. (English) Zbl 07392185

Summary: Non-prismatic beams are widely employed in several engineering fields, e.g., wind turbines, rotor blades, aircraft wings, and arched bridges. While analytical solutions for variable cross-section beams are desirable, a model describing all stress components for beams with general variation of their cross-section under generalised loading remains an open and important problem to solve. To partly address this issue, we propose an analytical solution for stress recovery of untwisted, asymmetric, non-prismatic beams with smooth and continuous taper shape under general loading, considering plane stress conditions for isotropic materials undergoing small strains. The methodology follows Jourawski’s formulation, including the effect of asymmetric variable cross-section, with internal forces as known variables. We confirm the non-triviality of the stress field of non-prismatic beams, i.e., the dependency on all internal forces and beam geometry to shear and transverse stress distributions. As a particular novelty, the new formulation for transverse direct stress includes internal forces derivatives, resulting in greater accuracy than state-of-the-art models for distributed loading conditions. Also, closed-form solutions are introduced for non-prismatic and linearly tapered, generally asymmetric beams, both with rectangular cross-sections. For validation purposes, we consider three different practical beam models: a symmetric and an asymmetric, both linearly tapered, and an arched beam. The results, checked against commercial finite element analysis, show that the proposed model predicts the stress-field of non-prismatic beams under distributed loads with good levels of accuracy. Traction-free boundary condition requirements are naturally satisfied on the beam surfaces.


74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
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[1] Ai, Q.; Weaver, P. M., Simplified analytical model for tapered sandwich beams using variable stiffness materials, J. Sandw. Struct. Mater., 19, 1, 3-25 (2017)
[2] Al-Gahtani, H.; Khan, M., Exact analysis of nonprismatic beams, J. Eng. Mech., 124, 11, 1290-1293 (1998)
[3] Balduzzi, G.; Aminbaghai, M.; Auricchio, F.; Füssl, J., Planar Timoshenko-like model for multilayer non-prismatic beams, Int. J. Mech. Mater. Des., 14, 1, 51-70 (2018)
[4] Balduzzi, G.; Aminbaghai, M.; Sacco, E.; Füssl, J.; Eberhardsteiner, J.; Auricchio, F., Non-prismatic beams: a simple and effective Timoshenko-like model, Int. J. Solids Struct., 90, 236-250 (2016)
[5] Balduzzi, G.; Hochreiner, G.; Füssl, J., Stress recovery from one dimensional models for tapered bi-symmetric thin-walled I beams: Deficiencies in modern engineering tools and procedures, Thin-Walled Struct., 119, 934-945 (2017)
[6] Balduzzi, G.; Hochreiner, G.; Füssl, J.; Auricchio, F., Serviceability analysis of non-prismatic timber beams: derivation and validation of new and effective straightforward formulas, Open J. Civ. Eng., 7, 1, 32-62 (2017)
[7] Balduzzi, G.; Morganti, S.; Auricchio, F.; Reali, A., Non-prismatic Timoshenko-like beam model: Numerical solution via isogeometric collocation, Comput. Math. Appl., 74, 7, 1531-1541 (2017) · Zbl 1394.65060
[8] Beltempo, A.; Balduzzi, G.; Alfano, G.; Auricchio, F., Analytical derivation of a general 2D non-prismatic beam model based on the Hellinger-Reissner principle, Eng. Struct., 101, 88-98 (2015)
[9] Bertolini, P.; Eder, M.; Taglialegne, L.; Valvo, P., Stresses in constant tapered beams with thin-walled rectangular and circular cross sections, Thin-Walled Struct., 137, 527-540 (2019)
[10] Bertolini, P.; Taglialegne, L., Analytical solution of the stresses in doubly tapered box girders, Eur. J. Mech. A Solids, 81, Article 103969 pp. (2020) · Zbl 1475.74075
[11] Bleich, F., Stahlhochbauten: ihre Theorie, Berechnung und bauliche Gestaltung, Band. 1 (1932), Springer-Verlag
[12] Blodgett, O. W., Design of Welded Structures (1966), James F. Lincoln Arc Welding Foundation: James F. Lincoln Arc Welding Foundation Cleveland
[13] Boley, B. A., On the accuracy of the Bernoulli-Euler theory for beams of variable section, J. Appl. Mech., 30(3), 373-378 (1963) · Zbl 0127.40501
[14] Bruhns, O. T., Advanced Mechanics of Solids (2003), Springer
[15] Carothers, S., XXVI.—Plane strain in a wedge, with applications to masonry Dams, Proc. Roy. Soc. Edinburgh, 33, 292-306 (1914) · JFM 44.0933.02
[16] Doeva, O., Khaneh Masjedi, P., Weaver, P.M., 2020a. Exact solution for the deflection of composite beams under non-uniformly distributed loads. In: AIAA Scitech 2020 Forum. p. 0245. · Zbl 1475.74076
[17] Doeva, O.; Masjedi, P. K.; Weaver, P. M., Static deflection of fully coupled composite timoshenko beams: An exact analytical solution, Eur. J. Mech. A Solids, 81, Article 103975 pp. (2020) · Zbl 1475.74076
[18] Fertis, D. G.; Keene, M. E., Elastic and inelastic analysis of nonprismatic members, J. Struct. Eng., 116, 2, 475-489 (1990)
[19] Filippi, M.; Pagani, A.; Carrera, E., Accurate nonlinear dynamics and mode aberration of rotating blades, J. Appl. Mech., 85, 11 (2018)
[20] Gimena, L.; Gimena, F.; Gonzaga, P., Structural analysis of a curved beam element defined in global coordinates, Eng. Struct., 30, 11, 3355-3364 (2008)
[21] Gimena, F.; Gonzaga, P.; Gimena, L., 3D-curved beam element with varying cross-sectional area under generalized loads, Eng. Struct., 30, 2, 404-411 (2008)
[22] Hodges, D.; Ho, J.; Yu, W., The effect of taper on section constants for in-plane deformation of an isotropic strip, J. Mech. Mater. Struct., 3, 3, 425-440 (2008)
[23] Hodges, D.; Rajagopal, A.; Ho, J.; Yu, W., Stress and strain recovery for the in-plane deformation of an isotropic tapered strip-beam, J. Mech. Mater. Struct., 5, 6, 963-975 (2011)
[24] Jourawski, D., Sur la résistance d’un corps prismatique et d’une pièce composée en bois ou en tôle de fer à une force perpendiculaire à leur longueur, (Annales des Ponts et Chaussées, Vol. 12 (1856)), 328-351
[25] Krahula, J. L., Shear formula for beams of variable cross section, AIAA J., 13, 10, 1390-1391 (1975)
[26] Masjedi, P. K.; Maheri, A.; Weaver, P. M., Large deflection of functionally graded porous beams based on a geometrically exact theory with a fully intrinsic formulation, Appl. Math. Model., 76, 938-957 (2019) · Zbl 1481.74444
[27] Masjedi, P. K.; Ovesy, H. R., Large deflection analysis of geometrically exact spatial beams under conservative and nonconservative loads using intrinsic equations, Acta Mech., 226, 6, 1689-1706 (2015) · Zbl 1325.74086
[28] Masjedi, P. K.; Ovesy, H., Chebyshev collocation method for static intrinsic equations of geometrically exact beams, Int. J. Solids Struct., 54, 183-191 (2015)
[29] Masjedi, P. K.; Weaver, P. M., Analytical solution for the fully coupled static response of variable stiffness composite beams, Appl. Math. Model., 81, 16-36 (2020)
[30] Masjedi, P. K.; Weaver, P. M., Variable stiffness composite beams subject to non-uniformly distributed loads: An analytical solution, Compos. Struct., Article 112975 pp. (2020)
[31] Mercuri, V.; Balduzzi, G.; Asprone, D.; Auricchio, F., Structural analysis of non-prismatic beams: Critical issues, accurate stress recovery, and analytical definition of the Finite Element (FE) stiffness matrix, Eng. Struct., 213, Article 110252 pp. (2020)
[32] Michell, J., The stress in an aelotropic elastic solid with an infinite plane boundary, Proc. Lond. Math. Soc., 1, 1, 247-257 (1900) · JFM 31.0760.04
[33] Murín, J.; Kutiš, V., 3D-beam element with continuous variation of the cross-sectional area, Comput. Struct., 80, 3-4, 329-338 (2002)
[34] Patni, M.; Minera Rebulla, S. A.; Weaver, P. M.; Pirrera, A., Efficient modelling of beam-like structures with general non-prismatic, curved geometry, Comput. Struct. (2020)
[35] Rajagopal, A.; Hodges, D. H., Asymptotic approach to oblique cross-sectional analysis of beams, J. Appl. Mech., 81, 3 (2014)
[36] Rezaiee-Pajand, M.; Karimipour, A., Stress analysis by two cuboid isoparametric elements, Eur. J. Comput. Mech., 373-410 (2019)
[37] Rezaiee-Pajand, M.; Karimipour, A., Analytical scheme for solid stress analysis, Int. J. Appl. Mech., 12, 06, Article 2050071 pp. (2020)
[38] Rezaiee-Pajand, M.; Karimipour, A., Three stress-based triangular elements, Eng. Comput., 36, 4, 1325-1345 (2020)
[39] Rezaiee-Pajand, M.; Karimipour, A., Two rectangular elements bbased on analytical functions, Adv. Comput. Des., 5, 2, 147-175 (2020)
[40] Romano, F., Deflections of Timoshenko beam with varying cross-section, Int. J. Mech. Sci., 38, 8-9, 1017-1035 (1996) · Zbl 0869.73035
[41] Romano, F.; Zingone, G., Deflections of beams with varying rectangular cross section, J. Eng. Mech., 118, 10, 2128-2134 (1992) · Zbl 0825.73281
[42] Sokolnikoff, I. S.; Redheffer, R. M., Mathematics of Physics and Modern Engineering (1958), McGraw-Hill · Zbl 0081.27601
[43] Taglialegne, L., Analytical Study of Stress Fields in Wind Turbine Blades, 1-181 (2018), Architecture, Civil Engineering and Environmental Engineering. Universities of Florence: Architecture, Civil Engineering and Environmental Engineering. Universities of Florence Perugia and Pisa - TU Braunschweig, (Ph.D. thesis)
[44] Thomas, M. A.; Hallett, S. R.; Weaver, P. M., Design considerations for variable stiffness, doubly curved composite plates, Compos. Struct., Article 112170 pp. (2020)
[45] Timoshenko, S. P.; Goodier, J. N., Theory of Elasticity (1951), McGraw-Hill · Zbl 0045.26402
[46] Timoshenko, S. P.; Young, D. H., Theory of Structures (1965), McGraw-Hill: McGraw-Hill New York
[47] Trahair, N.; Ansourian, P., In-plane behaviour of web-tapered beams, Eng. Struct., 108, 47-52 (2016)
[48] Vu-Quoc, L.; Léger, P., Efficient evaluation of the flexibility of tapered I-beams accounting for shear deformations, Internat. J. Numer. Methods Engrg., 33, 3, 553-566 (1992)
[49] Weeger, O.; Yeung, S. K.; Dunn, M. L., Fully isogeometric modeling and analysis of nonlinear 3D beams with spatially varying geometric and material parameters, Comput. Methods Appl. Mech. Engrg., 342, 95-115 (2018) · Zbl 1440.74191
[50] Wong, F. T.; Gunawan, J.; Agusta, K.; Herryanto, H.; Tanaya, L. S., On the derivation of exact solutions of a tapered cantilever Timoshenko beam, Civ. Eng. Dimens., 21, 2, 89-96 (2019)
[51] Yildiz, S.; Ikikardaslar, K. T.; Khan, H., Theoretical and computational analysis of circular cantilever tapered beams, Pract. Period. Struct. Des. Constr., 25, 1, Article 05019006 pp. (2020)
[52] Zhou, M.; Fu, H.; An, L., Distribution and properties of shear stress in elastic beams with variable cross section: Theoretical analysis and finite element modelling, KSCE J. Civ. Eng., 1-15 (2020)
[53] Zhou, M.; Zhang, J.; Zhong, J.; Zhao, Y., Shear stress calculation and distribution in variable cross sections of box girders with corrugated steel webs, J. Struct. Eng., 142, 6, 1-10 (2016)
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