Keywords: nonuniform torsion, inelastic torsion, warping, warping shear stresses, distributed plasticity, boundary element method, bar, beam, twist.

In this paper a boundary element method is developed for the inelastic nonuniform torsional problem of simply or multiply connected cylindrical bars of arbitrary cross section taking into account the effect of warping shear stresses. The bar is subjected to arbitrarily distributed or concentrated torsional loading along its length, while its edges are subjected to the most general torsional boundary conditions. To the authors' knowledge, research efforts with respect to the aforementioned problem concern cross sections of special geometry or ignore warping shear stresses with the exception of Gruttmann et al. [1] who exploit a warping shear stress distribution arising from the introduction of an independent warping parameter in the displacement field of the bar. However, since this theory is analogous to the Timoshenko beam theory of shear - bending loading conditions, it does not satisfy local equilibrium equations under elastic or inelastic conditions [2]. According to the proposed method a displacement based formulation is developed and inelastic redistribution is modeled through a distributed plasticity model exploiting three dimensional material constitutive laws and numerical integration over the cross sections. An incremental iterative solution strategy is adopted to restore global equilibrium along with an efficient iterative process to integrate the inelastic rate equations. Three boundary value problems with respect to the variable along the bar axis angle of twist, to the primary and to the secondary warping functions are formulated and solved employing the boundary element method, which has not yet been employed for the numerical analysis of the problem at hand. The first boundary value problem requires finite differences of plastic terms exclusively at the bar ends, if the effect of warping shear stresses is ignored. The rest are formulated by exploiting local equilibrium considerations under elastic conditions. An I-shaped cross section bar is worked out and good agreement is achieved between existing studies when warping shear stresses are ignored. When these stresses are taken into account, a slight decrease of torsional rigidity and of ultimate torsional loading that the bar can undergo is observed, while alteration of the magnitude and distribution of stress resultants with which bars resist torsional loading is noted.

1

F. Gruttmann, R. Sauer, W. Wagner, "Theory and numerics of three-dimensional beams with elastoplastic material behaviour", International Journal for Numerical Methods in Engineering, 48, 1675-1702, 2000. doi:10.1002/1097-0207(20000830)48:12<1675::AID-NME957>3.3.CO;2-Y

2

J.C. Simo, K.D. Hjelmstad, R.L. Taylor, "Numerical formulations of elasto-viscoplastic response of beams accounting for the effect of shear", Computer Methods in Applied Mechanics and Engineering, 42(3), 301-330, 1984. doi:10.1016/0045-7825(84)90011-2

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