A simple matrix expression is obtained for the strain components of a beam in which the displacements and rotations are large. The only restrictions are on the magnitudes of the strain and of the local rotation, a newly-identified kinematical quantity. The local rotation is defined as the change of orientation of material elements relative to the change of orientation of the beam reference triad. The vectors and tensors in the theory are resolved along orthogonal triads of base vectors centered along the undeformed and deformed beam reference axes, so Cartesian tensor notation is used. Although a curvilinear coordinate system is natural to the beam problem, the complications usually associated with its use are circumvented. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of beams with both open and closed cross sections in applications of the theory. The theory is used to obtain the kinematical relations for coupled bending, torsion, extension, shear deformation, and warping of an initially curved and twisted beam.

A general nonlinear theory for thin shells of arbitrary midsurface geometry is formulated in terms of a finite rotation vector and a stress-function vector. Compatibility equations, equilibrium equations, and boundary conditions are derived which are valid for shells undergoing arbitrarily large rotations and strains. For problems admitting a potential energy functional, a variational principle is formulated. The simplifications implied by small extensional strains are discussed. The theory contains, as special cases, Reissner’s equations for the axisymmetric deformation of shells of revolution, and the Sanders-Koiter linear shell theory.

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