Mechanical plane problems of the straight beams with deformable protect fixed section of a finite length

V. N. Paimushin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2022), pp. 89-96.

Résumé

On the example of a semi-infinite beam with fixed section of finite dimensions on one of the front faces, it is shown that in order to study static and dynamic deformation processes, it is necessary to take into account the transformation of the types of the stress-strain state and the mathematical models used to describe them. Transformation takes place when crossing the border from a non-fixed area to a fixed one. The Kirchhoff–Love model does not allow taking into account the deformability of the fixed section of the beam, and when using the simplest refined shear model of S.P. Tymoshenko, its transformation is possible by fixing the site only on one of the front faces. Under the terms of using the described models and their combinations, the kinematic and force conditions for the conjugation of the fixed and non-fixed sections are formulated. On the basis of the derived proposals, an exact analytical solution is found for the simplest linear problem of the bending of a beam with its cantilever fixation. It is shown that taking into account the deformability of the fixing section having a finite length is especially important for thin-walled structural elements made of composite materials.

Détail

Keywords: beam, plane problem, fixed section, Timoshenko model, static loading
Mots-clés : equations of motion.

Bibliographie

[1] Ambartsumyan S. A., Obschaya teoriya anizotropnykh obolochek, Nauka, M., 1974

[2] Ambartsumyan S. A., Teoriya anizotropnykh plastin. Prochnost, ustoichivost i kolebaniya, Nauka, M., 1987 | MR

[3] Reddy J. N., “A simple higher-order theory for laminated composite plates”, Appl. Mech., 51 (1984), 745–752 | DOI | Zbl

[4] Librescu L., “Refined geometrically non-linear theories of anisotropic laminated shells”, Quart. Appl. Math., 45 (1987), 1–22 | DOI | MR | Zbl

[5] Schmidt R., Reddy J. N., “A refined small strain and moderate rotation theory of elastic anisotropic shells”, J. Appl. Mech., 55 (1988), 611–617 | DOI | Zbl

[6] Librescu L., Schmidt R., “Refined theories of elastic anisotropic shells accounting for small strains and moderate rotations”, Int. J. Nonlinear Mech., 23 (1988), 217–229 | DOI | Zbl

[7] Reddy J. N., “A general non-linear third-order theory of plates with moderate thickness”, Int. J. Nonlinear Mech., 25 (1990), 677–686 | DOI | Zbl

[8] Librescu L., Schmidt R., “Substantiation of a shear-deformable theory of anisotropic composite laminated shells accounting for the interlaminate continuity conditions”, Int. J. Eng. Sci., 29 (1991), 669–683 | DOI | MR | Zbl

[9] Başar Y., Ding Y., Schultz R., “Refined shear-deformation models for composite laminates with finite rotations”, Int. J. Solids Struct., 30 (1993), 2611–2638 | DOI | Zbl

[10] Gruttmann F., Wagner W., “A linear quadrilateral shell element with fast stiffness computation”, Comp. Meth. Appl. Mech. Eng., 194 (2005), 4279–4300 | DOI | Zbl

[11] Gruttmann F., Wagner W., “Structural analysis of composite laminates using a mixed hybrid shell element”, Comput. Mech., 37 (2006), 479–497 | DOI | Zbl

[12] Schmidt R., Vu. T.D., “Nonlinear dynamic FE simulation of smart piezolaminated structures based on first- and third-order transverse shear deformation theory”, Adv. Materials Research, 79–82 (2009), 1313–1316 | DOI

[13] Yankovskii A. P., “Critical Analysis of the Equations of Statics in the Bending Theories of Composite Plates Obtained on the Basis of Variational Principles of Elasticity Theory $1$. General Theories of High Order”, Mech. of Composite Materials, 56:3 (2020), 271–290 | DOI

[14] Yankovskii A. P., “Critical Analysis of the Equations of Statics in the Bending Theories of Composite Plates Obtained on the Basis of Variational Principles of Elasticity Theory $2$. Particular Low-Order Theories”, Mech. of Composite Materials, 56:4 (2020), 437–454 | DOI

[15] Reissner E., “On the theory of bending of elastic plates”, J. Math. and Phys., 23:4 (1944), 184–191 | DOI | MR | Zbl

[16] Mindlin R. D., “Thickness-shear and flexural vibrations of crystal plates”, J. Appl. Phys., 23:3 (1951), 316–323 | DOI | MR

[17] Reissner E., “A consistent treatment of transverse shear deformations in laminated anisotropic plates”, AIAA J., 10:5 (1972), 716–718 | DOI

[18] Malmeister A. K., Tamuzh V. P., Teters G. A., Soprotivlenie polimernykh i kompozitnykh materialov, Zinatne, Riga, 1980

[19] Reddy J. N., “A refined nonlinear theory of plates with transverse shear deformation”, Int. J. of Solids and Structures, 250:9 (1984), 881–896 | DOI

[20] Nemirovskii Yu. V., Reznikov B. S., Prochnost elementov konstruktsii iz kompozitnykh materialov, Nauka, Novosibirsk, 1986

[21] Andreev A. N., Uprugost i termouprugost sloistykh kompozitnykh obolochek. Matematicheskaya model i nekotorye aspekty chislennogo analiza, Palmarium Academic Publishing, Saarbrucken (Deutschland), 2013

[22] Thai C. H., “Analysis of laminated composite plates using higher-order shear deformation plate theory and mode-based smoother discrete shear gap method”, Appl. Math. Modeling, 36:11 (2012), 5657–5677 | DOI | MR | Zbl

[23] Mau S., “A refined laminated plates theory”, J. Appl. Mech., 40:2 (1973), 606–607 | DOI

[24] Christensen R., Lo K., Wu E., “A high-order theory of plate deformation. Part 1: homogeneous plates”, J. Appl. Mech., 44:7 (1977), 663–668 | Zbl

[25] Andreev A. N., Nemirovskii Yu. V., Mnogosloinye anizotropnye obolochki i plastiny. Izgib, ustoichivost i kolebaniya, Nauka, Novosibirsk, 2001