Theoretical and computational aspects of vector-like parametrization of three-dimensional finite rotations, which uses only three rotation parameters, are examined in detail in this work. The relationship of the proposed parametrization with the intrinsic representation of finite rotations (via an orthogonal matrix) is clearly identified. Careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of Reissner's non-linear beam theory. Pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2-node three-dimensional beam element. A number of numerical simulations in three-dimensional finite rotation analysis are presented in order to illustrate the proposed approach.KEY WORDS: three-dimensional finite rotations; parametrization; computational procedure *One of the popular choices for three rotation parameters are the Euler angles, which have the well-known nonuniqueness problem (e.g. see Reference 2, p. 144)

SUM MARYWe discuss both linear and geometrically non-linear finite element analysis of elastic beams, taking into account the shear deformation. In linear analysis, a novel shallow beam element formulaton is consistently derived, and the end result is more suitable for the finite element implementation than earlier attempts. The element is very resourceful for an explanation of membrane and shear locking phenomena and exploration of their possible remedies. In addition, it sheds some light on locking phenomena in non-linear analysis. In non-linear analysis, we discuss the finite element implementation of the finite strain beam theory of Reissner.A. IBRAHIMBEGOVIC AND F. FREY (iv) Finally, we explore finite element formulation of Reissner's beam and demonstrate that considering initially curved configuration we can increase the aCCUrdCy compared with an equivalent but initially straight beam element of Simo et a].'An outline of the paper is as follows. In Section 2, we give a new formulation of a shallow curved beam derived by the consistent linearization of Reissner's beam theory. In Section 3, we discuss the finite element interpolations which eliminate the shear and membrane locking. In Section 4, we briefly recapitulate the pertinent equations of Reissner's beam theory, but using the Biot-type stress and energy-conjugate strain measures. In Section 5, it is discussed haw the same interpolations can avoid locking in the case of non-linear kinematics. Several numerical examples are given in Section 6. Conclusions are drawn in Section 7. TWO-DIMENSIONAL CURVED SHALLOW BEAM: LINEAR KINEMATICS

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A consistent formulation of the geometrically linear shell theory with drilling rotations is obtained by the consistent linearization of the geometrically non-linear shell theory considered in Parts I and I1 of this work.It was also shown that the same formulation can be recovered by linearizing the governing variational principle for the three-dimensional geometrically non-linear continuum with independent rotation field. In the finite element implementation of the presented shell theory, relying on the modified method of incompatible modes, we were able to construct a four-node shell element which delivers a very high-level performance. In order to simplify finite element implementation, a shallow reference configuration is assumed over each shell finite element. This approach does not impair the element performance for the present four-node element. The results obtained herein match those obtained with the state-of-the-art implementations based on the classical shell theory, over the complete set of standard benchmark problems.

SUMMARYWe discuss a geometrically non-linear method of incompatible modes. The model problem chosen for the discussion is the finite elasticity with independent rotations. The conditions which ensure the convergence of the method and the methodology to construct incompatible modes are prebented. A detailed derivation of variational equations and their linearized form is given for a two-dimensional plane problem. A couple of geometrically non-linear two-dimensional elements with independent rotational freedoms arc proposed based on the presented methodology. The elements exhibit a very satisfying performance over a set of problems in finite elasticity.