# zbMATH — the first resource for mathematics

Static analysis of Timoshenko beam resting on elastic half-plane based on the coupling of locking-free finite elements and boundary integral. (English) Zbl 1271.74271
Summary: Making use of a mixed variational formulation including the Green function of the soil and assuming as independent fields both the structure displacements and contact pressure, a finite element (FE) model is derived for the static analysis of a foundation beam resting on an elastic half-plane. The Timoshenko beam model is adopted to describe low slenderness and to impose displacement compatibility between beam and half-plane without requiring the continuity of the first-order derivative of surface displacements enforced by Euler-Bernoulli beam. Numerical results are obtained by using locking-free Hermite polynomials for the Timoshenko beam and constant reaction over the soil. Foundation beams for many load configurations illustrate accuracy and convergence of the proposed formulation. Moreover, the different behaviour of the Euler-Bernoulli and Timoshenko beam models is thoroughly discussed. Rectangular pipe loaded by a force on the upper beam exemplifies the straightforward coupling of the foundation FE with a structure described by usual FEs.

##### MSC:
 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74S05 Finite element methods applied to problems in solid mechanics 74L10 Soil and rock mechanics
Full Text:
##### References:
 [1] Wang YH, Tham LG, Cheung YK (2005) Beams and plates on elastic foundations: a review. Prog Struct Eng Mater 7(4): 174–182 [2] Brebbia CA, Georgiou P (1979) Combination of boundary and finite elements in elastostatics. Appl Math Model 3(3): 212–220 · Zbl 0406.73009 [3] Mendonca AV, Paiva JB (2003) An elastostatic FEM/BEM analysis of vertically loaded raft and piled raft foundation. Eng Anal Bound Elem 27(9): 919–933 · Zbl 1060.74654 [4] Gonzalez JA, Park KC, Felippa CA (2007) FEM and BEM coupling in elastostatics using localized Lagrange multipliers. Int J Numer Methods Eng 69(10): 2058–2074 · Zbl 1194.74401 [5] Ganguly S, Layton JB, Balakrishna C (2000) Symmetric coupling of multi-zone curved Galerkin boundary elements with finite elements in elasticity. Int J Numer Methods Eng 48(5): 633–654 · Zbl 0986.74073 [6] Leung KL, Zavareh PB, Beskos DE (1995) 2-D elastostatic analysis by a symmetrical BEM/FEM scheme. Eng Anal Bound Elem 15(1): 67–78 [7] Springhetti R, Novati G, Margonari M (2006) Weak coupling of the symmetric Galerkin BEM with FEM for potential and elastostatic problems. Comput Model Eng Sci 13(1): 67–80 · Zbl 1357.74064 [8] Lu YY, Belytschko T, Liu WK (1991) A variationally coupled FE-BE method for elasticity and fracture-mechanics. Comput Methods Appl Mech Eng 85(1): 21–37 · Zbl 0764.73085 [9] Zeng X, Kallivokas LF, Bielak J (1993) A symmetric variational finite-element boundary integral-equation coupling method. Comput Struct 46(6): 995–1000 · Zbl 0776.73078 [10] Polizzotto C, Zito M (1994) Variational formulations for coupled BE/FE methods in elastostatics. ZAMM 74(11): 533–543 · Zbl 0819.73067 [11] Cheung YK, Zienkiewicz OC (1965) Plates and tanks on elastic foundations–an application of finite element method. Int J Solids Struct 1(4): 451–461 [12] Rajapakse RKND, Selvadurai APS (1986) On the performance of Mindlin plate elements in modelling plate-elastic medium interaction: a comparative study. Int J Numer Methods Eng 23(7): 1229–1244 · Zbl 0646.73028 [13] Booker JR, Balaam NP, Davis EH (1985) The behaviour of an elastic, non-homogeneous half-space. Part I: line load and point loads. Int J Numer Anal Methods Geomech 9(4): 353–367 · Zbl 0569.73106 [14] Wang YH, Ni J, Cheung YK (2000) Plate on non-homogeneous elastic half-space analysed by FEM. Struct Eng Mech 9(2): 127–139 [15] Stark RF (2001) Integration of singularities in FE/BE analyses of soil-foundation interaction with non-homogeneous elastic soils. Meccanica 36(4): 329–350 · Zbl 1038.74049 [16] Guarracino F, Minutolo V, Nunziante L (1992) A simple analysis of soil-structure interaction by BEM-FEM coupling. Eng Anal Bound Elem 10(4): 283–289 [17] Bode C, Hirschauer R, Savidis SA (2002) Soil-structure interaction in the time domain using halfspace Green’s functions. Soil Dyn Earthq Eng 22(4): 283–295 [18] Kikuchi N (1980) Beam bending problems on a Pasternak foundation using reciprocal variational-inequalities. Q Appl Math 38(1): 91–108 · Zbl 0452.73094 [19] Bielak J, Stephan E (1983) A modified Galerkin procedure for bending of beams on elastic foundations. SIAM J Sci Stat Comput 4(2): 340–352 · Zbl 0541.73089 [20] Kikuchi N, Oden J (1988) Contact problems in elasticity. A study of variational inequalities and finite element methods. SIAM, Philadelphia · Zbl 0685.73002 [21] Alliney S, Tralli A, Alessandri C (1990) Boundary variational formulations and numerical solution techniques for unilateral contact problems. Comput Mech 6(4): 247–257 · Zbl 0719.73039 [22] Yuan RL, Wang LS (1991) Generalized variational principle of plates on elastic foundation. J Appl Mech ASME 58(4): 1001–1004 · Zbl 0745.73019 [23] Gurtin ME, Sternberg E (1961) Theorems in linear elastostatics for exterior domains. Arch Ration Mech Anal 8: 99–119 · Zbl 0101.17001 [24] Narayanaswami R, Adelman HM (1974) Inclusion of transverse shear deformation in finite element displacement formulations. AIAA J 12(11): 1613–1614 [25] Przemieniecki JS (1968) Theory of matrix structural analysis. McGraw-Hill, New York · Zbl 0177.53201 [26] Eisenberger M (1994) Derivation of shape functions for an exact 4 DOF Timoshenko beam element. Commun Numer Methods Eng 10(9): 673–681 · Zbl 0808.73069 [27] Reddy JN (1997) On locking-free shear deformable beam finite elements. Comput Methods Appl Mech Eng 149(1–4): 113–132 · Zbl 0918.73131 [28] Ortuzar JM, Samartin A (1998) Some consistent finite element formulations of 1-D beam models: a comparative study. Adv Eng Softw 29(7–9): 667–678 · Zbl 05469514 [29] Mukherjee S, Reddy JN, Krishnamoorthy CS (2001) Convergence properties and derivative extraction of the superconvergent Timoshenko beam finite element. Comput Methods Appl Mech Eng 190(26–27): 3475–3500 · Zbl 1022.74047 [30] Friedman Z, Kosmatka JB (1993) An improved two-node Timoshenko beam finite element. Comput Struct 47(3): 473–481 · Zbl 0775.73249 [31] Kosmatka JB (1995) An improved two-node finite element for stability and natural frequencies of axial-loaded Timoshenko beams. Comput Struct 57(1): 141–149 · Zbl 0918.73150 [32] Minghini F, Tullini N, Laudiero F (2007) Locking-free finite elements for shear deformable orthotropic thin-walled beams. Int J Numer Methods Eng 72(7): 808–834 · Zbl 1194.74446 [33] Minghini F, Tullini N, Laudiero F (2008) Buckling analysis of FRP pultruded frames using locking-free finite-elements. Thin-Walled Struct 46(3): 223–241 [34] Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New York · Zbl 0788.73002 [35] Bathe KJ (1996) Finite element procedures. Prentice Hal, New Jersey [36] Bathe KJ (2001) The inf–sup condition and its evaluation for mixed finite element methods. Comput Struct 79: 243–252 [37] Cowper GR (1966) The shear coefficient in Timoshenko’s beam theory. J Appl Mech ASME 33: 335–340 · Zbl 0151.37901 [38] Tullini N, Savoia M (1999) Elasticity interior solution for orthotropic strips and the accuracy of beam theories. J Appl Mech ASME 66(2): 368–373 [39] Johnson KL (1985) Contact mechanics. Cambridge University Press, Cambridge · Zbl 0599.73108 [40] Hsiao GC, Wendland WL (1977) A finite element method for some integral equations of the first kind. J Math Anal Appl 58(3): 449–481 · Zbl 0352.45016 [41] Costabel M (1988) Boundary integral operators on Lipschitz domains: elementary results. SIAM J Math Anal 19: 613–626 · Zbl 0644.35037 [42] Biot MA (1937) Bending of an infinite beam on an elastic foundation. J Appl Mech 4: A1–A7 [43] Vesic AB (1961) Bending of beams on isotropic elastic medium. J Eng Mech Div ASCE 87(EM2): 35–53 [44] Muskhelishvili NI (1963) Some basic problems of the mathematical theory of elasticity. Noordhoff Ltd, Groningen · Zbl 0124.17404 [45] Sneddon IN (1951) Fourier transforms. McGraw-Hill, New York [46] Truman E, Sackfield A, Hills DA (1995) Contact mechanics of wedge and cone indenters. Int J Mech Sci 37(3): 261–275 · Zbl 0819.73061 [47] Villaggio P (1977) Qualitative methods in elasticity. Noordhoof International Publishing, Leyden · Zbl 0374.73001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.