A Displacement Reconstruction Strategy for Long, Slender Structures from Limited Strain Measurements and Its Application to Underground Pipeline Monitoring

Abstract

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.

An Improved Shape Reconstruction Methodology for Long Rod Like Structures Using Cosserat Kinematics- Including the Poisson’s Effect

“…This paper extends the theory of shape sensing developed by Todd et al [1] and Chadha and Todd [2][3] to include Poisson's effect along with curvature, shear, torsion and axial deformation. This formulation relaxes Euler-Bernoulli's rigid crosssection.…”

“…We consider that the initial unstrained configuration of the beam Ω ⊂ ℝ to be straight. Section [2.1] of [2][3] details the geometric description of the beam configuration. The inclusion of Poisson's effect does not change the position vector of the midcurve and the curve framing by the director triad {𝐝 (ξ )}; all it changes is the position vector of the material 2 point in the deformed state, which we shall explore further.…”

“…The inclusion of Poisson's effect does not change the position vector of the midcurve and the curve framing by the director triad {𝐝 (ξ )}; all it changes is the position vector of the material 2 point in the deformed state, which we shall explore further. As a matter of completion, we detail the necessary geometry, in same sense as section [2.1] of [2][3].…”

“…Researchers have used various curve framing approaches to capture curvature effects. We maintain a body-centered frame in the unsheared cross-section (as mentioned in Todd et al [1]) and the orthogonal director triad for a general case to capture torsion and shear (as mentioned in Chadha and Todd [2][3]).…”

The mathematical theory of a higher-order geometrically-exact beam with a deforming cross-section

“…Interestingly, further development of beam theory continues to date. The advanced and versatile applications of beam theory to numerous areas like deformation of bio-polymers [2,2,3], biological structures [4], shape-sensing [5,6,7,8], robotics, multibody dynamics [9], composite structures [10], contact problems [11], thermal problems [12,13], micro and nanostructures used in MEMS and NEMS etc., necessitates further development and refinement of this theory. We first perform a relevant literature review in the next few paragraphs.…”