Error in the determination of the deformed shape of prismatic beams using the double integration of curvature

Abstract

The deformed shape is a consequence of loading the structure and it is defined by the shape of the centroid line of the beam after deformation. The deformed shape is a universal parameter of beam-like structures. It is correlated with the curvature of the cross-section; therefore, any unusual behavior that affects the curvature is reflected through the deformed shape. Excessive deformations cause user discomfort, damage to adjacent structural members, and may ultimately lead to issues in structural safety. However, direct long-term monitoring of the deformed shape in real-life settings is challenging, and an alternative is indirect determination of the deformed shape based on curvature monitoring. The challenge of the latter is an accurate evaluation of error in the deformed shape determination, which is directly correlated with the number of sensors needed to achieve the desired accuracy. The aim of this paper is to study the deformed shape evaluated by numerical double integration of the monitored curvature distribution along the beam, and create a method to predict the associated errors and suggest the number of sensors needed to achieve the desired accuracy. The error due to the accuracy in the curvature measurement is evaluated within the scope of this work. Additionally, the error due to the numerical integration is evaluated. This error depends on the load case (i.e., the shape of the curvature diagram), the magnitude of curvature, and the density of the sensor network. The method is tested on a laboratory specimen and a real structure. In a laboratory setting, the double integration is in excellent agreement with the beam theory solution which was within the predicted error limits of the numerical integration. Consistent results are also achieved on a real structure—Streicker Bridge on Princeton University campus.

Strain Gauges Based 3D Shape Monitoring of Beam Structures Using Finite Width Gauge Model

“…In the case of beam monitoring with strain gauges placed parallel to the beam, when the length of the strain gauge is small compared to the length of the beam, the variation of the axial strain along the length of the gauge is neglected. Moreover, when the plane of bending is orthogonal to the gauge the axial strain is constant along the width of the beam, such as in bridge deformation monitoring [16]. Consequently, as the strain field is supposed uniform under the strain gauge, the averaging by the gauge is ignored [16,17].…”

“…Moreover, when the plane of bending is orthogonal to the gauge the axial strain is constant along the width of the beam, such as in bridge deformation monitoring [16]. Consequently, as the strain field is supposed uniform under the strain gauge, the averaging by the gauge is ignored [16,17]. For some circular beam shape monitoring, such as needle shape reconstruction, the strain sensor width is not taken into account and the strain measurements are supposed to be point-related [27,15,5].…”

“…In some case, the positions of the loads on a structure are restricted or can be anticipated. It is then possible to formulate hypothesis on the direction of deformations which allow to reduce the number of strain gauges necessary for shape reconstruction [16]. As no restrictive hypothesis have been formulated on the direction of the deformation of the beam, it is thus necessary to place three strain gauges per instrumented cross section.…”

“…The topic of using strain measures for beam shape monitoring has been widely covered in the litterature on both theoretical [10,11,12,13,14] and applied aspects, such as needles deflections during medical intervention [4,5,15] or bridge deformation over time [16]. The strain measures are acquired through technologies such as strain gauges [10,13,17], or fiber Bragg gratings [16,18,19,20,21] and are used to obtain deformations of the beam. Depending on the orientations of the sensors these measures can be used to recover the different deformations of the beam such as bending [12], torsion [22] and shearing and elongation [14].…”

“…The strain measures are thus used as inputs in a reconstruction method to obtain the structure deformed shape. The reconstruction methods proposed in the literature are either two dimensional [10,11,16,17,18,23] or three dimensional [12,14,19,20,21] depending on the deformation hypothesis made.…”