This paper extends the approach for determining the three-dimensional global displaced shape of slender structures from a limited set of scalar surface strain measurements. It is an exhaustive approach that captures the effect of curvature, shear, torsion, and elongation. The theory developed provides both a determination of the uniaxial strain (in a given direction) anywhere in the structure and the deformed shape, given a set of strain values. The approach utilizes Cosserat rod theory and exploits a localized linearization approach that helps to obtain a local basis function set for the displacement solution in the Cosserat frame. For the assumed deformed shape (both the midcurve and the crosssectional orientation), the uniaxial value of strain in any given direction is obtained analytically, and this strain model is the basis used to predict the shape via an approximate local linearized solution strategy. Error analysis due to noise in measured strain values and in uncertainty in the proximal boundary condition is performed showing uniform convergence with increased sensor count.

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This paper investigates the variational formulation and numerical solution of a higher-order, geometrically exact Cosserat type beam with a deforming cross-section, instigated from generalized kinematics presented in earlier works. The generalizations include the effects of a fully-coupled Poisson's and warping deformations in addition to other deformation modes in Simo-Reissner beam kinematics.The kinematics at hand renders the deformation map to be a function of not only the configuration of the beam but also on the elements of the tangent space of the beam's configuration (axial strain vector, curvature, warping amplitude, and their derivatives). This complicates the process of deriving the balance laws and exploring the variational formulation of the beam, at the same time, make it worthwhile. The weak and strong form is derived for the dynamic case considering a general boundary.We restrict ourselves to linear small-strain elastic constitutive law and the static case for numerical implementation. The finite element modeling of this beam has higher regularity requirements. The matrix (discretized) form of the equation of motion is derived. Finally, numerical simulations comparing various beam models are presented.

In this paper, we discuss about reconstructing the global deformed shape of slender structures such as pipelines, tethers, or cables from a limited set of scalar surface strain measurements. We present a comprehensive approach that captures the effect of curvature, shear, torsion, and axial deformation. Our primary focus is to demonstrate the applicability of the approach to aid in damage detection algorithms. This theory utilizes Cosserat rod theory and exploit localized linearization approach that helps to obtain local basis function set for the displacement solution in director frame. The uniaxial strain vector and the surface strain for the Cosserat beam incorporating the abovementioned effects are obtained and used to develop the reverse algorithm to reconstruct global shape of the structure. Error analysis due to noise in measured strain values is performed and results are discussed.